Subgroups — ΓL(1,𝔽₈)×D₂₅₅ #24924225

Order 10710.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₁₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	ΓL(1,𝔽₈)×D₈₅	—	3	1	no	maximal
3	C₇×D₂₅₅	C₃	1	1	yes	maximal, characteristic, complemented
4	ΓL(1,𝔽₈)×D₅₁	—	5	1	no	maximal, Hall
5	C₃×C₅×C₇×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
6	C₅×C₁₇×ΓL(1,𝔽₈)	—	2	1	no	—
7	C₅×C₁₇×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
8	C₃×D₂₅₅	—	7	1	no	maximal, Hall
9	C₇×D₈₅	—	3	1	no	Hall
10	C₃×C₁₇×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, complemented, Hall
11	C₃²×C₅×C₁₇	—	7	1	no	Hall
12	C₇×D₅₁	—	5	1	no	—
13	ΓL(1,𝔽₈)×D₁₇	—	15	1	no	—
14	ΓL(1,𝔽₈)×D₁₅	—	17	1	no	maximal, Hall
15	C₅×C₇×C₁₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
16	D₂₅₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
17	C₃×D₈₅	—	21	1	no	—
18	C₃×C₇×C₁₇	C₃×D₅	1	1	yes	characteristic, complemented
19	C₁₇×ΓL(1,𝔽₈)	—	2	1	no	—
20	C₁₇×ΓL(1,𝔽₈)	D₁₅	1	1	yes	characteristic, complemented
21	C₃×C₅×ΓL(1,𝔽₈)	D₁₇	1	1	yes	characteristic, complemented, Hall
22	C₃×D₅₁	—	35	1	no	Hall
23	C₃×C₅×C₁₇	—	7	1	no	—
24	C₃×C₅×C₁₇	—	14	1	no	—
25	C₃×C₅×C₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
26	C₇×D₁₇	—	15	1	no	Hall
27	C₇×D₁₅	—	17	1	no	—
28	D₅×ΓL(1,𝔽₈)	—	51	1	no	—
29	D₈₅	—	3	1	no	Hall
30	C₃²×C₁₇	—	7	1	no	Hall
31	D₃×ΓL(1,𝔽₈)	—	85	1	no	Hall
32	C₇×C₁₇	C₃×D₁₅	1	1	yes	characteristic, complemented, Hall
33	C₃×C₅×C₇	C₃×D₁₇	1	1	yes	characteristic, complemented
34	C₅×ΓL(1,𝔽₈)	—	2	1	no	—
35	C₅×ΓL(1,𝔽₈)	D₅₁	1	1	yes	characteristic, complemented
36	D₅₁	—	5	1	no	—
37	C₃×D₁₇	—	105	1	no	—
38	C₃×D₁₅	—	119	1	no	Hall
39	C₅×C₁₇	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
40	C₇×D₅	—	51	1	no	Hall
41	C₃×ΓL(1,𝔽₈)	D₈₅	1	1	yes	characteristic, complemented, Hall
42	C₃×C₁₇	—	7	1	no	—
43	C₃×C₁₇	—	14	1	no	—
44	C₃×C₁₇	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
45	C₃²×C₅	—	7	1	no	Hall
46	D₃×C₇	—	85	1	no	—
47	C₂×ΓL(1,𝔽₈)	—	255	1	no	—
48	C₅×C₇	C₃×D₅₁	1	1	yes	characteristic, complemented, Hall
49	D₁₇	—	15	1	no	Hall
50	D₁₅	—	17	1	no	—
51	C₃×D₅	—	357	1	no	—
52	C₃×C₇	C₃×D₈₅	1	1	yes	characteristic, complemented
53	ΓL(1,𝔽₈)	—	2	1	no	—
54	ΓL(1,𝔽₈)	D₂₅₅	1	1	yes	characteristic, direct product factor
55	C₃×D₃	—	595	1	no	Hall
56	C₁₇	ΓL(1,𝔽₈)×D₁₅	1	1	yes	characteristic, complemented, Sylow
57	C₃×C₅	—	7	1	no	—
58	C₃×C₅	—	14	1	no	—
59	C₃×C₅	ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, complemented
60	C₂×C₇	—	255	1	no	Hall
61	D₅	—	51	1	no	Hall
62	C₃²	—	7	1	no	Sylow
63	C₇	C₃×D₂₅₅	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	1785	1	no	—
65	D₃	—	85	1	no	—
66	C₅	ΓL(1,𝔽₈)×D₅₁	1	1	yes	characteristic, complemented, Sylow
67	C₃	—	7	1	no	—
68	C₃	—	14	1	no	—
69	C₃	ΓL(1,𝔽₈)×D₈₅	1	1	yes	characteristic, complemented
70	C₂	—	255	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₇×C₁₇	C₂×C₃
LCS	G ▷ C₃×C₅×C₇×C₁₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₁₇×ΓL(1,𝔽₈)	C₂	semidirect
3. C₇×D₂₅₅	C₃	semidirect
5. C₃×C₅×C₇×C₁₇	C₂×C₃	semidirect
7. C₅×C₁₇×ΓL(1,𝔽₈)	D₃	semidirect
10. C₃×C₁₇×ΓL(1,𝔽₈)	D₅	semidirect
15. C₅×C₇×C₁₇	C₃×D₃	semidirect
16. D₂₅₅	ΓL(1,𝔽₈)	direct (N × H)
18. C₃×C₇×C₁₇	C₃×D₅	semidirect
20. C₁₇×ΓL(1,𝔽₈)	D₁₅	semidirect
21. C₃×C₅×ΓL(1,𝔽₈)	D₁₇	semidirect
25. C₃×C₅×C₁₇	C₂×ΓL(1,𝔽₈)	semidirect
32. C₇×C₁₇	C₃×D₁₅	semidirect
33. C₃×C₅×C₇	C₃×D₁₇	semidirect
35. C₅×ΓL(1,𝔽₈)	D₅₁	semidirect
39. C₅×C₁₇	D₃×ΓL(1,𝔽₈)	semidirect
41. C₃×ΓL(1,𝔽₈)	D₈₅	semidirect
44. C₃×C₁₇	D₅×ΓL(1,𝔽₈)	semidirect
48. C₅×C₇	C₃×D₅₁	semidirect
52. C₃×C₇	C₃×D₈₅	semidirect
54. ΓL(1,𝔽₈)	D₂₅₅	direct (N × H)
56. C₁₇	ΓL(1,𝔽₈)×D₁₅	semidirect
59. C₃×C₅	ΓL(1,𝔽₈)×D₁₇	semidirect
63. C₇	C₃×D₂₅₅	semidirect
66. C₅	ΓL(1,𝔽₈)×D₅₁	semidirect
69. C₃	ΓL(1,𝔽₈)×D₈₅	semidirect

Subgroups — ΓL(1,𝔽8)×D255 #24924225

Conjugacy classes of subgroups

Complements

Subgroups — ΓL(1,𝔽₈)×D₂₅₅ #24924225