Subgroups — C₃×C₄×C₈×C₁₂₅ #72190409

Order 12000.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄²×C₁₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈×C₁₂₅	C₂	1	2	yes	maximal
3	C₄×C₈×C₁₂₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃×C₄×C₁₂₅	C₄	1	2	yes	—
5	C₂×C₃×C₄×C₁₂₅	C₂²	1	1	yes	characteristic
6	C₃×C₈×C₁₂₅	C₄	1	4	yes	direct product factor
7	C₃×C₄×C₈×C₂₅	C₅	1	1	yes	maximal, characteristic
8	C₂×C₈×C₁₂₅	C₂×C₃	1	2	yes	—
9	C₄²×C₁₂₅	C₂×C₃	1	1	yes	characteristic
10	C₂²×C₃×C₁₂₅	C₂×C₄	1	1	yes	characteristic
11	C₃×C₄×C₁₂₅	C₈	1	4	yes	direct product factor
12	C₃×C₄×C₁₂₅	C₂×C₄	1	2	yes	—
13	C₂×C₃×C₈×C₂₅	C₂×C₅	1	2	yes	—
14	C₃×C₄²×C₂₅	C₂×C₅	1	1	yes	characteristic
15	C₂×C₄×C₁₂₅	C₃×C₄	1	2	yes	—
16	C₂×C₄×C₁₂₅	C₂²×C₃	1	1	yes	characteristic
17	C₈×C₁₂₅	C₃×C₄	1	4	yes	direct product factor
18	C₄×C₈×C₂₅	C₃×C₅	1	1	yes	characteristic
19	C₂×C₃×C₁₂₅	C₄²	1	1	yes	characteristic
20	C₂×C₃×C₁₂₅	C₂×C₈	1	2	yes	—
21	C₂×C₃×C₄×C₂₅	C₄×C₅	1	2	yes	—
22	C₂×C₃×C₄×C₂₅	C₂²×C₅	1	1	yes	characteristic
23	C₃×C₈×C₂₅	C₄×C₅	1	4	yes	—
24	C₂²×C₁₂₅	C₂×C₃×C₄	1	1	yes	characteristic
25	C₄×C₁₂₅	C₃×C₈	1	4	yes	direct product factor
26	C₄×C₁₂₅	C₂×C₃×C₄	1	2	yes	—
27	C₃×C₄×C₅×C₈	C₂₅	1	1	yes	characteristic
28	C₂×C₈×C₂₅	C₂×C₃×C₅	1	2	yes	—
29	C₄²×C₂₅	C₂×C₃×C₅	1	1	yes	characteristic
30	C₃×C₁₂₅	C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
31	C₂²×C₃×C₂₅	C₂×C₄×C₅	1	1	yes	characteristic
32	C₃×C₄×C₂₅	C₅×C₈	1	4	yes	—
33	C₃×C₄×C₂₅	C₂×C₄×C₅	1	2	yes	—
34	C₂×C₁₂₅	C₃×C₄²	1	1	yes	characteristic
35	C₂×C₁₂₅	C₂×C₃×C₈	1	2	yes	—
36	C₂×C₃×C₅×C₈	C₂×C₂₅	1	2	yes	—
37	C₃×C₄²×C₅	C₂×C₂₅	1	1	yes	characteristic
38	C₂×C₄×C₂₅	C₃×C₄×C₅	1	2	yes	—
39	C₂×C₄×C₂₅	C₂²×C₃×C₅	1	1	yes	characteristic
40	C₈×C₂₅	C₃×C₄×C₅	1	4	yes	—
41	C₄×C₅×C₈	C₃×C₂₅	1	1	yes	characteristic
42	C₂×C₃×C₂₅	C₄²×C₅	1	1	yes	characteristic
43	C₂×C₃×C₂₅	C₂×C₅×C₈	1	2	yes	—
44	C₁₂₅	C₃×C₄×C₈	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₃×C₄×C₅	C₄×C₂₅	1	2	yes	—
46	C₂×C₃×C₄×C₅	C₂²×C₂₅	1	1	yes	characteristic
47	C₃×C₅×C₈	C₄×C₂₅	1	4	yes	—
48	C₂²×C₂₅	C₂×C₃×C₄×C₅	1	1	yes	characteristic
49	C₄×C₂₅	C₃×C₅×C₈	1	4	yes	—
50	C₄×C₂₅	C₂×C₃×C₄×C₅	1	2	yes	—
51	C₃×C₄×C₈	C₁₂₅	1	1	yes	characteristic, direct product factor, Hall
52	C₂×C₅×C₈	C₂×C₃×C₂₅	1	2	yes	—
53	C₄²×C₅	C₂×C₃×C₂₅	1	1	yes	characteristic
54	C₃×C₂₅	C₄×C₅×C₈	1	1	yes	characteristic
55	C₂²×C₃×C₅	C₂×C₄×C₂₅	1	1	yes	characteristic
56	C₃×C₄×C₅	C₈×C₂₅	1	4	yes	—
57	C₃×C₄×C₅	C₂×C₄×C₂₅	1	2	yes	—
58	C₂×C₂₅	C₃×C₄²×C₅	1	1	yes	characteristic
59	C₂×C₂₅	C₂×C₃×C₅×C₈	1	2	yes	—
60	C₂×C₃×C₈	C₂×C₁₂₅	1	2	yes	—
61	C₃×C₄²	C₂×C₁₂₅	1	1	yes	characteristic
62	C₂×C₄×C₅	C₃×C₄×C₂₅	1	2	yes	—
63	C₂×C₄×C₅	C₂²×C₃×C₂₅	1	1	yes	characteristic
64	C₅×C₈	C₃×C₄×C₂₅	1	4	yes	—
65	C₄×C₈	C₃×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃×C₅	C₄²×C₂₅	1	1	yes	characteristic
67	C₂×C₃×C₅	C₂×C₈×C₂₅	1	2	yes	—
68	C₂₅	C₃×C₄×C₅×C₈	1	1	yes	characteristic
69	C₂×C₃×C₄	C₄×C₁₂₅	1	2	yes	—
70	C₂×C₃×C₄	C₂²×C₁₂₅	1	1	yes	characteristic
71	C₃×C₈	C₄×C₁₂₅	1	4	yes	direct product factor
72	C₂²×C₅	C₂×C₃×C₄×C₂₅	1	1	yes	characteristic
73	C₄×C₅	C₃×C₈×C₂₅	1	4	yes	—
74	C₄×C₅	C₂×C₃×C₄×C₂₅	1	2	yes	—
75	C₂×C₈	C₂×C₃×C₁₂₅	1	2	yes	—
76	C₄²	C₂×C₃×C₁₂₅	1	1	yes	characteristic
77	C₃×C₅	C₄×C₈×C₂₅	1	1	yes	characteristic
78	C₂²×C₃	C₂×C₄×C₁₂₅	1	1	yes	characteristic
79	C₃×C₄	C₈×C₁₂₅	1	4	yes	direct product factor
80	C₃×C₄	C₂×C₄×C₁₂₅	1	2	yes	—
81	C₂×C₅	C₃×C₄²×C₂₅	1	1	yes	characteristic
82	C₂×C₅	C₂×C₃×C₈×C₂₅	1	2	yes	—
83	C₂×C₄	C₃×C₄×C₁₂₅	1	2	yes	—
84	C₂×C₄	C₂²×C₃×C₁₂₅	1	1	yes	characteristic
85	C₈	C₃×C₄×C₁₂₅	1	4	yes	direct product factor
86	C₂×C₃	C₄²×C₁₂₅	1	1	yes	characteristic
87	C₂×C₃	C₂×C₈×C₁₂₅	1	2	yes	—
88	C₅	C₃×C₄×C₈×C₂₅	1	1	yes	characteristic
89	C₂²	C₂×C₃×C₄×C₁₂₅	1	1	yes	characteristic
90	C₄	C₃×C₈×C₁₂₅	1	4	yes	direct product factor
91	C₄	C₂×C₃×C₄×C₁₂₅	1	2	yes	—
92	C₃	C₄×C₈×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
93	C₂	C₂×C₃×C₈×C₁₂₅	1	2	yes	—
94	C₂	C₃×C₄²×C₁₂₅	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄×C₂₅	C₂²×C₃×C₅
socle, prime radical	C₂²×C₃×C₅	C₂×C₄×C₂₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₈×C₁₂₅	C₃	direct (N × H)
6. C₃×C₈×C₁₂₅	C₄	direct (N × H)
11. C₃×C₄×C₁₂₅	C₈	direct (N × H)
17. C₈×C₁₂₅	C₃×C₄	direct (N × H)
25. C₄×C₁₂₅	C₃×C₈	direct (N × H)
30. C₃×C₁₂₅	C₄×C₈	direct (N × H)
44. C₁₂₅	C₃×C₄×C₈	direct (N × H)
51. C₃×C₄×C₈	C₁₂₅	direct (N × H)
65. C₄×C₈	C₃×C₁₂₅	direct (N × H)
71. C₃×C₈	C₄×C₁₂₅	direct (N × H)
79. C₃×C₄	C₈×C₁₂₅	direct (N × H)
85. C₈	C₃×C₄×C₁₂₅	direct (N × H)
90. C₄	C₃×C₈×C₁₂₅	direct (N × H)
92. C₃	C₄×C₈×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
C₃×C₄²×C₁₂₅	2
C₂×C₃×C₈×C₁₂₅	2
C₄×C₈×C₁₂₅	3
C₃×C₄×C₈×C₂₅	5

Subgroups — C3×C4×C8×C125 #72190409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₈×C₁₂₅ #72190409