Subgroups — C₄₁×#61885 #23838086

Order 10496.

1814 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3507	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#559	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3396	C₂	1	4	yes	maximal, complemented
4	C₄₁×#3394	C₂	1	4	yes	maximal, complemented
5	C₄₁×#3365	C₂	1	4	yes	maximal, complemented
6	C₄₁×#3361	C₂	1	4	yes	maximal, complemented
7	C₄₁×#3211	C₂	1	4	yes	maximal, complemented
8	C₄₁×#3208	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#3205	C₂	1	1	yes	maximal, characteristic
10	C₄₁×#3105	C₂	1	4	yes	maximal, complemented
11	C₄₁×#3104	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#3103	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄₁×#2982	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄₁×#559	—	2	2	no	—
15	C₄₁×#559	—	2	2	no	—
16	C₄₁×#559	C₂²	1	4	yes	—
17	C₄₁×#559	C₂²	1	4	yes	—
18	C₄₁×#559	C₂²	1	2	yes	complemented
19	C₄₁×#557	C₂²	1	2	yes	complemented
20	C₄₁×#556	—	2	2	no	—
21	C₄₁×#556	C₂²	1	2	yes	complemented
22	C₄₁×#554	C₂²	1	2	yes	complemented
23	C₄₁×#552	—	2	2	no	—
24	C₄₁×#539	—	2	2	no	—
25	C₄₁×#536	—	2	1	no	—
26	C₄₁×#533	C₂²	1	1	yes	characteristic
27	C₄₁×#532	C₂²	1	1	yes	characteristic
28	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
29	C₄₁×#529	C₂²	1	4	yes	—
30	C₂×#124×C₄₁	—	2	1	no	—
31	C₂×#124×C₄₁	C₂²	1	2	yes	—
32	C₂×#123×C₄₁	—	2	2	no	—
33	C₄₁×#520	—	2	1	no	—
34	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₄₁	—	2	1	no	—
37	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
38	C₄₁×#495	C₂²	1	8	yes	complemented
39	C₄₁×#493	C₂²	1	2	yes	—
40	C₄₁×#492	C₂²	1	2	yes	—
41	C₄₁×#488	C₂²	1	8	yes	complemented
42	C₄₁×#486	C₂²	1	2	yes	—
43	C₄₁×#486	C₂²	1	2	yes	—
44	C₄₁×#484	C₂²	1	2	yes	—
45	C₄₁×#481	C₂²	1	2	yes	—
46	C₄₁×#478	C₂²	1	8	yes	complemented
47	C₄₁×#477	C₂²	1	2	yes	complemented
48	C₄₁×#475	C₂²	1	2	yes	complemented
49	C₄₁×#464	C₂²	1	8	yes	complemented
50	C₄₁×#463	C₂²	1	8	yes	complemented
51	C₄₁×#463	C₂²	1	8	yes	complemented
52	C₄₁×#462	C₂²	1	2	yes	complemented
53	C₄₁×#460	C₂²	1	4	yes	—
54	C₄₁×#460	C₂²	1	2	yes	complemented
55	C₄₁×#459	C₂²	1	4	yes	—
56	C₄₁×#451	C₂²	1	2	yes	—
57	C₄₁×#450	C₂²	1	4	yes	—
58	C₄₁×#448	C₂²	1	2	yes	—
59	C₄₁×#447	C₂²	1	4	yes	—
60	C₈×Q₈×C₄₁	C₂²	1	4	yes	complemented
61	C₈×Q₈×C₄₁	C₂²	1	4	yes	complemented
62	C₄×Q₁₆×C₄₁	C₂²	1	4	yes	complemented
63	C₄×SD₁₆×C₄₁	C₂²	1	4	yes	complemented
64	C₄×SD₁₆×C₄₁	C₂²	1	8	yes	complemented
65	C₄×D₈×C₄₁	C₂²	1	8	yes	complemented
66	C₄₁×#431	C₂²	1	1	yes	characteristic
67	C₄₁×#431	C₂²	1	1	yes	characteristic
68	C₄₁×#431	C₂²	1	4	yes	—
69	C₂×#41×C₄₁	—	2	1	no	—
70	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
71	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₄₁	—	2	1	no	—
73	C₂²×D₄×C₄₁	—	2	1	no	—
74	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₄₁	—	2	2	no	—
76	C₂×Q₁₆×C₄₁	C₂³	1	4	yes	—
77	C₂×SD₁₆×C₄₁	C₂³	1	4	yes	—
78	C₂×SD₁₆×C₄₁	C₂³	1	8	yes	—
79	C₂×D₈×C₄₁	C₂³	1	8	yes	—
80	C₄⋊Q₈×C₄₁	—	2	2	no	—
81	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
82	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
83	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
84	#127×C₄₁	—	2	1	no	—
85	#127×C₄₁	—	2	1	no	—
86	#127×C₄₁	C₂³	1	2	yes	—
87	#127×C₄₁	C₂³	1	2	yes	—
88	#127×C₄₁	C₂³	1	1	yes	characteristic
89	#126×C₄₁	—	2	4	no	—
90	#126×C₄₁	—	2	4	no	—
91	#125×C₄₁	—	2	4	no	—
92	#125×C₄₁	—	2	1	no	—
93	#125×C₄₁	C₂³	1	4	yes	—
94	#124×C₄₁	—	2	4	no	—
95	#124×C₄₁	—	2	4	no	—
96	#124×C₄₁	—	2	2	no	—
97	#124×C₄₁	—	2	4	no	—
98	#124×C₄₁	—	2	1	no	—
99	#124×C₄₁	C₂³	1	8	yes	—
100	#124×C₄₁	C₂³	1	4	yes	—
101	#123×C₄₁	—	2	4	no	—
102	#123×C₄₁	—	2	8	no	—
103	#123×C₄₁	—	2	8	no	—
104	#123×C₄₁	—	2	4	no	—
105	C₂²⋊Q₈×C₄₁	—	2	2	no	—
106	C₂²⋊Q₈×C₄₁	—	2	2	no	—
107	C₂²⋊Q₈×C₄₁	—	2	1	no	—
108	C₂²⋊Q₈×C₄₁	—	2	4	no	—
109	C₂²⋊Q₈×C₄₁	C₂³	1	2	yes	—
110	#121×C₄₁	—	2	1	no	—
111	#121×C₄₁	C₂³	1	2	yes	—
112	C₄×Q₈×C₄₁	—	2	2	no	—
113	C₄×Q₈×C₄₁	—	2	1	no	—
114	C₄×Q₈×C₄₁	—	2	1	no	—
115	C₄×Q₈×C₄₁	—	2	1	no	—
116	C₄×Q₈×C₄₁	D₄	1	2	yes	—
117	C₄×Q₈×C₄₁	D₄	1	2	yes	—
118	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
119	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
120	C₄×Q₈×C₄₁	C₂³	1	4	yes	—
121	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
122	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
123	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
124	C₄×D₄×C₄₁	—	2	4	no	—
125	C₄×D₄×C₄₁	—	2	4	no	—
126	C₄×D₄×C₄₁	—	2	4	no	—
127	C₄×D₄×C₄₁	—	2	2	no	—
128	C₄×D₄×C₄₁	—	2	4	no	—
129	C₄×D₄×C₄₁	—	2	1	no	—
130	C₄×D₄×C₄₁	C₂³	1	8	yes	—
131	C₄×D₄×C₄₁	C₂³	1	4	yes	—
132	C₄×D₄×C₄₁	C₂³	1	2	yes	—
133	#117×C₄₁	—	2	1	no	—
134	#117×C₄₁	—	2	2	no	—
135	#117×C₄₁	—	2	4	no	—
136	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
137	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
138	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
139	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
140	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
141	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
142	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
143	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
144	C₂×C₄²×C₄₁	—	2	1	no	—
145	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
146	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
147	#107×C₄₁	C₂³	1	4	yes	—
148	#106×C₄₁	C₂³	1	4	yes	—
149	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
150	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
151	C₄⋊C₈×C₄₁	C₂³	1	8	yes	—
152	#103×C₄₁	C₂³	1	4	yes	—
153	#103×C₄₁	C₂³	1	4	yes	—
154	#103×C₄₁	C₂³	1	4	yes	—
155	#102×C₄₁	C₂³	1	4	yes	—
156	#102×C₄₁	C₂³	1	8	yes	—
157	#102×C₄₁	C₂³	1	8	yes	—
158	C₂²⋊C₈×C₄₁	C₂³	1	2	yes	—
159	C₂²⋊C₈×C₄₁	C₂³	1	2	yes	—
160	C₄×C₈×C₄₁	C₂³	1	8	yes	—
161	C₄×C₈×C₄₁	C₂³	1	2	yes	—
162	C₄×C₈×C₄₁	C₂³	1	2	yes	—
163	C₂⁴×C₄₁	—	2	2	no	—
164	#41×C₄₁	—	4	4	no	—
165	C₂×Q₈×C₄₁	—	2	4	no	—
166	C₂×Q₈×C₄₁	—	2	1	no	—
167	C₂×Q₈×C₄₁	—	2	1	no	—
168	C₂×Q₈×C₄₁	—	2	1	no	—
169	C₂×Q₈×C₄₁	—	4	2	no	—
170	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
171	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
172	C₂×Q₈×C₄₁	C₂⁴	1	2	yes	—
173	C₂×Q₈×C₄₁	C₂⁴	1	2	yes	—
174	C₂×D₄×C₄₁	—	2	2	no	—
175	C₂×D₄×C₄₁	—	2	1	no	—
176	C₂×D₄×C₄₁	—	2	2	no	—
177	C₂×D₄×C₄₁	—	2	2	no	—
178	C₂×D₄×C₄₁	—	2	2	no	—
179	C₂×D₄×C₄₁	—	2	1	no	—
180	C₂×D₄×C₄₁	—	2	2	no	—
181	C₂×D₄×C₄₁	—	2	2	no	—
182	C₂×D₄×C₄₁	—	4	4	no	—
183	C₂×D₄×C₄₁	—	4	4	no	—
184	C₂×D₄×C₄₁	—	8	2	no	—
185	C₂×D₄×C₄₁	C₂⁴	1	4	yes	—
186	C₂×D₄×C₄₁	C₂⁴	1	2	yes	—
187	C₂²×C₄×C₄₁	—	2	1	no	—
188	C₂²×C₄×C₄₁	—	2	1	no	—
189	C₂²×C₄×C₄₁	—	2	2	no	—
190	C₂²×C₄×C₄₁	—	2	4	no	—
191	C₂²×C₄×C₄₁	—	2	8	no	—
192	C₂²×C₄×C₄₁	—	8	4	no	—
193	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
194	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
198	Q₁₆×C₄₁	—	4	4	no	—
199	SD₁₆×C₄₁	—	4	4	no	—
200	SD₁₆×C₄₁	—	4	8	no	—
201	D₈×C₄₁	—	4	8	no	—
202	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
203	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
204	C₄⋊C₄×C₄₁	—	2	1	no	—
205	C₄⋊C₄×C₄₁	—	2	4	no	—
206	C₄⋊C₄×C₄₁	—	2	2	no	—
207	C₄⋊C₄×C₄₁	—	2	2	no	—
208	C₄⋊C₄×C₄₁	—	2	1	no	—
209	C₄⋊C₄×C₄₁	—	2	1	no	—
210	C₄⋊C₄×C₄₁	—	2	2	no	—
211	C₄⋊C₄×C₄₁	—	2	4	no	—
212	C₄⋊C₄×C₄₁	—	2	1	no	—
213	C₄⋊C₄×C₄₁	—	2	1	no	—
214	C₄⋊C₄×C₄₁	—	2	2	no	—
215	C₄⋊C₄×C₄₁	—	2	4	no	—
216	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
217	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
218	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
219	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
220	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
221	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
222	C₄⋊C₄×C₄₁	C₂⁴	1	2	yes	—
223	C₂²⋊C₄×C₄₁	—	2	2	no	—
224	C₂²⋊C₄×C₄₁	—	2	8	no	—
225	C₂²⋊C₄×C₄₁	—	2	8	no	—
226	C₂²⋊C₄×C₄₁	—	2	8	no	—
227	C₂²⋊C₄×C₄₁	—	2	2	no	—
228	C₂²⋊C₄×C₄₁	—	2	4	no	—
229	C₂²⋊C₄×C₄₁	—	2	4	no	—
230	C₂²⋊C₄×C₄₁	—	2	2	no	—
231	C₂²⋊C₄×C₄₁	—	2	4	no	—
232	C₄²×C₄₁	—	2	2	no	—
233	C₄²×C₄₁	—	2	2	no	—
234	C₄²×C₄₁	—	2	4	no	—
235	C₄²×C₄₁	—	2	2	no	—
236	C₄²×C₄₁	C₂×D₄	1	4	yes	—
237	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
238	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
239	C₄²×C₄₁	C₂⁴	1	4	yes	—
240	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
241	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
242	C₂³×C₄₁	—	2	2	no	—
243	C₂³×C₄₁	—	2	4	no	—
244	C₂³×C₄₁	—	4	4	no	—
245	C₂³×C₄₁	—	4	4	no	—
246	C₂³×C₄₁	—	4	4	no	—
247	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
248	Q₈×C₄₁	—	2	4	no	—
249	Q₈×C₄₁	—	2	4	no	—
250	Q₈×C₄₁	—	2	4	no	—
251	Q₈×C₄₁	—	2	2	no	—
252	Q₈×C₄₁	—	2	2	no	—
253	Q₈×C₄₁	—	4	2	no	—
254	Q₈×C₄₁	—	4	2	no	—
255	Q₈×C₄₁	—	8	1	no	—
256	D₄×C₄₁	—	2	4	no	—
257	D₄×C₄₁	—	2	8	no	—
258	D₄×C₄₁	—	4	4	no	—
259	D₄×C₄₁	—	4	4	no	—
260	D₄×C₄₁	—	4	4	no	—
261	D₄×C₄₁	—	4	4	no	—
262	D₄×C₄₁	—	4	2	no	—
263	D₄×C₄₁	—	8	1	no	—
264	D₄×C₄₁	—	8	2	no	—
265	D₄×C₄₁	—	8	2	no	—
266	C₂×C₄×C₄₁	—	2	4	no	—
267	C₂×C₄×C₄₁	—	2	2	no	—
268	C₂×C₄×C₄₁	—	2	2	no	—
269	C₂×C₄×C₄₁	—	2	2	no	—
270	C₂×C₄×C₄₁	—	2	2	no	—
271	C₂×C₄×C₄₁	—	2	2	no	—
272	C₂×C₄×C₄₁	—	2	2	no	—
273	C₂×C₄×C₄₁	—	4	2	no	—
274	C₂×C₄×C₄₁	—	4	1	no	—
275	C₂×C₄×C₄₁	—	4	2	no	—
276	C₂×C₄×C₄₁	—	4	2	no	—
277	C₂×C₄×C₄₁	—	4	4	no	—
278	C₂×C₄×C₄₁	—	8	4	no	—
279	C₂×C₄×C₄₁	—	8	8	no	—
280	C₂×C₄×C₄₁	#135	1	4	yes	—
281	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
282	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
284	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
289	C₈×C₄₁	—	2	4	no	—
290	C₈×C₄₁	—	2	4	no	—
291	#61885	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
292	C₂²×C₄₁	—	2	1	no	—
293	C₂²×C₄₁	—	2	1	no	—
294	C₂²×C₄₁	—	4	4	no	—
295	C₂²×C₄₁	—	4	2	no	—
296	C₂²×C₄₁	—	4	4	no	—
297	C₂²×C₄₁	—	4	2	no	—
298	C₂²×C₄₁	—	4	2	no	—
299	C₂²×C₄₁	—	4	4	no	—
300	C₂²×C₄₁	—	8	8	no	—
301	C₂²×C₄₁	C₂×#135	1	2	yes	—
302	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
303	C₄×C₄₁	—	2	4	no	—
304	C₄×C₄₁	—	2	1	no	—
305	C₄×C₄₁	—	2	1	no	—
306	C₄×C₄₁	—	2	1	no	—
307	C₄×C₄₁	—	2	4	no	—
308	C₄×C₄₁	—	4	2	no	—
309	C₄×C₄₁	—	4	2	no	—
310	C₄×C₄₁	—	4	2	no	—
311	C₄×C₄₁	—	4	2	no	—
312	C₄×C₄₁	—	4	2	no	—
313	C₄×C₄₁	C₂×#135	1	2	yes	—
314	C₄×C₄₁	C₂×#135	1	2	yes	—
315	C₄×C₄₁	#585	1	2	yes	—
316	#3507	C₂×C₄₁	1	1	yes	characteristic, complemented
317	C₂×#559	C₂×C₄₁	1	1	yes	characteristic, complemented
318	#3396	C₂×C₄₁	1	4	yes	complemented
319	#3394	C₂×C₄₁	1	4	yes	complemented
320	#3365	C₂×C₄₁	1	4	yes	complemented
321	#3361	C₂×C₄₁	1	4	yes	complemented
322	#3211	C₂×C₄₁	1	4	yes	complemented
323	#3208	C₂×C₄₁	1	1	yes	characteristic, complemented
324	#3205	C₂×C₄₁	1	1	yes	characteristic
325	#3105	C₂×C₄₁	1	4	yes	complemented
326	#3104	C₂×C₄₁	1	1	yes	characteristic, complemented
327	#3103	C₂×C₄₁	1	1	yes	characteristic, complemented
328	#2982	C₂×C₄₁	1	1	yes	characteristic, complemented
329	C₂×C₄₁	—	2	2	no	—
330	C₂×C₄₁	—	8	2	no	—
331	C₂×C₄₁	—	8	4	no	—
332	C₂×C₄₁	#3487	1	1	yes	characteristic
333	C₂×C₄₁	C₂²×#135	1	1	yes	characteristic
334	C₂×C₄₁	#3585	1	1	yes	characteristic
335	#559	—	2	2	no	—
336	#559	—	2	2	no	—
337	#559	C₂²×C₄₁	1	4	yes	—
338	#559	C₂²×C₄₁	1	4	yes	—
339	#559	C₂²×C₄₁	1	2	yes	complemented
340	#557	C₂²×C₄₁	1	2	yes	complemented
341	#556	—	2	2	no	—
342	#556	C₂²×C₄₁	1	2	yes	complemented
343	#554	C₂²×C₄₁	1	2	yes	complemented
344	#552	—	2	2	no	—
345	#539	—	2	2	no	—
346	#536	—	2	1	no	—
347	#533	C₂²×C₄₁	1	1	yes	characteristic
348	#532	C₂²×C₄₁	1	1	yes	characteristic
349	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
350	#529	C₂²×C₄₁	1	4	yes	—
351	C₂×#124	—	2	1	no	—
352	C₂×#124	C₂²×C₄₁	1	2	yes	—
353	C₂×#123	—	2	2	no	—
354	#520	—	2	1	no	—
355	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
356	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
357	C₂×C₄×D₄	—	2	1	no	—
358	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
359	#495	C₂²×C₄₁	1	8	yes	complemented
360	#493	C₂²×C₄₁	1	2	yes	—
361	#492	C₂²×C₄₁	1	2	yes	—
362	#488	C₂²×C₄₁	1	8	yes	complemented
363	#486	C₂²×C₄₁	1	2	yes	—
364	#486	C₂²×C₄₁	1	2	yes	—
365	#484	C₂²×C₄₁	1	2	yes	—
366	#481	C₂²×C₄₁	1	2	yes	—
367	#478	C₂²×C₄₁	1	8	yes	complemented
368	#477	C₂²×C₄₁	1	2	yes	complemented
369	#475	C₂²×C₄₁	1	2	yes	complemented
370	#464	C₂²×C₄₁	1	8	yes	complemented
371	#463	C₂²×C₄₁	1	8	yes	complemented
372	#463	C₂²×C₄₁	1	8	yes	complemented
373	#462	C₂²×C₄₁	1	2	yes	complemented
374	#460	C₂²×C₄₁	1	4	yes	—
375	#460	C₂²×C₄₁	1	2	yes	complemented
376	#459	C₂²×C₄₁	1	4	yes	—
377	#451	C₂²×C₄₁	1	2	yes	—
378	#450	C₂²×C₄₁	1	4	yes	—
379	#448	C₂²×C₄₁	1	2	yes	—
380	#447	C₂²×C₄₁	1	4	yes	—
381	C₈×Q₈	C₂²×C₄₁	1	4	yes	complemented
382	C₈×Q₈	C₂²×C₄₁	1	4	yes	complemented
383	C₄×Q₁₆	C₂²×C₄₁	1	4	yes	complemented
384	C₄×SD₁₆	C₂²×C₄₁	1	4	yes	complemented
385	C₄×SD₁₆	C₂²×C₄₁	1	8	yes	complemented
386	C₄×D₈	C₂²×C₄₁	1	8	yes	complemented
387	#431	C₂²×C₄₁	1	1	yes	characteristic
388	#431	C₂²×C₄₁	1	1	yes	characteristic
389	#431	C₂²×C₄₁	1	4	yes	—
390	C₄₁	#61885	1	1	yes	characteristic, direct product factor, Sylow
391	C₂×#41	—	2	1	no	—
392	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
393	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
394	C₂²×D₄	—	2	1	no	—
395	C₂²×D₄	—	2	1	no	—
396	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
397	C₂³×C₄	—	2	2	no	—
398	C₂×Q₁₆	C₂³×C₄₁	1	4	yes	—
399	C₂×SD₁₆	C₂³×C₄₁	1	4	yes	—
400	C₂×SD₁₆	C₂³×C₄₁	1	8	yes	—
401	C₂×D₈	C₂³×C₄₁	1	8	yes	—
402	C₄⋊Q₈	—	2	2	no	—
403	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
404	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
405	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
406	#127	—	2	1	no	—
407	#127	—	2	1	no	—
408	#127	C₂³×C₄₁	1	2	yes	—
409	#127	C₂³×C₄₁	1	2	yes	—
410	#127	C₂³×C₄₁	1	1	yes	characteristic
411	#126	—	2	4	no	—
412	#126	—	2	4	no	—
413	#125	—	2	4	no	—
414	#125	—	2	1	no	—
415	#125	C₂³×C₄₁	1	4	yes	—
416	#124	—	2	4	no	—
417	#124	—	2	4	no	—
418	#124	—	2	2	no	—
419	#124	—	2	4	no	—
420	#124	—	2	1	no	—
421	#124	C₂³×C₄₁	1	8	yes	—
422	#124	C₂³×C₄₁	1	4	yes	—
423	#123	—	2	4	no	—
424	#123	—	2	8	no	—
425	#123	—	2	8	no	—
426	#123	—	2	4	no	—
427	C₂²⋊Q₈	—	2	2	no	—
428	C₂²⋊Q₈	—	2	2	no	—
429	C₂²⋊Q₈	—	2	1	no	—
430	C₂²⋊Q₈	—	2	4	no	—
431	C₂²⋊Q₈	C₂³×C₄₁	1	2	yes	—
432	#121	—	2	1	no	—
433	#121	C₂³×C₄₁	1	2	yes	—
434	C₄×Q₈	—	2	2	no	—
435	C₄×Q₈	—	2	1	no	—
436	C₄×Q₈	—	2	1	no	—
437	C₄×Q₈	—	2	1	no	—
438	C₄×Q₈	D₄×C₄₁	1	2	yes	—
439	C₄×Q₈	D₄×C₄₁	1	2	yes	—
440	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
441	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
442	C₄×Q₈	C₂³×C₄₁	1	4	yes	—
443	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
444	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
445	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
446	C₄×D₄	—	2	4	no	—
447	C₄×D₄	—	2	4	no	—
448	C₄×D₄	—	2	4	no	—
449	C₄×D₄	—	2	2	no	—
450	C₄×D₄	—	2	4	no	—
451	C₄×D₄	—	2	1	no	—
452	C₄×D₄	C₂³×C₄₁	1	8	yes	—
453	C₄×D₄	C₂³×C₄₁	1	4	yes	—
454	C₄×D₄	C₂³×C₄₁	1	2	yes	—
455	#117	—	2	1	no	—
456	#117	—	2	2	no	—
457	#117	—	2	4	no	—
458	C₂×C₄⋊C₄	—	2	1	no	—
459	C₂×C₄⋊C₄	—	2	2	no	—
460	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
461	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
462	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
463	C₂×C₂²⋊C₄	—	2	2	no	—
464	C₂×C₂²⋊C₄	—	2	2	no	—
465	C₂×C₂²⋊C₄	—	2	2	no	—
466	C₂×C₄²	—	2	1	no	—
467	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
468	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
469	#107	C₂³×C₄₁	1	4	yes	—
470	#106	C₂³×C₄₁	1	4	yes	—
471	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
472	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
473	C₄⋊C₈	C₂³×C₄₁	1	8	yes	—
474	#103	C₂³×C₄₁	1	4	yes	—
475	#103	C₂³×C₄₁	1	4	yes	—
476	#103	C₂³×C₄₁	1	4	yes	—
477	#102	C₂³×C₄₁	1	4	yes	—
478	#102	C₂³×C₄₁	1	8	yes	—
479	#102	C₂³×C₄₁	1	8	yes	—
480	C₂²⋊C₈	C₂³×C₄₁	1	2	yes	—
481	C₂²⋊C₈	C₂³×C₄₁	1	2	yes	—
482	C₄×C₈	C₂³×C₄₁	1	8	yes	—
483	C₄×C₈	C₂³×C₄₁	1	2	yes	—
484	C₄×C₈	C₂³×C₄₁	1	2	yes	—
485	C₂⁴	—	2	2	no	—
486	#41	—	4	4	no	—
487	C₂×Q₈	—	2	4	no	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×Q₈	—	2	1	no	—
490	C₂×Q₈	—	2	1	no	—
491	C₂×Q₈	—	4	2	no	—
492	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
493	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
494	C₂×Q₈	C₂⁴×C₄₁	1	2	yes	—
495	C₂×Q₈	C₂⁴×C₄₁	1	2	yes	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	2	no	—
499	C₂×D₄	—	2	2	no	—
500	C₂×D₄	—	2	2	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	2	no	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	—	4	4	no	—
505	C₂×D₄	—	4	4	no	—
506	C₂×D₄	—	8	2	no	—
507	C₂×D₄	C₂⁴×C₄₁	1	4	yes	—
508	C₂×D₄	C₂⁴×C₄₁	1	2	yes	—
509	C₂²×C₄	—	2	1	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	2	no	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	2	8	no	—
514	C₂²×C₄	—	8	4	no	—
515	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
516	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
517	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
518	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
519	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
520	Q₁₆	—	4	4	no	—
521	SD₁₆	—	4	4	no	—
522	SD₁₆	—	4	8	no	—
523	D₈	—	4	8	no	—
524	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
525	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	4	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	2	1	no	—
531	C₄⋊C₄	—	2	1	no	—
532	C₄⋊C₄	—	2	2	no	—
533	C₄⋊C₄	—	2	4	no	—
534	C₄⋊C₄	—	2	1	no	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	4	no	—
538	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
539	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
540	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
541	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
542	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
543	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
544	C₄⋊C₄	C₂⁴×C₄₁	1	2	yes	—
545	C₂²⋊C₄	—	2	2	no	—
546	C₂²⋊C₄	—	2	8	no	—
547	C₂²⋊C₄	—	2	8	no	—
548	C₂²⋊C₄	—	2	8	no	—
549	C₂²⋊C₄	—	2	2	no	—
550	C₂²⋊C₄	—	2	4	no	—
551	C₂²⋊C₄	—	2	4	no	—
552	C₂²⋊C₄	—	2	2	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₄²	—	2	2	no	—
555	C₄²	—	2	2	no	—
556	C₄²	—	2	4	no	—
557	C₄²	—	2	2	no	—
558	C₄²	C₂×D₄×C₄₁	1	4	yes	—
559	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
560	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
561	C₄²	C₂⁴×C₄₁	1	4	yes	—
562	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
563	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
564	C₂³	—	2	2	no	—
565	C₂³	—	2	4	no	—
566	C₂³	—	4	4	no	—
567	C₂³	—	4	4	no	—
568	C₂³	—	4	4	no	—
569	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
570	Q₈	—	2	4	no	—
571	Q₈	—	2	4	no	—
572	Q₈	—	2	4	no	—
573	Q₈	—	2	2	no	—
574	Q₈	—	2	2	no	—
575	Q₈	—	4	2	no	—
576	Q₈	—	4	2	no	—
577	Q₈	—	8	1	no	—
578	D₄	—	2	4	no	—
579	D₄	—	2	8	no	—
580	D₄	—	4	4	no	—
581	D₄	—	4	4	no	—
582	D₄	—	4	4	no	—
583	D₄	—	4	4	no	—
584	D₄	—	4	2	no	—
585	D₄	—	8	2	no	—
586	D₄	—	8	2	no	—
587	D₄	—	8	1	no	—
588	C₂×C₄	—	2	4	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	4	2	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	4	4	no	—
600	C₂×C₄	—	8	8	no	—
601	C₂×C₄	—	8	4	no	—
602	C₂×C₄	#135×C₄₁	1	4	yes	—
603	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
604	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
605	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
606	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
607	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
608	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
609	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
610	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
611	C₈	—	2	4	no	—
612	C₈	—	2	4	no	—
613	C₂²	—	2	1	no	—
614	C₂²	—	2	1	no	—
615	C₂²	—	4	2	no	—
616	C₂²	—	4	4	no	—
617	C₂²	—	4	4	no	—
618	C₂²	—	4	2	no	—
619	C₂²	—	4	4	no	—
620	C₂²	—	4	2	no	—
621	C₂²	—	8	8	no	—
622	C₂²	C₂×#135×C₄₁	1	2	yes	—
623	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
624	C₄	—	2	4	no	—
625	C₄	—	2	4	no	—
626	C₄	—	2	1	no	—
627	C₄	—	2	1	no	—
628	C₄	—	2	1	no	—
629	C₄	—	4	2	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	2	no	—
632	C₄	—	4	2	no	—
633	C₄	—	4	2	no	—
634	C₄	C₂×#135×C₄₁	1	2	yes	—
635	C₄	C₂×#135×C₄₁	1	2	yes	—
636	C₄	C₄₁×#585	1	2	yes	—
637	C₂	—	2	2	no	—
638	C₂	—	8	4	no	—
639	C₂	—	8	2	no	—
640	C₂	C₄₁×#3487	1	1	yes	characteristic
641	C₂	C₂²×#135×C₄₁	1	1	yes	characteristic
642	C₂	C₄₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
prime radical	C₄₁×#3205	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄×Q₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3507	C₂	semidirect
2. C₂×C₄₁×#559	C₂	semidirect
3. C₄₁×#3396	C₂	semidirect
4. C₄₁×#3394	C₂	semidirect
5. C₄₁×#3365	C₂	semidirect
6. C₄₁×#3361	C₂	semidirect
7. C₄₁×#3211	C₂	semidirect
8. C₄₁×#3208	C₂	semidirect
10. C₄₁×#3105	C₂	semidirect
11. C₄₁×#3104	C₂	semidirect
12. C₄₁×#3103	C₂	semidirect
13. C₄₁×#2982	C₂	semidirect
18. C₄₁×#559	C₂²	semidirect
19. C₄₁×#557	C₂²	semidirect
21. C₄₁×#556	C₂²	semidirect
22. C₄₁×#554	C₂²	semidirect
38. C₄₁×#495	C₂²	semidirect
41. C₄₁×#488	C₂²	semidirect
46. C₄₁×#478	C₂²	semidirect
47. C₄₁×#477	C₂²	semidirect
48. C₄₁×#475	C₂²	semidirect
49. C₄₁×#464	C₂²	semidirect
50. C₄₁×#463	C₂²	semidirect
51. C₄₁×#463	C₂²	semidirect
52. C₄₁×#462	C₂²	semidirect
54. C₄₁×#460	C₂²	semidirect
60. C₈×Q₈×C₄₁	C₂²	semidirect
61. C₈×Q₈×C₄₁	C₂²	semidirect
62. C₄×Q₁₆×C₄₁	C₂²	semidirect
63. C₄×SD₁₆×C₄₁	C₂²	semidirect
64. C₄×SD₁₆×C₄₁	C₂²	semidirect
65. C₄×D₈×C₄₁	C₂²	semidirect
291. #61885	C₄₁	direct (N × H)
316. #3507	C₂×C₄₁	semidirect
317. C₂×#559	C₂×C₄₁	semidirect
318. #3396	C₂×C₄₁	semidirect
319. #3394	C₂×C₄₁	semidirect
320. #3365	C₂×C₄₁	semidirect
321. #3361	C₂×C₄₁	semidirect
322. #3211	C₂×C₄₁	semidirect
323. #3208	C₂×C₄₁	semidirect
325. #3105	C₂×C₄₁	semidirect
326. #3104	C₂×C₄₁	semidirect
327. #3103	C₂×C₄₁	semidirect
328. #2982	C₂×C₄₁	semidirect
339. #559	C₂²×C₄₁	semidirect
340. #557	C₂²×C₄₁	semidirect
342. #556	C₂²×C₄₁	semidirect
343. #554	C₂²×C₄₁	semidirect
359. #495	C₂²×C₄₁	semidirect
362. #488	C₂²×C₄₁	semidirect
367. #478	C₂²×C₄₁	semidirect
368. #477	C₂²×C₄₁	semidirect
369. #475	C₂²×C₄₁	semidirect
370. #464	C₂²×C₄₁	semidirect
371. #463	C₂²×C₄₁	semidirect
372. #463	C₂²×C₄₁	semidirect
373. #462	C₂²×C₄₁	semidirect
375. #460	C₂²×C₄₁	semidirect
381. C₈×Q₈	C₂²×C₄₁	semidirect
382. C₈×Q₈	C₂²×C₄₁	semidirect
383. C₄×Q₁₆	C₂²×C₄₁	semidirect
384. C₄×SD₁₆	C₂²×C₄₁	semidirect
385. C₄×SD₁₆	C₂²×C₄₁	semidirect
386. C₄×D₈	C₂²×C₄₁	semidirect
390. C₄₁	#61885	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#3507	2
C₂×C₄₁×#559	2
C₄₁×#3396	2
C₄₁×#3394	2
C₄₁×#3365	2
C₄₁×#3361	2
C₄₁×#3211	2
C₄₁×#3208	2
C₄₁×#3205	2
C₄₁×#3105	2
C₄₁×#3104	2
C₄₁×#3103	2
C₄₁×#2982	2
#61885	41

Subgroups — C41×#61885 #23838086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#61885 #23838086