Subgroups — C₃²×C₈×C₉×C₃₇ #72201250

Order 23976.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₉×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₈×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃×C₈×C₉×C₃₇	C₃	1	12	yes	maximal, direct product factor
4	C₂×C₃²×C₉×C₃₇	C₄	1	1	yes	characteristic
5	C₃³×C₄×C₃₇	C₂×C₃	1	1	yes	characteristic
6	C₃×C₄×C₉×C₃₇	C₂×C₃	1	12	yes	—
7	C₃²×C₈×C₃₇	C₉	1	9	yes	direct product factor
8	C₃²×C₈×C₃₇	C₃²	1	4	yes	—
9	C₈×C₉×C₃₇	C₃²	1	9	yes	direct product factor
10	C₃²×C₉×C₃₇	C₈	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₃³×C₃₇	C₃×C₄	1	1	yes	characteristic
12	C₂×C₃×C₉×C₃₇	C₃×C₄	1	12	yes	—
13	C₃²×C₄×C₃₇	C₂×C₉	1	9	yes	—
14	C₃²×C₄×C₃₇	C₂×C₃²	1	4	yes	—
15	C₄×C₉×C₃₇	C₂×C₃²	1	9	yes	—
16	C₃³×C₃₇	C₃×C₈	1	1	yes	characteristic
17	C₃×C₉×C₃₇	C₃×C₈	1	12	yes	direct product factor
18	C₃×C₈×C₃₇	C₃×C₉	1	12	yes	direct product factor
19	C₃×C₈×C₃₇	C₃³	1	1	yes	characteristic
20	C₂×C₃²×C₃₇	C₄×C₉	1	9	yes	—
21	C₂×C₃²×C₃₇	C₃²×C₄	1	4	yes	—
22	C₂×C₉×C₃₇	C₃²×C₄	1	9	yes	—
23	C₃²×C₈×C₉	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₃×C₄×C₃₇	C₂×C₃×C₉	1	12	yes	—
25	C₃×C₄×C₃₇	C₂×C₃³	1	1	yes	characteristic
26	C₃²×C₃₇	C₈×C₉	1	9	yes	direct product factor
27	C₃²×C₃₇	C₃²×C₈	1	4	yes	—
28	C₉×C₃₇	C₃²×C₈	1	9	yes	direct product factor
29	C₃²×C₄×C₉	C₂×C₃₇	1	1	yes	characteristic
30	C₈×C₃₇	C₃²×C₉	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₃×C₃₇	C₃×C₄×C₉	1	12	yes	—
32	C₂×C₃×C₃₇	C₃³×C₄	1	1	yes	characteristic
33	C₃³×C₈	C₃×C₃₇	1	1	yes	characteristic
34	C₃×C₈×C₉	C₃×C₃₇	1	12	yes	direct product factor
35	C₂×C₃²×C₉	C₄×C₃₇	1	1	yes	characteristic
36	C₄×C₃₇	C₂×C₃²×C₉	1	1	yes	characteristic
37	C₃×C₃₇	C₃×C₈×C₉	1	12	yes	direct product factor
38	C₃×C₃₇	C₃³×C₈	1	1	yes	characteristic
39	C₃³×C₄	C₂×C₃×C₃₇	1	1	yes	characteristic
40	C₃×C₄×C₉	C₂×C₃×C₃₇	1	12	yes	—
41	C₃²×C₉	C₈×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₃₇	C₃²×C₄×C₉	1	1	yes	characteristic
43	C₃²×C₈	C₉×C₃₇	1	9	yes	direct product factor
44	C₃²×C₈	C₃²×C₃₇	1	4	yes	—
45	C₈×C₉	C₃²×C₃₇	1	9	yes	direct product factor
46	C₂×C₃³	C₃×C₄×C₃₇	1	1	yes	characteristic
47	C₂×C₃×C₉	C₃×C₄×C₃₇	1	12	yes	—
48	C₃₇	C₃²×C₈×C₉	1	1	yes	characteristic, direct product factor, Sylow
49	C₃²×C₄	C₂×C₉×C₃₇	1	9	yes	—
50	C₃²×C₄	C₂×C₃²×C₃₇	1	4	yes	—
51	C₄×C₉	C₂×C₃²×C₃₇	1	9	yes	—
52	C₃³	C₃×C₈×C₃₇	1	1	yes	characteristic
53	C₃×C₉	C₃×C₈×C₃₇	1	12	yes	direct product factor
54	C₃×C₈	C₃×C₉×C₃₇	1	12	yes	direct product factor
55	C₃×C₈	C₃³×C₃₇	1	1	yes	characteristic
56	C₂×C₃²	C₄×C₉×C₃₇	1	9	yes	—
57	C₂×C₃²	C₃²×C₄×C₃₇	1	4	yes	—
58	C₂×C₉	C₃²×C₄×C₃₇	1	9	yes	—
59	C₃×C₄	C₂×C₃×C₉×C₃₇	1	12	yes	—
60	C₃×C₄	C₂×C₃³×C₃₇	1	1	yes	characteristic
61	C₃²	C₈×C₉×C₃₇	1	9	yes	direct product factor
62	C₃²	C₃²×C₈×C₃₇	1	4	yes	—
63	C₉	C₃²×C₈×C₃₇	1	9	yes	direct product factor
64	C₈	C₃²×C₉×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₃	C₃×C₄×C₉×C₃₇	1	12	yes	—
66	C₂×C₃	C₃³×C₄×C₃₇	1	1	yes	characteristic
67	C₄	C₂×C₃²×C₉×C₃₇	1	1	yes	characteristic
68	C₃	C₃×C₈×C₉×C₃₇	1	12	yes	direct product factor
69	C₃	C₃³×C₈×C₃₇	1	1	yes	characteristic
70	C₂	C₃²×C₄×C₉×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₄	C₂×C₃³×C₃₇
socle, prime radical	C₂×C₃³×C₃₇	C₃×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₈×C₉×C₃₇	C₃	direct (N × H)
7. C₃²×C₈×C₃₇	C₉	direct (N × H)
9. C₈×C₉×C₃₇	C₃²	direct (N × H)
10. C₃²×C₉×C₃₇	C₈	direct (N × H)
17. C₃×C₉×C₃₇	C₃×C₈	direct (N × H)
18. C₃×C₈×C₃₇	C₃×C₉	direct (N × H)
23. C₃²×C₈×C₉	C₃₇	direct (N × H)
26. C₃²×C₃₇	C₈×C₉	direct (N × H)
28. C₉×C₃₇	C₃²×C₈	direct (N × H)
30. C₈×C₃₇	C₃²×C₉	direct (N × H)
34. C₃×C₈×C₉	C₃×C₃₇	direct (N × H)
37. C₃×C₃₇	C₃×C₈×C₉	direct (N × H)
41. C₃²×C₉	C₈×C₃₇	direct (N × H)
43. C₃²×C₈	C₉×C₃₇	direct (N × H)
45. C₈×C₉	C₃²×C₃₇	direct (N × H)
48. C₃₇	C₃²×C₈×C₉	direct (N × H)
53. C₃×C₉	C₃×C₈×C₃₇	direct (N × H)
54. C₃×C₈	C₃×C₉×C₃₇	direct (N × H)
61. C₃²	C₈×C₉×C₃₇	direct (N × H)
63. C₉	C₃²×C₈×C₃₇	direct (N × H)
64. C₈	C₃²×C₉×C₃₇	direct (N × H)
68. C₃	C₃×C₈×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
C₃²×C₄×C₉×C₃₇	2
C₃³×C₈×C₃₇	3
C₃×C₈×C₉×C₃₇	3
C₃²×C₈×C₉	37

Subgroups — C32×C8×C9×C37 #72201250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₈×C₉×C₃₇ #72201250