Finding maximal, solvable, primitive subgroups of a large group in GAP I am trying to find the maximal, solvable, primitive subgroups of a large group $N$ which is itself a subgroup of $GL(n,p)$ for $(n,p)=(4,3),(4,5),(4,7),(6,3),(10,3)$. However, GAP is too slow to run this except for in the first case.
I first tried testing all conjugacy classes of subgroups but this was too slow (except for (4,3) case). Then I tried iterating through the group using low index subgroups and only keeping subgroups of subgroups which were primitive but this was also too slow. The third thing I tried was the LowLayerSubgroups function but this was also very slow and outputted a single non-solvable group (the entire group) at completion.
 A: Since you are searching for small subgroups in rather large groups, this is something which probably needs some programming and will not just work with the turnkey function.
Some remarks/hints on how to proceed:

*

*I think for the groups you consider, working in a permutation representation (IsomorphismPermGroup, and using the isomorphism to the original matrix representation -- the inverse of it -- to test primitivity) is probably the best way.
Make sure you test for the properties first before testing for conjugacy. Probably many subgroups can be discarded quickly.


*I think that proceeding down through maximal subgroups is the way to go, discarding subgroups as soon as they are not irreducible or not primitive. Stop when the groups get solvable, these are your answers.


*GAP currently has maximal subgroups of SL but not of GL. If your group is the full GL, do the calculation in SL (and then just take normalizers of the results in GL to get the corresponding subgroups of GL).


*You can ignore maximal subgroups $M\le U$ that do not contain the radical of $U$ -- if such a subgroup is solvable, primitive, so will be $\langle rad(U),M\rangle$.
