Subgroups — #90632

Order 448.

411 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶	C₇	1	1	yes	maximal, characteristic, complemented, Sylow
2	AGL(1,𝔽₈)	—	8	2	no	maximal
3	C₂⁵	—	7	7	no	—
4	C₂⁵	—	7	2	no	—
5	C₂⁴	—	7	21	no	—
6	C₂⁴	—	7	21	no	—
7	C₂⁴	—	7	42	no	—
8	C₂⁴	—	7	7	no	—
9	C₂⁴	—	7	2	no	—
10	C₂³	—	7	21	no	—
11	C₂³	—	7	14	no	—
12	C₂³	—	7	14	no	—
13	C₂³	—	7	42	no	—
14	C₂³	—	7	42	no	—
15	C₂³	—	7	63	no	—
16	C₂³	—	7	3	no	—
17	C₂³	AGL(1,𝔽₈)	1	2	yes	complemented
18	C₇	—	64	1	no	Sylow
19	C₂²	—	7	2	no	—
20	C₂²	—	7	21	no	—
21	C₂²	—	7	21	no	—
22	C₂²	—	7	42	no	—
23	C₂²	—	7	7	no	—
24	C₂	—	7	7	no	—
25	C₂	—	7	2	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂⁶	C₇
LCS	G ▷ C₂⁶
UCS	C₁

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

N	H ≅ G/N	type
1. C₂⁶	C₇	semidirect
17. C₂³	AGL(1,𝔽₈)	semidirect

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

subgroup	index
C₂⁶	7
AGL(1,𝔽₈)	8