Computing Conjugacy Classes of Subgroups in GAP

Solution 1:

First, you may find helpful some functions like NormalSubgroups, IsomorphicSubgroups etc. which are listed in "How do I get the subgroups of my group?" entry from the GAP F.A.Q. There are also further functions not listed there, such as RepresentativesPerfectSubgroups, RepresentativesSimpleSubgroups, ConjugacyClassesMaximalSubgroups - see Chapter "Groups" from the GAP manual for more functions with the similar semantics.

Regarding p-subgroups, the manual entry for IsomorphicSubgroups suggests the same approach as in the Derek Holt's comment above: "To find p-subgroups it is often faster to compute the subgroup lattice of the Sylow subgroup".

Furthermore, LatticeByCyclicExtension and SubgroupsSolvableGroup accept optional arguments which allow to put restrictions on computed subgroups. In the latter case functions SizeConsiderFunction and ExactSizeConsiderFunction may be used. All four functions mentioned in this paragraph are documented in the Chapter "Groups", type e.g. ?SubgroupsSolvableGroup to see their descriptions in GAP.

As for tables of marks mentioned by Jack Schmidt, see Chapter "Tables of Marks" and in particular Section "Accessing Subgroups via Tables of Marks" - but that will work only for groups whose tables are contained in the tables of marks library.