Subgroups — C₇⋊(C₇⋊C₈) #87635

Order 392.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇²	C₄	1	1	yes	characteristic
3	C₇⋊C₈	—	7	8	no	maximal
4	C₇²	C₈	1	1	yes	characteristic, complemented, Sylow
5	C₄×C₇	D₇	1	8	yes	—
6	C₂×C₇	Dic₇	1	8	yes	—
7	C₈	—	49	1	no	Sylow
8	C₇	C₇⋊C₈	1	8	yes	complemented
9	C₄	C₇⋊D₇	1	1	yes	characteristic
10	C₂	C₇⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	C₇⋊D₇
commutator, hypocenter	C₇²	C₈
Fitting, cosocle	C₄×C₇²	C₂
socle, prime radical	C₂×C₇²	C₄
LCS	G ▷ C₇²
UCS	C₁ ◁ C₄

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

N	H ≅ G/N	type
4. C₇²	C₈	semidirect
8. C₇	C₇⋊C₈	semidirect

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

subgroup	index
C₄×C₇²	2
C₇⋊C₈	7