Subgroups — C₇⋊C₃₂ #6369

Order 224.

12 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₇×C₈	C₄	1	1	yes	characteristic
3	C₃₂	—	7	1	no	maximal, Sylow
4	C₄×C₇	C₈	1	1	yes	characteristic
5	C₁₆	D₇	1	1	yes	characteristic
6	C₂×C₇	C₁₆	1	1	yes	characteristic
7	C₈	Dic₇	1	1	yes	characteristic
8	C₇	C₃₂	1	1	yes	characteristic, complemented, Sylow
9	C₄	C₇⋊C₈	1	1	yes	characteristic
10	C₂	C₇⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₁₆	D₇
commutator, hypocenter	C₇	C₃₂
Fitting, cosocle	C₇×C₁₆	C₂
socle, prime radical	C₂×C₇	C₁₆
LCS	G ▷ C₇
UCS	C₁ ◁ C₁₆

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

N	H ≅ G/N	type
8. C₇	C₃₂	semidirect

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

subgroup	index
C₇×C₁₆	2
C₃₂	7