Subgroups — C₂×C₄²×C₇×C₁₂₅ #72205019

Order 28000.

432 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×C₁₂₅	C₂	1	3	yes	maximal
2	C₄²×C₇×C₁₂₅	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₇×C₁₂₅	C₂²	1	1	yes	characteristic
4	C₂×C₄×C₇×C₁₂₅	C₄	1	12	yes	direct product factor
5	C₂×C₄×C₇×C₁₂₅	C₂²	1	6	yes	—
6	C₂×C₄²×C₇×C₂₅	C₅	1	1	yes	maximal, characteristic
7	C₂×C₄²×C₁₂₅	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₇×C₁₂₅	C₂×C₄	1	6	yes	—
9	C₂²×C₇×C₁₂₅	C₂³	1	1	yes	characteristic
10	C₄×C₇×C₁₂₅	C₂×C₄	1	12	yes	direct product factor
11	C₂²×C₄×C₇×C₂₅	C₂×C₅	1	3	yes	—
12	C₄²×C₇×C₂₅	C₂×C₅	1	4	yes	—
13	C₂²×C₄×C₁₂₅	C₂×C₇	1	3	yes	—
14	C₄²×C₁₂₅	C₂×C₇	1	4	yes	direct product factor
15	C₂×C₇×C₁₂₅	C₄²	1	4	yes	direct product factor
16	C₂×C₇×C₁₂₅	C₂²×C₄	1	3	yes	—
17	C₂³×C₇×C₂₅	C₂²×C₅	1	1	yes	characteristic
18	C₂×C₄×C₇×C₂₅	C₄×C₅	1	12	yes	—
19	C₂×C₄×C₇×C₂₅	C₂²×C₅	1	6	yes	—
20	C₂×C₄²×C₅×C₇	C₂₅	1	1	yes	characteristic
21	C₂³×C₁₂₅	C₂²×C₇	1	1	yes	characteristic
22	C₂×C₄×C₁₂₅	C₄×C₇	1	12	yes	direct product factor
23	C₂×C₄×C₁₂₅	C₂²×C₇	1	6	yes	—
24	C₇×C₁₂₅	C₂×C₄²	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₄²×C₂₅	C₅×C₇	1	1	yes	characteristic
26	C₂²×C₇×C₂₅	C₂×C₄×C₅	1	6	yes	—
27	C₂²×C₇×C₂₅	C₂³×C₅	1	1	yes	characteristic
28	C₄×C₇×C₂₅	C₂×C₄×C₅	1	12	yes	—
29	C₂²×C₄×C₅×C₇	C₂×C₂₅	1	3	yes	—
30	C₄²×C₅×C₇	C₂×C₂₅	1	4	yes	—
31	C₂²×C₁₂₅	C₂×C₄×C₇	1	6	yes	—
32	C₂²×C₁₂₅	C₂³×C₇	1	1	yes	characteristic
33	C₄×C₁₂₅	C₂×C₄×C₇	1	12	yes	direct product factor
34	C₂²×C₄×C₂₅	C₂×C₅×C₇	1	3	yes	—
35	C₄²×C₂₅	C₂×C₅×C₇	1	4	yes	—
36	C₂×C₇×C₂₅	C₄²×C₅	1	4	yes	—
37	C₂×C₇×C₂₅	C₂²×C₄×C₅	1	3	yes	—
38	C₂³×C₅×C₇	C₂²×C₂₅	1	1	yes	characteristic
39	C₂×C₄×C₅×C₇	C₄×C₂₅	1	12	yes	—
40	C₂×C₄×C₅×C₇	C₂²×C₂₅	1	6	yes	—
41	C₂×C₁₂₅	C₄²×C₇	1	4	yes	direct product factor
42	C₂×C₁₂₅	C₂²×C₄×C₇	1	3	yes	—
43	C₂×C₄²×C₇	C₁₂₅	1	1	yes	characteristic, direct product factor, Hall
44	C₂³×C₂₅	C₂²×C₅×C₇	1	1	yes	characteristic
45	C₂×C₄×C₂₅	C₄×C₅×C₇	1	12	yes	—
46	C₂×C₄×C₂₅	C₂²×C₅×C₇	1	6	yes	—
47	C₇×C₂₅	C₂×C₄²×C₅	1	1	yes	characteristic
48	C₂×C₄²×C₅	C₇×C₂₅	1	1	yes	characteristic
49	C₂²×C₅×C₇	C₂×C₄×C₂₅	1	6	yes	—
50	C₂²×C₅×C₇	C₂³×C₂₅	1	1	yes	characteristic
51	C₄×C₅×C₇	C₂×C₄×C₂₅	1	12	yes	—
52	C₁₂₅	C₂×C₄²×C₇	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²×C₄×C₇	C₂×C₁₂₅	1	3	yes	—
54	C₄²×C₇	C₂×C₁₂₅	1	4	yes	direct product factor
55	C₂²×C₂₅	C₂×C₄×C₅×C₇	1	6	yes	—
56	C₂²×C₂₅	C₂³×C₅×C₇	1	1	yes	characteristic
57	C₄×C₂₅	C₂×C₄×C₅×C₇	1	12	yes	—
58	C₂²×C₄×C₅	C₂×C₇×C₂₅	1	3	yes	—
59	C₄²×C₅	C₂×C₇×C₂₅	1	4	yes	—
60	C₂×C₅×C₇	C₄²×C₂₅	1	4	yes	—
61	C₂×C₅×C₇	C₂²×C₄×C₂₅	1	3	yes	—
62	C₂³×C₇	C₂²×C₁₂₅	1	1	yes	characteristic
63	C₂×C₄×C₇	C₄×C₁₂₅	1	12	yes	direct product factor
64	C₂×C₄×C₇	C₂²×C₁₂₅	1	6	yes	—
65	C₂×C₂₅	C₄²×C₅×C₇	1	4	yes	—
66	C₂×C₂₅	C₂²×C₄×C₅×C₇	1	3	yes	—
67	C₂³×C₅	C₂²×C₇×C₂₅	1	1	yes	characteristic
68	C₂×C₄×C₅	C₄×C₇×C₂₅	1	12	yes	—
69	C₂×C₄×C₅	C₂²×C₇×C₂₅	1	6	yes	—
70	C₅×C₇	C₂×C₄²×C₂₅	1	1	yes	characteristic
71	C₂×C₄²	C₇×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²×C₇	C₂×C₄×C₁₂₅	1	6	yes	—
73	C₂²×C₇	C₂³×C₁₂₅	1	1	yes	characteristic
74	C₄×C₇	C₂×C₄×C₁₂₅	1	12	yes	direct product factor
75	C₂₅	C₂×C₄²×C₅×C₇	1	1	yes	characteristic
76	C₂²×C₅	C₂×C₄×C₇×C₂₅	1	6	yes	—
77	C₂²×C₅	C₂³×C₇×C₂₅	1	1	yes	characteristic
78	C₄×C₅	C₂×C₄×C₇×C₂₅	1	12	yes	—
79	C₂²×C₄	C₂×C₇×C₁₂₅	1	3	yes	—
80	C₄²	C₂×C₇×C₁₂₅	1	4	yes	direct product factor
81	C₂×C₇	C₄²×C₁₂₅	1	4	yes	direct product factor
82	C₂×C₇	C₂²×C₄×C₁₂₅	1	3	yes	—
83	C₂×C₅	C₄²×C₇×C₂₅	1	4	yes	—
84	C₂×C₅	C₂²×C₄×C₇×C₂₅	1	3	yes	—
85	C₂³	C₂²×C₇×C₁₂₅	1	1	yes	characteristic
86	C₂×C₄	C₄×C₇×C₁₂₅	1	12	yes	direct product factor
87	C₂×C₄	C₂²×C₇×C₁₂₅	1	6	yes	—
88	C₇	C₂×C₄²×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
89	C₅	C₂×C₄²×C₇×C₂₅	1	1	yes	characteristic
90	C₂²	C₂×C₄×C₇×C₁₂₅	1	6	yes	—
91	C₂²	C₂³×C₇×C₁₂₅	1	1	yes	characteristic
92	C₄	C₂×C₄×C₇×C₁₂₅	1	12	yes	direct product factor
93	C₂	C₄²×C₇×C₁₂₅	1	4	yes	direct product factor
94	C₂	C₂²×C₄×C₇×C₁₂₅	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₂₅	C₂³×C₅×C₇
socle, prime radical	C₂³×C₅×C₇	C₂²×C₂₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄²×C₇×C₁₂₅	C₂	direct (N × H)
4. C₂×C₄×C₇×C₁₂₅	C₄	direct (N × H)
7. C₂×C₄²×C₁₂₅	C₇	direct (N × H)
10. C₄×C₇×C₁₂₅	C₂×C₄	direct (N × H)
14. C₄²×C₁₂₅	C₂×C₇	direct (N × H)
15. C₂×C₇×C₁₂₅	C₄²	direct (N × H)
22. C₂×C₄×C₁₂₅	C₄×C₇	direct (N × H)
24. C₇×C₁₂₅	C₂×C₄²	direct (N × H)
33. C₄×C₁₂₅	C₂×C₄×C₇	direct (N × H)
41. C₂×C₁₂₅	C₄²×C₇	direct (N × H)
43. C₂×C₄²×C₇	C₁₂₅	direct (N × H)
52. C₁₂₅	C₂×C₄²×C₇	direct (N × H)
54. C₄²×C₇	C₂×C₁₂₅	direct (N × H)
63. C₂×C₄×C₇	C₄×C₁₂₅	direct (N × H)
71. C₂×C₄²	C₇×C₁₂₅	direct (N × H)
74. C₄×C₇	C₂×C₄×C₁₂₅	direct (N × H)
80. C₄²	C₂×C₇×C₁₂₅	direct (N × H)
81. C₂×C₇	C₄²×C₁₂₅	direct (N × H)
86. C₂×C₄	C₄×C₇×C₁₂₅	direct (N × H)
88. C₇	C₂×C₄²×C₁₂₅	direct (N × H)
92. C₄	C₂×C₄×C₇×C₁₂₅	direct (N × H)
93. C₂	C₄²×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
C₂²×C₄×C₇×C₁₂₅	2
C₄²×C₇×C₁₂₅	2
C₂×C₄²×C₇×C₂₅	5
C₂×C₄²×C₁₂₅	7

Subgroups — C2×C42×C7×C125 #72205019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×C₇×C₁₂₅ #72205019