Subgroups — ΓL(1,𝔽₈)×D₇₅₃ #44539439

Order 31626.

36 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×ΓL(1,𝔽₈)×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	ΓL(1,𝔽₈)×D₂₅₁	—	3	1	no	maximal
3	C₇×D₇₅₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×C₂₅₁	C₂×C₃	1	1	yes	characteristic, complemented
5	ΓL(1,𝔽₈)×C₂₅₁	—	2	1	no	—
6	ΓL(1,𝔽₈)×C₂₅₁	D₃	1	1	yes	characteristic, complemented
7	C₃×D₇₅₃	—	7	1	no	maximal, Hall
8	C₇×D₂₅₁	—	3	1	no	Hall
9	C₃²×C₂₅₁	—	7	1	no	Hall
10	C₇×C₂₅₁	C₃×D₃	1	1	yes	characteristic, complemented, Hall
11	D₇₅₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
12	C₃×D₂₅₁	—	21	1	no	—
13	C₃×C₂₅₁	—	7	1	no	—
14	C₃×C₂₅₁	—	14	1	no	—
15	C₃×C₂₅₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
16	D₂₅₁	—	3	1	no	Hall
17	C₂₅₁	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
18	D₃×ΓL(1,𝔽₈)	—	251	1	no	maximal, Hall
19	C₃×ΓL(1,𝔽₈)	D₂₅₁	1	1	yes	characteristic, complemented, Hall
20	D₃×C₇	—	251	1	no	—
21	C₂×ΓL(1,𝔽₈)	—	753	1	no	—
22	C₃×C₇	C₃×D₂₅₁	1	1	yes	characteristic, complemented
23	ΓL(1,𝔽₈)	—	2	1	no	—
24	ΓL(1,𝔽₈)	D₇₅₃	1	1	yes	characteristic, direct product factor
25	C₃×D₃	—	1757	1	no	Hall
26	C₂×C₇	—	753	1	no	Hall
27	C₃²	—	7	1	no	Sylow
28	C₇	C₃×D₇₅₃	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₃	—	5271	1	no	—
30	D₃	—	251	1	no	—
31	C₃	—	7	1	no	—
32	C₃	—	14	1	no	—
33	C₃	ΓL(1,𝔽₈)×D₂₅₁	1	1	yes	characteristic, complemented
34	C₂	—	753	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₂₅₁	C₂×C₃
LCS	G ▷ C₃×C₇×C₂₅₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×ΓL(1,𝔽₈)×C₂₅₁	C₂	semidirect
3. C₇×D₇₅₃	C₃	semidirect
4. C₃×C₇×C₂₅₁	C₂×C₃	semidirect
6. ΓL(1,𝔽₈)×C₂₅₁	D₃	semidirect
10. C₇×C₂₅₁	C₃×D₃	semidirect
11. D₇₅₃	ΓL(1,𝔽₈)	direct (N × H)
15. C₃×C₂₅₁	C₂×ΓL(1,𝔽₈)	semidirect
17. C₂₅₁	D₃×ΓL(1,𝔽₈)	semidirect
19. C₃×ΓL(1,𝔽₈)	D₂₅₁	semidirect
22. C₃×C₇	C₃×D₂₅₁	semidirect
24. ΓL(1,𝔽₈)	D₇₅₃	direct (N × H)
28. C₇	C₃×D₇₅₃	semidirect
33. C₃	ΓL(1,𝔽₈)×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₃×ΓL(1,𝔽₈)×C₂₅₁	2
ΓL(1,𝔽₈)×D₂₅₁	3
C₇×D₇₅₃	3
C₃×D₇₅₃	7
D₃×ΓL(1,𝔽₈)	251

Subgroups — ΓL(1,𝔽8)×D753 #44539439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₇₅₃ #44539439