Subgroups — C₂⁵×C₄²×C₆₁ #72208043

Order 31232.

259716 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×C₄×C₆₁	C₂	1	3	yes	maximal
2	C₂⁴×C₄²×C₆₁	C₂	1	124	yes	maximal, direct product factor
3	C₂⁷×C₆₁	C₂²	1	1	yes	characteristic
4	C₂⁵×C₄×C₆₁	C₄	1	192	yes	direct product factor
5	C₂⁵×C₄×C₆₁	C₂²	1	186	yes	—
6	C₂³×C₄²×C₆₁	C₂²	1	2480	yes	direct product factor
7	C₂⁶×C₆₁	C₂×C₄	1	96	yes	—
8	C₂⁶×C₆₁	C₂³	1	31	yes	—
9	C₂⁴×C₄×C₆₁	C₂×C₄	1	5952	yes	direct product factor
10	C₂⁴×C₄×C₆₁	C₂³	1	1860	yes	—
11	C₂²×C₄²×C₆₁	C₂³	1	9920	yes	direct product factor
12	C₂⁵×C₆₁	C₄²	1	1024	yes	direct product factor
13	C₂⁵×C₆₁	C₂²×C₄	1	1488	yes	—
14	C₂⁵×C₆₁	C₂⁴	1	155	yes	—
15	C₂³×C₄×C₆₁	C₂²×C₄	1	29760	yes	direct product factor
16	C₂³×C₄×C₆₁	C₂⁴	1	3720	yes	—
17	C₂×C₄²×C₆₁	C₂⁴	1	7936	yes	direct product factor
18	C₂⁴×C₆₁	C₂×C₄²	1	7936	yes	direct product factor
19	C₂⁴×C₆₁	C₂³×C₄	1	3720	yes	—
20	C₂⁴×C₆₁	C₂⁵	1	155	yes	—
21	C₂²×C₄×C₆₁	C₂³×C₄	1	29760	yes	direct product factor
22	C₂²×C₄×C₆₁	C₂⁵	1	1488	yes	—
23	C₄²×C₆₁	C₂⁵	1	1024	yes	direct product factor
24	C₂⁵×C₄²	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₆₁	C₂²×C₄²	1	9920	yes	direct product factor
26	C₂³×C₆₁	C₂⁴×C₄	1	1860	yes	—
27	C₂³×C₆₁	C₂⁶	1	31	yes	—
28	C₂×C₄×C₆₁	C₂⁴×C₄	1	5952	yes	direct product factor
29	C₂×C₄×C₆₁	C₂⁶	1	96	yes	—
30	C₂⁶×C₄	C₂×C₆₁	1	3	yes	—
31	C₂⁴×C₄²	C₂×C₆₁	1	124	yes	direct product factor
32	C₂²×C₆₁	C₂³×C₄²	1	2480	yes	direct product factor
33	C₂²×C₆₁	C₂⁵×C₄	1	186	yes	—
34	C₂²×C₆₁	C₂⁷	1	1	yes	characteristic
35	C₄×C₆₁	C₂⁵×C₄	1	192	yes	direct product factor
36	C₂⁷	C₂²×C₆₁	1	1	yes	characteristic
37	C₂⁵×C₄	C₄×C₆₁	1	192	yes	direct product factor
38	C₂⁵×C₄	C₂²×C₆₁	1	186	yes	—
39	C₂³×C₄²	C₂²×C₆₁	1	2480	yes	direct product factor
40	C₂×C₆₁	C₂⁴×C₄²	1	124	yes	direct product factor
41	C₂×C₆₁	C₂⁶×C₄	1	3	yes	—
42	C₂⁶	C₂×C₄×C₆₁	1	96	yes	—
43	C₂⁶	C₂³×C₆₁	1	31	yes	—
44	C₂⁴×C₄	C₂×C₄×C₆₁	1	5952	yes	direct product factor
45	C₂⁴×C₄	C₂³×C₆₁	1	1860	yes	—
46	C₂²×C₄²	C₂³×C₆₁	1	9920	yes	direct product factor
47	C₆₁	C₂⁵×C₄²	1	1	yes	characteristic, direct product factor, Sylow
48	C₂⁵	C₄²×C₆₁	1	1024	yes	direct product factor
49	C₂⁵	C₂²×C₄×C₆₁	1	1488	yes	—
50	C₂⁵	C₂⁴×C₆₁	1	155	yes	—
51	C₂³×C₄	C₂²×C₄×C₆₁	1	29760	yes	direct product factor
52	C₂³×C₄	C₂⁴×C₆₁	1	3720	yes	—
53	C₂×C₄²	C₂⁴×C₆₁	1	7936	yes	direct product factor
54	C₂⁴	C₂×C₄²×C₆₁	1	7936	yes	direct product factor
55	C₂⁴	C₂³×C₄×C₆₁	1	3720	yes	—
56	C₂⁴	C₂⁵×C₆₁	1	155	yes	—
57	C₂²×C₄	C₂³×C₄×C₆₁	1	29760	yes	direct product factor
58	C₂²×C₄	C₂⁵×C₆₁	1	1488	yes	—
59	C₄²	C₂⁵×C₆₁	1	1024	yes	direct product factor
60	C₂³	C₂²×C₄²×C₆₁	1	9920	yes	direct product factor
61	C₂³	C₂⁴×C₄×C₆₁	1	1860	yes	—
62	C₂³	C₂⁶×C₆₁	1	31	yes	—
63	C₂×C₄	C₂⁴×C₄×C₆₁	1	5952	yes	direct product factor
64	C₂×C₄	C₂⁶×C₆₁	1	96	yes	—
65	C₂²	C₂³×C₄²×C₆₁	1	2480	yes	direct product factor
66	C₂²	C₂⁵×C₄×C₆₁	1	186	yes	—
67	C₂²	C₂⁷×C₆₁	1	1	yes	characteristic
68	C₄	C₂⁵×C₄×C₆₁	1	192	yes	direct product factor
69	C₂	C₂⁴×C₄²×C₆₁	1	124	yes	direct product factor
70	C₂	C₂⁶×C₄×C₆₁	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂⁷×C₆₁
socle, prime radical	C₂⁷×C₆₁	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄²×C₆₁	C₂	direct (N × H)
4. C₂⁵×C₄×C₆₁	C₄	direct (N × H)
6. C₂³×C₄²×C₆₁	C₂²	direct (N × H)
9. C₂⁴×C₄×C₆₁	C₂×C₄	direct (N × H)
11. C₂²×C₄²×C₆₁	C₂³	direct (N × H)
12. C₂⁵×C₆₁	C₄²	direct (N × H)
15. C₂³×C₄×C₆₁	C₂²×C₄	direct (N × H)
17. C₂×C₄²×C₆₁	C₂⁴	direct (N × H)
18. C₂⁴×C₆₁	C₂×C₄²	direct (N × H)
21. C₂²×C₄×C₆₁	C₂³×C₄	direct (N × H)
23. C₄²×C₆₁	C₂⁵	direct (N × H)
24. C₂⁵×C₄²	C₆₁	direct (N × H)
25. C₂³×C₆₁	C₂²×C₄²	direct (N × H)
28. C₂×C₄×C₆₁	C₂⁴×C₄	direct (N × H)
31. C₂⁴×C₄²	C₂×C₆₁	direct (N × H)
32. C₂²×C₆₁	C₂³×C₄²	direct (N × H)
35. C₄×C₆₁	C₂⁵×C₄	direct (N × H)
37. C₂⁵×C₄	C₄×C₆₁	direct (N × H)
39. C₂³×C₄²	C₂²×C₆₁	direct (N × H)
40. C₂×C₆₁	C₂⁴×C₄²	direct (N × H)
44. C₂⁴×C₄	C₂×C₄×C₆₁	direct (N × H)
46. C₂²×C₄²	C₂³×C₆₁	direct (N × H)
47. C₆₁	C₂⁵×C₄²	direct (N × H)
48. C₂⁵	C₄²×C₆₁	direct (N × H)
51. C₂³×C₄	C₂²×C₄×C₆₁	direct (N × H)
53. C₂×C₄²	C₂⁴×C₆₁	direct (N × H)
54. C₂⁴	C₂×C₄²×C₆₁	direct (N × H)
57. C₂²×C₄	C₂³×C₄×C₆₁	direct (N × H)
59. C₄²	C₂⁵×C₆₁	direct (N × H)
60. C₂³	C₂²×C₄²×C₆₁	direct (N × H)
63. C₂×C₄	C₂⁴×C₄×C₆₁	direct (N × H)
65. C₂²	C₂³×C₄²×C₆₁	direct (N × H)
68. C₄	C₂⁵×C₄×C₆₁	direct (N × H)
69. C₂	C₂⁴×C₄²×C₆₁	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁶×C₄×C₆₁	2
C₂⁴×C₄²×C₆₁	2
C₂⁵×C₄²	61

Subgroups — C25×C42×C61 #72208043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×C₄²×C₆₁ #72208043