Subgroups — C₂²×C₃×C₅²×C₃₁⋊C₃ #42124979

Order 27900.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅²×C₃₁⋊C₃	C₂	direct (N × H)
2. C₂²×C₅²×C₃₁⋊C₃	C₃	direct (N × H)
3. C₂²×C₃×C₅²×C₃₁	C₃	semidirect
4. C₃×C₅²×C₃₁⋊C₃	C₂²	direct (N × H)
5. C₂²×C₃×C₅×C₃₁⋊C₃	C₅	direct (N × H)
6. C₂×C₅²×C₃₁⋊C₃	C₂×C₃	direct (N × H)
7. C₂×C₃×C₅²×C₃₁	C₂×C₃	semidirect
8. C₂²×C₅²×C₃₁	C₃²	semidirect
9. C₂×C₃×C₅×C₃₁⋊C₃	C₂×C₅	direct (N × H)
10. C₅²×C₃₁⋊C₃	C₂²×C₃	direct (N × H)
11. C₃×C₅²×C₃₁	C₂²×C₃	semidirect
12. C₂²×C₃×C₅×C₃₁	C₃×C₅	semidirect
13. C₂²×C₅×C₃₁⋊C₃	C₃×C₅	direct (N × H)
14. C₂×C₅²×C₃₁	C₂×C₃²	semidirect
15. C₃×C₅×C₃₁⋊C₃	C₂²×C₅	direct (N × H)
16. C₂²×C₃×C₃₁⋊C₃	C₅²	direct (N × H)
17. C₂×C₃×C₅×C₃₁	C₂×C₃×C₅	semidirect
18. C₂×C₅×C₃₁⋊C₃	C₂×C₃×C₅	direct (N × H)
20. C₅²×C₃₁	C₂²×C₃²	semidirect
21. C₂²×C₅×C₃₁	C₃²×C₅	semidirect
22. C₂×C₃×C₃₁⋊C₃	C₂×C₅²	direct (N × H)
23. C₃×C₅×C₃₁	C₂²×C₃×C₅	semidirect
24. C₅×C₃₁⋊C₃	C₂²×C₃×C₅	direct (N × H)
26. C₂²×C₃×C₃₁	C₃×C₅²	semidirect
27. C₂²×C₃₁⋊C₃	C₃×C₅²	direct (N × H)
28. C₂×C₅×C₃₁	C₂×C₃²×C₅	semidirect
30. C₂²×C₃×C₅²	C₃₁⋊C₃	direct (N × H)
31. C₃×C₃₁⋊C₃	C₂²×C₅²	direct (N × H)
33. C₂×C₃×C₃₁	C₂×C₃×C₅²	semidirect
34. C₂×C₃₁⋊C₃	C₂×C₃×C₅²	direct (N × H)
36. C₅×C₃₁	C₂²×C₃²×C₅	semidirect
38. C₂×C₃×C₅²	C₂×C₃₁⋊C₃	direct (N × H)
39. C₂²×C₃₁	C₃²×C₅²	semidirect
40. C₂²×C₅²	C₃×C₃₁⋊C₃	direct (N × H)
41. C₃×C₃₁	C₂²×C₃×C₅²	semidirect
42. C₃₁⋊C₃	C₂²×C₃×C₅²	direct (N × H)
45. C₃×C₅²	C₂²×C₃₁⋊C₃	direct (N × H)
46. C₂×C₃₁	C₂×C₃²×C₅²	semidirect
48. C₂²×C₃×C₅	C₅×C₃₁⋊C₃	direct (N × H)
49. C₂×C₅²	C₂×C₃×C₃₁⋊C₃	direct (N × H)
52. C₃₁	C₂²×C₃²×C₅²	semidirect
54. C₂×C₃×C₅	C₂×C₅×C₃₁⋊C₃	direct (N × H)
55. C₅²	C₂²×C₃×C₃₁⋊C₃	direct (N × H)
56. C₂²×C₅	C₃×C₅×C₃₁⋊C₃	direct (N × H)
59. C₃×C₅	C₂²×C₅×C₃₁⋊C₃	direct (N × H)
61. C₂²×C₃	C₅²×C₃₁⋊C₃	direct (N × H)
62. C₂×C₅	C₂×C₃×C₅×C₃₁⋊C₃	direct (N × H)
65. C₂×C₃	C₂×C₅²×C₃₁⋊C₃	direct (N × H)
66. C₅	C₂²×C₃×C₅×C₃₁⋊C₃	direct (N × H)
67. C₂²	C₃×C₅²×C₃₁⋊C₃	direct (N × H)
69. C₃	C₂²×C₅²×C₃₁⋊C₃	direct (N × H)
70. C₂	C₂×C₃×C₅²×C₃₁⋊C₃	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×C₃×C₅²×C₃₁⋊C₃	2
C₂²×C₅²×C₃₁⋊C₃	3
C₂²×C₃×C₅²×C₃₁	3
C₂²×C₃×C₅×C₃₁⋊C₃	5
C₂²×C₃²×C₅²	31

Subgroups — C22×C3×C52×C31⋊C3 #42124979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅²×C₃₁⋊C₃ #42124979