Subgroups — #26043804

Order 11700.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×C₃⋊(C₃⋊(C₅⋊D₅))	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×C₃⋊D₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₅⋊D₆₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃₉⋊(C₅⋊D₅)	—	3	4	no	maximal
5	C₃²×C₅²×C₁₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₆₅⋊(C₃⋊D₃)	—	5	6	no	maximal
7	C₁₃×C₃⋊(C₅⋊D₅)	—	3	4	no	—
8	C₅²×D₃₉	—	3	4	no	—
9	C₃×C₅⋊D₆₅	D₃	1	4	yes	complemented
10	D₁₃×C₅⋊D₅	—	9	1	no	Hall
11	C₁₃×C₃⋊D₁₅	—	5	6	no	—
12	C₅×C₃⋊D₃₉	D₅	1	6	yes	complemented
13	C₃²×D₆₅	—	5	6	no	—
14	C₃×C₅²×C₁₃	C₂×D₃	1	4	yes	complemented
15	C₃⋊D₃×C₅⋊D₅	—	13	1	no	maximal, Hall
16	D₁₅⋊D₁₃	—	15	24	no	—
17	C₅⋊D₆₅	C₃⋊D₃	1	1	yes	characteristic, complemented
18	C₁₃×C₅⋊D₅	—	9	1	no	—
19	C₅²×D₁₃	—	9	1	no	—
20	C₃²×C₅×C₁₃	C₂×D₅	1	6	yes	complemented
21	C₃⋊D₃×D₁₃	—	25	1	no	Hall
22	C₃⋊(C₃⋊(C₅⋊D₅))	D₁₃	1	1	yes	characteristic, complemented
23	C₅²×C₃⋊D₃	—	13	1	no	—
24	C₃²×C₅⋊D₅	—	13	1	no	—
25	C₁₃×D₁₅	—	15	24	no	—
26	C₅×D₃₉	—	3	24	no	—
27	C₃×D₆₅	—	5	24	no	—
28	C₅²×C₁₃	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
29	D₃×C₅⋊D₅	—	39	4	no	—
30	D₅×D₁₃	—	45	6	no	—
31	C₃⋊D₃₉	C₅⋊D₅	1	1	yes	characteristic, complemented
32	C₁₃×C₃⋊D₃	—	25	1	no	—
33	C₃²×D₁₃	—	25	1	no	—
34	C₃²×C₅²	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
35	C₃×C₅×C₁₃	D₃×D₅	1	24	yes	complemented
36	D₅×C₃⋊D₃	—	65	6	no	—
37	D₃×D₁₃	—	75	4	no	—
38	C₃⋊(C₅⋊D₅)	—	3	4	no	—
39	C₅²×D₃	—	39	4	no	—
40	C₃×C₅⋊D₅	—	13	4	no	—
41	D₆₅	—	5	6	no	—
42	C₅×D₁₃	—	9	6	no	—
43	D₅×C₁₃	—	45	6	no	—
44	C₃²×C₁₃	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
45	C₂×C₅⋊D₅	—	117	1	no	Hall
46	C₃⋊D₁₅	—	5	6	no	—
47	C₅×C₃⋊D₃	—	13	6	no	—
48	C₃²×D₅	—	65	6	no	—
49	D₃₉	—	3	4	no	—
50	C₃×D₁₃	—	25	4	no	—
51	D₃×C₁₃	—	75	4	no	—
52	C₃×C₅²	D₃×D₁₃	1	4	yes	complemented
53	C₅×C₁₃	D₅×C₃⋊D₃	1	6	yes	complemented
54	D₃×D₅	—	195	24	no	—
55	C₂×D₁₃	—	225	1	no	Hall
56	C₂×C₅²	—	117	1	no	—
57	C₅⋊D₅	—	9	1	no	—
58	C₅⋊D₅	—	13	1	no	—
59	C₃²×C₅	D₅×D₁₃	1	6	yes	complemented
60	C₃×C₁₃	D₃×C₅⋊D₅	1	4	yes	complemented
61	C₂×C₃⋊D₃	—	325	1	no	Hall
62	D₁₅	—	15	24	no	—
63	C₃×D₅	—	65	24	no	—
64	C₅×D₃	—	39	24	no	—
65	C₂×C₁₃	—	225	1	no	—
66	D₁₃	—	9	1	no	—
67	D₁₃	—	25	1	no	—
68	C₅²	C₃⋊D₃×D₁₃	1	1	yes	characteristic, complemented, Sylow
69	C₂×D₅	—	585	6	no	—
70	C₂×C₃²	—	325	1	no	—
71	C₃⋊D₃	—	13	1	no	—
72	C₃⋊D₃	—	25	1	no	—
73	C₃×C₅	D₁₅⋊D₁₃	1	24	yes	complemented
74	C₁₃	C₃⋊D₃×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₃	—	975	4	no	—
76	C₂×C₅	—	117	6	no	—
77	D₅	—	45	6	no	—
78	D₅	—	65	6	no	—
79	C₃²	D₁₃×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₃	—	325	4	no	—
81	D₃	—	39	4	no	—
82	D₃	—	75	4	no	—
83	C₅	D₆₅⋊(C₃⋊D₃)	1	6	yes	complemented
84	C₂²	—	2925	1	no	Sylow
85	C₃	D₃₉⋊(C₅⋊D₅)	1	4	yes	complemented
86	C₂	—	325	1	no	—
87	C₂	—	117	1	no	—
88	C₂	—	225	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₅²×C₁₃	C₂²
LCS	G ▷ C₃²×C₅²×C₁₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₁₃×C₃⋊(C₃⋊(C₅⋊D₅))	C₂	semidirect
2. C₅²×C₃⋊D₃₉	C₂	semidirect
3. C₃²×C₅⋊D₆₅	C₂	semidirect
5. C₃²×C₅²×C₁₃	C₂²	semidirect
9. C₃×C₅⋊D₆₅	D₃	semidirect
12. C₅×C₃⋊D₃₉	D₅	semidirect
14. C₃×C₅²×C₁₃	C₂×D₃	semidirect
17. C₅⋊D₆₅	C₃⋊D₃	semidirect
20. C₃²×C₅×C₁₃	C₂×D₅	semidirect
22. C₃⋊(C₃⋊(C₅⋊D₅))	D₁₃	semidirect
28. C₅²×C₁₃	C₂×C₃⋊D₃	semidirect
31. C₃⋊D₃₉	C₅⋊D₅	semidirect
34. C₃²×C₅²	C₂×D₁₃	semidirect
35. C₃×C₅×C₁₃	D₃×D₅	semidirect
44. C₃²×C₁₃	C₂×C₅⋊D₅	semidirect
52. C₃×C₅²	D₃×D₁₃	semidirect
53. C₅×C₁₃	D₅×C₃⋊D₃	semidirect
59. C₃²×C₅	D₅×D₁₃	semidirect
60. C₃×C₁₃	D₃×C₅⋊D₅	semidirect
68. C₅²	C₃⋊D₃×D₁₃	semidirect
73. C₃×C₅	D₁₅⋊D₁₃	semidirect
74. C₁₃	C₃⋊D₃×C₅⋊D₅	semidirect
79. C₃²	D₁₃×C₅⋊D₅	semidirect
83. C₅	D₆₅⋊(C₃⋊D₃)	semidirect
85. C₃	D₃₉⋊(C₅⋊D₅)	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×C₃⋊(C₃⋊(C₅⋊D₅))	2
C₅²×C₃⋊D₃₉	2
C₃²×C₅⋊D₆₅	2
D₃₉⋊(C₅⋊D₅)	3
D₆₅⋊(C₃⋊D₃)	5
C₃⋊D₃×C₅⋊D₅	13