Subgroups — C₂²×C₃×C₅²×ΓL(1,𝔽₈) #20263155

Order 6300.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅²×ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₅²×ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, direct product factor
3	C₂²×C₃×C₅²×C₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×ΓL(1,𝔽₈)	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₃×C₅²×C₇	C₂×C₃	1	3	yes	complemented
7	C₂×C₅²×ΓL(1,𝔽₈)	C₂×C₃	1	9	yes	direct product factor
8	C₂²×C₃²×C₅²	—	7	1	no	maximal, Hall
9	C₂²×C₅²×C₇	C₃²	1	1	yes	characteristic, complemented, Hall
10	C₂×C₃×C₅×ΓL(1,𝔽₈)	C₂×C₅	1	18	yes	direct product factor
11	C₃×C₅²×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
12	C₅²×ΓL(1,𝔽₈)	C₂²×C₃	1	3	yes	direct product factor
13	C₂×C₃²×C₅²	—	7	3	no	—
14	C₂²×C₃×C₅×C₇	C₃×C₅	1	6	yes	complemented
15	C₂²×C₅×ΓL(1,𝔽₈)	C₃×C₅	1	18	yes	direct product factor
16	C₂×C₅²×C₇	C₂×C₃²	1	3	yes	complemented
17	C₃×C₅×ΓL(1,𝔽₈)	C₂²×C₅	1	6	yes	direct product factor
18	C₂²×C₃×C₅²	—	7	3	no	—
19	C₂²×C₃×C₅²	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
20	C₂²×C₃×ΓL(1,𝔽₈)	C₅²	1	1	yes	characteristic, direct product factor, Hall
21	C₃²×C₅²	—	7	1	no	Hall
22	C₂×C₃×C₅×C₇	C₂×C₃×C₅	1	18	yes	complemented
23	C₂×C₅×ΓL(1,𝔽₈)	C₂×C₃×C₅	1	54	yes	direct product factor
24	C₂²×C₃²×C₅	—	7	6	no	—
25	C₅²×C₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
26	C₂×C₃×C₅²	—	7	9	no	—
27	C₂×C₃×C₅²	C₂×ΓL(1,𝔽₈)	1	3	yes	direct product factor
28	C₂²×C₅×C₇	C₃²×C₅	1	6	yes	complemented
29	C₂×C₃×ΓL(1,𝔽₈)	C₂×C₅²	1	3	yes	direct product factor
30	C₃×C₅×C₇	C₂²×C₃×C₅	1	6	yes	complemented
31	C₅×ΓL(1,𝔽₈)	C₂²×C₃×C₅	1	18	yes	direct product factor
32	C₂²×C₅²	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
33	C₂×C₃²×C₅	—	7	18	no	—
34	C₂²×C₃×C₇	C₃×C₅²	1	1	yes	characteristic, complemented
35	C₂²×ΓL(1,𝔽₈)	C₃×C₅²	1	3	yes	direct product factor
36	C₃×C₅²	—	7	3	no	—
37	C₃×C₅²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
38	C₂×C₅×C₇	C₂×C₃²×C₅	1	18	yes	complemented
39	C₃×ΓL(1,𝔽₈)	C₂²×C₅²	1	1	yes	characteristic, direct product factor, Hall
40	C₂²×C₃×C₅	—	7	18	no	—
41	C₂²×C₃×C₅	C₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
42	C₂×C₅²	C₂×C₃×ΓL(1,𝔽₈)	1	3	yes	direct product factor
43	C₃²×C₅	—	7	6	no	—
44	C₂×C₃×C₇	C₂×C₃×C₅²	1	3	yes	complemented
45	C₂×ΓL(1,𝔽₈)	C₂×C₃×C₅²	1	9	yes	direct product factor
46	C₂²×C₃²	—	7	1	no	Hall
47	C₅×C₇	C₂²×C₃²×C₅	1	6	yes	complemented
48	C₂×C₃×C₅	—	7	54	no	—
49	C₂×C₃×C₅	C₂×C₅×ΓL(1,𝔽₈)	1	18	yes	direct product factor
50	C₂²×C₇	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
51	C₅²	C₂²×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
52	C₃×C₇	C₂²×C₃×C₅²	1	1	yes	characteristic, complemented
53	ΓL(1,𝔽₈)	C₂²×C₃×C₅²	1	3	yes	direct product factor
54	C₂²×C₅	C₃×C₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
55	C₂×C₃²	—	7	3	no	—
56	C₃×C₅	—	7	18	no	—
57	C₃×C₅	C₂²×C₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
58	C₂×C₇	C₂×C₃²×C₅²	1	3	yes	complemented
59	C₂²×C₃	—	7	3	no	—
60	C₂²×C₃	C₅²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
61	C₂×C₅	C₂×C₃×C₅×ΓL(1,𝔽₈)	1	18	yes	direct product factor
62	C₃²	—	7	1	no	Sylow
63	C₇	C₂²×C₃²×C₅²	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	7	9	no	—
65	C₂×C₃	C₂×C₅²×ΓL(1,𝔽₈)	1	3	yes	direct product factor
66	C₅	C₂²×C₃×C₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
67	C₂²	C₃×C₅²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
68	C₃	—	7	3	no	—
69	C₃	C₂²×C₅²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
70	C₂	C₂×C₃×C₅²×ΓL(1,𝔽₈)	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₅²	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₂²×C₃²×C₅²
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅²×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₃×C₅²

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅²×ΓL(1,𝔽₈)	C₂	direct (N × H)
2. C₂²×C₅²×ΓL(1,𝔽₈)	C₃	direct (N × H)
3. C₂²×C₃×C₅²×C₇	C₃	semidirect
4. C₃×C₅²×ΓL(1,𝔽₈)	C₂²	direct (N × H)
5. C₂²×C₃×C₅×ΓL(1,𝔽₈)	C₅	direct (N × H)
6. C₂×C₃×C₅²×C₇	C₂×C₃	semidirect
7. C₂×C₅²×ΓL(1,𝔽₈)	C₂×C₃	direct (N × H)
9. C₂²×C₅²×C₇	C₃²	semidirect
10. C₂×C₃×C₅×ΓL(1,𝔽₈)	C₂×C₅	direct (N × H)
11. C₃×C₅²×C₇	C₂²×C₃	semidirect
12. C₅²×ΓL(1,𝔽₈)	C₂²×C₃	direct (N × H)
14. C₂²×C₃×C₅×C₇	C₃×C₅	semidirect
15. C₂²×C₅×ΓL(1,𝔽₈)	C₃×C₅	direct (N × H)
16. C₂×C₅²×C₇	C₂×C₃²	semidirect
17. C₃×C₅×ΓL(1,𝔽₈)	C₂²×C₅	direct (N × H)
19. C₂²×C₃×C₅²	ΓL(1,𝔽₈)	direct (N × H)
20. C₂²×C₃×ΓL(1,𝔽₈)	C₅²	direct (N × H)
22. C₂×C₃×C₅×C₇	C₂×C₃×C₅	semidirect
23. C₂×C₅×ΓL(1,𝔽₈)	C₂×C₃×C₅	direct (N × H)
25. C₅²×C₇	C₂²×C₃²	semidirect
27. C₂×C₃×C₅²	C₂×ΓL(1,𝔽₈)	direct (N × H)
28. C₂²×C₅×C₇	C₃²×C₅	semidirect
29. C₂×C₃×ΓL(1,𝔽₈)	C₂×C₅²	direct (N × H)
30. C₃×C₅×C₇	C₂²×C₃×C₅	semidirect
31. C₅×ΓL(1,𝔽₈)	C₂²×C₃×C₅	direct (N × H)
32. C₂²×C₅²	C₃×ΓL(1,𝔽₈)	direct (N × H)
34. C₂²×C₃×C₇	C₃×C₅²	semidirect
35. C₂²×ΓL(1,𝔽₈)	C₃×C₅²	direct (N × H)
37. C₃×C₅²	C₂²×ΓL(1,𝔽₈)	direct (N × H)
38. C₂×C₅×C₇	C₂×C₃²×C₅	semidirect
39. C₃×ΓL(1,𝔽₈)	C₂²×C₅²	direct (N × H)
41. C₂²×C₃×C₅	C₅×ΓL(1,𝔽₈)	direct (N × H)
42. C₂×C₅²	C₂×C₃×ΓL(1,𝔽₈)	direct (N × H)
44. C₂×C₃×C₇	C₂×C₃×C₅²	semidirect
45. C₂×ΓL(1,𝔽₈)	C₂×C₃×C₅²	direct (N × H)
47. C₅×C₇	C₂²×C₃²×C₅	semidirect
49. C₂×C₃×C₅	C₂×C₅×ΓL(1,𝔽₈)	direct (N × H)
50. C₂²×C₇	C₃²×C₅²	semidirect
51. C₅²	C₂²×C₃×ΓL(1,𝔽₈)	direct (N × H)
52. C₃×C₇	C₂²×C₃×C₅²	semidirect
53. ΓL(1,𝔽₈)	C₂²×C₃×C₅²	direct (N × H)
54. C₂²×C₅	C₃×C₅×ΓL(1,𝔽₈)	direct (N × H)
57. C₃×C₅	C₂²×C₅×ΓL(1,𝔽₈)	direct (N × H)
58. C₂×C₇	C₂×C₃²×C₅²	semidirect
60. C₂²×C₃	C₅²×ΓL(1,𝔽₈)	direct (N × H)
61. C₂×C₅	C₂×C₃×C₅×ΓL(1,𝔽₈)	direct (N × H)
63. C₇	C₂²×C₃²×C₅²	semidirect
65. C₂×C₃	C₂×C₅²×ΓL(1,𝔽₈)	direct (N × H)
66. C₅	C₂²×C₃×C₅×ΓL(1,𝔽₈)	direct (N × H)
67. C₂²	C₃×C₅²×ΓL(1,𝔽₈)	direct (N × H)
69. C₃	C₂²×C₅²×ΓL(1,𝔽₈)	direct (N × H)
70. C₂	C₂×C₃×C₅²×ΓL(1,𝔽₈)	direct (N × H)

Subgroups — C22×C3×C52×ΓL(1,𝔽8) #20263155

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×C₃×C₅²×ΓL(1,𝔽₈) #20263155