Subgroups — C₄₁×#10283 #23786484

Order 10496.

626 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3227	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#3206	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2868	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1962	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#532	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₄₁	—	2	2	no	—
10	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#498	—	2	1	no	—
13	C₄₁×#492	—	2	1	no	—
14	C₄₁×#487	—	2	1	no	—
15	C₄₁×#486	—	2	1	no	—
16	C₄₁×#481	—	2	2	no	—
17	C₄₁×#476	—	2	2	no	—
18	C₄₁×#461	—	2	2	no	—
19	C₄₁×#448	—	2	2	no	—
20	C₄₁×#431	C₂²	1	1	yes	characteristic
21	C₄₁×#431	C₂²	1	1	yes	characteristic
22	C₄₁×#423	C₂²	1	1	yes	characteristic
23	C₂×#104×C₄₁	—	2	2	no	—
24	C₄₁×#384	—	2	2	no	—
25	C₄₁×#384	—	2	2	no	—
26	C₄³×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#333	—	2	2	no	—
28	C₄₁×#332	—	2	2	no	—
29	C₂×#41×C₄₁	—	2	1	no	—
30	C₂²×Q₈×C₄₁	—	2	1	no	—
31	C₂²×Q₈×C₄₁	D₄	1	2	yes	—
32	C₂×SD₁₆×C₄₁	—	2	4	no	—
33	C₂×M₁₆×C₄₁	—	2	1	no	—
34	C₄⋊Q₈×C₄₁	—	2	1	no	—
35	C₄⋊Q₈×C₄₁	—	2	1	no	—
36	C₄⋊Q₈×C₄₁	—	2	1	no	—
37	C₄⋊Q₈×C₄₁	—	2	1	no	—
38	C₄⋊Q₈×C₄₁	—	2	1	no	—
39	C₄⋊Q₈×C₄₁	—	2	1	no	—
40	C₄⋊Q₈×C₄₁	—	2	1	no	—
41	C₄⋊Q₈×C₄₁	—	2	1	no	—
42	C₄⋊Q₈×C₄₁	—	4	8	no	—
43	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
44	#127×C₄₁	D₄	1	1	yes	characteristic
45	#124×C₄₁	—	2	1	no	—
46	#121×C₄₁	—	2	1	no	—
47	#121×C₄₁	—	2	1	no	—
48	C₄×D₄×C₄₁	—	2	1	no	—
49	C₄×D₄×C₄₁	—	2	1	no	—
50	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
51	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
55	C₂×C₄²×C₄₁	—	2	1	no	—
56	C₂×C₄²×C₄₁	—	2	1	no	—
57	C₂×C₄²×C₄₁	D₄	1	2	yes	—
58	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
59	#106×C₄₁	—	2	4	no	—
60	C₄⋊C₈×C₄₁	—	2	1	no	—
61	C₄⋊C₈×C₄₁	—	2	1	no	—
62	C₄⋊C₈×C₄₁	D₄	1	2	yes	—
63	C₄⋊C₈×C₄₁	D₄	1	2	yes	—
64	#104×C₄₁	—	4	4	no	—
65	#103×C₄₁	—	2	4	no	—
66	#102×C₄₁	—	2	4	no	—
67	C₂²⋊C₈×C₄₁	—	2	1	no	—
68	C₂²⋊C₈×C₄₁	—	2	1	no	—
69	C₄×C₈×C₄₁	—	2	1	no	—
70	C₄×C₈×C₄₁	—	2	1	no	—
71	#41×C₄₁	—	4	2	no	—
72	#41×C₄₁	—	4	2	no	—
73	C₂×Q₈×C₄₁	—	2	2	no	—
74	C₂×Q₈×C₄₁	—	2	1	no	—
75	C₂×Q₈×C₄₁	—	2	2	no	—
76	C₂×Q₈×C₄₁	—	4	4	no	—
77	C₂×Q₈×C₄₁	—	4	4	no	—
78	C₂×Q₈×C₄₁	—	4	1	no	—
79	C₂×Q₈×C₄₁	—	4	1	no	—
80	C₂×D₄×C₄₁	—	2	1	no	—
81	C₂×D₄×C₄₁	—	2	1	no	—
82	C₂×D₄×C₄₁	—	2	1	no	—
83	C₂×D₄×C₄₁	—	4	1	no	—
84	C₂²×C₄×C₄₁	—	2	2	no	—
85	C₂²×C₄×C₄₁	—	2	2	no	—
86	C₂²×C₄×C₄₁	—	2	2	no	—
87	C₂²×C₄×C₄₁	—	4	1	no	—
88	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
91	SD₁₆×C₄₁	—	8	4	no	—
92	M₁₆×C₄₁	—	4	2	no	—
93	C₂×C₈×C₄₁	—	2	2	no	—
94	C₂×C₈×C₄₁	—	2	2	no	—
95	C₂×C₈×C₄₁	—	4	1	no	—
96	C₄⋊C₄×C₄₁	—	2	2	no	—
97	C₄⋊C₄×C₄₁	—	2	1	no	—
98	C₄⋊C₄×C₄₁	—	2	1	no	—
99	C₄⋊C₄×C₄₁	—	2	2	no	—
100	C₄⋊C₄×C₄₁	—	4	16	no	—
101	C₄⋊C₄×C₄₁	—	4	1	no	—
102	C₄⋊C₄×C₄₁	—	4	1	no	—
103	C₄⋊C₄×C₄₁	—	4	1	no	—
104	C₄⋊C₄×C₄₁	—	4	1	no	—
105	C₄⋊C₄×C₄₁	—	4	1	no	—
106	C₄⋊C₄×C₄₁	—	4	1	no	—
107	C₂²⋊C₄×C₄₁	—	4	1	no	—
108	C₂²⋊C₄×C₄₁	—	4	1	no	—
109	C₄²×C₄₁	—	2	4	no	—
110	C₄²×C₄₁	—	2	4	no	—
111	C₄²×C₄₁	—	2	2	no	—
112	C₄²×C₄₁	SD₁₆	1	4	yes	complemented
113	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₁	#41	1	1	yes	characteristic
117	C₂³×C₄₁	—	4	1	no	—
118	C₂³×C₄₁	#120	1	1	yes	characteristic
119	Q₈×C₄₁	—	4	4	no	—
120	Q₈×C₄₁	—	4	2	no	—
121	Q₈×C₄₁	—	8	2	no	—
122	Q₈×C₄₁	—	8	2	no	—
123	Q₈×C₄₁	—	8	2	no	—
124	D₄×C₄₁	—	4	2	no	—
125	D₄×C₄₁	—	4	2	no	—
126	D₄×C₄₁	—	4	2	no	—
127	D₄×C₄₁	—	8	2	no	—
128	C₂×C₄×C₄₁	—	2	4	no	—
129	C₂×C₄×C₄₁	—	2	4	no	—
130	C₂×C₄×C₄₁	—	2	4	no	—
131	C₂×C₄×C₄₁	—	2	2	no	—
132	C₂×C₄×C₄₁	—	2	2	no	—
133	C₂×C₄×C₄₁	—	4	4	no	—
134	C₂×C₄×C₄₁	—	4	4	no	—
135	C₂×C₄×C₄₁	—	4	1	no	—
136	C₂×C₄×C₄₁	—	4	2	no	—
137	C₂×C₄×C₄₁	—	4	2	no	—
138	C₂×C₄×C₄₁	—	8	1	no	—
139	C₂×C₄×C₄₁	—	8	1	no	—
140	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
141	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
142	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
143	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
144	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₁	#127	1	1	yes	characteristic
146	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
147	C₂×C₄×C₄₁	#137	1	2	yes	—
148	C₈×C₄₁	—	4	2	no	—
149	C₈×C₄₁	—	4	2	no	—
150	C₈×C₄₁	—	8	1	no	—
151	#10283	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₄₁	—	2	1	no	—
153	C₂²×C₄₁	—	2	1	no	—
154	C₂²×C₄₁	—	8	1	no	—
155	C₂²×C₄₁	—	8	1	no	—
156	C₂²×C₄₁	—	8	1	no	—
157	C₂²×C₄₁	#392	1	1	yes	characteristic
158	C₂²×C₄₁	#448	1	2	yes	—
159	C₄×C₄₁	—	2	8	no	—
160	C₄×C₄₁	—	2	1	no	—
161	C₄×C₄₁	—	2	1	no	—
162	C₄×C₄₁	—	8	2	no	—
163	C₄×C₄₁	—	8	2	no	—
164	C₄×C₄₁	—	8	1	no	—
165	C₄×C₄₁	#453	1	2	yes	—
166	C₄×C₄₁	#456	1	2	yes	—
167	C₄×C₄₁	#461	1	2	yes	—
168	C₄×C₄₁	#492	1	1	yes	characteristic
169	C₄×C₄₁	#497	1	1	yes	characteristic
170	#3227	C₂×C₄₁	1	1	yes	characteristic
171	#3206	C₂×C₄₁	1	1	yes	characteristic
172	#2868	C₂×C₄₁	1	1	yes	characteristic, complemented
173	#1962	C₂×C₄₁	1	1	yes	characteristic
174	#1549	C₂×C₄₁	1	1	yes	characteristic, complemented
175	#1548	C₂×C₄₁	1	1	yes	characteristic, complemented
176	#1276	C₂×C₄₁	1	1	yes	characteristic, complemented
177	C₂×C₄₁	—	2	2	no	—
178	C₂×C₄₁	—	16	1	no	—
179	C₂×C₄₁	#2002	1	1	yes	characteristic
180	C₂×C₄₁	#2005	1	1	yes	characteristic
181	C₂×C₄₁	#2013	1	1	yes	characteristic
182	#532	C₂²×C₄₁	1	1	yes	characteristic
183	C₂×C₄⋊Q₈	—	2	2	no	—
184	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
185	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
186	#498	—	2	1	no	—
187	#492	—	2	1	no	—
188	#487	—	2	1	no	—
189	#486	—	2	1	no	—
190	#481	—	2	2	no	—
191	#476	—	2	2	no	—
192	#461	—	2	2	no	—
193	#448	—	2	2	no	—
194	#431	C₂²×C₄₁	1	1	yes	characteristic
195	#431	C₂²×C₄₁	1	1	yes	characteristic
196	#423	C₂²×C₄₁	1	1	yes	characteristic
197	C₂×#104	—	2	2	no	—
198	#384	—	2	2	no	—
199	#384	—	2	2	no	—
200	C₄³	C₂²×C₄₁	1	1	yes	characteristic
201	#333	—	2	2	no	—
202	#332	—	2	2	no	—
203	C₄₁	#10283	1	1	yes	characteristic, direct product factor, Sylow
204	C₂×#41	—	2	1	no	—
205	C₂²×Q₈	—	2	1	no	—
206	C₂²×Q₈	D₄×C₄₁	1	2	yes	—
207	C₂×SD₁₆	—	2	4	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	4	8	no	—
218	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
219	#127	D₄×C₄₁	1	1	yes	characteristic
220	#124	—	2	1	no	—
221	#121	—	2	1	no	—
222	#121	—	2	1	no	—
223	C₄×D₄	—	2	1	no	—
224	C₄×D₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	8	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
230	C₂×C₄²	—	2	1	no	—
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	D₄×C₄₁	1	2	yes	—
233	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
234	#106	—	2	4	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	D₄×C₄₁	1	2	yes	—
238	C₄⋊C₈	D₄×C₄₁	1	2	yes	—
239	#104	—	4	4	no	—
240	#103	—	2	4	no	—
241	#102	—	2	4	no	—
242	C₂²⋊C₈	—	2	1	no	—
243	C₂²⋊C₈	—	2	1	no	—
244	C₄×C₈	—	2	1	no	—
245	C₄×C₈	—	2	1	no	—
246	#41	—	4	2	no	—
247	#41	—	4	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	4	4	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
266	SD₁₆	—	8	4	no	—
267	M₁₆	—	4	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	4	1	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	16	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	1	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	2	4	no	—
286	C₄²	—	2	2	no	—
287	C₄²	SD₁₆×C₄₁	1	4	yes	complemented
288	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
289	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
290	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
291	C₄²	#41×C₄₁	1	1	yes	characteristic
292	C₂³	—	4	1	no	—
293	C₂³	#120×C₄₁	1	1	yes	characteristic
294	Q₈	—	4	4	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	8	2	no	—
297	Q₈	—	8	2	no	—
298	Q₈	—	8	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	8	2	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	1	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
316	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
317	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
318	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
319	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
320	C₂×C₄	#127×C₄₁	1	1	yes	characteristic
321	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
322	C₂×C₄	#137×C₄₁	1	2	yes	—
323	C₈	—	4	2	no	—
324	C₈	—	4	2	no	—
325	C₈	—	8	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	C₄₁×#392	1	1	yes	characteristic
332	C₂²	C₄₁×#448	1	2	yes	—
333	C₄	—	2	8	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	1	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	2	no	—
339	C₄	C₄₁×#453	1	2	yes	—
340	C₄	C₄₁×#456	1	2	yes	—
341	C₄	C₄₁×#461	1	2	yes	—
342	C₄	C₄₁×#492	1	1	yes	characteristic
343	C₄	C₄₁×#497	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	16	1	no	—
346	C₂	C₄₁×#2002	1	1	yes	characteristic
347	C₂	C₄₁×#2005	1	1	yes	characteristic
348	C₂	C₄₁×#2013	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
prime radical	C₄₁×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#2868	C₂	semidirect
5. C₄₁×#1549	C₂	semidirect
6. C₄₁×#1548	C₂	semidirect
7. C₄₁×#1276	C₂	semidirect
112. C₄²×C₄₁	SD₁₆	semidirect
151. #10283	C₄₁	direct (N × H)
172. #2868	C₂×C₄₁	semidirect
174. #1549	C₂×C₄₁	semidirect
175. #1548	C₂×C₄₁	semidirect
176. #1276	C₂×C₄₁	semidirect
203. C₄₁	#10283	direct (N × H)
287. C₄²	SD₁₆×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3227	2
C₄₁×#3206	2
C₄₁×#2868	2
C₄₁×#1962	2
C₄₁×#1549	2
C₄₁×#1548	2
C₄₁×#1276	2
#10283	41

Subgroups — C41×#10283 #23786484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#10283 #23786484