Subgroups — C₄×C₅²×C₈×C₃₁ #72202091

Order 24800.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅²×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×C₈×C₃₁	C₂	1	2	yes	maximal
3	C₂×C₄×C₅²×C₃₁	C₄	1	2	yes	—
4	C₂×C₄×C₅²×C₃₁	C₂²	1	1	yes	characteristic
5	C₅²×C₈×C₃₁	C₄	1	4	yes	direct product factor
6	C₄×C₅×C₈×C₃₁	C₅	1	6	yes	maximal, direct product factor
7	C₄²×C₅×C₃₁	C₂×C₅	1	6	yes	—
8	C₂×C₅×C₈×C₃₁	C₂×C₅	1	12	yes	—
9	C₂²×C₅²×C₃₁	C₂×C₄	1	1	yes	characteristic
10	C₄×C₅²×C₃₁	C₈	1	4	yes	direct product factor
11	C₄×C₅²×C₃₁	C₂×C₄	1	2	yes	—
12	C₂×C₅²×C₃₁	C₄²	1	1	yes	characteristic
13	C₂×C₅²×C₃₁	C₂×C₈	1	2	yes	—
14	C₂×C₄×C₅×C₃₁	C₄×C₅	1	12	yes	—
15	C₂×C₄×C₅×C₃₁	C₂²×C₅	1	6	yes	—
16	C₅×C₈×C₃₁	C₄×C₅	1	24	yes	direct product factor
17	C₄×C₈×C₃₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
18	C₄×C₅²×C₈	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅²×C₃₁	C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
20	C₂²×C₅×C₃₁	C₂×C₄×C₅	1	6	yes	—
21	C₄×C₅×C₃₁	C₅×C₈	1	24	yes	direct product factor
22	C₄×C₅×C₃₁	C₂×C₄×C₅	1	12	yes	—
23	C₂×C₈×C₃₁	C₂×C₅²	1	2	yes	—
24	C₄²×C₃₁	C₂×C₅²	1	1	yes	characteristic
25	C₂×C₅²×C₈	C₂×C₃₁	1	2	yes	—
26	C₄²×C₅²	C₂×C₃₁	1	1	yes	characteristic
27	C₂×C₅×C₃₁	C₄²×C₅	1	6	yes	—
28	C₂×C₅×C₃₁	C₂×C₅×C₈	1	12	yes	—
29	C₂×C₄×C₃₁	C₄×C₅²	1	2	yes	—
30	C₂×C₄×C₃₁	C₂²×C₅²	1	1	yes	characteristic
31	C₈×C₃₁	C₄×C₅²	1	4	yes	direct product factor
32	C₂×C₄×C₅²	C₄×C₃₁	1	2	yes	—
33	C₂×C₄×C₅²	C₂²×C₃₁	1	1	yes	characteristic
34	C₅²×C₈	C₄×C₃₁	1	4	yes	direct product factor
35	C₄×C₅×C₈	C₅×C₃₁	1	6	yes	direct product factor
36	C₅×C₃₁	C₄×C₅×C₈	1	6	yes	direct product factor
37	C₂²×C₃₁	C₂×C₄×C₅²	1	1	yes	characteristic
38	C₄×C₃₁	C₅²×C₈	1	4	yes	direct product factor
39	C₄×C₃₁	C₂×C₄×C₅²	1	2	yes	—
40	C₂²×C₅²	C₂×C₄×C₃₁	1	1	yes	characteristic
41	C₄×C₅²	C₈×C₃₁	1	4	yes	direct product factor
42	C₄×C₅²	C₂×C₄×C₃₁	1	2	yes	—
43	C₂×C₅×C₈	C₂×C₅×C₃₁	1	12	yes	—
44	C₄²×C₅	C₂×C₅×C₃₁	1	6	yes	—
45	C₂×C₃₁	C₄²×C₅²	1	1	yes	characteristic
46	C₂×C₃₁	C₂×C₅²×C₈	1	2	yes	—
47	C₂×C₅²	C₄²×C₃₁	1	1	yes	characteristic
48	C₂×C₅²	C₂×C₈×C₃₁	1	2	yes	—
49	C₂×C₄×C₅	C₄×C₅×C₃₁	1	12	yes	—
50	C₂×C₄×C₅	C₂²×C₅×C₃₁	1	6	yes	—
51	C₅×C₈	C₄×C₅×C₃₁	1	24	yes	direct product factor
52	C₄×C₈	C₅²×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
53	C₃₁	C₄×C₅²×C₈	1	1	yes	characteristic, direct product factor, Sylow
54	C₅²	C₄×C₈×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₅	C₂×C₄×C₅×C₃₁	1	6	yes	—
56	C₄×C₅	C₅×C₈×C₃₁	1	24	yes	direct product factor
57	C₄×C₅	C₂×C₄×C₅×C₃₁	1	12	yes	—
58	C₂×C₈	C₂×C₅²×C₃₁	1	2	yes	—
59	C₄²	C₂×C₅²×C₃₁	1	1	yes	characteristic
60	C₂×C₅	C₂×C₅×C₈×C₃₁	1	12	yes	—
61	C₂×C₅	C₄²×C₅×C₃₁	1	6	yes	—
62	C₂×C₄	C₄×C₅²×C₃₁	1	2	yes	—
63	C₂×C₄	C₂²×C₅²×C₃₁	1	1	yes	characteristic
64	C₈	C₄×C₅²×C₃₁	1	4	yes	direct product factor
65	C₅	C₄×C₅×C₈×C₃₁	1	6	yes	direct product factor
66	C₂²	C₂×C₄×C₅²×C₃₁	1	1	yes	characteristic
67	C₄	C₅²×C₈×C₃₁	1	4	yes	direct product factor
68	C₄	C₂×C₄×C₅²×C₃₁	1	2	yes	—
69	C₂	C₂×C₅²×C₈×C₃₁	1	2	yes	—
70	C₂	C₄²×C₅²×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄	C₂²×C₅²×C₃₁
socle, prime radical	C₂²×C₅²×C₃₁	C₂×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅²×C₈×C₃₁	C₄	direct (N × H)
6. C₄×C₅×C₈×C₃₁	C₅	direct (N × H)
10. C₄×C₅²×C₃₁	C₈	direct (N × H)
16. C₅×C₈×C₃₁	C₄×C₅	direct (N × H)
17. C₄×C₈×C₃₁	C₅²	direct (N × H)
18. C₄×C₅²×C₈	C₃₁	direct (N × H)
19. C₅²×C₃₁	C₄×C₈	direct (N × H)
21. C₄×C₅×C₃₁	C₅×C₈	direct (N × H)
31. C₈×C₃₁	C₄×C₅²	direct (N × H)
34. C₅²×C₈	C₄×C₃₁	direct (N × H)
35. C₄×C₅×C₈	C₅×C₃₁	direct (N × H)
36. C₅×C₃₁	C₄×C₅×C₈	direct (N × H)
38. C₄×C₃₁	C₅²×C₈	direct (N × H)
41. C₄×C₅²	C₈×C₃₁	direct (N × H)
51. C₅×C₈	C₄×C₅×C₃₁	direct (N × H)
52. C₄×C₈	C₅²×C₃₁	direct (N × H)
53. C₃₁	C₄×C₅²×C₈	direct (N × H)
54. C₅²	C₄×C₈×C₃₁	direct (N × H)
56. C₄×C₅	C₅×C₈×C₃₁	direct (N × H)
64. C₈	C₄×C₅²×C₃₁	direct (N × H)
65. C₅	C₄×C₅×C₈×C₃₁	direct (N × H)
67. C₄	C₅²×C₈×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
C₄²×C₅²×C₃₁	2
C₂×C₅²×C₈×C₃₁	2
C₄×C₅×C₈×C₃₁	5
C₄×C₅²×C₈	31

Subgroups — C4×C52×C8×C31 #72202091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅²×C₈×C₃₁ #72202091