Subgroups — C₃₁×(C₃⋊D₃)⋊(C₅⋊D₅) #42124944

Order 27900.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₁×C₃⋊(C₅⋊D₅)	C₂	1	2	yes	maximal, complemented
2	C₅²×C₃⋊D₃×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	D₃×C₃₁×C₅⋊D₅	—	3	2	no	maximal
4	C₃²×C₅²×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃₁×(C₃⋊D₃)⋊D₅	—	5	6	no	maximal
6	C₃₁×C₃⋊(C₅⋊D₅)	D₃	1	2	yes	complemented
7	C₅²×D₃×C₃₁	—	3	2	no	—
8	C₅²×D₃×C₃₁	—	6	1	no	—
9	C₃×C₃₁×C₅⋊D₅	—	3	2	no	—
10	C₂×C₃₁×C₅⋊D₅	—	9	1	no	Hall
11	C₅×C₃⋊D₃×C₃₁	D₅	1	6	yes	complemented
12	C₃×D₁₅×C₃₁	—	5	12	no	—
13	C₃×C₅²×C₃₁	—	2	1	no	—
14	C₃×C₅²×C₃₁	C₂×D₃	1	2	yes	complemented
15	D₃×D₅×C₃₁	—	15	12	no	—
16	C₂×C₅²×C₃₁	—	9	1	no	—
17	C₃₁×C₅⋊D₅	—	3	2	no	—
18	C₃²×C₅×C₃₁	C₂×D₅	1	6	yes	complemented
19	D₃²×C₃₁	—	25	1	no	Hall
20	D₁₅×C₃₁	—	5	12	no	—
21	C₅×D₃×C₃₁	—	3	12	no	—
22	C₅×D₃×C₃₁	—	6	6	no	—
23	C₃×D₅×C₃₁	—	15	12	no	—
24	(C₃⋊D₃)⋊(C₅⋊D₅)	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅²×C₃₁	D₃²	1	1	yes	characteristic, complemented, Hall
26	C₂×D₅×C₃₁	—	45	6	no	—
27	C₃⋊D₃×C₃₁	C₅⋊D₅	1	1	yes	characteristic, complemented
28	C₃×D₃×C₃₁	—	25	2	no	—
29	C₃×C₅×C₃₁	—	2	6	no	—
30	C₃×C₅×C₃₁	D₃×D₅	1	12	yes	complemented
31	C₅²×C₃⋊D₃	C₂×C₃₁	1	1	yes	characteristic, complemented
32	C₃×C₃⋊(C₅⋊D₅)	C₂×C₃₁	1	2	yes	complemented
33	C₂×D₃×C₃₁	—	75	2	no	—
34	C₂×C₅×C₃₁	—	9	6	no	—
35	D₅×C₃₁	—	15	12	no	—
36	D₃×C₅⋊D₅	—	3	2	no	—
37	C₃²×C₃₁	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
38	C₃²×C₅²	C₂²×C₃₁	1	1	yes	characteristic, complemented, Hall
39	C₂×C₃×C₃₁	—	75	2	no	—
40	D₃×C₃₁	—	3	2	no	—
41	D₃×C₃₁	—	6	1	no	—
42	D₃×C₃₁	—	25	2	no	—
43	(C₃⋊D₃)⋊D₅	—	5	6	no	—
44	C₅×C₃₁	(C₃⋊D₃)⋊D₅	1	6	yes	complemented
45	C₃⋊(C₅⋊D₅)	D₃×C₃₁	1	2	yes	complemented
46	C₅²×D₃	—	3	2	no	—
47	C₅²×D₃	—	6	1	no	—
48	C₃×C₅⋊D₅	—	3	2	no	—
49	C₂²×C₃₁	—	225	1	no	Hall
50	C₂×C₅⋊D₅	—	9	1	no	Hall
51	C₃×C₃₁	—	2	1	no	—
52	C₃×C₃₁	D₃×C₅⋊D₅	1	2	yes	complemented
53	C₅×C₃⋊D₃	D₅×C₃₁	1	6	yes	complemented
54	C₃×D₁₅	—	5	12	no	—
55	C₃×C₅²	—	2	1	no	—
56	C₃×C₅²	C₂×D₃×C₃₁	1	2	yes	complemented
57	C₂×C₃₁	—	9	1	no	—
58	C₂×C₃₁	—	75	2	no	—
59	D₃×D₅	—	15	12	no	—
60	C₂×C₅²	—	9	1	no	—
61	C₅⋊D₅	—	3	2	no	—
62	C₃²×C₅	C₂×D₅×C₃₁	1	6	yes	complemented
63	D₃²	—	25	1	no	Hall
64	C₃₁	(C₃⋊D₃)⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor, Sylow
65	D₁₅	—	5	12	no	—
66	C₃×D₅	—	15	12	no	—
67	C₅×D₃	—	3	12	no	—
68	C₅×D₃	—	6	6	no	—
69	C₅²	D₃²×C₃₁	1	1	yes	characteristic, complemented, Sylow
70	C₂×D₅	—	45	6	no	—
71	C₃⋊D₃	C₃₁×C₅⋊D₅	1	1	yes	characteristic, complemented
72	C₃×D₃	—	25	2	no	—
73	C₃×C₅	—	2	6	no	—
74	C₃×C₅	D₃×D₅×C₃₁	1	12	yes	complemented
75	C₂×D₃	—	75	2	no	—
76	C₂×C₅	—	9	6	no	—
77	D₅	—	15	12	no	—
78	C₃²	C₂×C₃₁×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃	—	75	2	no	—
80	D₃	—	3	2	no	—
81	D₃	—	6	1	no	—
82	D₃	—	25	2	no	—
83	C₅	C₃₁×(C₃⋊D₃)⋊D₅	1	6	yes	complemented
84	C₂²	—	225	1	no	Sylow
85	C₃	—	2	1	no	—
86	C₃	D₃×C₃₁×C₅⋊D₅	1	2	yes	complemented
87	C₂	—	9	1	no	—
88	C₂	—	75	2	no	—

	subgroup	G/H
center, hypercenter	C₃₁	(C₃⋊D₃)⋊(C₅⋊D₅)
commutator, cosocle, hypocenter	C₃²×C₅²	C₂²×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₂²
LCS	G ▷ C₃²×C₅²
UCS	C₁ ◁ C₃₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₃₁×C₃⋊(C₅⋊D₅)	C₂	semidirect
2. C₅²×C₃⋊D₃×C₃₁	C₂	semidirect
4. C₃²×C₅²×C₃₁	C₂²	semidirect
6. C₃₁×C₃⋊(C₅⋊D₅)	D₃	semidirect
11. C₅×C₃⋊D₃×C₃₁	D₅	semidirect
14. C₃×C₅²×C₃₁	C₂×D₃	semidirect
18. C₃²×C₅×C₃₁	C₂×D₅	semidirect
24. (C₃⋊D₃)⋊(C₅⋊D₅)	C₃₁	direct (N × H)
25. C₅²×C₃₁	D₃²	semidirect
27. C₃⋊D₃×C₃₁	C₅⋊D₅	semidirect
30. C₃×C₅×C₃₁	D₃×D₅	semidirect
31. C₅²×C₃⋊D₃	C₂×C₃₁	semidirect
32. C₃×C₃⋊(C₅⋊D₅)	C₂×C₃₁	semidirect
37. C₃²×C₃₁	C₂×C₅⋊D₅	semidirect
38. C₃²×C₅²	C₂²×C₃₁	semidirect
44. C₅×C₃₁	(C₃⋊D₃)⋊D₅	semidirect
45. C₃⋊(C₅⋊D₅)	D₃×C₃₁	semidirect
52. C₃×C₃₁	D₃×C₅⋊D₅	semidirect
53. C₅×C₃⋊D₃	D₅×C₃₁	semidirect
56. C₃×C₅²	C₂×D₃×C₃₁	semidirect
62. C₃²×C₅	C₂×D₅×C₃₁	semidirect
64. C₃₁	(C₃⋊D₃)⋊(C₅⋊D₅)	direct (N × H)
69. C₅²	D₃²×C₃₁	semidirect
71. C₃⋊D₃	C₃₁×C₅⋊D₅	semidirect
74. C₃×C₅	D₃×D₅×C₃₁	semidirect
78. C₃²	C₂×C₃₁×C₅⋊D₅	semidirect
83. C₅	C₃₁×(C₃⋊D₃)⋊D₅	semidirect
86. C₃	D₃×C₃₁×C₅⋊D₅	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃₁×C₃⋊(C₅⋊D₅)	2
C₅²×C₃⋊D₃×C₃₁	2
D₃×C₃₁×C₅⋊D₅	3
C₃₁×(C₃⋊D₃)⋊D₅	5
(C₃⋊D₃)⋊(C₅⋊D₅)	31

Subgroups — C31×(C3⋊D3)⋊(C5⋊D5) #42124944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×(C₃⋊D₃)⋊(C₅⋊D₅) #42124944