Subgroups — C₇×C₅⋊(ΓL(1,𝔽₈)⋊D₁₃) #34062119

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×D₄₅₅	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	C₅⋊(ΓL(1,𝔽₈)⋊D₁₃)	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×C₅⋊AGL(1,𝔽₇)	—	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	C₇×D₉₁	—	5	1	no	Hall
12	D₄₅₅	C₃×C₇	1	1	yes	characteristic, complemented
13	C₇×D₆₅	—	7	1	no	—
14	C₅×C₇×ΓL(1,𝔽₈)	D₁₃	1	1	yes	characteristic, complemented, Hall
15	C₇²×C₁₃	C₃×D₅	1	1	yes	characteristic, complemented, Hall
16	C₃×C₇×D₁₃	—	35	1	no	—
17	ΓL(1,𝔽₈)⋊D₁₃	—	5	1	no	—
18	C₇×D₃₅	—	13	1	no	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×D₆₅	—	7	1	no	Hall
23	C₇×AGL(1,𝔽₇)	—	65	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	C₃×D₁₃	1	1	yes	characteristic, complemented, Hall
27	C₃×C₇×D₅	—	91	1	no	—
28	C₅⋊AGL(1,𝔽₇)	—	13	1	no	—
29	C₃×C₅×C₁₃	—	7	1	no	Hall
30	D₉₁	—	5	1	no	—
31	C₇×D₁₃	—	105	1	no	—
32	C₇×ΓL(1,𝔽₈)	D₆₅	1	1	yes	characteristic, complemented, Hall
33	D₆₅	—	21	1	no	Hall
34	C₃×C₅×C₇	—	7	1	no	—
35	C₅×ΓL(1,𝔽₈)	C₇×D₁₃	1	1	yes	characteristic, complemented
36	C₇×D₇	—	65	1	no	Hall
37	C₇×C₁₃	—	6	1	no	—
38	C₇×C₁₃	C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
39	C₇×C₁₃	C₃×C₇×D₅	1	1	yes	characteristic, complemented
40	C₃×D₁₃	—	35	1	no	Hall
41	D₃₅	—	13	1	no	—
42	C₇×D₅	—	273	1	no	—
43	C₅×C₁₃	C₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
44	C₇²	C₃×D₆₅	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃×C₇	—	455	1	no	—
46	AGL(1,𝔽₇)	—	65	1	no	—
47	C₃×C₁₃	—	7	1	no	Hall
48	C₅×C₇	—	6	1	no	—
49	C₅×C₇	ΓL(1,𝔽₈)⋊D₁₃	1	1	yes	characteristic, complemented
50	C₅×C₇	C₃×C₇×D₁₃	1	1	yes	characteristic, complemented
51	C₃×D₅	—	91	1	no	Hall
52	D₁₃	—	35	1	no	Hall
53	C₃×C₇	—	7	1	no	—
54	ΓL(1,𝔽₈)	C₇×D₆₅	1	1	yes	characteristic, complemented
55	C₃×C₅	—	7	1	no	Hall
56	C₂×C₇	—	455	1	no	—
57	D₇	—	65	1	no	—
58	C₁₃	C₇×C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
59	D₅	—	273	1	no	Hall
60	C₇	—	6	1	no	—
61	C₇	C₃×C₇×D₆₅	1	1	yes	characteristic, complemented
62	C₇	C₅⋊(ΓL(1,𝔽₈)⋊D₁₃)	1	1	yes	characteristic, direct product factor
63	C₂×C₃	—	455	1	no	Hall
64	C₅	C₇×ΓL(1,𝔽₈)⋊D₁₃	1	1	yes	characteristic, complemented, Sylow
65	C₃	—	7	1	no	Sylow
66	C₂	—	455	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	C₅⋊(ΓL(1,𝔽₈)⋊D₁₃)
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. C₇×D₄₅₅	C₃	semidirect
4. C₅×C₇²×C₁₃	C₂×C₃	semidirect
5. C₅⋊(ΓL(1,𝔽₈)⋊D₁₃)	C₇	direct (N × H)
7. C₇×C₁₃×ΓL(1,𝔽₈)	D₅	semidirect
10. C₅×C₁₃×ΓL(1,𝔽₈)	C₂×C₇	semidirect
12. D₄₅₅	C₃×C₇	semidirect
14. C₅×C₇×ΓL(1,𝔽₈)	D₁₃	semidirect
15. C₇²×C₁₃	C₃×D₅	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	C₃×D₁₃	semidirect
32. C₇×ΓL(1,𝔽₈)	D₆₅	semidirect
35. C₅×ΓL(1,𝔽₈)	C₇×D₁₃	semidirect
38. C₇×C₁₃	C₅⋊AGL(1,𝔽₇)	semidirect
39. C₇×C₁₃	C₃×C₇×D₅	semidirect
43. C₅×C₁₃	C₇×AGL(1,𝔽₇)	semidirect
44. C₇²	C₃×D₆₅	semidirect
49. C₅×C₇	ΓL(1,𝔽₈)⋊D₁₃	semidirect
50. C₅×C₇	C₃×C₇×D₁₃	semidirect
54. ΓL(1,𝔽₈)	C₇×D₆₅	semidirect
58. C₁₃	C₇×C₅⋊AGL(1,𝔽₇)	semidirect
61. C₇	C₃×C₇×D₆₅	semidirect
62. C₇	C₅⋊(ΓL(1,𝔽₈)⋊D₁₃)	direct (N × H)
64. 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
C₇×D₄₅₅	3
C₇×ΓL(1,𝔽₈)⋊D₁₃	5
C₅⋊(ΓL(1,𝔽₈)⋊D₁₃)	7
C₃×C₇×D₆₅	7
C₇×C₅⋊AGL(1,𝔽₇)	13

Subgroups — C7×C5⋊(ΓL(1,𝔽8)⋊D13) #34062119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₅⋊(ΓL(1,𝔽₈)⋊D₁₃) #34062119