Subgroups — C₇×AGL(1,𝔽₇)×C₇₉ #37551113

Order 23226.

34 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	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	C₇	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃×C₇×C₇₉	—	7	1	no	maximal
6	C₃×C₇×C₇₉	—	7	1	no	—
7	ΓL(1,𝔽₈)×C₇₉	C₂×C₇	1	1	yes	characteristic, complemented
8	C₂×C₇×C₇₉	—	7	1	no	—
9	D₇×C₇₉	C₃×C₇	1	1	yes	characteristic, complemented
10	C₇×C₇₉	—	6	1	no	—
11	C₇×C₇₉	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
12	C₇×C₇₉	C₂×C₃×C₇	1	1	yes	characteristic, complemented
13	C₂×C₃×C₇₉	—	7	1	no	Hall
14	C₇×AGL(1,𝔽₇)	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₃×C₇₉	—	7	1	no	Hall
16	C₂×C₇₉	—	7	1	no	Hall
17	C₇×ΓL(1,𝔽₈)	C₂×C₇₉	1	1	yes	characteristic, complemented, Hall
18	C₇×D₇	C₃×C₇₉	1	1	yes	characteristic, complemented, Hall
19	C₇₉	C₇×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
20	C₇²	C₂×C₃×C₇₉	1	1	yes	characteristic, complemented, Sylow
21	C₂×C₃×C₇	—	7	1	no	—
22	AGL(1,𝔽₇)	C₇×C₇₉	1	1	yes	characteristic, direct product factor
23	C₃×C₇	—	7	1	no	—
24	ΓL(1,𝔽₈)	C₂×C₇×C₇₉	1	1	yes	characteristic, complemented
25	C₂×C₇	—	7	1	no	—
26	D₇	C₃×C₇×C₇₉	1	1	yes	characteristic, complemented
27	C₇	—	6	1	no	—
28	C₇	C₂×C₃×C₇×C₇₉	1	1	yes	characteristic, complemented
29	C₇	AGL(1,𝔽₇)×C₇₉	1	1	yes	characteristic, direct product factor
30	C₂×C₃	—	7	1	no	Hall
31	C₃	—	7	1	no	Sylow
32	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇×C₇₉	AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₇×C₇₉
Frattini	C₁	G
Fitting, socle	C₇²×C₇₉	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₇×C₇₉

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. C₇×D₇×C₇₉	C₃	semidirect
3. C₇²×C₇₉	C₂×C₃	semidirect
4. AGL(1,𝔽₇)×C₇₉	C₇	direct (N × H)
7. ΓL(1,𝔽₈)×C₇₉	C₂×C₇	semidirect
9. D₇×C₇₉	C₃×C₇	semidirect
11. C₇×C₇₉	AGL(1,𝔽₇)	direct (N × H)
12. C₇×C₇₉	C₂×C₃×C₇	semidirect
14. C₇×AGL(1,𝔽₇)	C₇₉	direct (N × H)
17. C₇×ΓL(1,𝔽₈)	C₂×C₇₉	semidirect
18. C₇×D₇	C₃×C₇₉	semidirect
19. C₇₉	C₇×AGL(1,𝔽₇)	direct (N × H)
20. C₇²	C₂×C₃×C₇₉	semidirect
22. AGL(1,𝔽₇)	C₇×C₇₉	direct (N × H)
24. ΓL(1,𝔽₈)	C₂×C₇×C₇₉	semidirect
26. D₇	C₃×C₇×C₇₉	semidirect
28. C₇	C₂×C₃×C₇×C₇₉	semidirect
29. C₇	AGL(1,𝔽₇)×C₇₉	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
C₇×D₇×C₇₉	3
AGL(1,𝔽₇)×C₇₉	7
C₂×C₃×C₇×C₇₉	7
C₇×AGL(1,𝔽₇)	79

Subgroups — C7×AGL(1,𝔽7)×C79 #37551113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×AGL(1,𝔽₇)×C₇₉ #37551113