Subgroups — C₂²×C₃×C₄²×C₁₂₅ #72201286

Order 24000.

1992 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₁₂₅	C₂	1	3	yes	maximal
2	C₂×C₃×C₄²×C₁₂₅	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄²×C₁₂₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂⁴×C₃×C₁₂₅	C₂²	1	1	yes	characteristic
5	C₂²×C₃×C₄×C₁₂₅	C₄	1	24	yes	direct product factor
6	C₂²×C₃×C₄×C₁₂₅	C₂²	1	18	yes	—
7	C₃×C₄²×C₁₂₅	C₂²	1	16	yes	direct product factor
8	C₂²×C₃×C₄²×C₂₅	C₅	1	1	yes	maximal, characteristic
9	C₂³×C₄×C₁₂₅	C₂×C₃	1	3	yes	—
10	C₂×C₄²×C₁₂₅	C₂×C₃	1	12	yes	direct product factor
11	C₂³×C₃×C₁₂₅	C₂×C₄	1	12	yes	—
12	C₂³×C₃×C₁₂₅	C₂³	1	3	yes	—
13	C₂×C₃×C₄×C₁₂₅	C₂×C₄	1	72	yes	direct product factor
14	C₂×C₃×C₄×C₁₂₅	C₂³	1	12	yes	—
15	C₂³×C₃×C₄×C₂₅	C₂×C₅	1	3	yes	—
16	C₂×C₃×C₄²×C₂₅	C₂×C₅	1	12	yes	—
17	C₂⁴×C₁₂₅	C₂²×C₃	1	1	yes	characteristic
18	C₂²×C₄×C₁₂₅	C₃×C₄	1	24	yes	direct product factor
19	C₂²×C₄×C₁₂₅	C₂²×C₃	1	18	yes	—
20	C₄²×C₁₂₅	C₂²×C₃	1	16	yes	direct product factor
21	C₂²×C₄²×C₂₅	C₃×C₅	1	1	yes	characteristic
22	C₂²×C₃×C₁₂₅	C₄²	1	16	yes	direct product factor
23	C₂²×C₃×C₁₂₅	C₂²×C₄	1	18	yes	—
24	C₂²×C₃×C₁₂₅	C₂⁴	1	1	yes	characteristic
25	C₃×C₄×C₁₂₅	C₂²×C₄	1	24	yes	direct product factor
26	C₂⁴×C₃×C₂₅	C₂²×C₅	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₂₅	C₄×C₅	1	24	yes	—
28	C₂²×C₃×C₄×C₂₅	C₂²×C₅	1	18	yes	—
29	C₃×C₄²×C₂₅	C₂²×C₅	1	16	yes	—
30	C₂³×C₁₂₅	C₂×C₃×C₄	1	12	yes	—
31	C₂³×C₁₂₅	C₂³×C₃	1	3	yes	—
32	C₂×C₄×C₁₂₅	C₂×C₃×C₄	1	72	yes	direct product factor
33	C₂×C₄×C₁₂₅	C₂³×C₃	1	12	yes	—
34	C₂²×C₃×C₄²×C₅	C₂₅	1	1	yes	characteristic
35	C₂³×C₄×C₂₅	C₂×C₃×C₅	1	3	yes	—
36	C₂×C₄²×C₂₅	C₂×C₃×C₅	1	12	yes	—
37	C₂×C₃×C₁₂₅	C₂×C₄²	1	12	yes	direct product factor
38	C₂×C₃×C₁₂₅	C₂³×C₄	1	3	yes	—
39	C₂³×C₃×C₂₅	C₂×C₄×C₅	1	12	yes	—
40	C₂³×C₃×C₂₅	C₂³×C₅	1	3	yes	—
41	C₂×C₃×C₄×C₂₅	C₂×C₄×C₅	1	72	yes	—
42	C₂×C₃×C₄×C₂₅	C₂³×C₅	1	12	yes	—
43	C₂²×C₁₂₅	C₃×C₄²	1	16	yes	direct product factor
44	C₂²×C₁₂₅	C₂²×C₃×C₄	1	18	yes	—
45	C₂²×C₁₂₅	C₂⁴×C₃	1	1	yes	characteristic
46	C₄×C₁₂₅	C₂²×C₃×C₄	1	24	yes	direct product factor
47	C₂³×C₃×C₄×C₅	C₂×C₂₅	1	3	yes	—
48	C₂×C₃×C₄²×C₅	C₂×C₂₅	1	12	yes	—
49	C₂⁴×C₂₅	C₂²×C₃×C₅	1	1	yes	characteristic
50	C₂²×C₄×C₂₅	C₃×C₄×C₅	1	24	yes	—
51	C₂²×C₄×C₂₅	C₂²×C₃×C₅	1	18	yes	—
52	C₄²×C₂₅	C₂²×C₃×C₅	1	16	yes	—
53	C₃×C₁₂₅	C₂²×C₄²	1	1	yes	characteristic, direct product factor, Hall
54	C₂²×C₄²×C₅	C₃×C₂₅	1	1	yes	characteristic
55	C₂²×C₃×C₂₅	C₄²×C₅	1	16	yes	—
56	C₂²×C₃×C₂₅	C₂²×C₄×C₅	1	18	yes	—
57	C₂²×C₃×C₂₅	C₂⁴×C₅	1	1	yes	characteristic
58	C₃×C₄×C₂₅	C₂²×C₄×C₅	1	24	yes	—
59	C₂×C₁₂₅	C₂×C₃×C₄²	1	12	yes	direct product factor
60	C₂×C₁₂₅	C₂³×C₃×C₄	1	3	yes	—
61	C₂⁴×C₃×C₅	C₂²×C₂₅	1	1	yes	characteristic
62	C₂²×C₃×C₄×C₅	C₄×C₂₅	1	24	yes	—
63	C₂²×C₃×C₄×C₅	C₂²×C₂₅	1	18	yes	—
64	C₃×C₄²×C₅	C₂²×C₂₅	1	16	yes	—
65	C₂³×C₂₅	C₂×C₃×C₄×C₅	1	12	yes	—
66	C₂³×C₂₅	C₂³×C₃×C₅	1	3	yes	—
67	C₂×C₄×C₂₅	C₂×C₃×C₄×C₅	1	72	yes	—
68	C₂×C₄×C₂₅	C₂³×C₃×C₅	1	12	yes	—
69	C₂²×C₃×C₄²	C₁₂₅	1	1	yes	characteristic, direct product factor, Hall
70	C₂³×C₄×C₅	C₂×C₃×C₂₅	1	3	yes	—
71	C₂×C₄²×C₅	C₂×C₃×C₂₅	1	12	yes	—
72	C₂×C₃×C₂₅	C₂×C₄²×C₅	1	12	yes	—
73	C₂×C₃×C₂₅	C₂³×C₄×C₅	1	3	yes	—
74	C₁₂₅	C₂²×C₃×C₄²	1	1	yes	characteristic, direct product factor, Sylow
75	C₂³×C₃×C₅	C₂×C₄×C₂₅	1	12	yes	—
76	C₂³×C₃×C₅	C₂³×C₂₅	1	3	yes	—
77	C₂×C₃×C₄×C₅	C₂×C₄×C₂₅	1	72	yes	—
78	C₂×C₃×C₄×C₅	C₂³×C₂₅	1	12	yes	—
79	C₂²×C₂₅	C₃×C₄²×C₅	1	16	yes	—
80	C₂²×C₂₅	C₂²×C₃×C₄×C₅	1	18	yes	—
81	C₂²×C₂₅	C₂⁴×C₃×C₅	1	1	yes	characteristic
82	C₄×C₂₅	C₂²×C₃×C₄×C₅	1	24	yes	—
83	C₂³×C₃×C₄	C₂×C₁₂₅	1	3	yes	—
84	C₂×C₃×C₄²	C₂×C₁₂₅	1	12	yes	direct product factor
85	C₂⁴×C₅	C₂²×C₃×C₂₅	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₃×C₄×C₂₅	1	24	yes	—
87	C₂²×C₄×C₅	C₂²×C₃×C₂₅	1	18	yes	—
88	C₄²×C₅	C₂²×C₃×C₂₅	1	16	yes	—
89	C₃×C₂₅	C₂²×C₄²×C₅	1	1	yes	characteristic
90	C₂²×C₄²	C₃×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
91	C₂²×C₃×C₅	C₄²×C₂₅	1	16	yes	—
92	C₂²×C₃×C₅	C₂²×C₄×C₂₅	1	18	yes	—
93	C₂²×C₃×C₅	C₂⁴×C₂₅	1	1	yes	characteristic
94	C₃×C₄×C₅	C₂²×C₄×C₂₅	1	24	yes	—
95	C₂×C₂₅	C₂×C₃×C₄²×C₅	1	12	yes	—
96	C₂×C₂₅	C₂³×C₃×C₄×C₅	1	3	yes	—
97	C₂⁴×C₃	C₂²×C₁₂₅	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₄×C₁₂₅	1	24	yes	direct product factor
99	C₂²×C₃×C₄	C₂²×C₁₂₅	1	18	yes	—
100	C₃×C₄²	C₂²×C₁₂₅	1	16	yes	direct product factor
101	C₂³×C₅	C₂×C₃×C₄×C₂₅	1	12	yes	—
102	C₂³×C₅	C₂³×C₃×C₂₅	1	3	yes	—
103	C₂×C₄×C₅	C₂×C₃×C₄×C₂₅	1	72	yes	—
104	C₂×C₄×C₅	C₂³×C₃×C₂₅	1	12	yes	—
105	C₂³×C₄	C₂×C₃×C₁₂₅	1	3	yes	—
106	C₂×C₄²	C₂×C₃×C₁₂₅	1	12	yes	direct product factor
107	C₂×C₃×C₅	C₂×C₄²×C₂₅	1	12	yes	—
108	C₂×C₃×C₅	C₂³×C₄×C₂₅	1	3	yes	—
109	C₂₅	C₂²×C₃×C₄²×C₅	1	1	yes	characteristic
110	C₂³×C₃	C₂×C₄×C₁₂₅	1	12	yes	—
111	C₂³×C₃	C₂³×C₁₂₅	1	3	yes	—
112	C₂×C₃×C₄	C₂×C₄×C₁₂₅	1	72	yes	direct product factor
113	C₂×C₃×C₄	C₂³×C₁₂₅	1	12	yes	—
114	C₂²×C₅	C₃×C₄²×C₂₅	1	16	yes	—
115	C₂²×C₅	C₂²×C₃×C₄×C₂₅	1	18	yes	—
116	C₂²×C₅	C₂⁴×C₃×C₂₅	1	1	yes	characteristic
117	C₄×C₅	C₂²×C₃×C₄×C₂₅	1	24	yes	—
118	C₂⁴	C₂²×C₃×C₁₂₅	1	1	yes	characteristic
119	C₂²×C₄	C₃×C₄×C₁₂₅	1	24	yes	direct product factor
120	C₂²×C₄	C₂²×C₃×C₁₂₅	1	18	yes	—
121	C₄²	C₂²×C₃×C₁₂₅	1	16	yes	direct product factor
122	C₃×C₅	C₂²×C₄²×C₂₅	1	1	yes	characteristic
123	C₂²×C₃	C₄²×C₁₂₅	1	16	yes	direct product factor
124	C₂²×C₃	C₂²×C₄×C₁₂₅	1	18	yes	—
125	C₂²×C₃	C₂⁴×C₁₂₅	1	1	yes	characteristic
126	C₃×C₄	C₂²×C₄×C₁₂₅	1	24	yes	direct product factor
127	C₂×C₅	C₂×C₃×C₄²×C₂₅	1	12	yes	—
128	C₂×C₅	C₂³×C₃×C₄×C₂₅	1	3	yes	—
129	C₂³	C₂×C₃×C₄×C₁₂₅	1	12	yes	—
130	C₂³	C₂³×C₃×C₁₂₅	1	3	yes	—
131	C₂×C₄	C₂×C₃×C₄×C₁₂₅	1	72	yes	direct product factor
132	C₂×C₄	C₂³×C₃×C₁₂₅	1	12	yes	—
133	C₂×C₃	C₂×C₄²×C₁₂₅	1	12	yes	direct product factor
134	C₂×C₃	C₂³×C₄×C₁₂₅	1	3	yes	—
135	C₅	C₂²×C₃×C₄²×C₂₅	1	1	yes	characteristic
136	C₂²	C₃×C₄²×C₁₂₅	1	16	yes	direct product factor
137	C₂²	C₂²×C₃×C₄×C₁₂₅	1	18	yes	—
138	C₂²	C₂⁴×C₃×C₁₂₅	1	1	yes	characteristic
139	C₄	C₂²×C₃×C₄×C₁₂₅	1	24	yes	direct product factor
140	C₃	C₂²×C₄²×C₁₂₅	1	1	yes	characteristic, direct product factor, Sylow
141	C₂	C₂×C₃×C₄²×C₁₂₅	1	12	yes	direct product factor
142	C₂	C₂³×C₃×C₄×C₁₂₅	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₂₅	C₂⁴×C₃×C₅
socle, prime radical	C₂⁴×C₃×C₅	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. C₂×C₃×C₄²×C₁₂₅	C₂	direct (N × H)
3. C₂²×C₄²×C₁₂₅	C₃	direct (N × H)
5. C₂²×C₃×C₄×C₁₂₅	C₄	direct (N × H)
7. C₃×C₄²×C₁₂₅	C₂²	direct (N × H)
10. C₂×C₄²×C₁₂₅	C₂×C₃	direct (N × H)
13. C₂×C₃×C₄×C₁₂₅	C₂×C₄	direct (N × H)
18. C₂²×C₄×C₁₂₅	C₃×C₄	direct (N × H)
20. C₄²×C₁₂₅	C₂²×C₃	direct (N × H)
22. C₂²×C₃×C₁₂₅	C₄²	direct (N × H)
25. C₃×C₄×C₁₂₅	C₂²×C₄	direct (N × H)
32. C₂×C₄×C₁₂₅	C₂×C₃×C₄	direct (N × H)
37. C₂×C₃×C₁₂₅	C₂×C₄²	direct (N × H)
43. C₂²×C₁₂₅	C₃×C₄²	direct (N × H)
46. C₄×C₁₂₅	C₂²×C₃×C₄	direct (N × H)
53. C₃×C₁₂₅	C₂²×C₄²	direct (N × H)
59. C₂×C₁₂₅	C₂×C₃×C₄²	direct (N × H)
69. C₂²×C₃×C₄²	C₁₂₅	direct (N × H)
74. C₁₂₅	C₂²×C₃×C₄²	direct (N × H)
84. C₂×C₃×C₄²	C₂×C₁₂₅	direct (N × H)
90. C₂²×C₄²	C₃×C₁₂₅	direct (N × H)
98. C₂²×C₃×C₄	C₄×C₁₂₅	direct (N × H)
100. C₃×C₄²	C₂²×C₁₂₅	direct (N × H)
106. C₂×C₄²	C₂×C₃×C₁₂₅	direct (N × H)
112. C₂×C₃×C₄	C₂×C₄×C₁₂₅	direct (N × H)
119. C₂²×C₄	C₃×C₄×C₁₂₅	direct (N × H)
121. C₄²	C₂²×C₃×C₁₂₅	direct (N × H)
123. C₂²×C₃	C₄²×C₁₂₅	direct (N × H)
126. C₃×C₄	C₂²×C₄×C₁₂₅	direct (N × H)
131. C₂×C₄	C₂×C₃×C₄×C₁₂₅	direct (N × H)
133. C₂×C₃	C₂×C₄²×C₁₂₅	direct (N × H)
136. C₂²	C₃×C₄²×C₁₂₅	direct (N × H)
139. C₄	C₂²×C₃×C₄×C₁₂₅	direct (N × H)
140. C₃	C₂²×C₄²×C₁₂₅	direct (N × H)
141. C₂	C₂×C₃×C₄²×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
C₂³×C₃×C₄×C₁₂₅	2
C₂×C₃×C₄²×C₁₂₅	2
C₂²×C₄²×C₁₂₅	3
C₂²×C₃×C₄²×C₂₅	5

Subgroups — C22×C3×C42×C125 #72201286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄²×C₁₂₅ #72201286