Subgroups — C₂×C₃²×C₄×C₉×C₃₇ #72201255

Order 23976.

800 conjugacy classes of subgroups.

The subgroup lattice has 108 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₉×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄×C₉×C₃₇	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃³×C₄×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×C₉×C₃₇	C₃	1	12	yes	maximal, direct product factor
5	C₂×C₃²×C₉×C₃₇	C₄	1	2	yes	direct product factor
6	C₂×C₃²×C₉×C₃₇	C₂²	1	1	yes	characteristic
7	C₂²×C₃³×C₃₇	C₂×C₃	1	1	yes	characteristic
8	C₂²×C₃×C₉×C₃₇	C₂×C₃	1	12	yes	—
9	C₃³×C₄×C₃₇	C₂×C₃	1	2	yes	—
10	C₃×C₄×C₉×C₃₇	C₂×C₃	1	24	yes	direct product factor
11	C₂×C₃²×C₄×C₃₇	C₉	1	9	yes	direct product factor
12	C₂×C₃²×C₄×C₃₇	C₃²	1	4	yes	—
13	C₂×C₄×C₉×C₃₇	C₃²	1	9	yes	direct product factor
14	C₃²×C₉×C₃₇	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
15	C₂×C₃³×C₃₇	C₃×C₄	1	2	yes	—
16	C₂×C₃³×C₃₇	C₂²×C₃	1	1	yes	characteristic
17	C₂×C₃×C₉×C₃₇	C₃×C₄	1	24	yes	direct product factor
18	C₂×C₃×C₉×C₃₇	C₂²×C₃	1	12	yes	—
19	C₂²×C₃²×C₃₇	C₂×C₉	1	9	yes	—
20	C₂²×C₃²×C₃₇	C₂×C₃²	1	4	yes	—
21	C₃²×C₄×C₃₇	C₂×C₉	1	18	yes	direct product factor
22	C₃²×C₄×C₃₇	C₂×C₃²	1	8	yes	—
23	C₂²×C₉×C₃₇	C₂×C₃²	1	9	yes	—
24	C₄×C₉×C₃₇	C₂×C₃²	1	18	yes	direct product factor
25	C₃³×C₃₇	C₂×C₃×C₄	1	1	yes	characteristic
26	C₃×C₉×C₃₇	C₂×C₃×C₄	1	12	yes	direct product factor
27	C₂×C₃×C₄×C₃₇	C₃×C₉	1	12	yes	direct product factor
28	C₂×C₃×C₄×C₃₇	C₃³	1	1	yes	characteristic
29	C₂×C₃²×C₃₇	C₄×C₉	1	18	yes	direct product factor
30	C₂×C₃²×C₃₇	C₂²×C₉	1	9	yes	—
31	C₂×C₃²×C₃₇	C₃²×C₄	1	8	yes	—
32	C₂×C₃²×C₃₇	C₂²×C₃²	1	4	yes	—
33	C₂×C₉×C₃₇	C₃²×C₄	1	18	yes	direct product factor
34	C₂×C₉×C₃₇	C₂²×C₃²	1	9	yes	—
35	C₂×C₃²×C₄×C₉	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
36	C₂²×C₃×C₃₇	C₂×C₃×C₉	1	12	yes	—
37	C₂²×C₃×C₃₇	C₂×C₃³	1	1	yes	characteristic
38	C₃×C₄×C₃₇	C₂×C₃×C₉	1	24	yes	direct product factor
39	C₃×C₄×C₃₇	C₂×C₃³	1	2	yes	—
40	C₃²×C₃₇	C₂×C₄×C₉	1	9	yes	direct product factor
41	C₃²×C₃₇	C₂×C₃²×C₄	1	4	yes	—
42	C₉×C₃₇	C₂×C₃²×C₄	1	9	yes	direct product factor
43	C₂²×C₃²×C₉	C₂×C₃₇	1	1	yes	characteristic
44	C₃²×C₄×C₉	C₂×C₃₇	1	2	yes	direct product factor
45	C₂×C₄×C₃₇	C₃²×C₉	1	1	yes	characteristic, direct product factor, Hall
46	C₂×C₃×C₃₇	C₃×C₄×C₉	1	24	yes	direct product factor
47	C₂×C₃×C₃₇	C₂²×C₃×C₉	1	12	yes	—
48	C₂×C₃×C₃₇	C₃³×C₄	1	2	yes	—
49	C₂×C₃×C₃₇	C₂²×C₃³	1	1	yes	characteristic
50	C₂×C₃³×C₄	C₃×C₃₇	1	1	yes	characteristic
51	C₂×C₃×C₄×C₉	C₃×C₃₇	1	12	yes	direct product factor
52	C₂×C₃²×C₉	C₄×C₃₇	1	2	yes	direct product factor
53	C₂×C₃²×C₉	C₂²×C₃₇	1	1	yes	characteristic
54	C₂²×C₃₇	C₂×C₃²×C₉	1	1	yes	characteristic
55	C₄×C₃₇	C₂×C₃²×C₉	1	2	yes	direct product factor
56	C₃×C₃₇	C₂×C₃×C₄×C₉	1	12	yes	direct product factor
57	C₃×C₃₇	C₂×C₃³×C₄	1	1	yes	characteristic
58	C₂²×C₃³	C₂×C₃×C₃₇	1	1	yes	characteristic
59	C₃³×C₄	C₂×C₃×C₃₇	1	2	yes	—
60	C₂²×C₃×C₉	C₂×C₃×C₃₇	1	12	yes	—
61	C₃×C₄×C₉	C₂×C₃×C₃₇	1	24	yes	direct product factor
62	C₃²×C₉	C₂×C₄×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×C₃₇	C₃²×C₄×C₉	1	2	yes	direct product factor
64	C₂×C₃₇	C₂²×C₃²×C₉	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₉×C₃₇	1	9	yes	direct product factor
66	C₂×C₃²×C₄	C₃²×C₃₇	1	4	yes	—
67	C₂×C₄×C₉	C₃²×C₃₇	1	9	yes	direct product factor
68	C₂×C₃³	C₃×C₄×C₃₇	1	2	yes	—
69	C₂×C₃³	C₂²×C₃×C₃₇	1	1	yes	characteristic
70	C₂×C₃×C₉	C₃×C₄×C₃₇	1	24	yes	direct product factor
71	C₂×C₃×C₉	C₂²×C₃×C₃₇	1	12	yes	—
72	C₃₇	C₂×C₃²×C₄×C₉	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²×C₃²	C₂×C₉×C₃₇	1	9	yes	—
74	C₂²×C₃²	C₂×C₃²×C₃₇	1	4	yes	—
75	C₃²×C₄	C₂×C₉×C₃₇	1	18	yes	direct product factor
76	C₃²×C₄	C₂×C₃²×C₃₇	1	8	yes	—
77	C₂²×C₉	C₂×C₃²×C₃₇	1	9	yes	—
78	C₄×C₉	C₂×C₃²×C₃₇	1	18	yes	direct product factor
79	C₃³	C₂×C₃×C₄×C₃₇	1	1	yes	characteristic
80	C₃×C₉	C₂×C₃×C₄×C₃₇	1	12	yes	direct product factor
81	C₂×C₃×C₄	C₃×C₉×C₃₇	1	12	yes	direct product factor
82	C₂×C₃×C₄	C₃³×C₃₇	1	1	yes	characteristic
83	C₂×C₃²	C₄×C₉×C₃₇	1	18	yes	direct product factor
84	C₂×C₃²	C₂²×C₉×C₃₇	1	9	yes	—
85	C₂×C₃²	C₃²×C₄×C₃₇	1	8	yes	—
86	C₂×C₃²	C₂²×C₃²×C₃₇	1	4	yes	—
87	C₂×C₉	C₃²×C₄×C₃₇	1	18	yes	direct product factor
88	C₂×C₉	C₂²×C₃²×C₃₇	1	9	yes	—
89	C₂²×C₃	C₂×C₃×C₉×C₃₇	1	12	yes	—
90	C₂²×C₃	C₂×C₃³×C₃₇	1	1	yes	characteristic
91	C₃×C₄	C₂×C₃×C₉×C₃₇	1	24	yes	direct product factor
92	C₃×C₄	C₂×C₃³×C₃₇	1	2	yes	—
93	C₃²	C₂×C₄×C₉×C₃₇	1	9	yes	direct product factor
94	C₃²	C₂×C₃²×C₄×C₃₇	1	4	yes	—
95	C₉	C₂×C₃²×C₄×C₃₇	1	9	yes	direct product factor
96	C₂×C₄	C₃²×C₉×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×C₃	C₃×C₄×C₉×C₃₇	1	24	yes	direct product factor
98	C₂×C₃	C₃³×C₄×C₃₇	1	2	yes	—
99	C₂×C₃	C₂²×C₃×C₉×C₃₇	1	12	yes	—
100	C₂×C₃	C₂²×C₃³×C₃₇	1	1	yes	characteristic
101	C₂²	C₂×C₃²×C₉×C₃₇	1	1	yes	characteristic
102	C₄	C₂×C₃²×C₉×C₃₇	1	2	yes	direct product factor
103	C₃	C₂×C₃×C₄×C₉×C₃₇	1	12	yes	direct product factor
104	C₃	C₂×C₃³×C₄×C₃₇	1	1	yes	characteristic
105	C₂	C₃²×C₄×C₉×C₃₇	1	2	yes	direct product factor
106	C₂	C₂²×C₃²×C₉×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃	C₂²×C₃³×C₃₇
socle, prime radical	C₂²×C₃³×C₃₇	C₂×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄×C₉×C₃₇	C₂	direct (N × H)
4. C₂×C₃×C₄×C₉×C₃₇	C₃	direct (N × H)
5. C₂×C₃²×C₉×C₃₇	C₄	direct (N × H)
10. C₃×C₄×C₉×C₃₇	C₂×C₃	direct (N × H)
11. C₂×C₃²×C₄×C₃₇	C₉	direct (N × H)
13. C₂×C₄×C₉×C₃₇	C₃²	direct (N × H)
14. C₃²×C₉×C₃₇	C₂×C₄	direct (N × H)
17. C₂×C₃×C₉×C₃₇	C₃×C₄	direct (N × H)
21. C₃²×C₄×C₃₇	C₂×C₉	direct (N × H)
24. C₄×C₉×C₃₇	C₂×C₃²	direct (N × H)
26. C₃×C₉×C₃₇	C₂×C₃×C₄	direct (N × H)
27. C₂×C₃×C₄×C₃₇	C₃×C₉	direct (N × H)
29. C₂×C₃²×C₃₇	C₄×C₉	direct (N × H)
33. C₂×C₉×C₃₇	C₃²×C₄	direct (N × H)
35. C₂×C₃²×C₄×C₉	C₃₇	direct (N × H)
38. C₃×C₄×C₃₇	C₂×C₃×C₉	direct (N × H)
40. C₃²×C₃₇	C₂×C₄×C₉	direct (N × H)
42. C₉×C₃₇	C₂×C₃²×C₄	direct (N × H)
44. C₃²×C₄×C₉	C₂×C₃₇	direct (N × H)
45. C₂×C₄×C₃₇	C₃²×C₉	direct (N × H)
46. C₂×C₃×C₃₇	C₃×C₄×C₉	direct (N × H)
51. C₂×C₃×C₄×C₉	C₃×C₃₇	direct (N × H)
52. C₂×C₃²×C₉	C₄×C₃₇	direct (N × H)
55. C₄×C₃₇	C₂×C₃²×C₉	direct (N × H)
56. C₃×C₃₇	C₂×C₃×C₄×C₉	direct (N × H)
61. C₃×C₄×C₉	C₂×C₃×C₃₇	direct (N × H)
62. C₃²×C₉	C₂×C₄×C₃₇	direct (N × H)
63. C₂×C₃₇	C₃²×C₄×C₉	direct (N × H)
65. C₂×C₃²×C₄	C₉×C₃₇	direct (N × H)
67. C₂×C₄×C₉	C₃²×C₃₇	direct (N × H)
70. C₂×C₃×C₉	C₃×C₄×C₃₇	direct (N × H)
72. C₃₇	C₂×C₃²×C₄×C₉	direct (N × H)
75. C₃²×C₄	C₂×C₉×C₃₇	direct (N × H)
78. C₄×C₉	C₂×C₃²×C₃₇	direct (N × H)
80. C₃×C₉	C₂×C₃×C₄×C₃₇	direct (N × H)
81. C₂×C₃×C₄	C₃×C₉×C₃₇	direct (N × H)
83. C₂×C₃²	C₄×C₉×C₃₇	direct (N × H)
87. C₂×C₉	C₃²×C₄×C₃₇	direct (N × H)
91. C₃×C₄	C₂×C₃×C₉×C₃₇	direct (N × H)
93. C₃²	C₂×C₄×C₉×C₃₇	direct (N × H)
95. C₉	C₂×C₃²×C₄×C₃₇	direct (N × H)
96. C₂×C₄	C₃²×C₉×C₃₇	direct (N × H)
97. C₂×C₃	C₃×C₄×C₉×C₃₇	direct (N × H)
102. C₄	C₂×C₃²×C₉×C₃₇	direct (N × H)
103. C₃	C₂×C₃×C₄×C₉×C₃₇	direct (N × H)
105. C₂	C₃²×C₄×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
C₃²×C₄×C₉×C₃₇	2
C₂×C₃³×C₄×C₃₇	3
C₂×C₃×C₄×C₉×C₃₇	3
C₂×C₃²×C₄×C₉	37

Subgroups — C2×C32×C4×C9×C37 #72201255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₄×C₉×C₃₇ #72201255