Subgroups — H(3,𝔽₃) #81

Order 27.

11 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²	C₃	1	4	yes	maximal, complemented
2	C₃	—	3	4	no	—
3	C₃	C₃²	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃	C₃²
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ G
LECS	G ▷ C₃ ▷ C₁

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

N	H ≅ G/N	type
1. C₃²	C₃	semidirect

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

subgroup	index
C₃²	3