Subgroups — C₄₁×#8441 #23784642

Order 10496.

864 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3483	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1962	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#104×C₄₁	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1525	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1489	C₂	1	2	yes	maximal, complemented
6	Q₈²×C₄₁	C₄	1	1	yes	characteristic, complemented
7	C₄₁×#555	—	2	2	no	—
8	C₄₁×#555	—	2	2	no	—
9	C₄₁×#555	C₄	1	1	yes	characteristic, complemented
10	C₄₁×#550	—	2	2	no	—
11	C₄₁×#550	C₄	1	2	yes	complemented
12	C₄₁×#532	—	2	2	no	—
13	C₄₁×#532	—	2	2	no	—
14	C₄₁×#532	C₂²	1	1	yes	characteristic
15	C₄×#41×C₄₁	—	2	2	no	—
16	C₄×#41×C₄₁	C₂²	1	2	yes	—
17	C₄₁×#423	C₂²	1	1	yes	characteristic
18	C₂×#104×C₄₁	—	2	2	no	—
19	C₂×#104×C₄₁	—	2	2	no	—
20	C₄×M₁₆×C₄₁	C₂²	1	2	yes	—
21	C₄³×C₄₁	C₂²	1	1	yes	characteristic
22	C₄₁×#339	—	2	2	no	—
23	C₄₁×#329	—	2	2	no	—
24	C₄₁×Q₈⋊C₈	—	2	2	no	—
25	C₄₁×D₄⋊C₈	—	2	2	no	—
26	C₂×#41×C₄₁	—	2	2	no	—
27	C₂×#41×C₄₁	—	2	2	no	—
28	C₂×#41×C₄₁	D₄	1	2	yes	—
29	C₂×#41×C₄₁	D₄	1	2	yes	—
30	C₂×M₁₆×C₄₁	—	2	2	no	—
31	C₂×M₁₆×C₄₁	—	2	1	no	—
32	C₄⋊Q₈×C₄₁	—	2	2	no	—
33	C₄⋊Q₈×C₄₁	—	2	2	no	—
34	C₄⋊Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
35	#127×C₄₁	—	2	2	no	—
36	#127×C₄₁	—	2	2	no	—
37	#127×C₄₁	C₂×C₄	1	1	yes	characteristic
38	#124×C₄₁	—	2	2	no	—
39	#124×C₄₁	—	2	4	no	—
40	#124×C₄₁	—	2	2	no	—
41	#124×C₄₁	D₄	1	2	yes	—
42	#121×C₄₁	—	2	4	no	—
43	#121×C₄₁	—	2	4	no	—
44	#121×C₄₁	—	2	2	no	—
45	#121×C₄₁	—	2	4	no	—
46	#121×C₄₁	—	2	2	no	—
47	#121×C₄₁	—	2	2	no	—
48	C₄×Q₈×C₄₁	—	2	2	no	—
49	C₄×Q₈×C₄₁	C₂×C₄	1	2	yes	—
50	C₄×Q₈×C₄₁	D₄	1	2	yes	—
51	C₄×D₄×C₄₁	—	2	4	no	—
52	C₄×D₄×C₄₁	—	2	4	no	—
53	C₄×D₄×C₄₁	—	2	2	no	—
54	C₄×D₄×C₄₁	—	2	2	no	—
55	C₄×D₄×C₄₁	—	2	2	no	—
56	C₄×D₄×C₄₁	—	2	2	no	—
57	C₄×D₄×C₄₁	C₂×C₄	1	2	yes	—
58	C₄×D₄×C₄₁	D₄	1	2	yes	—
59	#117×C₄₁	—	2	4	no	—
60	#117×C₄₁	—	2	2	no	—
61	#117×C₄₁	—	2	2	no	—
62	#117×C₄₁	D₄	1	2	yes	—
63	C₂×C₄²×C₄₁	—	2	2	no	—
64	C₂×C₄²×C₄₁	—	2	2	no	—
65	C₂×C₄²×C₄₁	—	2	2	no	—
66	C₂×C₄²×C₄₁	—	2	1	no	—
67	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₄₁	—	2	2	no	—
69	#104×C₄₁	—	2	8	no	—
70	#104×C₄₁	—	4	4	no	—
71	#97×C₄₁	—	2	2	no	—
72	C₄×C₈×C₄₁	—	2	2	no	—
73	#41×C₄₁	—	2	4	no	—
74	#41×C₄₁	—	4	2	no	—
75	#41×C₄₁	—	4	2	no	—
76	#41×C₄₁	—	4	2	no	—
77	#41×C₄₁	—	4	4	no	—
78	#41×C₄₁	—	4	4	no	—
79	#41×C₄₁	—	8	2	no	—
80	#41×C₄₁	—	8	2	no	—
81	C₂×Q₈×C₄₁	—	2	2	no	—
82	C₂×Q₈×C₄₁	C₂²⋊C₄	1	2	yes	—
83	C₂×Q₈×C₄₁	C₂²⋊C₄	1	2	yes	—
84	C₂×D₄×C₄₁	—	2	4	no	—
85	C₂×D₄×C₄₁	—	2	2	no	—
86	C₂×D₄×C₄₁	—	2	2	no	—
87	C₂×D₄×C₄₁	—	2	4	no	—
88	C₂×D₄×C₄₁	—	2	2	no	—
89	C₂×D₄×C₄₁	—	2	2	no	—
90	C₂×D₄×C₄₁	C₂²⋊C₄	1	2	yes	—
91	C₂×D₄×C₄₁	C₂²⋊C₄	1	2	yes	—
92	C₂²×C₄×C₄₁	—	2	4	no	—
93	C₂²×C₄×C₄₁	—	2	2	no	—
94	C₂²×C₄×C₄₁	—	2	2	no	—
95	C₂²×C₄×C₄₁	—	2	2	no	—
96	C₂²×C₄×C₄₁	—	2	2	no	—
97	C₂²×C₄×C₄₁	—	4	1	no	—
98	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
100	M₁₆×C₄₁	—	2	4	no	—
101	M₁₆×C₄₁	—	4	2	no	—
102	M₁₆×C₄₁	—	4	2	no	—
103	C₂×C₈×C₄₁	—	4	2	no	—
104	C₂×C₈×C₄₁	—	4	1	no	—
105	C₄⋊C₄×C₄₁	—	2	4	no	—
106	C₄⋊C₄×C₄₁	—	2	2	no	—
107	C₄⋊C₄×C₄₁	—	2	2	no	—
108	C₄⋊C₄×C₄₁	—	2	2	no	—
109	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
110	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
111	C₂²⋊C₄×C₄₁	—	2	4	no	—
112	C₂²⋊C₄×C₄₁	—	2	4	no	—
113	C₂²⋊C₄×C₄₁	—	2	2	no	—
114	C₂²⋊C₄×C₄₁	—	2	2	no	—
115	C₂²⋊C₄×C₄₁	—	2	4	no	—
116	C₂²⋊C₄×C₄₁	—	2	2	no	—
117	C₄²×C₄₁	—	2	2	no	—
118	C₄²×C₄₁	—	2	2	no	—
119	C₄²×C₄₁	—	2	2	no	—
120	C₄²×C₄₁	—	2	2	no	—
121	C₄²×C₄₁	—	2	2	no	—
122	C₄²×C₄₁	—	2	4	no	—
123	C₄²×C₄₁	—	2	2	no	—
124	C₄²×C₄₁	—	4	2	no	—
125	C₄²×C₄₁	—	4	1	no	—
126	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₄₁	C₂×D₄	1	2	yes	—
129	C₂³×C₄₁	—	2	2	no	—
130	C₂³×C₄₁	—	2	2	no	—
131	C₂³×C₄₁	#120	1	1	yes	characteristic
132	Q₈×C₄₁	—	2	2	no	—
133	Q₈×C₄₁	—	4	2	no	—
134	Q₈×C₄₁	—	8	2	no	—
135	Q₈×C₄₁	#104	1	4	yes	complemented
136	D₄×C₄₁	—	2	2	no	—
137	D₄×C₄₁	—	2	2	no	—
138	D₄×C₄₁	—	4	4	no	—
139	D₄×C₄₁	—	4	2	no	—
140	D₄×C₄₁	—	4	8	no	—
141	D₄×C₄₁	—	4	2	no	—
142	D₄×C₄₁	—	4	2	no	—
143	D₄×C₄₁	—	8	2	no	—
144	D₄×C₄₁	—	8	2	no	—
145	D₄×C₄₁	—	8	4	no	—
146	C₂×C₄×C₄₁	—	2	2	no	—
147	C₂×C₄×C₄₁	—	2	2	no	—
148	C₂×C₄×C₄₁	—	2	2	no	—
149	C₂×C₄×C₄₁	—	2	1	no	—
150	C₂×C₄×C₄₁	—	2	1	no	—
151	C₂×C₄×C₄₁	—	2	2	no	—
152	C₂×C₄×C₄₁	—	2	2	no	—
153	C₂×C₄×C₄₁	—	2	2	no	—
154	C₂×C₄×C₄₁	—	4	4	no	—
155	C₂×C₄×C₄₁	—	4	8	no	—
156	C₂×C₄×C₄₁	—	4	2	no	—
157	C₂×C₄×C₄₁	—	4	4	no	—
158	C₂×C₄×C₄₁	—	4	2	no	—
159	C₂×C₄×C₄₁	—	4	2	no	—
160	C₂×C₄×C₄₁	—	4	2	no	—
161	C₂×C₄×C₄₁	—	4	2	no	—
162	C₂×C₄×C₄₁	—	4	2	no	—
163	C₂×C₄×C₄₁	—	4	2	no	—
164	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
166	C₂×C₄×C₄₁	#120	1	2	yes	—
167	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
168	C₈×C₄₁	—	4	4	no	—
169	C₈×C₄₁	—	8	1	no	—
170	#8441	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂²×C₄₁	—	2	1	no	—
172	C₂²×C₄₁	—	2	2	no	—
173	C₂²×C₄₁	—	4	2	no	—
174	C₂²×C₄₁	—	4	2	no	—
175	C₂²×C₄₁	—	4	2	no	—
176	C₂²×C₄₁	—	4	4	no	—
177	C₂²×C₄₁	—	4	2	no	—
178	C₂²×C₄₁	#379	1	1	yes	characteristic
179	C₄×C₄₁	—	2	1	no	—
180	C₄×C₄₁	—	2	2	no	—
181	C₄×C₄₁	—	2	4	no	—
182	C₄×C₄₁	—	4	4	no	—
183	C₄×C₄₁	—	4	2	no	—
184	C₄×C₄₁	—	4	2	no	—
185	C₄×C₄₁	—	4	2	no	—
186	C₄×C₄₁	C₂×#104	1	2	yes	—
187	C₄×C₄₁	#421	1	2	yes	—
188	C₄×C₄₁	#453	1	1	yes	characteristic
189	C₄×C₄₁	#455	1	1	yes	characteristic
190	#3483	C₂×C₄₁	1	1	yes	characteristic
191	#1962	C₂×C₄₁	1	1	yes	characteristic, complemented
192	C₄×#104	C₂×C₄₁	1	2	yes	complemented
193	#1525	C₂×C₄₁	1	1	yes	characteristic, complemented
194	#1489	C₂×C₄₁	1	2	yes	complemented
195	C₂×C₄₁	—	4	1	no	—
196	C₂×C₄₁	—	8	2	no	—
197	C₂×C₄₁	—	8	2	no	—
198	C₂×C₄₁	#1789	1	2	yes	—
199	C₂×C₄₁	#1792	1	1	yes	characteristic
200	Q₈²	C₄×C₄₁	1	1	yes	characteristic, complemented
201	#555	—	2	2	no	—
202	#555	—	2	2	no	—
203	#555	C₄×C₄₁	1	1	yes	characteristic, complemented
204	#550	—	2	2	no	—
205	#550	C₄×C₄₁	1	2	yes	complemented
206	#532	—	2	2	no	—
207	#532	—	2	2	no	—
208	#532	C₂²×C₄₁	1	1	yes	characteristic
209	C₄×#41	—	2	2	no	—
210	C₄×#41	C₂²×C₄₁	1	2	yes	—
211	#423	C₂²×C₄₁	1	1	yes	characteristic
212	C₂×#104	—	2	2	no	—
213	C₂×#104	—	2	2	no	—
214	C₄×M₁₆	C₂²×C₄₁	1	2	yes	—
215	C₄³	C₂²×C₄₁	1	1	yes	characteristic
216	#339	—	2	2	no	—
217	#329	—	2	2	no	—
218	Q₈⋊C₈	—	2	2	no	—
219	D₄⋊C₈	—	2	2	no	—
220	C₄₁	#8441	1	1	yes	characteristic, direct product factor, Sylow
221	C₂×#41	—	2	2	no	—
222	C₂×#41	—	2	2	no	—
223	C₂×#41	D₄×C₄₁	1	2	yes	—
224	C₂×#41	D₄×C₄₁	1	2	yes	—
225	C₂×M₁₆	—	2	2	no	—
226	C₂×M₁₆	—	2	1	no	—
227	C₄⋊Q₈	—	2	2	no	—
228	C₄⋊Q₈	—	2	2	no	—
229	C₄⋊Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
230	#127	—	2	2	no	—
231	#127	—	2	2	no	—
232	#127	C₂×C₄×C₄₁	1	1	yes	characteristic
233	#124	—	2	2	no	—
234	#124	—	2	4	no	—
235	#124	—	2	2	no	—
236	#124	D₄×C₄₁	1	2	yes	—
237	#121	—	2	4	no	—
238	#121	—	2	4	no	—
239	#121	—	2	2	no	—
240	#121	—	2	4	no	—
241	#121	—	2	2	no	—
242	#121	—	2	2	no	—
243	C₄×Q₈	—	2	2	no	—
244	C₄×Q₈	C₂×C₄×C₄₁	1	2	yes	—
245	C₄×Q₈	D₄×C₄₁	1	2	yes	—
246	C₄×D₄	—	2	4	no	—
247	C₄×D₄	—	2	4	no	—
248	C₄×D₄	—	2	2	no	—
249	C₄×D₄	—	2	2	no	—
250	C₄×D₄	—	2	2	no	—
251	C₄×D₄	—	2	2	no	—
252	C₄×D₄	C₂×C₄×C₄₁	1	2	yes	—
253	C₄×D₄	D₄×C₄₁	1	2	yes	—
254	#117	—	2	4	no	—
255	#117	—	2	2	no	—
256	#117	—	2	2	no	—
257	#117	D₄×C₄₁	1	2	yes	—
258	C₂×C₄²	—	2	2	no	—
259	C₂×C₄²	—	2	2	no	—
260	C₂×C₄²	—	2	2	no	—
261	C₂×C₄²	—	2	1	no	—
262	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
263	C₄⋊C₈	—	2	2	no	—
264	#104	—	2	8	no	—
265	#104	—	4	4	no	—
266	#97	—	2	2	no	—
267	C₄×C₈	—	2	2	no	—
268	#41	—	2	4	no	—
269	#41	—	4	2	no	—
270	#41	—	4	2	no	—
271	#41	—	4	2	no	—
272	#41	—	4	4	no	—
273	#41	—	4	4	no	—
274	#41	—	8	2	no	—
275	#41	—	8	2	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	C₂²⋊C₄×C₄₁	1	2	yes	—
278	C₂×Q₈	C₂²⋊C₄×C₄₁	1	2	yes	—
279	C₂×D₄	—	2	4	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	4	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	C₂²⋊C₄×C₄₁	1	2	yes	—
286	C₂×D₄	C₂²⋊C₄×C₄₁	1	2	yes	—
287	C₂²×C₄	—	2	4	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
294	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
295	M₁₆	—	2	4	no	—
296	M₁₆	—	4	2	no	—
297	M₁₆	—	4	2	no	—
298	C₂×C₈	—	4	2	no	—
299	C₂×C₈	—	4	1	no	—
300	C₄⋊C₄	—	2	4	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
305	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
306	C₂²⋊C₄	—	2	4	no	—
307	C₂²⋊C₄	—	2	4	no	—
308	C₂²⋊C₄	—	2	2	no	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	2	4	no	—
311	C₂²⋊C₄	—	2	2	no	—
312	C₄²	—	2	2	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	4	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	4	2	no	—
320	C₄²	—	4	1	no	—
321	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
322	C₄²	C₂×D₄×C₄₁	1	2	yes	—
323	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
324	C₂³	—	2	2	no	—
325	C₂³	—	2	2	no	—
326	C₂³	#120×C₄₁	1	1	yes	characteristic
327	Q₈	—	2	2	no	—
328	Q₈	—	4	2	no	—
329	Q₈	—	8	2	no	—
330	Q₈	#104×C₄₁	1	4	yes	complemented
331	D₄	—	2	2	no	—
332	D₄	—	2	2	no	—
333	D₄	—	4	4	no	—
334	D₄	—	4	2	no	—
335	D₄	—	4	8	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	8	2	no	—
339	D₄	—	8	2	no	—
340	D₄	—	8	4	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	4	no	—
350	C₂×C₄	—	4	8	no	—
351	C₂×C₄	—	4	4	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
360	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
361	C₂×C₄	#120×C₄₁	1	2	yes	—
362	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
363	C₈	—	4	4	no	—
364	C₈	—	8	1	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	4	no	—
369	C₂²	—	4	2	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	C₄₁×#379	1	1	yes	characteristic
373	C₄	—	2	2	no	—
374	C₄	—	2	4	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	4	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	2	no	—
379	C₄	—	4	2	no	—
380	C₄	C₂×#104×C₄₁	1	2	yes	—
381	C₄	C₄₁×#421	1	2	yes	—
382	C₄	C₄₁×#453	1	1	yes	characteristic
383	C₄	C₄₁×#455	1	1	yes	characteristic
384	C₂	—	4	1	no	—
385	C₂	—	8	2	no	—
386	C₂	—	8	2	no	—
387	C₂	C₄₁×#1789	1	2	yes	—
388	C₂	C₄₁×#1792	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	#120
commutator	C₄²	C₂²×C₄×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
socle	C₂²×C₄₁	#379
prime radical	C₄₁×#3483	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#1962	C₂	semidirect
3. C₄×#104×C₄₁	C₂	semidirect
4. C₄₁×#1525	C₂	semidirect
5. C₄₁×#1489	C₂	semidirect
6. Q₈²×C₄₁	C₄	semidirect
9. C₄₁×#555	C₄	semidirect
11. C₄₁×#550	C₄	semidirect
135. Q₈×C₄₁	#104	semidirect
170. #8441	C₄₁	direct (N × H)
191. #1962	C₂×C₄₁	semidirect
192. C₄×#104	C₂×C₄₁	semidirect
193. #1525	C₂×C₄₁	semidirect
194. #1489	C₂×C₄₁	semidirect
200. Q₈²	C₄×C₄₁	semidirect
203. #555	C₄×C₄₁	semidirect
205. #550	C₄×C₄₁	semidirect
220. C₄₁	#8441	direct (N × H)
330. Q₈	#104×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3483	2
C₄₁×#1962	2
C₄×#104×C₄₁	2
C₄₁×#1525	2
C₄₁×#1489	2
#8441	41

Subgroups — C41×#8441 #23784642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#8441 #23784642