Subgroups — C₂²×C₃×C₇×C₉×C₃₁ #72200804

Order 23436.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×C₉×C₃₁	C₂	1	3	yes	maximal, direct product factor
2	C₃×C₇×C₉×C₃₁	C₂²	1	1	yes	characteristic, direct product factor, Hall
3	C₂²×C₃×C₉×C₃₁	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃²×C₇×C₃₁	C₃	1	1	yes	maximal, characteristic
5	C₂²×C₇×C₉×C₃₁	C₃	1	3	yes	maximal, direct product factor
6	C₂×C₃²×C₇×C₃₁	C₂×C₃	1	3	yes	—
7	C₂×C₇×C₉×C₃₁	C₂×C₃	1	9	yes	direct product factor
8	C₂²×C₃×C₇×C₃₁	C₉	1	3	yes	direct product factor
9	C₂²×C₃×C₇×C₃₁	C₃²	1	1	yes	characteristic
10	C₃²×C₇×C₃₁	C₂²×C₃	1	1	yes	characteristic
11	C₇×C₉×C₃₁	C₂²×C₃	1	3	yes	direct product factor
12	C₂×C₃×C₉×C₃₁	C₂×C₇	1	3	yes	direct product factor
13	C₂×C₃×C₇×C₃₁	C₂×C₉	1	9	yes	direct product factor
14	C₂×C₃×C₇×C₃₁	C₂×C₃²	1	3	yes	—
15	C₂²×C₃²×C₃₁	C₃×C₇	1	1	yes	characteristic
16	C₂²×C₉×C₃₁	C₃×C₇	1	3	yes	direct product factor
17	C₂²×C₇×C₃₁	C₃×C₉	1	1	yes	characteristic, direct product factor, Hall
18	C₃×C₉×C₃₁	C₂²×C₇	1	1	yes	characteristic, direct product factor, Hall
19	C₂²×C₃×C₇×C₉	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₃×C₇×C₃₁	C₂²×C₉	1	3	yes	direct product factor
21	C₃×C₇×C₃₁	C₂²×C₃²	1	1	yes	characteristic
22	C₂×C₃²×C₃₁	C₂×C₃×C₇	1	3	yes	—
23	C₂×C₉×C₃₁	C₂×C₃×C₇	1	9	yes	direct product factor
24	C₂×C₇×C₃₁	C₂×C₃×C₉	1	3	yes	direct product factor
25	C₂×C₃×C₇×C₉	C₂×C₃₁	1	3	yes	direct product factor
26	C₂²×C₃×C₃₁	C₇×C₉	1	3	yes	direct product factor
27	C₂²×C₃×C₃₁	C₃²×C₇	1	1	yes	characteristic
28	C₃²×C₃₁	C₂²×C₃×C₇	1	1	yes	characteristic
29	C₉×C₃₁	C₂²×C₃×C₇	1	3	yes	direct product factor
30	C₂²×C₃²×C₇	C₃×C₃₁	1	1	yes	characteristic
31	C₂²×C₇×C₉	C₃×C₃₁	1	3	yes	direct product factor
32	C₇×C₃₁	C₂²×C₃×C₉	1	1	yes	characteristic, direct product factor, Hall
33	C₃×C₇×C₉	C₂²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₃×C₃₁	C₂×C₇×C₉	1	9	yes	direct product factor
35	C₂×C₃×C₃₁	C₂×C₃²×C₇	1	3	yes	—
36	C₂×C₃²×C₇	C₂×C₃×C₃₁	1	3	yes	—
37	C₂×C₇×C₉	C₂×C₃×C₃₁	1	9	yes	direct product factor
38	C₂²×C₃₁	C₃×C₇×C₉	1	1	yes	characteristic, direct product factor, Hall
39	C₂²×C₃×C₉	C₇×C₃₁	1	1	yes	characteristic, direct product factor, Hall
40	C₃×C₃₁	C₂²×C₇×C₉	1	3	yes	direct product factor
41	C₃×C₃₁	C₂²×C₃²×C₇	1	1	yes	characteristic
42	C₂²×C₃×C₇	C₉×C₃₁	1	3	yes	direct product factor
43	C₂²×C₃×C₇	C₃²×C₃₁	1	1	yes	characteristic
44	C₃²×C₇	C₂²×C₃×C₃₁	1	1	yes	characteristic
45	C₇×C₉	C₂²×C₃×C₃₁	1	3	yes	direct product factor
46	C₂×C₃₁	C₂×C₃×C₇×C₉	1	3	yes	direct product factor
47	C₂×C₃×C₉	C₂×C₇×C₃₁	1	3	yes	direct product factor
48	C₂×C₃×C₇	C₂×C₉×C₃₁	1	9	yes	direct product factor
49	C₂×C₃×C₇	C₂×C₃²×C₃₁	1	3	yes	—
50	C₂²×C₃²	C₃×C₇×C₃₁	1	1	yes	characteristic
51	C₂²×C₉	C₃×C₇×C₃₁	1	3	yes	direct product factor
52	C₃₁	C₂²×C₃×C₇×C₉	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²×C₇	C₃×C₉×C₃₁	1	1	yes	characteristic, direct product factor, Hall
54	C₃×C₉	C₂²×C₇×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
55	C₃×C₇	C₂²×C₉×C₃₁	1	3	yes	direct product factor
56	C₃×C₇	C₂²×C₃²×C₃₁	1	1	yes	characteristic
57	C₂×C₃²	C₂×C₃×C₇×C₃₁	1	3	yes	—
58	C₂×C₉	C₂×C₃×C₇×C₃₁	1	9	yes	direct product factor
59	C₂×C₇	C₂×C₃×C₉×C₃₁	1	3	yes	direct product factor
60	C₂²×C₃	C₇×C₉×C₃₁	1	3	yes	direct product factor
61	C₂²×C₃	C₃²×C₇×C₃₁	1	1	yes	characteristic
62	C₃²	C₂²×C₃×C₇×C₃₁	1	1	yes	characteristic
63	C₉	C₂²×C₃×C₇×C₃₁	1	3	yes	direct product factor
64	C₇	C₂²×C₃×C₉×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₃	C₂×C₇×C₉×C₃₁	1	9	yes	direct product factor
66	C₂×C₃	C₂×C₃²×C₇×C₃₁	1	3	yes	—
67	C₂²	C₃×C₇×C₉×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
68	C₃	C₂²×C₇×C₉×C₃₁	1	3	yes	direct product factor
69	C₃	C₂²×C₃²×C₇×C₃₁	1	1	yes	characteristic
70	C₂	C₂×C₃×C₇×C₉×C₃₁	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₂²×C₃²×C₇×C₃₁
socle, prime radical	C₂²×C₃²×C₇×C₃₁	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₇×C₉×C₃₁	C₂	direct (N × H)
2. C₃×C₇×C₉×C₃₁	C₂²	direct (N × H)
3. C₂²×C₃×C₉×C₃₁	C₇	direct (N × H)
5. C₂²×C₇×C₉×C₃₁	C₃	direct (N × H)
7. C₂×C₇×C₉×C₃₁	C₂×C₃	direct (N × H)
8. C₂²×C₃×C₇×C₃₁	C₉	direct (N × H)
11. C₇×C₉×C₃₁	C₂²×C₃	direct (N × H)
12. C₂×C₃×C₉×C₃₁	C₂×C₇	direct (N × H)
13. C₂×C₃×C₇×C₃₁	C₂×C₉	direct (N × H)
16. C₂²×C₉×C₃₁	C₃×C₇	direct (N × H)
17. C₂²×C₇×C₃₁	C₃×C₉	direct (N × H)
18. C₃×C₉×C₃₁	C₂²×C₇	direct (N × H)
19. C₂²×C₃×C₇×C₉	C₃₁	direct (N × H)
20. C₃×C₇×C₃₁	C₂²×C₉	direct (N × H)
23. C₂×C₉×C₃₁	C₂×C₃×C₇	direct (N × H)
24. C₂×C₇×C₃₁	C₂×C₃×C₉	direct (N × H)
25. C₂×C₃×C₇×C₉	C₂×C₃₁	direct (N × H)
26. C₂²×C₃×C₃₁	C₇×C₉	direct (N × H)
29. C₉×C₃₁	C₂²×C₃×C₇	direct (N × H)
31. C₂²×C₇×C₉	C₃×C₃₁	direct (N × H)
32. C₇×C₃₁	C₂²×C₃×C₉	direct (N × H)
33. C₃×C₇×C₉	C₂²×C₃₁	direct (N × H)
34. C₂×C₃×C₃₁	C₂×C₇×C₉	direct (N × H)
37. C₂×C₇×C₉	C₂×C₃×C₃₁	direct (N × H)
38. C₂²×C₃₁	C₃×C₇×C₉	direct (N × H)
39. C₂²×C₃×C₉	C₇×C₃₁	direct (N × H)
40. C₃×C₃₁	C₂²×C₇×C₉	direct (N × H)
42. C₂²×C₃×C₇	C₉×C₃₁	direct (N × H)
45. C₇×C₉	C₂²×C₃×C₃₁	direct (N × H)
46. C₂×C₃₁	C₂×C₃×C₇×C₉	direct (N × H)
47. C₂×C₃×C₉	C₂×C₇×C₃₁	direct (N × H)
48. C₂×C₃×C₇	C₂×C₉×C₃₁	direct (N × H)
51. C₂²×C₉	C₃×C₇×C₃₁	direct (N × H)
52. C₃₁	C₂²×C₃×C₇×C₉	direct (N × H)
53. C₂²×C₇	C₃×C₉×C₃₁	direct (N × H)
54. C₃×C₉	C₂²×C₇×C₃₁	direct (N × H)
55. C₃×C₇	C₂²×C₉×C₃₁	direct (N × H)
58. C₂×C₉	C₂×C₃×C₇×C₃₁	direct (N × H)
59. C₂×C₇	C₂×C₃×C₉×C₃₁	direct (N × H)
60. C₂²×C₃	C₇×C₉×C₃₁	direct (N × H)
63. C₉	C₂²×C₃×C₇×C₃₁	direct (N × H)
64. C₇	C₂²×C₃×C₉×C₃₁	direct (N × H)
65. C₂×C₃	C₂×C₇×C₉×C₃₁	direct (N × H)
67. C₂²	C₃×C₇×C₉×C₃₁	direct (N × H)
68. C₃	C₂²×C₇×C₉×C₃₁	direct (N × H)
70. C₂	C₂×C₃×C₇×C₉×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×C₃×C₇×C₉×C₃₁	2
C₂²×C₃×C₉×C₃₁	7
C₂²×C₃²×C₇×C₃₁	3
C₂²×C₇×C₉×C₃₁	3
C₂²×C₃×C₇×C₉	31

Subgroups — C22×C3×C7×C9×C31 #72200804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₇×C₉×C₃₁ #72200804