Subgroups — C₄₁×#14694 #23790895

Order 10496.

7034 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#120×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#386	C₂	1	6	yes	maximal, complemented
4	C₂×C₄₁×#379	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×C₂²⋊#95	C₂	1	6	yes	maximal, complemented
6	C₂⁶×C₄₁	C₂²	1	1	yes	characteristic
7	C₂³×D₄×C₄₁	C₂²	1	3	yes	—
8	C₂⁴×C₄×C₄₁	—	2	4	no	—
9	C₂⁴×C₄×C₄₁	—	2	24	no	—
10	C₂⁴×C₄×C₄₁	C₂²	1	6	yes	—
11	C₂×#120×C₄₁	—	2	4	no	—
12	C₂×#120×C₄₁	—	2	4	no	—
13	C₂×#120×C₄₁	C₄	1	8	yes	complemented
14	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	6	yes	—
15	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	3	yes	—
16	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	complemented
17	C₄₁×#386	—	2	24	no	—
18	C₄₁×#379	—	2	4	no	—
19	C₂×#95×C₄₁	C₂²	1	12	yes	complemented
20	C₂⁵×C₄₁	—	2	12	no	—
21	C₂⁵×C₄₁	—	2	12	no	—
22	C₂⁵×C₄₁	C₂×C₄	1	4	yes	—
23	C₂⁵×C₄₁	D₄	1	4	yes	—
24	C₂⁵×C₄₁	D₄	1	4	yes	—
25	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
26	C₂⁵×C₄₁	C₂³	1	3	yes	—
27	C₂²×D₄×C₄₁	—	2	6	no	—
28	C₂²×D₄×C₄₁	—	2	6	no	—
29	C₂²×D₄×C₄₁	—	2	12	no	—
30	C₂²×D₄×C₄₁	—	2	12	no	—
31	C₂²×D₄×C₄₁	C₂×C₄	1	12	yes	complemented
32	C₂³×C₄×C₄₁	—	2	96	no	—
33	C₂³×C₄×C₄₁	—	2	128	no	—
34	C₂³×C₄×C₄₁	—	2	12	no	—
35	C₂³×C₄×C₄₁	—	2	12	no	—
36	C₂³×C₄×C₄₁	—	2	48	no	—
37	C₂³×C₄×C₄₁	D₄	1	24	yes	complemented
38	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
39	C₂³×C₄×C₄₁	C₂³	1	3	yes	—
40	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
41	#120×C₄₁	—	2	32	no	—
42	C₂×C₂²⋊C₄×C₄₁	—	2	6	no	—
43	C₂×C₂²⋊C₄×C₄₁	—	2	6	no	—
44	C₂×C₂²⋊C₄×C₄₁	—	2	12	no	—
45	C₂×C₂²⋊C₄×C₄₁	—	2	12	no	—
46	C₂×C₂²⋊C₄×C₄₁	—	2	12	no	—
47	C₂×C₂²⋊C₄×C₄₁	—	2	24	no	—
48	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	12	yes	complemented
49	#95×C₄₁	—	2	24	no	—
50	C₂⁴×C₄₁	—	2	32	no	—
51	C₂⁴×C₄₁	—	2	96	no	—
52	C₂⁴×C₄₁	—	2	48	no	—
53	C₂⁴×C₄₁	—	2	24	no	—
54	C₂⁴×C₄₁	—	2	48	no	—
55	C₂⁴×C₄₁	—	2	24	no	—
56	C₂⁴×C₄₁	—	2	12	no	—
57	C₂⁴×C₄₁	—	2	12	no	—
58	C₂⁴×C₄₁	—	2	6	no	—
59	C₂⁴×C₄₁	—	2	1	no	—
60	C₂⁴×C₄₁	—	2	1	no	—
61	C₂⁴×C₄₁	—	2	6	no	—
62	C₂⁴×C₄₁	C₂²⋊C₄	1	16	yes	complemented
63	C₂⁴×C₄₁	C₂²×C₄	1	6	yes	—
64	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	complemented
65	C₂⁴×C₄₁	C₂×D₄	1	6	yes	—
66	C₂⁴×C₄₁	C₂×D₄	1	6	yes	—
67	C₂⁴×C₄₁	C₂×D₄	1	12	yes	—
68	C₂⁴×C₄₁	#41	1	12	yes	—
69	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
70	C₂×D₄×C₄₁	—	2	24	no	—
71	C₂×D₄×C₄₁	—	2	48	no	—
72	C₂×D₄×C₄₁	—	4	12	no	—
73	C₂×D₄×C₄₁	—	4	24	no	—
74	C₂×D₄×C₄₁	—	4	12	no	—
75	C₂²×C₄×C₄₁	—	2	256	no	—
76	C₂²×C₄×C₄₁	—	2	96	no	—
77	C₂²×C₄×C₄₁	—	2	96	no	—
78	C₂²×C₄×C₄₁	—	2	48	no	—
79	C₂²×C₄×C₄₁	—	2	4	no	—
80	C₂²×C₄×C₄₁	—	2	12	no	—
81	C₂²×C₄×C₄₁	—	2	96	no	—
82	C₂²×C₄×C₄₁	—	2	6	no	—
83	C₂²×C₄×C₄₁	—	2	3	no	—
84	C₂²×C₄×C₄₁	—	2	3	no	—
85	C₂²×C₄×C₄₁	—	2	6	no	—
86	C₂²×C₄×C₄₁	—	2	12	no	—
87	C₂²×C₄×C₄₁	—	2	4	no	—
88	C₂²×C₄×C₄₁	—	2	12	no	—
89	C₂²×C₄×C₄₁	C₂²×C₄	1	6	yes	complemented
90	C₂²×C₄×C₄₁	C₂×D₄	1	24	yes	—
91	C₂²⋊C₄×C₄₁	—	2	24	no	—
92	C₂²⋊C₄×C₄₁	—	4	24	no	—
93	C₂²⋊C₄×C₄₁	—	4	16	no	—
94	C₂³×C₄₁	—	2	32	no	—
95	C₂³×C₄₁	—	2	128	no	—
96	C₂³×C₄₁	—	2	96	no	—
97	C₂³×C₄₁	—	2	128	no	—
98	C₂³×C₄₁	—	2	24	no	—
99	C₂³×C₄₁	—	2	24	no	—
100	C₂³×C₄₁	—	2	12	no	—
101	C₂³×C₄₁	—	2	24	no	—
102	C₂³×C₄₁	—	2	12	no	—
103	C₂³×C₄₁	—	2	24	no	—
104	C₂³×C₄₁	—	2	48	no	—
105	C₂³×C₄₁	—	2	96	no	—
106	C₂³×C₄₁	—	2	2	no	—
107	C₂³×C₄₁	—	2	12	no	—
108	C₂³×C₄₁	—	4	4	no	—
109	C₂³×C₄₁	—	4	3	no	—
110	C₂³×C₄₁	—	4	6	no	—
111	C₂³×C₄₁	—	4	3	no	—
112	C₂³×C₄₁	—	4	12	no	—
113	C₂³×C₄₁	C₂×C₂²⋊C₄	1	12	yes	—
114	C₂³×C₄₁	C₄×D₄	1	24	yes	—
115	C₂³×C₄₁	#120	1	12	yes	—
116	C₂³×C₄₁	#121	1	24	yes	—
117	C₂³×C₄₁	#123	1	12	yes	—
118	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
119	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
120	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
121	C₂³×C₄₁	C₂²×D₄	1	6	yes	—
122	C₂³×C₄₁	C₂×#41	1	6	yes	—
123	D₄×C₄₁	—	4	48	no	—
124	C₂×C₄×C₄₁	—	2	192	no	—
125	C₂×C₄×C₄₁	—	2	24	no	—
126	C₂×C₄×C₄₁	—	2	24	no	—
127	C₂×C₄×C₄₁	—	2	6	no	—
128	C₂×C₄×C₄₁	—	2	12	no	—
129	C₂×C₄×C₄₁	—	4	3	no	—
130	C₂×C₄×C₄₁	—	4	6	no	—
131	C₂×C₄×C₄₁	—	4	3	no	—
132	C₂×C₄×C₄₁	—	4	4	no	—
133	C₂×C₄×C₄₁	—	4	12	no	—
134	C₂×C₄×C₄₁	—	4	12	no	—
135	C₂×C₄×C₄₁	#120	1	16	yes	complemented
136	#14694	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₄₁	—	2	128	no	—
138	C₂²×C₄₁	—	2	12	no	—
139	C₂²×C₄₁	—	2	48	no	—
140	C₂²×C₄₁	—	2	12	no	—
141	C₂²×C₄₁	—	2	24	no	—
142	C₂²×C₄₁	—	2	24	no	—
143	C₂²×C₄₁	—	2	48	no	—
144	C₂²×C₄₁	—	4	6	no	—
145	C₂²×C₄₁	—	4	8	no	—
146	C₂²×C₄₁	—	8	1	no	—
147	C₂²×C₄₁	—	8	4	no	—
148	C₂²×C₄₁	—	8	3	no	—
149	C₂²×C₄₁	#386	1	24	yes	complemented
150	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₄₁	C₂×C₄×D₄	1	6	yes	—
152	C₂²×C₄₁	C₂×#120	1	4	yes	—
153	C₂²×C₄₁	C₂×#120	1	3	yes	—
154	C₂²×C₄₁	C₂×#121	1	6	yes	—
155	C₂²×C₄₁	C₂×#123	1	3	yes	—
156	C₂²×C₄₁	#525	1	12	yes	—
157	C₄×C₄₁	—	2	16	no	—
158	C₄×C₄₁	—	4	6	no	—
159	C₄×C₄₁	—	8	4	no	—
160	C₂⁵×C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
161	C₂²×#120	C₂×C₄₁	1	1	yes	characteristic
162	C₂×#386	C₂×C₄₁	1	6	yes	complemented
163	C₂×#379	C₂×C₄₁	1	1	yes	characteristic, complemented
164	C₂²⋊#95	C₂×C₄₁	1	6	yes	complemented
165	C₂×C₄₁	—	2	24	no	—
166	C₂×C₄₁	—	8	2	no	—
167	C₂×C₄₁	C₂×#386	1	3	yes	—
168	C₂×C₄₁	C₄×#120	1	4	yes	—
169	C₂×C₄₁	#2846	1	1	yes	characteristic
170	C₂×C₄₁	#2847	1	4	yes	—
171	C₂×C₄₁	#2849	1	3	yes	—
172	C₂⁶	C₂²×C₄₁	1	1	yes	characteristic
173	C₂³×D₄	C₂²×C₄₁	1	3	yes	—
174	C₂⁴×C₄	—	2	4	no	—
175	C₂⁴×C₄	—	2	24	no	—
176	C₂⁴×C₄	C₂²×C₄₁	1	6	yes	—
177	C₂×#120	—	2	4	no	—
178	C₂×#120	—	2	4	no	—
179	C₂×#120	C₄×C₄₁	1	8	yes	complemented
180	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	6	yes	—
181	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	3	yes	—
182	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	complemented
183	#386	—	2	24	no	—
184	#379	—	2	4	no	—
185	C₂×#95	C₂²×C₄₁	1	12	yes	complemented
186	C₄₁	#14694	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁵	—	2	12	no	—
188	C₂⁵	—	2	12	no	—
189	C₂⁵	C₂×C₄×C₄₁	1	4	yes	—
190	C₂⁵	D₄×C₄₁	1	4	yes	—
191	C₂⁵	D₄×C₄₁	1	4	yes	—
192	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
193	C₂⁵	C₂³×C₄₁	1	3	yes	—
194	C₂²×D₄	—	2	6	no	—
195	C₂²×D₄	—	2	6	no	—
196	C₂²×D₄	—	2	12	no	—
197	C₂²×D₄	—	2	12	no	—
198	C₂²×D₄	C₂×C₄×C₄₁	1	12	yes	complemented
199	C₂³×C₄	—	2	96	no	—
200	C₂³×C₄	—	2	128	no	—
201	C₂³×C₄	—	2	12	no	—
202	C₂³×C₄	—	2	12	no	—
203	C₂³×C₄	—	2	48	no	—
204	C₂³×C₄	D₄×C₄₁	1	24	yes	complemented
205	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
206	C₂³×C₄	C₂³×C₄₁	1	3	yes	—
207	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
208	#120	—	2	32	no	—
209	C₂×C₂²⋊C₄	—	2	6	no	—
210	C₂×C₂²⋊C₄	—	2	6	no	—
211	C₂×C₂²⋊C₄	—	2	12	no	—
212	C₂×C₂²⋊C₄	—	2	12	no	—
213	C₂×C₂²⋊C₄	—	2	12	no	—
214	C₂×C₂²⋊C₄	—	2	24	no	—
215	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	12	yes	complemented
216	#95	—	2	24	no	—
217	C₂⁴	—	2	32	no	—
218	C₂⁴	—	2	96	no	—
219	C₂⁴	—	2	48	no	—
220	C₂⁴	—	2	24	no	—
221	C₂⁴	—	2	48	no	—
222	C₂⁴	—	2	24	no	—
223	C₂⁴	—	2	12	no	—
224	C₂⁴	—	2	12	no	—
225	C₂⁴	—	2	6	no	—
226	C₂⁴	—	2	1	no	—
227	C₂⁴	—	2	1	no	—
228	C₂⁴	—	2	6	no	—
229	C₂⁴	C₂²⋊C₄×C₄₁	1	16	yes	complemented
230	C₂⁴	C₂²×C₄×C₄₁	1	6	yes	—
231	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	complemented
232	C₂⁴	C₂×D₄×C₄₁	1	6	yes	—
233	C₂⁴	C₂×D₄×C₄₁	1	6	yes	—
234	C₂⁴	C₂×D₄×C₄₁	1	12	yes	—
235	C₂⁴	#41×C₄₁	1	12	yes	—
236	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
237	C₂×D₄	—	2	24	no	—
238	C₂×D₄	—	2	48	no	—
239	C₂×D₄	—	4	12	no	—
240	C₂×D₄	—	4	24	no	—
241	C₂×D₄	—	4	12	no	—
242	C₂²×C₄	—	2	256	no	—
243	C₂²×C₄	—	2	96	no	—
244	C₂²×C₄	—	2	96	no	—
245	C₂²×C₄	—	2	48	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	12	no	—
248	C₂²×C₄	—	2	96	no	—
249	C₂²×C₄	—	2	6	no	—
250	C₂²×C₄	—	2	3	no	—
251	C₂²×C₄	—	2	3	no	—
252	C₂²×C₄	—	2	6	no	—
253	C₂²×C₄	—	2	12	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	12	no	—
256	C₂²×C₄	C₂²×C₄×C₄₁	1	6	yes	complemented
257	C₂²×C₄	C₂×D₄×C₄₁	1	24	yes	—
258	C₂²⋊C₄	—	2	24	no	—
259	C₂²⋊C₄	—	4	24	no	—
260	C₂²⋊C₄	—	4	16	no	—
261	C₂³	—	2	32	no	—
262	C₂³	—	2	128	no	—
263	C₂³	—	2	128	no	—
264	C₂³	—	2	96	no	—
265	C₂³	—	2	24	no	—
266	C₂³	—	2	12	no	—
267	C₂³	—	2	24	no	—
268	C₂³	—	2	24	no	—
269	C₂³	—	2	12	no	—
270	C₂³	—	2	24	no	—
271	C₂³	—	2	48	no	—
272	C₂³	—	2	96	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	12	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	3	no	—
277	C₂³	—	4	6	no	—
278	C₂³	—	4	3	no	—
279	C₂³	—	4	12	no	—
280	C₂³	C₂×C₂²⋊C₄×C₄₁	1	12	yes	—
281	C₂³	C₄×D₄×C₄₁	1	24	yes	—
282	C₂³	#120×C₄₁	1	12	yes	—
283	C₂³	#121×C₄₁	1	24	yes	—
284	C₂³	#123×C₄₁	1	12	yes	—
285	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
286	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
287	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
288	C₂³	C₂²×D₄×C₄₁	1	6	yes	—
289	C₂³	C₂×#41×C₄₁	1	6	yes	—
290	D₄	—	4	48	no	—
291	C₂×C₄	—	2	192	no	—
292	C₂×C₄	—	2	24	no	—
293	C₂×C₄	—	2	24	no	—
294	C₂×C₄	—	2	6	no	—
295	C₂×C₄	—	2	12	no	—
296	C₂×C₄	—	4	3	no	—
297	C₂×C₄	—	4	6	no	—
298	C₂×C₄	—	4	3	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	12	no	—
301	C₂×C₄	—	4	12	no	—
302	C₂×C₄	#120×C₄₁	1	16	yes	complemented
303	C₂²	—	2	128	no	—
304	C₂²	—	2	24	no	—
305	C₂²	—	2	48	no	—
306	C₂²	—	2	12	no	—
307	C₂²	—	2	48	no	—
308	C₂²	—	2	12	no	—
309	C₂²	—	2	24	no	—
310	C₂²	—	4	6	no	—
311	C₂²	—	4	8	no	—
312	C₂²	—	8	1	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	3	no	—
315	C₂²	C₄₁×#386	1	24	yes	complemented
316	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
317	C₂²	C₂×C₄×D₄×C₄₁	1	6	yes	—
318	C₂²	C₂×#120×C₄₁	1	3	yes	—
319	C₂²	C₂×#120×C₄₁	1	4	yes	—
320	C₂²	C₂×#121×C₄₁	1	6	yes	—
321	C₂²	C₂×#123×C₄₁	1	3	yes	—
322	C₂²	C₄₁×#525	1	12	yes	—
323	C₄	—	2	16	no	—
324	C₄	—	4	6	no	—
325	C₄	—	8	4	no	—
326	C₂	—	2	24	no	—
327	C₂	—	8	2	no	—
328	C₂	C₂×C₄₁×#386	1	3	yes	—
329	C₂	C₄×#120×C₄₁	1	4	yes	—
330	C₂	C₄₁×#2846	1	1	yes	characteristic
331	C₂	C₄₁×#2847	1	4	yes	—
332	C₂	C₄₁×#2849	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₄₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂⁴	C₂⁴×C₄₁
prime radical	C₂²×#120×C₄₁	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁵×C₄×C₄₁	C₂	semidirect
3. C₂×C₄₁×#386	C₂	semidirect
4. C₂×C₄₁×#379	C₂	semidirect
5. C₄₁×C₂²⋊#95	C₂	semidirect
13. C₂×#120×C₄₁	C₄	semidirect
16. C₂²×C₂²⋊C₄×C₄₁	C₂²	semidirect
19. C₂×#95×C₄₁	C₂²	semidirect
31. C₂²×D₄×C₄₁	C₂×C₄	semidirect
37. C₂³×C₄×C₄₁	D₄	semidirect
48. C₂×C₂²⋊C₄×C₄₁	C₂×C₄	semidirect
62. C₂⁴×C₄₁	C₂²⋊C₄	semidirect
64. C₂⁴×C₄₁	C₂²×C₄	semidirect
89. C₂²×C₄×C₄₁	C₂²×C₄	semidirect
135. C₂×C₄×C₄₁	#120	semidirect
136. #14694	C₄₁	direct (N × H)
149. C₂²×C₄₁	#386	semidirect
160. C₂⁵×C₄	C₂×C₄₁	semidirect
162. C₂×#386	C₂×C₄₁	semidirect
163. C₂×#379	C₂×C₄₁	semidirect
164. C₂²⋊#95	C₂×C₄₁	semidirect
179. C₂×#120	C₄×C₄₁	semidirect
182. C₂²×C₂²⋊C₄	C₂²×C₄₁	semidirect
185. C₂×#95	C₂²×C₄₁	semidirect
186. C₄₁	#14694	direct (N × H)
198. C₂²×D₄	C₂×C₄×C₄₁	semidirect
204. C₂³×C₄	D₄×C₄₁	semidirect
215. C₂×C₂²⋊C₄	C₂×C₄×C₄₁	semidirect
229. C₂⁴	C₂²⋊C₄×C₄₁	semidirect
231. C₂⁴	C₂²×C₄×C₄₁	semidirect
256. C₂²×C₄	C₂²×C₄×C₄₁	semidirect
302. C₂×C₄	#120×C₄₁	semidirect
315. C₂²	C₄₁×#386	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁵×C₄×C₄₁	2
C₂²×#120×C₄₁	2
C₂×C₄₁×#386	2
C₂×C₄₁×#379	2
C₄₁×C₂²⋊#95	2
#14694	41

Subgroups — C41×#14694 #23790895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#14694 #23790895