Subgroups — C₅×ΓL(1,𝔽₈)×D₁₅₁ #44540265

Order 31710.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₅	ΓL(1,𝔽₈)×D₁₅₁
commutator, cosocle, hypocenter	C₇×C₁₅₁	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₅₁	C₂×C₃
LCS	G ▷ C₇×C₁₅₁
UCS	C₁ ◁ C₅

Complements

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

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

Maximal subgroups

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

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

Subgroups — C5×ΓL(1,𝔽8)×D151 #44540265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×ΓL(1,𝔽₈)×D₁₅₁ #44540265