Subgroups — C₅₃×(C₃⋊D₃)⋊D₁₁ #35230607

Order 20988.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×D₃₃	C₂	1	2	yes	maximal, complemented
2	C₁₁×C₃⋊D₃×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	D₃×D₁₁×C₅₃	—	3	2	no	maximal
4	C₃²×C₁₁×C₅₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₅₃×D₃₃	D₃	1	2	yes	complemented
6	D₃×C₁₁×C₅₃	—	3	2	no	—
7	D₃×C₁₁×C₅₃	—	6	1	no	—
8	C₃×D₁₁×C₅₃	—	3	2	no	—
9	C₂×D₁₁×C₅₃	—	9	1	no	Hall
10	D₃²×C₅₃	—	11	1	no	maximal, Hall
11	C₃×C₁₁×C₅₃	—	2	1	no	—
12	C₃×C₁₁×C₅₃	C₂×D₃	1	2	yes	complemented
13	C₂×C₁₁×C₅₃	—	9	1	no	—
14	D₁₁×C₅₃	—	3	2	no	—
15	C₃⋊D₃×C₅₃	D₁₁	1	1	yes	characteristic, complemented
16	C₃×D₃×C₅₃	—	11	2	no	—
17	C₂×D₃×C₅₃	—	33	2	no	—
18	C₁₁×C₅₃	D₃²	1	1	yes	characteristic, complemented, Hall
19	C₃²×C₅₃	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
20	(C₃⋊D₃)⋊D₁₁	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₃×C₅₃	—	33	2	no	—
22	D₃×C₅₃	—	3	2	no	—
23	D₃×C₅₃	—	6	1	no	—
24	D₃×C₅₃	—	11	2	no	—
25	C₂²×C₅₃	—	99	1	no	Hall
26	C₁₁×C₃⋊D₃	C₂×C₅₃	1	1	yes	characteristic, complemented
27	C₃×D₃₃	C₂×C₅₃	1	2	yes	complemented
28	C₃×C₅₃	—	2	1	no	—
29	C₃×C₅₃	D₃×D₁₁	1	2	yes	complemented
30	D₃×D₁₁	—	3	2	no	—
31	C₂×C₅₃	—	9	1	no	—
32	C₂×C₅₃	—	33	2	no	—
33	C₃²×C₁₁	C₂²×C₅₃	1	1	yes	characteristic, complemented, Hall
34	D₃₃	D₃×C₅₃	1	2	yes	complemented
35	C₃×D₁₁	—	3	2	no	—
36	D₃×C₁₁	—	3	2	no	—
37	D₃×C₁₁	—	6	1	no	—
38	C₅₃	(C₃⋊D₃)⋊D₁₁	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×D₁₁	—	9	1	no	Hall
40	D₃²	—	11	1	no	Hall
41	C₃×C₁₁	—	2	1	no	—
42	C₃×C₁₁	C₂×D₃×C₅₃	1	2	yes	complemented
43	C₂×C₁₁	—	9	1	no	—
44	D₁₁	—	3	2	no	—
45	C₃⋊D₃	D₁₁×C₅₃	1	1	yes	characteristic, complemented
46	C₃×D₃	—	11	2	no	—
47	C₂×D₃	—	33	2	no	—
48	C₁₁	D₃²×C₅₃	1	1	yes	characteristic, complemented, Sylow
49	C₃²	C₂×D₁₁×C₅₃	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃	—	33	2	no	—
51	D₃	—	3	2	no	—
52	D₃	—	6	1	no	—
53	D₃	—	11	2	no	—
54	C₂²	—	99	1	no	Sylow
55	C₃	—	2	1	no	—
56	C₃	D₃×D₁₁×C₅₃	1	2	yes	complemented
57	C₂	—	9	1	no	—
58	C₂	—	33	2	no	—

	subgroup	G/H
center, hypercenter	C₅₃	(C₃⋊D₃)⋊D₁₁
commutator, cosocle, hypocenter	C₃²×C₁₁	C₂²×C₅₃
Frattini	C₁	G
Fitting, socle	C₃²×C₁₁×C₅₃	C₂²
LCS	G ▷ C₃²×C₁₁
UCS	C₁ ◁ C₅₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₅₃×D₃₃	C₂	semidirect
2. C₁₁×C₃⋊D₃×C₅₃	C₂	semidirect
4. C₃²×C₁₁×C₅₃	C₂²	semidirect
5. C₅₃×D₃₃	D₃	semidirect
12. C₃×C₁₁×C₅₃	C₂×D₃	semidirect
15. C₃⋊D₃×C₅₃	D₁₁	semidirect
18. C₁₁×C₅₃	D₃²	semidirect
19. C₃²×C₅₃	C₂×D₁₁	semidirect
20. (C₃⋊D₃)⋊D₁₁	C₅₃	direct (N × H)
26. C₁₁×C₃⋊D₃	C₂×C₅₃	semidirect
27. C₃×D₃₃	C₂×C₅₃	semidirect
29. C₃×C₅₃	D₃×D₁₁	semidirect
33. C₃²×C₁₁	C₂²×C₅₃	semidirect
34. D₃₃	D₃×C₅₃	semidirect
38. C₅₃	(C₃⋊D₃)⋊D₁₁	direct (N × H)
42. C₃×C₁₁	C₂×D₃×C₅₃	semidirect
45. C₃⋊D₃	D₁₁×C₅₃	semidirect
48. C₁₁	D₃²×C₅₃	semidirect
49. C₃²	C₂×D₁₁×C₅₃	semidirect
56. C₃	D₃×D₁₁×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅₃×D₃₃	2
C₁₁×C₃⋊D₃×C₅₃	2
D₃×D₁₁×C₅₃	3
D₃²×C₅₃	11
(C₃⋊D₃)⋊D₁₁	53

Subgroups — C53×(C3⋊D3)⋊D11 #35230607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×(C₃⋊D₃)⋊D₁₁ #35230607