Subgroups — (C₂²⋊C₉)⋊C₂ #618

Order 72.

18 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#67	—	3	1	no	maximal
3	D₉	—	4	1	no	maximal
4	C₂²×C₃	D₃	1	1	yes	characteristic
5	C₂×D₃	—	3	1	no	—
6	Dic₃	—	3	1	no	—
7	C₉	—	4	1	no	Sylow
8	D₄	—	9	1	no	Sylow
9	C₂×C₃	—	3	1	no	—
10	D₃	—	6	1	no	—
11	C₂²	—	9	1	no	—
12	C₂²	D₉	1	1	yes	characteristic, complemented
13	C₄	—	9	1	no	—
14	C₃	S₄	1	1	yes	characteristic
15	C₂	—	3	1	no	—
16	C₂	—	18	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²⋊C₉	C₂
Frattini	C₃	S₄
Fitting, socle	C₂²×C₃	D₃
LCS	G ▷ C₂²⋊C₉
UCS	C₁

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

N	H ≅ G/N	type
1. C₂²⋊C₉	C₂	semidirect
12. C₂²	D₉	semidirect

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