Subgroups — C₂×C₁₇×ΓL(1,𝔽₈)×C₄₇ #46776751

Order 33558.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×ΓL(1,𝔽₈)×C₄₇	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₇×C₁₇×C₄₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₁₇×C₄₇	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₁₇×C₄₇	—	7	1	no	maximal, Hall
5	C₃×C₁₇×C₄₇	—	7	1	no	Hall
6	C₂×ΓL(1,𝔽₈)×C₄₇	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₁₇×C₄₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
8	ΓL(1,𝔽₈)×C₄₇	C₂×C₁₇	1	1	yes	characteristic, direct product factor, Hall
9	C₁₇×C₄₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₁₇×ΓL(1,𝔽₈)	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₇×C₄₇	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
12	C₁₇×ΓL(1,𝔽₈)	C₂×C₄₇	1	1	yes	characteristic, direct product factor, Hall
13	C₇×C₄₇	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₄₇	—	7	1	no	Hall
15	C₂×C₇×C₁₇	C₃×C₄₇	1	1	yes	characteristic, complemented, Hall
16	C₃×C₄₇	—	7	1	no	Hall
17	C₇×C₁₇	C₂×C₃×C₄₇	1	1	yes	characteristic, complemented, Hall
18	C₂×C₃×C₁₇	—	7	1	no	Hall
19	C₂×C₄₇	C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
20	C₃×C₁₇	—	7	1	no	Hall
21	C₄₇	C₂×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
22	C₂×ΓL(1,𝔽₈)	C₁₇×C₄₇	1	1	yes	characteristic, direct product factor, Hall
23	C₂×C₁₇	ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, direct product factor, Hall
24	ΓL(1,𝔽₈)	C₂×C₁₇×C₄₇	1	1	yes	characteristic, direct product factor, Hall
25	C₁₇	C₂×ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, direct product factor, Sylow
26	C₂×C₇	C₃×C₁₇×C₄₇	1	1	yes	characteristic, complemented, Hall
27	C₇	C₂×C₃×C₁₇×C₄₇	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃	—	7	1	no	Hall
29	C₃	—	7	1	no	Sylow
30	C₂	C₁₇×ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₁₇×C₄₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₁₇×C₄₇
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₇×C₄₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₁₇×C₄₇

Complements

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

N	H ≅ G/N	type
1. C₁₇×ΓL(1,𝔽₈)×C₄₇	C₂	direct (N × H)
2. C₂×C₇×C₁₇×C₄₇	C₃	semidirect
3. C₇×C₁₇×C₄₇	C₂×C₃	semidirect
6. C₂×ΓL(1,𝔽₈)×C₄₇	C₁₇	direct (N × H)
7. C₂×C₁₇×C₄₇	ΓL(1,𝔽₈)	direct (N × H)
8. ΓL(1,𝔽₈)×C₄₇	C₂×C₁₇	direct (N × H)
9. C₁₇×C₄₇	C₂×ΓL(1,𝔽₈)	direct (N × H)
10. C₂×C₁₇×ΓL(1,𝔽₈)	C₄₇	direct (N × H)
11. C₂×C₇×C₄₇	C₃×C₁₇	semidirect
12. C₁₇×ΓL(1,𝔽₈)	C₂×C₄₇	direct (N × H)
13. C₇×C₄₇	C₂×C₃×C₁₇	semidirect
15. C₂×C₇×C₁₇	C₃×C₄₇	semidirect
17. C₇×C₁₇	C₂×C₃×C₄₇	semidirect
19. C₂×C₄₇	C₁₇×ΓL(1,𝔽₈)	direct (N × H)
21. C₄₇	C₂×C₁₇×ΓL(1,𝔽₈)	direct (N × H)
22. C₂×ΓL(1,𝔽₈)	C₁₇×C₄₇	direct (N × H)
23. C₂×C₁₇	ΓL(1,𝔽₈)×C₄₇	direct (N × H)
24. ΓL(1,𝔽₈)	C₂×C₁₇×C₄₇	direct (N × H)
25. C₁₇	C₂×ΓL(1,𝔽₈)×C₄₇	direct (N × H)
26. C₂×C₇	C₃×C₁₇×C₄₇	semidirect
27. C₇	C₂×C₃×C₁₇×C₄₇	semidirect
30. C₂	C₁₇×ΓL(1,𝔽₈)×C₄₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₇×ΓL(1,𝔽₈)×C₄₇	2
C₂×C₇×C₁₇×C₄₇	3
C₂×C₃×C₁₇×C₄₇	7
C₂×ΓL(1,𝔽₈)×C₄₇	17
C₂×C₁₇×ΓL(1,𝔽₈)	47

Subgroups — C2×C17×ΓL(1,𝔽8)×C47 #46776751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×ΓL(1,𝔽₈)×C₄₇ #46776751