Subgroups — C₇×ΓL(1,𝔽₈)×D₃₇ #24925943

Order 10878.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×ΓL(1,𝔽₈)×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇²×D₃₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇²×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₃×C₇×D₃₇	—	7	1	no	maximal
5	ΓL(1,𝔽₈)×D₃₇	C₇	1	1	yes	maximal, characteristic, direct product factor
6	C₃×C₇×C₃₇	—	7	1	no	—
7	ΓL(1,𝔽₈)×C₃₇	C₂×C₇	1	1	yes	characteristic, complemented
8	C₇×D₃₇	—	3	2	no	—
9	C₇×D₃₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
10	C₇×D₃₇	C₃×C₇	1	1	yes	characteristic, complemented
11	C₂×C₇×ΓL(1,𝔽₈)	—	37	1	no	maximal, Hall
12	C₇×C₃₇	—	3	2	no	—
13	C₇×C₃₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
14	C₇×C₃₇	C₂×C₃×C₇	1	1	yes	characteristic, complemented
15	C₃×D₃₇	—	7	1	no	Hall
16	C₇×ΓL(1,𝔽₈)	D₃₇	1	1	yes	characteristic, direct product factor, Hall
17	C₃×C₃₇	—	7	1	no	Hall
18	C₂×C₇²	—	37	1	no	Hall
19	D₃₇	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
20	C₇²	C₃×D₃₇	1	1	yes	characteristic, complemented, Sylow
21	C₂×C₃×C₇	—	259	1	no	—
22	C₂×ΓL(1,𝔽₈)	—	37	1	no	—
23	C₃₇	C₂×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
24	C₃×C₇	—	7	1	no	—
25	ΓL(1,𝔽₈)	C₇×D₃₇	1	1	yes	characteristic, direct product factor
26	C₂×C₇	—	37	1	no	—
27	C₂×C₇	—	37	1	no	—
28	C₂×C₇	—	111	2	no	—
29	C₇	—	3	2	no	—
30	C₇	ΓL(1,𝔽₈)×D₃₇	1	1	yes	characteristic, direct product factor
31	C₇	C₃×C₇×D₃₇	1	1	yes	characteristic, complemented
32	C₂×C₃	—	259	1	no	Hall
33	C₃	—	7	1	no	Sylow
34	C₂	—	37	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	ΓL(1,𝔽₈)×D₃₇
commutator, cosocle, hypocenter	C₇×C₃₇	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₇²×C₃₇	C₂×C₃
LCS	G ▷ C₇×C₃₇
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₇×ΓL(1,𝔽₈)×C₃₇	C₂	semidirect
2. C₇²×D₃₇	C₃	semidirect
3. C₇²×C₃₇	C₂×C₃	semidirect
5. ΓL(1,𝔽₈)×D₃₇	C₇	direct (N × H)
7. ΓL(1,𝔽₈)×C₃₇	C₂×C₇	semidirect
9. C₇×D₃₇	ΓL(1,𝔽₈)	direct (N × H)
10. C₇×D₃₇	C₃×C₇	semidirect
13. C₇×C₃₇	C₂×ΓL(1,𝔽₈)	semidirect
14. C₇×C₃₇	C₂×C₃×C₇	semidirect
16. C₇×ΓL(1,𝔽₈)	D₃₇	direct (N × H)
19. D₃₇	C₇×ΓL(1,𝔽₈)	direct (N × H)
20. C₇²	C₃×D₃₇	semidirect
23. C₃₇	C₂×C₇×ΓL(1,𝔽₈)	semidirect
25. ΓL(1,𝔽₈)	C₇×D₃₇	direct (N × H)
30. C₇	ΓL(1,𝔽₈)×D₃₇	direct (N × H)
31. C₇	C₃×C₇×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₇×ΓL(1,𝔽₈)×C₃₇	2
C₇²×D₃₇	3
C₃×C₇×D₃₇	7
ΓL(1,𝔽₈)×D₃₇	7
C₂×C₇×ΓL(1,𝔽₈)	37

Subgroups — C7×ΓL(1,𝔽8)×D37 #24925943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×ΓL(1,𝔽₈)×D₃₇ #24925943