Subgroups — C₄₁×#29185 #23805386

Order 10496.

590 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2647	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2330	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2088	C₂	1	2	yes	maximal
4	C₄₁×#2088	C₂	1	2	yes	maximal
5	C₄₁×#1992	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#1991	C₂	1	1	yes	maximal, characteristic
7	C₄₁×#1920	C₂	1	2	yes	maximal
8	C₄₁×#1919	C₂	1	2	yes	maximal
9	C₄₁×#1844	C₂	1	1	yes	maximal, characteristic
10	C₄₁×#1836	C₂	1	1	yes	maximal, characteristic
11	C₄₁×#1768	C₂	1	1	yes	maximal, characteristic
12	C₂×#125×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×#106×C₄₁	C₂²	1	2	yes	—
14	C₂×#106×C₄₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
19	C₄₁×#400	—	2	2	no	—
20	C₄₁×#398	—	2	2	no	—
21	C₄₁×#389	—	2	2	no	—
22	C₄₁×#389	C₂²	1	2	yes	—
23	C₄₁×#389	C₂²	1	2	yes	—
24	C₄₁×#389	C₂²	1	2	yes	—
25	C₄₁×#387	—	2	2	no	—
26	C₄₁×#384	C₂²	1	1	yes	characteristic
27	C₄₁×#384	C₂²	1	1	yes	characteristic
28	C₄₁×#382	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
30	C₄₁×#376	C₂²	1	1	yes	characteristic
31	C₄₁×#340	C₂²	1	2	yes	—
32	C₄₁×#340	C₂²	1	2	yes	—
33	C₄₁×#340	C₂²	1	2	yes	—
34	C₄₁×#340	C₂²	1	2	yes	—
35	C₄₁×#340	C₂²	1	2	yes	—
36	C₄₁×#340	C₂²	1	2	yes	—
37	C₄₁×#336	C₂²	1	1	yes	characteristic
38	C₄₁×#336	C₂²	1	1	yes	characteristic
39	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
43	#125×C₄₁	—	2	2	no	—
44	#125×C₄₁	Q₈	1	4	yes	—
45	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₁	—	2	2	no	—
56	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
59	#106×C₄₁	—	2	4	no	—
60	#106×C₄₁	—	2	4	no	—
61	C₄⋊C₈×C₄₁	—	2	2	no	—
62	C₄⋊C₈×C₄₁	—	2	2	no	—
63	#97×C₄₁	—	2	2	no	—
64	C₄×C₈×C₄₁	—	2	2	no	—
65	#95×C₄₁	—	2	2	no	—
66	#95×C₄₁	—	2	2	no	—
67	#95×C₄₁	—	2	2	no	—
68	#95×C₄₁	—	2	2	no	—
69	#95×C₄₁	—	2	2	no	—
70	#95×C₄₁	D₄	1	2	yes	—
71	#95×C₄₁	D₄	1	2	yes	—
72	C₂²×C₄×C₄₁	—	2	2	no	—
73	C₂²×C₄×C₄₁	—	2	2	no	—
74	C₂²×C₄×C₄₁	—	2	2	no	—
75	C₂²×C₄×C₄₁	—	2	2	no	—
76	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₄₁	—	2	1	no	—
82	C₂×C₈×C₄₁	—	2	1	no	—
83	C₂×C₈×C₄₁	—	2	1	no	—
84	C₂×C₈×C₄₁	—	2	1	no	—
85	C₂×C₈×C₄₁	—	2	1	no	—
86	C₂×C₈×C₄₁	—	2	1	no	—
87	C₂×C₈×C₄₁	—	2	1	no	—
88	C₂×C₈×C₄₁	—	2	1	no	—
89	C₂×C₈×C₄₁	—	2	1	no	—
90	C₂×C₈×C₄₁	—	2	1	no	—
91	C₂×C₈×C₄₁	#41	1	2	yes	—
92	C₂×C₈×C₄₁	#41	1	2	yes	—
93	C₄⋊C₄×C₄₁	—	2	8	no	—
94	C₄⋊C₄×C₄₁	—	2	2	no	—
95	C₄⋊C₄×C₄₁	—	2	2	no	—
96	C₄⋊C₄×C₄₁	—	2	2	no	—
97	C₄⋊C₄×C₄₁	—	2	2	no	—
98	C₄⋊C₄×C₄₁	—	2	2	no	—
99	C₄⋊C₄×C₄₁	—	2	2	no	—
100	C₄⋊C₄×C₄₁	—	2	2	no	—
101	C₄⋊C₄×C₄₁	—	8	2	no	—
102	C₄⋊C₄×C₄₁	—	8	2	no	—
103	C₄⋊C₄×C₄₁	—	8	2	no	—
104	C₄⋊C₄×C₄₁	—	8	2	no	—
105	C₄⋊C₄×C₄₁	—	8	2	no	—
106	C₄⋊C₄×C₄₁	C₂×Q₈	1	4	yes	—
107	C₄²×C₄₁	—	2	1	no	—
108	C₄²×C₄₁	—	2	1	no	—
109	C₄²×C₄₁	—	2	1	no	—
110	C₄²×C₄₁	—	2	1	no	—
111	C₄²×C₄₁	—	8	2	no	—
112	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
113	C₄²×C₄₁	#41	1	2	yes	—
114	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₁	—	2	2	no	—
116	C₂×C₄×C₄₁	—	2	2	no	—
117	C₂×C₄×C₄₁	—	2	4	no	—
118	C₂×C₄×C₄₁	—	2	1	no	—
119	C₂×C₄×C₄₁	—	2	1	no	—
120	C₂×C₄×C₄₁	—	2	2	no	—
121	C₂×C₄×C₄₁	—	2	2	no	—
122	C₂×C₄×C₄₁	—	2	2	no	—
123	C₂×C₄×C₄₁	—	2	2	no	—
124	C₂×C₄×C₄₁	—	2	2	no	—
125	C₂×C₄×C₄₁	—	4	2	no	—
126	C₂×C₄×C₄₁	—	4	2	no	—
127	C₂×C₄×C₄₁	—	4	2	no	—
128	C₂×C₄×C₄₁	—	4	2	no	—
129	C₂×C₄×C₄₁	—	4	2	no	—
130	C₂×C₄×C₄₁	—	4	2	no	—
131	C₂×C₄×C₄₁	—	4	2	no	—
132	C₂×C₄×C₄₁	—	4	2	no	—
133	C₂×C₄×C₄₁	—	4	2	no	—
134	C₂×C₄×C₄₁	—	4	2	no	—
135	C₂×C₄×C₄₁	—	4	2	no	—
136	C₂×C₄×C₄₁	C₂²⋊Q₈	1	4	yes	—
137	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
143	C₈×C₄₁	—	2	2	no	—
144	C₈×C₄₁	—	2	2	no	—
145	C₈×C₄₁	—	4	1	no	—
146	C₈×C₄₁	—	4	1	no	—
147	#29185	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₄₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
149	C₂²×C₄₁	#532	1	1	yes	characteristic
150	C₂²×C₄₁	#536	1	1	yes	characteristic
151	C₂²×C₄₁	#577	1	2	yes	—
152	C₂²×C₄₁	#577	1	2	yes	—
153	C₄×C₄₁	—	2	4	no	—
154	C₄×C₄₁	—	4	4	no	—
155	C₄×C₄₁	—	4	1	no	—
156	C₄×C₄₁	—	4	1	no	—
157	C₄×C₄₁	—	4	2	no	—
158	C₄×C₄₁	—	8	2	no	—
159	C₄×C₄₁	—	8	2	no	—
160	C₄×C₄₁	—	8	2	no	—
161	C₄×C₄₁	#552	1	1	yes	characteristic
162	C₄×C₄₁	#552	1	1	yes	characteristic
163	C₄×C₄₁	#557	1	1	yes	characteristic
164	C₄×C₄₁	#557	1	1	yes	characteristic
165	#2647	C₂×C₄₁	1	1	yes	characteristic
166	#2330	C₂×C₄₁	1	1	yes	characteristic
167	#2088	C₂×C₄₁	1	2	yes	—
168	#2088	C₂×C₄₁	1	2	yes	—
169	#1992	C₂×C₄₁	1	1	yes	characteristic
170	#1991	C₂×C₄₁	1	1	yes	characteristic
171	#1920	C₂×C₄₁	1	2	yes	—
172	#1919	C₂×C₄₁	1	2	yes	—
173	#1844	C₂×C₄₁	1	1	yes	characteristic
174	#1836	C₂×C₄₁	1	1	yes	characteristic
175	#1768	C₂×C₄₁	1	1	yes	characteristic
176	C₂×C₄₁	#2565	1	1	yes	characteristic
177	C₂×C₄₁	#3084	1	2	yes	—
178	C₂×C₄₁	#3174	1	1	yes	characteristic
179	C₂×C₄₁	#3174	1	1	yes	characteristic
180	C₂×C₄₁	#3244	1	2	yes	—
181	C₂×#125	C₂²×C₄₁	1	1	yes	characteristic
182	C₂×#106	C₂²×C₄₁	1	2	yes	—
183	C₂×#106	C₂²×C₄₁	1	2	yes	—
184	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
185	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
186	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
187	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
188	#400	—	2	2	no	—
189	#398	—	2	2	no	—
190	#389	—	2	2	no	—
191	#389	C₂²×C₄₁	1	2	yes	—
192	#389	C₂²×C₄₁	1	2	yes	—
193	#389	C₂²×C₄₁	1	2	yes	—
194	#387	—	2	2	no	—
195	#384	C₂²×C₄₁	1	1	yes	characteristic
196	#384	C₂²×C₄₁	1	1	yes	characteristic
197	#382	C₂²×C₄₁	1	1	yes	characteristic
198	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
199	#376	C₂²×C₄₁	1	1	yes	characteristic
200	#340	C₂²×C₄₁	1	2	yes	—
201	#340	C₂²×C₄₁	1	2	yes	—
202	#340	C₂²×C₄₁	1	2	yes	—
203	#340	C₂²×C₄₁	1	2	yes	—
204	#340	C₂²×C₄₁	1	2	yes	—
205	#340	C₂²×C₄₁	1	2	yes	—
206	#336	C₂²×C₄₁	1	1	yes	characteristic
207	#336	C₂²×C₄₁	1	1	yes	characteristic
208	C₄₁	#29185	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
213	#125	—	2	2	no	—
214	#125	Q₈×C₄₁	1	4	yes	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
222	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
225	C₂×C₄²	—	2	2	no	—
226	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
228	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
229	#106	—	2	4	no	—
230	#106	—	2	4	no	—
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	—	2	2	no	—
233	#97	—	2	2	no	—
234	C₄×C₈	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	D₄×C₄₁	1	2	yes	—
241	#95	D₄×C₄₁	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
250	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	#41×C₄₁	1	2	yes	—
262	C₂×C₈	#41×C₄₁	1	2	yes	—
263	C₄⋊C₄	—	2	8	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	8	2	no	—
272	C₄⋊C₄	—	8	2	no	—
273	C₄⋊C₄	—	8	2	no	—
274	C₄⋊C₄	—	8	2	no	—
275	C₄⋊C₄	—	8	2	no	—
276	C₄⋊C₄	C₂×Q₈×C₄₁	1	4	yes	—
277	C₄²	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	8	2	no	—
282	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
283	C₄²	#41×C₄₁	1	2	yes	—
284	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	C₂²⋊Q₈×C₄₁	1	4	yes	—
307	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
308	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
310	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
311	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
312	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	2	2	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₂²	C₂×C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
318	C₂²	C₄₁×#532	1	1	yes	characteristic
319	C₂²	C₄₁×#536	1	1	yes	characteristic
320	C₂²	C₄₁×#577	1	2	yes	—
321	C₂²	C₄₁×#577	1	2	yes	—
322	C₄	—	2	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	C₄₁×#552	1	1	yes	characteristic
331	C₄	C₄₁×#552	1	1	yes	characteristic
332	C₄	C₄₁×#557	1	1	yes	characteristic
333	C₄	C₄₁×#557	1	1	yes	characteristic
334	C₂	C₄₁×#2565	1	1	yes	characteristic
335	C₂	C₄₁×#3084	1	2	yes	—
336	C₂	C₄₁×#3174	1	1	yes	characteristic
337	C₂	C₄₁×#3174	1	1	yes	characteristic
338	C₂	C₄₁×#3244	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
147. #29185	C₄₁	direct (N × H)
208. C₄₁	#29185	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#2647	2
C₄₁×#2330	2
C₄₁×#2088	2
C₄₁×#1992	2
C₄₁×#1991	2
C₄₁×#1920	2
C₄₁×#1919	2
C₄₁×#1844	2
C₄₁×#1836	2
C₄₁×#1768	2
#29185	41

Subgroups — C41×#29185 #23805386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#29185 #23805386