Subgroups — #49137789

Order 36300.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×C₁₁⋊D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₁×C₃⋊(C₅⋊D₅₅)	C₂	1	2	yes	maximal, complemented
3	#27130989	—	3	1	no	maximal, Hall
4	C₃×C₅²×C₁₁²	C₂²	1	1	yes	characteristic, complemented, Hall
5	#20300106	—	5	6	no	maximal
6	C₁₁×C₅⋊D₅₅	—	3	2	no	—
7	C₅²×C₁₁⋊D₁₁	D₃	1	1	yes	characteristic, complemented
8	C₁₁×D₁₆₅	—	5	12	no	—
9	C₃×C₅×C₁₁⋊D₁₁	D₅	1	6	yes	complemented
10	D₁₁×C₃⋊(C₅⋊D₅)	—	11	2	no	maximal
11	C₅²×C₁₁²	C₂×D₃	1	1	yes	characteristic, complemented, Hall
12	D₅₅⋊D₁₁	—	15	6	no	—
13	C₃×C₅×C₁₁²	C₂×D₅	1	6	yes	complemented
14	C₃⋊(C₅⋊D₅₅)	D₁₁	1	2	yes	complemented
15	C₁₁×C₃⋊(C₅⋊D₅)	—	11	2	no	—
16	C₃×C₅²×D₁₁	—	11	2	no	—
17	C₃×C₅²×D₁₁	—	22	5	no	—
18	D₃₃⋊D₁₁	—	25	1	no	Hall
19	C₁₁×D₅₅	—	15	12	no	—
20	C₅×C₁₁⋊D₁₁	D₁₅	1	6	yes	complemented
21	D₁₁×C₅⋊D₅	—	33	2	no	—
22	C₃×C₅²×C₁₁	—	2	5	no	—
23	C₃×C₅²×C₁₁	C₂×D₁₁	1	2	yes	complemented
24	C₁₁×D₃₃	—	25	2	no	—
25	C₃×C₁₁⋊D₁₁	C₅⋊D₅	1	1	yes	characteristic, complemented
26	D₁₁×D₁₅	—	55	12	no	—
27	C₅×C₁₁²	C₂×D₁₅	1	6	yes	complemented
28	C₅⋊D₅₅	—	3	2	no	—
29	C₁₁×C₅⋊D₅	—	33	2	no	—
30	C₅²×D₁₁	—	11	2	no	—
31	C₅²×D₁₁	—	22	5	no	—
32	D₁₁²	—	75	1	no	Hall
33	C₃×C₁₁²	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
34	D₁₆₅	—	5	12	no	—
35	C₁₁×D₁₅	—	55	12	no	—
36	C₃×C₅×D₁₁	—	11	12	no	—
37	C₃×C₅×D₁₁	—	22	30	no	—
38	C₂×C₃⋊(C₅⋊D₅)	—	121	1	no	Hall
39	C₅²×C₁₁	—	2	5	no	—
40	C₅²×C₁₁	D₃×D₁₁	1	2	yes	complemented
41	C₁₁⋊D₁₁	C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, complemented
42	C₁₁×D₁₁	—	75	2	no	—
43	D₅×D₁₁	—	165	12	no	—
44	C₃×C₅×C₁₁	—	2	30	no	—
45	C₃×C₅×C₁₁	D₅×D₁₁	1	12	yes	complemented
46	C₂×C₃×C₅²	—	121	1	no	—
47	C₃⋊(C₅⋊D₅)	—	11	2	no	—
48	D₃×D₁₁	—	275	2	no	—
49	C₁₁²	C₂×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, complemented, Sylow
50	D₅₅	—	15	12	no	—
51	C₅×D₁₁	—	11	12	no	—
52	C₅×D₁₁	—	22	30	no	—
53	D₅×C₁₁	—	165	12	no	—
54	C₂×C₅⋊D₅	—	363	1	no	Hall
55	C₃×C₅²	D₁₁²	1	1	yes	characteristic, complemented, Hall
56	D₃₃	—	25	2	no	—
57	C₃×D₁₁	—	11	2	no	—
58	C₃×D₁₁	—	22	5	no	—
59	D₃×C₁₁	—	275	2	no	—
60	C₂×D₁₅	—	605	6	no	—
61	C₅×C₁₁	—	2	30	no	—
62	C₅×C₁₁	D₁₁×D₁₅	1	12	yes	complemented
63	C₂×C₅²	—	121	1	no	—
64	C₅⋊D₅	—	33	2	no	—
65	C₂×D₁₁	—	825	2	no	—
66	C₃×C₁₁	—	2	5	no	—
67	C₃×C₁₁	D₁₁×C₅⋊D₅	1	2	yes	complemented
68	C₂×C₃×C₅	—	121	6	no	—
69	D₁₅	—	55	12	no	—
70	C₅²	D₃₃⋊D₁₁	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₁₁	—	825	2	no	—
72	D₁₁	—	11	2	no	—
73	D₁₁	—	22	5	no	—
74	D₁₁	—	75	2	no	—
75	C₂×D₅	—	1815	6	no	—
76	C₃×C₅	D₅₅⋊D₁₁	1	6	yes	complemented
77	C₂×D₃	—	3025	1	no	Hall
78	C₁₁	—	2	5	no	—
79	C₁₁	D₁₁×C₃⋊(C₅⋊D₅)	1	2	yes	complemented
80	C₂×C₅	—	121	6	no	—
81	D₅	—	165	12	no	—
82	C₂×C₃	—	121	1	no	—
83	D₃	—	275	2	no	—
84	C₅	#20300106	1	6	yes	complemented
85	C₂²	—	9075	1	no	Sylow
86	C₃	#27130989	1	1	yes	characteristic, complemented, Sylow
87	C₂	—	825	2	no	—
88	C₂	—	121	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×C₁₁⋊D₁₁	C₂	semidirect
2. C₁₁×C₃⋊(C₅⋊D₅₅)	C₂	semidirect
4. C₃×C₅²×C₁₁²	C₂²	semidirect
7. C₅²×C₁₁⋊D₁₁	D₃	semidirect
9. C₃×C₅×C₁₁⋊D₁₁	D₅	semidirect
11. C₅²×C₁₁²	C₂×D₃	semidirect
13. C₃×C₅×C₁₁²	C₂×D₅	semidirect
14. C₃⋊(C₅⋊D₅₅)	D₁₁	semidirect
20. C₅×C₁₁⋊D₁₁	D₁₅	semidirect
23. C₃×C₅²×C₁₁	C₂×D₁₁	semidirect
25. C₃×C₁₁⋊D₁₁	C₅⋊D₅	semidirect
27. C₅×C₁₁²	C₂×D₁₅	semidirect
33. C₃×C₁₁²	C₂×C₅⋊D₅	semidirect
40. C₅²×C₁₁	D₃×D₁₁	semidirect
41. C₁₁⋊D₁₁	C₃⋊(C₅⋊D₅)	semidirect
45. C₃×C₅×C₁₁	D₅×D₁₁	semidirect
49. C₁₁²	C₂×C₃⋊(C₅⋊D₅)	semidirect
55. C₃×C₅²	D₁₁²	semidirect
62. C₅×C₁₁	D₁₁×D₁₅	semidirect
67. C₃×C₁₁	D₁₁×C₅⋊D₅	semidirect
70. C₅²	D₃₃⋊D₁₁	semidirect
76. C₃×C₅	D₅₅⋊D₁₁	semidirect
79. C₁₁	D₁₁×C₃⋊(C₅⋊D₅)	semidirect
84. C₅	#20300106	semidirect
86. C₃	#27130989	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅²×C₁₁⋊D₁₁	2
C₁₁×C₃⋊(C₅⋊D₅₅)	2
#27130989	3
#20300106	5
D₁₁×C₃⋊(C₅⋊D₅)	11