Subgroups — H(3,𝔽₇) #66803

Order 343.

19 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²	C₇	1	8	yes	maximal, complemented
2	C₇	—	7	8	no	—
3	C₇	C₇²	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₇	C₇²
LCS	G ▷ C₇ ▷ C₁
UCS	C₁ ◁ C₇ ◁ G
LECS	G ▷ C₇ ▷ C₁

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

N	H ≅ G/N	type
1. C₇²	C₇	semidirect

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

subgroup	index
C₇²	7