Subgroups — C₂²×C₅²×C₁₇×ΓL(1,𝔽₈) #49088039

Order 35700.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×C₁₇×ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₅²×C₇×C₁₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅²×C₁₇×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂²×C₅×C₁₇×ΓL(1,𝔽₈)	C₅	1	6	yes	maximal, direct product factor
5	C₂×C₅²×C₇×C₁₇	C₂×C₃	1	3	yes	complemented
6	C₂²×C₃×C₅²×C₁₇	—	7	1	no	maximal, Hall
7	C₂×C₅×C₁₇×ΓL(1,𝔽₈)	C₂×C₅	1	18	yes	direct product factor
8	C₅²×C₇×C₁₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₅²×C₁₇	—	7	3	no	—
10	C₂²×C₅×C₇×C₁₇	C₃×C₅	1	6	yes	complemented
11	C₂²×C₅²×ΓL(1,𝔽₈)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₅×C₁₇×ΓL(1,𝔽₈)	C₂²×C₅	1	6	yes	direct product factor
13	C₂²×C₅²×C₁₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
14	C₂²×C₁₇×ΓL(1,𝔽₈)	C₅²	1	1	yes	characteristic, direct product factor, Hall
15	C₃×C₅²×C₁₇	—	7	1	no	Hall
16	C₂×C₅×C₇×C₁₇	C₂×C₃×C₅	1	18	yes	complemented
17	C₂×C₅²×ΓL(1,𝔽₈)	C₂×C₁₇	1	3	yes	direct product factor
18	C₂²×C₃×C₅×C₁₇	—	7	6	no	—
19	C₂×C₅²×C₁₇	C₂×ΓL(1,𝔽₈)	1	3	yes	direct product factor
20	C₂×C₁₇×ΓL(1,𝔽₈)	C₂×C₅²	1	3	yes	direct product factor
21	C₂²×C₅²×C₇	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
22	C₅×C₇×C₁₇	C₂²×C₃×C₅	1	6	yes	complemented
23	C₅²×ΓL(1,𝔽₈)	C₂²×C₁₇	1	1	yes	characteristic, direct product factor, Hall
24	C₂×C₃×C₅×C₁₇	—	7	18	no	—
25	C₂²×C₇×C₁₇	C₃×C₅²	1	1	yes	characteristic, complemented, Hall
26	C₅²×C₁₇	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
27	C₂²×C₅×ΓL(1,𝔽₈)	C₅×C₁₇	1	6	yes	direct product factor
28	C₁₇×ΓL(1,𝔽₈)	C₂²×C₅²	1	1	yes	characteristic, direct product factor, Hall
29	C₂×C₅²×C₇	C₂×C₃×C₁₇	1	3	yes	complemented
30	C₂²×C₅×C₁₇	C₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
31	C₂²×C₃×C₅²	—	7	1	no	Hall
32	C₃×C₅×C₁₇	—	7	6	no	—
33	C₂×C₇×C₁₇	C₂×C₃×C₅²	1	3	yes	complemented
34	C₂×C₅×ΓL(1,𝔽₈)	C₂×C₅×C₁₇	1	18	yes	direct product factor
35	C₂²×C₃×C₁₇	—	7	1	no	Hall
36	C₅²×C₇	C₂²×C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
37	C₂×C₅×C₁₇	C₂×C₅×ΓL(1,𝔽₈)	1	18	yes	direct product factor
38	C₂×C₃×C₅²	—	7	3	no	—
39	C₂²×C₅×C₇	C₃×C₅×C₁₇	1	6	yes	complemented
40	C₇×C₁₇	C₂²×C₃×C₅²	1	1	yes	characteristic, complemented, Hall
41	C₅×ΓL(1,𝔽₈)	C₂²×C₅×C₁₇	1	6	yes	direct product factor
42	C₂×C₃×C₁₇	—	7	3	no	—
43	C₂²×C₅²	C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
44	C₅×C₁₇	C₂²×C₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
45	C₂²×ΓL(1,𝔽₈)	C₅²×C₁₇	1	1	yes	characteristic, direct product factor, Hall
46	C₃×C₅²	—	7	1	no	Hall
47	C₂×C₅×C₇	C₂×C₃×C₅×C₁₇	1	18	yes	complemented
48	C₂²×C₁₇	C₅²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
49	C₂²×C₃×C₅	—	7	6	no	—
50	C₃×C₁₇	—	7	1	no	Hall
51	C₂×C₅²	C₂×C₁₇×ΓL(1,𝔽₈)	1	3	yes	direct product factor
52	C₂×ΓL(1,𝔽₈)	C₂×C₅²×C₁₇	1	3	yes	direct product factor
53	C₅×C₇	C₂²×C₃×C₅×C₁₇	1	6	yes	complemented
54	C₂×C₁₇	C₂×C₅²×ΓL(1,𝔽₈)	1	3	yes	direct product factor
55	C₂×C₃×C₅	—	7	18	no	—
56	C₂²×C₇	C₃×C₅²×C₁₇	1	1	yes	characteristic, complemented, Hall
57	C₅²	C₂²×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
58	ΓL(1,𝔽₈)	C₂²×C₅²×C₁₇	1	1	yes	characteristic, direct product factor, Hall
59	C₂²×C₅	C₅×C₁₇×ΓL(1,𝔽₈)	1	6	yes	direct product factor
60	C₁₇	C₂²×C₅²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
61	C₃×C₅	—	7	6	no	—
62	C₂×C₇	C₂×C₃×C₅²×C₁₇	1	3	yes	complemented
63	C₂²×C₃	—	7	1	no	Hall
64	C₂×C₅	C₂×C₅×C₁₇×ΓL(1,𝔽₈)	1	18	yes	direct product factor
65	C₇	C₂²×C₃×C₅²×C₁₇	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	7	3	no	—
67	C₅	C₂²×C₅×C₁₇×ΓL(1,𝔽₈)	1	6	yes	direct product factor
68	C₂²	C₅²×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
69	C₃	—	7	1	no	Sylow
70	C₂	C₂×C₅²×C₁₇×ΓL(1,𝔽₈)	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₅²×C₁₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂²×C₃×C₅²×C₁₇
Frattini	C₁	G
Fitting, socle	C₂²×C₅²×C₇×C₁₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₅²×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×C₁₇×ΓL(1,𝔽₈)	C₂	direct (N × H)
2. C₂²×C₅²×C₇×C₁₇	C₃	semidirect
3. C₅²×C₁₇×ΓL(1,𝔽₈)	C₂²	direct (N × H)
4. C₂²×C₅×C₁₇×ΓL(1,𝔽₈)	C₅	direct (N × H)
5. C₂×C₅²×C₇×C₁₇	C₂×C₃	semidirect
7. C₂×C₅×C₁₇×ΓL(1,𝔽₈)	C₂×C₅	direct (N × H)
8. C₅²×C₇×C₁₇	C₂²×C₃	semidirect
10. C₂²×C₅×C₇×C₁₇	C₃×C₅	semidirect
11. C₂²×C₅²×ΓL(1,𝔽₈)	C₁₇	direct (N × H)
12. C₅×C₁₇×ΓL(1,𝔽₈)	C₂²×C₅	direct (N × H)
13. C₂²×C₅²×C₁₇	ΓL(1,𝔽₈)	direct (N × H)
14. C₂²×C₁₇×ΓL(1,𝔽₈)	C₅²	direct (N × H)
16. C₂×C₅×C₇×C₁₇	C₂×C₃×C₅	semidirect
17. C₂×C₅²×ΓL(1,𝔽₈)	C₂×C₁₇	direct (N × H)
19. C₂×C₅²×C₁₇	C₂×ΓL(1,𝔽₈)	direct (N × H)
20. C₂×C₁₇×ΓL(1,𝔽₈)	C₂×C₅²	direct (N × H)
21. C₂²×C₅²×C₇	C₃×C₁₇	semidirect
22. C₅×C₇×C₁₇	C₂²×C₃×C₅	semidirect
23. C₅²×ΓL(1,𝔽₈)	C₂²×C₁₇	direct (N × H)
25. C₂²×C₇×C₁₇	C₃×C₅²	semidirect
26. C₅²×C₁₇	C₂²×ΓL(1,𝔽₈)	direct (N × H)
27. C₂²×C₅×ΓL(1,𝔽₈)	C₅×C₁₇	direct (N × H)
28. C₁₇×ΓL(1,𝔽₈)	C₂²×C₅²	direct (N × H)
29. C₂×C₅²×C₇	C₂×C₃×C₁₇	semidirect
30. C₂²×C₅×C₁₇	C₅×ΓL(1,𝔽₈)	direct (N × H)
33. C₂×C₇×C₁₇	C₂×C₃×C₅²	semidirect
34. C₂×C₅×ΓL(1,𝔽₈)	C₂×C₅×C₁₇	direct (N × H)
36. C₅²×C₇	C₂²×C₃×C₁₇	semidirect
37. C₂×C₅×C₁₇	C₂×C₅×ΓL(1,𝔽₈)	direct (N × H)
39. C₂²×C₅×C₇	C₃×C₅×C₁₇	semidirect
40. C₇×C₁₇	C₂²×C₃×C₅²	semidirect
41. C₅×ΓL(1,𝔽₈)	C₂²×C₅×C₁₇	direct (N × H)
43. C₂²×C₅²	C₁₇×ΓL(1,𝔽₈)	direct (N × H)
44. C₅×C₁₇	C₂²×C₅×ΓL(1,𝔽₈)	direct (N × H)
45. C₂²×ΓL(1,𝔽₈)	C₅²×C₁₇	direct (N × H)
47. C₂×C₅×C₇	C₂×C₃×C₅×C₁₇	semidirect
48. C₂²×C₁₇	C₅²×ΓL(1,𝔽₈)	direct (N × H)
51. C₂×C₅²	C₂×C₁₇×ΓL(1,𝔽₈)	direct (N × H)
52. C₂×ΓL(1,𝔽₈)	C₂×C₅²×C₁₇	direct (N × H)
53. C₅×C₇	C₂²×C₃×C₅×C₁₇	semidirect
54. C₂×C₁₇	C₂×C₅²×ΓL(1,𝔽₈)	direct (N × H)
56. C₂²×C₇	C₃×C₅²×C₁₇	semidirect
57. C₅²	C₂²×C₁₇×ΓL(1,𝔽₈)	direct (N × H)
58. ΓL(1,𝔽₈)	C₂²×C₅²×C₁₇	direct (N × H)
59. C₂²×C₅	C₅×C₁₇×ΓL(1,𝔽₈)	direct (N × H)
60. C₁₇	C₂²×C₅²×ΓL(1,𝔽₈)	direct (N × H)
62. C₂×C₇	C₂×C₃×C₅²×C₁₇	semidirect
64. C₂×C₅	C₂×C₅×C₁₇×ΓL(1,𝔽₈)	direct (N × H)
65. C₇	C₂²×C₃×C₅²×C₁₇	semidirect
67. C₅	C₂²×C₅×C₁₇×ΓL(1,𝔽₈)	direct (N × H)
68. C₂²	C₅²×C₁₇×ΓL(1,𝔽₈)	direct (N × H)
70. C₂	C₂×C₅²×C₁₇×ΓL(1,𝔽₈)	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅²×C₁₇×ΓL(1,𝔽₈)	2
C₂²×C₅²×C₇×C₁₇	3
C₂²×C₅×C₁₇×ΓL(1,𝔽₈)	5
C₂²×C₃×C₅²×C₁₇	7
C₂²×C₅²×ΓL(1,𝔽₈)	17

Subgroups — C22×C52×C17×ΓL(1,𝔽8) #49088039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅²×C₁₇×ΓL(1,𝔽₈) #49088039