Subgroups — C₅×ΓL(1,𝔽₈)×D₅₁ #24924221

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	C₅×ΓL(1,𝔽₈)×D₁₇	—	3	1	no	maximal
3	C₅×C₇×D₅₁	C₃	1	1	yes	maximal, characteristic, complemented
4	ΓL(1,𝔽₈)×D₅₁	C₅	1	1	yes	maximal, characteristic, direct product factor, 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×C₅×D₅₁	—	7	1	no	maximal, Hall
9	C₅×C₇×D₁₇	—	3	1	no	Hall
10	C₃×C₁₇×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic, complemented, Hall
11	C₃²×C₅×C₁₇	—	7	1	no	Hall
12	C₇×D₅₁	C₃×C₅	1	1	yes	characteristic, complemented
13	ΓL(1,𝔽₈)×D₁₇	—	3	1	no	—
14	C₅×D₃×ΓL(1,𝔽₈)	—	17	1	no	maximal, Hall
15	C₅×C₇×C₁₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
16	C₅×D₅₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
17	C₃×C₅×D₁₇	—	21	1	no	—
18	C₃×C₇×C₁₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
19	C₁₇×ΓL(1,𝔽₈)	—	2	1	no	—
20	C₁₇×ΓL(1,𝔽₈)	C₅×D₃	1	1	yes	characteristic, complemented
21	C₃×C₅×ΓL(1,𝔽₈)	D₁₇	1	1	yes	characteristic, complemented, Hall
22	C₃×D₅₁	—	7	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	—	3	1	no	Hall
27	C₅×D₃×C₇	—	17	1	no	—
28	C₂×C₅×ΓL(1,𝔽₈)	—	51	1	no	—
29	C₅×D₁₇	—	3	1	no	Hall
30	C₃²×C₁₇	—	7	1	no	Hall
31	D₃×ΓL(1,𝔽₈)	—	17	1	no	Hall
32	C₇×C₁₇	C₃×C₅×D₃	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, direct product factor
36	D₅₁	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
37	C₃×D₁₇	—	21	1	no	—
38	C₃×C₅×D₃	—	119	1	no	Hall
39	C₅×C₁₇	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
40	C₂×C₅×C₇	—	51	1	no	Hall
41	C₃×ΓL(1,𝔽₈)	C₅×D₁₇	1	1	yes	characteristic, complemented, Hall
42	C₃×C₁₇	—	7	1	no	—
43	C₃×C₁₇	—	14	1	no	—
44	C₃×C₁₇	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
45	C₃²×C₅	—	7	1	no	Hall
46	D₃×C₇	—	17	1	no	—
47	C₂×ΓL(1,𝔽₈)	—	51	1	no	—
48	C₅×C₇	C₃×D₅₁	1	1	yes	characteristic, complemented, Hall
49	D₁₇	—	3	1	no	Hall
50	C₂×C₃×C₅	—	357	1	no	—
51	C₅×D₃	—	17	1	no	—
52	C₃×C₇	C₃×C₅×D₁₇	1	1	yes	characteristic, complemented
53	ΓL(1,𝔽₈)	—	2	1	no	—
54	ΓL(1,𝔽₈)	C₅×D₅₁	1	1	yes	characteristic, direct product factor
55	C₃×D₃	—	119	1	no	Hall
56	C₁₇	C₅×D₃×ΓL(1,𝔽₈)	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	—	51	1	no	Hall
61	C₂×C₅	—	51	1	no	Hall
62	C₃²	—	7	1	no	Sylow
63	C₇	C₃×C₅×D₅₁	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	357	1	no	—
65	D₃	—	17	1	no	—
66	C₅	ΓL(1,𝔽₈)×D₅₁	1	1	yes	characteristic, direct product factor, Sylow
67	C₃	—	7	1	no	—
68	C₃	—	14	1	no	—
69	C₃	C₅×ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, complemented
70	C₂	—	51	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅	ΓL(1,𝔽₈)×D₅₁
commutator, cosocle, hypocenter	C₃×C₇×C₁₇	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₁₇	C₂×C₃
LCS	G ▷ C₃×C₇×C₁₇
UCS	C₁ ◁ C₅

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. C₅×C₇×D₅₁	C₃	semidirect
4. ΓL(1,𝔽₈)×D₅₁	C₅	direct (N × H)
5. C₃×C₅×C₇×C₁₇	C₂×C₃	semidirect
7. C₅×C₁₇×ΓL(1,𝔽₈)	D₃	semidirect
10. C₃×C₁₇×ΓL(1,𝔽₈)	C₂×C₅	semidirect
12. C₇×D₅₁	C₃×C₅	semidirect
15. C₅×C₇×C₁₇	C₃×D₃	semidirect
16. C₅×D₅₁	ΓL(1,𝔽₈)	direct (N × H)
18. C₃×C₇×C₁₇	C₂×C₃×C₅	semidirect
20. C₁₇×ΓL(1,𝔽₈)	C₅×D₃	semidirect
21. C₃×C₅×ΓL(1,𝔽₈)	D₁₇	semidirect
25. C₃×C₅×C₁₇	C₂×ΓL(1,𝔽₈)	semidirect
32. C₇×C₁₇	C₃×C₅×D₃	semidirect
33. C₃×C₅×C₇	C₃×D₁₇	semidirect
35. C₅×ΓL(1,𝔽₈)	D₅₁	direct (N × H)
36. D₅₁	C₅×ΓL(1,𝔽₈)	direct (N × H)
39. C₅×C₁₇	D₃×ΓL(1,𝔽₈)	semidirect
41. C₃×ΓL(1,𝔽₈)	C₅×D₁₇	semidirect
44. C₃×C₁₇	C₂×C₅×ΓL(1,𝔽₈)	semidirect
48. C₅×C₇	C₃×D₅₁	semidirect
52. C₃×C₇	C₃×C₅×D₁₇	semidirect
54. ΓL(1,𝔽₈)	C₅×D₅₁	direct (N × H)
56. C₁₇	C₅×D₃×ΓL(1,𝔽₈)	semidirect
59. C₃×C₅	ΓL(1,𝔽₈)×D₁₇	semidirect
63. C₇	C₃×C₅×D₅₁	semidirect
66. C₅	ΓL(1,𝔽₈)×D₅₁	direct (N × H)
69. C₃	C₅×ΓL(1,𝔽₈)×D₁₇	semidirect

Subgroups — C5×ΓL(1,𝔽8)×D51 #24924221

Conjugacy classes of subgroups

Complements

Subgroups — C₅×ΓL(1,𝔽₈)×D₅₁ #24924221