Subgroups — C₂×C₅²×D₃×C₁₁² #49137824

Order 36300.

1120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×D₃×C₁₁²	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₅²×C₁₁²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅²×C₁₁²	—	3	1	no	maximal, Hall
4	C₃×C₅²×C₁₁²	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×D₃×C₁₁²	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₅²×C₁₁²	—	3	2	no	—
7	C₂×C₅²×C₁₁²	D₃	1	1	yes	characteristic, direct product factor
8	C₅×D₃×C₁₁²	C₂×C₅	1	12	yes	direct product factor
9	C₂×C₃×C₅×C₁₁²	C₂×C₅	1	6	yes	complemented
10	C₂×C₅²×D₃×C₁₁	C₁₁	1	12	yes	maximal, direct product factor
11	C₅²×C₁₁²	C₂×D₃	1	1	yes	characteristic, direct product factor, Hall
12	C₂²×C₅×C₁₁²	—	3	6	no	—
13	C₃×C₅×C₁₁²	C₂²×C₅	1	6	yes	complemented
14	C₂×C₃×C₅²×C₁₁	C₂×C₁₁	1	12	yes	complemented
15	C₅²×D₃×C₁₁	C₂×C₁₁	1	24	yes	direct product factor
16	C₂×D₃×C₁₁²	C₅²	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₅×C₁₁²	—	3	12	no	—
18	C₂×C₅×C₁₁²	C₅×D₃	1	6	yes	direct product factor
19	C₂²×C₅²×C₁₁	—	3	12	no	—
20	C₃×C₅²×C₁₁	C₂²×C₁₁	1	12	yes	complemented
21	C₂×C₃×C₁₁²	C₂×C₅²	1	1	yes	characteristic, complemented
22	D₃×C₁₁²	C₂×C₅²	1	2	yes	direct product factor
23	C₂×C₅×D₃×C₁₁	C₅×C₁₁	1	72	yes	direct product factor
24	C₅×C₁₁²	C₂×C₅×D₃	1	6	yes	direct product factor
25	C₂×C₅²×C₁₁	—	3	24	no	—
26	C₂×C₅²×C₁₁	D₃×C₁₁	1	12	yes	direct product factor
27	C₂²×C₁₁²	—	3	1	no	Hall
28	C₃×C₁₁²	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
29	C₂×C₃×C₅×C₁₁	C₂×C₅×C₁₁	1	72	yes	complemented
30	C₅×D₃×C₁₁	C₂×C₅×C₁₁	1	144	yes	direct product factor
31	C₂×C₅²×D₃	C₁₁²	1	1	yes	characteristic, direct product factor, Hall
32	C₅²×C₁₁	C₂×D₃×C₁₁	1	12	yes	direct product factor
33	C₂×C₁₁²	—	3	2	no	—
34	C₂×C₁₁²	C₅²×D₃	1	1	yes	characteristic, direct product factor
35	C₂²×C₅×C₁₁	—	3	72	no	—
36	C₃×C₅×C₁₁	C₂²×C₅×C₁₁	1	72	yes	complemented
37	C₂×C₃×C₅²	C₂×C₁₁²	1	1	yes	characteristic, complemented
38	C₅²×D₃	C₂×C₁₁²	1	2	yes	direct product factor
39	C₂×D₃×C₁₁	C₅²×C₁₁	1	12	yes	direct product factor
40	C₁₁²	C₂×C₅²×D₃	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₅×C₁₁	—	3	144	no	—
42	C₂×C₅×C₁₁	C₅×D₃×C₁₁	1	72	yes	direct product factor
43	C₂²×C₅²	—	3	1	no	Hall
44	C₃×C₅²	C₂²×C₁₁²	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₁₁	C₂×C₅²×C₁₁	1	12	yes	complemented
46	D₃×C₁₁	C₂×C₅²×C₁₁	1	24	yes	direct product factor
47	C₂×C₅×D₃	C₅×C₁₁²	1	6	yes	direct product factor
48	C₅×C₁₁	C₂×C₅×D₃×C₁₁	1	72	yes	direct product factor
49	C₂×C₅²	—	3	2	no	—
50	C₂×C₅²	D₃×C₁₁²	1	1	yes	characteristic, direct product factor
51	C₂²×C₁₁	—	3	12	no	—
52	C₃×C₁₁	C₂²×C₅²×C₁₁	1	12	yes	complemented
53	C₂×C₃×C₅	C₂×C₅×C₁₁²	1	6	yes	complemented
54	C₅×D₃	C₂×C₅×C₁₁²	1	12	yes	direct product factor
55	C₅²	C₂×D₃×C₁₁²	1	1	yes	characteristic, direct product factor, Sylow
56	C₂×C₁₁	—	3	24	no	—
57	C₂×C₁₁	C₅²×D₃×C₁₁	1	12	yes	direct product factor
58	C₂²×C₅	—	3	6	no	—
59	C₃×C₅	C₂²×C₅×C₁₁²	1	6	yes	complemented
60	C₂×D₃	C₅²×C₁₁²	1	1	yes	characteristic, direct product factor, Hall
61	C₁₁	C₂×C₅²×D₃×C₁₁	1	12	yes	direct product factor
62	C₂×C₅	—	3	12	no	—
63	C₂×C₅	C₅×D₃×C₁₁²	1	6	yes	direct product factor
64	C₂×C₃	C₂×C₅²×C₁₁²	1	1	yes	characteristic, complemented
65	D₃	C₂×C₅²×C₁₁²	1	2	yes	direct product factor
66	C₅	C₂×C₅×D₃×C₁₁²	1	6	yes	direct product factor
67	C₂²	—	3	1	no	Sylow
68	C₃	C₂²×C₅²×C₁₁²	1	1	yes	characteristic, complemented, Sylow
69	C₂	—	3	2	no	—
70	C₂	C₅²×D₃×C₁₁²	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅²×C₁₁²	D₃
commutator, cosocle, hypocenter	C₃	C₂²×C₅²×C₁₁²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₁₁²	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₅²×C₁₁²

Complements

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

N	H ≅ G/N	type
1. C₅²×D₃×C₁₁²	C₂	direct (N × H)
2. C₂×C₃×C₅²×C₁₁²	C₂	semidirect
4. C₃×C₅²×C₁₁²	C₂²	semidirect
5. C₂×C₅×D₃×C₁₁²	C₅	direct (N × H)
7. C₂×C₅²×C₁₁²	D₃	direct (N × H)
8. C₅×D₃×C₁₁²	C₂×C₅	direct (N × H)
9. C₂×C₃×C₅×C₁₁²	C₂×C₅	semidirect
10. C₂×C₅²×D₃×C₁₁	C₁₁	direct (N × H)
11. C₅²×C₁₁²	C₂×D₃	direct (N × H)
13. C₃×C₅×C₁₁²	C₂²×C₅	semidirect
14. C₂×C₃×C₅²×C₁₁	C₂×C₁₁	semidirect
15. C₅²×D₃×C₁₁	C₂×C₁₁	direct (N × H)
16. C₂×D₃×C₁₁²	C₅²	direct (N × H)
18. C₂×C₅×C₁₁²	C₅×D₃	direct (N × H)
20. C₃×C₅²×C₁₁	C₂²×C₁₁	semidirect
21. C₂×C₃×C₁₁²	C₂×C₅²	semidirect
22. D₃×C₁₁²	C₂×C₅²	direct (N × H)
23. C₂×C₅×D₃×C₁₁	C₅×C₁₁	direct (N × H)
24. C₅×C₁₁²	C₂×C₅×D₃	direct (N × H)
26. C₂×C₅²×C₁₁	D₃×C₁₁	direct (N × H)
28. C₃×C₁₁²	C₂²×C₅²	semidirect
29. C₂×C₃×C₅×C₁₁	C₂×C₅×C₁₁	semidirect
30. C₅×D₃×C₁₁	C₂×C₅×C₁₁	direct (N × H)
31. C₂×C₅²×D₃	C₁₁²	direct (N × H)
32. C₅²×C₁₁	C₂×D₃×C₁₁	direct (N × H)
34. C₂×C₁₁²	C₅²×D₃	direct (N × H)
36. C₃×C₅×C₁₁	C₂²×C₅×C₁₁	semidirect
37. C₂×C₃×C₅²	C₂×C₁₁²	semidirect
38. C₅²×D₃	C₂×C₁₁²	direct (N × H)
39. C₂×D₃×C₁₁	C₅²×C₁₁	direct (N × H)
40. C₁₁²	C₂×C₅²×D₃	direct (N × H)
42. C₂×C₅×C₁₁	C₅×D₃×C₁₁	direct (N × H)
44. C₃×C₅²	C₂²×C₁₁²	semidirect
45. C₂×C₃×C₁₁	C₂×C₅²×C₁₁	semidirect
46. D₃×C₁₁	C₂×C₅²×C₁₁	direct (N × H)
47. C₂×C₅×D₃	C₅×C₁₁²	direct (N × H)
48. C₅×C₁₁	C₂×C₅×D₃×C₁₁	direct (N × H)
50. C₂×C₅²	D₃×C₁₁²	direct (N × H)
52. C₃×C₁₁	C₂²×C₅²×C₁₁	semidirect
53. C₂×C₃×C₅	C₂×C₅×C₁₁²	semidirect
54. C₅×D₃	C₂×C₅×C₁₁²	direct (N × H)
55. C₅²	C₂×D₃×C₁₁²	direct (N × H)
57. C₂×C₁₁	C₅²×D₃×C₁₁	direct (N × H)
59. C₃×C₅	C₂²×C₅×C₁₁²	semidirect
60. C₂×D₃	C₅²×C₁₁²	direct (N × H)
61. C₁₁	C₂×C₅²×D₃×C₁₁	direct (N × H)
63. C₂×C₅	C₅×D₃×C₁₁²	direct (N × H)
64. C₂×C₃	C₂×C₅²×C₁₁²	semidirect
65. D₃	C₂×C₅²×C₁₁²	direct (N × H)
66. C₅	C₂×C₅×D₃×C₁₁²	direct (N × H)
68. C₃	C₂²×C₅²×C₁₁²	semidirect
70. C₂	C₅²×D₃×C₁₁²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×D₃×C₁₁²	2
C₂×C₃×C₅²×C₁₁²	2
C₂²×C₅²×C₁₁²	3
C₂×C₅×D₃×C₁₁²	5
C₂×C₅²×D₃×C₁₁	11

Subgroups — C2×C52×D3×C112 #49137824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×D₃×C₁₁² #49137824