Subgroups — C₁₃×C₅⋊(ΓL(1,𝔽₈)⋊D₇) #34062150

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	C₁₃×C₅⋊(C₇⋊D₇)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₃×ΓL(1,𝔽₈)⋊D₇	—	5	1	no	maximal, Hall
4	C₅×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	C₁₃×C₅⋊AGL(1,𝔽₇)	—	7	1	no	maximal
6	C₃×C₁₃×D₃₅	—	7	1	no	maximal
7	C₇×C₁₃×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, complemented, Hall
8	C₅⋊(ΓL(1,𝔽₈)⋊D₇)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×C₇×C₁₃	—	7	1	no	—
10	C₅×C₁₃×ΓL(1,𝔽₈)	D₇	1	1	yes	characteristic, complemented
11	C₁₃×C₇⋊D₇	—	5	1	no	Hall
12	C₁₃×D₃₅	—	7	1	no	—
13	C₁₃×D₃₅	—	7	1	no	—
14	C₁₃×D₃₅	—	21	2	no	—
15	C₅×C₇×ΓL(1,𝔽₈)	C₂×C₁₃	1	1	yes	characteristic, complemented, Hall
16	C₇²×C₁₃	C₃×D₅	1	1	yes	characteristic, complemented, Hall
17	C₃×C₁₃×D₇	—	35	1	no	—
18	C₁₃×AGL(1,𝔽₇)	—	35	1	no	—
19	C₅⋊(C₇⋊D₇)	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
20	C₅×C₇×C₁₃	—	3	2	no	—
21	C₅×C₇×C₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
22	C₅×C₇×C₁₃	C₃×D₇	1	1	yes	characteristic, complemented
23	C₃×D₅×C₁₃	—	49	1	no	Hall
24	ΓL(1,𝔽₈)⋊D₇	—	5	1	no	Hall
25	C₃×C₇×C₁₃	—	7	1	no	—
26	C₁₃×ΓL(1,𝔽₈)	D₃₅	1	1	yes	characteristic, complemented
27	C₅×C₇²	C₂×C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
28	C₃×D₃₅	—	7	1	no	—
29	C₅⋊AGL(1,𝔽₇)	—	7	1	no	—
30	C₃×C₅×C₁₃	—	7	1	no	Hall
31	C₁₃×D₇	—	35	1	no	—
32	C₁₃×D₇	—	35	1	no	—
33	C₁₃×D₇	—	105	2	no	—
34	C₇×ΓL(1,𝔽₈)	D₅×C₁₃	1	1	yes	characteristic, complemented, Hall
35	D₅×C₁₃	—	49	1	no	Hall
36	C₃×C₅×C₇	—	7	1	no	—
37	C₅×ΓL(1,𝔽₈)	C₁₃×D₇	1	1	yes	characteristic, complemented
38	C₇⋊D₇	—	5	1	no	Hall
39	C₇×C₁₃	—	3	2	no	—
40	C₇×C₁₃	C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
41	C₇×C₁₃	C₃×D₃₅	1	1	yes	characteristic, complemented
42	C₂×C₃×C₁₃	—	245	1	no	Hall
43	D₃₅	—	7	1	no	—
44	D₃₅	—	7	1	no	—
45	D₃₅	—	21	2	no	—
46	C₅×C₁₃	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented, Hall
47	C₇²	C₃×D₅×C₁₃	1	1	yes	characteristic, complemented, Sylow
48	C₃×D₇	—	35	1	no	—
49	AGL(1,𝔽₇)	—	35	1	no	—
50	C₃×C₁₃	—	7	1	no	Hall
51	C₅×C₇	—	3	2	no	—
52	C₅×C₇	C₁₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
53	C₅×C₇	C₃×C₁₃×D₇	1	1	yes	characteristic, complemented
54	C₃×D₅	—	49	1	no	Hall
55	C₂×C₁₃	—	245	1	no	Hall
56	C₃×C₇	—	7	1	no	—
57	ΓL(1,𝔽₈)	C₁₃×D₃₅	1	1	yes	characteristic, complemented
58	C₃×C₅	—	7	1	no	Hall
59	D₇	—	35	1	no	—
60	D₇	—	35	1	no	—
61	D₇	—	105	2	no	—
62	C₁₃	C₅⋊(ΓL(1,𝔽₈)⋊D₇)	1	1	yes	characteristic, direct product factor, Sylow
63	D₅	—	49	1	no	Hall
64	C₇	—	3	2	no	—
65	C₇	C₃×C₁₃×D₃₅	1	1	yes	characteristic, complemented
66	C₇	C₁₃×C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
67	C₂×C₃	—	245	1	no	Hall
68	C₅	C₁₃×ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented, Sylow
69	C₃	—	7	1	no	Sylow
70	C₂	—	245	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₃	C₅⋊(ΓL(1,𝔽₈)⋊D₇)
commutator, cosocle, hypocenter	C₅×C₇²	C₂×C₃×C₁₃
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₁₃	C₂×C₃
LCS	G ▷ C₅×C₇²
UCS	C₁ ◁ 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. C₁₃×C₅⋊(C₇⋊D₇)	C₃	semidirect
4. C₅×C₇²×C₁₃	C₂×C₃	semidirect
7. C₇×C₁₃×ΓL(1,𝔽₈)	D₅	semidirect
8. C₅⋊(ΓL(1,𝔽₈)⋊D₇)	C₁₃	direct (N × H)
10. C₅×C₁₃×ΓL(1,𝔽₈)	D₇	semidirect
15. C₅×C₇×ΓL(1,𝔽₈)	C₂×C₁₃	semidirect
16. C₇²×C₁₃	C₃×D₅	semidirect
19. C₅⋊(C₇⋊D₇)	C₃×C₁₃	semidirect
21. C₅×C₇×C₁₃	AGL(1,𝔽₇)	semidirect
22. C₅×C₇×C₁₃	C₃×D₇	semidirect
26. C₁₃×ΓL(1,𝔽₈)	D₃₅	semidirect
27. C₅×C₇²	C₂×C₃×C₁₃	semidirect
34. C₇×ΓL(1,𝔽₈)	D₅×C₁₃	semidirect
37. C₅×ΓL(1,𝔽₈)	C₁₃×D₇	semidirect
40. C₇×C₁₃	C₅⋊AGL(1,𝔽₇)	semidirect
41. C₇×C₁₃	C₃×D₃₅	semidirect
46. C₅×C₁₃	ΓL(1,𝔽₈)⋊D₇	semidirect
47. C₇²	C₃×D₅×C₁₃	semidirect
52. C₅×C₇	C₁₃×AGL(1,𝔽₇)	semidirect
53. C₅×C₇	C₃×C₁₃×D₇	semidirect
57. ΓL(1,𝔽₈)	C₁₃×D₃₅	semidirect
62. C₁₃	C₅⋊(ΓL(1,𝔽₈)⋊D₇)	direct (N × H)
65. C₇	C₃×C₁₃×D₃₅	semidirect
66. C₇	C₁₃×C₅⋊AGL(1,𝔽₇)	semidirect
68. C₅	C₁₃×ΓL(1,𝔽₈)⋊D₇	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
C₁₃×C₅⋊(C₇⋊D₇)	3
C₁₃×ΓL(1,𝔽₈)⋊D₇	5
C₁₃×C₅⋊AGL(1,𝔽₇)	7
C₃×C₁₃×D₃₅	7
C₅⋊(ΓL(1,𝔽₈)⋊D₇)	13

Subgroups — C13×C5⋊(ΓL(1,𝔽8)⋊D7) #34062150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×C₅⋊(ΓL(1,𝔽₈)⋊D₇) #34062150