Subgroups — ΓL(1,𝔽₈)×D₁₅₁⋊C₅ #44540293

Order 31710.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×C₁₅₁⋊C₅	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×D₁₅₁⋊C₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	ΓL(1,𝔽₈)×D₁₅₁	C₅	1	1	yes	maximal, characteristic, complemented, Hall
4	C₇×C₁₅₁⋊C₅	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	C₃×D₁₅₁⋊C₅	—	7	1	no	maximal, Hall
6	ΓL(1,𝔽₈)×C₁₅₁	C₂×C₅	1	1	yes	characteristic, complemented, Hall
7	C₃×C₁₅₁⋊C₅	—	7	1	no	Hall
8	C₇×D₁₅₁	C₃×C₅	1	1	yes	characteristic, complemented, Hall
9	D₁₅₁⋊C₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
10	C₇×C₁₅₁	C₂×C₃×C₅	1	1	yes	characteristic, complemented, Hall
11	C₃×D₁₅₁	—	7	1	no	Hall
12	C₁₅₁⋊C₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
13	C₃×C₁₅₁	—	7	1	no	Hall
14	D₁₅₁	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
15	C₂×C₅×ΓL(1,𝔽₈)	—	151	1	no	maximal, Hall
16	C₁₅₁	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
17	C₅×ΓL(1,𝔽₈)	—	151	1	no	Hall
18	C₂×C₅×C₇	—	151	1	no	Hall
19	C₂×ΓL(1,𝔽₈)	—	151	1	no	Hall
20	C₅×C₇	—	151	1	no	Hall
21	C₂×C₃×C₅	—	1057	1	no	Hall
22	ΓL(1,𝔽₈)	D₁₅₁⋊C₅	1	1	yes	characteristic, direct product factor, Hall
23	C₃×C₅	—	1057	1	no	Hall
24	C₂×C₇	—	151	1	no	Hall
25	C₂×C₅	—	151	1	no	Hall
26	C₇	C₃×D₁₅₁⋊C₅	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₃	—	1057	1	no	Hall
28	C₅	—	151	1	no	Sylow
29	C₃	—	7	1	no	Sylow
30	C₂	—	151	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₅₁	C₂×C₃×C₅
LCS	G ▷ C₇×C₁₅₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×C₁₅₁⋊C₅	C₂	semidirect
2. C₇×D₁₅₁⋊C₅	C₃	semidirect
3. ΓL(1,𝔽₈)×D₁₅₁	C₅	semidirect
4. C₇×C₁₅₁⋊C₅	C₂×C₃	semidirect
6. ΓL(1,𝔽₈)×C₁₅₁	C₂×C₅	semidirect
8. C₇×D₁₅₁	C₃×C₅	semidirect
9. D₁₅₁⋊C₅	ΓL(1,𝔽₈)	direct (N × H)
10. C₇×C₁₅₁	C₂×C₃×C₅	semidirect
12. C₁₅₁⋊C₅	C₂×ΓL(1,𝔽₈)	semidirect
14. D₁₅₁	C₅×ΓL(1,𝔽₈)	semidirect
16. C₁₅₁	C₂×C₅×ΓL(1,𝔽₈)	semidirect
22. ΓL(1,𝔽₈)	D₁₅₁⋊C₅	direct (N × H)
26. C₇	C₃×D₁₅₁⋊C₅	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×C₁₅₁⋊C₅	2
C₇×D₁₅₁⋊C₅	3
ΓL(1,𝔽₈)×D₁₅₁	5
C₃×D₁₅₁⋊C₅	7
C₂×C₅×ΓL(1,𝔽₈)	151

Subgroups — ΓL(1,𝔽8)×D151⋊C5 #44540293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₁₅₁⋊C₅ #44540293