Subgroups — C₄₁×#23464 #23799665

Order 10496.

1146 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	C₄₁×#3458	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#419	C₂	1	2	yes	maximal
3	C₄₁×#2310	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2028	C₂	1	8	yes	maximal, complemented
5	C₄₁×#1742	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#1733	C₂	1	2	yes	maximal
7	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
9	C₄₁×#544	—	2	4	no	—
10	C₄₁×#544	C₂²	1	4	yes	—
11	C₄₁×#543	—	2	2	no	—
12	C₄₁×#533	—	2	4	no	—
13	C₄₁×#533	—	2	2	no	—
14	C₂×#117×C₄₁	C₂²	1	2	yes	—
15	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#419	—	2	4	no	—
18	C₄₁×#419	C₂²	1	8	yes	—
19	C₂×#103×C₄₁	—	2	4	no	—
20	C₄₁×#391	—	2	2	no	—
21	C₄₁×#391	C₂²	1	4	yes	—
22	C₄₁×#381	—	2	2	no	—
23	C₄₁×#343	C₂²	1	4	yes	—
24	C₄₁×#339	C₂²	1	8	yes	—
25	C₂²×Q₈×C₄₁	—	2	1	no	—
26	C₂²×Q₈×C₄₁	—	2	2	no	—
27	C₂²×Q₈×C₄₁	—	2	2	no	—
28	C₂²×Q₈×C₄₁	—	2	2	no	—
29	C₂²×Q₈×C₄₁	—	2	4	no	—
30	C₂²×Q₈×C₄₁	Q₈	1	4	yes	complemented
31	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
32	C₂³×C₄×C₄₁	—	2	1	no	—
33	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₄₁	—	2	2	no	—
35	C₂×M₁₆×C₄₁	—	2	4	no	—
36	C₂×M₁₆×C₄₁	—	2	4	no	—
37	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
38	C₂²×C₈×C₄₁	—	2	2	no	—
39	C₄⋊Q₈×C₄₁	—	2	4	no	—
40	C₄⋊Q₈×C₄₁	—	2	8	no	—
41	#125×C₄₁	—	2	4	no	—
42	#125×C₄₁	—	2	8	no	—
43	C₂²⋊Q₈×C₄₁	—	2	16	no	—
44	C₂²⋊Q₈×C₄₁	—	2	8	no	—
45	C₄×Q₈×C₄₁	—	2	8	no	—
46	#117×C₄₁	—	2	2	no	—
47	#117×C₄₁	—	2	2	no	—
48	#117×C₄₁	—	2	2	no	—
49	#117×C₄₁	—	2	2	no	—
50	#117×C₄₁	—	2	4	no	—
51	#117×C₄₁	D₄	1	4	yes	—
52	#117×C₄₁	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₄₁	D₄	1	4	yes	—
55	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
58	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
59	C₂×C₄²×C₄₁	—	2	2	no	—
60	C₂×C₄²×C₄₁	D₄	1	2	yes	—
61	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
62	#103×C₄₁	—	2	16	no	—
63	#95×C₄₁	—	2	8	no	—
64	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
65	C₂×Q₈×C₄₁	—	2	8	no	—
66	C₂×Q₈×C₄₁	—	2	4	no	—
67	C₂×Q₈×C₄₁	—	2	2	no	—
68	C₂×Q₈×C₄₁	—	2	8	no	—
69	C₂×Q₈×C₄₁	—	2	8	no	—
70	C₂×Q₈×C₄₁	—	2	4	no	—
71	C₂×Q₈×C₄₁	—	2	4	no	—
72	C₂×Q₈×C₄₁	—	4	1	no	—
73	C₂×Q₈×C₄₁	—	4	1	no	—
74	C₂×Q₈×C₄₁	—	4	2	no	—
75	C₂×Q₈×C₄₁	—	4	4	no	—
76	C₂×Q₈×C₄₁	—	4	4	no	—
77	C₂×Q₈×C₄₁	C₂×Q₈	1	4	yes	—
78	C₂²×C₄×C₄₁	—	2	4	no	—
79	C₂²×C₄×C₄₁	—	2	2	no	—
80	C₂²×C₄×C₄₁	—	2	4	no	—
81	C₂²×C₄×C₄₁	—	2	2	no	—
82	C₂²×C₄×C₄₁	—	4	2	no	—
83	C₂²×C₄×C₄₁	—	4	2	no	—
84	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₄₁	#41	1	4	yes	—
90	C₂²×C₄×C₄₁	#41	1	2	yes	—
91	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
92	M₁₆×C₄₁	—	2	4	no	—
93	M₁₆×C₄₁	—	2	4	no	—
94	M₁₆×C₄₁	—	4	4	no	—
95	C₂×C₈×C₄₁	—	2	4	no	—
96	C₂×C₈×C₄₁	—	4	4	no	—
97	C₄⋊C₄×C₄₁	—	2	16	no	—
98	C₄⋊C₄×C₄₁	—	2	8	no	—
99	C₄⋊C₄×C₄₁	—	2	4	no	—
100	C₄⋊C₄×C₄₁	—	2	4	no	—
101	C₄⋊C₄×C₄₁	—	4	2	no	—
102	C₄⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
103	C₂²⋊C₄×C₄₁	—	2	4	no	—
104	C₂²⋊C₄×C₄₁	—	2	4	no	—
105	C₂²⋊C₄×C₄₁	—	2	4	no	—
106	C₂²⋊C₄×C₄₁	—	4	2	no	—
107	C₄²×C₄₁	—	2	4	no	—
108	C₄²×C₄₁	—	2	2	no	—
109	C₄²×C₄₁	—	2	2	no	—
110	C₄²×C₄₁	—	2	4	no	—
111	C₄²×C₄₁	—	2	4	no	—
112	C₂³×C₄₁	—	4	2	no	—
113	C₂³×C₄₁	#120	1	2	yes	—
114	C₂³×C₄₁	C₂²⋊Q₈	1	2	yes	—
115	C₂³×C₄₁	#124	1	2	yes	—
116	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
117	Q₈×C₄₁	—	2	8	no	—
118	Q₈×C₄₁	—	4	4	no	—
119	Q₈×C₄₁	—	4	4	no	—
120	Q₈×C₄₁	—	4	4	no	—
121	Q₈×C₄₁	—	8	8	no	—
122	C₂×C₄×C₄₁	—	2	4	no	—
123	C₂×C₄×C₄₁	—	2	8	no	—
124	C₂×C₄×C₄₁	—	2	4	no	—
125	C₂×C₄×C₄₁	—	4	2	no	—
126	C₂×C₄×C₄₁	—	4	2	no	—
127	C₂×C₄×C₄₁	—	4	8	no	—
128	C₂×C₄×C₄₁	—	4	4	no	—
129	C₂×C₄×C₄₁	—	4	4	no	—
130	C₂×C₄×C₄₁	—	4	2	no	—
131	C₂×C₄×C₄₁	—	8	4	no	—
132	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
133	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
134	C₂×C₄×C₄₁	C₂²⋊Q₈	1	4	yes	—
135	C₂×C₄×C₄₁	C₂²⋊Q₈	1	4	yes	—
136	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄×C₄₁	#124	1	4	yes	—
138	C₂×C₄×C₄₁	#124	1	4	yes	—
139	C₂×C₄×C₄₁	#124	1	2	yes	—
140	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
141	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
143	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
144	C₈×C₄₁	—	4	4	no	—
145	#23464	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₄₁	—	2	2	no	—
147	C₂²×C₄₁	—	2	4	no	—
148	C₂²×C₄₁	—	4	2	no	—
149	C₂²×C₄₁	—	4	2	no	—
150	C₂²×C₄₁	#393	1	4	yes	—
151	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
152	C₂²×C₄₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
153	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
154	C₄×C₄₁	—	4	4	no	—
155	C₄×C₄₁	—	4	4	no	—
156	C₄×C₄₁	—	4	8	no	—
157	C₄×C₄₁	—	8	2	no	—
158	C₄×C₄₁	#393	1	2	yes	—
159	C₄×C₄₁	#393	1	2	yes	—
160	C₄×C₄₁	C₂×C₂²⋊Q₈	1	2	yes	—
161	C₄×C₄₁	C₂×#124	1	2	yes	—
162	#3458	C₂×C₄₁	1	1	yes	characteristic
163	C₂×#419	C₂×C₄₁	1	2	yes	—
164	#2310	C₂×C₄₁	1	1	yes	characteristic
165	#2028	C₂×C₄₁	1	8	yes	complemented
166	#1742	C₂×C₄₁	1	1	yes	characteristic
167	#1733	C₂×C₄₁	1	2	yes	—
168	C₂×C₄₁	—	2	1	no	—
169	C₂×C₄₁	—	2	1	no	—
170	C₂×C₄₁	—	2	4	no	—
171	C₂×C₄₁	C₂×#393	1	1	yes	characteristic
172	C₂×C₄₁	#3022	1	2	yes	—
173	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
174	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
175	#544	—	2	4	no	—
176	#544	C₂²×C₄₁	1	4	yes	—
177	#543	—	2	2	no	—
178	#533	—	2	4	no	—
179	#533	—	2	2	no	—
180	C₂×#117	C₂²×C₄₁	1	2	yes	—
181	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
182	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
183	#419	—	2	4	no	—
184	#419	C₂²×C₄₁	1	8	yes	—
185	C₂×#103	—	2	4	no	—
186	#391	—	2	2	no	—
187	#391	C₂²×C₄₁	1	4	yes	—
188	#381	—	2	2	no	—
189	#343	C₂²×C₄₁	1	4	yes	—
190	#339	C₂²×C₄₁	1	8	yes	—
191	C₄₁	#23464	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²×Q₈	—	2	1	no	—
193	C₂²×Q₈	—	2	2	no	—
194	C₂²×Q₈	—	2	2	no	—
195	C₂²×Q₈	—	2	2	no	—
196	C₂²×Q₈	—	2	4	no	—
197	C₂²×Q₈	Q₈×C₄₁	1	4	yes	complemented
198	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
199	C₂³×C₄	—	2	1	no	—
200	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	2	4	no	—
203	C₂×M₁₆	—	2	4	no	—
204	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
205	C₂²×C₈	—	2	2	no	—
206	C₄⋊Q₈	—	2	4	no	—
207	C₄⋊Q₈	—	2	8	no	—
208	#125	—	2	4	no	—
209	#125	—	2	8	no	—
210	C₂²⋊Q₈	—	2	16	no	—
211	C₂²⋊Q₈	—	2	8	no	—
212	C₄×Q₈	—	2	8	no	—
213	#117	—	2	2	no	—
214	#117	—	2	2	no	—
215	#117	—	2	2	no	—
216	#117	—	2	2	no	—
217	#117	—	2	4	no	—
218	#117	D₄×C₄₁	1	4	yes	—
219	#117	C₂³×C₄₁	1	4	yes	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	D₄×C₄₁	1	4	yes	—
222	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
223	C₂×C₂²⋊C₄	—	2	2	no	—
224	C₂×C₂²⋊C₄	—	2	1	no	—
225	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	D₄×C₄₁	1	2	yes	—
228	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
229	#103	—	2	16	no	—
230	#95	—	2	8	no	—
231	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
232	C₂×Q₈	—	2	8	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	8	no	—
236	C₂×Q₈	—	2	8	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	—	4	4	no	—
244	C₂×Q₈	C₂×Q₈×C₄₁	1	4	yes	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
252	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
255	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
256	C₂²×C₄	#41×C₄₁	1	4	yes	—
257	C₂²×C₄	#41×C₄₁	1	2	yes	—
258	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
259	M₁₆	—	2	4	no	—
260	M₁₆	—	2	4	no	—
261	M₁₆	—	4	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	4	no	—
264	C₄⋊C₄	—	2	16	no	—
265	C₄⋊C₄	—	2	8	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₂²⋊C₄	—	2	4	no	—
272	C₂²⋊C₄	—	2	4	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₄²	—	2	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	2	4	no	—
279	C₂³	—	4	2	no	—
280	C₂³	#120×C₄₁	1	2	yes	—
281	C₂³	C₂²⋊Q₈×C₄₁	1	2	yes	—
282	C₂³	#124×C₄₁	1	2	yes	—
283	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
284	Q₈	—	2	8	no	—
285	Q₈	—	4	4	no	—
286	Q₈	—	4	4	no	—
287	Q₈	—	4	4	no	—
288	Q₈	—	8	8	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	8	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	8	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
300	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
301	C₂×C₄	C₂²⋊Q₈×C₄₁	1	4	yes	—
302	C₂×C₄	C₂²⋊Q₈×C₄₁	1	4	yes	—
303	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
304	C₂×C₄	#124×C₄₁	1	4	yes	—
305	C₂×C₄	#124×C₄₁	1	4	yes	—
306	C₂×C₄	#124×C₄₁	1	2	yes	—
307	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
308	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
310	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
311	C₈	—	4	4	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	C₄₁×#393	1	4	yes	—
317	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
318	C₂²	C₂×C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
319	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
320	C₄	—	4	4	no	—
321	C₄	—	4	4	no	—
322	C₄	—	4	8	no	—
323	C₄	—	8	2	no	—
324	C₄	C₄₁×#393	1	2	yes	—
325	C₄	C₄₁×#393	1	2	yes	—
326	C₄	C₂×C₂²⋊Q₈×C₄₁	1	2	yes	—
327	C₄	C₂×#124×C₄₁	1	2	yes	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	2	4	no	—
331	C₂	C₂×C₄₁×#393	1	1	yes	characteristic
332	C₂	C₄₁×#3022	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#2028	C₂	semidirect
30. C₂²×Q₈×C₄₁	Q₈	semidirect
145. #23464	C₄₁	direct (N × H)
165. #2028	C₂×C₄₁	semidirect
191. C₄₁	#23464	direct (N × H)
197. C₂²×Q₈	Q₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3458	2
C₂×C₄₁×#419	2
C₄₁×#2310	2
C₄₁×#2028	2
C₄₁×#1742	2
C₄₁×#1733	2
#23464	41

Subgroups — C41×#23464 #23799665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23464 #23799665