Subgroups — C₃×C₄×C₅×C₁₆×C₂₅ #72201270

Order 24000.

812 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×C₈×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₁₆×C₂₅	C₂	1	2	yes	maximal
3	C₄×C₅×C₁₆×C₂₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₄²×C₅×C₂₅	C₄	1	1	yes	characteristic
5	C₂×C₃×C₅×C₈×C₂₅	C₄	1	1	yes	characteristic
6	C₂×C₃×C₅×C₈×C₂₅	C₂²	1	1	yes	characteristic
7	C₃×C₅×C₁₆×C₂₅	C₄	1	4	yes	direct product factor
8	C₃×C₄×C₅²×C₁₆	C₅	1	1	yes	maximal, characteristic
9	C₃×C₄×C₁₆×C₂₅	C₅	1	5	yes	maximal, direct product factor
10	C₄×C₅×C₈×C₂₅	C₂×C₃	1	1	yes	characteristic
11	C₂×C₅×C₁₆×C₂₅	C₂×C₃	1	2	yes	—
12	C₂×C₃×C₄×C₅×C₂₅	C₈	1	2	yes	—
13	C₂×C₃×C₄×C₅×C₂₅	C₂×C₄	1	1	yes	characteristic
14	C₃×C₅×C₈×C₂₅	C₈	1	2	yes	—
15	C₃×C₅×C₈×C₂₅	C₂×C₄	1	2	yes	—
16	C₃×C₄×C₅²×C₈	C₂×C₅	1	1	yes	characteristic
17	C₃×C₄×C₈×C₂₅	C₂×C₅	1	5	yes	—
18	C₂×C₃×C₅²×C₁₆	C₂×C₅	1	2	yes	—
19	C₂×C₃×C₁₆×C₂₅	C₂×C₅	1	10	yes	—
20	C₂×C₅×C₈×C₂₅	C₃×C₄	1	1	yes	characteristic
21	C₂×C₅×C₈×C₂₅	C₂²×C₃	1	1	yes	characteristic
22	C₄²×C₅×C₂₅	C₃×C₄	1	1	yes	characteristic
23	C₅×C₁₆×C₂₅	C₃×C₄	1	4	yes	direct product factor
24	C₄×C₅²×C₁₆	C₃×C₅	1	1	yes	characteristic
25	C₄×C₁₆×C₂₅	C₃×C₅	1	5	yes	direct product factor
26	C₂²×C₃×C₅×C₂₅	C₂×C₈	1	1	yes	characteristic
27	C₃×C₄×C₅×C₂₅	C₁₆	1	4	yes	direct product factor
28	C₃×C₄×C₅×C₂₅	C₄²	1	1	yes	characteristic
29	C₃×C₄×C₅×C₂₅	C₂×C₈	1	1	yes	characteristic
30	C₂×C₃×C₅²×C₈	C₄×C₅	1	1	yes	characteristic
31	C₂×C₃×C₅²×C₈	C₂²×C₅	1	1	yes	characteristic
32	C₃×C₄²×C₅²	C₄×C₅	1	1	yes	characteristic
33	C₃×C₅²×C₁₆	C₄×C₅	1	4	yes	—
34	C₂×C₃×C₈×C₂₅	C₄×C₅	1	5	yes	—
35	C₂×C₃×C₈×C₂₅	C₂²×C₅	1	5	yes	—
36	C₃×C₄²×C₂₅	C₄×C₅	1	5	yes	—
37	C₃×C₁₆×C₂₅	C₄×C₅	1	20	yes	direct product factor
38	C₂×C₄×C₅×C₂₅	C₃×C₈	1	2	yes	—
39	C₂×C₄×C₅×C₂₅	C₂×C₃×C₄	1	1	yes	characteristic
40	C₅×C₈×C₂₅	C₃×C₈	1	2	yes	—
41	C₅×C₈×C₂₅	C₂×C₃×C₄	1	2	yes	—
42	C₃×C₄×C₅×C₁₆	C₂₅	1	5	yes	direct product factor
43	C₃×C₄×C₅×C₁₆	C₅²	1	1	yes	characteristic
44	C₂×C₅²×C₁₆	C₂×C₃×C₅	1	2	yes	—
45	C₄×C₅²×C₈	C₂×C₃×C₅	1	1	yes	characteristic
46	C₂×C₁₆×C₂₅	C₂×C₃×C₅	1	10	yes	—
47	C₄×C₈×C₂₅	C₂×C₃×C₅	1	5	yes	—
48	C₂×C₃×C₅×C₂₅	C₄×C₈	1	1	yes	characteristic
49	C₂×C₃×C₅×C₂₅	C₂×C₁₆	1	2	yes	—
50	C₂×C₃×C₄×C₅²	C₅×C₈	1	2	yes	—
51	C₂×C₃×C₄×C₅²	C₂×C₄×C₅	1	1	yes	characteristic
52	C₃×C₅²×C₈	C₅×C₈	1	2	yes	—
53	C₃×C₅²×C₈	C₂×C₄×C₅	1	2	yes	—
54	C₂×C₃×C₄×C₂₅	C₅×C₈	1	10	yes	—
55	C₂×C₃×C₄×C₂₅	C₂×C₄×C₅	1	5	yes	—
56	C₃×C₈×C₂₅	C₅×C₈	1	10	yes	—
57	C₃×C₈×C₂₅	C₂×C₄×C₅	1	10	yes	—
58	C₂²×C₅×C₂₅	C₂×C₃×C₈	1	1	yes	characteristic
59	C₄×C₅×C₂₅	C₃×C₁₆	1	4	yes	direct product factor
60	C₄×C₅×C₂₅	C₃×C₄²	1	1	yes	characteristic
61	C₄×C₅×C₂₅	C₂×C₃×C₈	1	1	yes	characteristic
62	C₂×C₃×C₅×C₁₆	C₂×C₂₅	1	10	yes	—
63	C₂×C₃×C₅×C₁₆	C₂×C₅²	1	2	yes	—
64	C₃×C₄×C₅×C₈	C₂×C₂₅	1	5	yes	—
65	C₃×C₄×C₅×C₈	C₂×C₅²	1	1	yes	characteristic
66	C₂×C₅²×C₈	C₃×C₄×C₅	1	1	yes	characteristic
67	C₂×C₅²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
68	C₄²×C₅²	C₃×C₄×C₅	1	1	yes	characteristic
69	C₅²×C₁₆	C₃×C₄×C₅	1	4	yes	—
70	C₂×C₈×C₂₅	C₃×C₄×C₅	1	5	yes	—
71	C₂×C₈×C₂₅	C₂²×C₃×C₅	1	5	yes	—
72	C₄²×C₂₅	C₃×C₄×C₅	1	5	yes	—
73	C₁₆×C₂₅	C₃×C₄×C₅	1	20	yes	direct product factor
74	C₃×C₅×C₂₅	C₄×C₁₆	1	1	yes	characteristic, direct product factor, Hall
75	C₄×C₅×C₁₆	C₃×C₂₅	1	5	yes	direct product factor
76	C₄×C₅×C₁₆	C₃×C₅²	1	1	yes	characteristic
77	C₂²×C₃×C₅²	C₂×C₅×C₈	1	1	yes	characteristic
78	C₃×C₄×C₅²	C₅×C₁₆	1	4	yes	—
79	C₃×C₄×C₅²	C₄²×C₅	1	1	yes	characteristic
80	C₃×C₄×C₅²	C₂×C₅×C₈	1	1	yes	characteristic
81	C₂²×C₃×C₂₅	C₂×C₅×C₈	1	5	yes	—
82	C₃×C₄×C₂₅	C₅×C₁₆	1	20	yes	direct product factor
83	C₃×C₄×C₂₅	C₄²×C₅	1	5	yes	—
84	C₃×C₄×C₂₅	C₂×C₅×C₈	1	5	yes	—
85	C₂×C₅×C₂₅	C₃×C₄×C₈	1	1	yes	characteristic
86	C₂×C₅×C₂₅	C₂×C₃×C₁₆	1	2	yes	—
87	C₂×C₃×C₅×C₈	C₄×C₂₅	1	5	yes	—
88	C₂×C₃×C₅×C₈	C₂²×C₂₅	1	5	yes	—
89	C₂×C₃×C₅×C₈	C₄×C₅²	1	1	yes	characteristic
90	C₂×C₃×C₅×C₈	C₂²×C₅²	1	1	yes	characteristic
91	C₃×C₄²×C₅	C₄×C₂₅	1	5	yes	—
92	C₃×C₄²×C₅	C₄×C₅²	1	1	yes	characteristic
93	C₃×C₅×C₁₆	C₄×C₂₅	1	20	yes	direct product factor
94	C₃×C₅×C₁₆	C₄×C₅²	1	4	yes	—
95	C₂×C₄×C₅²	C₃×C₅×C₈	1	2	yes	—
96	C₂×C₄×C₅²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
97	C₅²×C₈	C₃×C₅×C₈	1	2	yes	—
98	C₅²×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
99	C₂×C₄×C₂₅	C₃×C₅×C₈	1	10	yes	—
100	C₂×C₄×C₂₅	C₂×C₃×C₄×C₅	1	5	yes	—
101	C₈×C₂₅	C₃×C₅×C₈	1	10	yes	—
102	C₈×C₂₅	C₂×C₃×C₄×C₅	1	10	yes	—
103	C₃×C₄×C₁₆	C₅×C₂₅	1	1	yes	characteristic, direct product factor, Hall
104	C₂×C₅×C₁₆	C₂×C₃×C₂₅	1	10	yes	—
105	C₂×C₅×C₁₆	C₂×C₃×C₅²	1	2	yes	—
106	C₄×C₅×C₈	C₂×C₃×C₂₅	1	5	yes	—
107	C₄×C₅×C₈	C₂×C₃×C₅²	1	1	yes	characteristic
108	C₂×C₃×C₅²	C₄×C₅×C₈	1	1	yes	characteristic
109	C₂×C₃×C₅²	C₂×C₅×C₁₆	1	2	yes	—
110	C₂×C₃×C₂₅	C₄×C₅×C₈	1	5	yes	—
111	C₂×C₃×C₂₅	C₂×C₅×C₁₆	1	10	yes	—
112	C₅×C₂₅	C₃×C₄×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
113	C₂×C₃×C₄×C₅	C₈×C₂₅	1	10	yes	—
114	C₂×C₃×C₄×C₅	C₂×C₄×C₂₅	1	5	yes	—
115	C₂×C₃×C₄×C₅	C₅²×C₈	1	2	yes	—
116	C₂×C₃×C₄×C₅	C₂×C₄×C₅²	1	1	yes	characteristic
117	C₃×C₅×C₈	C₈×C₂₅	1	10	yes	—
118	C₃×C₅×C₈	C₂×C₄×C₂₅	1	10	yes	—
119	C₃×C₅×C₈	C₅²×C₈	1	2	yes	—
120	C₃×C₅×C₈	C₂×C₄×C₅²	1	2	yes	—
121	C₂²×C₅²	C₂×C₃×C₅×C₈	1	1	yes	characteristic
122	C₄×C₅²	C₃×C₅×C₁₆	1	4	yes	—
123	C₄×C₅²	C₃×C₄²×C₅	1	1	yes	characteristic
124	C₄×C₅²	C₂×C₃×C₅×C₈	1	1	yes	characteristic
125	C₂²×C₂₅	C₂×C₃×C₅×C₈	1	5	yes	—
126	C₄×C₂₅	C₃×C₅×C₁₆	1	20	yes	direct product factor
127	C₄×C₂₅	C₃×C₄²×C₅	1	5	yes	—
128	C₄×C₂₅	C₂×C₃×C₅×C₈	1	5	yes	—
129	C₂×C₃×C₁₆	C₂×C₅×C₂₅	1	2	yes	—
130	C₃×C₄×C₈	C₂×C₅×C₂₅	1	1	yes	characteristic
131	C₂×C₅×C₈	C₃×C₄×C₂₅	1	5	yes	—
132	C₂×C₅×C₈	C₂²×C₃×C₂₅	1	5	yes	—
133	C₂×C₅×C₈	C₃×C₄×C₅²	1	1	yes	characteristic
134	C₂×C₅×C₈	C₂²×C₃×C₅²	1	1	yes	characteristic
135	C₄²×C₅	C₃×C₄×C₂₅	1	5	yes	—
136	C₄²×C₅	C₃×C₄×C₅²	1	1	yes	characteristic
137	C₅×C₁₆	C₃×C₄×C₂₅	1	20	yes	direct product factor
138	C₅×C₁₆	C₃×C₄×C₅²	1	4	yes	—
139	C₃×C₅²	C₄×C₅×C₁₆	1	1	yes	characteristic
140	C₃×C₂₅	C₄×C₅×C₁₆	1	5	yes	direct product factor
141	C₄×C₁₆	C₃×C₅×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×C₃×C₅	C₂×C₈×C₂₅	1	5	yes	—
143	C₂²×C₃×C₅	C₂×C₅²×C₈	1	1	yes	characteristic
144	C₃×C₄×C₅	C₁₆×C₂₅	1	20	yes	direct product factor
145	C₃×C₄×C₅	C₄²×C₂₅	1	5	yes	—
146	C₃×C₄×C₅	C₂×C₈×C₂₅	1	5	yes	—
147	C₃×C₄×C₅	C₅²×C₁₆	1	4	yes	—
148	C₃×C₄×C₅	C₄²×C₅²	1	1	yes	characteristic
149	C₃×C₄×C₅	C₂×C₅²×C₈	1	1	yes	characteristic
150	C₂×C₅²	C₃×C₄×C₅×C₈	1	1	yes	characteristic
151	C₂×C₅²	C₂×C₃×C₅×C₁₆	1	2	yes	—
152	C₂×C₂₅	C₃×C₄×C₅×C₈	1	5	yes	—
153	C₂×C₂₅	C₂×C₃×C₅×C₁₆	1	10	yes	—
154	C₂×C₃×C₈	C₄×C₅×C₂₅	1	1	yes	characteristic
155	C₂×C₃×C₈	C₂²×C₅×C₂₅	1	1	yes	characteristic
156	C₃×C₄²	C₄×C₅×C₂₅	1	1	yes	characteristic
157	C₃×C₁₆	C₄×C₅×C₂₅	1	4	yes	direct product factor
158	C₂×C₄×C₅	C₃×C₈×C₂₅	1	10	yes	—
159	C₂×C₄×C₅	C₂×C₃×C₄×C₂₅	1	5	yes	—
160	C₂×C₄×C₅	C₃×C₅²×C₈	1	2	yes	—
161	C₂×C₄×C₅	C₂×C₃×C₄×C₅²	1	1	yes	characteristic
162	C₅×C₈	C₃×C₈×C₂₅	1	10	yes	—
163	C₅×C₈	C₂×C₃×C₄×C₂₅	1	10	yes	—
164	C₅×C₈	C₃×C₅²×C₈	1	2	yes	—
165	C₅×C₈	C₂×C₃×C₄×C₅²	1	2	yes	—
166	C₂×C₁₆	C₂×C₃×C₅×C₂₅	1	2	yes	—
167	C₄×C₈	C₂×C₃×C₅×C₂₅	1	1	yes	characteristic
168	C₂×C₃×C₅	C₄×C₈×C₂₅	1	5	yes	—
169	C₂×C₃×C₅	C₂×C₁₆×C₂₅	1	10	yes	—
170	C₂×C₃×C₅	C₄×C₅²×C₈	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×C₅²×C₁₆	1	2	yes	—
172	C₅²	C₃×C₄×C₅×C₁₆	1	1	yes	characteristic
173	C₂₅	C₃×C₄×C₅×C₁₆	1	5	yes	direct product factor
174	C₂×C₃×C₄	C₅×C₈×C₂₅	1	2	yes	—
175	C₂×C₃×C₄	C₂×C₄×C₅×C₂₅	1	1	yes	characteristic
176	C₃×C₈	C₅×C₈×C₂₅	1	2	yes	—
177	C₃×C₈	C₂×C₄×C₅×C₂₅	1	2	yes	—
178	C₂²×C₅	C₂×C₃×C₈×C₂₅	1	5	yes	—
179	C₂²×C₅	C₂×C₃×C₅²×C₈	1	1	yes	characteristic
180	C₄×C₅	C₃×C₁₆×C₂₅	1	20	yes	direct product factor
181	C₄×C₅	C₃×C₄²×C₂₅	1	5	yes	—
182	C₄×C₅	C₂×C₃×C₈×C₂₅	1	5	yes	—
183	C₄×C₅	C₃×C₅²×C₁₆	1	4	yes	—
184	C₄×C₅	C₃×C₄²×C₅²	1	1	yes	characteristic
185	C₄×C₅	C₂×C₃×C₅²×C₈	1	1	yes	characteristic
186	C₂×C₈	C₃×C₄×C₅×C₂₅	1	1	yes	characteristic
187	C₂×C₈	C₂²×C₃×C₅×C₂₅	1	1	yes	characteristic
188	C₄²	C₃×C₄×C₅×C₂₅	1	1	yes	characteristic
189	C₁₆	C₃×C₄×C₅×C₂₅	1	4	yes	direct product factor
190	C₃×C₅	C₄×C₁₆×C₂₅	1	5	yes	direct product factor
191	C₃×C₅	C₄×C₅²×C₁₆	1	1	yes	characteristic
192	C₂²×C₃	C₂×C₅×C₈×C₂₅	1	1	yes	characteristic
193	C₃×C₄	C₅×C₁₆×C₂₅	1	4	yes	direct product factor
194	C₃×C₄	C₄²×C₅×C₂₅	1	1	yes	characteristic
195	C₃×C₄	C₂×C₅×C₈×C₂₅	1	1	yes	characteristic
196	C₂×C₅	C₂×C₃×C₁₆×C₂₅	1	10	yes	—
197	C₂×C₅	C₂×C₃×C₅²×C₁₆	1	2	yes	—
198	C₂×C₅	C₃×C₄×C₈×C₂₅	1	5	yes	—
199	C₂×C₅	C₃×C₄×C₅²×C₈	1	1	yes	characteristic
200	C₂×C₄	C₃×C₅×C₈×C₂₅	1	2	yes	—
201	C₂×C₄	C₂×C₃×C₄×C₅×C₂₅	1	1	yes	characteristic
202	C₈	C₃×C₅×C₈×C₂₅	1	2	yes	—
203	C₈	C₂×C₃×C₄×C₅×C₂₅	1	2	yes	—
204	C₂×C₃	C₂×C₅×C₁₆×C₂₅	1	2	yes	—
205	C₂×C₃	C₄×C₅×C₈×C₂₅	1	1	yes	characteristic
206	C₅	C₃×C₄×C₁₆×C₂₅	1	5	yes	direct product factor
207	C₅	C₃×C₄×C₅²×C₁₆	1	1	yes	characteristic
208	C₂²	C₂×C₃×C₅×C₈×C₂₅	1	1	yes	characteristic
209	C₄	C₃×C₅×C₁₆×C₂₅	1	4	yes	direct product factor
210	C₄	C₂×C₃×C₅×C₈×C₂₅	1	1	yes	characteristic
211	C₄	C₃×C₄²×C₅×C₂₅	1	1	yes	characteristic
212	C₃	C₄×C₅×C₁₆×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
213	C₂	C₂×C₃×C₅×C₁₆×C₂₅	1	2	yes	—
214	C₂	C₃×C₄×C₅×C₈×C₂₅	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₅×C₈	C₂²×C₃×C₅²
socle, prime radical	C₂²×C₃×C₅²	C₂×C₅×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₅×C₁₆×C₂₅	C₃	direct (N × H)
7. C₃×C₅×C₁₆×C₂₅	C₄	direct (N × H)
9. C₃×C₄×C₁₆×C₂₅	C₅	direct (N × H)
23. C₅×C₁₆×C₂₅	C₃×C₄	direct (N × H)
25. C₄×C₁₆×C₂₅	C₃×C₅	direct (N × H)
27. C₃×C₄×C₅×C₂₅	C₁₆	direct (N × H)
37. C₃×C₁₆×C₂₅	C₄×C₅	direct (N × H)
42. C₃×C₄×C₅×C₁₆	C₂₅	direct (N × H)
59. C₄×C₅×C₂₅	C₃×C₁₆	direct (N × H)
73. C₁₆×C₂₅	C₃×C₄×C₅	direct (N × H)
74. C₃×C₅×C₂₅	C₄×C₁₆	direct (N × H)
75. C₄×C₅×C₁₆	C₃×C₂₅	direct (N × H)
82. C₃×C₄×C₂₅	C₅×C₁₆	direct (N × H)
93. C₃×C₅×C₁₆	C₄×C₂₅	direct (N × H)
103. C₃×C₄×C₁₆	C₅×C₂₅	direct (N × H)
112. C₅×C₂₅	C₃×C₄×C₁₆	direct (N × H)
126. C₄×C₂₅	C₃×C₅×C₁₆	direct (N × H)
137. C₅×C₁₆	C₃×C₄×C₂₅	direct (N × H)
140. C₃×C₂₅	C₄×C₅×C₁₆	direct (N × H)
141. C₄×C₁₆	C₃×C₅×C₂₅	direct (N × H)
144. C₃×C₄×C₅	C₁₆×C₂₅	direct (N × H)
157. C₃×C₁₆	C₄×C₅×C₂₅	direct (N × H)
173. C₂₅	C₃×C₄×C₅×C₁₆	direct (N × H)
180. C₄×C₅	C₃×C₁₆×C₂₅	direct (N × H)
189. C₁₆	C₃×C₄×C₅×C₂₅	direct (N × H)
190. C₃×C₅	C₄×C₁₆×C₂₅	direct (N × H)
193. C₃×C₄	C₅×C₁₆×C₂₅	direct (N × H)
206. C₅	C₃×C₄×C₁₆×C₂₅	direct (N × H)
209. C₄	C₃×C₅×C₁₆×C₂₅	direct (N × H)
212. C₃	C₄×C₅×C₁₆×C₂₅	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×C₄×C₅×C₈×C₂₅	2
C₂×C₃×C₅×C₁₆×C₂₅	2
C₄×C₅×C₁₆×C₂₅	3
C₃×C₄×C₅²×C₁₆	5
C₃×C₄×C₁₆×C₂₅	5

Subgroups — C3×C4×C5×C16×C25 #72201270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₅×C₁₆×C₂₅ #72201270