Subgroups — #3862

Order 147.

10 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
2	ΓL(1,𝔽₈)	—	7	2	no	maximal
3	C₇	—	3	2	no	—
4	C₇	ΓL(1,𝔽₈)	1	2	yes	complemented
5	C₃	—	49	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²	C₃
LCS	G ▷ C₇²
UCS	C₁

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

N	H ≅ G/N	type
1. C₇²	C₃	semidirect
4. C₇	ΓL(1,𝔽₈)	semidirect

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

subgroup	index
C₇²	3
ΓL(1,𝔽₈)	7