Subgroups — C₄₁×#22808 #23799009

Order 10496.

782 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	C₂×C₄×Q₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2355	C₂	1	2	yes	maximal
3	C₄₁×#1969	C₂	1	2	yes	maximal
4	C₄₁×#1928	C₂	1	4	yes	maximal
5	C₄₁×#1845	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#1803	C₂	1	2	yes	maximal
7	C₄×#107×C₄₁	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₄₁	C₂²	1	2	yes	—
11	C₄×Q₁₆×C₄₁	C₄	1	8	yes	complemented
12	C₂×#107×C₄₁	C₂²	1	2	yes	—
13	C₂×#107×C₄₁	C₂²	1	2	yes	—
14	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×#103×C₄₁	C₂²	1	4	yes	—
17	C₂×#103×C₄₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
20	C₄₁×#391	C₂²	1	4	yes	—
21	C₄₁×#387	—	2	4	no	—
22	C₄₁×#384	C₂²	1	2	yes	—
23	C₄₁×#382	—	2	2	no	—
24	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
25	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
26	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#340	C₂²	1	4	yes	—
28	C₄₁×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
30	C₂×Q₁₆×C₄₁	—	2	2	no	—
31	C₂×Q₁₆×C₄₁	—	2	2	no	—
32	C₂×Q₁₆×C₄₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₄₁	—	2	4	no	—
37	C₄×Q₈×C₄₁	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
45	C₂×C₄²×C₄₁	—	2	2	no	—
46	C₂×C₄²×C₄₁	—	2	4	no	—
47	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
48	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
49	#107×C₄₁	—	2	4	no	—
50	#107×C₄₁	—	4	2	no	—
51	#107×C₄₁	C₂×C₄	1	4	yes	—
52	C₄⋊C₈×C₄₁	—	2	2	no	—
53	#103×C₄₁	—	4	4	no	—
54	#103×C₄₁	C₂×C₄	1	8	yes	—
55	C₄×C₈×C₄₁	—	2	4	no	—
56	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
57	#95×C₄₁	—	2	4	no	—
58	#95×C₄₁	—	2	4	no	—
59	#95×C₄₁	—	2	4	no	—
60	#95×C₄₁	D₄	1	2	yes	—
61	C₂×Q₈×C₄₁	—	2	2	no	—
62	C₂×Q₈×C₄₁	—	2	2	no	—
63	C₂×Q₈×C₄₁	—	2	2	no	—
64	C₂×Q₈×C₄₁	—	2	2	no	—
65	C₂×Q₈×C₄₁	—	2	2	no	—
66	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₄₁	—	2	2	no	—
68	C₂²×C₄×C₄₁	—	2	2	no	—
69	C₂²×C₄×C₄₁	—	2	4	no	—
70	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
75	Q₁₆×C₄₁	—	2	8	no	—
76	C₂×C₈×C₄₁	—	2	1	no	—
77	C₂×C₈×C₄₁	—	2	1	no	—
78	C₂×C₈×C₄₁	—	2	2	no	—
79	C₂×C₈×C₄₁	—	2	2	no	—
80	C₂×C₈×C₄₁	—	2	2	no	—
81	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₁	#41	1	2	yes	—
84	C₂×C₈×C₄₁	#41	1	2	yes	—
85	C₄⋊C₄×C₄₁	—	2	8	no	—
86	C₄⋊C₄×C₄₁	—	2	2	no	—
87	C₄⋊C₄×C₄₁	—	2	2	no	—
88	C₄⋊C₄×C₄₁	—	2	4	no	—
89	C₄⋊C₄×C₄₁	—	2	4	no	—
90	C₄⋊C₄×C₄₁	—	4	4	no	—
91	C₄⋊C₄×C₄₁	—	8	2	no	—
92	C₄⋊C₄×C₄₁	Q₁₆	1	4	yes	complemented
93	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
94	C₄²×C₄₁	—	2	8	no	—
95	C₄²×C₄₁	—	2	2	no	—
96	C₄²×C₄₁	—	2	2	no	—
97	C₄²×C₄₁	—	8	4	no	—
98	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
99	C₄²×C₄₁	#41	1	2	yes	—
100	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₄₁	—	2	4	no	—
102	Q₈×C₄₁	—	2	4	no	—
103	Q₈×C₄₁	—	4	2	no	—
104	Q₈×C₄₁	—	4	2	no	—
105	C₂×C₄×C₄₁	—	2	2	no	—
106	C₂×C₄×C₄₁	—	2	2	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	2	no	—
112	C₂×C₄×C₄₁	—	2	2	no	—
113	C₂×C₄×C₄₁	—	2	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	—	4	4	no	—
119	C₂×C₄×C₄₁	—	4	4	no	—
120	C₂×C₄×C₄₁	—	4	4	no	—
121	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₄₁	C₂×Q₁₆	1	4	yes	—
124	C₂×C₄×C₄₁	C₂×Q₁₆	1	2	yes	—
125	C₂×C₄×C₄₁	#135	1	2	yes	—
126	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
132	C₈×C₄₁	—	2	2	no	—
133	C₈×C₄₁	—	4	2	no	—
134	C₈×C₄₁	#117	1	4	yes	—
135	#22808	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₄₁	#547	1	1	yes	characteristic
138	C₂²×C₄₁	#550	1	1	yes	characteristic
139	C₂²×C₄₁	C₂²×Q₁₆	1	1	yes	characteristic
140	C₂²×C₄₁	C₂×#135	1	1	yes	characteristic
141	C₂²×C₄₁	#576	1	1	yes	characteristic
142	C₂²×C₄₁	#578	1	1	yes	characteristic
143	C₄×C₄₁	—	2	4	no	—
144	C₄×C₄₁	—	2	2	no	—
145	C₄×C₄₁	—	4	2	no	—
146	C₄×C₄₁	—	4	4	no	—
147	C₄×C₄₁	—	4	4	no	—
148	C₄×C₄₁	—	8	4	no	—
149	C₄×C₄₁	C₄×Q₁₆	1	4	yes	—
150	C₄×C₄₁	C₂×#117	1	1	yes	characteristic
151	C₄×C₄₁	#520	1	1	yes	characteristic
152	C₄×C₄₁	#555	1	1	yes	characteristic
153	C₄×C₄₁	#556	1	1	yes	characteristic
154	C₂×C₄×Q₁₆	C₂×C₄₁	1	1	yes	characteristic
155	#2355	C₂×C₄₁	1	2	yes	—
156	#1969	C₂×C₄₁	1	2	yes	—
157	#1928	C₂×C₄₁	1	4	yes	—
158	#1845	C₂×C₄₁	1	1	yes	characteristic
159	#1803	C₂×C₄₁	1	2	yes	—
160	C₄×#107	C₂×C₄₁	1	2	yes	—
161	C₈×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
162	C₂×C₄₁	#2354	1	1	yes	characteristic
163	C₂×C₄₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
164	C₂×C₄₁	#2951	1	1	yes	characteristic
165	C₂×C₄₁	#3338	1	1	yes	characteristic
166	C₂×C₄₁	#3342	1	1	yes	characteristic
167	C₂×C₄₁	#3395	1	1	yes	characteristic
168	C₂×C₄₁	#3396	1	1	yes	characteristic
169	C₂²×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
170	C₂×C₄×Q₈	C₂²×C₄₁	1	2	yes	—
171	C₄×Q₁₆	C₄×C₄₁	1	8	yes	complemented
172	C₂×#107	C₂²×C₄₁	1	2	yes	—
173	C₂×#107	C₂²×C₄₁	1	2	yes	—
174	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
175	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
176	C₂×#103	C₂²×C₄₁	1	4	yes	—
177	C₂×#103	C₂²×C₄₁	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
180	#391	C₂²×C₄₁	1	4	yes	—
181	#387	—	2	4	no	—
182	#384	C₂²×C₄₁	1	2	yes	—
183	#382	—	2	2	no	—
184	C₄×C₄⋊C₄	—	2	2	no	—
185	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
186	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
187	#340	C₂²×C₄₁	1	4	yes	—
188	#336	C₂²×C₄₁	1	2	yes	—
189	C₄₁	#22808	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
191	C₂×Q₁₆	—	2	2	no	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×Q₁₆	C₂×C₄×C₄₁	1	4	yes	—
194	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
195	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
197	C₄×Q₈	—	2	4	no	—
198	C₄×Q₈	C₂×C₄×C₄₁	1	8	yes	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
203	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
205	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	2	4	no	—
208	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
209	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
210	#107	—	2	4	no	—
211	#107	—	4	2	no	—
212	#107	C₂×C₄×C₄₁	1	4	yes	—
213	C₄⋊C₈	—	2	2	no	—
214	#103	—	4	4	no	—
215	#103	C₂×C₄×C₄₁	1	8	yes	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
218	#95	—	2	4	no	—
219	#95	—	2	4	no	—
220	#95	—	2	4	no	—
221	#95	D₄×C₄₁	1	2	yes	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
236	Q₁₆	—	2	8	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
243	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
244	C₂×C₈	#41×C₄₁	1	2	yes	—
245	C₂×C₈	#41×C₄₁	1	2	yes	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	8	2	no	—
253	C₄⋊C₄	Q₁₆×C₄₁	1	4	yes	complemented
254	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
255	C₄²	—	2	8	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	8	4	no	—
259	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
260	C₄²	#41×C₄₁	1	2	yes	—
261	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
262	Q₈	—	2	4	no	—
263	Q₈	—	2	4	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	4	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	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	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
283	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
284	C₂×C₄	C₂×Q₁₆×C₄₁	1	4	yes	—
285	C₂×C₄	C₂×Q₁₆×C₄₁	1	2	yes	—
286	C₂×C₄	#135×C₄₁	1	2	yes	—
287	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
288	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
289	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
290	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
291	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
292	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	4	2	no	—
295	C₈	#117×C₄₁	1	4	yes	—
296	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
297	C₂²	C₄₁×#547	1	1	yes	characteristic
298	C₂²	C₄₁×#550	1	1	yes	characteristic
299	C₂²	C₂²×Q₁₆×C₄₁	1	1	yes	characteristic
300	C₂²	C₂×#135×C₄₁	1	1	yes	characteristic
301	C₂²	C₄₁×#576	1	1	yes	characteristic
302	C₂²	C₄₁×#578	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	4	no	—
307	C₄	—	4	4	no	—
308	C₄	—	8	4	no	—
309	C₄	C₄×Q₁₆×C₄₁	1	4	yes	—
310	C₄	C₂×#117×C₄₁	1	1	yes	characteristic
311	C₄	C₄₁×#520	1	1	yes	characteristic
312	C₄	C₄₁×#555	1	1	yes	characteristic
313	C₄	C₄₁×#556	1	1	yes	characteristic
314	C₂	C₄₁×#2354	1	1	yes	characteristic
315	C₂	C₂×C₄×Q₁₆×C₄₁	1	1	yes	characteristic
316	C₂	C₄₁×#2951	1	1	yes	characteristic
317	C₂	C₄₁×#3338	1	1	yes	characteristic
318	C₂	C₄₁×#3342	1	1	yes	characteristic
319	C₂	C₄₁×#3395	1	1	yes	characteristic
320	C₂	C₄₁×#3396	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
11. C₄×Q₁₆×C₄₁	C₄	semidirect
92. C₄⋊C₄×C₄₁	Q₁₆	semidirect
135. #22808	C₄₁	direct (N × H)
171. C₄×Q₁₆	C₄×C₄₁	semidirect
189. C₄₁	#22808	direct (N × H)
253. C₄⋊C₄	Q₁₆×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₄₁	2
C₄₁×#2355	2
C₄₁×#1969	2
C₄₁×#1928	2
C₄₁×#1845	2
C₄₁×#1803	2
C₄×#107×C₄₁	2
C₈×C₄⋊C₄×C₄₁	2
#22808	41

Subgroups — C41×#22808 #23799009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22808 #23799009