Subgroups — C₂³×C₃²×C₉×C₃₇ #72201260

Order 23976.

1600 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₉×C₃₇	C₂	1	7	yes	maximal, direct product factor
2	C₂³×C₃³×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₂³×C₃×C₉×C₃₇	C₃	1	12	yes	maximal, direct product factor
4	C₂×C₃²×C₉×C₃₇	C₂²	1	7	yes	direct product factor
5	C₂²×C₃³×C₃₇	C₂×C₃	1	7	yes	—
6	C₂²×C₃×C₉×C₃₇	C₂×C₃	1	84	yes	direct product factor
7	C₂³×C₃²×C₃₇	C₉	1	9	yes	direct product factor
8	C₂³×C₃²×C₃₇	C₃²	1	4	yes	—
9	C₂³×C₉×C₃₇	C₃²	1	9	yes	direct product factor
10	C₃²×C₉×C₃₇	C₂³	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₃³×C₃₇	C₂²×C₃	1	7	yes	—
12	C₂×C₃×C₉×C₃₇	C₂²×C₃	1	84	yes	direct product factor
13	C₂²×C₃²×C₃₇	C₂×C₉	1	63	yes	direct product factor
14	C₂²×C₃²×C₃₇	C₂×C₃²	1	28	yes	—
15	C₂²×C₉×C₃₇	C₂×C₃²	1	63	yes	direct product factor
16	C₃³×C₃₇	C₂³×C₃	1	1	yes	characteristic
17	C₃×C₉×C₃₇	C₂³×C₃	1	12	yes	direct product factor
18	C₂³×C₃×C₃₇	C₃×C₉	1	12	yes	direct product factor
19	C₂³×C₃×C₃₇	C₃³	1	1	yes	characteristic
20	C₂×C₃²×C₃₇	C₂²×C₉	1	63	yes	direct product factor
21	C₂×C₃²×C₃₇	C₂²×C₃²	1	28	yes	—
22	C₂×C₉×C₃₇	C₂²×C₃²	1	63	yes	direct product factor
23	C₂³×C₃²×C₉	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂²×C₃×C₃₇	C₂×C₃×C₉	1	84	yes	direct product factor
25	C₂²×C₃×C₃₇	C₂×C₃³	1	7	yes	—
26	C₃²×C₃₇	C₂³×C₉	1	9	yes	direct product factor
27	C₃²×C₃₇	C₂³×C₃²	1	4	yes	—
28	C₉×C₃₇	C₂³×C₃²	1	9	yes	direct product factor
29	C₂²×C₃²×C₉	C₂×C₃₇	1	7	yes	direct product factor
30	C₂³×C₃₇	C₃²×C₉	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₃×C₃₇	C₂²×C₃×C₉	1	84	yes	direct product factor
32	C₂×C₃×C₃₇	C₂²×C₃³	1	7	yes	—
33	C₂³×C₃³	C₃×C₃₇	1	1	yes	characteristic
34	C₂³×C₃×C₉	C₃×C₃₇	1	12	yes	direct product factor
35	C₂×C₃²×C₉	C₂²×C₃₇	1	7	yes	direct product factor
36	C₂²×C₃₇	C₂×C₃²×C₉	1	7	yes	direct product factor
37	C₃×C₃₇	C₂³×C₃×C₉	1	12	yes	direct product factor
38	C₃×C₃₇	C₂³×C₃³	1	1	yes	characteristic
39	C₂²×C₃³	C₂×C₃×C₃₇	1	7	yes	—
40	C₂²×C₃×C₉	C₂×C₃×C₃₇	1	84	yes	direct product factor
41	C₃²×C₉	C₂³×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₃₇	C₂²×C₃²×C₉	1	7	yes	direct product factor
43	C₂³×C₃²	C₉×C₃₇	1	9	yes	direct product factor
44	C₂³×C₃²	C₃²×C₃₇	1	4	yes	—
45	C₂³×C₉	C₃²×C₃₇	1	9	yes	direct product factor
46	C₂×C₃³	C₂²×C₃×C₃₇	1	7	yes	—
47	C₂×C₃×C₉	C₂²×C₃×C₃₇	1	84	yes	direct product factor
48	C₃₇	C₂³×C₃²×C₉	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²×C₃²	C₂×C₉×C₃₇	1	63	yes	direct product factor
50	C₂²×C₃²	C₂×C₃²×C₃₇	1	28	yes	—
51	C₂²×C₉	C₂×C₃²×C₃₇	1	63	yes	direct product factor
52	C₃³	C₂³×C₃×C₃₇	1	1	yes	characteristic
53	C₃×C₉	C₂³×C₃×C₃₇	1	12	yes	direct product factor
54	C₂³×C₃	C₃×C₉×C₃₇	1	12	yes	direct product factor
55	C₂³×C₃	C₃³×C₃₇	1	1	yes	characteristic
56	C₂×C₃²	C₂²×C₉×C₃₇	1	63	yes	direct product factor
57	C₂×C₃²	C₂²×C₃²×C₃₇	1	28	yes	—
58	C₂×C₉	C₂²×C₃²×C₃₇	1	63	yes	direct product factor
59	C₂²×C₃	C₂×C₃×C₉×C₃₇	1	84	yes	direct product factor
60	C₂²×C₃	C₂×C₃³×C₃₇	1	7	yes	—
61	C₃²	C₂³×C₉×C₃₇	1	9	yes	direct product factor
62	C₃²	C₂³×C₃²×C₃₇	1	4	yes	—
63	C₉	C₂³×C₃²×C₃₇	1	9	yes	direct product factor
64	C₂³	C₃²×C₉×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₃	C₂²×C₃×C₉×C₃₇	1	84	yes	direct product factor
66	C₂×C₃	C₂²×C₃³×C₃₇	1	7	yes	—
67	C₂²	C₂×C₃²×C₉×C₃₇	1	7	yes	direct product factor
68	C₃	C₂³×C₃×C₉×C₃₇	1	12	yes	direct product factor
69	C₃	C₂³×C₃³×C₃₇	1	1	yes	characteristic
70	C₂	C₂²×C₃²×C₉×C₃₇	1	7	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₂³×C₃³×C₃₇
socle, prime radical	C₂³×C₃³×C₃₇	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₉×C₃₇	C₂	direct (N × H)
3. C₂³×C₃×C₉×C₃₇	C₃	direct (N × H)
4. C₂×C₃²×C₉×C₃₇	C₂²	direct (N × H)
6. C₂²×C₃×C₉×C₃₇	C₂×C₃	direct (N × H)
7. C₂³×C₃²×C₃₇	C₉	direct (N × H)
9. C₂³×C₉×C₃₇	C₃²	direct (N × H)
10. C₃²×C₉×C₃₇	C₂³	direct (N × H)
12. C₂×C₃×C₉×C₃₇	C₂²×C₃	direct (N × H)
13. C₂²×C₃²×C₃₇	C₂×C₉	direct (N × H)
15. C₂²×C₉×C₃₇	C₂×C₃²	direct (N × H)
17. C₃×C₉×C₃₇	C₂³×C₃	direct (N × H)
18. C₂³×C₃×C₃₇	C₃×C₉	direct (N × H)
20. C₂×C₃²×C₃₇	C₂²×C₉	direct (N × H)
22. C₂×C₉×C₃₇	C₂²×C₃²	direct (N × H)
23. C₂³×C₃²×C₉	C₃₇	direct (N × H)
24. C₂²×C₃×C₃₇	C₂×C₃×C₉	direct (N × H)
26. C₃²×C₃₇	C₂³×C₉	direct (N × H)
28. C₉×C₃₇	C₂³×C₃²	direct (N × H)
29. C₂²×C₃²×C₉	C₂×C₃₇	direct (N × H)
30. C₂³×C₃₇	C₃²×C₉	direct (N × H)
31. C₂×C₃×C₃₇	C₂²×C₃×C₉	direct (N × H)
34. C₂³×C₃×C₉	C₃×C₃₇	direct (N × H)
35. C₂×C₃²×C₉	C₂²×C₃₇	direct (N × H)
36. C₂²×C₃₇	C₂×C₃²×C₉	direct (N × H)
37. C₃×C₃₇	C₂³×C₃×C₉	direct (N × H)
40. C₂²×C₃×C₉	C₂×C₃×C₃₇	direct (N × H)
41. C₃²×C₉	C₂³×C₃₇	direct (N × H)
42. C₂×C₃₇	C₂²×C₃²×C₉	direct (N × H)
43. C₂³×C₃²	C₉×C₃₇	direct (N × H)
45. C₂³×C₉	C₃²×C₃₇	direct (N × H)
47. C₂×C₃×C₉	C₂²×C₃×C₃₇	direct (N × H)
48. C₃₇	C₂³×C₃²×C₉	direct (N × H)
49. C₂²×C₃²	C₂×C₉×C₃₇	direct (N × H)
51. C₂²×C₉	C₂×C₃²×C₃₇	direct (N × H)
53. C₃×C₉	C₂³×C₃×C₃₇	direct (N × H)
54. C₂³×C₃	C₃×C₉×C₃₇	direct (N × H)
56. C₂×C₃²	C₂²×C₉×C₃₇	direct (N × H)
58. C₂×C₉	C₂²×C₃²×C₃₇	direct (N × H)
59. C₂²×C₃	C₂×C₃×C₉×C₃₇	direct (N × H)
61. C₃²	C₂³×C₉×C₃₇	direct (N × H)
63. C₉	C₂³×C₃²×C₃₇	direct (N × H)
64. C₂³	C₃²×C₉×C₃₇	direct (N × H)
65. C₂×C₃	C₂²×C₃×C₉×C₃₇	direct (N × H)
67. C₂²	C₂×C₃²×C₉×C₃₇	direct (N × H)
68. C₃	C₂³×C₃×C₉×C₃₇	direct (N × H)
70. C₂	C₂²×C₃²×C₉×C₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₉×C₃₇	2
C₂³×C₃³×C₃₇	3
C₂³×C₃×C₉×C₃₇	3
C₂³×C₃²×C₉	37

Subgroups — C23×C32×C9×C37 #72201260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃²×C₉×C₃₇ #72201260