Subgroups — C₂×C₄²×C₅²×C₃₁ #72202095

Order 24800.

864 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅²×C₃₁	C₂	1	3	yes	maximal
2	C₄²×C₅²×C₃₁	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₅²×C₃₁	C₂²	1	1	yes	characteristic
4	C₂×C₄×C₅²×C₃₁	C₄	1	12	yes	direct product factor
5	C₂×C₄×C₅²×C₃₁	C₂²	1	6	yes	—
6	C₂×C₄²×C₅×C₃₁	C₅	1	6	yes	maximal, direct product factor
7	C₂²×C₄×C₅×C₃₁	C₂×C₅	1	18	yes	—
8	C₄²×C₅×C₃₁	C₂×C₅	1	24	yes	direct product factor
9	C₂²×C₅²×C₃₁	C₂×C₄	1	6	yes	—
10	C₂²×C₅²×C₃₁	C₂³	1	1	yes	characteristic
11	C₄×C₅²×C₃₁	C₂×C₄	1	12	yes	direct product factor
12	C₂×C₅²×C₃₁	C₄²	1	4	yes	direct product factor
13	C₂×C₅²×C₃₁	C₂²×C₄	1	3	yes	—
14	C₂³×C₅×C₃₁	C₂²×C₅	1	6	yes	—
15	C₂×C₄×C₅×C₃₁	C₄×C₅	1	72	yes	direct product factor
16	C₂×C₄×C₅×C₃₁	C₂²×C₅	1	36	yes	—
17	C₂×C₄²×C₃₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₄²×C₅²	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅²×C₃₁	C₂×C₄²	1	1	yes	characteristic, direct product factor, Hall
20	C₂²×C₅×C₃₁	C₂×C₄×C₅	1	36	yes	—
21	C₂²×C₅×C₃₁	C₂³×C₅	1	6	yes	—
22	C₄×C₅×C₃₁	C₂×C₄×C₅	1	72	yes	direct product factor
23	C₂²×C₄×C₃₁	C₂×C₅²	1	3	yes	—
24	C₄²×C₃₁	C₂×C₅²	1	4	yes	direct product factor
25	C₂²×C₄×C₅²	C₂×C₃₁	1	3	yes	—
26	C₄²×C₅²	C₂×C₃₁	1	4	yes	direct product factor
27	C₂×C₅×C₃₁	C₄²×C₅	1	24	yes	direct product factor
28	C₂×C₅×C₃₁	C₂²×C₄×C₅	1	18	yes	—
29	C₂³×C₃₁	C₂²×C₅²	1	1	yes	characteristic
30	C₂×C₄×C₃₁	C₄×C₅²	1	12	yes	direct product factor
31	C₂×C₄×C₃₁	C₂²×C₅²	1	6	yes	—
32	C₂³×C₅²	C₂²×C₃₁	1	1	yes	characteristic
33	C₂×C₄×C₅²	C₄×C₃₁	1	12	yes	direct product factor
34	C₂×C₄×C₅²	C₂²×C₃₁	1	6	yes	—
35	C₂×C₄²×C₅	C₅×C₃₁	1	6	yes	direct product factor
36	C₅×C₃₁	C₂×C₄²×C₅	1	6	yes	direct product factor
37	C₂²×C₃₁	C₂×C₄×C₅²	1	6	yes	—
38	C₂²×C₃₁	C₂³×C₅²	1	1	yes	characteristic
39	C₄×C₃₁	C₂×C₄×C₅²	1	12	yes	direct product factor
40	C₂²×C₅²	C₂×C₄×C₃₁	1	6	yes	—
41	C₂²×C₅²	C₂³×C₃₁	1	1	yes	characteristic
42	C₄×C₅²	C₂×C₄×C₃₁	1	12	yes	direct product factor
43	C₂²×C₄×C₅	C₂×C₅×C₃₁	1	18	yes	—
44	C₄²×C₅	C₂×C₅×C₃₁	1	24	yes	direct product factor
45	C₂×C₃₁	C₄²×C₅²	1	4	yes	direct product factor
46	C₂×C₃₁	C₂²×C₄×C₅²	1	3	yes	—
47	C₂×C₅²	C₄²×C₃₁	1	4	yes	direct product factor
48	C₂×C₅²	C₂²×C₄×C₃₁	1	3	yes	—
49	C₂³×C₅	C₂²×C₅×C₃₁	1	6	yes	—
50	C₂×C₄×C₅	C₄×C₅×C₃₁	1	72	yes	direct product factor
51	C₂×C₄×C₅	C₂²×C₅×C₃₁	1	36	yes	—
52	C₂×C₄²	C₅²×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
53	C₃₁	C₂×C₄²×C₅²	1	1	yes	characteristic, direct product factor, Sylow
54	C₅²	C₂×C₄²×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₅	C₂×C₄×C₅×C₃₁	1	36	yes	—
56	C₂²×C₅	C₂³×C₅×C₃₁	1	6	yes	—
57	C₄×C₅	C₂×C₄×C₅×C₃₁	1	72	yes	direct product factor
58	C₂²×C₄	C₂×C₅²×C₃₁	1	3	yes	—
59	C₄²	C₂×C₅²×C₃₁	1	4	yes	direct product factor
60	C₂×C₅	C₄²×C₅×C₃₁	1	24	yes	direct product factor
61	C₂×C₅	C₂²×C₄×C₅×C₃₁	1	18	yes	—
62	C₂³	C₂²×C₅²×C₃₁	1	1	yes	characteristic
63	C₂×C₄	C₄×C₅²×C₃₁	1	12	yes	direct product factor
64	C₂×C₄	C₂²×C₅²×C₃₁	1	6	yes	—
65	C₅	C₂×C₄²×C₅×C₃₁	1	6	yes	direct product factor
66	C₂²	C₂×C₄×C₅²×C₃₁	1	6	yes	—
67	C₂²	C₂³×C₅²×C₃₁	1	1	yes	characteristic
68	C₄	C₂×C₄×C₅²×C₃₁	1	12	yes	direct product factor
69	C₂	C₄²×C₅²×C₃₁	1	4	yes	direct product factor
70	C₂	C₂²×C₄×C₅²×C₃₁	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂³×C₅²×C₃₁
socle, prime radical	C₂³×C₅²×C₃₁	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄²×C₅²×C₃₁	C₂	direct (N × H)
4. C₂×C₄×C₅²×C₃₁	C₄	direct (N × H)
6. C₂×C₄²×C₅×C₃₁	C₅	direct (N × H)
8. C₄²×C₅×C₃₁	C₂×C₅	direct (N × H)
11. C₄×C₅²×C₃₁	C₂×C₄	direct (N × H)
12. C₂×C₅²×C₃₁	C₄²	direct (N × H)
15. C₂×C₄×C₅×C₃₁	C₄×C₅	direct (N × H)
17. C₂×C₄²×C₃₁	C₅²	direct (N × H)
18. C₂×C₄²×C₅²	C₃₁	direct (N × H)
19. C₅²×C₃₁	C₂×C₄²	direct (N × H)
22. C₄×C₅×C₃₁	C₂×C₄×C₅	direct (N × H)
24. C₄²×C₃₁	C₂×C₅²	direct (N × H)
26. C₄²×C₅²	C₂×C₃₁	direct (N × H)
27. C₂×C₅×C₃₁	C₄²×C₅	direct (N × H)
30. C₂×C₄×C₃₁	C₄×C₅²	direct (N × H)
33. C₂×C₄×C₅²	C₄×C₃₁	direct (N × H)
35. C₂×C₄²×C₅	C₅×C₃₁	direct (N × H)
36. C₅×C₃₁	C₂×C₄²×C₅	direct (N × H)
39. C₄×C₃₁	C₂×C₄×C₅²	direct (N × H)
42. C₄×C₅²	C₂×C₄×C₃₁	direct (N × H)
44. C₄²×C₅	C₂×C₅×C₃₁	direct (N × H)
45. C₂×C₃₁	C₄²×C₅²	direct (N × H)
47. C₂×C₅²	C₄²×C₃₁	direct (N × H)
50. C₂×C₄×C₅	C₄×C₅×C₃₁	direct (N × H)
52. C₂×C₄²	C₅²×C₃₁	direct (N × H)
53. C₃₁	C₂×C₄²×C₅²	direct (N × H)
54. C₅²	C₂×C₄²×C₃₁	direct (N × H)
57. C₄×C₅	C₂×C₄×C₅×C₃₁	direct (N × H)
59. C₄²	C₂×C₅²×C₃₁	direct (N × H)
60. C₂×C₅	C₄²×C₅×C₃₁	direct (N × H)
63. C₂×C₄	C₄×C₅²×C₃₁	direct (N × H)
65. C₅	C₂×C₄²×C₅×C₃₁	direct (N × H)
68. C₄	C₂×C₄×C₅²×C₃₁	direct (N × H)
69. C₂	C₄²×C₅²×C₃₁	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×C₄×C₅²×C₃₁	2
C₄²×C₅²×C₃₁	2
C₂×C₄²×C₅×C₃₁	5
C₂×C₄²×C₅²	31

Subgroups — C2×C42×C52×C31 #72202095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×C₅²×C₃₁ #72202095