Subgroups — C₂×ΓL(1,𝔽₈)×D₉₁ #21439739

Order 7644.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₁₃×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	ΓL(1,𝔽₈)×D₉₁	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×D₉₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₇×C₁₃×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
6	C₇×D₉₁	C₂×C₃	1	2	yes	complemented
7	C₂×C₃×D₉₁	—	7	1	no	maximal
8	C₂×ΓL(1,𝔽₈)×D₁₃	—	7	1	no	maximal
9	C₇²×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×D₇×ΓL(1,𝔽₈)	—	13	1	no	maximal, Hall
11	C₂×C₃×C₇×C₁₃	—	7	1	no	—
12	C₃×D₉₁	—	7	2	no	—
13	C₂×C₁₃×ΓL(1,𝔽₈)	D₇	1	1	yes	characteristic, complemented
14	ΓL(1,𝔽₈)×D₁₃	—	7	2	no	—
15	C₂×D₉₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
16	C₂×C₇×D₁₃	—	7	1	no	—
17	C₂×C₇×ΓL(1,𝔽₈)	D₁₃	1	1	yes	characteristic, complemented
18	D₇×ΓL(1,𝔽₈)	—	13	2	no	—
19	C₃×C₇×C₁₃	—	7	1	no	—
20	C₁₃×ΓL(1,𝔽₈)	C₂×D₇	1	1	yes	characteristic, complemented
21	C₂×C₇×D₇	—	13	1	no	Hall
22	C₂×C₇×C₁₃	—	6	1	no	—
23	C₂×C₇×C₁₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	C₂×C₇×C₁₃	C₃×D₇	1	1	yes	characteristic, complemented
25	D₉₁	C₂×ΓL(1,𝔽₈)	1	2	yes	direct product factor
26	C₇×D₁₃	—	7	2	no	—
27	C₂×C₃×D₁₃	—	49	1	no	Hall
28	C₇×ΓL(1,𝔽₈)	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
29	C₂×C₇²	C₃×D₁₃	1	1	yes	characteristic, complemented
30	C₇×D₇	—	13	2	no	—
31	C₇×C₁₃	—	6	1	no	—
32	C₇×C₁₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
33	C₇×C₁₃	C₂×C₃×D₇	1	1	yes	characteristic, complemented
34	C₂×C₃×D₇	—	91	1	no	—
35	C₂²×ΓL(1,𝔽₈)	—	91	1	no	—
36	C₂×C₃×C₁₃	—	7	1	no	—
37	C₃×D₁₃	—	49	2	no	—
38	C₂×D₁₃	—	7	1	no	Hall
39	C₇²	C₂×C₃×D₁₃	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₃×C₇	—	7	1	no	—
41	C₃×D₇	—	91	2	no	—
42	C₂×ΓL(1,𝔽₈)	—	91	2	no	—
43	C₂×ΓL(1,𝔽₈)	D₉₁	1	1	yes	characteristic, direct product factor
44	C₃×C₁₃	—	7	1	no	Hall
45	C₂²×C₇	—	91	1	no	—
46	C₂×D₇	—	13	1	no	—
47	C₂×C₁₃	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
48	D₁₃	—	7	2	no	—
49	C₃×C₇	—	7	1	no	—
50	ΓL(1,𝔽₈)	C₂×D₉₁	1	1	yes	characteristic, direct product factor
51	C₂×C₇	—	6	1	no	—
52	C₂×C₇	—	91	2	no	—
53	C₂×C₇	ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented
54	C₂×C₇	C₃×D₉₁	1	1	yes	characteristic, complemented
55	D₇	—	13	2	no	—
56	C₁₃	C₂×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₃	—	637	1	no	Hall
58	C₇	—	6	1	no	—
59	C₇	C₂×ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented
60	C₇	C₂×C₃×D₉₁	1	1	yes	characteristic, complemented
61	C₂×C₃	—	7	1	no	—
62	C₂×C₃	—	637	2	no	—
63	C₂²	—	91	1	no	Sylow
64	C₃	—	7	1	no	Sylow
65	C₂	—	91	2	no	—
66	C₂	ΓL(1,𝔽₈)×D₉₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₈)×D₉₁
commutator, hypocenter	C₇²×C₁₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₁₃	C₂×C₃
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×C₁₃×ΓL(1,𝔽₈)	C₂	semidirect
2. ΓL(1,𝔽₈)×D₉₁	C₂	direct (N × H)
3. C₂×C₇×D₉₁	C₃	semidirect
4. C₇×C₁₃×ΓL(1,𝔽₈)	C₂²	semidirect
5. C₂×C₇²×C₁₃	C₂×C₃	semidirect
6. C₇×D₉₁	C₂×C₃	semidirect
9. C₇²×C₁₃	C₂²×C₃	semidirect
13. C₂×C₁₃×ΓL(1,𝔽₈)	D₇	semidirect
15. C₂×D₉₁	ΓL(1,𝔽₈)	direct (N × H)
17. C₂×C₇×ΓL(1,𝔽₈)	D₁₃	semidirect
20. C₁₃×ΓL(1,𝔽₈)	C₂×D₇	semidirect
23. C₂×C₇×C₁₃	C₂×ΓL(1,𝔽₈)	semidirect
24. C₂×C₇×C₁₃	C₃×D₇	semidirect
25. D₉₁	C₂×ΓL(1,𝔽₈)	direct (N × H)
28. C₇×ΓL(1,𝔽₈)	C₂×D₁₃	semidirect
29. C₂×C₇²	C₃×D₁₃	semidirect
32. C₇×C₁₃	C₂²×ΓL(1,𝔽₈)	semidirect
33. C₇×C₁₃	C₂×C₃×D₇	semidirect
39. C₇²	C₂×C₃×D₁₃	semidirect
43. C₂×ΓL(1,𝔽₈)	D₉₁	direct (N × H)
47. C₂×C₁₃	D₇×ΓL(1,𝔽₈)	semidirect
50. ΓL(1,𝔽₈)	C₂×D₉₁	direct (N × H)
53. C₂×C₇	ΓL(1,𝔽₈)×D₁₃	semidirect
54. C₂×C₇	C₃×D₉₁	semidirect
56. C₁₃	C₂×D₇×ΓL(1,𝔽₈)	semidirect
59. C₇	C₂×ΓL(1,𝔽₈)×D₁₃	semidirect
60. C₇	C₂×C₃×D₉₁	semidirect
66. C₂	ΓL(1,𝔽₈)×D₉₁	direct (N × H)

Subgroups — C2×ΓL(1,𝔽8)×D91 #21439739

Conjugacy classes of subgroups

Complements

Subgroups — C₂×ΓL(1,𝔽₈)×D₉₁ #21439739