Subgroups — C₂²×C₅²×C₁₁×ΓL(1,𝔽₈) #37528736

Order 23100.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×C₁₁×ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₅²×C₇×C₁₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅²×C₁₁×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂²×C₅×C₁₁×ΓL(1,𝔽₈)	C₅	1	6	yes	maximal, direct product factor
5	C₂×C₅²×C₇×C₁₁	C₂×C₃	1	3	yes	complemented
6	C₂²×C₃×C₅²×C₁₁	—	7	1	no	maximal, Hall
7	C₂×C₅×C₁₁×ΓL(1,𝔽₈)	C₂×C₅	1	18	yes	direct product factor
8	C₂²×C₅²×ΓL(1,𝔽₈)	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅²×C₇×C₁₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×C₃×C₅²×C₁₁	—	7	3	no	—
11	C₂²×C₅×C₇×C₁₁	C₃×C₅	1	6	yes	complemented
12	C₅×C₁₁×ΓL(1,𝔽₈)	C₂²×C₅	1	6	yes	direct product factor
13	C₂²×C₅²×C₁₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₅²×ΓL(1,𝔽₈)	C₂×C₁₁	1	3	yes	direct product factor
15	C₂²×C₁₁×ΓL(1,𝔽₈)	C₅²	1	1	yes	characteristic, direct product factor, Hall
16	C₃×C₅²×C₁₁	—	7	1	no	Hall
17	C₂×C₅×C₇×C₁₁	C₂×C₃×C₅	1	18	yes	complemented
18	C₂²×C₅²×C₇	C₃×C₁₁	1	1	yes	characteristic, complemented, Hall
19	C₂²×C₃×C₅×C₁₁	—	7	6	no	—
20	C₂×C₅²×C₁₁	C₂×ΓL(1,𝔽₈)	1	3	yes	direct product factor
21	C₅²×ΓL(1,𝔽₈)	C₂²×C₁₁	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₁₁×ΓL(1,𝔽₈)	C₂×C₅²	1	3	yes	direct product factor
23	C₂²×C₅×ΓL(1,𝔽₈)	C₅×C₁₁	1	6	yes	direct product factor
24	C₅×C₇×C₁₁	C₂²×C₃×C₅	1	6	yes	complemented
25	C₂×C₅²×C₇	C₂×C₃×C₁₁	1	3	yes	complemented
26	C₂×C₃×C₅×C₁₁	—	7	18	no	—
27	C₂²×C₇×C₁₁	C₃×C₅²	1	1	yes	characteristic, complemented, Hall
28	C₂²×C₃×C₅²	—	7	1	no	Hall
29	C₅²×C₁₁	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
30	C₁₁×ΓL(1,𝔽₈)	C₂²×C₅²	1	1	yes	characteristic, direct product factor, Hall
31	C₂²×C₅×C₁₁	C₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
32	C₂×C₅×ΓL(1,𝔽₈)	C₂×C₅×C₁₁	1	18	yes	direct product factor
33	C₅²×C₇	C₂²×C₃×C₁₁	1	1	yes	characteristic, complemented, Hall
34	C₃×C₅×C₁₁	—	7	6	no	—
35	C₂×C₇×C₁₁	C₂×C₃×C₅²	1	3	yes	complemented
36	C₂×C₃×C₅²	—	7	3	no	—
37	C₂²×C₅×C₇	C₃×C₅×C₁₁	1	6	yes	complemented
38	C₂²×C₃×C₁₁	—	7	1	no	Hall
39	C₂×C₅×C₁₁	C₂×C₅×ΓL(1,𝔽₈)	1	18	yes	direct product factor
40	C₅×ΓL(1,𝔽₈)	C₂²×C₅×C₁₁	1	6	yes	direct product factor
41	C₂²×C₅²	C₁₁×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
42	C₂²×ΓL(1,𝔽₈)	C₅²×C₁₁	1	1	yes	characteristic, direct product factor, Hall
43	C₇×C₁₁	C₂²×C₃×C₅²	1	1	yes	characteristic, complemented, Hall
44	C₃×C₅²	—	7	1	no	Hall
45	C₂×C₅×C₇	C₂×C₃×C₅×C₁₁	1	18	yes	complemented
46	C₂×C₃×C₁₁	—	7	3	no	—
47	C₂²×C₃×C₅	—	7	6	no	—
48	C₅×C₁₁	C₂²×C₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
49	C₂×C₅²	C₂×C₁₁×ΓL(1,𝔽₈)	1	3	yes	direct product factor
50	C₂²×C₁₁	C₅²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
51	C₂×ΓL(1,𝔽₈)	C₂×C₅²×C₁₁	1	3	yes	direct product factor
52	C₅×C₇	C₂²×C₃×C₅×C₁₁	1	6	yes	complemented
53	C₃×C₁₁	—	7	1	no	Hall
54	C₂×C₃×C₅	—	7	18	no	—
55	C₂²×C₇	C₃×C₅²×C₁₁	1	1	yes	characteristic, complemented, Hall
56	C₅²	C₂²×C₁₁×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×C₁₁	C₂×C₅²×ΓL(1,𝔽₈)	1	3	yes	direct product factor
58	ΓL(1,𝔽₈)	C₂²×C₅²×C₁₁	1	1	yes	characteristic, direct product factor, Hall
59	C₂²×C₅	C₅×C₁₁×ΓL(1,𝔽₈)	1	6	yes	direct product factor
60	C₃×C₅	—	7	6	no	—
61	C₂×C₇	C₂×C₃×C₅²×C₁₁	1	3	yes	complemented
62	C₂²×C₃	—	7	1	no	Hall
63	C₁₁	C₂²×C₅²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₅	C₂×C₅×C₁₁×ΓL(1,𝔽₈)	1	18	yes	direct product factor
65	C₇	C₂²×C₃×C₅²×C₁₁	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	7	3	no	—
67	C₅	C₂²×C₅×C₁₁×ΓL(1,𝔽₈)	1	6	yes	direct product factor
68	C₂²	C₅²×C₁₁×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
69	C₃	—	7	1	no	Sylow
70	C₂	C₂×C₅²×C₁₁×ΓL(1,𝔽₈)	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₅²×C₁₁	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂²×C₃×C₅²×C₁₁
Frattini	C₁	G
Fitting, socle	C₂²×C₅²×C₇×C₁₁	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₅²×C₁₁

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×C₁₁×ΓL(1,𝔽₈)	C₂	direct (N × H)
2. C₂²×C₅²×C₇×C₁₁	C₃	semidirect
3. C₅²×C₁₁×ΓL(1,𝔽₈)	C₂²	direct (N × H)
4. C₂²×C₅×C₁₁×ΓL(1,𝔽₈)	C₅	direct (N × H)
5. C₂×C₅²×C₇×C₁₁	C₂×C₃	semidirect
7. C₂×C₅×C₁₁×ΓL(1,𝔽₈)	C₂×C₅	direct (N × H)
8. C₂²×C₅²×ΓL(1,𝔽₈)	C₁₁	direct (N × H)
9. C₅²×C₇×C₁₁	C₂²×C₃	semidirect
11. C₂²×C₅×C₇×C₁₁	C₃×C₅	semidirect
12. C₅×C₁₁×ΓL(1,𝔽₈)	C₂²×C₅	direct (N × H)
13. C₂²×C₅²×C₁₁	ΓL(1,𝔽₈)	direct (N × H)
14. C₂×C₅²×ΓL(1,𝔽₈)	C₂×C₁₁	direct (N × H)
15. C₂²×C₁₁×ΓL(1,𝔽₈)	C₅²	direct (N × H)
17. C₂×C₅×C₇×C₁₁	C₂×C₃×C₅	semidirect
18. C₂²×C₅²×C₇	C₃×C₁₁	semidirect
20. C₂×C₅²×C₁₁	C₂×ΓL(1,𝔽₈)	direct (N × H)
21. C₅²×ΓL(1,𝔽₈)	C₂²×C₁₁	direct (N × H)
22. C₂×C₁₁×ΓL(1,𝔽₈)	C₂×C₅²	direct (N × H)
23. C₂²×C₅×ΓL(1,𝔽₈)	C₅×C₁₁	direct (N × H)
24. C₅×C₇×C₁₁	C₂²×C₃×C₅	semidirect
25. C₂×C₅²×C₇	C₂×C₃×C₁₁	semidirect
27. C₂²×C₇×C₁₁	C₃×C₅²	semidirect
29. C₅²×C₁₁	C₂²×ΓL(1,𝔽₈)	direct (N × H)
30. C₁₁×ΓL(1,𝔽₈)	C₂²×C₅²	direct (N × H)
31. C₂²×C₅×C₁₁	C₅×ΓL(1,𝔽₈)	direct (N × H)
32. C₂×C₅×ΓL(1,𝔽₈)	C₂×C₅×C₁₁	direct (N × H)
33. C₅²×C₇	C₂²×C₃×C₁₁	semidirect
35. C₂×C₇×C₁₁	C₂×C₃×C₅²	semidirect
37. C₂²×C₅×C₇	C₃×C₅×C₁₁	semidirect
39. C₂×C₅×C₁₁	C₂×C₅×ΓL(1,𝔽₈)	direct (N × H)
40. C₅×ΓL(1,𝔽₈)	C₂²×C₅×C₁₁	direct (N × H)
41. C₂²×C₅²	C₁₁×ΓL(1,𝔽₈)	direct (N × H)
42. C₂²×ΓL(1,𝔽₈)	C₅²×C₁₁	direct (N × H)
43. C₇×C₁₁	C₂²×C₃×C₅²	semidirect
45. C₂×C₅×C₇	C₂×C₃×C₅×C₁₁	semidirect
48. C₅×C₁₁	C₂²×C₅×ΓL(1,𝔽₈)	direct (N × H)
49. C₂×C₅²	C₂×C₁₁×ΓL(1,𝔽₈)	direct (N × H)
50. C₂²×C₁₁	C₅²×ΓL(1,𝔽₈)	direct (N × H)
51. C₂×ΓL(1,𝔽₈)	C₂×C₅²×C₁₁	direct (N × H)
52. C₅×C₇	C₂²×C₃×C₅×C₁₁	semidirect
55. C₂²×C₇	C₃×C₅²×C₁₁	semidirect
56. C₅²	C₂²×C₁₁×ΓL(1,𝔽₈)	direct (N × H)
57. C₂×C₁₁	C₂×C₅²×ΓL(1,𝔽₈)	direct (N × H)
58. ΓL(1,𝔽₈)	C₂²×C₅²×C₁₁	direct (N × H)
59. C₂²×C₅	C₅×C₁₁×ΓL(1,𝔽₈)	direct (N × H)
61. C₂×C₇	C₂×C₃×C₅²×C₁₁	semidirect
63. C₁₁	C₂²×C₅²×ΓL(1,𝔽₈)	direct (N × H)
64. C₂×C₅	C₂×C₅×C₁₁×ΓL(1,𝔽₈)	direct (N × H)
65. C₇	C₂²×C₃×C₅²×C₁₁	semidirect
67. C₅	C₂²×C₅×C₁₁×ΓL(1,𝔽₈)	direct (N × H)
68. C₂²	C₅²×C₁₁×ΓL(1,𝔽₈)	direct (N × H)
70. C₂	C₂×C₅²×C₁₁×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×C₁₁×ΓL(1,𝔽₈)	2
C₂²×C₅²×C₇×C₁₁	3
C₂²×C₅×C₁₁×ΓL(1,𝔽₈)	5
C₂²×C₃×C₅²×C₁₁	7
C₂²×C₅²×ΓL(1,𝔽₈)	11

Subgroups — C22×C52×C11×ΓL(1,𝔽8) #37528736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅²×C₁₁×ΓL(1,𝔽₈) #37528736