Subgroups — #49137784

Order 36300.

672 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₁²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₃⋊(C₅⋊(C₁₁⋊D₁₁))	C₂	1	2	yes	maximal, complemented
3	(C₅⋊D₅)⋊(C₁₁⋊D₁₁)	—	3	1	no	maximal, Hall
4	C₃×C₅²×C₁₁²	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₅×C₃⋊(C₁₁⋊D₁₁)	—	5	2	no	maximal
6	C₁₁²×C₅⋊D₅	D₃	1	1	yes	characteristic, complemented
7	C₅×C₅⋊(C₁₁⋊D₁₁)	—	3	2	no	—
8	C₃⋊(C₅⋊(C₁₁⋊D₁₁))	D₅	1	2	yes	complemented
9	C₅×C₃⋊(C₁₁⋊D₁₁)	—	5	2	no	—
10	C₃×D₅×C₁₁²	—	5	2	no	—
11	C₃×D₅×C₁₁²	—	10	2	no	—
12	#15693217	—	11	12	no	maximal
13	C₅²×C₁₁²	C₂×D₃	1	1	yes	characteristic, complemented, Hall
14	D₅×C₁₁⋊D₁₁	—	15	2	no	—
15	C₃×C₅×C₁₁²	—	2	2	no	—
16	C₃×C₅×C₁₁²	C₂×D₅	1	2	yes	complemented
17	C₅×D₁₆₅	—	11	24	no	—
18	C₃×C₁₁×C₅⋊D₅	D₁₁	1	12	yes	complemented
19	C₂×C₃⋊(C₁₁⋊D₁₁)	—	25	1	no	Hall
20	C₅⋊(C₁₁⋊D₁₁)	—	3	2	no	—
21	D₅×C₁₁²	—	5	2	no	—
22	D₅×C₁₁²	—	10	2	no	—
23	C₅×C₁₁⋊D₁₁	—	15	2	no	—
24	(C₅⋊D₅)⋊D₁₁	—	33	12	no	—
25	C₃×C₅²×C₁₁	C₂×D₁₁	1	12	yes	complemented
26	C₂×C₃×C₁₁²	—	25	1	no	—
27	C₃⋊(C₁₁⋊D₁₁)	—	5	2	no	—
28	D₅×D₃₃	—	55	24	no	—
29	C₅×C₁₁²	—	2	2	no	—
30	C₅×C₁₁²	D₃×D₅	1	2	yes	complemented
31	C₁₁×C₅⋊D₅	D₃₃	1	12	yes	complemented
32	C₅×D₅₅	—	33	24	no	—
33	C₂×C₁₁⋊D₁₁	—	75	1	no	Hall
34	C₃×C₁₁²	D₅²	1	1	yes	characteristic, complemented, Hall
35	D₁₆₅	—	11	24	no	—
36	C₅×D₃₃	—	55	24	no	—
37	C₃×D₅×C₁₁	—	5	24	no	—
38	C₃×D₅×C₁₁	—	10	24	no	—
39	D₁₅⋊D₅	—	121	1	no	Hall
40	C₅²×C₁₁	C₂×D₃₃	1	12	yes	complemented
41	C₂×C₁₁²	—	25	1	no	—
42	C₁₁⋊D₁₁	—	15	2	no	—
43	D₅×D₁₁	—	165	24	no	—
44	C₃×C₅×C₁₁	—	2	24	no	—
45	C₃×C₅×C₁₁	D₅×D₁₁	1	24	yes	complemented
46	C₅×D₁₅	—	121	2	no	—
47	C₃×C₅⋊D₅	C₁₁⋊D₁₁	1	1	yes	characteristic, complemented
48	C₂×D₃₃	—	275	12	no	—
49	C₁₁²	D₁₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
50	D₅₅	—	33	24	no	—
51	C₅×D₁₁	—	165	24	no	—
52	D₅×C₁₁	—	5	24	no	—
53	D₅×C₁₁	—	10	24	no	—
54	D₅²	—	363	1	no	Hall
55	C₃×C₅²	C₂×C₁₁⋊D₁₁	1	1	yes	characteristic, complemented, Hall
56	C₂×C₃×C₁₁	—	25	12	no	—
57	D₃₃	—	55	24	no	—
58	D₃×D₅	—	605	2	no	—
59	C₅×C₁₁	—	2	24	no	—
60	C₅×C₁₁	D₅×D₃₃	1	24	yes	complemented
61	C₅⋊D₅	C₃⋊(C₁₁⋊D₁₁)	1	1	yes	characteristic, complemented
62	C₅×D₅	—	363	2	no	—
63	C₂×D₁₁	—	825	12	no	—
64	C₃×C₁₁	(C₅⋊D₅)⋊D₁₁	1	12	yes	complemented
65	D₁₅	—	121	2	no	—
66	C₃×D₅	—	5	2	no	—
67	C₃×D₅	—	10	2	no	—
68	C₅×D₃	—	605	2	no	—
69	C₅²	C₂×C₃⋊(C₁₁⋊D₁₁)	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₁₁	—	25	12	no	—
71	D₁₁	—	165	24	no	—
72	C₂×D₅	—	1815	2	no	—
73	C₃×C₅	—	2	2	no	—
74	C₃×C₅	D₅×C₁₁⋊D₁₁	1	2	yes	complemented
75	C₂×D₃	—	3025	1	no	Hall
76	C₁₁	#15693217	1	12	yes	complemented
77	C₂×C₅	—	1815	2	no	—
78	D₅	—	5	2	no	—
79	D₅	—	10	2	no	—
80	D₅	—	363	2	no	—
81	C₂×C₃	—	25	1	no	—
82	D₃	—	605	2	no	—
83	C₅	—	2	2	no	—
84	C₅	D₅×C₃⋊(C₁₁⋊D₁₁)	1	2	yes	complemented
85	C₂²	—	9075	1	no	Sylow
86	C₃	(C₅⋊D₅)⋊(C₁₁⋊D₁₁)	1	1	yes	characteristic, complemented, Sylow
87	C₂	—	1815	2	no	—
88	C₂	—	25	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×C₁₁²×C₅⋊D₅	C₂	semidirect
2. C₅×C₃⋊(C₅⋊(C₁₁⋊D₁₁))	C₂	semidirect
4. C₃×C₅²×C₁₁²	C₂²	semidirect
6. C₁₁²×C₅⋊D₅	D₃	semidirect
8. C₃⋊(C₅⋊(C₁₁⋊D₁₁))	D₅	semidirect
13. C₅²×C₁₁²	C₂×D₃	semidirect
16. C₃×C₅×C₁₁²	C₂×D₅	semidirect
18. C₃×C₁₁×C₅⋊D₅	D₁₁	semidirect
25. C₃×C₅²×C₁₁	C₂×D₁₁	semidirect
30. C₅×C₁₁²	D₃×D₅	semidirect
31. C₁₁×C₅⋊D₅	D₃₃	semidirect
34. C₃×C₁₁²	D₅²	semidirect
40. C₅²×C₁₁	C₂×D₃₃	semidirect
45. C₃×C₅×C₁₁	D₅×D₁₁	semidirect
47. C₃×C₅⋊D₅	C₁₁⋊D₁₁	semidirect
49. C₁₁²	D₁₅⋊D₅	semidirect
55. C₃×C₅²	C₂×C₁₁⋊D₁₁	semidirect
60. C₅×C₁₁	D₅×D₃₃	semidirect
61. C₅⋊D₅	C₃⋊(C₁₁⋊D₁₁)	semidirect
64. C₃×C₁₁	(C₅⋊D₅)⋊D₁₁	semidirect
69. C₅²	C₂×C₃⋊(C₁₁⋊D₁₁)	semidirect
74. C₃×C₅	D₅×C₁₁⋊D₁₁	semidirect
76. C₁₁	#15693217	semidirect
84. C₅	D₅×C₃⋊(C₁₁⋊D₁₁)	semidirect
86. C₃	(C₅⋊D₅)⋊(C₁₁⋊D₁₁)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₁₁²×C₅⋊D₅	2
C₅×C₃⋊(C₅⋊(C₁₁⋊D₁₁))	2
(C₅⋊D₅)⋊(C₁₁⋊D₁₁)	3
D₅×C₃⋊(C₁₁⋊D₁₁)	5
#15693217	11