Subgroups — C₇×ΓL(1,𝔽₈)×D₆₅ #34062125

Order 19110.

80 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	C₇²×D₆₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×ΓL(1,𝔽₈)×D₁₃	—	5	1	no	maximal, Hall
4	C₅×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	ΓL(1,𝔽₈)×D₆₅	C₇	1	1	yes	maximal, characteristic, direct product factor
6	C₃×C₇×D₆₅	—	7	1	no	maximal
7	C₇×C₁₃×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, complemented, Hall
8	C₇×D₅×ΓL(1,𝔽₈)	—	13	1	no	maximal, Hall
9	C₃×C₅×C₇×C₁₃	—	7	1	no	—
10	C₅×C₁₃×ΓL(1,𝔽₈)	C₂×C₇	1	1	yes	characteristic, complemented
11	C₇²×D₁₃	—	5	1	no	Hall
12	C₇×D₆₅	—	3	2	no	—
13	C₇×D₆₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
14	C₇×D₆₅	C₃×C₇	1	1	yes	characteristic, complemented
15	C₅×C₇×ΓL(1,𝔽₈)	D₁₃	1	1	yes	characteristic, complemented, Hall
16	C₇²×C₁₃	C₃×D₅	1	1	yes	characteristic, complemented, Hall
17	C₃×C₇×D₁₃	—	35	1	no	—
18	ΓL(1,𝔽₈)×D₁₃	—	5	1	no	—
19	C₇²×D₅	—	13	1	no	Hall
20	C₅×C₇×C₁₃	—	3	2	no	—
21	C₅×C₇×C₁₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₅×C₇×C₁₃	C₂×C₃×C₇	1	1	yes	characteristic, complemented
23	C₃×D₆₅	—	7	1	no	Hall
24	C₂×C₇×ΓL(1,𝔽₈)	—	65	1	no	Hall
25	C₃×C₇×C₁₃	—	7	1	no	—
26	C₁₃×ΓL(1,𝔽₈)	C₇×D₅	1	1	yes	characteristic, complemented
27	C₅×C₇²	C₃×D₁₃	1	1	yes	characteristic, complemented, Hall
28	C₃×C₇×D₅	—	91	1	no	—
29	D₅×ΓL(1,𝔽₈)	—	13	1	no	—
30	C₃×C₅×C₁₃	—	7	1	no	Hall
31	C₇×D₁₃	—	5	1	no	—
32	C₇×D₁₃	—	5	1	no	—
33	C₇×D₁₃	—	15	2	no	—
34	C₇×ΓL(1,𝔽₈)	D₆₅	1	1	yes	characteristic, direct product factor, Hall
35	D₆₅	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
36	C₃×C₅×C₇	—	7	1	no	—
37	C₅×ΓL(1,𝔽₈)	C₇×D₁₃	1	1	yes	characteristic, complemented
38	C₂×C₇²	—	65	1	no	Hall
39	C₇×C₁₃	—	3	2	no	—
40	C₇×C₁₃	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
41	C₇×C₁₃	C₃×C₇×D₅	1	1	yes	characteristic, complemented
42	C₃×D₁₃	—	35	1	no	Hall
43	C₇×D₅	—	13	1	no	—
44	C₇×D₅	—	13	1	no	—
45	C₇×D₅	—	39	2	no	—
46	C₅×C₁₃	C₂×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
47	C₇²	C₃×D₆₅	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₃×C₇	—	455	1	no	—
49	C₂×ΓL(1,𝔽₈)	—	65	1	no	—
50	C₃×C₁₃	—	7	1	no	Hall
51	C₅×C₇	—	3	2	no	—
52	C₅×C₇	ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented
53	C₅×C₇	C₃×C₇×D₁₃	1	1	yes	characteristic, complemented
54	C₃×D₅	—	91	1	no	Hall
55	D₁₃	—	5	1	no	Hall
56	C₃×C₇	—	7	1	no	—
57	ΓL(1,𝔽₈)	C₇×D₆₅	1	1	yes	characteristic, direct product factor
58	C₃×C₅	—	7	1	no	Hall
59	C₂×C₇	—	65	1	no	—
60	C₂×C₇	—	65	1	no	—
61	C₂×C₇	—	195	2	no	—
62	C₁₃	C₇×D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
63	D₅	—	13	1	no	Hall
64	C₇	—	3	2	no	—
65	C₇	C₃×C₇×D₆₅	1	1	yes	characteristic, complemented
66	C₇	ΓL(1,𝔽₈)×D₆₅	1	1	yes	characteristic, direct product factor
67	C₂×C₃	—	455	1	no	Hall
68	C₅	C₇×ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented, Sylow
69	C₃	—	7	1	no	Sylow
70	C₂	—	65	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	ΓL(1,𝔽₈)×D₆₅
commutator, cosocle, hypocenter	C₅×C₇×C₁₃	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₁₃	C₂×C₃
LCS	G ▷ C₅×C₇×C₁₃
UCS	C₁ ◁ C₇

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. C₇²×D₆₅	C₃	semidirect
4. C₅×C₇²×C₁₃	C₂×C₃	semidirect
5. ΓL(1,𝔽₈)×D₆₅	C₇	direct (N × H)
7. C₇×C₁₃×ΓL(1,𝔽₈)	D₅	semidirect
10. C₅×C₁₃×ΓL(1,𝔽₈)	C₂×C₇	semidirect
13. C₇×D₆₅	ΓL(1,𝔽₈)	direct (N × H)
14. C₇×D₆₅	C₃×C₇	semidirect
15. C₅×C₇×ΓL(1,𝔽₈)	D₁₃	semidirect
16. C₇²×C₁₃	C₃×D₅	semidirect
21. C₅×C₇×C₁₃	C₂×ΓL(1,𝔽₈)	semidirect
22. C₅×C₇×C₁₃	C₂×C₃×C₇	semidirect
26. C₁₃×ΓL(1,𝔽₈)	C₇×D₅	semidirect
27. C₅×C₇²	C₃×D₁₃	semidirect
34. C₇×ΓL(1,𝔽₈)	D₆₅	direct (N × H)
35. D₆₅	C₇×ΓL(1,𝔽₈)	direct (N × H)
37. C₅×ΓL(1,𝔽₈)	C₇×D₁₃	semidirect
40. C₇×C₁₃	D₅×ΓL(1,𝔽₈)	semidirect
41. C₇×C₁₃	C₃×C₇×D₅	semidirect
46. C₅×C₁₃	C₂×C₇×ΓL(1,𝔽₈)	semidirect
47. C₇²	C₃×D₆₅	semidirect
52. C₅×C₇	ΓL(1,𝔽₈)×D₁₃	semidirect
53. C₅×C₇	C₃×C₇×D₁₃	semidirect
57. ΓL(1,𝔽₈)	C₇×D₆₅	direct (N × H)
62. C₁₃	C₇×D₅×ΓL(1,𝔽₈)	semidirect
65. C₇	C₃×C₇×D₆₅	semidirect
66. C₇	ΓL(1,𝔽₈)×D₆₅	direct (N × H)
68. C₅	C₇×ΓL(1,𝔽₈)×D₁₃	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
C₇²×D₆₅	3
C₇×ΓL(1,𝔽₈)×D₁₃	5
ΓL(1,𝔽₈)×D₆₅	7
C₃×C₇×D₆₅	7
C₇×D₅×ΓL(1,𝔽₈)	13

Subgroups — C7×ΓL(1,𝔽8)×D65 #34062125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×ΓL(1,𝔽₈)×D₆₅ #34062125