Subgroups — C₂×C₃×D₁₁×C₆₁⋊C₃ #37563518

Order 24156.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₁₁×C₆₁⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₁₁×C₆₁⋊C₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×D₁₁×C₆₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₁×C₆₁⋊C₃	C₃	1	3	yes	maximal, direct product factor
5	C₃×C₁₁×C₆₁⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₁₁×C₆₁⋊C₃	C₂×C₃	1	6	yes	direct product factor
7	C₃×D₁₁×C₆₁	C₂×C₃	1	2	yes	complemented
8	C₂×C₁₁×C₆₁⋊C₃	C₂×C₃	1	3	yes	complemented
9	C₂×C₃×C₁₁×C₆₁	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×D₁₁×C₆₁	C₃²	1	1	yes	characteristic, complemented, Hall
11	C₂²×C₃×C₆₁⋊C₃	—	11	1	no	maximal, Hall
12	C₃×C₁₁×C₆₁	C₂²×C₃	1	1	yes	characteristic, complemented
13	C₁₁×C₆₁⋊C₃	C₂²×C₃	1	3	yes	complemented
14	C₂×C₁₁×C₆₁	C₂×C₃²	1	1	yes	characteristic, complemented
15	D₁₁×C₆₁	C₂×C₃²	1	2	yes	complemented
16	C₂×C₃×C₆₁⋊C₃	—	11	2	no	—
17	C₂×C₃×C₆₁⋊C₃	D₁₁	1	1	yes	characteristic, direct product factor
18	C₂²×C₃×C₆₁	—	11	1	no	—
19	C₂²×C₆₁⋊C₃	—	11	3	no	—
20	C₁₁×C₆₁	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
21	C₃×C₆₁⋊C₃	C₂×D₁₁	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₃²×D₁₁	—	61	1	no	maximal, Hall
23	C₂×C₃×C₆₁	—	11	2	no	—
24	C₂×C₃×C₆₁	C₃×D₁₁	1	1	yes	characteristic, complemented
25	C₂×C₆₁⋊C₃	—	11	6	no	—
26	C₂×C₆₁⋊C₃	C₃×D₁₁	1	3	yes	direct product factor
27	C₂²×C₆₁	—	11	1	no	Hall
28	C₂×C₃²×C₁₁	—	61	1	no	—
29	C₃²×D₁₁	—	61	2	no	—
30	C₃×C₆₁	C₂×C₃×D₁₁	1	1	yes	characteristic, complemented
31	C₆₁⋊C₃	C₂×C₃×D₁₁	1	3	yes	direct product factor
32	C₂×C₃×D₁₁	—	61	3	no	—
33	C₂×C₃×D₁₁	C₆₁⋊C₃	1	1	yes	characteristic, direct product factor
34	C₂×C₆₁	—	11	2	no	—
35	C₂×C₆₁	C₃²×D₁₁	1	1	yes	characteristic, complemented
36	C₃²×C₁₁	—	61	1	no	Hall
37	C₂×C₃×C₁₁	—	61	3	no	—
38	C₂×C₃×C₁₁	C₂×C₆₁⋊C₃	1	1	yes	characteristic, complemented
39	C₃×D₁₁	—	61	6	no	—
40	C₃×D₁₁	C₂×C₆₁⋊C₃	1	2	yes	direct product factor
41	C₆₁	C₂×C₃²×D₁₁	1	1	yes	characteristic, complemented, Sylow
42	C₂×D₁₁	C₃×C₆₁⋊C₃	1	1	yes	characteristic, direct product factor, Hall
43	C₂²×C₃²	—	671	1	no	Hall
44	C₃×C₁₁	—	61	3	no	—
45	C₃×C₁₁	C₂²×C₆₁⋊C₃	1	1	yes	characteristic, complemented
46	C₂×C₁₁	C₂×C₃×C₆₁⋊C₃	1	1	yes	characteristic, complemented
47	D₁₁	C₂×C₃×C₆₁⋊C₃	1	2	yes	direct product factor
48	C₂×C₃²	—	61	1	no	—
49	C₂×C₃²	—	671	2	no	—
50	C₂²×C₃	—	11	1	no	—
51	C₂²×C₃	—	671	3	no	—
52	C₁₁	C₂²×C₃×C₆₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
53	C₃²	—	61	1	no	Sylow
54	C₂×C₃	—	11	2	no	—
55	C₂×C₃	—	61	3	no	—
56	C₂×C₃	—	671	6	no	—
57	C₂×C₃	D₁₁×C₆₁⋊C₃	1	1	yes	characteristic, direct product factor
58	C₂²	—	11	1	no	Sylow
59	C₃	—	61	3	no	—
60	C₃	C₂×D₁₁×C₆₁⋊C₃	1	1	yes	characteristic, direct product factor
61	C₂	—	11	2	no	—
62	C₂	C₃×D₁₁×C₆₁⋊C₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	D₁₁×C₆₁⋊C₃
commutator, cosocle, hypocenter	C₁₁×C₆₁	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₁×C₆₁	C₂×C₃
LCS	G ▷ C₁₁×C₆₁
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₁₁×C₆₁⋊C₃	C₂	semidirect
2. C₃×D₁₁×C₆₁⋊C₃	C₂	direct (N × H)
3. C₂×C₃×D₁₁×C₆₁	C₃	semidirect
4. C₂×D₁₁×C₆₁⋊C₃	C₃	direct (N × H)
5. C₃×C₁₁×C₆₁⋊C₃	C₂²	semidirect
6. D₁₁×C₆₁⋊C₃	C₂×C₃	direct (N × H)
7. C₃×D₁₁×C₆₁	C₂×C₃	semidirect
8. C₂×C₁₁×C₆₁⋊C₃	C₂×C₃	semidirect
9. C₂×C₃×C₁₁×C₆₁	C₂×C₃	semidirect
10. C₂×D₁₁×C₆₁	C₃²	semidirect
12. C₃×C₁₁×C₆₁	C₂²×C₃	semidirect
13. C₁₁×C₆₁⋊C₃	C₂²×C₃	semidirect
14. C₂×C₁₁×C₆₁	C₂×C₃²	semidirect
15. D₁₁×C₆₁	C₂×C₃²	semidirect
17. C₂×C₃×C₆₁⋊C₃	D₁₁	direct (N × H)
20. C₁₁×C₆₁	C₂²×C₃²	semidirect
21. C₃×C₆₁⋊C₃	C₂×D₁₁	direct (N × H)
24. C₂×C₃×C₆₁	C₃×D₁₁	semidirect
26. C₂×C₆₁⋊C₃	C₃×D₁₁	direct (N × H)
30. C₃×C₆₁	C₂×C₃×D₁₁	semidirect
31. C₆₁⋊C₃	C₂×C₃×D₁₁	direct (N × H)
33. C₂×C₃×D₁₁	C₆₁⋊C₃	direct (N × H)
35. C₂×C₆₁	C₃²×D₁₁	semidirect
38. C₂×C₃×C₁₁	C₂×C₆₁⋊C₃	semidirect
40. C₃×D₁₁	C₂×C₆₁⋊C₃	direct (N × H)
41. C₆₁	C₂×C₃²×D₁₁	semidirect
42. C₂×D₁₁	C₃×C₆₁⋊C₃	direct (N × H)
45. C₃×C₁₁	C₂²×C₆₁⋊C₃	semidirect
46. C₂×C₁₁	C₂×C₃×C₆₁⋊C₃	semidirect
47. D₁₁	C₂×C₃×C₆₁⋊C₃	direct (N × H)
52. C₁₁	C₂²×C₃×C₆₁⋊C₃	semidirect
57. C₂×C₃	D₁₁×C₆₁⋊C₃	direct (N × H)
60. C₃	C₂×D₁₁×C₆₁⋊C₃	direct (N × H)
62. C₂	C₃×D₁₁×C₆₁⋊C₃	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×C₃×C₁₁×C₆₁⋊C₃	2
C₃×D₁₁×C₆₁⋊C₃	2
C₂×C₃×D₁₁×C₆₁	3
C₂×D₁₁×C₆₁⋊C₃	3
C₂²×C₃×C₆₁⋊C₃	11
C₂×C₃²×D₁₁	61

Subgroups — C2×C3×D11×C61⋊C3 #37563518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₁₁×C₆₁⋊C₃ #37563518