Subgroups — C₃²×Dic₁₁×C₆₁ #37563501

Order 24156.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₁₁×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₃×Dic₁₁×C₆₁	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₁₁×C₆₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₁₁×C₆₁	C₂×C₃	1	4	yes	—
5	Dic₁₁×C₆₁	C₃²	1	1	yes	characteristic, direct product factor, Hall
6	C₃²×C₄×C₆₁	—	11	1	no	maximal, Hall
7	C₃×C₁₁×C₆₁	C₃×C₄	1	4	yes	complemented
8	C₂×C₁₁×C₆₁	C₂×C₃²	1	1	yes	characteristic
9	C₂×C₃²×C₆₁	D₁₁	1	1	yes	characteristic
10	C₃×C₄×C₆₁	—	11	4	no	—
11	C₁₁×C₆₁	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
12	C₃²×C₆₁	Dic₁₁	1	1	yes	characteristic, direct product factor, Hall
13	C₃²×Dic₁₁	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₃×C₆₁	C₃×D₁₁	1	4	yes	—
15	C₄×C₆₁	—	11	1	no	Hall
16	C₂×C₃²×C₁₁	C₂×C₆₁	1	1	yes	characteristic
17	C₃×C₆₁	C₃×Dic₁₁	1	4	yes	direct product factor
18	C₃×Dic₁₁	C₃×C₆₁	1	4	yes	direct product factor
19	C₂×C₆₁	C₃²×D₁₁	1	1	yes	characteristic
20	C₃²×C₁₁	C₄×C₆₁	1	1	yes	characteristic, complemented, Hall
21	C₂×C₃×C₁₁	C₂×C₃×C₆₁	1	4	yes	—
22	C₆₁	C₃²×Dic₁₁	1	1	yes	characteristic, direct product factor, Sylow
23	Dic₁₁	C₃²×C₆₁	1	1	yes	characteristic, direct product factor, Hall
24	C₃²×C₄	—	11	1	no	Hall
25	C₃×C₁₁	C₃×C₄×C₆₁	1	4	yes	complemented
26	C₂×C₁₁	C₂×C₃²×C₆₁	1	1	yes	characteristic
27	C₂×C₃²	D₁₁×C₆₁	1	1	yes	characteristic
28	C₃×C₄	—	11	4	no	—
29	C₁₁	C₃²×C₄×C₆₁	1	1	yes	characteristic, complemented, Sylow
30	C₃²	Dic₁₁×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₃	C₃×D₁₁×C₆₁	1	4	yes	—
32	C₄	—	11	1	no	Sylow
33	C₃	C₃×Dic₁₁×C₆₁	1	4	yes	direct product factor
34	C₂	C₃²×D₁₁×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃²×C₆₁	D₁₁
commutator, hypocenter	C₁₁	C₃²×C₄×C₆₁
Frattini	C₂	C₃²×D₁₁×C₆₁
Fitting, socle, prime radical	C₂×C₃²×C₁₁×C₆₁	C₂
cosocle	C₂×C₁₁	C₂×C₃²×C₆₁
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₂×C₃²×C₆₁

Complements

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

N	H ≅ G/N	type
2. C₃×Dic₁₁×C₆₁	C₃	direct (N × H)
3. C₃²×C₁₁×C₆₁	C₄	semidirect
5. Dic₁₁×C₆₁	C₃²	direct (N × H)
7. C₃×C₁₁×C₆₁	C₃×C₄	semidirect
11. C₁₁×C₆₁	C₃²×C₄	semidirect
12. C₃²×C₆₁	Dic₁₁	direct (N × H)
13. C₃²×Dic₁₁	C₆₁	direct (N × H)
17. C₃×C₆₁	C₃×Dic₁₁	direct (N × H)
18. C₃×Dic₁₁	C₃×C₆₁	direct (N × H)
20. C₃²×C₁₁	C₄×C₆₁	semidirect
22. C₆₁	C₃²×Dic₁₁	direct (N × H)
23. Dic₁₁	C₃²×C₆₁	direct (N × H)
25. C₃×C₁₁	C₃×C₄×C₆₁	semidirect
29. C₁₁	C₃²×C₄×C₆₁	semidirect
30. C₃²	Dic₁₁×C₆₁	direct (N × H)
33. C₃	C₃×Dic₁₁×C₆₁	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×C₃²×C₁₁×C₆₁	2
C₃×Dic₁₁×C₆₁	3
C₃²×C₄×C₆₁	11
C₃²×Dic₁₁	61

Subgroups — C32×Dic11×C61 #37563501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×Dic₁₁×C₆₁ #37563501