Subgroups — H(3,𝔽₅) #1249

Order 125.

15 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²	C₅	1	6	yes	maximal, complemented
2	C₅	—	5	6	no	—
3	C₅	C₅²	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₅	C₅²
LCS	G ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₅ ◁ G
LECS	G ▷ C₅ ▷ C₁

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

N	H ≅ G/N	type
1. C₅²	C₅	semidirect

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

subgroup	index
C₅²	5