Subgroups — C₂×C₃×C₅×C₁₇×ΓL(1,𝔽₈) #24924226

Order 10710.

96 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, direct product factor, Hall
2	C₂×C₅×C₁₇×ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, direct product factor
3	C₂×C₃×C₅×C₇×C₁₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₁₇×ΓL(1,𝔽₈)	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)	C₂×C₃	1	3	yes	direct product factor
7	C₂×C₃²×C₅×C₁₇	—	7	1	no	maximal, Hall
8	C₂×C₅×C₇×C₁₇	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₁₇×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic, direct product factor, Hall
10	C₃²×C₅×C₁₇	—	7	1	no	Hall
11	C₂×C₃×C₇×C₁₇	C₃×C₅	1	1	yes	characteristic, complemented
12	C₂×C₁₇×ΓL(1,𝔽₈)	C₃×C₅	1	3	yes	direct product factor
13	C₂×C₃×C₅×ΓL(1,𝔽₈)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₅×C₇×C₁₇	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
15	C₂×C₃×C₅×C₁₇	—	7	3	no	—
16	C₂×C₃×C₅×C₁₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
17	C₃×C₇×C₁₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
18	C₁₇×ΓL(1,𝔽₈)	C₂×C₃×C₅	1	3	yes	direct product factor
19	C₃×C₅×ΓL(1,𝔽₈)	C₂×C₁₇	1	1	yes	characteristic, direct product factor, Hall
20	C₂×C₃²×C₁₇	—	7	1	no	Hall
21	C₃×C₅×C₁₇	—	7	3	no	—
22	C₃×C₅×C₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
23	C₂×C₇×C₁₇	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
24	C₂×C₃×C₅×C₇	C₃×C₁₇	1	1	yes	characteristic, complemented
25	C₂×C₅×ΓL(1,𝔽₈)	C₃×C₁₇	1	3	yes	direct product factor
26	C₂×C₅×C₁₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
27	C₃²×C₁₇	—	7	1	no	Hall
28	C₂×C₃×ΓL(1,𝔽₈)	C₅×C₁₇	1	1	yes	characteristic, direct product factor, Hall
29	C₇×C₁₇	C₂×C₃²×C₅	1	1	yes	characteristic, complemented, Hall
30	C₃×C₅×C₇	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
31	C₅×ΓL(1,𝔽₈)	C₂×C₃×C₁₇	1	3	yes	direct product factor
32	C₂×C₃×C₁₇	—	7	3	no	—
33	C₂×C₃×C₁₇	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
34	C₂×C₃²×C₅	—	7	1	no	Hall
35	C₅×C₁₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
36	C₂×C₅×C₇	C₃²×C₁₇	1	1	yes	characteristic, complemented, Hall
37	C₃×ΓL(1,𝔽₈)	C₂×C₅×C₁₇	1	1	yes	characteristic, direct product factor, Hall
38	C₃×C₁₇	—	7	3	no	—
39	C₃×C₁₇	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
40	C₃²×C₅	—	7	1	no	Hall
41	C₂×C₃×C₇	C₃×C₅×C₁₇	1	1	yes	characteristic, complemented
42	C₂×ΓL(1,𝔽₈)	C₃×C₅×C₁₇	1	3	yes	direct product factor
43	C₅×C₇	C₂×C₃²×C₁₇	1	1	yes	characteristic, complemented, Hall
44	C₂×C₁₇	C₃×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
45	C₂×C₃×C₅	—	7	3	no	—
46	C₂×C₃×C₅	C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
47	C₃×C₇	C₂×C₃×C₅×C₁₇	1	1	yes	characteristic, complemented
48	ΓL(1,𝔽₈)	C₂×C₃×C₅×C₁₇	1	3	yes	direct product factor
49	C₂×C₃²	—	7	1	no	Hall
50	C₁₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
51	C₃×C₅	—	7	3	no	—
52	C₃×C₅	C₂×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
53	C₂×C₇	C₃²×C₅×C₁₇	1	1	yes	characteristic, complemented, Hall
54	C₂×C₅	C₃×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
55	C₃²	—	7	1	no	Sylow
56	C₇	C₂×C₃²×C₅×C₁₇	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₃	—	7	3	no	—
58	C₂×C₃	C₅×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
59	C₅	C₂×C₃×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
60	C₃	—	7	3	no	—
61	C₃	C₂×C₅×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
62	C₂	C₃×C₅×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅×C₁₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃²×C₅×C₁₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₇×C₁₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₅×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₁₇×ΓL(1,𝔽₈)	C₂	direct (N × H)
2. C₂×C₅×C₁₇×ΓL(1,𝔽₈)	C₃	direct (N × H)
3. C₂×C₃×C₅×C₇×C₁₇	C₃	semidirect
4. C₂×C₃×C₁₇×ΓL(1,𝔽₈)	C₅	direct (N × H)
5. C₃×C₅×C₇×C₁₇	C₂×C₃	semidirect
6. C₅×C₁₇×ΓL(1,𝔽₈)	C₂×C₃	direct (N × H)
8. C₂×C₅×C₇×C₁₇	C₃²	semidirect
9. C₃×C₁₇×ΓL(1,𝔽₈)	C₂×C₅	direct (N × H)
11. C₂×C₃×C₇×C₁₇	C₃×C₅	semidirect
12. C₂×C₁₇×ΓL(1,𝔽₈)	C₃×C₅	direct (N × H)
13. C₂×C₃×C₅×ΓL(1,𝔽₈)	C₁₇	direct (N × H)
14. C₅×C₇×C₁₇	C₂×C₃²	semidirect
16. C₂×C₃×C₅×C₁₇	ΓL(1,𝔽₈)	direct (N × H)
17. C₃×C₇×C₁₇	C₂×C₃×C₅	semidirect
18. C₁₇×ΓL(1,𝔽₈)	C₂×C₃×C₅	direct (N × H)
19. C₃×C₅×ΓL(1,𝔽₈)	C₂×C₁₇	direct (N × H)
22. C₃×C₅×C₁₇	C₂×ΓL(1,𝔽₈)	direct (N × H)
23. C₂×C₇×C₁₇	C₃²×C₅	semidirect
24. C₂×C₃×C₅×C₇	C₃×C₁₇	semidirect
25. C₂×C₅×ΓL(1,𝔽₈)	C₃×C₁₇	direct (N × H)
26. C₂×C₅×C₁₇	C₃×ΓL(1,𝔽₈)	direct (N × H)
28. C₂×C₃×ΓL(1,𝔽₈)	C₅×C₁₇	direct (N × H)
29. C₇×C₁₇	C₂×C₃²×C₅	semidirect
30. C₃×C₅×C₇	C₂×C₃×C₁₇	semidirect
31. C₅×ΓL(1,𝔽₈)	C₂×C₃×C₁₇	direct (N × H)
33. C₂×C₃×C₁₇	C₅×ΓL(1,𝔽₈)	direct (N × H)
35. C₅×C₁₇	C₂×C₃×ΓL(1,𝔽₈)	direct (N × H)
36. C₂×C₅×C₇	C₃²×C₁₇	semidirect
37. C₃×ΓL(1,𝔽₈)	C₂×C₅×C₁₇	direct (N × H)
39. C₃×C₁₇	C₂×C₅×ΓL(1,𝔽₈)	direct (N × H)
41. C₂×C₃×C₇	C₃×C₅×C₁₇	semidirect
42. C₂×ΓL(1,𝔽₈)	C₃×C₅×C₁₇	direct (N × H)
43. C₅×C₇	C₂×C₃²×C₁₇	semidirect
44. C₂×C₁₇	C₃×C₅×ΓL(1,𝔽₈)	direct (N × H)
46. C₂×C₃×C₅	C₁₇×ΓL(1,𝔽₈)	direct (N × H)
47. C₃×C₇	C₂×C₃×C₅×C₁₇	semidirect
48. ΓL(1,𝔽₈)	C₂×C₃×C₅×C₁₇	direct (N × H)
50. C₁₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	direct (N × H)
52. C₃×C₅	C₂×C₁₇×ΓL(1,𝔽₈)	direct (N × H)
53. C₂×C₇	C₃²×C₅×C₁₇	semidirect
54. C₂×C₅	C₃×C₁₇×ΓL(1,𝔽₈)	direct (N × H)
56. C₇	C₂×C₃²×C₅×C₁₇	semidirect
58. C₂×C₃	C₅×C₁₇×ΓL(1,𝔽₈)	direct (N × H)
59. C₅	C₂×C₃×C₁₇×ΓL(1,𝔽₈)	direct (N × H)
61. C₃	C₂×C₅×C₁₇×ΓL(1,𝔽₈)	direct (N × H)
62. C₂	C₃×C₅×C₁₇×ΓL(1,𝔽₈)	direct (N × H)

Subgroups — C2×C3×C5×C17×ΓL(1,𝔽8) #24924226

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₃×C₅×C₁₇×ΓL(1,𝔽₈) #24924226