Subgroups beyond an index bound (awaiting review)

In cases where we do not compute the whole subgroup lattice of a group, we still attempt to compute all subgroups up to a certain index bound. However, we often still want to store the following subgroups, even if they happen to fall beyond the index bound.

• Since Sylow subgroups are the unique subgroup of their order up to conjugacy, we can compute their labels a priori.
• We aim to compute the lattice of normal subgroups even when we cannot find all subgroups. In this case, we can use the same scheme described above applied to the lattice or normal subgroups to obtain a label with the letter \texttt{.N} appended; such labels are used for non-Sylow normal subgroups beyond the index bound.
• We store complements of normal subgroups even beyond the index bound so that we can describe groups as semidirect products. Given a normal subgroup with label $\texttt{n.i.m.a.c.N}$, we label its complements $\texttt{n.i.m.a.c.N1}$, $\texttt{n.i.m.a.c.N2}$, $\dots$. Unfortunately, these labels are not canonical since they depend on the order that complements are produced. Also note that a subgroup can appear as a complement of multiple normal subgroups; in this case we just label it based on the first normal subgroup where it arises.
• For non-Sylow, non-normal maximal subgroups beyond the index bound, we apply the original labeling scheme restricted to maximal subgroups, and append the letter $\texttt{.M}$.
• We also keep core-free subgroups beyond the index bound rather than discarding them, since these yield transitive permutation representations. The labels here are just $\texttt{n.i.m.CFk}$, where $\texttt{k}$ is a counter arbitrarily running over the core-free subgroups of a given index.