Subgroups — #87658

Order 392.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5997	C₂	1	3	yes	maximal
2	C₇⋊D₇	C₂²	1	1	yes	characteristic
3	C₇²	Q₈	1	1	yes	characteristic, complemented, Sylow
4	D₇	—	28	2	no	—
5	Q₈	—	49	1	no	maximal, Sylow
6	C₇	—	4	2	no	—
7	C₄	—	49	3	no	—
8	C₂	—	49	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, prime radical, cosocle	C₇⋊D₇	C₂²
Fitting, socle, hypocenter	C₇²	Q₈
LCS	G ▷ C₇⋊D₇ ▷ C₇²
UCS	C₁

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

N	H ≅ G/N	type
3. C₇²	Q₈	semidirect

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

subgroup	index
#5997	2
Q₈	49