Subgroups — C₃×C₅²×Dic₃×C₃₁ #42124663

Order 27900.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅²×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₅²×Dic₃×C₃₁	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₄×C₅²×C₃₁	—	3	1	no	maximal
4	C₃²×C₅²×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×C₅×Dic₃×C₃₁	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₃×C₅²×C₃₁	—	2	1	no	—
7	C₂×C₃×C₅²×C₃₁	D₃	1	1	yes	characteristic
8	C₂×C₃×C₅²×C₃₁	C₂×C₃	1	1	yes	characteristic
9	C₄×C₅²×C₃₁	—	3	1	no	Hall
10	C₂×C₃²×C₅×C₃₁	C₂×C₅	1	6	yes	—
11	C₃×C₅²×C₃₁	—	2	1	no	—
12	C₃×C₅²×C₃₁	Dic₃	1	1	yes	characteristic, direct product factor
13	C₃×C₅²×C₃₁	C₃×C₄	1	1	yes	characteristic, complemented
14	C₃×C₄×C₅×C₃₁	—	3	6	no	—
15	C₅×Dic₃×C₃₁	C₃×C₅	1	6	yes	direct product factor
16	C₂×C₅²×C₃₁	C₃×D₃	1	1	yes	characteristic
17	C₃²×C₅×C₃₁	C₄×C₅	1	6	yes	complemented
18	C₃×Dic₃×C₃₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₃×C₅×C₃₁	—	2	6	no	—
20	C₂×C₃×C₅×C₃₁	C₅×D₃	1	6	yes	—
21	C₂×C₃×C₅×C₃₁	C₂×C₃×C₅	1	6	yes	—
22	C₃×C₅²×Dic₃	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅²×C₃₁	C₃×Dic₃	1	1	yes	characteristic, direct product factor, Hall
24	C₄×C₅×C₃₁	—	3	6	no	—
25	C₂×C₃²×C₃₁	C₂×C₅²	1	1	yes	characteristic
26	C₃×C₅×C₃₁	—	2	6	no	—
27	C₃×C₅×C₃₁	C₅×Dic₃	1	6	yes	direct product factor
28	C₃×C₅×C₃₁	C₃×C₄×C₅	1	6	yes	complemented
29	C₂×C₃²×C₅²	C₂×C₃₁	1	1	yes	characteristic
30	C₃×C₄×C₃₁	—	3	1	no	—
31	Dic₃×C₃₁	C₃×C₅²	1	1	yes	characteristic, direct product factor
32	C₂×C₅×C₃₁	C₃×C₅×D₃	1	6	yes	—
33	C₃×C₄×C₅²	—	3	1	no	—
34	C₅²×Dic₃	C₃×C₃₁	1	1	yes	characteristic, direct product factor
35	C₃²×C₃₁	C₄×C₅²	1	1	yes	characteristic, complemented, Hall
36	C₃²×C₅²	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
37	C₂×C₃×C₃₁	—	2	1	no	—
38	C₂×C₃×C₃₁	C₅²×D₃	1	1	yes	characteristic
39	C₂×C₃×C₃₁	C₂×C₃×C₅²	1	1	yes	characteristic
40	C₃×C₅×Dic₃	C₅×C₃₁	1	6	yes	direct product factor
41	C₅×C₃₁	C₃×C₅×Dic₃	1	6	yes	direct product factor
42	C₂×C₃×C₅²	—	2	1	no	—
43	C₂×C₃×C₅²	D₃×C₃₁	1	1	yes	characteristic
44	C₂×C₃×C₅²	C₂×C₃×C₃₁	1	1	yes	characteristic
45	C₄×C₃₁	—	3	1	no	Hall
46	C₄×C₅²	—	3	1	no	Hall
47	C₃×C₃₁	—	2	1	no	—
48	C₃×C₃₁	C₅²×Dic₃	1	1	yes	characteristic, direct product factor
49	C₃×C₃₁	C₃×C₄×C₅²	1	1	yes	characteristic, complemented
50	C₂×C₃²×C₅	C₂×C₅×C₃₁	1	6	yes	—
51	C₃×C₅²	—	2	1	no	—
52	C₃×C₅²	Dic₃×C₃₁	1	1	yes	characteristic, direct product factor
53	C₃×C₅²	C₃×C₄×C₃₁	1	1	yes	characteristic, complemented
54	C₂×C₃₁	C₃×C₅²×D₃	1	1	yes	characteristic
55	C₃×C₄×C₅	—	3	6	no	—
56	C₅×Dic₃	C₃×C₅×C₃₁	1	6	yes	direct product factor
57	C₂×C₅²	C₃×D₃×C₃₁	1	1	yes	characteristic
58	C₃²×C₅	C₄×C₅×C₃₁	1	6	yes	complemented
59	C₃×Dic₃	C₅²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
60	C₃₁	C₃×C₅²×Dic₃	1	1	yes	characteristic, direct product factor, Sylow
61	C₂×C₃×C₅	—	2	6	no	—
62	C₂×C₃×C₅	C₅×D₃×C₃₁	1	6	yes	—
63	C₂×C₃×C₅	C₂×C₃×C₅×C₃₁	1	6	yes	—
64	C₅²	C₃×Dic₃×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
65	C₄×C₅	—	3	6	no	—
66	C₂×C₃²	C₂×C₅²×C₃₁	1	1	yes	characteristic
67	C₃×C₅	—	2	6	no	—
68	C₃×C₅	C₅×Dic₃×C₃₁	1	6	yes	direct product factor
69	C₃×C₅	C₃×C₄×C₅×C₃₁	1	6	yes	complemented
70	C₃×C₄	—	3	1	no	—
71	Dic₃	C₃×C₅²×C₃₁	1	1	yes	characteristic, direct product factor
72	C₂×C₅	C₃×C₅×D₃×C₃₁	1	6	yes	—
73	C₃²	C₄×C₅²×C₃₁	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₃	—	2	1	no	—
75	C₂×C₃	C₂×C₃×C₅²×C₃₁	1	1	yes	characteristic
76	C₂×C₃	C₅²×D₃×C₃₁	1	1	yes	characteristic
77	C₅	C₃×C₅×Dic₃×C₃₁	1	6	yes	direct product factor
78	C₄	—	3	1	no	Sylow
79	C₃	—	2	1	no	—
80	C₃	C₃×C₄×C₅²×C₃₁	1	1	yes	characteristic, complemented
81	C₃	C₅²×Dic₃×C₃₁	1	1	yes	characteristic, direct product factor
82	C₂	C₃×C₅²×D₃×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅²×C₃₁	D₃
commutator, hypocenter	C₃	C₃×C₄×C₅²×C₃₁
Frattini	C₂	C₃×C₅²×D₃×C₃₁
Fitting, socle, prime radical	C₂×C₃²×C₅²×C₃₁	C₂
cosocle	C₂×C₃	C₂×C₃×C₅²×C₃₁
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₅²×C₃₁

Complements

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

N	H ≅ G/N	type
2. C₅²×Dic₃×C₃₁	C₃	direct (N × H)
4. C₃²×C₅²×C₃₁	C₄	semidirect
5. C₃×C₅×Dic₃×C₃₁	C₅	direct (N × H)
12. C₃×C₅²×C₃₁	Dic₃	direct (N × H)
13. C₃×C₅²×C₃₁	C₃×C₄	semidirect
15. C₅×Dic₃×C₃₁	C₃×C₅	direct (N × H)
17. C₃²×C₅×C₃₁	C₄×C₅	semidirect
18. C₃×Dic₃×C₃₁	C₅²	direct (N × H)
22. C₃×C₅²×Dic₃	C₃₁	direct (N × H)
23. C₅²×C₃₁	C₃×Dic₃	direct (N × H)
27. C₃×C₅×C₃₁	C₅×Dic₃	direct (N × H)
28. C₃×C₅×C₃₁	C₃×C₄×C₅	semidirect
31. Dic₃×C₃₁	C₃×C₅²	direct (N × H)
34. C₅²×Dic₃	C₃×C₃₁	direct (N × H)
35. C₃²×C₃₁	C₄×C₅²	semidirect
36. C₃²×C₅²	C₄×C₃₁	semidirect
40. C₃×C₅×Dic₃	C₅×C₃₁	direct (N × H)
41. C₅×C₃₁	C₃×C₅×Dic₃	direct (N × H)
48. C₃×C₃₁	C₅²×Dic₃	direct (N × H)
49. C₃×C₃₁	C₃×C₄×C₅²	semidirect
52. C₃×C₅²	Dic₃×C₃₁	direct (N × H)
53. C₃×C₅²	C₃×C₄×C₃₁	semidirect
56. C₅×Dic₃	C₃×C₅×C₃₁	direct (N × H)
58. C₃²×C₅	C₄×C₅×C₃₁	semidirect
59. C₃×Dic₃	C₅²×C₃₁	direct (N × H)
60. C₃₁	C₃×C₅²×Dic₃	direct (N × H)
64. C₅²	C₃×Dic₃×C₃₁	direct (N × H)
68. C₃×C₅	C₅×Dic₃×C₃₁	direct (N × H)
69. C₃×C₅	C₃×C₄×C₅×C₃₁	semidirect
71. Dic₃	C₃×C₅²×C₃₁	direct (N × H)
73. C₃²	C₄×C₅²×C₃₁	semidirect
77. C₅	C₃×C₅×Dic₃×C₃₁	direct (N × H)
80. C₃	C₃×C₄×C₅²×C₃₁	semidirect
81. C₃	C₅²×Dic₃×C₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₅²×C₃₁	2
C₅²×Dic₃×C₃₁	3
C₃×C₄×C₅²×C₃₁	3
C₃×C₅×Dic₃×C₃₁	5
C₃×C₅²×Dic₃	31

Subgroups — C3×C52×Dic3×C31 #42124663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×Dic₃×C₃₁ #42124663