Subgroups — C₇×ΓL(1,𝔽₈)×D₇₉ #37551112

Order 23226.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×ΓL(1,𝔽₈)×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇²×D₇₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇²×C₇₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	ΓL(1,𝔽₈)×D₇₉	C₇	1	1	yes	maximal, characteristic, direct product factor
5	C₃×C₇×D₇₉	—	7	1	no	maximal
6	C₃×C₇×C₇₉	—	7	1	no	—
7	ΓL(1,𝔽₈)×C₇₉	C₂×C₇	1	1	yes	characteristic, complemented
8	C₇×D₇₉	—	3	2	no	—
9	C₇×D₇₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
10	C₇×D₇₉	C₃×C₇	1	1	yes	characteristic, complemented
11	C₇×C₇₉	—	3	2	no	—
12	C₇×C₇₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
13	C₇×C₇₉	C₂×C₃×C₇	1	1	yes	characteristic, complemented
14	C₃×D₇₉	—	7	1	no	Hall
15	C₂×C₇×ΓL(1,𝔽₈)	—	79	1	no	maximal, Hall
16	C₃×C₇₉	—	7	1	no	Hall
17	D₇₉	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
18	C₇×ΓL(1,𝔽₈)	D₇₉	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₇²	—	79	1	no	Hall
20	C₇₉	C₂×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
21	C₇²	C₃×D₇₉	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₃×C₇	—	553	1	no	—
23	C₂×ΓL(1,𝔽₈)	—	79	1	no	—
24	C₃×C₇	—	7	1	no	—
25	ΓL(1,𝔽₈)	C₇×D₇₉	1	1	yes	characteristic, direct product factor
26	C₂×C₇	—	79	1	no	—
27	C₂×C₇	—	79	1	no	—
28	C₂×C₇	—	237	2	no	—
29	C₇	—	3	2	no	—
30	C₇	C₃×C₇×D₇₉	1	1	yes	characteristic, complemented
31	C₇	ΓL(1,𝔽₈)×D₇₉	1	1	yes	characteristic, direct product factor
32	C₂×C₃	—	553	1	no	Hall
33	C₃	—	7	1	no	Sylow
34	C₂	—	79	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	ΓL(1,𝔽₈)×D₇₉
commutator, cosocle, hypocenter	C₇×C₇₉	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₇²×C₇₉	C₂×C₃
LCS	G ▷ C₇×C₇₉
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₇×ΓL(1,𝔽₈)×C₇₉	C₂	semidirect
2. C₇²×D₇₉	C₃	semidirect
3. C₇²×C₇₉	C₂×C₃	semidirect
4. ΓL(1,𝔽₈)×D₇₉	C₇	direct (N × H)
7. ΓL(1,𝔽₈)×C₇₉	C₂×C₇	semidirect
9. C₇×D₇₉	ΓL(1,𝔽₈)	direct (N × H)
10. C₇×D₇₉	C₃×C₇	semidirect
12. C₇×C₇₉	C₂×ΓL(1,𝔽₈)	semidirect
13. C₇×C₇₉	C₂×C₃×C₇	semidirect
17. D₇₉	C₇×ΓL(1,𝔽₈)	direct (N × H)
18. C₇×ΓL(1,𝔽₈)	D₇₉	direct (N × H)
20. C₇₉	C₂×C₇×ΓL(1,𝔽₈)	semidirect
21. C₇²	C₃×D₇₉	semidirect
25. ΓL(1,𝔽₈)	C₇×D₇₉	direct (N × H)
30. C₇	C₃×C₇×D₇₉	semidirect
31. C₇	ΓL(1,𝔽₈)×D₇₉	direct (N × H)

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)×C₇₉	2
C₇²×D₇₉	3
ΓL(1,𝔽₈)×D₇₉	7
C₃×C₇×D₇₉	7
C₂×C₇×ΓL(1,𝔽₈)	79

Subgroups — C7×ΓL(1,𝔽8)×D79 #37551112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×ΓL(1,𝔽₈)×D₇₉ #37551112