Subgroups — C₂×C₄₁×#2047 #23799720

Order 10496.

1330 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	C₂³×Q₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#424	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#419	C₂	1	2	yes	maximal
4	C₂²×#101×C₄₁	C₂	1	2	yes	maximal
5	C₄₁×#2047	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄₁×#341	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₄₁	C₂²	1	2	yes	—
8	C₂²×Q₁₆×C₄₁	—	2	8	no	—
9	C₂²×Q₁₆×C₄₁	—	2	2	no	—
10	C₂²×Q₁₆×C₄₁	—	2	2	no	—
11	C₂²×Q₁₆×C₄₁	C₂²	1	4	yes	—
12	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
14	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#424	—	2	2	no	—
17	C₄₁×#424	C₂²	1	4	yes	—
18	C₂×C₄⋊C₈×C₄₁	—	2	2	no	—
19	C₄₁×#419	—	2	4	no	—
20	C₄₁×#419	C₂²	1	8	yes	—
21	C₂×#103×C₄₁	—	2	4	no	—
22	C₂×#101×C₄₁	—	2	8	no	—
23	C₂×#101×C₄₁	C₂²	1	8	yes	—
24	C₄₁×#341	C₂²	1	4	yes	—
25	C₂²×Q₈×C₄₁	—	2	4	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	—	2	2	no	—
31	C₂²×Q₈×C₄₁	—	2	2	no	—
32	C₂²×Q₈×C₄₁	—	2	4	no	—
33	C₂²×Q₈×C₄₁	D₄	1	8	yes	—
34	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
35	C₂³×C₄×C₄₁	—	2	2	no	—
36	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
37	C₂×Q₁₆×C₄₁	—	2	16	no	—
38	C₂×Q₁₆×C₄₁	—	2	16	no	—
39	C₂×Q₁₆×C₄₁	—	2	16	no	—
40	C₂×Q₁₆×C₄₁	—	2	8	no	—
41	C₂×M₁₆×C₄₁	—	2	2	no	—
42	C₂×M₁₆×C₄₁	—	2	1	no	—
43	C₂×M₁₆×C₄₁	—	2	4	no	—
44	C₂×M₁₆×C₄₁	—	2	4	no	—
45	C₂×M₁₆×C₄₁	—	2	4	no	—
46	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
47	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₄₁	—	2	4	no	—
49	C₂²×C₈×C₄₁	—	2	2	no	—
50	C₂²×C₈×C₄₁	—	2	1	no	—
51	C₂²×C₈×C₄₁	D₄	1	2	yes	—
52	C₂²×C₈×C₄₁	D₄	1	2	yes	—
53	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
54	#117×C₄₁	—	2	1	no	—
55	#117×C₄₁	—	2	1	no	—
56	#117×C₄₁	D₄	1	2	yes	—
57	#117×C₄₁	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
60	C₂×C₄²×C₄₁	—	2	1	no	—
61	C₄⋊C₈×C₄₁	—	2	8	no	—
62	#103×C₄₁	—	2	16	no	—
63	#101×C₄₁	—	2	16	no	—
64	C₂⁴×C₄₁	#41	1	1	yes	characteristic
65	C₂×Q₈×C₄₁	—	2	8	no	—
66	C₂×Q₈×C₄₁	—	2	4	no	—
67	C₂×Q₈×C₄₁	—	2	8	no	—
68	C₂×Q₈×C₄₁	—	2	8	no	—
69	C₂×Q₈×C₄₁	—	2	8	no	—
70	C₂×Q₈×C₄₁	—	2	8	no	—
71	C₂×Q₈×C₄₁	—	2	4	no	—
72	C₂×Q₈×C₄₁	—	2	4	no	—
73	C₂×Q₈×C₄₁	—	2	4	no	—
74	C₂×Q₈×C₄₁	—	2	4	no	—
75	C₂×Q₈×C₄₁	—	4	2	no	—
76	C₂×Q₈×C₄₁	—	4	2	no	—
77	C₂×Q₈×C₄₁	—	4	4	no	—
78	C₂×Q₈×C₄₁	—	4	4	no	—
79	C₂×Q₈×C₄₁	—	4	4	no	—
80	C₂×Q₈×C₄₁	—	4	4	no	—
81	C₂×Q₈×C₄₁	—	4	4	no	—
82	C₂×Q₈×C₄₁	C₂×D₄	1	8	yes	—
83	C₂²×C₄×C₄₁	—	2	2	no	—
84	C₂²×C₄×C₄₁	—	2	8	no	—
85	C₂²×C₄×C₄₁	—	2	4	no	—
86	C₂²×C₄×C₄₁	—	4	2	no	—
87	C₂²×C₄×C₄₁	—	4	2	no	—
88	C₂²×C₄×C₄₁	—	4	4	no	—
89	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
90	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
95	C₂²×C₄×C₄₁	#41	1	4	yes	—
96	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
97	Q₁₆×C₄₁	—	2	32	no	—
98	M₁₆×C₄₁	—	2	4	no	—
99	M₁₆×C₄₁	—	2	4	no	—
100	M₁₆×C₄₁	—	2	4	no	—
101	M₁₆×C₄₁	—	4	2	no	—
102	M₁₆×C₄₁	—	4	4	no	—
103	C₂×C₈×C₄₁	—	2	4	no	—
104	C₂×C₈×C₄₁	—	2	4	no	—
105	C₂×C₈×C₄₁	—	2	4	no	—
106	C₂×C₈×C₄₁	—	2	4	no	—
107	C₂×C₈×C₄₁	—	2	2	no	—
108	C₂×C₈×C₄₁	—	4	2	no	—
109	C₂×C₈×C₄₁	—	4	4	no	—
110	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
111	C₄⋊C₄×C₄₁	—	2	2	no	—
112	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
113	C₂²⋊C₄×C₄₁	—	2	2	no	—
114	C₂²⋊C₄×C₄₁	—	2	2	no	—
115	C₄²×C₄₁	—	2	2	no	—
116	C₄²×C₄₁	—	2	2	no	—
117	C₂³×C₄₁	—	2	4	no	—
118	C₂³×C₄₁	#121	1	1	yes	characteristic
119	C₂³×C₄₁	#121	1	1	yes	characteristic
120	C₂³×C₄₁	#123	1	4	yes	—
121	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
122	Q₈×C₄₁	—	2	8	no	—
123	Q₈×C₄₁	—	2	8	no	—
124	Q₈×C₄₁	—	4	4	no	—
125	Q₈×C₄₁	—	4	4	no	—
126	Q₈×C₄₁	—	4	8	no	—
127	Q₈×C₄₁	—	4	8	no	—
128	C₂×C₄×C₄₁	—	2	4	no	—
129	C₂×C₄×C₄₁	—	2	8	no	—
130	C₂×C₄×C₄₁	—	4	4	no	—
131	C₂×C₄×C₄₁	—	4	4	no	—
132	C₂×C₄×C₄₁	—	4	8	no	—
133	C₂×C₄×C₄₁	—	4	8	no	—
134	C₂×C₄×C₄₁	—	4	4	no	—
135	C₂×C₄×C₄₁	#120	1	2	yes	—
136	C₂×C₄×C₄₁	#120	1	2	yes	—
137	C₂×C₄×C₄₁	#121	1	4	yes	—
138	C₂×C₄×C₄₁	#121	1	4	yes	—
139	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₁	#123	1	4	yes	—
142	C₂×C₄×C₄₁	#124	1	2	yes	—
143	C₂×C₄×C₄₁	#124	1	2	yes	—
144	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
145	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
148	C₈×C₄₁	—	2	4	no	—
149	C₈×C₄₁	—	4	2	no	—
150	C₈×C₄₁	—	4	4	no	—
151	C₂×#2047	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂²×C₄₁	—	2	2	no	—
153	C₂²×C₄₁	—	2	2	no	—
154	C₂²×C₄₁	—	2	8	no	—
155	C₂²×C₄₁	#394	1	2	yes	—
156	C₂²×C₄₁	#394	1	2	yes	—
157	C₂²×C₄₁	#473	1	4	yes	—
158	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
159	C₂²×C₄₁	C₂×#123	1	2	yes	—
160	C₄×C₄₁	—	4	8	no	—
161	C₄×C₄₁	—	4	4	no	—
162	C₄×C₄₁	#394	1	2	yes	—
163	C₄×C₄₁	#394	1	2	yes	—
164	C₄×C₄₁	C₂×#120	1	1	yes	characteristic
165	C₄×C₄₁	C₂×#121	1	2	yes	—
166	C₄×C₄₁	C₂×#124	1	1	yes	characteristic
167	C₂³×Q₁₆	C₂×C₄₁	1	1	yes	characteristic
168	C₂×#424	C₂×C₄₁	1	1	yes	characteristic
169	C₂×#419	C₂×C₄₁	1	2	yes	—
170	C₂²×#101	C₂×C₄₁	1	2	yes	—
171	#2047	C₂×C₄₁	1	8	yes	direct product factor
172	C₂×#341	C₂×C₄₁	1	1	yes	characteristic
173	C₂×C₄₁	—	2	2	no	—
174	C₂×C₄₁	—	2	2	no	—
175	C₂×C₄₁	#2047	1	4	yes	direct product factor
176	C₂×C₄₁	C₂×#394	1	1	yes	characteristic
177	C₂×C₄₁	C₂×#473	1	2	yes	—
178	C₂³×Q₈	C₂²×C₄₁	1	2	yes	—
179	C₂²×Q₁₆	—	2	8	no	—
180	C₂²×Q₁₆	—	2	2	no	—
181	C₂²×Q₁₆	—	2	2	no	—
182	C₂²×Q₁₆	C₂²×C₄₁	1	4	yes	—
183	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
184	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
185	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
186	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
187	#424	—	2	2	no	—
188	#424	C₂²×C₄₁	1	4	yes	—
189	C₂×C₄⋊C₈	—	2	2	no	—
190	#419	—	2	4	no	—
191	#419	C₂²×C₄₁	1	8	yes	—
192	C₂×#103	—	2	4	no	—
193	C₂×#101	—	2	8	no	—
194	C₂×#101	C₂²×C₄₁	1	8	yes	—
195	#341	C₂²×C₄₁	1	4	yes	—
196	C₄₁	C₂×#2047	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²×Q₈	—	2	4	no	—
198	C₂²×Q₈	—	2	2	no	—
199	C₂²×Q₈	—	2	2	no	—
200	C₂²×Q₈	—	2	2	no	—
201	C₂²×Q₈	—	2	4	no	—
202	C₂²×Q₈	—	2	2	no	—
203	C₂²×Q₈	—	2	2	no	—
204	C₂²×Q₈	—	2	4	no	—
205	C₂²×Q₈	D₄×C₄₁	1	8	yes	—
206	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
207	C₂³×C₄	—	2	2	no	—
208	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
209	C₂×Q₁₆	—	2	16	no	—
210	C₂×Q₁₆	—	2	16	no	—
211	C₂×Q₁₆	—	2	16	no	—
212	C₂×Q₁₆	—	2	8	no	—
213	C₂×M₁₆	—	2	2	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	2	4	no	—
216	C₂×M₁₆	—	2	4	no	—
217	C₂×M₁₆	—	2	4	no	—
218	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
219	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
220	C₂²×C₈	—	2	4	no	—
221	C₂²×C₈	—	2	2	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	D₄×C₄₁	1	2	yes	—
224	C₂²×C₈	D₄×C₄₁	1	2	yes	—
225	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
226	#117	—	2	1	no	—
227	#117	—	2	1	no	—
228	#117	D₄×C₄₁	1	2	yes	—
229	#117	C₂³×C₄₁	1	2	yes	—
230	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
231	C₂×C₂²⋊C₄	—	2	1	no	—
232	C₂×C₄²	—	2	1	no	—
233	C₄⋊C₈	—	2	8	no	—
234	#103	—	2	16	no	—
235	#101	—	2	16	no	—
236	C₂⁴	#41×C₄₁	1	1	yes	characteristic
237	C₂×Q₈	—	2	8	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	2	8	no	—
240	C₂×Q₈	—	2	8	no	—
241	C₂×Q₈	—	2	8	no	—
242	C₂×Q₈	—	2	8	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	—	4	2	no	—
249	C₂×Q₈	—	4	4	no	—
250	C₂×Q₈	—	4	4	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	4	4	no	—
253	C₂×Q₈	—	4	4	no	—
254	C₂×Q₈	C₂×D₄×C₄₁	1	8	yes	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	8	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	4	4	no	—
261	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
262	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
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	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
267	C₂²×C₄	#41×C₄₁	1	4	yes	—
268	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
269	Q₁₆	—	2	32	no	—
270	M₁₆	—	2	4	no	—
271	M₁₆	—	2	4	no	—
272	M₁₆	—	2	4	no	—
273	M₁₆	—	4	2	no	—
274	M₁₆	—	4	4	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	—	4	4	no	—
282	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₂³	—	2	4	no	—
290	C₂³	#121×C₄₁	1	1	yes	characteristic
291	C₂³	#121×C₄₁	1	1	yes	characteristic
292	C₂³	#123×C₄₁	1	4	yes	—
293	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
294	Q₈	—	2	8	no	—
295	Q₈	—	2	8	no	—
296	Q₈	—	4	4	no	—
297	Q₈	—	4	4	no	—
298	Q₈	—	4	8	no	—
299	Q₈	—	4	8	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	8	no	—
305	C₂×C₄	—	4	8	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	#120×C₄₁	1	2	yes	—
308	C₂×C₄	#120×C₄₁	1	2	yes	—
309	C₂×C₄	#121×C₄₁	1	4	yes	—
310	C₂×C₄	#121×C₄₁	1	4	yes	—
311	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
312	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
313	C₂×C₄	#123×C₄₁	1	4	yes	—
314	C₂×C₄	#124×C₄₁	1	2	yes	—
315	C₂×C₄	#124×C₄₁	1	2	yes	—
316	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
317	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
319	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₈	—	4	4	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	8	no	—
326	C₂²	C₄₁×#394	1	2	yes	—
327	C₂²	C₄₁×#394	1	2	yes	—
328	C₂²	C₄₁×#473	1	4	yes	—
329	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
330	C₂²	C₂×#123×C₄₁	1	2	yes	—
331	C₄	—	4	4	no	—
332	C₄	—	4	8	no	—
333	C₄	C₄₁×#394	1	2	yes	—
334	C₄	C₄₁×#394	1	2	yes	—
335	C₄	C₂×#120×C₄₁	1	1	yes	characteristic
336	C₄	C₂×#121×C₄₁	1	2	yes	—
337	C₄	C₂×#124×C₄₁	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	2	2	no	—
340	C₂	C₄₁×#2047	1	4	yes	direct product factor
341	C₂	C₂×C₄₁×#394	1	1	yes	characteristic
342	C₂	C₂×C₄₁×#473	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂⁴×C₄₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₄₁×#2047	C₂	direct (N × H)
151. C₂×#2047	C₄₁	direct (N × H)
171. #2047	C₂×C₄₁	direct (N × H)
175. C₂×C₄₁	#2047	direct (N × H)
196. C₄₁	C₂×#2047	direct (N × H)
340. C₂	C₄₁×#2047	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₄₁	2
C₂×C₄₁×#424	2
C₂×C₄₁×#419	2
C₂²×#101×C₄₁	2
C₄₁×#2047	2
C₂×C₄₁×#341	2
C₂×#2047	41

Subgroups — C2×C41×#2047 #23799720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2047 #23799720