The database currently contains 544,831 groups from many different sources, the largest of which is $S_{47}$ of order $47!$. In addition, it contains 275,379,753 of their subgroups and 39,933,457 of their irreducible complex characters.
The database of finite groups consists of groups, subgroups, characters and conjugacy classes. The set of groups included contains:
-
All groups of order up to 2000, except those whose order is larger than 500 and divisible by 128,
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All groups with a transitive permutation representation of degree up to 47, except those of degree 32 and order between 512 and $4 \times 10^{10}$,
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All groups with an arbitrary permutation representation of degree up to 15,
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All groups with a rational matrix representation of dimension up to 6,
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Finite groups of Lie type up to certain bounds,
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Chevalley groups $E(6,2)$, $E(7,2)$, $F(4,2)$, $G(2,2)$, $G(2,3)$, $G(2,5)$, ${}^2B(2,2)$, ${}^2B(2,8)$, ${}^2B(2,32)$, ${}^3D(4,2)$, ${}^2E(6,2)$, ${}^2F(4,2)$, ${}^2F(4,2)'$, ${}^2G(2,3)$, ${}^2G(2,27)$,
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Subgroups of $\GL(2,\Z/N)$ for $N$ up to 124, except for $N=80,96,104,112, 120$
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Subgroups of $\GL(2,\mathbb{F}_q)$ for $q$ up to 1000 (except $\mathbb{F}_{512}$), subgroups of $\GL(3, \mathbb{F}_q)$ for $q$ up to 13, subgroups of $\GL(4, \mathbb{F}_q)$ for $q$ up to 5, subgroups of $\GL(5,2)$,
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Perfect groups of order up to 50,000,
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Sporadic simple groups of order up to $10^{15}$, except $ON$ and $Suz$.
For each group, the lattice of subgroups, complex and rational character tables, representatives for the conjugacy classes, automorphism and outer automorphism group, rank, Schur multiplier and various other quantities were computed, subject to limits on time, memory and errors arising in Magma. For space reasons, complex and rational character tables are never stored if larger than $511 \times 511$, and subgroups are only computed up to automorphism if there are more than 128 conjugacy classes of subgroups. For time reasons, there are some large groups where subgroups are only computed up to conjugacy and not up to automorphism. All normal subgroups are computed when possible, except that normal subgroups of size roughly the square root of the group order are omitted when there are more than 4096 normal subgroups.
Distribution of Solvability as a function of order
| order factorization type | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $1$ | $p$ | $p^2$ | $p^{3-6}$ | $p^{7+}$ | $pq,pqr,\ldots$ | $p^2q,p^2q^2$ | $p^3q,p^4q$ | $p^{3+}q^2$ | $p^{3+}q^{3+}$ | $p^{5+}q$ | $p^{1,2}q^{1,2}r^{1,2}\cdots$ | $p^{3+}qr\cdots$ | other | Total | ||
|
solvability type
|
cyclic | 1 | 303 | 14 | 12 | 3 | 1315 | 221 | 156 | 29 | 10 | 65 | 602 | 549 | 268 | 3548 |
| 100.00% | 100.00% | 31.11% | 1.22% | 0.00% | 33.53% | 18.54% | 5.03% | 0.07% | 0.01% | 0.07% | 5.62% | 0.96% | 0.39% | 0.65% | ||
| abelian and metacyclic, not cyclic | 31 | 24 | 31 | 316 | 163 | 220 | 113 | 115 | 501 | 154 | 362 | 2030 | ||||
| 68.89% | 2.44% | 0.05% | 26.51% | 5.25% | 0.54% | 0.06% | 0.12% | 4.68% | 0.27% | 0.52% | 0.37% | |||||
| abelian, not metacyclic | 37 | 52 | 167 | 111 | 108 | 189 | 183 | 94 | 941 | |||||||
| 3.76% | 0.08% | 5.38% | 0.27% | 0.06% | 0.19% | 0.32% | 0.14% | 0.17% | ||||||||
| nilpotent and metacyclic, not abelian | 52 | 91 | 309 | 220 | 205 | 435 | 391 | 178 | 1881 | |||||||
| 5.29% | 0.13% | 9.96% | 0.54% | 0.11% | 0.45% | 0.69% | 0.26% | 0.35% | ||||||||
| nilpotent and metabelian, not abelian or metacyclic | 858 | 66326 | 238 | 1408 | 2044 | 6516 | 919 | 159 | 78468 | |||||||
| 87.28% | 96.95% | 7.67% | 3.47% | 1.06% | 6.72% | 1.61% | 0.23% | 14.40% | ||||||||
| nilpotent, not metabelian | 1910 | 8 | 142 | 2060 | ||||||||||||
| 2.79% | 0.02% | 0.15% | 0.38% | |||||||||||||
| metacyclic, not nilpotent | 2607 | 567 | 1121 | 763 | 672 | 1384 | 7452 | 7467 | 6582 | 28615 | ||||||
| 66.47% | 47.57% | 36.13% | 1.88% | 0.35% | 1.43% | 69.54% | 13.10% | 9.49% | 5.25% | |||||||
| metabelian and supersolvable, not nilpotent or metacyclic | 66 | 868 | 19651 | 41773 | 33138 | 1284 | 34164 | 12056 | 143000 | |||||||
| 5.54% | 27.97% | 48.45% | 21.72% | 34.17% | 11.98% | 59.95% | 17.38% | 26.25% | ||||||||
| metabelian and monomial, not supersolvable | 22 | 20 | 1197 | 3385 | 371 | 570 | 851 | 2912 | 9328 | |||||||
| 1.85% | 0.64% | 2.95% | 1.76% | 0.38% | 5.32% | 1.49% | 4.20% | 1.71% | ||||||||
| metabelian, not monomial | 8 | 67 | 1 | 2 | 78 | |||||||||||
| 0.02% | 0.03% | 0.00% | 0.00% | 0.01% | ||||||||||||
| supersolvable, not nilpotent or metabelian | 47 | 1725 | 8146 | 1795 | 187 | 569 | 12469 | |||||||||
| 1.51% | 4.25% | 4.24% | 1.85% | 0.33% | 0.82% | 2.29% | ||||||||||
| monomial, not supersolvable or metabelian | 4 | 2649 | 7828 | 1716 | 108 | 902 | 3176 | 16383 | ||||||||
| 0.13% | 6.53% | 4.07% | 1.77% | 1.01% | 1.58% | 4.58% | 3.01% | |||||||||
| solvable, not monomial or metabelian | 9 | 1061 | 5720 | 1356 | 1247 | 1130 | 10523 | |||||||||
| 0.29% | 2.62% | 2.97% | 1.40% | 2.19% | 1.63% | 1.93% | ||||||||||
| not solvable | 193 | 8160 | 23728 | 32081 | ||||||||||||
| 1.80% | 14.32% | 34.21% | 5.89% | |||||||||||||
| metabelian, not supersolvable, unknown if monomial | 1 | 266 | 307 | 572 | 6 | 117 | 416 | 1685 | ||||||||
| 0.03% | 0.66% | 0.16% | 0.59% | 0.06% | 0.21% | 0.60% | 0.31% | |||||||||
| solvable, not metabelian, unknown if monomial | 11243 | 121905 | 49171 | 1692 | 17730 | 201741 | ||||||||||
| 27.72% | 63.40% | 50.71% | 2.97% | 25.56% | 37.03% | |||||||||||
| Total | 1 | 303 | 45 | 983 | 68413 | 3922 | 1192 | 3103 | 40559 | 192283 | 96966 | 10716 | 56983 | 69362 | 544831 | |
| 0.00% | 0.06% | 0.01% | 0.18% | 12.56% | 0.72% | 0.22% | 0.57% | 7.44% | 35.29% | 17.80% | 1.97% | 10.46% | 12.73% | |||
Distribution of nilpotency class as a function of order
| order factorization type | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $1$ | $p$ | $p^2$ | $p^{3-6}$ | $p^{7+}$ | $pq,pqr,\ldots$ | $p^2q,p^2q^2$ | $p^3q,p^4q$ | $p^{3+}q^2$ | $p^{3+}q^{3+}$ | $p^{5+}q$ | $p^{1,2}q^{1,2}r^{1,2}\cdots$ | $p^{3+}qr\cdots$ | other | Total | ||
|
nilpotency class
|
not | 2607 | 655 | 2070 | 38563 | 189803 | 89504 | 9613 | 54787 | 68301 | 455903 | |||||
| 66.47% | 54.95% | 66.71% | 95.08% | 98.71% | 92.30% | 89.71% | 96.15% | 98.47% | 83.68% | |||||||
| 0 | 1 | 1 | ||||||||||||||
| 100.00% | 0.00% | |||||||||||||||
| 1 | 303 | 45 | 73 | 86 | 1315 | 537 | 486 | 360 | 231 | 369 | 1103 | 886 | 724 | 6518 | ||
| 100.00% | 100.00% | 7.43% | 0.13% | 33.53% | 45.05% | 15.66% | 0.89% | 0.12% | 0.38% | 10.29% | 1.55% | 1.04% | 1.20% | |||
| 2 | 338 | 33218 | 402 | 805 | 1153 | 2878 | 722 | 236 | 39752 | |||||||
| 34.38% | 48.56% | 12.96% | 1.98% | 0.60% | 2.97% | 1.27% | 0.34% | 7.30% | ||||||||
| 3 | 457 | 25343 | 145 | 654 | 918 | 3091 | 461 | 78 | 31147 | |||||||
| 46.49% | 37.04% | 4.67% | 1.61% | 0.48% | 3.19% | 0.81% | 0.11% | 5.72% | ||||||||
| 4 | 104 | 7107 | 140 | 162 | 793 | 98 | 14 | 8418 | ||||||||
| 10.58% | 10.39% | 0.35% | 0.08% | 0.82% | 0.17% | 0.02% | 1.55% | |||||||||
| 5 | 11 | 1900 | 31 | 16 | 204 | 19 | 4 | 2185 | ||||||||
| 1.12% | 2.78% | 0.08% | 0.01% | 0.21% | 0.03% | 0.01% | 0.40% | |||||||||
| 6 | 440 | 3 | 99 | 6 | 2 | 550 | ||||||||||
| 0.64% | 0.01% | 0.10% | 0.01% | 0.00% | 0.10% | |||||||||||
| 7 | 196 | 2 | 25 | 3 | 2 | 228 | ||||||||||
| 0.29% | 0.00% | 0.03% | 0.01% | 0.00% | 0.04% | |||||||||||
| 8 | 112 | 1 | 3 | 1 | 1 | 118 | ||||||||||
| 0.16% | 0.00% | 0.00% | 0.00% | 0.00% | 0.02% | |||||||||||
| 9 | 11 | 11 | ||||||||||||||
| 0.02% | 0.00% | |||||||||||||||
| Total | 1 | 303 | 45 | 983 | 68413 | 3922 | 1192 | 3103 | 40559 | 192283 | 96966 | 10716 | 56983 | 69362 | 544831 | |
| 0.00% | 0.06% | 0.01% | 0.18% | 12.56% | 0.72% | 0.22% | 0.57% | 7.44% | 35.29% | 17.80% | 1.97% | 10.46% | 12.73% | |||
Distribution of rank as a function of order
| order factorization type | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $1$ | $p$ | $p^2$ | $p^{3-6}$ | $p^{7+}$ | $pq,pqr,\ldots$ | $p^2q,p^2q^2$ | $p^3q,p^4q$ | $p^{3+}q^2$ | $p^{3+}q^{3+}$ | $p^{5+}q$ | $p^{1,2}q^{1,2}r^{1,2}\cdots$ | $p^{3+}qr\cdots$ | other | Total | ||
|
rank
|
0 | 1 | 1 | |||||||||||||
| 100.00% | 0.00% | |||||||||||||||
| 1 | 303 | 14 | 12 | 3 | 1315 | 221 | 156 | 29 | 10 | 65 | 602 | 549 | 268 | 3547 | ||
| 100.00% | 31.11% | 1.22% | 0.00% | 33.74% | 18.60% | 5.03% | 0.10% | 0.01% | 0.14% | 5.89% | 1.16% | 0.89% | 1.13% | |||
| 2 | 31 | 256 | 1427 | 2582 | 928 | 2154 | 6488 | 13991 | 8780 | 9013 | 16665 | 19119 | 81434 | |||
| 68.89% | 26.04% | 2.09% | 66.26% | 78.11% | 69.46% | 22.58% | 19.45% | 18.36% | 88.14% | 35.21% | 63.51% | 25.93% | ||||
| 3 | 529 | 9871 | 39 | 701 | 14554 | 33917 | 18417 | 611 | 19581 | 9640 | 107860 | |||||
| 53.81% | 14.43% | 3.28% | 22.61% | 50.66% | 47.16% | 38.51% | 5.97% | 41.37% | 32.02% | 34.34% | ||||||
| 4 | 168 | 25380 | 88 | 7234 | 23042 | 18031 | 10116 | 1042 | 85101 | |||||||
| 17.09% | 37.10% | 2.84% | 25.18% | 32.04% | 37.70% | 21.37% | 3.46% | 27.10% | ||||||||
| 5 | 16 | 30576 | 2 | 404 | 925 | 2424 | 392 | 35 | 34774 | |||||||
| 1.63% | 44.69% | 0.06% | 1.41% | 1.29% | 5.07% | 0.83% | 0.12% | 11.07% | ||||||||
| 6 | 2 | 901 | 22 | 30 | 101 | 17 | 1 | 1074 | ||||||||
| 0.20% | 1.32% | 0.08% | 0.04% | 0.21% | 0.04% | 0.00% | 0.34% | |||||||||
| 7 | 193 | 9 | 5 | 207 | ||||||||||||
| 0.28% | 0.02% | 0.01% | 0.07% | |||||||||||||
| 8 | 43 | 3 | 4 | 50 | ||||||||||||
| 0.06% | 0.01% | 0.01% | 0.02% | |||||||||||||
| 9 | 15 | 15 | ||||||||||||||
| 0.02% | 0.00% | |||||||||||||||
| 10 | 3 | 3 | ||||||||||||||
| 0.00% | 0.00% | |||||||||||||||
| 12 | 1 | 1 | ||||||||||||||
| 0.00% | 0.00% | |||||||||||||||
| Total | 1 | 303 | 45 | 983 | 68413 | 3897 | 1188 | 3101 | 28731 | 71915 | 47830 | 10226 | 47329 | 30105 | 314067 | |
| 0.00% | 0.10% | 0.01% | 0.31% | 21.78% | 1.24% | 0.38% | 0.99% | 9.15% | 22.90% | 15.23% | 3.26% | 15.07% | 9.59% | |||
Distribution of derived length among solvable groups as a function of order
| order factorization type | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $1$ | $p$ | $p^2$ | $p^{3-6}$ | $p^{7+}$ | $pq,pqr,\ldots$ | $p^2q,p^2q^2$ | $p^3q,p^4q$ | $p^{3+}q^2$ | $p^{3+}q^{3+}$ | $p^{5+}q$ | $p^{1,2}q^{1,2}r^{1,2}\cdots$ | $p^{3+}qr\cdots$ | other | Total | ||
|
derived length
|
0 | 1 | 1 | |||||||||||||
| 100.00% | 0.00% | |||||||||||||||
| 1 | 303 | 45 | 73 | 86 | 1315 | 537 | 486 | 360 | 231 | 369 | 1103 | 886 | 724 | 6518 | ||
| 100.00% | 100.00% | 7.43% | 0.13% | 33.53% | 45.05% | 15.66% | 0.89% | 0.12% | 0.38% | 10.48% | 1.81% | 1.59% | 1.27% | |||
| 2 | 910 | 66417 | 2607 | 655 | 2557 | 23513 | 48453 | 42417 | 9312 | 43909 | 22305 | 263055 | ||||
| 92.57% | 97.08% | 66.47% | 54.95% | 82.40% | 57.97% | 25.20% | 43.74% | 88.49% | 89.94% | 48.88% | 51.30% | |||||
| 3 | 1902 | 58 | 9427 | 46801 | 34597 | 108 | 2584 | 7332 | 102809 | |||||||
| 2.78% | 1.87% | 23.24% | 24.34% | 35.68% | 1.03% | 5.29% | 16.07% | 20.05% | ||||||||
| 4 | 8 | 2 | 6563 | 77776 | 19182 | 1444 | 8591 | 113566 | ||||||||
| 0.01% | 0.06% | 16.18% | 40.45% | 19.78% | 2.96% | 18.83% | 22.15% | |||||||||
| 5 | 696 | 16811 | 401 | 5409 | 23317 | |||||||||||
| 1.72% | 8.74% | 0.41% | 11.85% | 4.55% | ||||||||||||
| 6 | 1982 | 1257 | 3239 | |||||||||||||
| 1.03% | 2.75% | 0.63% | ||||||||||||||
| 7 | 221 | 16 | 237 | |||||||||||||
| 0.11% | 0.04% | 0.05% | ||||||||||||||
| 8 | 8 | 8 | ||||||||||||||
| 0.00% | 0.00% | |||||||||||||||
| Total | 1 | 303 | 45 | 983 | 68413 | 3922 | 1192 | 3103 | 40559 | 192283 | 96966 | 10523 | 48823 | 45634 | 512750 | |
| 0.00% | 0.06% | 0.01% | 0.19% | 13.34% | 0.76% | 0.23% | 0.61% | 7.91% | 37.50% | 18.91% | 2.05% | 9.52% | 8.90% | |||
Distribution of automorphism group order as a function of order for nonabelian groups
| order factorization type | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $p^{3-6}$ | $p^{7+}$ | $pq,pqr,\ldots$ | $p^2q,p^2q^2$ | $p^3q,p^4q$ | $p^{3+}q^2$ | $p^{3+}q^{3+}$ | $p^{5+}q$ | $p^{1,2}q^{1,2}r^{1,2}\cdots$ | $p^{3+}qr\cdots$ | other | Total | ||
|
aut order
|
1-7 | 1 | 1 | ||||||||||
| 0.04% | 0.00% | ||||||||||||
| 8-32 | 10 | 2 | 7 | 10 | 29 | ||||||||
| 1.10% | 0.08% | 1.07% | 0.38% | 0.01% | |||||||||
| 33-128 | 47 | 2 | 17 | 12 | 57 | 24 | 29 | 27 | 16 | 231 | |||
| 5.16% | 0.00% | 0.65% | 1.83% | 2.18% | 0.06% | 0.03% | 0.28% | 0.03% | 0.04% | ||||
| 129-512 | 139 | 173 | 63 | 21 | 135 | 178 | 82 | 399 | 163 | 269 | 77 | 1699 | |
| 15.27% | 0.25% | 2.42% | 3.21% | 5.16% | 0.44% | 0.04% | 0.43% | 1.70% | 0.48% | 0.11% | 0.32% | ||
| 513-2048 | 89 | 2262 | 246 | 31 | 287 | 1096 | 788 | 2622 | 609 | 2039 | 1069 | 11138 | |
| 9.78% | 3.31% | 9.44% | 4.73% | 10.97% | 2.73% | 0.41% | 2.80% | 6.34% | 3.64% | 1.56% | 2.09% | ||
| 2049-8192 | 84 | 9115 | 343 | 48 | 392 | 3817 | 5485 | 9518 | 1067 | 6997 | 3909 | 40775 | |
| 9.23% | 13.34% | 13.16% | 7.33% | 14.98% | 9.51% | 2.88% | 10.15% | 11.11% | 12.48% | 5.71% | 7.64% | ||
| 8193-65536 | 223 | 42181 | 672 | 139 | 634 | 11727 | 28962 | 20878 | 2127 | 23915 | 10631 | 142089 | |
| 24.51% | 61.73% | 25.78% | 21.22% | 24.23% | 29.21% | 15.21% | 22.28% | 22.14% | 42.67% | 15.53% | 26.63% | ||
| 65537- | 318 | 14594 | 1263 | 397 | 1102 | 23308 | 155052 | 60282 | 5614 | 22810 | 52780 | 337520 | |
| 34.95% | 21.36% | 48.45% | 60.61% | 42.11% | 58.05% | 81.45% | 64.32% | 58.44% | 40.70% | 77.09% | 63.27% | ||
| Total | 910 | 68327 | 2607 | 655 | 2617 | 40150 | 190369 | 93728 | 9607 | 56046 | 68466 | 533482 | |
| 0.17% | 12.81% | 0.49% | 0.12% | 0.49% | 7.53% | 35.68% | 17.57% | 1.80% | 10.51% | 12.83% | |||
Distribution of outer aut. group order as a function of order for nonabelian groups
| order factorization type | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $p^{3-6}$ | $p^{7+}$ | $pq,pqr,\ldots$ | $p^2q,p^2q^2$ | $p^3q,p^4q$ | $p^{3+}q^2$ | $p^{3+}q^{3+}$ | $p^{5+}q$ | $p^{1,2}q^{1,2}r^{1,2}\cdots$ | $p^{3+}qr\cdots$ | other | Total | ||
|
outer order
|
1 | 39 | 1 | 3 | 43 | 671 | 17 | 33 | 377 | 1784 | 2968 | ||
| 1.50% | 0.15% | 0.11% | 0.11% | 0.35% | 0.02% | 0.34% | 0.67% | 2.61% | 0.56% | ||||
| 2-7 | 18 | 22 | 262 | 28 | 77 | 781 | 27156 | 463 | 586 | 1935 | 20240 | 51568 | |
| 1.98% | 0.03% | 10.05% | 4.27% | 2.94% | 1.95% | 14.26% | 0.49% | 6.10% | 3.45% | 29.56% | 9.67% | ||
| 8-32 | 161 | 2084 | 421 | 66 | 268 | 8406 | 79517 | 12275 | 1251 | 11317 | 20399 | 136165 | |
| 17.69% | 3.05% | 16.15% | 10.08% | 10.24% | 20.94% | 41.77% | 13.10% | 13.02% | 20.19% | 29.79% | 25.52% | ||
| 33-128 | 171 | 7544 | 654 | 140 | 618 | 11325 | 36236 | 22221 | 2028 | 15305 | 9910 | 106152 | |
| 18.79% | 11.04% | 25.09% | 21.37% | 23.61% | 28.21% | 19.03% | 23.71% | 21.11% | 27.31% | 14.47% | 19.90% | ||
| 129-512 | 177 | 12076 | 928 | 216 | 932 | 9281 | 22679 | 37841 | 2215 | 17278 | 6559 | 110182 | |
| 19.45% | 17.67% | 35.60% | 32.98% | 35.61% | 23.12% | 11.91% | 40.37% | 23.06% | 30.83% | 9.58% | 20.65% | ||
| 513-2048 | 137 | 30283 | 173 | 79 | 504 | 5040 | 12191 | 13212 | 1519 | 6246 | 4569 | 73953 | |
| 15.05% | 44.32% | 6.64% | 12.06% | 19.26% | 12.55% | 6.40% | 14.10% | 15.81% | 11.14% | 6.67% | 13.86% | ||
| 2049-8192 | 98 | 8476 | 91 | 57 | 124 | 2561 | 6586 | 4141 | 1062 | 2159 | 2734 | 28089 | |
| 10.77% | 12.41% | 3.49% | 8.70% | 4.74% | 6.38% | 3.46% | 4.42% | 11.05% | 3.85% | 3.99% | 5.27% | ||
| 8193-65536 | 88 | 4411 | 39 | 56 | 57 | 1906 | 2681 | 2100 | 770 | 1044 | 1609 | 14761 | |
| 9.67% | 6.46% | 1.50% | 8.55% | 2.18% | 4.75% | 1.41% | 2.24% | 8.01% | 1.86% | 2.35% | 2.77% | ||
| 65537- | 60 | 3431 | 12 | 34 | 807 | 2652 | 1458 | 143 | 385 | 662 | 9644 | ||
| 6.59% | 5.02% | 1.83% | 1.30% | 2.01% | 1.39% | 1.56% | 1.49% | 0.69% | 0.97% | 1.81% | |||
| Total | 910 | 68327 | 2607 | 655 | 2617 | 40150 | 190369 | 93728 | 9607 | 56046 | 68466 | 533482 | |
| 0.17% | 12.81% | 0.49% | 0.12% | 0.49% | 7.53% | 35.68% | 17.57% | 1.80% | 10.51% | 12.83% | |||