Subgroups — C₂³×C₄²×C₅×C₃₇ #72201032

Order 23680.

6000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₅×C₃₇	C₂	1	3	yes	maximal
2	C₂²×C₄²×C₅×C₃₇	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₄×C₅×C₃₇	C₄	1	48	yes	direct product factor
4	C₂³×C₄×C₅×C₃₇	C₂²	1	42	yes	—
5	C₂⁵×C₅×C₃₇	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₅×C₃₇	C₂²	1	112	yes	direct product factor
7	C₂⁴×C₅×C₃₇	C₂×C₄	1	24	yes	—
8	C₂⁴×C₅×C₃₇	C₂³	1	7	yes	—
9	C₂²×C₄×C₅×C₃₇	C₂×C₄	1	336	yes	direct product factor
10	C₂²×C₄×C₅×C₃₇	C₂³	1	84	yes	—
11	C₄²×C₅×C₃₇	C₂³	1	64	yes	direct product factor
12	C₂³×C₄²×C₃₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂⁴×C₄×C₃₇	C₂×C₅	1	3	yes	—
14	C₂²×C₄²×C₃₇	C₂×C₅	1	28	yes	direct product factor
15	C₂³×C₅×C₃₇	C₄²	1	64	yes	direct product factor
16	C₂³×C₅×C₃₇	C₂²×C₄	1	84	yes	—
17	C₂³×C₅×C₃₇	C₂⁴	1	7	yes	—
18	C₂×C₄×C₅×C₃₇	C₂²×C₄	1	336	yes	direct product factor
19	C₂×C₄×C₅×C₃₇	C₂⁴	1	24	yes	—
20	C₂⁵×C₃₇	C₂²×C₅	1	1	yes	characteristic
21	C₂³×C₄×C₃₇	C₄×C₅	1	48	yes	direct product factor
22	C₂³×C₄×C₃₇	C₂²×C₅	1	42	yes	—
23	C₂×C₄²×C₃₇	C₂²×C₅	1	112	yes	direct product factor
24	C₂²×C₅×C₃₇	C₂×C₄²	1	112	yes	direct product factor
25	C₂²×C₅×C₃₇	C₂³×C₄	1	42	yes	—
26	C₂²×C₅×C₃₇	C₂⁵	1	1	yes	characteristic
27	C₄×C₅×C₃₇	C₂³×C₄	1	48	yes	direct product factor
28	C₂³×C₄²×C₅	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₂⁴×C₃₇	C₂×C₄×C₅	1	24	yes	—
30	C₂⁴×C₃₇	C₂³×C₅	1	7	yes	—
31	C₂²×C₄×C₃₇	C₂×C₄×C₅	1	336	yes	direct product factor
32	C₂²×C₄×C₃₇	C₂³×C₅	1	84	yes	—
33	C₄²×C₃₇	C₂³×C₅	1	64	yes	direct product factor
34	C₂×C₅×C₃₇	C₂²×C₄²	1	28	yes	direct product factor
35	C₂×C₅×C₃₇	C₂⁴×C₄	1	3	yes	—
36	C₂⁴×C₄×C₅	C₂×C₃₇	1	3	yes	—
37	C₂²×C₄²×C₅	C₂×C₃₇	1	28	yes	direct product factor
38	C₂³×C₃₇	C₄²×C₅	1	64	yes	direct product factor
39	C₂³×C₃₇	C₂²×C₄×C₅	1	84	yes	—
40	C₂³×C₃₇	C₂⁴×C₅	1	7	yes	—
41	C₂×C₄×C₃₇	C₂²×C₄×C₅	1	336	yes	direct product factor
42	C₂×C₄×C₃₇	C₂⁴×C₅	1	24	yes	—
43	C₅×C₃₇	C₂³×C₄²	1	1	yes	characteristic, direct product factor, Hall
44	C₂⁵×C₅	C₂²×C₃₇	1	1	yes	characteristic
45	C₂³×C₄×C₅	C₄×C₃₇	1	48	yes	direct product factor
46	C₂³×C₄×C₅	C₂²×C₃₇	1	42	yes	—
47	C₂×C₄²×C₅	C₂²×C₃₇	1	112	yes	direct product factor
48	C₂²×C₃₇	C₂×C₄²×C₅	1	112	yes	direct product factor
49	C₂²×C₃₇	C₂³×C₄×C₅	1	42	yes	—
50	C₂²×C₃₇	C₂⁵×C₅	1	1	yes	characteristic
51	C₄×C₃₇	C₂³×C₄×C₅	1	48	yes	direct product factor
52	C₂³×C₄²	C₅×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
53	C₂⁴×C₅	C₂×C₄×C₃₇	1	24	yes	—
54	C₂⁴×C₅	C₂³×C₃₇	1	7	yes	—
55	C₂²×C₄×C₅	C₂×C₄×C₃₇	1	336	yes	direct product factor
56	C₂²×C₄×C₅	C₂³×C₃₇	1	84	yes	—
57	C₄²×C₅	C₂³×C₃₇	1	64	yes	direct product factor
58	C₂×C₃₇	C₂²×C₄²×C₅	1	28	yes	direct product factor
59	C₂×C₃₇	C₂⁴×C₄×C₅	1	3	yes	—
60	C₂⁴×C₄	C₂×C₅×C₃₇	1	3	yes	—
61	C₂²×C₄²	C₂×C₅×C₃₇	1	28	yes	direct product factor
62	C₂³×C₅	C₄²×C₃₇	1	64	yes	direct product factor
63	C₂³×C₅	C₂²×C₄×C₃₇	1	84	yes	—
64	C₂³×C₅	C₂⁴×C₃₇	1	7	yes	—
65	C₂×C₄×C₅	C₂²×C₄×C₃₇	1	336	yes	direct product factor
66	C₂×C₄×C₅	C₂⁴×C₃₇	1	24	yes	—
67	C₃₇	C₂³×C₄²×C₅	1	1	yes	characteristic, direct product factor, Sylow
68	C₂⁵	C₂²×C₅×C₃₇	1	1	yes	characteristic
69	C₂³×C₄	C₄×C₅×C₃₇	1	48	yes	direct product factor
70	C₂³×C₄	C₂²×C₅×C₃₇	1	42	yes	—
71	C₂×C₄²	C₂²×C₅×C₃₇	1	112	yes	direct product factor
72	C₂²×C₅	C₂×C₄²×C₃₇	1	112	yes	direct product factor
73	C₂²×C₅	C₂³×C₄×C₃₇	1	42	yes	—
74	C₂²×C₅	C₂⁵×C₃₇	1	1	yes	characteristic
75	C₄×C₅	C₂³×C₄×C₃₇	1	48	yes	direct product factor
76	C₂⁴	C₂×C₄×C₅×C₃₇	1	24	yes	—
77	C₂⁴	C₂³×C₅×C₃₇	1	7	yes	—
78	C₂²×C₄	C₂×C₄×C₅×C₃₇	1	336	yes	direct product factor
79	C₂²×C₄	C₂³×C₅×C₃₇	1	84	yes	—
80	C₄²	C₂³×C₅×C₃₇	1	64	yes	direct product factor
81	C₂×C₅	C₂²×C₄²×C₃₇	1	28	yes	direct product factor
82	C₂×C₅	C₂⁴×C₄×C₃₇	1	3	yes	—
83	C₂³	C₄²×C₅×C₃₇	1	64	yes	direct product factor
84	C₂³	C₂²×C₄×C₅×C₃₇	1	84	yes	—
85	C₂³	C₂⁴×C₅×C₃₇	1	7	yes	—
86	C₂×C₄	C₂²×C₄×C₅×C₃₇	1	336	yes	direct product factor
87	C₂×C₄	C₂⁴×C₅×C₃₇	1	24	yes	—
88	C₅	C₂³×C₄²×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²	C₂×C₄²×C₅×C₃₇	1	112	yes	direct product factor
90	C₂²	C₂⁵×C₅×C₃₇	1	1	yes	characteristic
91	C₂²	C₂³×C₄×C₅×C₃₇	1	42	yes	—
92	C₄	C₂³×C₄×C₅×C₃₇	1	48	yes	direct product factor
93	C₂	C₂²×C₄²×C₅×C₃₇	1	28	yes	direct product factor
94	C₂	C₂⁴×C₄×C₅×C₃₇	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂⁵×C₅×C₃₇
socle, prime radical	C₂⁵×C₅×C₃₇	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄²×C₅×C₃₇	C₂	direct (N × H)
3. C₂³×C₄×C₅×C₃₇	C₄	direct (N × H)
6. C₂×C₄²×C₅×C₃₇	C₂²	direct (N × H)
9. C₂²×C₄×C₅×C₃₇	C₂×C₄	direct (N × H)
11. C₄²×C₅×C₃₇	C₂³	direct (N × H)
12. C₂³×C₄²×C₃₇	C₅	direct (N × H)
14. C₂²×C₄²×C₃₇	C₂×C₅	direct (N × H)
15. C₂³×C₅×C₃₇	C₄²	direct (N × H)
18. C₂×C₄×C₅×C₃₇	C₂²×C₄	direct (N × H)
21. C₂³×C₄×C₃₇	C₄×C₅	direct (N × H)
23. C₂×C₄²×C₃₇	C₂²×C₅	direct (N × H)
24. C₂²×C₅×C₃₇	C₂×C₄²	direct (N × H)
27. C₄×C₅×C₃₇	C₂³×C₄	direct (N × H)
28. C₂³×C₄²×C₅	C₃₇	direct (N × H)
31. C₂²×C₄×C₃₇	C₂×C₄×C₅	direct (N × H)
33. C₄²×C₃₇	C₂³×C₅	direct (N × H)
34. C₂×C₅×C₃₇	C₂²×C₄²	direct (N × H)
37. C₂²×C₄²×C₅	C₂×C₃₇	direct (N × H)
38. C₂³×C₃₇	C₄²×C₅	direct (N × H)
41. C₂×C₄×C₃₇	C₂²×C₄×C₅	direct (N × H)
43. C₅×C₃₇	C₂³×C₄²	direct (N × H)
45. C₂³×C₄×C₅	C₄×C₃₇	direct (N × H)
47. C₂×C₄²×C₅	C₂²×C₃₇	direct (N × H)
48. C₂²×C₃₇	C₂×C₄²×C₅	direct (N × H)
51. C₄×C₃₇	C₂³×C₄×C₅	direct (N × H)
52. C₂³×C₄²	C₅×C₃₇	direct (N × H)
55. C₂²×C₄×C₅	C₂×C₄×C₃₇	direct (N × H)
57. C₄²×C₅	C₂³×C₃₇	direct (N × H)
58. C₂×C₃₇	C₂²×C₄²×C₅	direct (N × H)
61. C₂²×C₄²	C₂×C₅×C₃₇	direct (N × H)
62. C₂³×C₅	C₄²×C₃₇	direct (N × H)
65. C₂×C₄×C₅	C₂²×C₄×C₃₇	direct (N × H)
67. C₃₇	C₂³×C₄²×C₅	direct (N × H)
69. C₂³×C₄	C₄×C₅×C₃₇	direct (N × H)
71. C₂×C₄²	C₂²×C₅×C₃₇	direct (N × H)
72. C₂²×C₅	C₂×C₄²×C₃₇	direct (N × H)
75. C₄×C₅	C₂³×C₄×C₃₇	direct (N × H)
78. C₂²×C₄	C₂×C₄×C₅×C₃₇	direct (N × H)
80. C₄²	C₂³×C₅×C₃₇	direct (N × H)
81. C₂×C₅	C₂²×C₄²×C₃₇	direct (N × H)
83. C₂³	C₄²×C₅×C₃₇	direct (N × H)
86. C₂×C₄	C₂²×C₄×C₅×C₃₇	direct (N × H)
88. C₅	C₂³×C₄²×C₃₇	direct (N × H)
89. C₂²	C₂×C₄²×C₅×C₃₇	direct (N × H)
92. C₄	C₂³×C₄×C₅×C₃₇	direct (N × H)
93. C₂	C₂²×C₄²×C₅×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
C₂⁴×C₄×C₅×C₃₇	2
C₂²×C₄²×C₅×C₃₇	2
C₂³×C₄²×C₃₇	5
C₂³×C₄²×C₅	37

Subgroups — C23×C42×C5×C37 #72201032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄²×C₅×C₃₇ #72201032