Subgroups — C₅²×D₃²×C₃₁ #42124923

Order 27900.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₅²×C₃₁	D₃²
commutator, cosocle, hypocenter	C₃²	C₂²×C₅²×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₅²×C₃₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×D₃×C₃₁	C₂	semidirect
2. C₅²×C₃⋊D₃×C₃₁	C₂	semidirect
4. C₃²×C₅²×C₃₁	C₂²	semidirect
5. C₅×D₃²×C₃₁	C₅	direct (N × H)
8. C₅²×D₃×C₃₁	D₃	direct (N × H)
11. C₅×C₃⋊D₃×C₃₁	C₂×C₅	semidirect
12. C₃×C₅×D₃×C₃₁	C₂×C₅	semidirect
14. C₃×C₅²×C₃₁	C₂×D₃	semidirect
18. C₃²×C₅×C₃₁	C₂²×C₅	semidirect
19. D₃²×C₃₁	C₅²	direct (N × H)
23. C₅×D₃×C₃₁	C₅×D₃	direct (N × H)
24. C₅²×D₃²	C₃₁	direct (N × H)
25. C₅²×C₃₁	D₃²	direct (N × H)
27. C₃⋊D₃×C₃₁	C₂×C₅²	semidirect
28. C₃×D₃×C₃₁	C₂×C₅²	semidirect
30. C₃×C₅×C₃₁	C₂×C₅×D₃	semidirect
31. C₅²×C₃⋊D₃	C₂×C₃₁	semidirect
32. C₃×C₅²×D₃	C₂×C₃₁	semidirect
37. C₃²×C₃₁	C₂²×C₅²	semidirect
38. C₃²×C₅²	C₂²×C₃₁	semidirect
42. D₃×C₃₁	C₅²×D₃	direct (N × H)
43. C₅×D₃²	C₅×C₃₁	direct (N × H)
44. C₅×C₃₁	C₅×D₃²	direct (N × H)
48. C₅²×D₃	D₃×C₃₁	direct (N × H)
52. C₃×C₃₁	C₂×C₅²×D₃	semidirect
53. C₅×C₃⋊D₃	C₂×C₅×C₃₁	semidirect
54. C₃×C₅×D₃	C₂×C₅×C₃₁	semidirect
56. C₃×C₅²	C₂×D₃×C₃₁	semidirect
62. C₃²×C₅	C₂²×C₅×C₃₁	semidirect
63. D₃²	C₅²×C₃₁	direct (N × H)
64. C₃₁	C₅²×D₃²	direct (N × H)
68. C₅×D₃	C₅×D₃×C₃₁	direct (N × H)
69. C₅²	D₃²×C₃₁	direct (N × H)
71. C₃⋊D₃	C₂×C₅²×C₃₁	semidirect
72. C₃×D₃	C₂×C₅²×C₃₁	semidirect
74. C₃×C₅	C₂×C₅×D₃×C₃₁	semidirect
78. C₃²	C₂²×C₅²×C₃₁	semidirect
82. D₃	C₅²×D₃×C₃₁	direct (N × H)
83. C₅	C₅×D₃²×C₃₁	direct (N × H)
86. C₃	C₂×C₅²×D₃×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅²×D₃×C₃₁	2
C₅²×C₃⋊D₃×C₃₁	2
C₂×C₅²×D₃×C₃₁	3
C₅×D₃²×C₃₁	5
C₅²×D₃²	31

Subgroups — C52×D32×C31 #42124923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×D₃²×C₃₁ #42124923