Subgroups — C₂×ΓL(1,𝔽₈)×D₁₁₉ #23756227

Order 9996.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₁₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	ΓL(1,𝔽₈)×D₁₁₉	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×D₁₁₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₇×C₁₇×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇²×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
6	C₇×D₁₁₉	C₂×C₃	1	2	yes	complemented
7	C₂×C₃×D₁₁₉	—	7	1	no	maximal
8	C₂×ΓL(1,𝔽₈)×D₁₇	—	7	1	no	maximal
9	C₇²×C₁₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×C₃×C₇×C₁₇	—	7	1	no	—
11	C₃×D₁₁₉	—	7	2	no	—
12	C₂×C₁₇×ΓL(1,𝔽₈)	D₇	1	1	yes	characteristic, complemented
13	ΓL(1,𝔽₈)×D₁₇	—	7	2	no	—
14	C₂×D₇×ΓL(1,𝔽₈)	—	17	1	no	maximal, Hall
15	C₂×D₁₁₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
16	C₂×C₇×D₁₇	—	7	1	no	—
17	C₃×C₇×C₁₇	—	7	1	no	—
18	C₁₇×ΓL(1,𝔽₈)	C₂×D₇	1	1	yes	characteristic, complemented
19	C₂×C₇×ΓL(1,𝔽₈)	D₁₇	1	1	yes	characteristic, complemented
20	D₇×ΓL(1,𝔽₈)	—	17	2	no	—
21	C₂×C₇×C₁₇	—	6	1	no	—
22	C₂×C₇×C₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
23	C₂×C₇×C₁₇	C₃×D₇	1	1	yes	characteristic, complemented
24	D₁₁₉	C₂×ΓL(1,𝔽₈)	1	2	yes	direct product factor
25	C₇×D₁₇	—	7	2	no	—
26	C₂×C₃×D₁₇	—	49	1	no	Hall
27	C₂×C₇×D₇	—	17	1	no	Hall
28	C₇×ΓL(1,𝔽₈)	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
29	C₇×C₁₇	—	6	1	no	—
30	C₇×C₁₇	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
31	C₇×C₁₇	C₂×C₃×D₇	1	1	yes	characteristic, complemented
32	C₂×C₃×C₁₇	—	7	1	no	—
33	C₃×D₁₇	—	49	2	no	—
34	C₂×C₇²	C₃×D₁₇	1	1	yes	characteristic, complemented
35	C₇×D₇	—	17	2	no	—
36	C₂×C₃×D₇	—	119	1	no	—
37	C₂²×ΓL(1,𝔽₈)	—	119	1	no	—
38	C₂×D₁₇	—	7	1	no	Hall
39	C₃×C₁₇	—	7	1	no	Hall
40	C₇²	C₂×C₃×D₁₇	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃×C₇	—	7	1	no	—
42	C₃×D₇	—	119	2	no	—
43	C₂×ΓL(1,𝔽₈)	—	119	2	no	—
44	C₂×ΓL(1,𝔽₈)	D₁₁₉	1	1	yes	characteristic, direct product factor
45	C₂×C₁₇	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
46	D₁₇	—	7	2	no	—
47	C₂²×C₇	—	119	1	no	—
48	C₂×D₇	—	17	1	no	—
49	C₃×C₇	—	7	1	no	—
50	ΓL(1,𝔽₈)	C₂×D₁₁₉	1	1	yes	characteristic, direct product factor
51	C₁₇	C₂×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₇	—	6	1	no	—
53	C₂×C₇	—	119	2	no	—
54	C₂×C₇	ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, complemented
55	C₂×C₇	C₃×D₁₁₉	1	1	yes	characteristic, complemented
56	D₇	—	17	2	no	—
57	C₂²×C₃	—	833	1	no	Hall
58	C₇	—	6	1	no	—
59	C₇	C₂×ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, complemented
60	C₇	C₂×C₃×D₁₁₉	1	1	yes	characteristic, complemented
61	C₂×C₃	—	7	1	no	—
62	C₂×C₃	—	833	2	no	—
63	C₂²	—	119	1	no	Sylow
64	C₃	—	7	1	no	Sylow
65	C₂	—	119	2	no	—
66	C₂	ΓL(1,𝔽₈)×D₁₁₉	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₈)×D₁₁₉
commutator, cosocle, hypocenter	C₇²×C₁₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₁₇	C₂×C₃
LCS	G ▷ C₇²×C₁₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×C₁₇×ΓL(1,𝔽₈)	C₂	semidirect
2. ΓL(1,𝔽₈)×D₁₁₉	C₂	direct (N × H)
3. C₂×C₇×D₁₁₉	C₃	semidirect
4. C₇×C₁₇×ΓL(1,𝔽₈)	C₂²	semidirect
5. C₂×C₇²×C₁₇	C₂×C₃	semidirect
6. C₇×D₁₁₉	C₂×C₃	semidirect
9. C₇²×C₁₇	C₂²×C₃	semidirect
12. C₂×C₁₇×ΓL(1,𝔽₈)	D₇	semidirect
15. C₂×D₁₁₉	ΓL(1,𝔽₈)	direct (N × H)
18. C₁₇×ΓL(1,𝔽₈)	C₂×D₇	semidirect
19. C₂×C₇×ΓL(1,𝔽₈)	D₁₇	semidirect
22. C₂×C₇×C₁₇	C₂×ΓL(1,𝔽₈)	semidirect
23. C₂×C₇×C₁₇	C₃×D₇	semidirect
24. D₁₁₉	C₂×ΓL(1,𝔽₈)	direct (N × H)
28. C₇×ΓL(1,𝔽₈)	C₂×D₁₇	semidirect
30. C₇×C₁₇	C₂²×ΓL(1,𝔽₈)	semidirect
31. C₇×C₁₇	C₂×C₃×D₇	semidirect
34. C₂×C₇²	C₃×D₁₇	semidirect
40. C₇²	C₂×C₃×D₁₇	semidirect
44. C₂×ΓL(1,𝔽₈)	D₁₁₉	direct (N × H)
45. C₂×C₁₇	D₇×ΓL(1,𝔽₈)	semidirect
50. ΓL(1,𝔽₈)	C₂×D₁₁₉	direct (N × H)
51. C₁₇	C₂×D₇×ΓL(1,𝔽₈)	semidirect
54. C₂×C₇	ΓL(1,𝔽₈)×D₁₇	semidirect
55. C₂×C₇	C₃×D₁₁₉	semidirect
59. C₇	C₂×ΓL(1,𝔽₈)×D₁₇	semidirect
60. C₇	C₂×C₃×D₁₁₉	semidirect
66. C₂	ΓL(1,𝔽₈)×D₁₁₉	direct (N × H)

Subgroups — C2×ΓL(1,𝔽8)×D119 #23756227

Conjugacy classes of subgroups

Complements

Subgroups — C₂×ΓL(1,𝔽₈)×D₁₁₉ #23756227