Subgroups — C₂×C₃×C₄×C₅²×C₅₃ #72208564

Order 31800.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅²×C₅₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₃×C₄×C₅×C₅₃	C₅	1	6	yes	maximal, direct product factor
3	C₂×C₄×C₅×C₅₃	C₃×C₅	1	6	yes	direct product factor
4	C₂²×C₃×C₅²×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₃×C₄×C₅²×C₅₃	C₂	1	2	yes	maximal, direct product factor
6	C₂×C₃×C₅²×C₅₃	C₄	1	2	yes	direct product factor
7	C₂×C₃×C₅²×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×C₅²×C₅₃	C₂×C₃	1	1	yes	characteristic
9	C₄×C₅²×C₅₃	C₂×C₃	1	2	yes	direct product factor
10	C₃×C₅²×C₅₃	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
11	C₂²×C₃×C₅×C₅₃	C₂×C₅	1	6	yes	—
12	C₃×C₄×C₅×C₅₃	C₂×C₅	1	12	yes	direct product factor
13	C₂×C₅²×C₅₃	C₃×C₄	1	2	yes	direct product factor
14	C₂×C₅²×C₅₃	C₂²×C₃	1	1	yes	characteristic
15	C₂×C₃×C₅×C₅₃	C₄×C₅	1	12	yes	direct product factor
16	C₂×C₃×C₅×C₅₃	C₂²×C₅	1	6	yes	—
17	C₅²×C₅₃	C₂×C₃×C₄	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃×C₄×C₅₃	C₅²	1	1	yes	characteristic, direct product factor, Hall
19	C₂²×C₅×C₅₃	C₂×C₃×C₅	1	6	yes	—
20	C₄×C₅×C₅₃	C₂×C₃×C₅	1	12	yes	direct product factor
21	C₃×C₅×C₅₃	C₂×C₄×C₅	1	6	yes	direct product factor
22	C₂²×C₃×C₅₃	C₂×C₅²	1	1	yes	characteristic
23	C₃×C₄×C₅₃	C₂×C₅²	1	2	yes	direct product factor
24	C₂×C₃×C₄×C₅²	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₅×C₅₃	C₃×C₄×C₅	1	12	yes	direct product factor
26	C₂×C₅×C₅₃	C₂²×C₃×C₅	1	6	yes	—
27	C₂×C₄×C₅₃	C₃×C₅²	1	1	yes	characteristic, direct product factor, Hall
28	C₂×C₃×C₅₃	C₄×C₅²	1	2	yes	direct product factor
29	C₂×C₃×C₅₃	C₂²×C₅²	1	1	yes	characteristic
30	C₂²×C₃×C₅²	C₂×C₅₃	1	1	yes	characteristic
31	C₃×C₄×C₅²	C₂×C₅₃	1	2	yes	direct product factor
32	C₅×C₅₃	C₂×C₃×C₄×C₅	1	6	yes	direct product factor
33	C₂²×C₅₃	C₂×C₃×C₅²	1	1	yes	characteristic
34	C₄×C₅₃	C₂×C₃×C₅²	1	2	yes	direct product factor
35	C₂×C₄×C₅²	C₃×C₅₃	1	1	yes	characteristic, direct product factor, Hall
36	C₃×C₅₃	C₂×C₄×C₅²	1	1	yes	characteristic, direct product factor, Hall
37	C₂×C₃×C₅²	C₄×C₅₃	1	2	yes	direct product factor
38	C₂×C₃×C₅²	C₂²×C₅₃	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	C₅×C₅₃	1	6	yes	direct product factor
40	C₂×C₅₃	C₃×C₄×C₅²	1	2	yes	direct product factor
41	C₂×C₅₃	C₂²×C₃×C₅²	1	1	yes	characteristic
42	C₂²×C₅²	C₂×C₃×C₅₃	1	1	yes	characteristic
43	C₄×C₅²	C₂×C₃×C₅₃	1	2	yes	direct product factor
44	C₃×C₅²	C₂×C₄×C₅₃	1	1	yes	characteristic, direct product factor, Hall
45	C₂²×C₃×C₅	C₂×C₅×C₅₃	1	6	yes	—
46	C₃×C₄×C₅	C₂×C₅×C₅₃	1	12	yes	direct product factor
47	C₅₃	C₂×C₃×C₄×C₅²	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×C₅²	C₃×C₄×C₅₃	1	2	yes	direct product factor
49	C₂×C₅²	C₂²×C₃×C₅₃	1	1	yes	characteristic
50	C₂×C₄×C₅	C₃×C₅×C₅₃	1	6	yes	direct product factor
51	C₂×C₃×C₅	C₄×C₅×C₅₃	1	12	yes	direct product factor
52	C₂×C₃×C₅	C₂²×C₅×C₅₃	1	6	yes	—
53	C₅²	C₂×C₃×C₄×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
54	C₂×C₃×C₄	C₅²×C₅₃	1	1	yes	characteristic, direct product factor, Hall
55	C₂²×C₅	C₂×C₃×C₅×C₅₃	1	6	yes	—
56	C₄×C₅	C₂×C₃×C₅×C₅₃	1	12	yes	direct product factor
57	C₃×C₅	C₂×C₄×C₅×C₅₃	1	6	yes	direct product factor
58	C₂²×C₃	C₂×C₅²×C₅₃	1	1	yes	characteristic
59	C₃×C₄	C₂×C₅²×C₅₃	1	2	yes	direct product factor
60	C₂×C₅	C₃×C₄×C₅×C₅₃	1	12	yes	direct product factor
61	C₂×C₅	C₂²×C₃×C₅×C₅₃	1	6	yes	—
62	C₂×C₄	C₃×C₅²×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×C₃	C₄×C₅²×C₅₃	1	2	yes	direct product factor
64	C₂×C₃	C₂²×C₅²×C₅₃	1	1	yes	characteristic
65	C₅	C₂×C₃×C₄×C₅×C₅₃	1	6	yes	direct product factor
66	C₂²	C₂×C₃×C₅²×C₅₃	1	1	yes	characteristic
67	C₄	C₂×C₃×C₅²×C₅₃	1	2	yes	direct product factor
68	C₃	C₂×C₄×C₅²×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
69	C₂	C₃×C₄×C₅²×C₅₃	1	2	yes	direct product factor
70	C₂	C₂²×C₃×C₅²×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂²×C₃×C₅²×C₅₃
socle, prime radical	C₂²×C₃×C₅²×C₅₃	C₂
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×C₅²×C₅₃	C₃	direct (N × H)
2. C₂×C₃×C₄×C₅×C₅₃	C₅	direct (N × H)
3. C₂×C₄×C₅×C₅₃	C₃×C₅	direct (N × H)
5. C₃×C₄×C₅²×C₅₃	C₂	direct (N × H)
6. C₂×C₃×C₅²×C₅₃	C₄	direct (N × H)
9. C₄×C₅²×C₅₃	C₂×C₃	direct (N × H)
10. C₃×C₅²×C₅₃	C₂×C₄	direct (N × H)
12. C₃×C₄×C₅×C₅₃	C₂×C₅	direct (N × H)
13. C₂×C₅²×C₅₃	C₃×C₄	direct (N × H)
15. C₂×C₃×C₅×C₅₃	C₄×C₅	direct (N × H)
17. C₅²×C₅₃	C₂×C₃×C₄	direct (N × H)
18. C₂×C₃×C₄×C₅₃	C₅²	direct (N × H)
20. C₄×C₅×C₅₃	C₂×C₃×C₅	direct (N × H)
21. C₃×C₅×C₅₃	C₂×C₄×C₅	direct (N × H)
23. C₃×C₄×C₅₃	C₂×C₅²	direct (N × H)
24. C₂×C₃×C₄×C₅²	C₅₃	direct (N × H)
25. C₂×C₅×C₅₃	C₃×C₄×C₅	direct (N × H)
27. C₂×C₄×C₅₃	C₃×C₅²	direct (N × H)
28. C₂×C₃×C₅₃	C₄×C₅²	direct (N × H)
31. C₃×C₄×C₅²	C₂×C₅₃	direct (N × H)
32. C₅×C₅₃	C₂×C₃×C₄×C₅	direct (N × H)
34. C₄×C₅₃	C₂×C₃×C₅²	direct (N × H)
35. C₂×C₄×C₅²	C₃×C₅₃	direct (N × H)
36. C₃×C₅₃	C₂×C₄×C₅²	direct (N × H)
37. C₂×C₃×C₅²	C₄×C₅₃	direct (N × H)
39. C₂×C₃×C₄×C₅	C₅×C₅₃	direct (N × H)
40. C₂×C₅₃	C₃×C₄×C₅²	direct (N × H)
43. C₄×C₅²	C₂×C₃×C₅₃	direct (N × H)
44. C₃×C₅²	C₂×C₄×C₅₃	direct (N × H)
46. C₃×C₄×C₅	C₂×C₅×C₅₃	direct (N × H)
47. C₅₃	C₂×C₃×C₄×C₅²	direct (N × H)
48. C₂×C₅²	C₃×C₄×C₅₃	direct (N × H)
50. C₂×C₄×C₅	C₃×C₅×C₅₃	direct (N × H)
51. C₂×C₃×C₅	C₄×C₅×C₅₃	direct (N × H)
53. C₅²	C₂×C₃×C₄×C₅₃	direct (N × H)
54. C₂×C₃×C₄	C₅²×C₅₃	direct (N × H)
56. C₄×C₅	C₂×C₃×C₅×C₅₃	direct (N × H)
57. C₃×C₅	C₂×C₄×C₅×C₅₃	direct (N × H)
59. C₃×C₄	C₂×C₅²×C₅₃	direct (N × H)
60. C₂×C₅	C₃×C₄×C₅×C₅₃	direct (N × H)
62. C₂×C₄	C₃×C₅²×C₅₃	direct (N × H)
63. C₂×C₃	C₄×C₅²×C₅₃	direct (N × H)
65. C₅	C₂×C₃×C₄×C₅×C₅₃	direct (N × H)
67. C₄	C₂×C₃×C₅²×C₅₃	direct (N × H)
68. C₃	C₂×C₄×C₅²×C₅₃	direct (N × H)
69. C₂	C₃×C₄×C₅²×C₅₃	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×C₅²×C₅₃	3
C₂×C₃×C₄×C₅×C₅₃	5
C₂²×C₃×C₅²×C₅₃	2
C₃×C₄×C₅²×C₅₃	2
C₂×C₃×C₄×C₅²	53

Subgroups — C2×C3×C4×C52×C53 #72208564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×C₅²×C₅₃ #72208564