Subgroups — #49147193

Order 36900.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅⋊D₂₀₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×C₃⋊D₁₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×C₃⋊(C₃⋊(C₅⋊D₅))	C₂	1	1	yes	maximal, characteristic, complemented
4	D₁₂₃⋊(C₅⋊D₅)	—	3	4	no	maximal
5	C₃²×C₅²×C₄₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₂₀₅⋊(C₃⋊D₃)	—	5	6	no	maximal
7	C₄₁×C₃⋊(C₅⋊D₅)	—	3	4	no	—
8	C₅²×D₁₂₃	—	3	4	no	—
9	C₃×C₅⋊D₂₀₅	D₃	1	4	yes	complemented
10	C₅⋊D₅×D₄₁	—	9	1	no	Hall
11	C₄₁×C₃⋊D₁₅	—	5	6	no	—
12	C₅×C₃⋊D₁₂₃	D₅	1	6	yes	complemented
13	C₃²×D₂₀₅	—	5	6	no	—
14	C₃×C₅²×C₄₁	C₂×D₃	1	4	yes	complemented
15	D₁₅⋊D₄₁	—	15	24	no	—
16	C₅⋊D₂₀₅	C₃⋊D₃	1	1	yes	characteristic, complemented
17	C₄₁×C₅⋊D₅	—	9	1	no	—
18	C₅²×D₄₁	—	9	1	no	—
19	C₃²×C₅×C₄₁	C₂×D₅	1	6	yes	complemented
20	C₃⋊D₃×D₄₁	—	25	1	no	Hall
21	D₁₅×C₄₁	—	15	24	no	—
22	C₅×D₁₂₃	—	3	24	no	—
23	C₃×D₂₀₅	—	5	24	no	—
24	C₅²×C₄₁	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
25	C₃⋊D₃×C₅⋊D₅	—	41	1	no	maximal, Hall
26	D₅×D₄₁	—	45	6	no	—
27	C₃⋊D₁₂₃	C₅⋊D₅	1	1	yes	characteristic, complemented
28	C₃⋊D₃×C₄₁	—	25	1	no	—
29	C₃²×D₄₁	—	25	1	no	—
30	C₃×C₅×C₄₁	D₃×D₅	1	24	yes	complemented
31	D₃×D₄₁	—	75	4	no	—
32	C₃⋊(C₃⋊(C₅⋊D₅))	D₄₁	1	1	yes	characteristic, complemented
33	C₅²×C₃⋊D₃	—	41	1	no	—
34	C₃²×C₅⋊D₅	—	41	1	no	—
35	D₂₀₅	—	5	6	no	—
36	C₅×D₄₁	—	9	6	no	—
37	D₅×C₄₁	—	45	6	no	—
38	C₃²×C₄₁	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
39	D₃×C₅⋊D₅	—	123	4	no	—
40	D₁₂₃	—	3	4	no	—
41	C₃×D₄₁	—	25	4	no	—
42	D₃×C₄₁	—	75	4	no	—
43	C₃²×C₅²	C₂×D₄₁	1	1	yes	characteristic, complemented, Hall
44	C₅×C₄₁	D₅×C₃⋊D₃	1	6	yes	complemented
45	D₅×C₃⋊D₃	—	205	6	no	—
46	C₂×D₄₁	—	225	1	no	Hall
47	C₃⋊(C₅⋊D₅)	—	3	4	no	—
48	C₅²×D₃	—	123	4	no	—
49	C₃×C₅⋊D₅	—	41	4	no	—
50	C₃×C₄₁	D₃×C₅⋊D₅	1	4	yes	complemented
51	C₂×C₅⋊D₅	—	369	1	no	Hall
52	C₃⋊D₁₅	—	5	6	no	—
53	C₅×C₃⋊D₃	—	41	6	no	—
54	C₃²×D₅	—	205	6	no	—
55	C₂×C₄₁	—	225	1	no	—
56	D₄₁	—	9	1	no	—
57	D₄₁	—	25	1	no	—
58	C₃×C₅²	D₃×D₄₁	1	4	yes	complemented
59	D₃×D₅	—	615	24	no	—
60	C₂×C₅²	—	369	1	no	—
61	C₅⋊D₅	—	9	1	no	—
62	C₅⋊D₅	—	41	1	no	—
63	C₃²×C₅	D₅×D₄₁	1	6	yes	complemented
64	C₄₁	C₃⋊D₃×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃⋊D₃	—	1025	1	no	Hall
66	D₁₅	—	15	24	no	—
67	C₃×D₅	—	205	24	no	—
68	C₅×D₃	—	123	24	no	—
69	C₅²	C₃⋊D₃×D₄₁	1	1	yes	characteristic, complemented, Sylow
70	C₂×D₅	—	1845	6	no	—
71	C₂×C₃²	—	1025	1	no	—
72	C₃⋊D₃	—	25	1	no	—
73	C₃⋊D₃	—	41	1	no	—
74	C₃×C₅	D₁₅⋊D₄₁	1	24	yes	complemented
75	C₂×D₃	—	3075	4	no	—
76	C₂×C₅	—	369	6	no	—
77	D₅	—	45	6	no	—
78	D₅	—	205	6	no	—
79	C₃²	C₅⋊D₅×D₄₁	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₃	—	1025	4	no	—
81	D₃	—	75	4	no	—
82	D₃	—	123	4	no	—
83	C₅	D₂₀₅⋊(C₃⋊D₃)	1	6	yes	complemented
84	C₂²	—	9225	1	no	Sylow
85	C₃	D₁₂₃⋊(C₅⋊D₅)	1	4	yes	complemented
86	C₂	—	1025	1	no	—
87	C₂	—	369	1	no	—
88	C₂	—	225	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₅²×C₄₁	C₂²
LCS	G ▷ C₃²×C₅²×C₄₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅⋊D₂₀₅	C₂	semidirect
2. C₅²×C₃⋊D₁₂₃	C₂	semidirect
3. C₄₁×C₃⋊(C₃⋊(C₅⋊D₅))	C₂	semidirect
5. C₃²×C₅²×C₄₁	C₂²	semidirect
9. C₃×C₅⋊D₂₀₅	D₃	semidirect
12. C₅×C₃⋊D₁₂₃	D₅	semidirect
14. C₃×C₅²×C₄₁	C₂×D₃	semidirect
16. C₅⋊D₂₀₅	C₃⋊D₃	semidirect
19. C₃²×C₅×C₄₁	C₂×D₅	semidirect
24. C₅²×C₄₁	C₂×C₃⋊D₃	semidirect
27. C₃⋊D₁₂₃	C₅⋊D₅	semidirect
30. C₃×C₅×C₄₁	D₃×D₅	semidirect
32. C₃⋊(C₃⋊(C₅⋊D₅))	D₄₁	semidirect
38. C₃²×C₄₁	C₂×C₅⋊D₅	semidirect
43. C₃²×C₅²	C₂×D₄₁	semidirect
44. C₅×C₄₁	D₅×C₃⋊D₃	semidirect
50. C₃×C₄₁	D₃×C₅⋊D₅	semidirect
58. C₃×C₅²	D₃×D₄₁	semidirect
63. C₃²×C₅	D₅×D₄₁	semidirect
64. C₄₁	C₃⋊D₃×C₅⋊D₅	semidirect
69. C₅²	C₃⋊D₃×D₄₁	semidirect
74. C₃×C₅	D₁₅⋊D₄₁	semidirect
79. C₃²	C₅⋊D₅×D₄₁	semidirect
83. C₅	D₂₀₅⋊(C₃⋊D₃)	semidirect
85. C₃	D₁₂₃⋊(C₅⋊D₅)	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅⋊D₂₀₅	2
C₅²×C₃⋊D₁₂₃	2
C₄₁×C₃⋊(C₃⋊(C₅⋊D₅))	2
D₁₂₃⋊(C₅⋊D₅)	3
D₂₀₅⋊(C₃⋊D₃)	5
C₃⋊D₃×C₅⋊D₅	41