Subgroups — #6729

Order 240.

25 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	SL(2,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
2	GL(2,𝔽₃)	—	5	1	no	maximal, Hall
3	#170	—	6	1	no	maximal
4	#67	—	10	1	no	maximal
5	SL(2,𝔽₃)	—	5	1	no	—
6	Dic₅	—	6	1	no	—
7	SD₁₆	—	15	1	no	Sylow
8	C₂²×C₃	—	10	1	no	—
9	C₂×D₃	—	10	1	no	—
10	Dic₃	—	10	1	no	—
11	C₂×C₅	—	6	1	no	—
12	Q₈	—	5	1	no	—
13	D₄	—	15	1	no	—
14	C₈	—	15	1	no	—
15	C₂×C₃	—	10	1	no	—
16	C₂×C₃	—	20	1	no	—
17	D₃	—	20	1	no	—
18	C₅	—	6	1	no	Sylow
19	C₂²	—	10	1	no	—
20	C₄	—	15	1	no	—
21	C₃	—	10	1	no	Sylow
22	C₂	—	20	1	no	—
23	C₂	S₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, Fitting, socle, hypercenter	C₂	S₅
commutator, cosocle, hypocenter	SL(2,𝔽₅)	C₂
LCS	G ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. SL(2,𝔽₅)	C₂	semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.