Subgroups — C₄₁×#8652 #23784853

Order 10496.

510 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#436	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#1802	C₂	1	2	yes	maximal
4	C₂×C₄₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1296	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₄₁	—	2	1	no	—
7	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
8	C₄₁×#436	—	2	2	no	—
9	C₄₁×#436	C₄	1	4	yes	complemented
10	C₂×#107×C₄₁	—	2	1	no	—
11	C₂×#107×C₄₁	—	2	1	no	—
12	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×#102×C₄₁	—	2	2	no	—
15	C₂×C₂²⋊C₈×C₄₁	—	2	1	no	—
16	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#384	C₂²	1	2	yes	—
18	C₄₁×#340	—	2	2	no	—
19	C₄₁×#334	—	2	2	no	—
20	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₄₁	—	2	1	no	—
22	C₂×M₁₆×C₄₁	—	2	4	no	—
23	C₂×M₁₆×C₄₁	—	4	2	no	—
24	C₂×M₁₆×C₄₁	—	4	2	no	—
25	C₂²×C₈×C₄₁	—	2	1	no	—
26	C₂²×C₈×C₄₁	—	2	2	no	—
27	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
29	C₄×D₄×C₄₁	—	2	1	no	—
30	C₄×D₄×C₄₁	—	2	1	no	—
31	C₄×D₄×C₄₁	C₂×C₄	1	2	yes	—
32	C₄×D₄×C₄₁	Q₈	1	2	yes	—
33	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
38	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
39	#107×C₄₁	—	4	2	no	—
40	#107×C₄₁	—	4	2	no	—
41	C₄⋊C₈×C₄₁	—	2	1	no	—
42	C₄⋊C₈×C₄₁	—	2	2	no	—
43	C₄⋊C₈×C₄₁	—	2	2	no	—
44	C₄⋊C₈×C₄₁	C₂×C₄	1	2	yes	—
45	#102×C₄₁	—	4	4	no	—
46	C₂²⋊C₈×C₄₁	—	2	4	no	—
47	C₄×C₈×C₄₁	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₄₁	D₄	1	2	yes	—
49	C₂⁴×C₄₁	—	2	1	no	—
50	C₂×D₄×C₄₁	—	2	1	no	—
51	C₂×D₄×C₄₁	—	2	2	no	—
52	C₂×D₄×C₄₁	—	2	2	no	—
53	C₂×D₄×C₄₁	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₄₁	—	2	1	no	—
55	C₂²×C₄×C₄₁	—	2	2	no	—
56	C₂²×C₄×C₄₁	—	2	2	no	—
57	C₂²×C₄×C₄₁	—	4	1	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	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
65	M₁₆×C₄₁	—	4	8	no	—
66	C₂×C₈×C₄₁	—	2	1	no	—
67	C₂×C₈×C₄₁	—	2	1	no	—
68	C₂×C₈×C₄₁	—	2	2	no	—
69	C₂×C₈×C₄₁	—	4	1	no	—
70	C₂×C₈×C₄₁	—	4	1	no	—
71	C₂×C₈×C₄₁	—	4	2	no	—
72	C₂×C₈×C₄₁	—	4	2	no	—
73	C₂×C₈×C₄₁	—	4	2	no	—
74	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₄₁	D₈	1	2	yes	—
77	C₂×C₈×C₄₁	SD₁₆	1	2	yes	—
78	C₄⋊C₄×C₄₁	—	2	1	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	—	8	2	no	—
84	C₄⋊C₄×C₄₁	C₄⋊C₄	1	2	yes	—
85	C₂²⋊C₄×C₄₁	—	2	2	no	—
86	C₂²⋊C₄×C₄₁	—	2	2	no	—
87	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₁	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₄₁	#41	1	1	yes	characteristic
91	C₂³×C₄₁	—	2	2	no	—
92	C₂³×C₄₁	—	4	1	no	—
93	C₂³×C₄₁	—	4	1	no	—
94	C₂³×C₄₁	—	4	1	no	—
95	C₂³×C₄₁	—	4	1	no	—
96	C₂³×C₄₁	—	4	1	no	—
97	C₂³×C₄₁	—	4	1	no	—
98	C₂³×C₄₁	#121	1	1	yes	characteristic
99	D₄×C₄₁	—	2	4	no	—
100	D₄×C₄₁	—	4	2	no	—
101	C₂×C₄×C₄₁	—	2	1	no	—
102	C₂×C₄×C₄₁	—	2	1	no	—
103	C₂×C₄×C₄₁	—	2	2	no	—
104	C₂×C₄×C₄₁	—	4	1	no	—
105	C₂×C₄×C₄₁	—	4	1	no	—
106	C₂×C₄×C₄₁	—	4	2	no	—
107	C₂×C₄×C₄₁	—	4	2	no	—
108	C₂×C₄×C₄₁	—	4	2	no	—
109	C₂×C₄×C₄₁	—	4	2	no	—
110	C₂×C₄×C₄₁	—	8	2	no	—
111	C₂×C₄×C₄₁	—	8	2	no	—
112	C₂×C₄×C₄₁	—	8	2	no	—
113	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₁	#117	1	1	yes	characteristic
116	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
117	C₂×C₄×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₄₁	#137	1	1	yes	characteristic
124	C₂×C₄×C₄₁	#137	1	2	yes	—
125	C₈×C₄₁	—	2	2	no	—
126	C₈×C₄₁	—	4	2	no	—
127	C₈×C₄₁	—	8	2	no	—
128	C₈×C₄₁	#102	1	4	yes	complemented
129	#8652	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₄₁	—	4	2	no	—
131	C₂²×C₄₁	—	4	2	no	—
132	C₂²×C₄₁	—	4	2	no	—
133	C₂²×C₄₁	—	8	1	no	—
134	C₂²×C₄₁	—	8	1	no	—
135	C₂²×C₄₁	—	8	1	no	—
136	C₂²×C₄₁	—	8	1	no	—
137	C₂²×C₄₁	#380	1	1	yes	characteristic
138	C₂²×C₄₁	#459	1	1	yes	characteristic
139	C₂²×C₄₁	#461	1	1	yes	characteristic
140	C₂²×C₄₁	#469	1	1	yes	characteristic
141	C₂²×C₄₁	#470	1	1	yes	characteristic
142	C₂²×C₄₁	#472	1	1	yes	characteristic
143	C₂²×C₄₁	#473	1	1	yes	characteristic
144	C₄×C₄₁	—	2	1	no	—
145	C₄×C₄₁	—	2	1	no	—
146	C₄×C₄₁	—	4	2	no	—
147	C₄×C₄₁	—	16	2	no	—
148	C₄×C₄₁	C₂×#102	1	1	yes	characteristic
149	C₄×C₄₁	#417	1	1	yes	characteristic
150	C₄×C₄₁	#428	1	2	yes	—
151	C₄×C₄₁	#468	1	2	yes	—
152	C₄×C₄₁	#474	1	1	yes	characteristic
153	C₄×C₄₁	#476	1	1	yes	characteristic
154	C₂×#436	C₂×C₄₁	1	1	yes	characteristic
155	#1845	C₂×C₄₁	1	1	yes	characteristic, complemented
156	#1802	C₂×C₄₁	1	2	yes	—
157	C₂×#334	C₂×C₄₁	1	1	yes	characteristic, complemented
158	#1296	C₂×C₄₁	1	2	yes	complemented
159	C₂×C₄₁	—	8	2	no	—
160	C₂×C₄₁	#1687	1	1	yes	characteristic
161	C₂×C₄₁	#1689	1	1	yes	characteristic
162	C₂×C₄₁	#1693	1	1	yes	characteristic
163	C₂×C₄₁	#1695	1	1	yes	characteristic
164	C₂×C₄₁	#1802	1	1	yes	characteristic
165	C₂×C₄₁	#1809	1	1	yes	characteristic
166	C₂×C₄₁	#1816	1	1	yes	characteristic
167	C₂²×M₁₆	—	2	1	no	—
168	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
169	#436	—	2	2	no	—
170	#436	C₄×C₄₁	1	4	yes	complemented
171	C₂×#107	—	2	1	no	—
172	C₂×#107	—	2	1	no	—
173	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
174	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
175	C₂×#102	—	2	2	no	—
176	C₂×C₂²⋊C₈	—	2	1	no	—
177	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
178	#384	C₂²×C₄₁	1	2	yes	—
179	#340	—	2	2	no	—
180	#334	—	2	2	no	—
181	C₄₁	#8652	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
183	C₂³×C₄	—	2	1	no	—
184	C₂×M₁₆	—	2	4	no	—
185	C₂×M₁₆	—	4	2	no	—
186	C₂×M₁₆	—	4	2	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
190	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
191	C₄×D₄	—	2	1	no	—
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	C₂×C₄×C₄₁	1	2	yes	—
194	C₄×D₄	Q₈×C₄₁	1	2	yes	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
199	C₂×C₂²⋊C₄	—	2	1	no	—
200	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
201	#107	—	4	2	no	—
202	#107	—	4	2	no	—
203	C₄⋊C₈	—	2	1	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	C₂×C₄×C₄₁	1	2	yes	—
207	#102	—	4	4	no	—
208	C₂²⋊C₈	—	2	4	no	—
209	C₄×C₈	C₂×C₄×C₄₁	1	2	yes	—
210	C₄×C₈	D₄×C₄₁	1	2	yes	—
211	C₂⁴	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	C₄⋊C₄×C₄₁	1	2	yes	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
226	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
227	M₁₆	—	4	8	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
237	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
238	C₂×C₈	D₈×C₄₁	1	2	yes	—
239	C₂×C₈	SD₁₆×C₄₁	1	2	yes	—
240	C₄⋊C₄	—	2	1	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	C₄⋊C₄	—	8	2	no	—
246	C₄⋊C₄	C₄⋊C₄×C₄₁	1	2	yes	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
250	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
251	C₄²	C₂×Q₈×C₄₁	1	1	yes	characteristic
252	C₄²	#41×C₄₁	1	1	yes	characteristic
253	C₂³	—	2	2	no	—
254	C₂³	—	4	1	no	—
255	C₂³	—	4	1	no	—
256	C₂³	—	4	1	no	—
257	C₂³	—	4	1	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	#121×C₄₁	1	1	yes	characteristic
261	D₄	—	2	4	no	—
262	D₄	—	4	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
277	C₂×C₄	#117×C₄₁	1	1	yes	characteristic
278	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
279	C₂×C₄	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
280	C₂×C₄	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
281	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
282	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
283	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
284	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
285	C₂×C₄	#137×C₄₁	1	1	yes	characteristic
286	C₂×C₄	#137×C₄₁	1	2	yes	—
287	C₈	—	2	2	no	—
288	C₈	—	4	2	no	—
289	C₈	—	8	2	no	—
290	C₈	#102×C₄₁	1	4	yes	complemented
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	8	1	no	—
295	C₂²	—	8	1	no	—
296	C₂²	—	8	1	no	—
297	C₂²	—	8	1	no	—
298	C₂²	C₄₁×#380	1	1	yes	characteristic
299	C₂²	C₄₁×#459	1	1	yes	characteristic
300	C₂²	C₄₁×#461	1	1	yes	characteristic
301	C₂²	C₄₁×#469	1	1	yes	characteristic
302	C₂²	C₄₁×#470	1	1	yes	characteristic
303	C₂²	C₄₁×#472	1	1	yes	characteristic
304	C₂²	C₄₁×#473	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	2	no	—
308	C₄	—	16	2	no	—
309	C₄	C₂×#102×C₄₁	1	1	yes	characteristic
310	C₄	C₄₁×#417	1	1	yes	characteristic
311	C₄	C₄₁×#428	1	2	yes	—
312	C₄	C₄₁×#468	1	2	yes	—
313	C₄	C₄₁×#474	1	1	yes	characteristic
314	C₄	C₄₁×#476	1	1	yes	characteristic
315	C₂	—	8	2	no	—
316	C₂	C₄₁×#1687	1	1	yes	characteristic
317	C₂	C₄₁×#1689	1	1	yes	characteristic
318	C₂	C₄₁×#1693	1	1	yes	characteristic
319	C₂	C₄₁×#1695	1	1	yes	characteristic
320	C₂	C₄₁×#1802	1	1	yes	characteristic
321	C₂	C₄₁×#1809	1	1	yes	characteristic
322	C₂	C₄₁×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	#121
commutator	C₄²	C₂²×C₄×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
prime radical	C₂²×D₄×C₄₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#1845	C₂	semidirect
4. C₂×C₄₁×#334	C₂	semidirect
5. C₄₁×#1296	C₂	semidirect
9. C₄₁×#436	C₄	semidirect
128. C₈×C₄₁	#102	semidirect
129. #8652	C₄₁	direct (N × H)
155. #1845	C₂×C₄₁	semidirect
157. C₂×#334	C₂×C₄₁	semidirect
158. #1296	C₂×C₄₁	semidirect
170. #436	C₄×C₄₁	semidirect
181. C₄₁	#8652	direct (N × H)
290. C₈	#102×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×#436	2
C₄₁×#1845	2
C₄₁×#1802	2
C₂×C₄₁×#334	2
C₄₁×#1296	2
#8652	41

Subgroups — C41×#8652 #23784853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#8652 #23784853