Subgroups — C₄₁×#9879 #23786080

Order 10496.

616 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#389	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1797	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1280	C₂	1	4	yes	maximal
5	C₂²×C₄⋊C₄×C₄₁	C₄	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₄₁	C₄	1	2	yes	—
7	C₂²×C₄⋊C₄×C₄₁	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₄₁	—	2	1	no	—
10	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
11	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₄₁	—	2	1	no	—
13	C₄₁×#389	—	2	1	no	—
14	C₄₁×#389	—	2	1	no	—
15	C₄₁×#389	—	2	1	no	—
16	C₄₁×#389	—	2	1	no	—
17	C₂×#95×C₄₁	C₂²	1	2	yes	—
18	C₄₁×#336	—	2	1	no	—
19	C₄₁×#336	—	2	1	no	—
20	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₄₁	—	2	2	no	—
26	C₂²×C₈×C₄₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₁	—	4	2	no	—
35	C₂×C₄⋊C₄×C₄₁	—	4	4	no	—
36	C₂×C₄⋊C₄×C₄₁	C₈	1	4	yes	complemented
37	C₂×C₄²×C₄₁	—	4	2	no	—
38	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
42	C₄⋊C₈×C₄₁	—	4	2	no	—
43	C₂²⋊C₈×C₄₁	—	2	2	no	—
44	C₂²⋊C₈×C₄₁	—	2	4	no	—
45	C₂²⋊C₈×C₄₁	Q₈	1	4	yes	—
46	C₄×C₈×C₄₁	—	4	2	no	—
47	#95×C₄₁	—	2	2	no	—
48	#95×C₄₁	—	2	2	no	—
49	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₁	—	2	2	no	—
51	C₂²×C₄×C₄₁	—	2	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	4	no	—
58	C₂²×C₄×C₄₁	—	4	1	no	—
59	C₂²×C₄×C₄₁	—	4	1	no	—
60	C₂²×C₄×C₄₁	—	4	1	no	—
61	C₂²×C₄×C₄₁	—	4	2	no	—
62	C₂²×C₄×C₄₁	—	4	2	no	—
63	C₂²×C₄×C₄₁	—	4	2	no	—
64	C₂²×C₄×C₄₁	—	4	2	no	—
65	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₁	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₄₁	C₂×C₈	1	4	yes	—
71	C₂²×C₄×C₄₁	M₁₆	1	2	yes	—
72	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₄₁	#41	1	2	yes	—
76	C₂×C₈×C₄₁	—	2	4	no	—
77	C₂×C₈×C₄₁	—	4	2	no	—
78	C₂×C₈×C₄₁	—	4	2	no	—
79	C₂×C₈×C₄₁	—	4	2	no	—
80	C₂×C₈×C₄₁	—	4	2	no	—
81	C₂×C₈×C₄₁	—	4	2	no	—
82	C₄⋊C₄×C₄₁	—	2	4	no	—
83	C₄⋊C₄×C₄₁	—	2	4	no	—
84	C₄⋊C₄×C₄₁	—	4	2	no	—
85	C₄⋊C₄×C₄₁	—	4	4	no	—
86	C₄⋊C₄×C₄₁	—	4	2	no	—
87	C₄⋊C₄×C₄₁	—	4	4	no	—
88	C₄²×C₄₁	—	2	2	no	—
89	C₄²×C₄₁	—	2	2	no	—
90	C₄²×C₄₁	—	4	1	no	—
91	C₄²×C₄₁	—	4	1	no	—
92	C₂³×C₄₁	—	2	1	no	—
93	C₂³×C₄₁	—	2	1	no	—
94	C₂³×C₄₁	—	2	1	no	—
95	C₂³×C₄₁	—	2	1	no	—
96	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₄₁	C₄×Q₈	1	2	yes	—
98	C₂³×C₄₁	C₂²×C₈	1	1	yes	characteristic
99	C₂³×C₄₁	C₂×M₁₆	1	1	yes	characteristic
100	C₂³×C₄₁	#131	1	2	yes	—
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	1	no	—
109	C₂×C₄×C₄₁	—	2	4	no	—
110	C₂×C₄×C₄₁	—	2	4	no	—
111	C₂×C₄×C₄₁	—	2	4	no	—
112	C₂×C₄×C₄₁	—	2	1	no	—
113	C₂×C₄×C₄₁	—	2	1	no	—
114	C₂×C₄×C₄₁	—	2	1	no	—
115	C₂×C₄×C₄₁	—	4	2	no	—
116	C₂×C₄×C₄₁	—	4	2	no	—
117	C₂×C₄×C₄₁	—	4	4	no	—
118	C₂×C₄×C₄₁	—	4	4	no	—
119	C₂×C₄×C₄₁	—	4	2	no	—
120	C₂×C₄×C₄₁	—	4	2	no	—
121	C₂×C₄×C₄₁	—	4	4	no	—
122	C₂×C₄×C₄₁	—	4	2	no	—
123	C₂×C₄×C₄₁	—	4	2	no	—
124	C₂×C₄×C₄₁	—	4	2	no	—
125	C₂×C₄×C₄₁	—	8	4	no	—
126	C₂×C₄×C₄₁	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₄₁	C₂²⋊C₈	1	2	yes	—
128	C₂×C₄×C₄₁	#99	1	2	yes	—
129	C₂×C₄×C₄₁	#100	1	2	yes	—
130	C₂×C₄×C₄₁	#101	1	2	yes	—
131	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₄₁	#124	1	1	yes	characteristic
133	C₂×C₄×C₄₁	C₄⋊Q₈	1	1	yes	characteristic
134	C₈×C₄₁	—	4	4	no	—
135	C₈×C₄₁	—	8	2	no	—
136	#9879	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₄₁	—	2	1	no	—
138	C₂²×C₄₁	—	2	1	no	—
139	C₂²×C₄₁	—	2	1	no	—
140	C₂²×C₄₁	—	2	1	no	—
141	C₂²×C₄₁	—	2	2	no	—
142	C₂²×C₄₁	—	2	2	no	—
143	C₂²×C₄₁	—	2	2	no	—
144	C₂²×C₄₁	—	2	2	no	—
145	C₂²×C₄₁	#391	1	1	yes	characteristic
146	C₂²×C₄₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₂²×C₄₁	#408	1	1	yes	characteristic
148	C₂²×C₄₁	C₂×#99	1	1	yes	characteristic
149	C₂²×C₄₁	#410	1	1	yes	characteristic
150	C₂²×C₄₁	C₂×#100	1	1	yes	characteristic
151	C₂²×C₄₁	C₂×#101	1	1	yes	characteristic
152	C₂²×C₄₁	C₈×Q₈	1	2	yes	—
153	C₂²×C₄₁	#446	1	2	yes	—
154	C₄×C₄₁	—	2	2	no	—
155	C₄×C₄₁	—	2	2	no	—
156	C₄×C₄₁	—	2	2	no	—
157	C₄×C₄₁	—	2	1	no	—
158	C₄×C₄₁	—	2	1	no	—
159	C₄×C₄₁	—	4	4	no	—
160	C₄×C₄₁	—	4	1	no	—
161	C₄×C₄₁	—	8	2	no	—
162	C₄×C₄₁	#324	1	4	yes	—
163	C₂×#389	C₂×C₄₁	1	1	yes	characteristic
164	#1797	C₂×C₄₁	1	1	yes	characteristic
165	C₄×C₂²⋊C₈	C₂×C₄₁	1	1	yes	characteristic
166	#1280	C₂×C₄₁	1	4	yes	—
167	C₂×C₄₁	—	2	2	no	—
168	C₂×C₄₁	—	2	2	no	—
169	C₂×C₄₁	C₂×#324	1	1	yes	characteristic
170	C₂×C₄₁	#1460	1	1	yes	characteristic
171	C₂×C₄₁	#1470	1	1	yes	characteristic
172	C₂×C₄₁	#1471	1	1	yes	characteristic
173	C₂×C₄₁	#1984	1	1	yes	characteristic
174	C₂×C₄₁	#1996	1	1	yes	characteristic
175	C₂×C₄₁	#1997	1	1	yes	characteristic
176	C₂²×C₄⋊C₄	C₄×C₄₁	1	1	yes	characteristic
177	C₂²×C₄⋊C₄	C₄×C₄₁	1	2	yes	—
178	C₂²×C₄⋊C₄	C₄×C₄₁	1	1	yes	characteristic
179	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
180	C₂×C₄⋊C₈	—	2	1	no	—
181	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
182	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
183	C₂×C₄×C₈	—	2	1	no	—
184	#389	—	2	1	no	—
185	#389	—	2	1	no	—
186	#389	—	2	1	no	—
187	#389	—	2	1	no	—
188	C₂×#95	C₂²×C₄₁	1	2	yes	—
189	#336	—	2	1	no	—
190	#336	—	2	1	no	—
191	C₄₁	#9879	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
193	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
194	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
195	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
196	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	4	2	no	—
207	C₂×C₄⋊C₄	—	4	4	no	—
208	C₂×C₄⋊C₄	C₈×C₄₁	1	4	yes	complemented
209	C₂×C₄²	—	4	2	no	—
210	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
211	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
212	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
213	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
214	C₄⋊C₈	—	4	2	no	—
215	C₂²⋊C₈	—	2	2	no	—
216	C₂²⋊C₈	—	2	4	no	—
217	C₂²⋊C₈	Q₈×C₄₁	1	4	yes	—
218	C₄×C₈	—	4	2	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
240	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₈×C₄₁	1	2	yes	—
242	C₂²×C₄	C₂×C₈×C₄₁	1	4	yes	—
243	C₂²×C₄	M₁₆×C₄₁	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
246	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
247	C₂²×C₄	#41×C₄₁	1	2	yes	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	—	4	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
269	C₂³	C₄×Q₈×C₄₁	1	2	yes	—
270	C₂³	C₂²×C₈×C₄₁	1	1	yes	characteristic
271	C₂³	C₂×M₁₆×C₄₁	1	1	yes	characteristic
272	C₂³	#131×C₄₁	1	2	yes	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₂²⋊C₈×C₄₁	1	2	yes	—
299	C₂×C₄	C₂²⋊C₈×C₄₁	1	2	yes	—
300	C₂×C₄	#99×C₄₁	1	2	yes	—
301	C₂×C₄	#100×C₄₁	1	2	yes	—
302	C₂×C₄	#101×C₄₁	1	2	yes	—
303	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
304	C₂×C₄	#124×C₄₁	1	1	yes	characteristic
305	C₂×C₄	C₄⋊Q₈×C₄₁	1	1	yes	characteristic
306	C₈	—	4	4	no	—
307	C₈	—	8	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	C₄₁×#391	1	1	yes	characteristic
317	C₂²	C₂×C₂²⋊C₈×C₄₁	1	1	yes	characteristic
318	C₂²	C₄₁×#408	1	1	yes	characteristic
319	C₂²	C₂×#99×C₄₁	1	1	yes	characteristic
320	C₂²	C₄₁×#410	1	1	yes	characteristic
321	C₂²	C₂×#100×C₄₁	1	1	yes	characteristic
322	C₂²	C₂×#101×C₄₁	1	1	yes	characteristic
323	C₂²	C₈×Q₈×C₄₁	1	2	yes	—
324	C₂²	C₄₁×#446	1	2	yes	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	4	no	—
331	C₄	—	4	1	no	—
332	C₄	—	8	2	no	—
333	C₄	C₄₁×#324	1	4	yes	—
334	C₂	—	2	2	no	—
335	C₂	—	2	2	no	—
336	C₂	C₂×C₄₁×#324	1	1	yes	characteristic
337	C₂	C₄₁×#1460	1	1	yes	characteristic
338	C₂	C₄₁×#1470	1	1	yes	characteristic
339	C₂	C₄₁×#1471	1	1	yes	characteristic
340	C₂	C₄₁×#1984	1	1	yes	characteristic
341	C₂	C₄₁×#1996	1	1	yes	characteristic
342	C₂	C₄₁×#1997	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₄₁
Frattini, cosocle	C₂³×C₄	C₂³×C₄₁
prime radical	C₂⁴×C₄₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ 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
36. C₂×C₄⋊C₄×C₄₁	C₈	semidirect
136. #9879	C₄₁	direct (N × H)
191. C₄₁	#9879	direct (N × H)
208. C₂×C₄⋊C₄	C₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#389	2
C₄₁×#1797	2
C₄×C₂²⋊C₈×C₄₁	2
C₄₁×#1280	2
#9879	41

Subgroups — C41×#9879 #23786080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#9879 #23786080