Subgroups — C₁₅₁×C₅⋊AGL(1,𝔽₇) #44540285

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	D₃₅×C₁₅₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	AGL(1,𝔽₇)×C₁₅₁	—	5	1	no	maximal, Hall
4	C₅×C₇×C₁₅₁	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	C₃×D₅×C₁₅₁	—	7	1	no	maximal, Hall
6	ΓL(1,𝔽₈)×C₁₅₁	D₅	1	1	yes	characteristic, complemented, Hall
7	C₃×C₅×C₁₅₁	—	7	1	no	Hall
8	D₇×C₁₅₁	—	5	1	no	Hall
9	D₅×C₁₅₁	—	7	1	no	Hall
10	C₇×C₁₅₁	C₃×D₅	1	1	yes	characteristic, complemented, Hall
11	C₂×C₃×C₁₅₁	—	35	1	no	Hall
12	C₅×C₁₅₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
13	C₃×C₁₅₁	—	7	1	no	Hall
14	C₂×C₁₅₁	—	35	1	no	Hall
15	C₅⋊AGL(1,𝔽₇)	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₁₅₁	C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
17	C₅×ΓL(1,𝔽₈)	C₂×C₁₅₁	1	1	yes	characteristic, complemented, Hall
18	D₃₅	C₃×C₁₅₁	1	1	yes	characteristic, complemented, Hall
19	AGL(1,𝔽₇)	—	5	1	no	Hall
20	C₅×C₇	C₂×C₃×C₁₅₁	1	1	yes	characteristic, complemented, Hall
21	C₃×D₅	—	7	1	no	Hall
22	ΓL(1,𝔽₈)	D₅×C₁₅₁	1	1	yes	characteristic, complemented, Hall
23	C₃×C₅	—	7	1	no	Hall
24	D₇	—	5	1	no	Hall
25	D₅	—	7	1	no	Hall
26	C₇	C₃×D₅×C₁₅₁	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₃	—	35	1	no	Hall
28	C₅	AGL(1,𝔽₇)×C₁₅₁	1	1	yes	characteristic, complemented, Sylow
29	C₃	—	7	1	no	Sylow
30	C₂	—	35	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₅₁	C₅⋊AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₅×C₇	C₂×C₃×C₁₅₁
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₅₁	C₂×C₃
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₁₅₁

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. D₃₅×C₁₅₁	C₃	semidirect
4. C₅×C₇×C₁₅₁	C₂×C₃	semidirect
6. ΓL(1,𝔽₈)×C₁₅₁	D₅	semidirect
10. C₇×C₁₅₁	C₃×D₅	semidirect
12. C₅×C₁₅₁	AGL(1,𝔽₇)	semidirect
15. C₅⋊AGL(1,𝔽₇)	C₁₅₁	direct (N × H)
16. C₁₅₁	C₅⋊AGL(1,𝔽₇)	direct (N × H)
17. C₅×ΓL(1,𝔽₈)	C₂×C₁₅₁	semidirect
18. D₃₅	C₃×C₁₅₁	semidirect
20. C₅×C₇	C₂×C₃×C₁₅₁	semidirect
22. ΓL(1,𝔽₈)	D₅×C₁₅₁	semidirect
26. C₇	C₃×D₅×C₁₅₁	semidirect
28. C₅	AGL(1,𝔽₇)×C₁₅₁	semidirect

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
D₃₅×C₁₅₁	3
AGL(1,𝔽₇)×C₁₅₁	5
C₃×D₅×C₁₅₁	7
C₅⋊AGL(1,𝔽₇)	151

Subgroups — C151×C5⋊AGL(1,𝔽7) #44540285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×C₅⋊AGL(1,𝔽₇) #44540285