Subgroups — D₇×ΓL(1,𝔽₈)×D₁₇ #23756199

Order 9996.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₁₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₇×C₁₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×ΓL(1,𝔽₈)×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₇×D₁₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₇×C₁₇×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇×D₁₁₉	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₇×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇²×D₁₇	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×D₇×D₁₇	—	7	1	no	maximal
10	C₂×ΓL(1,𝔽₈)×D₁₇	—	7	1	no	maximal
11	C₇²×C₁₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₃×D₁₁₉	—	7	1	no	—
13	C₃×D₇×C₁₇	—	7	1	no	—
14	C₃×C₇×D₁₇	—	7	1	no	—
15	C₂×C₁₇×ΓL(1,𝔽₈)	—	7	1	no	—
16	ΓL(1,𝔽₈)×D₁₇	—	7	1	no	—
17	ΓL(1,𝔽₈)×D₁₇	D₇	1	1	yes	characteristic, direct product factor
18	C₂×D₇×ΓL(1,𝔽₈)	—	17	1	no	maximal, Hall
19	C₂×C₇×D₁₇	—	7	1	no	—
20	D₇×D₁₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
21	C₃×C₇×C₁₇	—	7	1	no	—
22	C₁₇×ΓL(1,𝔽₈)	C₂×D₇	1	1	yes	characteristic, complemented
23	C₂×C₇×ΓL(1,𝔽₈)	—	17	1	no	—
24	D₇×ΓL(1,𝔽₈)	—	17	1	no	—
25	D₇×ΓL(1,𝔽₈)	D₁₇	1	1	yes	characteristic, direct product factor
26	C₂×C₇×C₁₇	—	7	1	no	—
27	D₁₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
28	C₇×D₁₇	—	6	1	no	—
29	C₇×D₁₇	—	7	1	no	—
30	C₇×D₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
31	C₇×D₁₇	C₃×D₇	1	1	yes	characteristic, complemented
32	D₇×C₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
33	C₂×C₃×D₁₇	—	49	1	no	Hall
34	C₂×C₇×D₇	—	17	1	no	Hall
35	C₇×ΓL(1,𝔽₈)	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
36	C₇×C₁₇	—	6	1	no	—
37	C₇×C₁₇	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
38	C₇×C₁₇	C₂×C₃×D₇	1	1	yes	characteristic, complemented
39	C₂×C₃×C₁₇	—	49	1	no	—
40	C₃×D₁₇	—	7	1	no	—
41	C₃×D₁₇	—	49	1	no	—
42	C₂×C₇²	—	17	1	no	—
43	C₇×D₇	—	17	1	no	—
44	C₇×D₇	C₃×D₁₇	1	1	yes	characteristic, complemented
45	C₂×C₃×D₇	—	119	1	no	—
46	C₂²×ΓL(1,𝔽₈)	—	119	1	no	—
47	C₂×D₁₇	—	7	1	no	Hall
48	C₃×C₁₇	—	7	1	no	Hall
49	C₇²	C₂×C₃×D₁₇	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃×C₇	—	119	1	no	—
51	C₃×D₇	—	7	1	no	—
52	C₃×D₇	—	119	1	no	—
53	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
54	C₂×ΓL(1,𝔽₈)	—	17	1	no	—
55	C₂×ΓL(1,𝔽₈)	—	119	1	no	—
56	C₂×C₁₇	—	7	1	no	—
57	D₁₇	—	7	1	no	—
58	D₁₇	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
59	C₂²×C₇	—	119	1	no	—
60	C₂×D₇	—	17	1	no	—
61	C₃×C₇	—	7	1	no	—
62	ΓL(1,𝔽₈)	D₇×D₁₇	1	1	yes	characteristic, direct product factor
63	C₁₇	C₂×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₇	—	7	1	no	—
65	C₂×C₇	—	17	1	no	—
66	C₂×C₇	—	17	1	no	—
67	C₂×C₇	—	102	1	no	—
68	C₂×C₇	—	119	1	no	—
69	D₇	—	17	1	no	—
70	D₇	ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, direct product factor
71	C₂²×C₃	—	833	1	no	Hall
72	C₇	—	6	1	no	—
73	C₇	C₂×ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, complemented
74	C₇	C₃×D₇×D₁₇	1	1	yes	characteristic, complemented
75	C₂×C₃	—	49	1	no	—
76	C₂×C₃	—	833	1	no	—
77	C₂×C₃	—	119	1	no	—
78	C₂²	—	119	1	no	Sylow
79	C₃	—	7	1	no	Sylow
80	C₂	—	7	1	no	—
81	C₂	—	17	1	no	—
82	C₂	—	119	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₁₇	C₂²×C₃
LCS	G ▷ C₇²×C₁₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×D₁₁₉	C₂	semidirect
2. D₇×C₁₇×ΓL(1,𝔽₈)	C₂	semidirect
3. C₇×ΓL(1,𝔽₈)×D₁₇	C₂	semidirect
4. C₇×D₇×D₁₇	C₃	semidirect
5. C₇×C₁₇×ΓL(1,𝔽₈)	C₂²	semidirect
6. C₇×D₁₁₉	C₂×C₃	semidirect
7. C₇×D₇×C₁₇	C₂×C₃	semidirect
8. C₇²×D₁₇	C₂×C₃	semidirect
11. C₇²×C₁₇	C₂²×C₃	semidirect
17. ΓL(1,𝔽₈)×D₁₇	D₇	direct (N × H)
20. D₇×D₁₇	ΓL(1,𝔽₈)	direct (N × H)
22. C₁₇×ΓL(1,𝔽₈)	C₂×D₇	semidirect
25. D₇×ΓL(1,𝔽₈)	D₁₇	direct (N × H)
27. D₁₁₉	C₂×ΓL(1,𝔽₈)	semidirect
30. C₇×D₁₇	C₂×ΓL(1,𝔽₈)	semidirect
31. C₇×D₁₇	C₃×D₇	semidirect
32. D₇×C₁₇	C₂×ΓL(1,𝔽₈)	semidirect
35. C₇×ΓL(1,𝔽₈)	C₂×D₁₇	semidirect
37. C₇×C₁₇	C₂²×ΓL(1,𝔽₈)	semidirect
38. C₇×C₁₇	C₂×C₃×D₇	semidirect
44. C₇×D₇	C₃×D₁₇	semidirect
49. C₇²	C₂×C₃×D₁₇	semidirect
58. D₁₇	D₇×ΓL(1,𝔽₈)	direct (N × H)
62. ΓL(1,𝔽₈)	D₇×D₁₇	direct (N × H)
63. C₁₇	C₂×D₇×ΓL(1,𝔽₈)	semidirect
70. D₇	ΓL(1,𝔽₈)×D₁₇	direct (N × H)
73. C₇	C₂×ΓL(1,𝔽₈)×D₁₇	semidirect
74. C₇	C₃×D₇×D₁₇	semidirect

Subgroups — D7×ΓL(1,𝔽8)×D17 #23756199

Conjugacy classes of subgroups

Complements

Subgroups — D₇×ΓL(1,𝔽₈)×D₁₇ #23756199