Subgroups — C₄₁×#22859 #23799060

Order 10496.

1086 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#451	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#441	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#103×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2360	C₂	1	1	yes	maximal, characteristic
5	Q₈×C₂²⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#1971	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1935	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1927	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1892	C₂	1	1	yes	maximal, characteristic
10	C₄₁×#1878	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1878	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#1865	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄₁×#1853	C₂	1	1	yes	maximal, characteristic
14	C₄₁×#1763	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₁×#1752	C₂	1	1	yes	maximal, characteristic
16	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
18	C₂²×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
25	C₄₁×#451	C₄	1	8	yes	complemented
26	C₄₁×#441	—	2	4	no	—
27	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×#103×C₄₁	—	2	4	no	—
30	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
33	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
34	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
35	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
36	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
37	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
39	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
40	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
41	C₄₁×#391	—	2	1	no	—
42	C₄₁×#391	—	2	1	no	—
43	C₄₁×#391	C₂²	1	1	yes	characteristic
44	C₄₁×#391	C₂²	1	1	yes	characteristic
45	C₄₁×#391	C₂²	1	1	yes	characteristic
46	C₄₁×#389	—	2	1	no	—
47	C₄₁×#389	C₂²	1	1	yes	characteristic
48	C₄₁×#387	—	2	1	no	—
49	C₄₁×#385	—	2	1	no	—
50	C₄₁×#380	—	2	1	no	—
51	C₄₁×#380	C₂²	1	1	yes	characteristic
52	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
53	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
54	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
55	C₄×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
56	C₄₁×#340	C₂²	1	1	yes	characteristic
57	C₄₁×#340	C₂²	1	1	yes	characteristic
58	C₄₁×#340	C₂²	1	1	yes	characteristic
59	C₄₁×#340	C₂²	1	1	yes	characteristic
60	C₄₁×#336	C₂²	1	1	yes	characteristic
61	C₄₁×#336	C₂²	1	1	yes	characteristic
62	C₂²×Q₈×C₄₁	—	2	1	no	—
63	C₂²×Q₈×C₄₁	—	2	2	no	—
64	C₂²×Q₈×C₄₁	—	2	2	no	—
65	C₂²×Q₈×C₄₁	—	2	2	no	—
66	C₂²×Q₈×C₄₁	—	4	1	no	—
67	C₂²×Q₈×C₄₁	—	4	1	no	—
68	C₂²×Q₈×C₄₁	C₂×C₄	1	4	yes	—
69	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
71	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₄₁	—	2	1	no	—
73	C₂³×C₄×C₄₁	—	2	1	no	—
74	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
75	C₂×Q₁₆×C₄₁	—	2	2	no	—
76	C₂×Q₁₆×C₄₁	—	2	2	no	—
77	C₂×Q₁₆×C₄₁	—	4	1	no	—
78	C₂×Q₁₆×C₄₁	—	4	1	no	—
79	C₂×Q₁₆×C₄₁	C₂×C₄	1	4	yes	complemented
80	C₂×Q₁₆×C₄₁	C₂×C₄	1	4	yes	complemented
81	C₂²×C₈×C₄₁	—	2	2	no	—
82	C₂²×C₈×C₄₁	—	2	2	no	—
83	C₂²×C₈×C₄₁	—	4	1	no	—
84	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
88	C₂²⋊Q₈×C₄₁	—	2	2	no	—
89	C₂²⋊Q₈×C₄₁	C₂×C₄	1	4	yes	—
90	C₄×Q₈×C₄₁	—	2	2	no	—
91	C₄×Q₈×C₄₁	—	2	2	no	—
92	C₄×Q₈×C₄₁	—	2	2	no	—
93	C₄×Q₈×C₄₁	—	4	1	no	—
94	C₄×Q₈×C₄₁	—	4	1	no	—
95	C₄×Q₈×C₄₁	—	4	1	no	—
96	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
97	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
98	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
100	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
101	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
102	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
103	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
104	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
110	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₄₁	D₄	1	1	yes	characteristic
112	C₂×C₂²⋊C₄×C₄₁	D₄	1	1	yes	characteristic
113	C₂×C₄²×C₄₁	—	2	1	no	—
114	C₂×C₄²×C₄₁	—	2	1	no	—
115	C₂×C₄²×C₄₁	—	2	1	no	—
116	C₂×C₄²×C₄₁	—	2	1	no	—
117	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
119	#106×C₄₁	—	2	2	no	—
120	#106×C₄₁	—	4	1	no	—
121	#103×C₄₁	—	2	2	no	—
122	#103×C₄₁	—	2	2	no	—
123	#103×C₄₁	—	2	2	no	—
124	#103×C₄₁	—	2	2	no	—
125	#103×C₄₁	—	4	1	no	—
126	#103×C₄₁	C₂×C₄	1	4	yes	complemented
127	#103×C₄₁	C₂×C₄	1	4	yes	complemented
128	#103×C₄₁	D₄	1	4	yes	—
129	C₂²⋊C₈×C₄₁	C₂×C₄	1	4	yes	—
130	#97×C₄₁	—	2	2	no	—
131	#97×C₄₁	—	4	1	no	—
132	#95×C₄₁	—	2	1	no	—
133	#95×C₄₁	—	2	1	no	—
134	#95×C₄₁	—	2	1	no	—
135	#95×C₄₁	—	2	1	no	—
136	#95×C₄₁	—	2	1	no	—
137	#95×C₄₁	—	2	1	no	—
138	#95×C₄₁	—	2	1	no	—
139	#95×C₄₁	—	2	1	no	—
140	#95×C₄₁	—	2	1	no	—
141	#95×C₄₁	D₄	1	1	yes	characteristic
142	#95×C₄₁	D₄	1	1	yes	characteristic
143	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
144	C₂×Q₈×C₄₁	—	2	4	no	—
145	C₂×Q₈×C₄₁	—	2	1	no	—
146	C₂×Q₈×C₄₁	—	2	1	no	—
147	C₂×Q₈×C₄₁	—	2	1	no	—
148	C₂×Q₈×C₄₁	—	2	1	no	—
149	C₂×Q₈×C₄₁	—	2	1	no	—
150	C₂×Q₈×C₄₁	—	2	8	no	—
151	C₂×Q₈×C₄₁	—	2	1	no	—
152	C₂×Q₈×C₄₁	—	2	1	no	—
153	C₂×Q₈×C₄₁	—	2	1	no	—
154	C₂×Q₈×C₄₁	—	2	1	no	—
155	C₂×Q₈×C₄₁	—	2	1	no	—
156	C₂×Q₈×C₄₁	—	2	1	no	—
157	C₂×Q₈×C₄₁	—	2	1	no	—
158	C₂×Q₈×C₄₁	—	2	1	no	—
159	C₂×Q₈×C₄₁	—	4	2	no	—
160	C₂×Q₈×C₄₁	—	4	2	no	—
161	C₂×Q₈×C₄₁	—	4	4	no	—
162	C₂×Q₈×C₄₁	—	4	4	no	—
163	C₂×Q₈×C₄₁	—	4	4	no	—
164	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
165	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
166	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
167	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
168	C₂×Q₈×C₄₁	#41	1	2	yes	—
169	C₂²×C₄×C₄₁	—	2	1	no	—
170	C₂²×C₄×C₄₁	—	2	1	no	—
171	C₂²×C₄×C₄₁	—	2	1	no	—
172	C₂²×C₄×C₄₁	—	2	1	no	—
173	C₂²×C₄×C₄₁	—	2	1	no	—
174	C₂²×C₄×C₄₁	—	2	4	no	—
175	C₂²×C₄×C₄₁	—	2	1	no	—
176	C₂²×C₄×C₄₁	—	2	1	no	—
177	C₂²×C₄×C₄₁	—	2	1	no	—
178	C₂²×C₄×C₄₁	—	2	1	no	—
179	C₂²×C₄×C₄₁	—	2	1	no	—
180	C₂²×C₄×C₄₁	—	2	1	no	—
181	C₂²×C₄×C₄₁	—	2	1	no	—
182	C₂²×C₄×C₄₁	—	4	2	no	—
183	C₂²×C₄×C₄₁	—	4	2	no	—
184	C₂²×C₄×C₄₁	—	4	2	no	—
185	C₂²×C₄×C₄₁	—	4	2	no	—
186	C₂²×C₄×C₄₁	—	8	1	no	—
187	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
188	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₄₁	#41	1	2	yes	—
194	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
195	Q₁₆×C₄₁	—	2	8	no	—
196	Q₁₆×C₄₁	—	4	4	no	—
197	C₂×C₈×C₄₁	—	2	1	no	—
198	C₂×C₈×C₄₁	—	2	1	no	—
199	C₂×C₈×C₄₁	—	2	1	no	—
200	C₂×C₈×C₄₁	—	2	1	no	—
201	C₂×C₈×C₄₁	—	2	1	no	—
202	C₂×C₈×C₄₁	—	2	1	no	—
203	C₂×C₈×C₄₁	—	2	1	no	—
204	C₂×C₈×C₄₁	—	2	1	no	—
205	C₂×C₈×C₄₁	—	2	1	no	—
206	C₂×C₈×C₄₁	—	4	4	no	—
207	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
208	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
209	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
210	C₄⋊C₄×C₄₁	—	2	4	no	—
211	C₄⋊C₄×C₄₁	—	2	1	no	—
212	C₄⋊C₄×C₄₁	—	2	1	no	—
213	C₄⋊C₄×C₄₁	—	2	1	no	—
214	C₄⋊C₄×C₄₁	—	2	1	no	—
215	C₄⋊C₄×C₄₁	—	2	1	no	—
216	C₄⋊C₄×C₄₁	—	2	1	no	—
217	C₄⋊C₄×C₄₁	—	2	1	no	—
218	C₄⋊C₄×C₄₁	—	2	1	no	—
219	C₄⋊C₄×C₄₁	—	4	2	no	—
220	C₄⋊C₄×C₄₁	—	4	2	no	—
221	C₄⋊C₄×C₄₁	—	4	2	no	—
222	C₄⋊C₄×C₄₁	—	4	2	no	—
223	C₄⋊C₄×C₄₁	—	8	1	no	—
224	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
225	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
226	C₂²⋊C₄×C₄₁	—	2	4	no	—
227	C₂²⋊C₄×C₄₁	—	2	2	no	—
228	C₂²⋊C₄×C₄₁	—	2	2	no	—
229	C₂²⋊C₄×C₄₁	—	4	2	no	—
230	C₄²×C₄₁	—	2	1	no	—
231	C₄²×C₄₁	—	2	1	no	—
232	C₄²×C₄₁	—	2	1	no	—
233	C₄²×C₄₁	—	2	1	no	—
234	C₄²×C₄₁	—	4	2	no	—
235	C₄²×C₄₁	—	4	2	no	—
236	C₄²×C₄₁	—	4	2	no	—
237	C₄²×C₄₁	—	8	1	no	—
238	C₂³×C₄₁	—	2	1	no	—
239	C₂³×C₄₁	—	2	1	no	—
240	C₂³×C₄₁	—	2	1	no	—
241	C₂³×C₄₁	—	2	1	no	—
242	C₂³×C₄₁	C₄×D₄	1	2	yes	—
243	C₂³×C₄₁	#137	1	2	yes	—
244	C₂³×C₄₁	#137	1	2	yes	—
245	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
246	Q₈×C₄₁	—	2	2	no	—
247	Q₈×C₄₁	—	2	2	no	—
248	Q₈×C₄₁	—	2	2	no	—
249	Q₈×C₄₁	—	2	2	no	—
250	Q₈×C₄₁	—	2	2	no	—
251	Q₈×C₄₁	—	2	2	no	—
252	Q₈×C₄₁	—	2	2	no	—
253	Q₈×C₄₁	—	4	8	no	—
254	Q₈×C₄₁	—	4	1	no	—
255	Q₈×C₄₁	—	4	1	no	—
256	Q₈×C₄₁	—	4	1	no	—
257	Q₈×C₄₁	—	4	1	no	—
258	Q₈×C₄₁	C₄×D₄	1	4	yes	—
259	C₂×C₄×C₄₁	—	2	2	no	—
260	C₂×C₄×C₄₁	—	2	2	no	—
261	C₂×C₄×C₄₁	—	2	1	no	—
262	C₂×C₄×C₄₁	—	2	2	no	—
263	C₂×C₄×C₄₁	—	2	1	no	—
264	C₂×C₄×C₄₁	—	2	2	no	—
265	C₂×C₄×C₄₁	—	2	2	no	—
266	C₂×C₄×C₄₁	—	2	2	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	1	no	—
272	C₂×C₄×C₄₁	—	2	1	no	—
273	C₂×C₄×C₄₁	—	2	2	no	—
274	C₂×C₄×C₄₁	—	2	1	no	—
275	C₂×C₄×C₄₁	—	2	1	no	—
276	C₂×C₄×C₄₁	—	2	1	no	—
277	C₂×C₄×C₄₁	—	2	1	no	—
278	C₂×C₄×C₄₁	—	2	1	no	—
279	C₂×C₄×C₄₁	—	2	1	no	—
280	C₂×C₄×C₄₁	—	2	1	no	—
281	C₂×C₄×C₄₁	—	2	1	no	—
282	C₂×C₄×C₄₁	—	2	1	no	—
283	C₂×C₄×C₄₁	—	2	1	no	—
284	C₂×C₄×C₄₁	—	4	8	no	—
285	C₂×C₄×C₄₁	—	4	1	no	—
286	C₂×C₄×C₄₁	—	4	1	no	—
287	C₂×C₄×C₄₁	—	4	1	no	—
288	C₂×C₄×C₄₁	—	4	1	no	—
289	C₂×C₄×C₄₁	—	4	1	no	—
290	C₂×C₄×C₄₁	—	4	1	no	—
291	C₂×C₄×C₄₁	—	4	1	no	—
292	C₂×C₄×C₄₁	—	4	1	no	—
293	C₂×C₄×C₄₁	—	4	1	no	—
294	C₂×C₄×C₄₁	—	4	1	no	—
295	C₂×C₄×C₄₁	—	4	1	no	—
296	C₂×C₄×C₄₁	—	4	1	no	—
297	C₂×C₄×C₄₁	—	4	1	no	—
298	C₂×C₄×C₄₁	—	4	1	no	—
299	C₂×C₄×C₄₁	—	4	1	no	—
300	C₂×C₄×C₄₁	—	4	1	no	—
301	C₂×C₄×C₄₁	—	4	1	no	—
302	C₂×C₄×C₄₁	—	8	4	no	—
303	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
304	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
305	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
307	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
308	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
309	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
310	C₈×C₄₁	—	2	2	no	—
311	C₈×C₄₁	—	2	2	no	—
312	C₈×C₄₁	—	2	2	no	—
313	C₈×C₄₁	—	4	1	no	—
314	#22859	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
315	C₂²×C₄₁	—	2	2	no	—
316	C₂²×C₄₁	—	2	2	no	—
317	C₂²×C₄₁	—	2	2	no	—
318	C₂²×C₄₁	—	2	2	no	—
319	C₂²×C₄₁	—	2	2	no	—
320	C₂²×C₄₁	—	2	2	no	—
321	C₂²×C₄₁	#441	1	4	yes	complemented
322	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
323	C₂²×C₄₁	D₄²	1	1	yes	characteristic
324	C₂²×C₄₁	#546	1	1	yes	characteristic
325	C₂²×C₄₁	C₂×#137	1	1	yes	characteristic
326	C₂²×C₄₁	C₂×#137	1	1	yes	characteristic
327	C₂²×C₄₁	#577	1	1	yes	characteristic
328	C₂²×C₄₁	#578	1	1	yes	characteristic
329	C₄×C₄₁	—	2	2	no	—
330	C₄×C₄₁	—	2	4	no	—
331	C₄×C₄₁	—	4	2	no	—
332	C₄×C₄₁	—	4	2	no	—
333	C₄×C₄₁	—	4	2	no	—
334	C₄×C₄₁	—	4	1	no	—
335	C₄×C₄₁	—	4	2	no	—
336	C₄×C₄₁	—	4	1	no	—
337	C₄×C₄₁	—	4	1	no	—
338	C₄×C₄₁	—	4	1	no	—
339	C₄×C₄₁	—	8	1	no	—
340	C₄×C₄₁	—	8	1	no	—
341	C₄×C₄₁	—	8	1	no	—
342	C₄×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
343	C₄×C₄₁	#518	1	1	yes	characteristic
344	C₄×C₄₁	#546	1	1	yes	characteristic
345	C₄×C₄₁	#551	1	1	yes	characteristic
346	C₂×#451	C₂×C₄₁	1	1	yes	characteristic
347	C₂×#441	C₂×C₄₁	1	1	yes	characteristic, complemented
348	C₂²×#103	C₂×C₄₁	1	1	yes	characteristic
349	#2360	C₂×C₄₁	1	1	yes	characteristic
350	Q₈×C₂²⋊C₄	C₂×C₄₁	1	1	yes	characteristic
351	#1971	C₂×C₄₁	1	1	yes	characteristic, complemented
352	#1935	C₂×C₄₁	1	1	yes	characteristic, complemented
353	#1927	C₂×C₄₁	1	1	yes	characteristic, complemented
354	#1892	C₂×C₄₁	1	1	yes	characteristic
355	#1878	C₂×C₄₁	1	1	yes	characteristic, complemented
356	#1878	C₂×C₄₁	1	1	yes	characteristic, complemented
357	#1865	C₂×C₄₁	1	1	yes	characteristic, complemented
358	#1853	C₂×C₄₁	1	1	yes	characteristic
359	#1763	C₂×C₄₁	1	1	yes	characteristic, complemented
360	#1752	C₂×C₄₁	1	1	yes	characteristic
361	C₂×C₄₁	—	2	4	no	—
362	C₂×C₄₁	#2358	1	1	yes	characteristic
363	C₂×C₄₁	C₂×#441	1	1	yes	characteristic
364	C₂×C₄₁	#2947	1	1	yes	characteristic
365	C₂×C₄₁	#3266	1	1	yes	characteristic
366	C₂×C₄₁	#3270	1	1	yes	characteristic
367	C₂×C₄₁	#3312	1	1	yes	characteristic
368	C₂×C₄₁	#3316	1	1	yes	characteristic
369	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
370	C₂²×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
371	C₂²×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
372	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
373	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
374	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
375	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
376	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
377	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
378	#451	C₄×C₄₁	1	8	yes	complemented
379	#441	—	2	4	no	—
380	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
381	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
382	C₂×#103	—	2	4	no	—
383	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
384	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
385	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
386	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
387	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
388	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
389	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
390	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
391	C₂×C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
392	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
393	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
394	#391	—	2	1	no	—
395	#391	—	2	1	no	—
396	#391	C₂²×C₄₁	1	1	yes	characteristic
397	#391	C₂²×C₄₁	1	1	yes	characteristic
398	#391	C₂²×C₄₁	1	1	yes	characteristic
399	#389	—	2	1	no	—
400	#389	C₂²×C₄₁	1	1	yes	characteristic
401	#387	—	2	1	no	—
402	#385	—	2	1	no	—
403	#380	—	2	1	no	—
404	#380	C₂²×C₄₁	1	1	yes	characteristic
405	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
406	C₄×C₂²⋊C₄	—	2	1	no	—
407	C₄×C₂²⋊C₄	—	2	1	no	—
408	C₄×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
409	#340	C₂²×C₄₁	1	1	yes	characteristic
410	#340	C₂²×C₄₁	1	1	yes	characteristic
411	#340	C₂²×C₄₁	1	1	yes	characteristic
412	#340	C₂²×C₄₁	1	1	yes	characteristic
413	#336	C₂²×C₄₁	1	1	yes	characteristic
414	#336	C₂²×C₄₁	1	1	yes	characteristic
415	C₄₁	#22859	1	1	yes	characteristic, direct product factor, Sylow
416	C₂²×Q₈	—	2	1	no	—
417	C₂²×Q₈	—	2	2	no	—
418	C₂²×Q₈	—	2	2	no	—
419	C₂²×Q₈	—	2	2	no	—
420	C₂²×Q₈	—	4	1	no	—
421	C₂²×Q₈	—	4	1	no	—
422	C₂²×Q₈	C₂×C₄×C₄₁	1	4	yes	—
423	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
424	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
425	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
426	C₂³×C₄	—	2	1	no	—
427	C₂³×C₄	—	2	1	no	—
428	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
429	C₂×Q₁₆	—	2	2	no	—
430	C₂×Q₁₆	—	2	2	no	—
431	C₂×Q₁₆	—	4	1	no	—
432	C₂×Q₁₆	—	4	1	no	—
433	C₂×Q₁₆	C₂×C₄×C₄₁	1	4	yes	complemented
434	C₂×Q₁₆	C₂×C₄×C₄₁	1	4	yes	complemented
435	C₂²×C₈	—	2	2	no	—
436	C₂²×C₈	—	2	2	no	—
437	C₂²×C₈	—	4	1	no	—
438	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
439	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
440	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
441	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
442	C₂²⋊Q₈	—	2	2	no	—
443	C₂²⋊Q₈	C₂×C₄×C₄₁	1	4	yes	—
444	C₄×Q₈	—	2	2	no	—
445	C₄×Q₈	—	2	2	no	—
446	C₄×Q₈	—	2	2	no	—
447	C₄×Q₈	—	4	1	no	—
448	C₄×Q₈	—	4	1	no	—
449	C₄×Q₈	—	4	1	no	—
450	C₂×C₄⋊C₄	—	2	1	no	—
451	C₂×C₄⋊C₄	—	2	2	no	—
452	C₂×C₄⋊C₄	—	2	1	no	—
453	C₂×C₄⋊C₄	—	2	1	no	—
454	C₂×C₄⋊C₄	—	2	1	no	—
455	C₂×C₄⋊C₄	—	2	1	no	—
456	C₂×C₄⋊C₄	—	2	1	no	—
457	C₂×C₄⋊C₄	—	4	1	no	—
458	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
459	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
460	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
461	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
462	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
463	C₂×C₂²⋊C₄	—	2	1	no	—
464	C₂×C₂²⋊C₄	—	2	1	no	—
465	C₂×C₂²⋊C₄	D₄×C₄₁	1	1	yes	characteristic
466	C₂×C₂²⋊C₄	D₄×C₄₁	1	1	yes	characteristic
467	C₂×C₄²	—	2	1	no	—
468	C₂×C₄²	—	2	1	no	—
469	C₂×C₄²	—	2	1	no	—
470	C₂×C₄²	—	2	1	no	—
471	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
472	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
473	#106	—	2	2	no	—
474	#106	—	4	1	no	—
475	#103	—	2	2	no	—
476	#103	—	2	2	no	—
477	#103	—	2	2	no	—
478	#103	—	2	2	no	—
479	#103	—	4	1	no	—
480	#103	C₂×C₄×C₄₁	1	4	yes	complemented
481	#103	C₂×C₄×C₄₁	1	4	yes	complemented
482	#103	D₄×C₄₁	1	4	yes	—
483	C₂²⋊C₈	C₂×C₄×C₄₁	1	4	yes	—
484	#97	—	2	2	no	—
485	#97	—	4	1	no	—
486	#95	—	2	1	no	—
487	#95	—	2	1	no	—
488	#95	—	2	1	no	—
489	#95	—	2	1	no	—
490	#95	—	2	1	no	—
491	#95	—	2	1	no	—
492	#95	—	2	1	no	—
493	#95	—	2	1	no	—
494	#95	—	2	1	no	—
495	#95	D₄×C₄₁	1	1	yes	characteristic
496	#95	D₄×C₄₁	1	1	yes	characteristic
497	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
498	C₂×Q₈	—	2	4	no	—
499	C₂×Q₈	—	2	1	no	—
500	C₂×Q₈	—	2	1	no	—
501	C₂×Q₈	—	2	1	no	—
502	C₂×Q₈	—	2	1	no	—
503	C₂×Q₈	—	2	1	no	—
504	C₂×Q₈	—	2	8	no	—
505	C₂×Q₈	—	2	1	no	—
506	C₂×Q₈	—	2	1	no	—
507	C₂×Q₈	—	2	1	no	—
508	C₂×Q₈	—	2	1	no	—
509	C₂×Q₈	—	2	1	no	—
510	C₂×Q₈	—	2	1	no	—
511	C₂×Q₈	—	2	1	no	—
512	C₂×Q₈	—	2	1	no	—
513	C₂×Q₈	—	4	2	no	—
514	C₂×Q₈	—	4	2	no	—
515	C₂×Q₈	—	4	4	no	—
516	C₂×Q₈	—	4	4	no	—
517	C₂×Q₈	—	4	4	no	—
518	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
519	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
520	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
521	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
522	C₂×Q₈	#41×C₄₁	1	2	yes	—
523	C₂²×C₄	—	2	1	no	—
524	C₂²×C₄	—	2	1	no	—
525	C₂²×C₄	—	2	1	no	—
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	—	2	4	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	1	no	—
531	C₂²×C₄	—	2	1	no	—
532	C₂²×C₄	—	2	1	no	—
533	C₂²×C₄	—	2	1	no	—
534	C₂²×C₄	—	2	1	no	—
535	C₂²×C₄	—	2	1	no	—
536	C₂²×C₄	—	4	2	no	—
537	C₂²×C₄	—	4	2	no	—
538	C₂²×C₄	—	4	2	no	—
539	C₂²×C₄	—	4	2	no	—
540	C₂²×C₄	—	8	1	no	—
541	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
542	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
543	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
544	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
545	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
546	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
547	C₂²×C₄	#41×C₄₁	1	2	yes	—
548	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
549	Q₁₆	—	2	8	no	—
550	Q₁₆	—	4	4	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	2	1	no	—
554	C₂×C₈	—	2	1	no	—
555	C₂×C₈	—	2	1	no	—
556	C₂×C₈	—	2	1	no	—
557	C₂×C₈	—	2	1	no	—
558	C₂×C₈	—	2	1	no	—
559	C₂×C₈	—	2	1	no	—
560	C₂×C₈	—	4	4	no	—
561	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
562	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
563	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
564	C₄⋊C₄	—	2	4	no	—
565	C₄⋊C₄	—	2	1	no	—
566	C₄⋊C₄	—	2	1	no	—
567	C₄⋊C₄	—	2	1	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	2	1	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	—	4	2	no	—
574	C₄⋊C₄	—	4	2	no	—
575	C₄⋊C₄	—	4	2	no	—
576	C₄⋊C₄	—	4	2	no	—
577	C₄⋊C₄	—	8	1	no	—
578	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
579	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
580	C₂²⋊C₄	—	2	4	no	—
581	C₂²⋊C₄	—	2	2	no	—
582	C₂²⋊C₄	—	2	2	no	—
583	C₂²⋊C₄	—	4	2	no	—
584	C₄²	—	2	1	no	—
585	C₄²	—	2	1	no	—
586	C₄²	—	2	1	no	—
587	C₄²	—	2	1	no	—
588	C₄²	—	4	2	no	—
589	C₄²	—	4	2	no	—
590	C₄²	—	4	2	no	—
591	C₄²	—	8	1	no	—
592	C₂³	—	2	1	no	—
593	C₂³	—	2	1	no	—
594	C₂³	—	2	1	no	—
595	C₂³	—	2	1	no	—
596	C₂³	C₄×D₄×C₄₁	1	2	yes	—
597	C₂³	#137×C₄₁	1	2	yes	—
598	C₂³	#137×C₄₁	1	2	yes	—
599	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
600	Q₈	—	2	2	no	—
601	Q₈	—	2	2	no	—
602	Q₈	—	2	2	no	—
603	Q₈	—	2	2	no	—
604	Q₈	—	2	2	no	—
605	Q₈	—	2	2	no	—
606	Q₈	—	2	2	no	—
607	Q₈	—	4	8	no	—
608	Q₈	—	4	1	no	—
609	Q₈	—	4	1	no	—
610	Q₈	—	4	1	no	—
611	Q₈	—	4	1	no	—
612	Q₈	C₄×D₄×C₄₁	1	4	yes	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	1	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	2	1	no	—
632	C₂×C₄	—	2	1	no	—
633	C₂×C₄	—	2	1	no	—
634	C₂×C₄	—	2	1	no	—
635	C₂×C₄	—	2	1	no	—
636	C₂×C₄	—	2	1	no	—
637	C₂×C₄	—	2	1	no	—
638	C₂×C₄	—	4	8	no	—
639	C₂×C₄	—	4	1	no	—
640	C₂×C₄	—	4	1	no	—
641	C₂×C₄	—	4	1	no	—
642	C₂×C₄	—	4	1	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	4	1	no	—
653	C₂×C₄	—	4	1	no	—
654	C₂×C₄	—	4	1	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	8	4	no	—
657	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
658	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
659	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
660	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
661	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
662	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
663	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
664	C₈	—	2	2	no	—
665	C₈	—	2	2	no	—
666	C₈	—	2	2	no	—
667	C₈	—	4	1	no	—
668	C₂²	—	2	2	no	—
669	C₂²	—	2	2	no	—
670	C₂²	—	2	2	no	—
671	C₂²	—	2	2	no	—
672	C₂²	—	2	2	no	—
673	C₂²	—	2	2	no	—
674	C₂²	C₄₁×#441	1	4	yes	complemented
675	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
676	C₂²	D₄²×C₄₁	1	1	yes	characteristic
677	C₂²	C₄₁×#546	1	1	yes	characteristic
678	C₂²	C₂×#137×C₄₁	1	1	yes	characteristic
679	C₂²	C₂×#137×C₄₁	1	1	yes	characteristic
680	C₂²	C₄₁×#577	1	1	yes	characteristic
681	C₂²	C₄₁×#578	1	1	yes	characteristic
682	C₄	—	2	2	no	—
683	C₄	—	2	4	no	—
684	C₄	—	4	1	no	—
685	C₄	—	4	1	no	—
686	C₄	—	4	2	no	—
687	C₄	—	4	2	no	—
688	C₄	—	4	2	no	—
689	C₄	—	4	1	no	—
690	C₄	—	4	2	no	—
691	C₄	—	4	1	no	—
692	C₄	—	8	1	no	—
693	C₄	—	8	1	no	—
694	C₄	—	8	1	no	—
695	C₄	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
696	C₄	C₄₁×#518	1	1	yes	characteristic
697	C₄	C₄₁×#546	1	1	yes	characteristic
698	C₄	C₄₁×#551	1	1	yes	characteristic
699	C₂	—	2	4	no	—
700	C₂	C₄₁×#2358	1	1	yes	characteristic
701	C₂	C₂×C₄₁×#441	1	1	yes	characteristic
702	C₂	C₄₁×#2947	1	1	yes	characteristic
703	C₂	C₄₁×#3266	1	1	yes	characteristic
704	C₂	C₄₁×#3270	1	1	yes	characteristic
705	C₂	C₄₁×#3312	1	1	yes	characteristic
706	C₂	C₄₁×#3316	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#441	C₂	semidirect
6. C₄₁×#1971	C₂	semidirect
7. C₄₁×#1935	C₂	semidirect
8. C₄₁×#1927	C₂	semidirect
10. C₄₁×#1878	C₂	semidirect
11. C₄₁×#1878	C₂	semidirect
12. C₄₁×#1865	C₂	semidirect
14. C₄₁×#1763	C₂	semidirect
25. C₄₁×#451	C₄	semidirect
79. C₂×Q₁₆×C₄₁	C₂×C₄	semidirect
80. C₂×Q₁₆×C₄₁	C₂×C₄	semidirect
126. #103×C₄₁	C₂×C₄	semidirect
127. #103×C₄₁	C₂×C₄	semidirect
314. #22859	C₄₁	direct (N × H)
321. C₂²×C₄₁	#441	semidirect
347. C₂×#441	C₂×C₄₁	semidirect
351. #1971	C₂×C₄₁	semidirect
352. #1935	C₂×C₄₁	semidirect
353. #1927	C₂×C₄₁	semidirect
355. #1878	C₂×C₄₁	semidirect
356. #1878	C₂×C₄₁	semidirect
357. #1865	C₂×C₄₁	semidirect
359. #1763	C₂×C₄₁	semidirect
378. #451	C₄×C₄₁	semidirect
415. C₄₁	#22859	direct (N × H)
433. C₂×Q₁₆	C₂×C₄×C₄₁	semidirect
434. C₂×Q₁₆	C₂×C₄×C₄₁	semidirect
480. #103	C₂×C₄×C₄₁	semidirect
481. #103	C₂×C₄×C₄₁	semidirect
674. C₂²	C₄₁×#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#451	2
C₂×C₄₁×#441	2
C₂²×#103×C₄₁	2
C₄₁×#2360	2
Q₈×C₂²⋊C₄×C₄₁	2
C₄₁×#1971	2
C₄₁×#1935	2
C₄₁×#1927	2
C₄₁×#1892	2
C₄₁×#1878	2
C₄₁×#1865	2
C₄₁×#1853	2
C₄₁×#1763	2
C₄₁×#1752	2
#22859	41

Subgroups — C41×#22859 #23799060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22859 #23799060