Subgroups — C₂²×C₇×C₉×C₁₂₅ #72208263

Order 31500.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₉×C₁₂₅	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₇×C₁₂₅	C₃	1	1	yes	maximal, characteristic
3	C₇×C₉×C₁₂₅	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₃×C₇×C₁₂₅	C₂×C₃	1	3	yes	—
5	C₂²×C₉×C₁₂₅	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₇×C₁₂₅	C₉	1	1	yes	characteristic, direct product factor, Hall
7	C₃×C₇×C₁₂₅	C₂²×C₃	1	1	yes	characteristic
8	C₂×C₉×C₁₂₅	C₂×C₇	1	3	yes	direct product factor
9	C₂²×C₇×C₉×C₂₅	C₅	1	1	yes	maximal, characteristic
10	C₂×C₇×C₉×C₂₅	C₂×C₅	1	3	yes	—
11	C₂²×C₃×C₇×C₂₅	C₃×C₅	1	1	yes	characteristic
12	C₂×C₇×C₁₂₅	C₂×C₉	1	3	yes	direct product factor
13	C₇×C₉×C₂₅	C₂²×C₅	1	1	yes	characteristic
14	C₂²×C₃×C₁₂₅	C₃×C₇	1	1	yes	characteristic
15	C₂²×C₅×C₇×C₉	C₂₅	1	1	yes	characteristic
16	C₉×C₁₂₅	C₂²×C₇	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₃×C₇×C₂₅	C₂×C₃×C₅	1	3	yes	—
18	C₂²×C₉×C₂₅	C₅×C₇	1	1	yes	characteristic
19	C₇×C₁₂₅	C₂²×C₉	1	1	yes	characteristic, direct product factor, Hall
20	C₂×C₃×C₁₂₅	C₂×C₃×C₇	1	3	yes	—
21	C₂²×C₇×C₂₅	C₅×C₉	1	1	yes	characteristic
22	C₂×C₅×C₇×C₉	C₂×C₂₅	1	3	yes	—
23	C₃×C₇×C₂₅	C₂²×C₃×C₅	1	1	yes	characteristic
24	C₂²×C₁₂₅	C₇×C₉	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₉×C₂₅	C₂×C₅×C₇	1	3	yes	—
26	C₂²×C₃×C₅×C₇	C₃×C₂₅	1	1	yes	characteristic
27	C₃×C₁₂₅	C₂²×C₃×C₇	1	1	yes	characteristic
28	C₂×C₇×C₂₅	C₂×C₅×C₉	1	3	yes	—
29	C₅×C₇×C₉	C₂²×C₂₅	1	1	yes	characteristic
30	C₂²×C₃×C₂₅	C₃×C₅×C₇	1	1	yes	characteristic
31	C₂²×C₇×C₉	C₁₂₅	1	1	yes	characteristic, direct product factor, Hall
32	C₂×C₁₂₅	C₂×C₇×C₉	1	3	yes	direct product factor
33	C₉×C₂₅	C₂²×C₅×C₇	1	1	yes	characteristic
34	C₂×C₃×C₅×C₇	C₂×C₃×C₂₅	1	3	yes	—
35	C₂²×C₅×C₉	C₇×C₂₅	1	1	yes	characteristic
36	C₇×C₂₅	C₂²×C₅×C₉	1	1	yes	characteristic
37	C₂×C₃×C₂₅	C₂×C₃×C₅×C₇	1	3	yes	—
38	C₂²×C₅×C₇	C₉×C₂₅	1	1	yes	characteristic
39	C₂×C₇×C₉	C₂×C₁₂₅	1	3	yes	direct product factor
40	C₁₂₅	C₂²×C₇×C₉	1	1	yes	characteristic, direct product factor, Sylow
41	C₃×C₅×C₇	C₂²×C₃×C₂₅	1	1	yes	characteristic
42	C₂²×C₂₅	C₅×C₇×C₉	1	1	yes	characteristic
43	C₂×C₅×C₉	C₂×C₇×C₂₅	1	3	yes	—
44	C₂²×C₃×C₇	C₃×C₁₂₅	1	1	yes	characteristic
45	C₃×C₂₅	C₂²×C₃×C₅×C₇	1	1	yes	characteristic
46	C₂×C₅×C₇	C₂×C₉×C₂₅	1	3	yes	—
47	C₇×C₉	C₂²×C₁₂₅	1	1	yes	characteristic, direct product factor, Hall
48	C₂²×C₃×C₅	C₃×C₇×C₂₅	1	1	yes	characteristic
49	C₂×C₂₅	C₂×C₅×C₇×C₉	1	3	yes	—
50	C₅×C₉	C₂²×C₇×C₂₅	1	1	yes	characteristic
51	C₂×C₃×C₇	C₂×C₃×C₁₂₅	1	3	yes	—
52	C₂²×C₉	C₇×C₁₂₅	1	1	yes	characteristic, direct product factor, Hall
53	C₅×C₇	C₂²×C₉×C₂₅	1	1	yes	characteristic
54	C₂×C₃×C₅	C₂×C₃×C₇×C₂₅	1	3	yes	—
55	C₂²×C₇	C₉×C₁₂₅	1	1	yes	characteristic, direct product factor, Hall
56	C₂₅	C₂²×C₅×C₇×C₉	1	1	yes	characteristic
57	C₃×C₇	C₂²×C₃×C₁₂₅	1	1	yes	characteristic
58	C₂²×C₅	C₇×C₉×C₂₅	1	1	yes	characteristic
59	C₂×C₉	C₂×C₇×C₁₂₅	1	3	yes	direct product factor
60	C₃×C₅	C₂²×C₃×C₇×C₂₅	1	1	yes	characteristic
61	C₂×C₇	C₂×C₉×C₁₂₅	1	3	yes	direct product factor
62	C₂²×C₃	C₃×C₇×C₁₂₅	1	1	yes	characteristic
63	C₂×C₅	C₂×C₇×C₉×C₂₅	1	3	yes	—
64	C₉	C₂²×C₇×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
65	C₇	C₂²×C₉×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃	C₂×C₃×C₇×C₁₂₅	1	3	yes	—
67	C₅	C₂²×C₇×C₉×C₂₅	1	1	yes	characteristic
68	C₂²	C₇×C₉×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
69	C₃	C₂²×C₃×C₇×C₁₂₅	1	1	yes	characteristic
70	C₂	C₂×C₇×C₉×C₁₂₅	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₂₅	C₂²×C₃×C₅×C₇
socle, prime radical	C₂²×C₃×C₅×C₇	C₃×C₂₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×C₉×C₁₂₅	C₂	direct (N × H)
3. C₇×C₉×C₁₂₅	C₂²	direct (N × H)
5. C₂²×C₉×C₁₂₅	C₇	direct (N × H)
6. C₂²×C₇×C₁₂₅	C₉	direct (N × H)
8. C₂×C₉×C₁₂₅	C₂×C₇	direct (N × H)
12. C₂×C₇×C₁₂₅	C₂×C₉	direct (N × H)
16. C₉×C₁₂₅	C₂²×C₇	direct (N × H)
19. C₇×C₁₂₅	C₂²×C₉	direct (N × H)
24. C₂²×C₁₂₅	C₇×C₉	direct (N × H)
31. C₂²×C₇×C₉	C₁₂₅	direct (N × H)
32. C₂×C₁₂₅	C₂×C₇×C₉	direct (N × H)
39. C₂×C₇×C₉	C₂×C₁₂₅	direct (N × H)
40. C₁₂₅	C₂²×C₇×C₉	direct (N × H)
47. C₇×C₉	C₂²×C₁₂₅	direct (N × H)
52. C₂²×C₉	C₇×C₁₂₅	direct (N × H)
55. C₂²×C₇	C₉×C₁₂₅	direct (N × H)
59. C₂×C₉	C₂×C₇×C₁₂₅	direct (N × H)
61. C₂×C₇	C₂×C₉×C₁₂₅	direct (N × H)
64. C₉	C₂²×C₇×C₁₂₅	direct (N × H)
65. C₇	C₂²×C₉×C₁₂₅	direct (N × H)
68. C₂²	C₇×C₉×C₁₂₅	direct (N × H)
70. C₂	C₂×C₇×C₉×C₁₂₅	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×C₇×C₉×C₁₂₅	2
C₂²×C₃×C₇×C₁₂₅	3
C₂²×C₉×C₁₂₅	7
C₂²×C₇×C₉×C₂₅	5

Subgroups — C22×C7×C9×C125 #72208263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×C₉×C₁₂₅ #72208263