Subgroups — C₂×C₃×C₄×C₂₅×C₃₁ #72196401

Order 18600.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂₅×C₃₁	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₃×C₄×C₅×C₃₁	C₅	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₂₅×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×C₂₅×C₃₁	C₂	1	2	yes	maximal, direct product factor
5	C₂×C₃×C₂₅×C₃₁	C₄	1	2	yes	direct product factor
6	C₂×C₃×C₂₅×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₂₅×C₃₁	C₂×C₃	1	1	yes	characteristic
8	C₄×C₂₅×C₃₁	C₂×C₃	1	2	yes	direct product factor
9	C₃×C₂₅×C₃₁	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
10	C₂²×C₃×C₅×C₃₁	C₂×C₅	1	1	yes	characteristic
11	C₃×C₄×C₅×C₃₁	C₂×C₅	1	2	yes	—
12	C₂×C₂₅×C₃₁	C₃×C₄	1	2	yes	direct product factor
13	C₂×C₂₅×C₃₁	C₂²×C₃	1	1	yes	characteristic
14	C₂×C₄×C₅×C₃₁	C₃×C₅	1	1	yes	characteristic
15	C₂×C₃×C₅×C₃₁	C₄×C₅	1	2	yes	—
16	C₂×C₃×C₅×C₃₁	C₂²×C₅	1	1	yes	characteristic
17	C₂₅×C₃₁	C₂×C₃×C₄	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃×C₄×C₃₁	C₂₅	1	1	yes	characteristic, direct product factor, Hall
19	C₂²×C₅×C₃₁	C₂×C₃×C₅	1	1	yes	characteristic
20	C₄×C₅×C₃₁	C₂×C₃×C₅	1	2	yes	—
21	C₂×C₃×C₄×C₂₅	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₃×C₅×C₃₁	C₂×C₄×C₅	1	1	yes	characteristic
23	C₂²×C₃×C₃₁	C₂×C₂₅	1	1	yes	characteristic
24	C₃×C₄×C₃₁	C₂×C₂₅	1	2	yes	direct product factor
25	C₂×C₅×C₃₁	C₃×C₄×C₅	1	2	yes	—
26	C₂×C₅×C₃₁	C₂²×C₃×C₅	1	1	yes	characteristic
27	C₂²×C₃×C₂₅	C₂×C₃₁	1	1	yes	characteristic
28	C₃×C₄×C₂₅	C₂×C₃₁	1	2	yes	direct product factor
29	C₂×C₄×C₃₁	C₃×C₂₅	1	1	yes	characteristic, direct product factor, Hall
30	C₂×C₄×C₂₅	C₃×C₃₁	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₃×C₃₁	C₄×C₂₅	1	2	yes	direct product factor
32	C₂×C₃×C₃₁	C₂²×C₂₅	1	1	yes	characteristic
33	C₅×C₃₁	C₂×C₃×C₄×C₅	1	1	yes	characteristic
34	C₂×C₃×C₂₅	C₄×C₃₁	1	2	yes	direct product factor
35	C₂×C₃×C₂₅	C₂²×C₃₁	1	1	yes	characteristic
36	C₂²×C₃₁	C₂×C₃×C₂₅	1	1	yes	characteristic
37	C₄×C₃₁	C₂×C₃×C₂₅	1	2	yes	direct product factor
38	C₂×C₃×C₄×C₅	C₅×C₃₁	1	1	yes	characteristic
39	C₂²×C₂₅	C₂×C₃×C₃₁	1	1	yes	characteristic
40	C₄×C₂₅	C₂×C₃×C₃₁	1	2	yes	direct product factor
41	C₃×C₃₁	C₂×C₄×C₂₅	1	1	yes	characteristic, direct product factor, Hall
42	C₃×C₂₅	C₂×C₄×C₃₁	1	1	yes	characteristic, direct product factor, Hall
43	C₂×C₃₁	C₃×C₄×C₂₅	1	2	yes	direct product factor
44	C₂×C₃₁	C₂²×C₃×C₂₅	1	1	yes	characteristic
45	C₂²×C₃×C₅	C₂×C₅×C₃₁	1	1	yes	characteristic
46	C₃×C₄×C₅	C₂×C₅×C₃₁	1	2	yes	—
47	C₂×C₂₅	C₃×C₄×C₃₁	1	2	yes	direct product factor
48	C₂×C₂₅	C₂²×C₃×C₃₁	1	1	yes	characteristic
49	C₂×C₄×C₅	C₃×C₅×C₃₁	1	1	yes	characteristic
50	C₃₁	C₂×C₃×C₄×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₃×C₅	C₄×C₅×C₃₁	1	2	yes	—
52	C₂×C₃×C₅	C₂²×C₅×C₃₁	1	1	yes	characteristic
53	C₂₅	C₂×C₃×C₄×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
54	C₂×C₃×C₄	C₂₅×C₃₁	1	1	yes	characteristic, direct product factor, Hall
55	C₂²×C₅	C₂×C₃×C₅×C₃₁	1	1	yes	characteristic
56	C₄×C₅	C₂×C₃×C₅×C₃₁	1	2	yes	—
57	C₃×C₅	C₂×C₄×C₅×C₃₁	1	1	yes	characteristic
58	C₂²×C₃	C₂×C₂₅×C₃₁	1	1	yes	characteristic
59	C₃×C₄	C₂×C₂₅×C₃₁	1	2	yes	direct product factor
60	C₂×C₅	C₃×C₄×C₅×C₃₁	1	2	yes	—
61	C₂×C₅	C₂²×C₃×C₅×C₃₁	1	1	yes	characteristic
62	C₂×C₄	C₃×C₂₅×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×C₃	C₄×C₂₅×C₃₁	1	2	yes	direct product factor
64	C₂×C₃	C₂²×C₂₅×C₃₁	1	1	yes	characteristic
65	C₅	C₂×C₃×C₄×C₅×C₃₁	1	1	yes	characteristic
66	C₂²	C₂×C₃×C₂₅×C₃₁	1	1	yes	characteristic
67	C₄	C₂×C₃×C₂₅×C₃₁	1	2	yes	direct product factor
68	C₃	C₂×C₄×C₂₅×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
69	C₂	C₃×C₄×C₂₅×C₃₁	1	2	yes	direct product factor
70	C₂	C₂²×C₃×C₂₅×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₅	C₂²×C₃×C₅×C₃₁
socle, prime radical	C₂²×C₃×C₅×C₃₁	C₂×C₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×C₂₅×C₃₁	C₃	direct (N × H)
4. C₃×C₄×C₂₅×C₃₁	C₂	direct (N × H)
5. C₂×C₃×C₂₅×C₃₁	C₄	direct (N × H)
8. C₄×C₂₅×C₃₁	C₂×C₃	direct (N × H)
9. C₃×C₂₅×C₃₁	C₂×C₄	direct (N × H)
12. C₂×C₂₅×C₃₁	C₃×C₄	direct (N × H)
17. C₂₅×C₃₁	C₂×C₃×C₄	direct (N × H)
18. C₂×C₃×C₄×C₃₁	C₂₅	direct (N × H)
21. C₂×C₃×C₄×C₂₅	C₃₁	direct (N × H)
24. C₃×C₄×C₃₁	C₂×C₂₅	direct (N × H)
28. C₃×C₄×C₂₅	C₂×C₃₁	direct (N × H)
29. C₂×C₄×C₃₁	C₃×C₂₅	direct (N × H)
30. C₂×C₄×C₂₅	C₃×C₃₁	direct (N × H)
31. C₂×C₃×C₃₁	C₄×C₂₅	direct (N × H)
34. C₂×C₃×C₂₅	C₄×C₃₁	direct (N × H)
37. C₄×C₃₁	C₂×C₃×C₂₅	direct (N × H)
40. C₄×C₂₅	C₂×C₃×C₃₁	direct (N × H)
41. C₃×C₃₁	C₂×C₄×C₂₅	direct (N × H)
42. C₃×C₂₅	C₂×C₄×C₃₁	direct (N × H)
43. C₂×C₃₁	C₃×C₄×C₂₅	direct (N × H)
47. C₂×C₂₅	C₃×C₄×C₃₁	direct (N × H)
50. C₃₁	C₂×C₃×C₄×C₂₅	direct (N × H)
53. C₂₅	C₂×C₃×C₄×C₃₁	direct (N × H)
54. C₂×C₃×C₄	C₂₅×C₃₁	direct (N × H)
59. C₃×C₄	C₂×C₂₅×C₃₁	direct (N × H)
62. C₂×C₄	C₃×C₂₅×C₃₁	direct (N × H)
63. C₂×C₃	C₄×C₂₅×C₃₁	direct (N × H)
67. C₄	C₂×C₃×C₂₅×C₃₁	direct (N × H)
68. C₃	C₂×C₄×C₂₅×C₃₁	direct (N × H)
69. C₂	C₃×C₄×C₂₅×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
C₂×C₄×C₂₅×C₃₁	3
C₂×C₃×C₄×C₅×C₃₁	5
C₂²×C₃×C₂₅×C₃₁	2
C₃×C₄×C₂₅×C₃₁	2
C₂×C₃×C₄×C₂₅	31

Subgroups — C2×C3×C4×C25×C31 #72196401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×C₂₅×C₃₁ #72196401