Subgroups — AGL(1,𝔽₈) #292

Order 56.

6 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³	C₇	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₇	—	8	1	no	maximal, Sylow
3	C₂²	—	7	1	no	—
4	C₂	—	7	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂³	C₇
LCS	G ▷ C₂³
UCS	C₁

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

N	H ≅ G/N	type
1. C₂³	C₇	semidirect

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

subgroup	index
C₂³	7
C₇	8