Subgroups — C₄₁×#7685 #23783886

Order 10496.

562 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2857	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1728	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1271	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₄₁	—	2	1	no	—
6	C₂×C₄⋊Q₈×C₄₁	C₄	1	1	yes	characteristic
7	C₂×C₂²⋊Q₈×C₄₁	—	2	1	no	—
8	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊C₈×C₄₁	—	2	1	no	—
10	C₂×C₄×C₈×C₄₁	—	2	1	no	—
11	C₄₁×#391	—	2	1	no	—
12	C₄₁×#389	—	2	1	no	—
13	C₄₁×#389	C₂²	1	2	yes	—
14	C₄₁×#384	C₄	1	1	yes	characteristic
15	C₄₁×#380	—	2	1	no	—
16	C₄₁×#380	C₄	1	2	yes	—
17	C₄₁×#336	—	2	1	no	—
18	C₄₁×#336	C₂²	1	2	yes	—
19	C₄₁×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₄₁	—	2	1	no	—
21	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₄₁	—	2	4	no	—
25	C₂×M₁₆×C₄₁	—	4	2	no	—
26	C₂×M₁₆×C₄₁	—	4	2	no	—
27	C₂²×C₈×C₄₁	—	2	2	no	—
28	C₂²×C₈×C₄₁	—	2	2	no	—
29	C₄⋊Q₈×C₄₁	—	4	2	no	—
30	C₂²⋊Q₈×C₄₁	—	4	2	no	—
31	C₂²⋊Q₈×C₄₁	—	4	2	no	—
32	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
34	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
35	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
38	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
39	C₂×C₄²×C₄₁	—	4	2	no	—
40	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
44	C₂²⋊C₈×C₄₁	—	2	4	no	—
45	C₄×C₈×C₄₁	—	4	2	no	—
46	#95×C₄₁	—	2	2	no	—
47	#95×C₄₁	—	2	2	no	—
48	C₂⁴×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Q₈×C₄₁	—	4	2	no	—
50	C₂×Q₈×C₄₁	—	4	2	no	—
51	C₂×Q₈×C₄₁	—	4	2	no	—
52	C₂²×C₄×C₄₁	—	2	2	no	—
53	C₂²×C₄×C₄₁	—	2	2	no	—
54	C₂²×C₄×C₄₁	—	2	2	no	—
55	C₂²×C₄×C₄₁	—	2	2	no	—
56	C₂²×C₄×C₄₁	—	2	2	no	—
57	C₂²×C₄×C₄₁	—	2	2	no	—
58	C₂²×C₄×C₄₁	—	2	2	no	—
59	C₂²×C₄×C₄₁	—	2	2	no	—
60	C₂²×C₄×C₄₁	—	4	1	no	—
61	C₂²×C₄×C₄₁	—	4	1	no	—
62	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₁	D₈	1	2	yes	—
66	C₂²×C₄×C₄₁	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
70	M₁₆×C₄₁	—	4	8	no	—
71	C₂×C₈×C₄₁	—	2	4	no	—
72	C₂×C₈×C₄₁	—	2	4	no	—
73	C₂×C₈×C₄₁	—	4	2	no	—
74	C₂×C₈×C₄₁	—	4	2	no	—
75	C₂×C₈×C₄₁	—	4	2	no	—
76	C₂×C₈×C₄₁	—	4	2	no	—
77	C₄⋊C₄×C₄₁	—	2	4	no	—
78	C₄⋊C₄×C₄₁	—	4	2	no	—
79	C₄⋊C₄×C₄₁	—	4	2	no	—
80	C₄⋊C₄×C₄₁	—	4	2	no	—
81	C₄⋊C₄×C₄₁	—	4	2	no	—
82	C₄⋊C₄×C₄₁	—	4	2	no	—
83	C₄⋊C₄×C₄₁	—	4	2	no	—
84	C₂²⋊C₄×C₄₁	—	4	4	no	—
85	C₄²×C₄₁	—	2	2	no	—
86	C₄²×C₄₁	—	2	2	no	—
87	C₄²×C₄₁	—	4	1	no	—
88	C₄²×C₄₁	—	4	1	no	—
89	C₂³×C₄₁	—	2	1	no	—
90	C₂³×C₄₁	—	2	1	no	—
91	C₂³×C₄₁	—	2	1	no	—
92	C₂³×C₄₁	—	2	1	no	—
93	C₂³×C₄₁	—	2	1	no	—
94	C₂³×C₄₁	—	2	1	no	—
95	C₂³×C₄₁	#102	1	2	yes	—
96	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	Q₈×C₄₁	—	8	4	no	—
98	C₂×C₄×C₄₁	—	2	1	no	—
99	C₂×C₄×C₄₁	—	2	1	no	—
100	C₂×C₄×C₄₁	—	2	1	no	—
101	C₂×C₄×C₄₁	—	2	1	no	—
102	C₂×C₄×C₄₁	—	2	1	no	—
103	C₂×C₄×C₄₁	—	2	1	no	—
104	C₂×C₄×C₄₁	—	2	1	no	—
105	C₂×C₄×C₄₁	—	2	1	no	—
106	C₂×C₄×C₄₁	—	2	1	no	—
107	C₂×C₄×C₄₁	—	2	1	no	—
108	C₂×C₄×C₄₁	—	2	4	no	—
109	C₂×C₄×C₄₁	—	2	4	no	—
110	C₂×C₄×C₄₁	—	4	4	no	—
111	C₂×C₄×C₄₁	—	4	4	no	—
112	C₂×C₄×C₄₁	—	4	2	no	—
113	C₂×C₄×C₄₁	—	4	2	no	—
114	C₂×C₄×C₄₁	—	4	2	no	—
115	C₂×C₄×C₄₁	—	4	2	no	—
116	C₂×C₄×C₄₁	—	4	2	no	—
117	C₂×C₄×C₄₁	—	4	2	no	—
118	C₂×C₄×C₄₁	#99	1	2	yes	—
119	C₂×C₄×C₄₁	#101	1	2	yes	—
120	C₂×C₄×C₄₁	#102	1	2	yes	—
121	C₂×C₄×C₄₁	#104	1	2	yes	—
122	C₂×C₄×C₄₁	#104	1	2	yes	—
123	C₂×C₄×C₄₁	#131	1	2	yes	—
124	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₄₁	#137	1	2	yes	—
127	C₈×C₄₁	—	4	4	no	—
128	C₈×C₄₁	—	4	4	no	—
129	#7685	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₄₁	—	2	2	no	—
131	C₂²×C₄₁	—	2	2	no	—
132	C₂²×C₄₁	—	2	2	no	—
133	C₂²×C₄₁	—	4	1	no	—
134	C₂²×C₄₁	—	4	1	no	—
135	C₂²×C₄₁	—	4	1	no	—
136	C₂²×C₄₁	—	4	1	no	—
137	C₂²×C₄₁	#408	1	1	yes	characteristic
138	C₂²×C₄₁	C₂×#99	1	1	yes	characteristic
139	C₂²×C₄₁	C₂×#101	1	1	yes	characteristic
140	C₂²×C₄₁	C₂×#102	1	1	yes	characteristic
141	C₂²×C₄₁	#419	1	1	yes	characteristic
142	C₂²×C₄₁	C₂×#104	1	1	yes	characteristic
143	C₂²×C₄₁	C₂×#104	1	1	yes	characteristic
144	C₄×C₄₁	—	2	2	no	—
145	C₄×C₄₁	—	2	2	no	—
146	C₄×C₄₁	—	2	1	no	—
147	C₄×C₄₁	—	2	1	no	—
148	C₄×C₄₁	—	2	2	no	—
149	C₄×C₄₁	—	2	2	no	—
150	C₄×C₄₁	—	8	2	no	—
151	C₄×C₄₁	—	8	2	no	—
152	C₄×C₄₁	#327	1	4	yes	—
153	#2857	C₂×C₄₁	1	1	yes	characteristic
154	C₄×C₂²⋊C₈	C₂×C₄₁	1	1	yes	characteristic
155	#1728	C₂×C₄₁	1	1	yes	characteristic
156	#1271	C₂×C₄₁	1	4	yes	complemented
157	C₂×C₄₁	—	4	2	no	—
158	C₂×C₄₁	#1472	1	1	yes	characteristic
159	C₂×C₄₁	#1485	1	1	yes	characteristic
160	C₂×C₄₁	#1492	1	1	yes	characteristic
161	C₂×C₄₁	C₂×#327	1	1	yes	characteristic
162	C₂×C₄₁	#1508	1	1	yes	characteristic
163	C₂×C₄₁	#1526	1	1	yes	characteristic
164	C₂×C₄₁	#1546	1	1	yes	characteristic
165	C₂²×M₁₆	—	2	1	no	—
166	C₂×C₄⋊Q₈	C₄×C₄₁	1	1	yes	characteristic
167	C₂×C₂²⋊Q₈	—	2	1	no	—
168	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
169	C₂×C₂²⋊C₈	—	2	1	no	—
170	C₂×C₄×C₈	—	2	1	no	—
171	#391	—	2	1	no	—
172	#389	—	2	1	no	—
173	#389	C₂²×C₄₁	1	2	yes	—
174	#384	C₄×C₄₁	1	1	yes	characteristic
175	#380	—	2	1	no	—
176	#380	C₄×C₄₁	1	2	yes	—
177	#336	—	2	1	no	—
178	#336	C₂²×C₄₁	1	2	yes	—
179	#336	C₂²×C₄₁	1	2	yes	—
180	C₄₁	#7685	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²×Q₈	—	2	1	no	—
182	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
183	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
184	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
185	C₂×M₁₆	—	2	4	no	—
186	C₂×M₁₆	—	4	2	no	—
187	C₂×M₁₆	—	4	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₄⋊Q₈	—	4	2	no	—
191	C₂²⋊Q₈	—	4	2	no	—
192	C₂²⋊Q₈	—	4	2	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₄²	—	4	2	no	—
201	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
202	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
203	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
205	C₂²⋊C₈	—	2	4	no	—
206	C₄×C₈	—	4	2	no	—
207	#95	—	2	2	no	—
208	#95	—	2	2	no	—
209	C₂⁴	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
226	C₂²×C₄	D₈×C₄₁	1	2	yes	—
227	C₂²×C₄	SD₁₆×C₄₁	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
231	M₁₆	—	4	8	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	4	1	no	—
249	C₄²	—	4	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	#102×C₄₁	1	2	yes	—
257	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
258	Q₈	—	8	4	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	#99×C₄₁	1	2	yes	—
280	C₂×C₄	#101×C₄₁	1	2	yes	—
281	C₂×C₄	#102×C₄₁	1	2	yes	—
282	C₂×C₄	#104×C₄₁	1	2	yes	—
283	C₂×C₄	#104×C₄₁	1	2	yes	—
284	C₂×C₄	#131×C₄₁	1	2	yes	—
285	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
286	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
287	C₂×C₄	#137×C₄₁	1	2	yes	—
288	C₈	—	4	4	no	—
289	C₈	—	4	4	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₄₁×#408	1	1	yes	characteristic
298	C₂²	C₂×#99×C₄₁	1	1	yes	characteristic
299	C₂²	C₂×#101×C₄₁	1	1	yes	characteristic
300	C₂²	C₂×#102×C₄₁	1	1	yes	characteristic
301	C₂²	C₄₁×#419	1	1	yes	characteristic
302	C₂²	C₂×#104×C₄₁	1	1	yes	characteristic
303	C₂²	C₂×#104×C₄₁	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	C₄₁×#327	1	4	yes	—
313	C₂	—	4	2	no	—
314	C₂	C₄₁×#1472	1	1	yes	characteristic
315	C₂	C₄₁×#1485	1	1	yes	characteristic
316	C₂	C₄₁×#1492	1	1	yes	characteristic
317	C₂	C₂×C₄₁×#327	1	1	yes	characteristic
318	C₂	C₄₁×#1508	1	1	yes	characteristic
319	C₂	C₄₁×#1526	1	1	yes	characteristic
320	C₂	C₄₁×#1546	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
prime radical	C₂⁴×C₄₁	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1271	C₂	semidirect
129. #7685	C₄₁	direct (N × H)
156. #1271	C₂×C₄₁	semidirect
180. C₄₁	#7685	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×#2857	2
C₄×C₂²⋊C₈×C₄₁	2
C₄₁×#1728	2
C₄₁×#1271	2
#7685	41

Subgroups — C41×#7685 #23783886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#7685 #23783886