Subgroups — C₂×C₄×C₅×C₂₃×C₂₅ #72200330

Order 23000.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₂₃×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₂₃×C₂₅	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅×C₂₃×C₂₅	C₄	1	2	yes	direct product factor
4	C₂×C₅×C₂₃×C₂₅	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅²×C₂₃	C₅	1	1	yes	maximal, characteristic
6	C₂×C₄×C₂₃×C₂₅	C₅	1	5	yes	maximal, direct product factor
7	C₅×C₂₃×C₂₅	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
8	C₂²×C₅²×C₂₃	C₂×C₅	1	1	yes	characteristic
9	C₄×C₅²×C₂₃	C₂×C₅	1	2	yes	—
10	C₂²×C₂₃×C₂₅	C₂×C₅	1	5	yes	—
11	C₄×C₂₃×C₂₅	C₂×C₅	1	10	yes	direct product factor
12	C₂×C₅²×C₂₃	C₄×C₅	1	2	yes	—
13	C₂×C₅²×C₂₃	C₂²×C₅	1	1	yes	characteristic
14	C₂×C₂₃×C₂₅	C₄×C₅	1	10	yes	direct product factor
15	C₂×C₂₃×C₂₅	C₂²×C₅	1	5	yes	—
16	C₂×C₄×C₅×C₂₅	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₄×C₅×C₂₃	C₂₅	1	5	yes	direct product factor
18	C₂×C₄×C₅×C₂₃	C₅²	1	1	yes	characteristic
19	C₅²×C₂₃	C₂×C₄×C₅	1	1	yes	characteristic
20	C₂₃×C₂₅	C₂×C₄×C₅	1	5	yes	direct product factor
21	C₂²×C₅×C₂₅	C₂×C₂₃	1	1	yes	characteristic
22	C₄×C₅×C₂₅	C₂×C₂₃	1	2	yes	direct product factor
23	C₂²×C₅×C₂₃	C₂×C₂₅	1	5	yes	—
24	C₂²×C₅×C₂₃	C₂×C₅²	1	1	yes	characteristic
25	C₄×C₅×C₂₃	C₂×C₂₅	1	10	yes	direct product factor
26	C₄×C₅×C₂₃	C₂×C₅²	1	2	yes	—
27	C₂×C₅×C₂₅	C₄×C₂₃	1	2	yes	direct product factor
28	C₂×C₅×C₂₅	C₂²×C₂₃	1	1	yes	characteristic
29	C₂×C₅×C₂₃	C₄×C₂₅	1	10	yes	direct product factor
30	C₂×C₅×C₂₃	C₂²×C₂₅	1	5	yes	—
31	C₂×C₅×C₂₃	C₄×C₅²	1	2	yes	—
32	C₂×C₅×C₂₃	C₂²×C₅²	1	1	yes	characteristic
33	C₂×C₄×C₅²	C₅×C₂₃	1	1	yes	characteristic
34	C₂×C₄×C₂₅	C₅×C₂₃	1	5	yes	direct product factor
35	C₂×C₄×C₂₃	C₅×C₂₅	1	1	yes	characteristic, direct product factor, Hall
36	C₅×C₂₅	C₂×C₄×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
37	C₅×C₂₃	C₂×C₄×C₂₅	1	5	yes	direct product factor
38	C₅×C₂₃	C₂×C₄×C₅²	1	1	yes	characteristic
39	C₂²×C₅²	C₂×C₅×C₂₃	1	1	yes	characteristic
40	C₄×C₅²	C₂×C₅×C₂₃	1	2	yes	—
41	C₂²×C₂₅	C₂×C₅×C₂₃	1	5	yes	—
42	C₄×C₂₅	C₂×C₅×C₂₃	1	10	yes	direct product factor
43	C₂²×C₂₃	C₂×C₅×C₂₅	1	1	yes	characteristic
44	C₄×C₂₃	C₂×C₅×C₂₅	1	2	yes	direct product factor
45	C₂×C₅²	C₄×C₅×C₂₃	1	2	yes	—
46	C₂×C₅²	C₂²×C₅×C₂₃	1	1	yes	characteristic
47	C₂×C₂₅	C₄×C₅×C₂₃	1	10	yes	direct product factor
48	C₂×C₂₅	C₂²×C₅×C₂₃	1	5	yes	—
49	C₂×C₂₃	C₄×C₅×C₂₅	1	2	yes	direct product factor
50	C₂×C₂₃	C₂²×C₅×C₂₅	1	1	yes	characteristic
51	C₂×C₄×C₅	C₂₃×C₂₅	1	5	yes	direct product factor
52	C₂×C₄×C₅	C₅²×C₂₃	1	1	yes	characteristic
53	C₅²	C₂×C₄×C₅×C₂₃	1	1	yes	characteristic
54	C₂₅	C₂×C₄×C₅×C₂₃	1	5	yes	direct product factor
55	C₂₃	C₂×C₄×C₅×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×C₅	C₂×C₂₃×C₂₅	1	5	yes	—
57	C₂²×C₅	C₂×C₅²×C₂₃	1	1	yes	characteristic
58	C₄×C₅	C₂×C₂₃×C₂₅	1	10	yes	direct product factor
59	C₄×C₅	C₂×C₅²×C₂₃	1	2	yes	—
60	C₂×C₅	C₄×C₂₃×C₂₅	1	10	yes	direct product factor
61	C₂×C₅	C₂²×C₂₃×C₂₅	1	5	yes	—
62	C₂×C₅	C₄×C₅²×C₂₃	1	2	yes	—
63	C₂×C₅	C₂²×C₅²×C₂₃	1	1	yes	characteristic
64	C₂×C₄	C₅×C₂₃×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
65	C₅	C₂×C₄×C₂₃×C₂₅	1	5	yes	direct product factor
66	C₅	C₂×C₄×C₅²×C₂₃	1	1	yes	characteristic
67	C₂²	C₂×C₅×C₂₃×C₂₅	1	1	yes	characteristic
68	C₄	C₂×C₅×C₂₃×C₂₅	1	2	yes	direct product factor
69	C₂	C₄×C₅×C₂₃×C₂₅	1	2	yes	direct product factor
70	C₂	C₂²×C₅×C₂₃×C₂₅	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₅	C₂²×C₅²×C₂₃
socle, prime radical	C₂²×C₅²×C₂₃	C₂×C₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×C₂₃×C₂₅	C₂	direct (N × H)
3. C₂×C₅×C₂₃×C₂₅	C₄	direct (N × H)
6. C₂×C₄×C₂₃×C₂₅	C₅	direct (N × H)
7. C₅×C₂₃×C₂₅	C₂×C₄	direct (N × H)
11. C₄×C₂₃×C₂₅	C₂×C₅	direct (N × H)
14. C₂×C₂₃×C₂₅	C₄×C₅	direct (N × H)
16. C₂×C₄×C₅×C₂₅	C₂₃	direct (N × H)
17. C₂×C₄×C₅×C₂₃	C₂₅	direct (N × H)
20. C₂₃×C₂₅	C₂×C₄×C₅	direct (N × H)
22. C₄×C₅×C₂₅	C₂×C₂₃	direct (N × H)
25. C₄×C₅×C₂₃	C₂×C₂₅	direct (N × H)
27. C₂×C₅×C₂₅	C₄×C₂₃	direct (N × H)
29. C₂×C₅×C₂₃	C₄×C₂₅	direct (N × H)
34. C₂×C₄×C₂₅	C₅×C₂₃	direct (N × H)
35. C₂×C₄×C₂₃	C₅×C₂₅	direct (N × H)
36. C₅×C₂₅	C₂×C₄×C₂₃	direct (N × H)
37. C₅×C₂₃	C₂×C₄×C₂₅	direct (N × H)
42. C₄×C₂₅	C₂×C₅×C₂₃	direct (N × H)
44. C₄×C₂₃	C₂×C₅×C₂₅	direct (N × H)
47. C₂×C₂₅	C₄×C₅×C₂₃	direct (N × H)
49. C₂×C₂₃	C₄×C₅×C₂₅	direct (N × H)
51. C₂×C₄×C₅	C₂₃×C₂₅	direct (N × H)
54. C₂₅	C₂×C₄×C₅×C₂₃	direct (N × H)
55. C₂₃	C₂×C₄×C₅×C₂₅	direct (N × H)
58. C₄×C₅	C₂×C₂₃×C₂₅	direct (N × H)
60. C₂×C₅	C₄×C₂₃×C₂₅	direct (N × H)
64. C₂×C₄	C₅×C₂₃×C₂₅	direct (N × H)
65. C₅	C₂×C₄×C₂₃×C₂₅	direct (N × H)
68. C₄	C₂×C₅×C₂₃×C₂₅	direct (N × H)
69. C₂	C₄×C₅×C₂₃×C₂₅	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×C₅×C₂₃×C₂₅	2
C₄×C₅×C₂₃×C₂₅	2
C₂×C₄×C₅²×C₂₃	5
C₂×C₄×C₂₃×C₂₅	5
C₂×C₄×C₅×C₂₅	23

Subgroups — C2×C4×C5×C23×C25 #72200330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×C₂₃×C₂₅ #72200330