Subgroups — C₇₉×#64423 #37551142

Order 23226.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₉×#3861	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×D₇×C₇₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇²×C₇₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	AGL(1,𝔽₇)×C₇₉	—	7	1	no	maximal
5	C₂×ΓL(1,𝔽₈)×C₇₉	—	7	1	no	maximal
6	ΓL(1,𝔽₈)×C₇₉	—	7	1	no	—
7	ΓL(1,𝔽₈)×C₇₉	—	7	1	no	—
8	ΓL(1,𝔽₈)×C₇₉	—	14	3	no	—
9	C₂×C₇×C₇₉	—	7	1	no	—
10	D₇×C₇₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
11	C₇×C₇₉	—	2	3	no	—
12	C₇×C₇₉	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
13	C₇×C₇₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
14	C₂×C₃×C₇₉	—	49	1	no	Hall
15	#64423	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₃×C₇₉	—	49	1	no	Hall
17	C₂×C₇₉	—	7	1	no	Hall
18	#3861	C₂×C₇₉	1	1	yes	characteristic, complemented, Hall
19	C₇×D₇	C₃×C₇₉	1	1	yes	characteristic, complemented, Hall
20	C₇₉	#64423	1	1	yes	characteristic, direct product factor, Sylow
21	C₇²	C₂×C₃×C₇₉	1	1	yes	characteristic, complemented, Sylow
22	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
23	AGL(1,𝔽₇)	—	7	1	no	—
24	ΓL(1,𝔽₈)	—	7	1	no	—
25	ΓL(1,𝔽₈)	—	7	1	no	—
26	ΓL(1,𝔽₈)	—	14	3	no	—
27	C₂×C₇	—	7	1	no	—
28	D₇	ΓL(1,𝔽₈)×C₇₉	1	1	yes	characteristic, complemented
29	C₇	—	2	3	no	—
30	C₇	C₂×ΓL(1,𝔽₈)×C₇₉	1	1	yes	characteristic, complemented
31	C₇	AGL(1,𝔽₇)×C₇₉	1	1	yes	characteristic, complemented
32	C₂×C₃	—	49	1	no	Hall
33	C₃	—	49	1	no	Sylow
34	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇₉	#64423
commutator, cosocle, hypocenter	C₇²	C₂×C₃×C₇₉
Frattini	C₁	G
Fitting, socle	C₇²×C₇₉	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₇₉

Complements

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

N	H ≅ G/N	type
1. C₇₉×#3861	C₂	semidirect
2. C₇×D₇×C₇₉	C₃	semidirect
3. C₇²×C₇₉	C₂×C₃	semidirect
10. D₇×C₇₉	ΓL(1,𝔽₈)	semidirect
12. C₇×C₇₉	AGL(1,𝔽₇)	semidirect
13. C₇×C₇₉	C₂×ΓL(1,𝔽₈)	semidirect
15. #64423	C₇₉	direct (N × H)
18. #3861	C₂×C₇₉	semidirect
19. C₇×D₇	C₃×C₇₉	semidirect
20. C₇₉	#64423	direct (N × H)
21. C₇²	C₂×C₃×C₇₉	semidirect
28. D₇	ΓL(1,𝔽₈)×C₇₉	semidirect
30. C₇	C₂×ΓL(1,𝔽₈)×C₇₉	semidirect
31. C₇	AGL(1,𝔽₇)×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₇₉×#3861	2
C₇×D₇×C₇₉	3
AGL(1,𝔽₇)×C₇₉	7
C₂×ΓL(1,𝔽₈)×C₇₉	7
#64423	79

Subgroups — C79×#64423 #37551142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#64423 #37551142