Subgroups — #87656

Order 392.

12 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5997	C₂	1	1	yes	maximal, characteristic
2	C₇⋊D₇	C₄	1	1	yes	characteristic
3	C₇²	C₈	1	1	yes	characteristic, complemented, Sylow
4	D₇	—	28	2	no	—
5	C₈	—	49	1	no	maximal, Sylow
6	C₇	—	4	2	no	—
7	C₄	—	49	1	no	—
8	C₂	—	49	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇²	C₈
prime radical	C₇⋊D₇	C₄
cosocle	#5997	C₂
LCS	G ▷ C₇²
UCS	C₁

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

N	H ≅ G/N	type
3. C₇²	C₈	semidirect

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

subgroup	index
#5997	2
C₈	49