Subgroups — S₅ #1224

Order 120.

19 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	A₅	C₂	1	1	yes	maximal, characteristic, complemented
2	S₄	—	5	1	no	maximal, Hall
3	AGL(1,𝔽₅)	—	6	1	no	maximal
4	C₂×D₃	—	10	1	no	maximal
5	A₄	—	5	1	no	—
6	D₅	—	6	1	no	—
7	D₄	—	15	1	no	Sylow
8	C₂×C₃	—	10	1	no	—
9	D₃	—	10	1	no	—
10	D₃	—	10	1	no	—
11	C₅	—	6	1	no	Sylow
12	C₂²	—	5	1	no	—
13	C₂²	—	15	1	no	—
14	C₄	—	15	1	no	—
15	C₃	—	10	1	no	Sylow
16	C₂	—	10	1	no	—
17	C₂	—	15	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, socle, cosocle, hypocenter	A₅	C₂
LCS	G ▷ A₅
UCS	C₁

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

N	H ≅ G/N	type
1. A₅	C₂	semidirect

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