Subgroups — C₂×C₄³×C₅×C₃₇ #72201031

Order 23680.

2544 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₅×C₃₇	C₂	1	7	yes	maximal
2	C₄³×C₅×C₃₇	C₂	1	8	yes	maximal, direct product factor
3	C₂³×C₄×C₅×C₃₇	C₂²	1	7	yes	—
4	C₂×C₄²×C₅×C₃₇	C₄	1	56	yes	direct product factor
5	C₂×C₄²×C₅×C₃₇	C₂²	1	28	yes	—
6	C₂⁴×C₅×C₃₇	C₂³	1	1	yes	characteristic
7	C₂²×C₄×C₅×C₃₇	C₂×C₄	1	84	yes	—
8	C₂²×C₄×C₅×C₃₇	C₂³	1	14	yes	—
9	C₄²×C₅×C₃₇	C₂×C₄	1	112	yes	direct product factor
10	C₂×C₄³×C₃₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₄²×C₃₇	C₂×C₅	1	7	yes	—
12	C₄³×C₃₇	C₂×C₅	1	8	yes	direct product factor
13	C₂³×C₅×C₃₇	C₂²×C₄	1	14	yes	—
14	C₂³×C₅×C₃₇	C₂⁴	1	1	yes	characteristic
15	C₂×C₄×C₅×C₃₇	C₄²	1	112	yes	direct product factor
16	C₂×C₄×C₅×C₃₇	C₂²×C₄	1	84	yes	—
17	C₂³×C₄×C₃₇	C₂²×C₅	1	7	yes	—
18	C₂×C₄²×C₃₇	C₄×C₅	1	56	yes	direct product factor
19	C₂×C₄²×C₃₇	C₂²×C₅	1	28	yes	—
20	C₂²×C₅×C₃₇	C₂×C₄²	1	28	yes	—
21	C₂²×C₅×C₃₇	C₂³×C₄	1	7	yes	—
22	C₄×C₅×C₃₇	C₂×C₄²	1	56	yes	direct product factor
23	C₂×C₄³×C₅	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂⁴×C₃₇	C₂³×C₅	1	1	yes	characteristic
25	C₂²×C₄×C₃₇	C₂×C₄×C₅	1	84	yes	—
26	C₂²×C₄×C₃₇	C₂³×C₅	1	14	yes	—
27	C₄²×C₃₇	C₂×C₄×C₅	1	112	yes	direct product factor
28	C₂×C₅×C₃₇	C₄³	1	8	yes	direct product factor
29	C₂×C₅×C₃₇	C₂²×C₄²	1	7	yes	—
30	C₂²×C₄²×C₅	C₂×C₃₇	1	7	yes	—
31	C₄³×C₅	C₂×C₃₇	1	8	yes	direct product factor
32	C₂³×C₃₇	C₂²×C₄×C₅	1	14	yes	—
33	C₂³×C₃₇	C₂⁴×C₅	1	1	yes	characteristic
34	C₂×C₄×C₃₇	C₄²×C₅	1	112	yes	direct product factor
35	C₂×C₄×C₃₇	C₂²×C₄×C₅	1	84	yes	—
36	C₅×C₃₇	C₂×C₄³	1	1	yes	characteristic, direct product factor, Hall
37	C₂³×C₄×C₅	C₂²×C₃₇	1	7	yes	—
38	C₂×C₄²×C₅	C₄×C₃₇	1	56	yes	direct product factor
39	C₂×C₄²×C₅	C₂²×C₃₇	1	28	yes	—
40	C₂²×C₃₇	C₂×C₄²×C₅	1	28	yes	—
41	C₂²×C₃₇	C₂³×C₄×C₅	1	7	yes	—
42	C₄×C₃₇	C₂×C₄²×C₅	1	56	yes	direct product factor
43	C₂×C₄³	C₅×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
44	C₂⁴×C₅	C₂³×C₃₇	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×C₄×C₃₇	1	84	yes	—
46	C₂²×C₄×C₅	C₂³×C₃₇	1	14	yes	—
47	C₄²×C₅	C₂×C₄×C₃₇	1	112	yes	direct product factor
48	C₂×C₃₇	C₄³×C₅	1	8	yes	direct product factor
49	C₂×C₃₇	C₂²×C₄²×C₅	1	7	yes	—
50	C₂²×C₄²	C₂×C₅×C₃₇	1	7	yes	—
51	C₄³	C₂×C₅×C₃₇	1	8	yes	direct product factor
52	C₂³×C₅	C₂²×C₄×C₃₇	1	14	yes	—
53	C₂³×C₅	C₂⁴×C₃₇	1	1	yes	characteristic
54	C₂×C₄×C₅	C₄²×C₃₇	1	112	yes	direct product factor
55	C₂×C₄×C₅	C₂²×C₄×C₃₇	1	84	yes	—
56	C₃₇	C₂×C₄³×C₅	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₄	C₂²×C₅×C₃₇	1	7	yes	—
58	C₂×C₄²	C₄×C₅×C₃₇	1	56	yes	direct product factor
59	C₂×C₄²	C₂²×C₅×C₃₇	1	28	yes	—
60	C₂²×C₅	C₂×C₄²×C₃₇	1	28	yes	—
61	C₂²×C₅	C₂³×C₄×C₃₇	1	7	yes	—
62	C₄×C₅	C₂×C₄²×C₃₇	1	56	yes	direct product factor
63	C₂⁴	C₂³×C₅×C₃₇	1	1	yes	characteristic
64	C₂²×C₄	C₂×C₄×C₅×C₃₇	1	84	yes	—
65	C₂²×C₄	C₂³×C₅×C₃₇	1	14	yes	—
66	C₄²	C₂×C₄×C₅×C₃₇	1	112	yes	direct product factor
67	C₂×C₅	C₄³×C₃₇	1	8	yes	direct product factor
68	C₂×C₅	C₂²×C₄²×C₃₇	1	7	yes	—
69	C₂³	C₂²×C₄×C₅×C₃₇	1	14	yes	—
70	C₂³	C₂⁴×C₅×C₃₇	1	1	yes	characteristic
71	C₂×C₄	C₄²×C₅×C₃₇	1	112	yes	direct product factor
72	C₂×C₄	C₂²×C₄×C₅×C₃₇	1	84	yes	—
73	C₅	C₂×C₄³×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²	C₂×C₄²×C₅×C₃₇	1	28	yes	—
75	C₂²	C₂³×C₄×C₅×C₃₇	1	7	yes	—
76	C₄	C₂×C₄²×C₅×C₃₇	1	56	yes	direct product factor
77	C₂	C₄³×C₅×C₃₇	1	8	yes	direct product factor
78	C₂	C₂²×C₄²×C₅×C₃₇	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂³	C₂⁴×C₅×C₃₇
socle, prime radical	C₂⁴×C₅×C₃₇	C₂³
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄³×C₅×C₃₇	C₂	direct (N × H)
4. C₂×C₄²×C₅×C₃₇	C₄	direct (N × H)
9. C₄²×C₅×C₃₇	C₂×C₄	direct (N × H)
10. C₂×C₄³×C₃₇	C₅	direct (N × H)
12. C₄³×C₃₇	C₂×C₅	direct (N × H)
15. C₂×C₄×C₅×C₃₇	C₄²	direct (N × H)
18. C₂×C₄²×C₃₇	C₄×C₅	direct (N × H)
22. C₄×C₅×C₃₇	C₂×C₄²	direct (N × H)
23. C₂×C₄³×C₅	C₃₇	direct (N × H)
27. C₄²×C₃₇	C₂×C₄×C₅	direct (N × H)
28. C₂×C₅×C₃₇	C₄³	direct (N × H)
31. C₄³×C₅	C₂×C₃₇	direct (N × H)
34. C₂×C₄×C₃₇	C₄²×C₅	direct (N × H)
36. C₅×C₃₇	C₂×C₄³	direct (N × H)
38. C₂×C₄²×C₅	C₄×C₃₇	direct (N × H)
42. C₄×C₃₇	C₂×C₄²×C₅	direct (N × H)
43. C₂×C₄³	C₅×C₃₇	direct (N × H)
47. C₄²×C₅	C₂×C₄×C₃₇	direct (N × H)
48. C₂×C₃₇	C₄³×C₅	direct (N × H)
51. C₄³	C₂×C₅×C₃₇	direct (N × H)
54. C₂×C₄×C₅	C₄²×C₃₇	direct (N × H)
56. C₃₇	C₂×C₄³×C₅	direct (N × H)
58. C₂×C₄²	C₄×C₅×C₃₇	direct (N × H)
62. C₄×C₅	C₂×C₄²×C₃₇	direct (N × H)
66. C₄²	C₂×C₄×C₅×C₃₇	direct (N × H)
67. C₂×C₅	C₄³×C₃₇	direct (N × H)
71. C₂×C₄	C₄²×C₅×C₃₇	direct (N × H)
73. C₅	C₂×C₄³×C₃₇	direct (N × H)
76. C₄	C₂×C₄²×C₅×C₃₇	direct (N × H)
77. C₂	C₄³×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
C₂²×C₄²×C₅×C₃₇	2
C₄³×C₅×C₃₇	2
C₂×C₄³×C₃₇	5
C₂×C₄³×C₅	37

Subgroups — C2×C43×C5×C37 #72201031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄³×C₅×C₃₇ #72201031