Subgroups — C₂×C₃×C₄×C₁₀₂₄ #72201817

Order 24576.

540 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₁₀₂₄	C₂	1	2	yes	maximal
3	C₃×C₄×C₁₀₂₄	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₁₀₂₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂×C₃×C₄×C₂₅₆	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₅₁₂	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅₁₂	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₅₁₂	C₄	1	2	yes	—
9	C₃×C₄×C₅₁₂	C₂²	1	2	yes	—
10	C₂×C₃×C₁₀₂₄	C₄	1	8	yes	direct product factor
11	C₂×C₃×C₁₀₂₄	C₂²	1	4	yes	—
12	C₂×C₄×C₅₁₂	C₂×C₃	1	1	yes	characteristic
13	C₂²×C₁₀₂₄	C₂×C₃	1	2	yes	—
14	C₄×C₁₀₂₄	C₂×C₃	1	4	yes	direct product factor
15	C₂×C₃×C₄×C₁₂₈	C₈	1	1	yes	characteristic
16	C₂²×C₃×C₂₅₆	C₈	1	1	yes	characteristic
17	C₂²×C₃×C₂₅₆	C₂×C₄	1	1	yes	characteristic
18	C₃×C₄×C₂₅₆	C₈	1	2	yes	—
19	C₃×C₄×C₂₅₆	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₅₁₂	C₈	1	4	yes	—
21	C₂×C₃×C₅₁₂	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₅₁₂	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₅₁₂	C₂×C₄	1	4	yes	—
24	C₂×C₃×C₅₁₂	C₂³	1	1	yes	characteristic
25	C₃×C₁₀₂₄	C₂×C₄	1	8	yes	direct product factor
26	C₂×C₄×C₂₅₆	C₃×C₄	1	1	yes	characteristic
27	C₂²×C₅₁₂	C₃×C₄	1	1	yes	characteristic
28	C₂²×C₅₁₂	C₂²×C₃	1	1	yes	characteristic
29	C₄×C₅₁₂	C₃×C₄	1	2	yes	—
30	C₄×C₅₁₂	C₂²×C₃	1	2	yes	—
31	C₂×C₁₀₂₄	C₃×C₄	1	8	yes	direct product factor
32	C₂×C₁₀₂₄	C₂²×C₃	1	4	yes	—
33	C₂×C₃×C₄×C₆₄	C₁₆	1	1	yes	characteristic
34	C₂²×C₃×C₁₂₈	C₁₆	1	1	yes	characteristic
35	C₂²×C₃×C₁₂₈	C₂×C₈	1	1	yes	characteristic
36	C₃×C₄×C₁₂₈	C₁₆	1	2	yes	—
37	C₃×C₄×C₁₂₈	C₂×C₈	1	2	yes	—
38	C₂×C₃×C₂₅₆	C₁₆	1	4	yes	—
39	C₂×C₃×C₂₅₆	C₄²	1	2	yes	—
40	C₂×C₃×C₂₅₆	C₂×C₈	1	1	yes	characteristic
41	C₂×C₃×C₂₅₆	C₂×C₈	1	2	yes	—
42	C₂×C₃×C₂₅₆	C₂×C₈	1	2	yes	—
43	C₂×C₃×C₂₅₆	C₂²×C₄	1	1	yes	characteristic
44	C₃×C₅₁₂	C₄²	1	2	yes	—
45	C₃×C₅₁₂	C₂×C₈	1	4	yes	—
46	C₃×C₅₁₂	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₂₈	C₃×C₈	1	1	yes	characteristic
48	C₂²×C₂₅₆	C₃×C₈	1	1	yes	characteristic
49	C₂²×C₂₅₆	C₂×C₃×C₄	1	1	yes	characteristic
50	C₄×C₂₅₆	C₃×C₈	1	2	yes	—
51	C₄×C₂₅₆	C₂×C₃×C₄	1	2	yes	—
52	C₂×C₅₁₂	C₃×C₈	1	4	yes	—
53	C₂×C₅₁₂	C₂×C₃×C₄	1	1	yes	characteristic
54	C₂×C₅₁₂	C₂×C₃×C₄	1	2	yes	—
55	C₂×C₅₁₂	C₂×C₃×C₄	1	4	yes	—
56	C₂×C₅₁₂	C₂³×C₃	1	1	yes	characteristic
57	C₁₀₂₄	C₂×C₃×C₄	1	8	yes	direct product factor
58	C₂²×C₃×C₆₄	C₃₂	1	1	yes	characteristic
59	C₂²×C₃×C₆₄	C₂×C₁₆	1	1	yes	characteristic
60	C₂×C₃×C₄×C₃₂	C₃₂	1	1	yes	characteristic
61	C₂×C₃×C₁₂₈	C₃₂	1	4	yes	—
62	C₂×C₃×C₁₂₈	C₄×C₈	1	2	yes	—
63	C₂×C₃×C₁₂₈	C₂×C₁₆	1	1	yes	characteristic
64	C₂×C₃×C₁₂₈	C₂×C₁₆	1	2	yes	—
65	C₂×C₃×C₁₂₈	C₂×C₁₆	1	2	yes	—
66	C₂×C₃×C₁₂₈	C₂²×C₈	1	1	yes	characteristic
67	C₃×C₄×C₆₄	C₃₂	1	2	yes	—
68	C₃×C₄×C₆₄	C₂×C₁₆	1	2	yes	—
69	C₃×C₂₅₆	C₄×C₈	1	2	yes	—
70	C₃×C₂₅₆	C₂×C₁₆	1	4	yes	—
71	C₃×C₂₅₆	C₂×C₄²	1	1	yes	characteristic
72	C₃×C₂₅₆	C₂²×C₈	1	1	yes	characteristic
73	C₂²×C₁₂₈	C₃×C₁₆	1	1	yes	characteristic
74	C₂²×C₁₂₈	C₂×C₃×C₈	1	1	yes	characteristic
75	C₂×C₄×C₆₄	C₃×C₁₆	1	1	yes	characteristic
76	C₂×C₂₅₆	C₃×C₁₆	1	4	yes	—
77	C₂×C₂₅₆	C₃×C₄²	1	2	yes	—
78	C₂×C₂₅₆	C₂×C₃×C₈	1	1	yes	characteristic
79	C₂×C₂₅₆	C₂×C₃×C₈	1	2	yes	—
80	C₂×C₂₅₆	C₂×C₃×C₈	1	2	yes	—
81	C₂×C₂₅₆	C₂²×C₃×C₄	1	1	yes	characteristic
82	C₄×C₁₂₈	C₃×C₁₆	1	2	yes	—
83	C₄×C₁₂₈	C₂×C₃×C₈	1	2	yes	—
84	C₅₁₂	C₃×C₄²	1	2	yes	—
85	C₅₁₂	C₂×C₃×C₈	1	4	yes	—
86	C₅₁₂	C₂²×C₃×C₄	1	2	yes	—
87	C₂²×C₃×C₃₂	C₆₄	1	1	yes	characteristic
88	C₂²×C₃×C₃₂	C₂×C₃₂	1	1	yes	characteristic
89	C₂×C₃×C₄×C₁₆	C₆₄	1	1	yes	characteristic
90	C₂×C₃×C₆₄	C₆₄	1	4	yes	—
91	C₂×C₃×C₆₄	C₄×C₁₆	1	2	yes	—
92	C₂×C₃×C₆₄	C₂×C₃₂	1	1	yes	characteristic
93	C₂×C₃×C₆₄	C₂×C₃₂	1	2	yes	—
94	C₂×C₃×C₆₄	C₂×C₃₂	1	2	yes	—
95	C₂×C₃×C₆₄	C₂²×C₁₆	1	1	yes	characteristic
96	C₃×C₄×C₃₂	C₆₄	1	2	yes	—
97	C₃×C₄×C₃₂	C₂×C₃₂	1	2	yes	—
98	C₃×C₁₂₈	C₄×C₁₆	1	2	yes	—
99	C₃×C₁₂₈	C₂×C₃₂	1	4	yes	—
100	C₃×C₁₂₈	C₂×C₄×C₈	1	1	yes	characteristic
101	C₃×C₁₂₈	C₂²×C₁₆	1	1	yes	characteristic
102	C₂²×C₆₄	C₃×C₃₂	1	1	yes	characteristic
103	C₂²×C₆₄	C₂×C₃×C₁₆	1	1	yes	characteristic
104	C₂×C₄×C₃₂	C₃×C₃₂	1	1	yes	characteristic
105	C₂×C₁₂₈	C₃×C₃₂	1	4	yes	—
106	C₂×C₁₂₈	C₃×C₄×C₈	1	2	yes	—
107	C₂×C₁₂₈	C₂×C₃×C₁₆	1	1	yes	characteristic
108	C₂×C₁₂₈	C₂×C₃×C₁₆	1	2	yes	—
109	C₂×C₁₂₈	C₂×C₃×C₁₆	1	2	yes	—
110	C₂×C₁₂₈	C₂²×C₃×C₈	1	1	yes	characteristic
111	C₄×C₆₄	C₃×C₃₂	1	2	yes	—
112	C₄×C₆₄	C₂×C₃×C₁₆	1	2	yes	—
113	C₂₅₆	C₃×C₄×C₈	1	2	yes	—
114	C₂₅₆	C₂×C₃×C₁₆	1	4	yes	—
115	C₂₅₆	C₂×C₃×C₄²	1	1	yes	characteristic
116	C₂₅₆	C₂²×C₃×C₈	1	1	yes	characteristic
117	C₂²×C₃×C₁₆	C₁₂₈	1	1	yes	characteristic
118	C₂²×C₃×C₁₆	C₂×C₆₄	1	1	yes	characteristic
119	C₂×C₃×C₄×C₈	C₁₂₈	1	1	yes	characteristic
120	C₂×C₃×C₃₂	C₁₂₈	1	4	yes	—
121	C₂×C₃×C₃₂	C₄×C₃₂	1	2	yes	—
122	C₂×C₃×C₃₂	C₂×C₆₄	1	1	yes	characteristic
123	C₂×C₃×C₃₂	C₂×C₆₄	1	2	yes	—
124	C₂×C₃×C₃₂	C₂×C₆₄	1	2	yes	—
125	C₂×C₃×C₃₂	C₂²×C₃₂	1	1	yes	characteristic
126	C₃×C₄×C₁₆	C₁₂₈	1	2	yes	—
127	C₃×C₄×C₁₆	C₂×C₆₄	1	2	yes	—
128	C₃×C₆₄	C₄×C₃₂	1	2	yes	—
129	C₃×C₆₄	C₂×C₆₄	1	4	yes	—
130	C₃×C₆₄	C₂×C₄×C₁₆	1	1	yes	characteristic
131	C₃×C₆₄	C₂²×C₃₂	1	1	yes	characteristic
132	C₂²×C₃₂	C₃×C₆₄	1	1	yes	characteristic
133	C₂²×C₃₂	C₂×C₃×C₃₂	1	1	yes	characteristic
134	C₂×C₄×C₁₆	C₃×C₆₄	1	1	yes	characteristic
135	C₂×C₆₄	C₃×C₆₄	1	4	yes	—
136	C₂×C₆₄	C₃×C₄×C₁₆	1	2	yes	—
137	C₂×C₆₄	C₂×C₃×C₃₂	1	1	yes	characteristic
138	C₂×C₆₄	C₂×C₃×C₃₂	1	2	yes	—
139	C₂×C₆₄	C₂×C₃×C₃₂	1	2	yes	—
140	C₂×C₆₄	C₂²×C₃×C₁₆	1	1	yes	characteristic
141	C₄×C₃₂	C₃×C₆₄	1	2	yes	—
142	C₄×C₃₂	C₂×C₃×C₃₂	1	2	yes	—
143	C₁₂₈	C₃×C₄×C₁₆	1	2	yes	—
144	C₁₂₈	C₂×C₃×C₃₂	1	4	yes	—
145	C₁₂₈	C₂×C₃×C₄×C₈	1	1	yes	characteristic
146	C₁₂₈	C₂²×C₃×C₁₆	1	1	yes	characteristic
147	C₂²×C₃×C₈	C₂₅₆	1	1	yes	characteristic
148	C₂²×C₃×C₈	C₂×C₁₂₈	1	1	yes	characteristic
149	C₂×C₃×C₄²	C₂₅₆	1	1	yes	characteristic
150	C₂×C₃×C₁₆	C₂₅₆	1	4	yes	—
151	C₂×C₃×C₁₆	C₄×C₆₄	1	2	yes	—
152	C₂×C₃×C₁₆	C₂×C₁₂₈	1	1	yes	characteristic
153	C₂×C₃×C₁₆	C₂×C₁₂₈	1	2	yes	—
154	C₂×C₃×C₁₆	C₂×C₁₂₈	1	2	yes	—
155	C₂×C₃×C₁₆	C₂²×C₆₄	1	1	yes	characteristic
156	C₃×C₄×C₈	C₂₅₆	1	2	yes	—
157	C₃×C₄×C₈	C₂×C₁₂₈	1	2	yes	—
158	C₃×C₃₂	C₄×C₆₄	1	2	yes	—
159	C₃×C₃₂	C₂×C₁₂₈	1	4	yes	—
160	C₃×C₃₂	C₂×C₄×C₃₂	1	1	yes	characteristic
161	C₃×C₃₂	C₂²×C₆₄	1	1	yes	characteristic
162	C₂²×C₁₆	C₃×C₁₂₈	1	1	yes	characteristic
163	C₂²×C₁₆	C₂×C₃×C₆₄	1	1	yes	characteristic
164	C₂×C₄×C₈	C₃×C₁₂₈	1	1	yes	characteristic
165	C₂×C₃₂	C₃×C₁₂₈	1	4	yes	—
166	C₂×C₃₂	C₃×C₄×C₃₂	1	2	yes	—
167	C₂×C₃₂	C₂×C₃×C₆₄	1	1	yes	characteristic
168	C₂×C₃₂	C₂×C₃×C₆₄	1	2	yes	—
169	C₂×C₃₂	C₂×C₃×C₆₄	1	2	yes	—
170	C₂×C₃₂	C₂²×C₃×C₃₂	1	1	yes	characteristic
171	C₄×C₁₆	C₃×C₁₂₈	1	2	yes	—
172	C₄×C₁₆	C₂×C₃×C₆₄	1	2	yes	—
173	C₆₄	C₃×C₄×C₃₂	1	2	yes	—
174	C₆₄	C₂×C₃×C₆₄	1	4	yes	—
175	C₆₄	C₂×C₃×C₄×C₁₆	1	1	yes	characteristic
176	C₆₄	C₂²×C₃×C₃₂	1	1	yes	characteristic
177	C₂²×C₃×C₄	C₅₁₂	1	2	yes	—
178	C₂²×C₃×C₄	C₂×C₂₅₆	1	1	yes	characteristic
179	C₂×C₃×C₈	C₅₁₂	1	4	yes	—
180	C₂×C₃×C₈	C₄×C₁₂₈	1	2	yes	—
181	C₂×C₃×C₈	C₂×C₂₅₆	1	1	yes	characteristic
182	C₂×C₃×C₈	C₂×C₂₅₆	1	2	yes	—
183	C₂×C₃×C₈	C₂×C₂₅₆	1	2	yes	—
184	C₂×C₃×C₈	C₂²×C₁₂₈	1	1	yes	characteristic
185	C₃×C₄²	C₅₁₂	1	2	yes	—
186	C₃×C₄²	C₂×C₂₅₆	1	2	yes	—
187	C₃×C₁₆	C₄×C₁₂₈	1	2	yes	—
188	C₃×C₁₆	C₂×C₂₅₆	1	4	yes	—
189	C₃×C₁₆	C₂×C₄×C₆₄	1	1	yes	characteristic
190	C₃×C₁₆	C₂²×C₁₂₈	1	1	yes	characteristic
191	C₂²×C₈	C₃×C₂₅₆	1	1	yes	characteristic
192	C₂²×C₈	C₂×C₃×C₁₂₈	1	1	yes	characteristic
193	C₂×C₄²	C₃×C₂₅₆	1	1	yes	characteristic
194	C₂×C₁₆	C₃×C₂₅₆	1	4	yes	—
195	C₂×C₁₆	C₃×C₄×C₆₄	1	2	yes	—
196	C₂×C₁₆	C₂×C₃×C₁₂₈	1	1	yes	characteristic
197	C₂×C₁₆	C₂×C₃×C₁₂₈	1	2	yes	—
198	C₂×C₁₆	C₂×C₃×C₁₂₈	1	2	yes	—
199	C₂×C₁₆	C₂²×C₃×C₆₄	1	1	yes	characteristic
200	C₄×C₈	C₃×C₂₅₆	1	2	yes	—
201	C₄×C₈	C₂×C₃×C₁₂₈	1	2	yes	—
202	C₃₂	C₃×C₄×C₆₄	1	2	yes	—
203	C₃₂	C₂×C₃×C₁₂₈	1	4	yes	—
204	C₃₂	C₂×C₃×C₄×C₃₂	1	1	yes	characteristic
205	C₃₂	C₂²×C₃×C₆₄	1	1	yes	characteristic
206	C₂³×C₃	C₂×C₅₁₂	1	1	yes	characteristic
207	C₂×C₃×C₄	C₁₀₂₄	1	8	yes	direct product factor
208	C₂×C₃×C₄	C₂×C₅₁₂	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₅₁₂	1	2	yes	—
210	C₂×C₃×C₄	C₂×C₅₁₂	1	4	yes	—
211	C₂×C₃×C₄	C₄×C₂₅₆	1	2	yes	—
212	C₂×C₃×C₄	C₂²×C₂₅₆	1	1	yes	characteristic
213	C₃×C₈	C₂×C₅₁₂	1	4	yes	—
214	C₃×C₈	C₄×C₂₅₆	1	2	yes	—
215	C₃×C₈	C₂²×C₂₅₆	1	1	yes	characteristic
216	C₃×C₈	C₂×C₄×C₁₂₈	1	1	yes	characteristic
217	C₂²×C₄	C₃×C₅₁₂	1	2	yes	—
218	C₂²×C₄	C₂×C₃×C₂₅₆	1	1	yes	characteristic
219	C₂×C₈	C₃×C₅₁₂	1	4	yes	—
220	C₂×C₈	C₂×C₃×C₂₅₆	1	1	yes	characteristic
221	C₂×C₈	C₂×C₃×C₂₅₆	1	2	yes	—
222	C₂×C₈	C₂×C₃×C₂₅₆	1	2	yes	—
223	C₂×C₈	C₃×C₄×C₁₂₈	1	2	yes	—
224	C₂×C₈	C₂²×C₃×C₁₂₈	1	1	yes	characteristic
225	C₄²	C₃×C₅₁₂	1	2	yes	—
226	C₄²	C₂×C₃×C₂₅₆	1	2	yes	—
227	C₁₆	C₂×C₃×C₂₅₆	1	4	yes	—
228	C₁₆	C₃×C₄×C₁₂₈	1	2	yes	—
229	C₁₆	C₂²×C₃×C₁₂₈	1	1	yes	characteristic
230	C₁₆	C₂×C₃×C₄×C₆₄	1	1	yes	characteristic
231	C₂²×C₃	C₂×C₁₀₂₄	1	4	yes	—
232	C₂²×C₃	C₄×C₅₁₂	1	2	yes	—
233	C₂²×C₃	C₂²×C₅₁₂	1	1	yes	characteristic
234	C₃×C₄	C₂×C₁₀₂₄	1	8	yes	direct product factor
235	C₃×C₄	C₄×C₅₁₂	1	2	yes	—
236	C₃×C₄	C₂²×C₅₁₂	1	1	yes	characteristic
237	C₃×C₄	C₂×C₄×C₂₅₆	1	1	yes	characteristic
238	C₂³	C₂×C₃×C₅₁₂	1	1	yes	characteristic
239	C₂×C₄	C₃×C₁₀₂₄	1	8	yes	direct product factor
240	C₂×C₄	C₂×C₃×C₅₁₂	1	1	yes	characteristic
241	C₂×C₄	C₂×C₃×C₅₁₂	1	2	yes	—
242	C₂×C₄	C₂×C₃×C₅₁₂	1	4	yes	—
243	C₂×C₄	C₃×C₄×C₂₅₆	1	2	yes	—
244	C₂×C₄	C₂²×C₃×C₂₅₆	1	1	yes	characteristic
245	C₈	C₂×C₃×C₅₁₂	1	4	yes	—
246	C₈	C₃×C₄×C₂₅₆	1	2	yes	—
247	C₈	C₂²×C₃×C₂₅₆	1	1	yes	characteristic
248	C₈	C₂×C₃×C₄×C₁₂₈	1	1	yes	characteristic
249	C₂×C₃	C₄×C₁₀₂₄	1	4	yes	direct product factor
250	C₂×C₃	C₂²×C₁₀₂₄	1	2	yes	—
251	C₂×C₃	C₂×C₄×C₅₁₂	1	1	yes	characteristic
252	C₂²	C₂×C₃×C₁₀₂₄	1	4	yes	—
253	C₂²	C₃×C₄×C₅₁₂	1	2	yes	—
254	C₂²	C₂²×C₃×C₅₁₂	1	1	yes	characteristic
255	C₄	C₂×C₃×C₁₀₂₄	1	8	yes	direct product factor
256	C₄	C₃×C₄×C₅₁₂	1	2	yes	—
257	C₄	C₂²×C₃×C₅₁₂	1	1	yes	characteristic
258	C₄	C₂×C₃×C₄×C₂₅₆	1	1	yes	characteristic
259	C₃	C₂×C₄×C₁₀₂₄	1	1	yes	characteristic, direct product factor, Sylow
260	C₂	C₃×C₄×C₁₀₂₄	1	4	yes	direct product factor
261	C₂	C₂²×C₃×C₁₀₂₄	1	2	yes	—
262	C₂	C₂×C₃×C₄×C₅₁₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₅₁₂	C₂³×C₃
socle, prime radical	C₂³×C₃	C₂×C₅₁₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₁₀₂₄	C₂	direct (N × H)
4. C₂×C₄×C₁₀₂₄	C₃	direct (N × H)
10. C₂×C₃×C₁₀₂₄	C₄	direct (N × H)
14. C₄×C₁₀₂₄	C₂×C₃	direct (N × H)
25. C₃×C₁₀₂₄	C₂×C₄	direct (N × H)
31. C₂×C₁₀₂₄	C₃×C₄	direct (N × H)
57. C₁₀₂₄	C₂×C₃×C₄	direct (N × H)
207. C₂×C₃×C₄	C₁₀₂₄	direct (N × H)
234. C₃×C₄	C₂×C₁₀₂₄	direct (N × H)
239. C₂×C₄	C₃×C₁₀₂₄	direct (N × H)
249. C₂×C₃	C₄×C₁₀₂₄	direct (N × H)
255. C₄	C₂×C₃×C₁₀₂₄	direct (N × H)
259. C₃	C₂×C₄×C₁₀₂₄	direct (N × H)
260. C₂	C₃×C₄×C₁₀₂₄	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×C₃×C₄×C₅₁₂	2
C₂²×C₃×C₁₀₂₄	2
C₃×C₄×C₁₀₂₄	2
C₂×C₄×C₁₀₂₄	3

Subgroups — C2×C3×C4×C1024 #72201817

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×C₁₀₂₄ #72201817