Subgroups — C₃²×C₄×C₇×C₁₂₅ #72208261

Order 31500.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₇×C₁₂₅	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₇×C₁₂₅	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₇×C₁₂₅	C₄	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₃×C₇×C₁₂₅	C₂×C₃	1	4	yes	—
5	C₃²×C₄×C₁₂₅	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₄×C₇×C₁₂₅	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃×C₇×C₁₂₅	C₃×C₄	1	4	yes	direct product factor
8	C₂×C₃²×C₁₂₅	C₂×C₇	1	1	yes	characteristic
9	C₃²×C₄×C₇×C₂₅	C₅	1	1	yes	maximal, characteristic
10	C₂×C₃²×C₇×C₂₅	C₂×C₅	1	1	yes	characteristic
11	C₃×C₄×C₇×C₂₅	C₃×C₅	1	4	yes	—
12	C₂×C₇×C₁₂₅	C₂×C₃²	1	1	yes	characteristic
13	C₃²×C₇×C₂₅	C₄×C₅	1	1	yes	characteristic
14	C₃×C₄×C₁₂₅	C₃×C₇	1	4	yes	direct product factor
15	C₃²×C₄×C₅×C₇	C₂₅	1	1	yes	characteristic
16	C₃²×C₁₂₅	C₄×C₇	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₃×C₇×C₂₅	C₂×C₃×C₅	1	4	yes	—
18	C₃²×C₄×C₂₅	C₅×C₇	1	1	yes	characteristic
19	C₇×C₁₂₅	C₃²×C₄	1	1	yes	characteristic, direct product factor, Hall
20	C₂×C₃×C₁₂₅	C₂×C₃×C₇	1	4	yes	—
21	C₄×C₇×C₂₅	C₃²×C₅	1	1	yes	characteristic
22	C₂×C₃²×C₅×C₇	C₂×C₂₅	1	1	yes	characteristic
23	C₃×C₇×C₂₅	C₃×C₄×C₅	1	4	yes	—
24	C₄×C₁₂₅	C₃²×C₇	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₃²×C₂₅	C₂×C₅×C₇	1	1	yes	characteristic
26	C₃×C₄×C₅×C₇	C₃×C₂₅	1	4	yes	—
27	C₃×C₁₂₅	C₃×C₄×C₇	1	4	yes	direct product factor
28	C₂×C₇×C₂₅	C₂×C₃²×C₅	1	1	yes	characteristic
29	C₃²×C₅×C₇	C₄×C₂₅	1	1	yes	characteristic
30	C₃×C₄×C₂₅	C₃×C₅×C₇	1	4	yes	—
31	C₃²×C₄×C₇	C₁₂₅	1	1	yes	characteristic, direct product factor, Hall
32	C₂×C₁₂₅	C₂×C₃²×C₇	1	1	yes	characteristic
33	C₃²×C₂₅	C₄×C₅×C₇	1	1	yes	characteristic
34	C₂×C₃×C₅×C₇	C₂×C₃×C₂₅	1	4	yes	—
35	C₃²×C₄×C₅	C₇×C₂₅	1	1	yes	characteristic
36	C₇×C₂₅	C₃²×C₄×C₅	1	1	yes	characteristic
37	C₂×C₃×C₂₅	C₂×C₃×C₅×C₇	1	4	yes	—
38	C₄×C₅×C₇	C₃²×C₂₅	1	1	yes	characteristic
39	C₂×C₃²×C₇	C₂×C₁₂₅	1	1	yes	characteristic
40	C₁₂₅	C₃²×C₄×C₇	1	1	yes	characteristic, direct product factor, Sylow
41	C₃×C₅×C₇	C₃×C₄×C₂₅	1	4	yes	—
42	C₄×C₂₅	C₃²×C₅×C₇	1	1	yes	characteristic
43	C₂×C₃²×C₅	C₂×C₇×C₂₅	1	1	yes	characteristic
44	C₃×C₄×C₇	C₃×C₁₂₅	1	4	yes	direct product factor
45	C₃×C₂₅	C₃×C₄×C₅×C₇	1	4	yes	—
46	C₂×C₅×C₇	C₂×C₃²×C₂₅	1	1	yes	characteristic
47	C₃²×C₇	C₄×C₁₂₅	1	1	yes	characteristic, direct product factor, Hall
48	C₃×C₄×C₅	C₃×C₇×C₂₅	1	4	yes	—
49	C₂×C₂₅	C₂×C₃²×C₅×C₇	1	1	yes	characteristic
50	C₃²×C₅	C₄×C₇×C₂₅	1	1	yes	characteristic
51	C₂×C₃×C₇	C₂×C₃×C₁₂₅	1	4	yes	—
52	C₃²×C₄	C₇×C₁₂₅	1	1	yes	characteristic, direct product factor, Hall
53	C₅×C₇	C₃²×C₄×C₂₅	1	1	yes	characteristic
54	C₂×C₃×C₅	C₂×C₃×C₇×C₂₅	1	4	yes	—
55	C₄×C₇	C₃²×C₁₂₅	1	1	yes	characteristic, direct product factor, Hall
56	C₂₅	C₃²×C₄×C₅×C₇	1	1	yes	characteristic
57	C₃×C₇	C₃×C₄×C₁₂₅	1	4	yes	direct product factor
58	C₄×C₅	C₃²×C₇×C₂₅	1	1	yes	characteristic
59	C₂×C₃²	C₂×C₇×C₁₂₅	1	1	yes	characteristic
60	C₃×C₅	C₃×C₄×C₇×C₂₅	1	4	yes	—
61	C₂×C₇	C₂×C₃²×C₁₂₅	1	1	yes	characteristic
62	C₃×C₄	C₃×C₇×C₁₂₅	1	4	yes	direct product factor
63	C₂×C₅	C₂×C₃²×C₇×C₂₅	1	1	yes	characteristic
64	C₃²	C₄×C₇×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
65	C₇	C₃²×C₄×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃	C₂×C₃×C₇×C₁₂₅	1	4	yes	—
67	C₅	C₃²×C₄×C₇×C₂₅	1	1	yes	characteristic
68	C₄	C₃²×C₇×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
69	C₃	C₃×C₄×C₇×C₁₂₅	1	4	yes	direct product factor
70	C₂	C₂×C₃²×C₇×C₁₂₅	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₂₅	C₂×C₃²×C₅×C₇
socle, prime radical	C₂×C₃²×C₅×C₇	C₂×C₂₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₇×C₁₂₅	C₃	direct (N × H)
3. C₃²×C₇×C₁₂₅	C₄	direct (N × H)
5. C₃²×C₄×C₁₂₅	C₇	direct (N × H)
6. C₄×C₇×C₁₂₅	C₃²	direct (N × H)
7. C₃×C₇×C₁₂₅	C₃×C₄	direct (N × H)
14. C₃×C₄×C₁₂₅	C₃×C₇	direct (N × H)
16. C₃²×C₁₂₅	C₄×C₇	direct (N × H)
19. C₇×C₁₂₅	C₃²×C₄	direct (N × H)
24. C₄×C₁₂₅	C₃²×C₇	direct (N × H)
27. C₃×C₁₂₅	C₃×C₄×C₇	direct (N × H)
31. C₃²×C₄×C₇	C₁₂₅	direct (N × H)
40. C₁₂₅	C₃²×C₄×C₇	direct (N × H)
44. C₃×C₄×C₇	C₃×C₁₂₅	direct (N × H)
47. C₃²×C₇	C₄×C₁₂₅	direct (N × H)
52. C₃²×C₄	C₇×C₁₂₅	direct (N × H)
55. C₄×C₇	C₃²×C₁₂₅	direct (N × H)
57. C₃×C₇	C₃×C₄×C₁₂₅	direct (N × H)
62. C₃×C₄	C₃×C₇×C₁₂₅	direct (N × H)
64. C₃²	C₄×C₇×C₁₂₅	direct (N × H)
65. C₇	C₃²×C₄×C₁₂₅	direct (N × H)
68. C₄	C₃²×C₇×C₁₂₅	direct (N × H)
69. C₃	C₃×C₄×C₇×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×C₃²×C₇×C₁₂₅	2
C₃×C₄×C₇×C₁₂₅	3
C₃²×C₄×C₁₂₅	7
C₃²×C₄×C₇×C₂₅	5

Subgroups — C32×C4×C7×C125 #72208261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₇×C₁₂₅ #72208261