Quiz:Subgroup structure of special linear group:SL(2,5)

For background, see subgroup structure of special linear group:SL(2,5).

Basic stuff

Summary table on the structure of subgroups: [SHOW MORE]

Quick summary

Item	Value
number of subgroups	76
number of conjugacy classes of subgroups	12
number of automorphism classes of subgroups	12
isomorphism classes of Sylow subgroups, corresponding fusion systems, and Sylow numbers	2-Sylow: quaternion group (order 8) with its non-inner fusion system (see non-inner fusion system for quaternion group), Sylow number 5 3-Sylow: cyclic group:Z3 with its non-inner fusion system, Sylow number 10 5-Sylow: cyclic group:Z5, Sylow number 6
Hall subgroups	Other than the whole group, trivial subgroup, and Sylow subgroups, there is a $\{2,3\}$-Hall subgroup of order 24 (SL(2,3) in SL(2,5)). There is no $\{2,5\}$-Hall subgroup or $\{3,5\}$-Hall subgroup.
maximal subgroups	There are maximal subgroups of order 12 (index 10), order 20 (index 6) and order 24 (index 5).
normal subgroups	The only proper nontrivial normal subgroup is center of special linear group:SL(2,5), which is isomorphic to cyclic group:Z2 and the quotient group is isomorphic to alternating group:A5.

For which of the following divisors of 120 does there not exist a subgroup of $SL(2,5)$ with that as order?

	10
	12
	20
	24
	30