Subgroups — C₄₁×#18076 #23794277

Order 10496.

656 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3171	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2969	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2935	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1594	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1590	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1588	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1586	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1582	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1581	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄₁×#1484	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄₁×#1483	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₁×#1455	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#131×C₄₁	—	2	1	no	—
17	C₄₁×#532	C₂²	1	1	yes	characteristic
18	C₄₁×#529	C₂²	1	1	yes	characteristic
19	C₄₁×#501	C₄	1	1	yes	characteristic
20	C₄₁×#496	C₄	1	1	yes	characteristic
21	C₄₁×#490	C₄	1	1	yes	characteristic
22	C₄₁×#489	C₄	1	1	yes	characteristic
23	C₄₁×#469	C₄	1	1	yes	characteristic
24	C₄₁×#469	C₄	1	1	yes	characteristic
25	C₄₁×#468	C₄	1	1	yes	characteristic
26	C₄₁×#468	C₄	1	1	yes	characteristic
27	C₄₁×#446	C₂²	1	1	yes	characteristic
28	C₈×Q₈×C₄₁	C₂²	1	1	yes	characteristic
29	C₄×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
30	C₄×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₄₁	C₂²	1	1	yes	characteristic
32	C₄×D₈×C₄₁	C₂²	1	1	yes	characteristic
33	C₄₁×#436	—	2	1	no	—
34	C₄₁×#436	C₂²	1	1	yes	characteristic
35	C₄₁×#436	C₂²	1	1	yes	characteristic
36	C₄₁×#436	C₂²	1	1	yes	characteristic
37	C₄₁×#435	—	2	1	no	—
38	C₄₁×#435	—	2	1	no	—
39	C₄₁×#435	—	2	1	no	—
40	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic
42	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic
43	C₄₁×#433	—	2	1	no	—
44	C₄₁×#431	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
46	C₄₁×#408	—	2	1	no	—
47	C₄₁×#405	—	2	1	no	—
48	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
49	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
50	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
51	C₄₁×#335	C₂²	1	1	yes	characteristic
52	C₄₁×#335	C₂²	1	1	yes	characteristic
53	C₄₁×#334	C₂²	1	1	yes	characteristic
54	C₄₁×#334	C₂²	1	1	yes	characteristic
55	C₄₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
56	C₄₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
57	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
58	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
59	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
60	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
61	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
62	C₄₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
63	C₄₁×#322	C₂²	1	1	yes	characteristic
64	C₄₁×#322	C₂²	1	1	yes	characteristic
65	C₄₁×#322	C₂²	1	1	yes	characteristic
66	C₈²×C₄₁	C₂²	1	1	yes	characteristic
67	C₂×#41×C₄₁	—	2	1	no	—
68	C₂×SD₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
69	C₂×SD₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
70	C₂×D₈×C₄₁	C₂×C₄	1	1	yes	characteristic
71	C₂×D₈×C₄₁	C₂×C₄	1	1	yes	characteristic
72	#131×C₄₁	—	4	4	no	—
73	C₂×M₁₆×C₄₁	—	2	1	no	—
74	C₂×M₁₆×C₄₁	—	2	1	no	—
75	C₂×M₁₆×C₄₁	—	2	1	no	—
76	C₂×M₁₆×C₄₁	—	2	1	no	—
77	C₂×M₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
78	C₂×M₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
79	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
80	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₄₁	—	2	1	no	—
82	C₂²×C₈×C₄₁	—	2	1	no	—
83	C₂²×C₈×C₄₁	—	2	1	no	—
84	C₂²×C₈×C₄₁	—	2	1	no	—
85	C₄⋊Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
86	#127×C₄₁	C₂×C₄	1	1	yes	characteristic
87	#126×C₄₁	—	2	1	no	—
88	#125×C₄₁	C₂×C₄	1	1	yes	characteristic
89	#124×C₄₁	—	2	1	no	—
90	#124×C₄₁	C₂×C₄	1	1	yes	characteristic
91	#123×C₄₁	—	2	1	no	—
92	C₂²⋊Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
93	#121×C₄₁	—	2	1	no	—
94	#121×C₄₁	C₂×C₄	1	1	yes	characteristic
95	#121×C₄₁	C₂×C₄	1	1	yes	characteristic
96	#121×C₄₁	C₂×C₄	1	1	yes	characteristic
97	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
98	C₄×D₄×C₄₁	—	2	1	no	—
99	C₄×D₄×C₄₁	—	2	1	no	—
100	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
101	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
102	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
103	#117×C₄₁	—	2	1	no	—
104	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
105	#107×C₄₁	C₂×C₄	1	1	yes	characteristic
106	#107×C₄₁	C₂×C₄	1	1	yes	characteristic
107	#106×C₄₁	C₂×C₄	1	1	yes	characteristic
108	#106×C₄₁	C₂×C₄	1	1	yes	characteristic
109	C₄⋊C₈×C₄₁	—	2	1	no	—
110	C₄⋊C₈×C₄₁	—	2	1	no	—
111	C₄⋊C₈×C₄₁	—	2	1	no	—
112	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
113	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
114	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
115	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
116	#103×C₄₁	C₂×C₄	1	1	yes	characteristic
117	#103×C₄₁	C₂×C₄	1	1	yes	characteristic
118	#102×C₄₁	C₂×C₄	1	1	yes	characteristic
119	#102×C₄₁	C₂×C₄	1	1	yes	characteristic
120	#102×C₄₁	C₂×C₄	1	1	yes	characteristic
121	#102×C₄₁	C₂×C₄	1	1	yes	characteristic
122	#102×C₄₁	C₂×C₄	1	1	yes	characteristic
123	#102×C₄₁	C₂×C₄	1	1	yes	characteristic
124	C₂²⋊C₈×C₄₁	—	2	1	no	—
125	C₂²⋊C₈×C₄₁	—	2	1	no	—
126	C₂²⋊C₈×C₄₁	—	2	1	no	—
127	C₂²⋊C₈×C₄₁	—	2	1	no	—
128	C₂²⋊C₈×C₄₁	—	2	1	no	—
129	C₂²⋊C₈×C₄₁	—	2	1	no	—
130	C₂²⋊C₈×C₄₁	—	2	1	no	—
131	#97×C₄₁	—	2	1	no	—
132	#97×C₄₁	—	2	1	no	—
133	#97×C₄₁	C₂×C₄	1	1	yes	characteristic
134	#97×C₄₁	C₂×C₄	1	1	yes	characteristic
135	#97×C₄₁	D₄	1	1	yes	characteristic
136	#97×C₄₁	D₄	1	1	yes	characteristic
137	C₄×C₈×C₄₁	—	2	1	no	—
138	C₄×C₈×C₄₁	—	2	1	no	—
139	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
145	#41×C₄₁	—	4	2	no	—
146	#41×C₄₁	—	4	2	no	—
147	C₂×Q₈×C₄₁	—	2	1	no	—
148	C₂×Q₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
149	C₂×D₄×C₄₁	—	2	1	no	—
150	C₂×D₄×C₄₁	—	2	1	no	—
151	C₂×D₄×C₄₁	—	2	1	no	—
152	C₂×D₄×C₄₁	—	2	1	no	—
153	C₂×D₄×C₄₁	—	2	1	no	—
154	C₂×D₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
155	C₂×D₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
156	C₂×D₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₄₁	—	2	1	no	—
158	C₂²×C₄×C₄₁	—	2	1	no	—
159	C₂²×C₄×C₄₁	—	2	1	no	—
160	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
163	SD₁₆×C₄₁	—	2	2	no	—
164	SD₁₆×C₄₁	—	4	1	no	—
165	D₈×C₄₁	—	2	2	no	—
166	D₈×C₄₁	—	4	1	no	—
167	M₁₆×C₄₁	—	2	2	no	—
168	M₁₆×C₄₁	—	2	2	no	—
169	M₁₆×C₄₁	—	4	1	no	—
170	M₁₆×C₄₁	—	4	1	no	—
171	M₁₆×C₄₁	—	4	2	no	—
172	M₁₆×C₄₁	—	4	2	no	—
173	M₁₆×C₄₁	—	4	2	no	—
174	M₁₆×C₄₁	—	4	2	no	—
175	C₂×C₈×C₄₁	—	2	1	no	—
176	C₂×C₈×C₄₁	—	2	1	no	—
177	C₂×C₈×C₄₁	—	2	1	no	—
178	C₂×C₈×C₄₁	—	2	1	no	—
179	C₂×C₈×C₄₁	—	2	1	no	—
180	C₂×C₈×C₄₁	—	2	1	no	—
181	C₂×C₈×C₄₁	—	4	2	no	—
182	C₂×C₈×C₄₁	—	4	2	no	—
183	C₂×C₈×C₄₁	—	4	2	no	—
184	C₂×C₈×C₄₁	—	4	2	no	—
185	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
186	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
187	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
188	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
189	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
190	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
191	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
192	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
193	C₄⋊C₄×C₄₁	—	2	1	no	—
194	C₄⋊C₄×C₄₁	—	2	1	no	—
195	C₄⋊C₄×C₄₁	—	2	1	no	—
196	C₄⋊C₄×C₄₁	—	2	1	no	—
197	C₄⋊C₄×C₄₁	—	2	1	no	—
198	C₄⋊C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
199	C₄⋊C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
200	C₄⋊C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
201	C₄⋊C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
202	C₂²⋊C₄×C₄₁	—	2	1	no	—
203	C₂²⋊C₄×C₄₁	—	2	1	no	—
204	C₂²⋊C₄×C₄₁	—	2	1	no	—
205	C₂²⋊C₄×C₄₁	—	2	1	no	—
206	C₂²⋊C₄×C₄₁	—	2	1	no	—
207	C₂²⋊C₄×C₄₁	—	2	1	no	—
208	C₂²⋊C₄×C₄₁	—	2	1	no	—
209	C₂²⋊C₄×C₄₁	—	2	1	no	—
210	C₂²⋊C₄×C₄₁	—	2	1	no	—
211	C₄²×C₄₁	—	2	1	no	—
212	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
213	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₄₁	#41	1	1	yes	characteristic
215	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
216	C₂³×C₄₁	—	2	1	no	—
217	C₂³×C₄₁	—	2	1	no	—
218	C₂³×C₄₁	—	2	1	no	—
219	C₂³×C₄₁	C₄×D₄	1	1	yes	characteristic
220	Q₈×C₄₁	—	2	1	no	—
221	Q₈×C₄₁	—	2	1	no	—
222	Q₈×C₄₁	—	4	2	no	—
223	D₄×C₄₁	—	2	1	no	—
224	D₄×C₄₁	—	2	1	no	—
225	D₄×C₄₁	—	2	1	no	—
226	D₄×C₄₁	—	2	1	no	—
227	D₄×C₄₁	—	2	1	no	—
228	D₄×C₄₁	—	2	1	no	—
229	D₄×C₄₁	—	4	2	no	—
230	D₄×C₄₁	—	4	2	no	—
231	D₄×C₄₁	—	4	2	no	—
232	D₄×C₄₁	—	8	1	no	—
233	D₄×C₄₁	—	8	1	no	—
234	C₂×C₄×C₄₁	—	2	1	no	—
235	C₂×C₄×C₄₁	—	2	1	no	—
236	C₂×C₄×C₄₁	—	2	1	no	—
237	C₂×C₄×C₄₁	—	2	1	no	—
238	C₂×C₄×C₄₁	—	2	1	no	—
239	C₂×C₄×C₄₁	—	2	1	no	—
240	C₂×C₄×C₄₁	—	2	1	no	—
241	C₂×C₄×C₄₁	—	2	1	no	—
242	C₂×C₄×C₄₁	—	2	1	no	—
243	C₂×C₄×C₄₁	—	4	1	no	—
244	C₂×C₄×C₄₁	—	4	1	no	—
245	C₂×C₄×C₄₁	—	4	1	no	—
246	C₂×C₄×C₄₁	—	4	1	no	—
247	C₂×C₄×C₄₁	—	4	1	no	—
248	C₂×C₄×C₄₁	—	4	1	no	—
249	C₂×C₄×C₄₁	—	4	1	no	—
250	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
251	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
252	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
253	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
254	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
256	C₈×C₄₁	—	2	2	no	—
257	C₈×C₄₁	—	2	1	no	—
258	C₈×C₄₁	—	2	1	no	—
259	C₈×C₄₁	—	2	1	no	—
260	C₈×C₄₁	—	2	1	no	—
261	C₈×C₄₁	—	2	1	no	—
262	C₈×C₄₁	—	2	1	no	—
263	C₈×C₄₁	—	4	1	no	—
264	C₈×C₄₁	—	4	1	no	—
265	C₈×C₄₁	—	4	1	no	—
266	C₈×C₄₁	—	4	1	no	—
267	C₈×C₄₁	—	4	1	no	—
268	C₈×C₄₁	#131	1	2	yes	complemented
269	C₈×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
270	#18076	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
271	C₂²×C₄₁	—	2	1	no	—
272	C₂²×C₄₁	—	2	1	no	—
273	C₂²×C₄₁	—	2	1	no	—
274	C₂²×C₄₁	—	4	1	no	—
275	C₂²×C₄₁	—	4	1	no	—
276	C₂²×C₄₁	—	4	1	no	—
277	C₂²×C₄₁	—	4	1	no	—
278	C₂²×C₄₁	—	4	1	no	—
279	C₂²×C₄₁	—	4	1	no	—
280	C₂²×C₄₁	—	4	1	no	—
281	C₂²×C₄₁	—	4	1	no	—
282	C₂²×C₄₁	—	4	1	no	—
283	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
284	C₄×C₄₁	—	2	1	no	—
285	C₄×C₄₁	—	2	1	no	—
286	C₄×C₄₁	—	2	1	no	—
287	C₄×C₄₁	—	2	1	no	—
288	C₄×C₄₁	—	4	1	no	—
289	C₄×C₄₁	—	4	1	no	—
290	C₄×C₄₁	—	4	1	no	—
291	C₄×C₄₁	—	4	1	no	—
292	C₄×C₄₁	—	4	1	no	—
293	C₄×C₄₁	C₂×#131	1	1	yes	characteristic
294	C₄×C₄₁	#568	1	1	yes	characteristic
295	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
296	C₄×C₄₁	#575	1	1	yes	characteristic
297	#3171	C₂×C₄₁	1	1	yes	characteristic
298	#2969	C₂×C₄₁	1	1	yes	characteristic, complemented
299	#2935	C₂×C₄₁	1	1	yes	characteristic, complemented
300	#1594	C₂×C₄₁	1	1	yes	characteristic, complemented
301	#1590	C₂×C₄₁	1	1	yes	characteristic, complemented
302	#1588	C₂×C₄₁	1	1	yes	characteristic, complemented
303	#1586	C₂×C₄₁	1	1	yes	characteristic, complemented
304	#1582	C₂×C₄₁	1	1	yes	characteristic, complemented
305	#1581	C₂×C₄₁	1	1	yes	characteristic, complemented
306	SD₁₆⋊C₈	C₂×C₄₁	1	1	yes	characteristic, complemented
307	C₈×D₈	C₂×C₄₁	1	1	yes	characteristic, complemented
308	#1565	C₂×C₄₁	1	1	yes	characteristic, complemented
309	#1484	C₂×C₄₁	1	1	yes	characteristic, complemented
310	#1483	C₂×C₄₁	1	1	yes	characteristic, complemented
311	#1455	C₂×C₄₁	1	1	yes	characteristic, complemented
312	C₂×C₄₁	—	4	1	no	—
313	C₂×C₄₁	—	8	1	no	—
314	C₂×C₄₁	—	8	1	no	—
315	C₂×C₄₁	—	8	1	no	—
316	C₂×C₄₁	#2935	1	1	yes	characteristic
317	C₂×C₄₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
318	C₂×C₄₁	#2956	1	1	yes	characteristic
319	C₂×#131	—	2	1	no	—
320	#532	C₂²×C₄₁	1	1	yes	characteristic
321	#529	C₂²×C₄₁	1	1	yes	characteristic
322	#501	C₄×C₄₁	1	1	yes	characteristic
323	#496	C₄×C₄₁	1	1	yes	characteristic
324	#490	C₄×C₄₁	1	1	yes	characteristic
325	#489	C₄×C₄₁	1	1	yes	characteristic
326	#469	C₄×C₄₁	1	1	yes	characteristic
327	#469	C₄×C₄₁	1	1	yes	characteristic
328	#468	C₄×C₄₁	1	1	yes	characteristic
329	#468	C₄×C₄₁	1	1	yes	characteristic
330	#446	C₂²×C₄₁	1	1	yes	characteristic
331	C₈×Q₈	C₂²×C₄₁	1	1	yes	characteristic
332	C₄×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
333	C₄×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
334	C₄×D₈	C₂²×C₄₁	1	1	yes	characteristic
335	C₄×D₈	C₂²×C₄₁	1	1	yes	characteristic
336	#436	—	2	1	no	—
337	#436	C₂²×C₄₁	1	1	yes	characteristic
338	#436	C₂²×C₄₁	1	1	yes	characteristic
339	#436	C₂²×C₄₁	1	1	yes	characteristic
340	#435	—	2	1	no	—
341	#435	—	2	1	no	—
342	#435	—	2	1	no	—
343	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic
344	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic
345	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic
346	#433	—	2	1	no	—
347	#431	C₂²×C₄₁	1	1	yes	characteristic
348	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
349	#408	—	2	1	no	—
350	#405	—	2	1	no	—
351	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
352	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
353	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
354	#335	C₂²×C₄₁	1	1	yes	characteristic
355	#335	C₂²×C₄₁	1	1	yes	characteristic
356	#334	C₂²×C₄₁	1	1	yes	characteristic
357	#334	C₂²×C₄₁	1	1	yes	characteristic
358	Q₈⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
359	Q₈⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
360	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
361	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
362	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
363	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
364	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
365	D₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
366	#322	C₂²×C₄₁	1	1	yes	characteristic
367	#322	C₂²×C₄₁	1	1	yes	characteristic
368	#322	C₂²×C₄₁	1	1	yes	characteristic
369	C₈²	C₂²×C₄₁	1	1	yes	characteristic
370	C₄₁	#18076	1	1	yes	characteristic, direct product factor, Sylow
371	C₂×#41	—	2	1	no	—
372	C₂×SD₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
373	C₂×SD₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
374	C₂×D₈	C₂×C₄×C₄₁	1	1	yes	characteristic
375	C₂×D₈	C₂×C₄×C₄₁	1	1	yes	characteristic
376	#131	—	4	4	no	—
377	C₂×M₁₆	—	2	1	no	—
378	C₂×M₁₆	—	2	1	no	—
379	C₂×M₁₆	—	2	1	no	—
380	C₂×M₁₆	—	2	1	no	—
381	C₂×M₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
382	C₂×M₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
383	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
384	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
385	C₂²×C₈	—	2	1	no	—
386	C₂²×C₈	—	2	1	no	—
387	C₂²×C₈	—	2	1	no	—
388	C₂²×C₈	—	2	1	no	—
389	C₄⋊Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
390	#127	C₂×C₄×C₄₁	1	1	yes	characteristic
391	#126	—	2	1	no	—
392	#125	C₂×C₄×C₄₁	1	1	yes	characteristic
393	#124	—	2	1	no	—
394	#124	C₂×C₄×C₄₁	1	1	yes	characteristic
395	#123	—	2	1	no	—
396	C₂²⋊Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
397	#121	—	2	1	no	—
398	#121	C₂×C₄×C₄₁	1	1	yes	characteristic
399	#121	C₂×C₄×C₄₁	1	1	yes	characteristic
400	#121	C₂×C₄×C₄₁	1	1	yes	characteristic
401	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
402	C₄×D₄	—	2	1	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
405	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
406	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
407	#117	—	2	1	no	—
408	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
409	#107	C₂×C₄×C₄₁	1	1	yes	characteristic
410	#107	C₂×C₄×C₄₁	1	1	yes	characteristic
411	#106	C₂×C₄×C₄₁	1	1	yes	characteristic
412	#106	C₂×C₄×C₄₁	1	1	yes	characteristic
413	C₄⋊C₈	—	2	1	no	—
414	C₄⋊C₈	—	2	1	no	—
415	C₄⋊C₈	—	2	1	no	—
416	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
417	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
418	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
419	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
420	#103	C₂×C₄×C₄₁	1	1	yes	characteristic
421	#103	C₂×C₄×C₄₁	1	1	yes	characteristic
422	#102	C₂×C₄×C₄₁	1	1	yes	characteristic
423	#102	C₂×C₄×C₄₁	1	1	yes	characteristic
424	#102	C₂×C₄×C₄₁	1	1	yes	characteristic
425	#102	C₂×C₄×C₄₁	1	1	yes	characteristic
426	#102	C₂×C₄×C₄₁	1	1	yes	characteristic
427	#102	C₂×C₄×C₄₁	1	1	yes	characteristic
428	C₂²⋊C₈	—	2	1	no	—
429	C₂²⋊C₈	—	2	1	no	—
430	C₂²⋊C₈	—	2	1	no	—
431	C₂²⋊C₈	—	2	1	no	—
432	C₂²⋊C₈	—	2	1	no	—
433	C₂²⋊C₈	—	2	1	no	—
434	C₂²⋊C₈	—	2	1	no	—
435	#97	—	2	1	no	—
436	#97	—	2	1	no	—
437	#97	C₂×C₄×C₄₁	1	1	yes	characteristic
438	#97	C₂×C₄×C₄₁	1	1	yes	characteristic
439	#97	D₄×C₄₁	1	1	yes	characteristic
440	#97	D₄×C₄₁	1	1	yes	characteristic
441	C₄×C₈	—	2	1	no	—
442	C₄×C₈	—	2	1	no	—
443	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
444	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
445	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
446	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
447	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
448	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
449	#41	—	4	2	no	—
450	#41	—	4	2	no	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
459	C₂×D₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
460	C₂×D₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
465	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
466	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
467	SD₁₆	—	2	2	no	—
468	SD₁₆	—	4	1	no	—
469	D₈	—	2	2	no	—
470	D₈	—	4	1	no	—
471	M₁₆	—	2	2	no	—
472	M₁₆	—	2	2	no	—
473	M₁₆	—	4	1	no	—
474	M₁₆	—	4	1	no	—
475	M₁₆	—	4	2	no	—
476	M₁₆	—	4	2	no	—
477	M₁₆	—	4	2	no	—
478	M₁₆	—	4	2	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	4	2	no	—
486	C₂×C₈	—	4	2	no	—
487	C₂×C₈	—	4	2	no	—
488	C₂×C₈	—	4	2	no	—
489	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
490	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
491	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
492	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
493	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
494	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
495	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
496	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
503	C₄⋊C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
504	C₄⋊C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
505	C₄⋊C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
506	C₂²⋊C₄	—	2	1	no	—
507	C₂²⋊C₄	—	2	1	no	—
508	C₂²⋊C₄	—	2	1	no	—
509	C₂²⋊C₄	—	2	1	no	—
510	C₂²⋊C₄	—	2	1	no	—
511	C₂²⋊C₄	—	2	1	no	—
512	C₂²⋊C₄	—	2	1	no	—
513	C₂²⋊C₄	—	2	1	no	—
514	C₂²⋊C₄	—	2	1	no	—
515	C₄²	—	2	1	no	—
516	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
517	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
518	C₄²	#41×C₄₁	1	1	yes	characteristic
519	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
520	C₂³	—	2	1	no	—
521	C₂³	—	2	1	no	—
522	C₂³	—	2	1	no	—
523	C₂³	C₄×D₄×C₄₁	1	1	yes	characteristic
524	Q₈	—	2	1	no	—
525	Q₈	—	2	1	no	—
526	Q₈	—	4	2	no	—
527	D₄	—	2	1	no	—
528	D₄	—	2	1	no	—
529	D₄	—	2	1	no	—
530	D₄	—	2	1	no	—
531	D₄	—	2	1	no	—
532	D₄	—	2	1	no	—
533	D₄	—	4	2	no	—
534	D₄	—	4	2	no	—
535	D₄	—	4	2	no	—
536	D₄	—	8	1	no	—
537	D₄	—	8	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
555	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
556	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
557	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
558	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
559	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
560	C₈	—	2	2	no	—
561	C₈	—	2	1	no	—
562	C₈	—	2	1	no	—
563	C₈	—	2	1	no	—
564	C₈	—	2	1	no	—
565	C₈	—	2	1	no	—
566	C₈	—	2	1	no	—
567	C₈	—	4	1	no	—
568	C₈	—	4	1	no	—
569	C₈	—	4	1	no	—
570	C₈	—	4	1	no	—
571	C₈	—	4	1	no	—
572	C₈	#131×C₄₁	1	2	yes	complemented
573	C₈	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
574	C₂²	—	2	1	no	—
575	C₂²	—	2	1	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	—	4	1	no	—
582	C₂²	—	4	1	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
587	C₄	—	2	1	no	—
588	C₄	—	2	1	no	—
589	C₄	—	2	1	no	—
590	C₄	—	2	1	no	—
591	C₄	—	4	1	no	—
592	C₄	—	4	1	no	—
593	C₄	—	4	1	no	—
594	C₄	—	4	1	no	—
595	C₄	—	4	1	no	—
596	C₄	C₂×#131×C₄₁	1	1	yes	characteristic
597	C₄	C₄₁×#568	1	1	yes	characteristic
598	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
599	C₄	C₄₁×#575	1	1	yes	characteristic
600	C₂	—	4	1	no	—
601	C₂	—	8	1	no	—
602	C₂	—	8	1	no	—
603	C₂	—	8	1	no	—
604	C₂	C₄₁×#2935	1	1	yes	characteristic
605	C₂	C₄×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
606	C₂	C₄₁×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₄²	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂×C₄×D₄
prime radical	C₄₁×#3171	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₄×M₁₆×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2969	C₂	semidirect
3. C₄₁×#2935	C₂	semidirect
4. C₄₁×#1594	C₂	semidirect
5. C₄₁×#1590	C₂	semidirect
6. C₄₁×#1588	C₂	semidirect
7. C₄₁×#1586	C₂	semidirect
8. C₄₁×#1582	C₂	semidirect
9. C₄₁×#1581	C₂	semidirect
10. C₄₁×SD₁₆⋊C₈	C₂	semidirect
11. C₈×D₈×C₄₁	C₂	semidirect
12. C₄₁×#1565	C₂	semidirect
13. C₄₁×#1484	C₂	semidirect
14. C₄₁×#1483	C₂	semidirect
15. C₄₁×#1455	C₂	semidirect
268. C₈×C₄₁	#131	semidirect
270. #18076	C₄₁	direct (N × H)
298. #2969	C₂×C₄₁	semidirect
299. #2935	C₂×C₄₁	semidirect
300. #1594	C₂×C₄₁	semidirect
301. #1590	C₂×C₄₁	semidirect
302. #1588	C₂×C₄₁	semidirect
303. #1586	C₂×C₄₁	semidirect
304. #1582	C₂×C₄₁	semidirect
305. #1581	C₂×C₄₁	semidirect
306. SD₁₆⋊C₈	C₂×C₄₁	semidirect
307. C₈×D₈	C₂×C₄₁	semidirect
308. #1565	C₂×C₄₁	semidirect
309. #1484	C₂×C₄₁	semidirect
310. #1483	C₂×C₄₁	semidirect
311. #1455	C₂×C₄₁	semidirect
370. C₄₁	#18076	direct (N × H)
572. C₈	#131×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3171	2
C₄₁×#2969	2
C₄₁×#2935	2
C₄₁×#1594	2
C₄₁×#1590	2
C₄₁×#1588	2
C₄₁×#1586	2
C₄₁×#1582	2
C₄₁×#1581	2
C₄₁×SD₁₆⋊C₈	2
C₈×D₈×C₄₁	2
C₄₁×#1565	2
C₄₁×#1484	2
C₄₁×#1483	2
C₄₁×#1455	2
#18076	41

Subgroups — C41×#18076 #23794277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#18076 #23794277