Subgroups — C₇×C₁₃×C₅⋊AGL(1,𝔽₇) #34062114

Order 19110.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×C₁₃×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₁₃×D₃₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₁₃×AGL(1,𝔽₇)	—	5	1	no	maximal, Hall
4	C₅×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	C₁₃×C₅⋊AGL(1,𝔽₇)	C₇	1	1	yes	maximal, characteristic, direct product factor
6	C₃×C₇×D₅×C₁₃	—	7	1	no	maximal
7	C₇×C₁₃×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, complemented, Hall
8	C₇×C₅⋊AGL(1,𝔽₇)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×C₇×C₁₃	—	7	1	no	—
10	C₅×C₁₃×ΓL(1,𝔽₈)	C₂×C₇	1	1	yes	characteristic, complemented
11	C₇×C₁₃×D₇	—	5	1	no	Hall
12	C₁₃×D₃₅	C₃×C₇	1	1	yes	characteristic, complemented
13	C₇×D₅×C₁₃	—	7	1	no	—
14	C₅×C₇×ΓL(1,𝔽₈)	C₂×C₁₃	1	1	yes	characteristic, complemented, Hall
15	C₇²×C₁₃	C₃×D₅	1	1	yes	characteristic, complemented, Hall
16	C₂×C₃×C₇×C₁₃	—	35	1	no	—
17	C₁₃×AGL(1,𝔽₇)	—	5	1	no	—
18	C₇×D₃₅	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
19	C₅×C₇×C₁₃	—	6	1	no	—
20	C₅×C₇×C₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
21	C₅×C₇×C₁₃	C₂×C₃×C₇	1	1	yes	characteristic, complemented
22	C₃×D₅×C₁₃	—	7	1	no	Hall
23	C₇×AGL(1,𝔽₇)	—	5	1	no	Hall
24	C₃×C₇×C₁₃	—	7	1	no	—
25	C₁₃×ΓL(1,𝔽₈)	C₇×D₅	1	1	yes	characteristic, complemented
26	C₅×C₇²	C₂×C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
27	C₃×C₇×D₅	—	7	1	no	—
28	C₅⋊AGL(1,𝔽₇)	C₇×C₁₃	1	1	yes	characteristic, direct product factor
29	C₃×C₅×C₁₃	—	7	1	no	Hall
30	C₂×C₇×C₁₃	—	35	1	no	—
31	C₁₃×D₇	—	5	1	no	—
32	C₇×ΓL(1,𝔽₈)	D₅×C₁₃	1	1	yes	characteristic, complemented, Hall
33	D₅×C₁₃	—	7	1	no	Hall
34	C₃×C₅×C₇	—	7	1	no	—
35	C₅×ΓL(1,𝔽₈)	C₂×C₇×C₁₃	1	1	yes	characteristic, complemented
36	C₇×D₇	—	5	1	no	Hall
37	C₇×C₁₃	—	6	1	no	—
38	C₇×C₁₃	C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
39	C₇×C₁₃	C₃×C₇×D₅	1	1	yes	characteristic, complemented
40	C₂×C₃×C₁₃	—	35	1	no	Hall
41	D₃₅	C₃×C₇×C₁₃	1	1	yes	characteristic, complemented
42	C₇×D₅	—	7	1	no	—
43	C₅×C₁₃	C₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
44	C₇²	C₃×D₅×C₁₃	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃×C₇	—	35	1	no	—
46	AGL(1,𝔽₇)	—	5	1	no	—
47	C₃×C₁₃	—	7	1	no	Hall
48	C₅×C₇	—	6	1	no	—
49	C₅×C₇	C₁₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
50	C₅×C₇	C₂×C₃×C₇×C₁₃	1	1	yes	characteristic, complemented
51	C₃×D₅	—	7	1	no	Hall
52	C₂×C₁₃	—	35	1	no	Hall
53	C₃×C₇	—	7	1	no	—
54	ΓL(1,𝔽₈)	C₇×D₅×C₁₃	1	1	yes	characteristic, complemented
55	C₃×C₅	—	7	1	no	Hall
56	C₂×C₇	—	35	1	no	—
57	D₇	—	5	1	no	—
58	C₁₃	C₇×C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
59	D₅	—	7	1	no	Hall
60	C₇	—	6	1	no	—
61	C₇	C₃×C₇×D₅×C₁₃	1	1	yes	characteristic, complemented
62	C₇	C₁₃×C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
63	C₂×C₃	—	35	1	no	Hall
64	C₅	C₇×C₁₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
65	C₃	—	7	1	no	Sylow
66	C₂	—	35	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇×C₁₃	C₅⋊AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₅×C₇	C₂×C₃×C₇×C₁₃
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₁₃	C₂×C₃
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₇×C₁₃

Complements

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

N	H ≅ G/N	type
1. C₅×C₇×C₁₃×ΓL(1,𝔽₈)	C₂	semidirect
2. C₇×C₁₃×D₃₅	C₃	semidirect
4. C₅×C₇²×C₁₃	C₂×C₃	semidirect
5. C₁₃×C₅⋊AGL(1,𝔽₇)	C₇	direct (N × H)
7. C₇×C₁₃×ΓL(1,𝔽₈)	D₅	semidirect
8. C₇×C₅⋊AGL(1,𝔽₇)	C₁₃	direct (N × H)
10. C₅×C₁₃×ΓL(1,𝔽₈)	C₂×C₇	semidirect
12. C₁₃×D₃₅	C₃×C₇	semidirect
14. C₅×C₇×ΓL(1,𝔽₈)	C₂×C₁₃	semidirect
15. C₇²×C₁₃	C₃×D₅	semidirect
18. C₇×D₃₅	C₃×C₁₃	semidirect
20. C₅×C₇×C₁₃	AGL(1,𝔽₇)	semidirect
21. C₅×C₇×C₁₃	C₂×C₃×C₇	semidirect
25. C₁₃×ΓL(1,𝔽₈)	C₇×D₅	semidirect
26. C₅×C₇²	C₂×C₃×C₁₃	semidirect
28. C₅⋊AGL(1,𝔽₇)	C₇×C₁₃	direct (N × H)
32. C₇×ΓL(1,𝔽₈)	D₅×C₁₃	semidirect
35. C₅×ΓL(1,𝔽₈)	C₂×C₇×C₁₃	semidirect
38. C₇×C₁₃	C₅⋊AGL(1,𝔽₇)	direct (N × H)
39. C₇×C₁₃	C₃×C₇×D₅	semidirect
41. D₃₅	C₃×C₇×C₁₃	semidirect
43. C₅×C₁₃	C₇×AGL(1,𝔽₇)	semidirect
44. C₇²	C₃×D₅×C₁₃	semidirect
49. C₅×C₇	C₁₃×AGL(1,𝔽₇)	semidirect
50. C₅×C₇	C₂×C₃×C₇×C₁₃	semidirect
54. ΓL(1,𝔽₈)	C₇×D₅×C₁₃	semidirect
58. C₁₃	C₇×C₅⋊AGL(1,𝔽₇)	direct (N × H)
61. C₇	C₃×C₇×D₅×C₁₃	semidirect
62. C₇	C₁₃×C₅⋊AGL(1,𝔽₇)	direct (N × H)
64. C₅	C₇×C₁₃×AGL(1,𝔽₇)	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₇×C₁₃×ΓL(1,𝔽₈)	2
C₇×C₁₃×D₃₅	3
C₇×C₁₃×AGL(1,𝔽₇)	5
C₁₃×C₅⋊AGL(1,𝔽₇)	7
C₃×C₇×D₅×C₁₃	7
C₇×C₅⋊AGL(1,𝔽₇)	13

Subgroups — C7×C13×C5⋊AGL(1,𝔽7) #34062114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₁₃×C₅⋊AGL(1,𝔽₇) #34062114