Subgroups — C₂×C₄₁×#2056 #23799739

Order 10496.

1486 conjugacy classes of subgroups.

The subgroup lattice has 336 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	C₂²×C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#103×C₄₁	C₂	1	2	yes	maximal
4	C₂×C₄₁×#391	C₂	1	2	yes	maximal
5	C₄₁×#2056	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄₁×#340	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	4	no	—
10	C₂²×Q₁₆×C₄₁	C₂²	1	4	yes	—
11	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
14	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₄₁	—	2	4	no	—
16	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
17	C₂×#103×C₄₁	—	2	8	no	—
18	C₂×#103×C₄₁	C₂²	1	8	yes	—
19	C₄₁×#391	—	2	4	no	—
20	C₄₁×#391	C₂²	1	8	yes	—
21	C₂×#95×C₄₁	—	2	2	no	—
22	C₂×#95×C₄₁	—	2	2	no	—
23	C₄₁×#340	C₂²	1	4	yes	—
24	C₂²×Q₈×C₄₁	—	2	2	no	—
25	C₂²×Q₈×C₄₁	—	2	4	no	—
26	C₂²×Q₈×C₄₁	—	2	4	no	—
27	C₂²×Q₈×C₄₁	—	2	4	no	—
28	C₂²×Q₈×C₄₁	—	2	4	no	—
29	C₂²×Q₈×C₄₁	—	2	4	no	—
30	C₂²×Q₈×C₄₁	D₄	1	8	yes	—
31	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
32	C₂³×C₄×C₄₁	—	2	2	no	—
33	C₂³×C₄×C₄₁	—	2	2	no	—
34	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
37	C₂×Q₁₆×C₄₁	—	2	32	no	—
38	C₂×Q₁₆×C₄₁	—	2	16	no	—
39	C₂×Q₁₆×C₄₁	—	2	8	no	—
40	C₂²×C₈×C₄₁	—	2	4	no	—
41	C₂²×C₈×C₄₁	—	2	4	no	—
42	C₂²×C₈×C₄₁	—	2	2	no	—
43	C₂²×C₈×C₄₁	D₄	1	4	yes	—
44	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
50	C₂×C₄²×C₄₁	—	2	2	no	—
51	C₂×C₄²×C₄₁	—	2	2	no	—
52	C₂×C₄²×C₄₁	D₄	1	2	yes	—
53	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
54	C₄⋊C₈×C₄₁	—	2	8	no	—
55	#103×C₄₁	—	2	16	no	—
56	#95×C₄₁	—	2	8	no	—
57	#95×C₄₁	—	2	8	no	—
58	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₄₁	—	2	8	no	—
60	C₂×Q₈×C₄₁	—	2	4	no	—
61	C₂×Q₈×C₄₁	—	2	16	no	—
62	C₂×Q₈×C₄₁	—	2	16	no	—
63	C₂×Q₈×C₄₁	—	2	4	no	—
64	C₂×Q₈×C₄₁	—	2	4	no	—
65	C₂×Q₈×C₄₁	—	2	4	no	—
66	C₂×Q₈×C₄₁	—	2	4	no	—
67	C₂×Q₈×C₄₁	—	4	4	no	—
68	C₂×Q₈×C₄₁	—	4	4	no	—
69	C₂×Q₈×C₄₁	—	4	8	no	—
70	C₂×Q₈×C₄₁	—	4	8	no	—
71	C₂×Q₈×C₄₁	C₂×D₄	1	8	yes	—
72	C₂²×C₄×C₄₁	—	2	2	no	—
73	C₂²×C₄×C₄₁	—	2	2	no	—
74	C₂²×C₄×C₄₁	—	2	2	no	—
75	C₂²×C₄×C₄₁	—	2	8	no	—
76	C₂²×C₄×C₄₁	—	2	4	no	—
77	C₂²×C₄×C₄₁	—	2	4	no	—
78	C₂²×C₄×C₄₁	—	2	4	no	—
79	C₂²×C₄×C₄₁	—	4	4	no	—
80	C₂²×C₄×C₄₁	—	4	4	no	—
81	C₂²×C₄×C₄₁	—	4	4	no	—
82	C₂²×C₄×C₄₁	—	4	4	no	—
83	C₂²×C₄×C₄₁	—	4	4	no	—
84	C₂²×C₄×C₄₁	Q₁₆	1	4	yes	complemented
85	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₁	#41	1	2	yes	—
91	C₂²×C₄×C₄₁	#41	1	4	yes	—
92	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
93	Q₁₆×C₄₁	—	2	32	no	—
94	C₂×C₈×C₄₁	—	2	8	no	—
95	C₂×C₈×C₄₁	—	2	4	no	—
96	C₂×C₈×C₄₁	—	2	4	no	—
97	C₂×C₈×C₄₁	—	2	2	no	—
98	C₂×C₈×C₄₁	—	4	4	no	—
99	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
100	C₄⋊C₄×C₄₁	—	2	4	no	—
101	C₄⋊C₄×C₄₁	—	2	4	no	—
102	C₄⋊C₄×C₄₁	—	4	4	no	—
103	C₄²×C₄₁	—	2	4	no	—
104	C₄²×C₄₁	—	2	2	no	—
105	C₄²×C₄₁	—	2	2	no	—
106	C₂³×C₄₁	#121	1	4	yes	—
107	C₂³×C₄₁	#121	1	2	yes	—
108	C₂³×C₄₁	C₂×Q₁₆	1	2	yes	—
109	C₂³×C₄₁	#135	1	2	yes	—
110	C₂³×C₄₁	#137	1	4	yes	—
111	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₄₁	—	2	8	no	—
113	Q₈×C₄₁	—	2	8	no	—
114	Q₈×C₄₁	—	4	4	no	—
115	Q₈×C₄₁	—	4	4	no	—
116	Q₈×C₄₁	—	4	16	no	—
117	C₂×C₄×C₄₁	—	2	2	no	—
118	C₂×C₄×C₄₁	—	2	2	no	—
119	C₂×C₄×C₄₁	—	2	8	no	—
120	C₂×C₄×C₄₁	—	2	2	no	—
121	C₂×C₄×C₄₁	—	2	8	no	—
122	C₂×C₄×C₄₁	—	2	4	no	—
123	C₂×C₄×C₄₁	—	4	4	no	—
124	C₂×C₄×C₄₁	—	4	4	no	—
125	C₂×C₄×C₄₁	—	4	16	no	—
126	C₂×C₄×C₄₁	—	4	4	no	—
127	C₂×C₄×C₄₁	—	4	4	no	—
128	C₂×C₄×C₄₁	—	4	4	no	—
129	C₂×C₄×C₄₁	—	4	4	no	—
130	C₂×C₄×C₄₁	—	8	8	no	—
131	C₂×C₄×C₄₁	#120	1	4	yes	—
132	C₂×C₄×C₄₁	#121	1	2	yes	—
133	C₂×C₄×C₄₁	#121	1	4	yes	—
134	C₂×C₄×C₄₁	#123	1	8	yes	—
135	C₂×C₄×C₄₁	#124	1	4	yes	—
136	C₂×C₄×C₄₁	C₂×Q₁₆	1	4	yes	complemented
137	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
141	C₈×C₄₁	—	2	4	no	—
142	C₈×C₄₁	—	4	2	no	—
143	C₂×#2056	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₄₁	#394	1	4	yes	—
145	C₂²×C₄₁	#462	1	8	yes	—
146	C₂²×C₄₁	#464	1	8	yes	—
147	C₂²×C₄₁	#467	1	4	yes	—
148	C₂²×C₄₁	#470	1	4	yes	—
149	C₂²×C₄₁	C₂×#121	1	2	yes	—
150	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
151	C₂²×C₄₁	C₂²×Q₁₆	1	1	yes	characteristic
152	C₂²×C₄₁	C₂×#135	1	1	yes	characteristic
153	C₂²×C₄₁	C₂×#137	1	2	yes	—
154	C₄×C₄₁	—	2	4	no	—
155	C₄×C₄₁	—	4	8	no	—
156	C₄×C₄₁	—	4	2	no	—
157	C₄×C₄₁	—	8	4	no	—
158	C₄×C₄₁	#394	1	4	yes	—
159	C₄×C₄₁	C₂×#120	1	1	yes	characteristic
160	C₄×C₄₁	C₂×#123	1	2	yes	—
161	C₄×C₄₁	C₂×#124	1	1	yes	characteristic
162	C₂³×Q₁₆	C₂×C₄₁	1	1	yes	characteristic
163	C₂²×C₄⋊C₈	C₂×C₄₁	1	1	yes	characteristic
164	C₂²×#103	C₂×C₄₁	1	2	yes	—
165	C₂×#391	C₂×C₄₁	1	2	yes	—
166	#2056	C₂×C₄₁	1	8	yes	direct product factor
167	C₂×#340	C₂×C₄₁	1	1	yes	characteristic
168	C₂×C₄₁	#2056	1	8	yes	direct product factor
169	C₂×C₄₁	C₂×#394	1	1	yes	characteristic
170	C₂×C₄₁	C₂×#462	1	2	yes	—
171	C₂×C₄₁	C₂×#464	1	2	yes	—
172	C₂×C₄₁	C₂×#467	1	1	yes	characteristic
173	C₂×C₄₁	C₂×#470	1	1	yes	characteristic
174	C₂³×Q₈	C₂²×C₄₁	1	2	yes	—
175	C₂²×Q₁₆	—	2	8	no	—
176	C₂²×Q₁₆	—	2	4	no	—
177	C₂²×Q₁₆	C₂²×C₄₁	1	4	yes	—
178	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
179	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
181	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
182	C₂×C₄⋊C₈	—	2	4	no	—
183	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
184	C₂×#103	—	2	8	no	—
185	C₂×#103	C₂²×C₄₁	1	8	yes	—
186	#391	—	2	4	no	—
187	#391	C₂²×C₄₁	1	8	yes	—
188	C₂×#95	—	2	2	no	—
189	C₂×#95	—	2	2	no	—
190	#340	C₂²×C₄₁	1	4	yes	—
191	C₄₁	C₂×#2056	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²×Q₈	—	2	2	no	—
193	C₂²×Q₈	—	2	4	no	—
194	C₂²×Q₈	—	2	4	no	—
195	C₂²×Q₈	—	2	4	no	—
196	C₂²×Q₈	—	2	4	no	—
197	C₂²×Q₈	—	2	4	no	—
198	C₂²×Q₈	D₄×C₄₁	1	8	yes	—
199	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
200	C₂³×C₄	—	2	2	no	—
201	C₂³×C₄	—	2	2	no	—
202	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
203	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
204	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
205	C₂×Q₁₆	—	2	32	no	—
206	C₂×Q₁₆	—	2	16	no	—
207	C₂×Q₁₆	—	2	8	no	—
208	C₂²×C₈	—	2	4	no	—
209	C₂²×C₈	—	2	4	no	—
210	C₂²×C₈	—	2	2	no	—
211	C₂²×C₈	D₄×C₄₁	1	4	yes	—
212	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
213	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
214	C₂×C₄⋊C₄	—	2	4	no	—
215	C₂×C₄⋊C₄	—	2	4	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	D₄×C₄₁	1	2	yes	—
221	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
222	C₄⋊C₈	—	2	8	no	—
223	#103	—	2	16	no	—
224	#95	—	2	8	no	—
225	#95	—	2	8	no	—
226	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
227	C₂×Q₈	—	2	8	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	2	16	no	—
230	C₂×Q₈	—	2	16	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	4	4	no	—
236	C₂×Q₈	—	4	4	no	—
237	C₂×Q₈	—	4	8	no	—
238	C₂×Q₈	—	4	8	no	—
239	C₂×Q₈	C₂×D₄×C₄₁	1	8	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	—	4	4	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	—	4	4	no	—
251	C₂²×C₄	—	4	4	no	—
252	C₂²×C₄	Q₁₆×C₄₁	1	4	yes	complemented
253	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
256	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
257	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
258	C₂²×C₄	#41×C₄₁	1	2	yes	—
259	C₂²×C₄	#41×C₄₁	1	4	yes	—
260	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
261	Q₁₆	—	2	32	no	—
262	C₂×C₈	—	2	8	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	4	4	no	—
267	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	4	4	no	—
271	C₄²	—	2	4	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₂³	#121×C₄₁	1	4	yes	—
275	C₂³	#121×C₄₁	1	2	yes	—
276	C₂³	C₂×Q₁₆×C₄₁	1	2	yes	—
277	C₂³	#135×C₄₁	1	2	yes	—
278	C₂³	#137×C₄₁	1	4	yes	—
279	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
280	Q₈	—	2	8	no	—
281	Q₈	—	2	8	no	—
282	Q₈	—	4	4	no	—
283	Q₈	—	4	4	no	—
284	Q₈	—	4	16	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	16	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	8	8	no	—
299	C₂×C₄	#120×C₄₁	1	4	yes	—
300	C₂×C₄	#121×C₄₁	1	2	yes	—
301	C₂×C₄	#121×C₄₁	1	4	yes	—
302	C₂×C₄	#123×C₄₁	1	8	yes	—
303	C₂×C₄	#124×C₄₁	1	4	yes	—
304	C₂×C₄	C₂×Q₁₆×C₄₁	1	4	yes	complemented
305	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
306	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
308	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
309	C₈	—	2	4	no	—
310	C₈	—	4	2	no	—
311	C₂²	C₄₁×#394	1	4	yes	—
312	C₂²	C₄₁×#462	1	8	yes	—
313	C₂²	C₄₁×#464	1	8	yes	—
314	C₂²	C₄₁×#467	1	4	yes	—
315	C₂²	C₄₁×#470	1	4	yes	—
316	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
317	C₂²	C₂×#121×C₄₁	1	2	yes	—
318	C₂²	C₂²×Q₁₆×C₄₁	1	1	yes	characteristic
319	C₂²	C₂×#135×C₄₁	1	1	yes	characteristic
320	C₂²	C₂×#137×C₄₁	1	2	yes	—
321	C₄	—	2	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	8	no	—
324	C₄	—	8	4	no	—
325	C₄	C₄₁×#394	1	4	yes	—
326	C₄	C₂×#120×C₄₁	1	1	yes	characteristic
327	C₄	C₂×#123×C₄₁	1	2	yes	—
328	C₄	C₂×#124×C₄₁	1	1	yes	characteristic
329	C₂	C₄₁×#2056	1	8	yes	direct product factor
330	C₂	C₂×C₄₁×#394	1	1	yes	characteristic
331	C₂	C₂×C₄₁×#462	1	2	yes	—
332	C₂	C₂×C₄₁×#464	1	2	yes	—
333	C₂	C₂×C₄₁×#467	1	1	yes	characteristic
334	C₂	C₂×C₄₁×#470	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₄₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₄₁×#2056	C₂	direct (N × H)
84. C₂²×C₄×C₄₁	Q₁₆	semidirect
136. C₂×C₄×C₄₁	C₂×Q₁₆	semidirect
143. C₂×#2056	C₄₁	direct (N × H)
166. #2056	C₂×C₄₁	direct (N × H)
168. C₂×C₄₁	#2056	direct (N × H)
191. C₄₁	C₂×#2056	direct (N × H)
252. C₂²×C₄	Q₁₆×C₄₁	semidirect
304. C₂×C₄	C₂×Q₁₆×C₄₁	semidirect
329. C₂	C₄₁×#2056	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
C₂²×C₄⋊C₈×C₄₁	2
C₂²×#103×C₄₁	2
C₂×C₄₁×#391	2
C₄₁×#2056	2
C₂×C₄₁×#340	2
C₂×#2056	41

Subgroups — C2×C41×#2056 #23799739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2056 #23799739