Subgroups — C₄₁×#21121 #23797322

Order 10496.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#440	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2295	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1928	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1927	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1921	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1918	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1877	C₂	1	1	yes	maximal, characteristic
9	C₄₁×#1877	C₂	1	1	yes	maximal, characteristic
10	C₄₁×#1864	C₂	1	1	yes	maximal, characteristic
11	C₄₁×#1862	C₂	1	1	yes	maximal, characteristic
12	C₄₁×#1776	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄₁×#1763	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄₁×#1762	C₂	1	1	yes	maximal, characteristic
15	C₄₁×#1745	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₄₁	—	2	1	no	—
22	C₄₁×#440	C₄	1	8	yes	complemented
23	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
33	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
34	C₄₁×#391	C₂²	1	1	yes	characteristic
35	C₄₁×#391	C₂²	1	1	yes	characteristic
36	C₄₁×#389	C₂²	1	1	yes	characteristic
37	C₄₁×#389	C₂²	1	1	yes	characteristic
38	C₄₁×#387	—	2	1	no	—
39	C₄₁×#387	—	2	1	no	—
40	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
41	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
45	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
46	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
47	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
48	C₄₁×#376	—	2	1	no	—
49	C₄³×C₄₁	C₂²	1	1	yes	characteristic
50	C₄₁×#340	C₂²	1	1	yes	characteristic
51	C₄₁×#340	C₂²	1	1	yes	characteristic
52	C₄₁×#340	C₂²	1	1	yes	characteristic
53	C₄₁×#340	C₂²	1	1	yes	characteristic
54	C₄₁×#340	C₂²	1	1	yes	characteristic
55	C₄₁×#340	C₂²	1	1	yes	characteristic
56	C₄₁×#336	C₂²	1	1	yes	characteristic
57	C₄₁×#336	C₂²	1	1	yes	characteristic
58	C₄₁×#336	C₂²	1	1	yes	characteristic
59	C₄₁×#336	C₂²	1	1	yes	characteristic
60	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
61	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₄₁	—	2	1	no	—
63	C₂³×C₄×C₄₁	—	2	1	no	—
64	C₂³×C₄×C₄₁	—	2	1	no	—
65	C₂×SD₁₆×C₄₁	—	2	2	no	—
66	C₂×SD₁₆×C₄₁	—	2	2	no	—
67	C₂×SD₁₆×C₄₁	C₂×C₄	1	4	yes	—
68	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₄₁	—	2	2	no	—
73	C₄×Q₈×C₄₁	C₂×C₄	1	4	yes	complemented
74	C₄×D₄×C₄₁	—	2	2	no	—
75	C₄×D₄×C₄₁	—	2	2	no	—
76	C₄×D₄×C₄₁	—	2	2	no	—
77	C₄×D₄×C₄₁	—	2	2	no	—
78	C₄×D₄×C₄₁	—	2	2	no	—
79	C₄×D₄×C₄₁	C₂×C₄	1	4	yes	—
80	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
92	C₂×C₄²×C₄₁	—	2	1	no	—
93	C₂×C₄²×C₄₁	—	2	1	no	—
94	C₂×C₄²×C₄₁	—	2	1	no	—
95	C₂×C₄²×C₄₁	—	2	1	no	—
96	C₂×C₄²×C₄₁	—	2	1	no	—
97	C₂×C₄²×C₄₁	—	4	4	no	—
98	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
105	#107×C₄₁	—	2	2	no	—
106	#107×C₄₁	C₂×C₄	1	4	yes	complemented
107	#106×C₄₁	—	2	2	no	—
108	#103×C₄₁	—	2	2	no	—
109	#103×C₄₁	—	2	2	no	—
110	#103×C₄₁	C₂×C₄	1	4	yes	complemented
111	#102×C₄₁	—	2	2	no	—
112	#102×C₄₁	—	2	2	no	—
113	#102×C₄₁	C₂×C₄	1	4	yes	—
114	#97×C₄₁	—	2	2	no	—
115	#97×C₄₁	C₂×C₄	1	4	yes	complemented
116	#95×C₄₁	—	2	1	no	—
117	#95×C₄₁	—	2	1	no	—
118	#95×C₄₁	—	2	1	no	—
119	#95×C₄₁	—	2	1	no	—
120	#95×C₄₁	—	2	1	no	—
121	#95×C₄₁	—	2	1	no	—
122	#95×C₄₁	—	2	1	no	—
123	#95×C₄₁	—	2	1	no	—
124	#95×C₄₁	—	2	1	no	—
125	#95×C₄₁	—	2	1	no	—
126	C₂⁴×C₄₁	—	2	1	no	—
127	C₂×Q₈×C₄₁	—	2	1	no	—
128	C₂×Q₈×C₄₁	—	2	1	no	—
129	C₂×Q₈×C₄₁	—	2	1	no	—
130	C₂×Q₈×C₄₁	—	2	1	no	—
131	C₂×Q₈×C₄₁	—	2	1	no	—
132	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
133	C₂×D₄×C₄₁	—	2	1	no	—
134	C₂×D₄×C₄₁	—	2	1	no	—
135	C₂×D₄×C₄₁	—	2	1	no	—
136	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
137	C₂×D₄×C₄₁	#41	1	2	yes	—
138	C₂×D₄×C₄₁	#41	1	2	yes	—
139	C₂²×C₄×C₄₁	—	2	4	no	—
140	C₂²×C₄×C₄₁	—	2	4	no	—
141	C₂²×C₄×C₄₁	—	2	1	no	—
142	C₂²×C₄×C₄₁	—	2	1	no	—
143	C₂²×C₄×C₄₁	—	2	1	no	—
144	C₂²×C₄×C₄₁	—	2	1	no	—
145	C₂²×C₄×C₄₁	—	2	1	no	—
146	C₂²×C₄×C₄₁	—	2	4	no	—
147	C₂²×C₄×C₄₁	—	2	1	no	—
148	C₂²×C₄×C₄₁	—	2	1	no	—
149	C₂²×C₄×C₄₁	—	4	2	no	—
150	C₂²×C₄×C₄₁	—	4	2	no	—
151	C₂²×C₄×C₄₁	—	4	2	no	—
152	C₂²×C₄×C₄₁	—	4	2	no	—
153	C₂²×C₄×C₄₁	—	4	2	no	—
154	C₂²×C₄×C₄₁	—	4	2	no	—
155	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
162	SD₁₆×C₄₁	—	2	8	no	—
163	C₂×C₈×C₄₁	—	2	1	no	—
164	C₂×C₈×C₄₁	—	2	1	no	—
165	C₂×C₈×C₄₁	—	2	1	no	—
166	C₂×C₈×C₄₁	—	2	1	no	—
167	C₂×C₈×C₄₁	—	2	1	no	—
168	C₂×C₈×C₄₁	—	2	1	no	—
169	C₂×C₈×C₄₁	—	2	1	no	—
170	C₂×C₈×C₄₁	—	2	1	no	—
171	C₂×C₈×C₄₁	—	2	1	no	—
172	C₂×C₈×C₄₁	—	2	1	no	—
173	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
174	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
175	C₄⋊C₄×C₄₁	—	2	4	no	—
176	C₄⋊C₄×C₄₁	—	2	1	no	—
177	C₄⋊C₄×C₄₁	—	2	1	no	—
178	C₄⋊C₄×C₄₁	—	2	1	no	—
179	C₄⋊C₄×C₄₁	—	2	1	no	—
180	C₄⋊C₄×C₄₁	—	2	1	no	—
181	C₄⋊C₄×C₄₁	—	2	1	no	—
182	C₄⋊C₄×C₄₁	—	2	1	no	—
183	C₄⋊C₄×C₄₁	—	2	1	no	—
184	C₄⋊C₄×C₄₁	—	4	2	no	—
185	C₄⋊C₄×C₄₁	—	4	2	no	—
186	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
187	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
188	C₄⋊C₄×C₄₁	#41	1	2	yes	—
189	C₄⋊C₄×C₄₁	#41	1	2	yes	—
190	C₂²⋊C₄×C₄₁	—	2	4	no	—
191	C₂²⋊C₄×C₄₁	—	2	4	no	—
192	C₂²⋊C₄×C₄₁	—	2	4	no	—
193	C₄²×C₄₁	—	2	4	no	—
194	C₄²×C₄₁	—	2	2	no	—
195	C₄²×C₄₁	—	2	2	no	—
196	C₄²×C₄₁	—	2	1	no	—
197	C₄²×C₄₁	—	2	1	no	—
198	C₄²×C₄₁	—	2	1	no	—
199	C₄²×C₄₁	—	2	1	no	—
200	C₄²×C₄₁	—	4	1	no	—
201	C₄²×C₄₁	—	4	1	no	—
202	C₄²×C₄₁	—	4	2	no	—
203	C₄²×C₄₁	—	4	2	no	—
204	C₄²×C₄₁	—	4	2	no	—
205	C₄²×C₄₁	—	4	2	no	—
206	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
207	C₄²×C₄₁	#41	1	2	yes	—
208	C₂³×C₄₁	—	2	2	no	—
209	C₂³×C₄₁	—	2	2	no	—
210	C₂³×C₄₁	—	2	2	no	—
211	C₂³×C₄₁	—	4	1	no	—
212	C₂³×C₄₁	—	4	1	no	—
213	C₂³×C₄₁	—	4	1	no	—
214	C₂³×C₄₁	—	4	1	no	—
215	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
216	Q₈×C₄₁	—	2	2	no	—
217	Q₈×C₄₁	—	2	2	no	—
218	Q₈×C₄₁	—	4	1	no	—
219	Q₈×C₄₁	—	4	1	no	—
220	D₄×C₄₁	—	2	2	no	—
221	D₄×C₄₁	—	2	2	no	—
222	D₄×C₄₁	—	2	2	no	—
223	D₄×C₄₁	#117	1	4	yes	—
224	C₂×C₄×C₄₁	—	2	1	no	—
225	C₂×C₄×C₄₁	—	2	1	no	—
226	C₂×C₄×C₄₁	—	2	1	no	—
227	C₂×C₄×C₄₁	—	2	1	no	—
228	C₂×C₄×C₄₁	—	2	1	no	—
229	C₂×C₄×C₄₁	—	2	1	no	—
230	C₂×C₄×C₄₁	—	2	1	no	—
231	C₂×C₄×C₄₁	—	2	1	no	—
232	C₂×C₄×C₄₁	—	2	2	no	—
233	C₂×C₄×C₄₁	—	2	2	no	—
234	C₂×C₄×C₄₁	—	2	2	no	—
235	C₂×C₄×C₄₁	—	2	1	no	—
236	C₂×C₄×C₄₁	—	2	1	no	—
237	C₂×C₄×C₄₁	—	2	2	no	—
238	C₂×C₄×C₄₁	—	2	2	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	8	no	—
244	C₂×C₄×C₄₁	—	4	8	no	—
245	C₂×C₄×C₄₁	—	4	8	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	—	4	1	no	—
251	C₂×C₄×C₄₁	—	4	1	no	—
252	C₂×C₄×C₄₁	—	4	1	no	—
253	C₂×C₄×C₄₁	—	4	1	no	—
254	C₂×C₄×C₄₁	—	4	1	no	—
255	C₂×C₄×C₄₁	—	4	1	no	—
256	C₂×C₄×C₄₁	—	4	1	no	—
257	C₂×C₄×C₄₁	—	4	1	no	—
258	C₂×C₄×C₄₁	—	4	1	no	—
259	C₂×C₄×C₄₁	—	4	1	no	—
260	C₂×C₄×C₄₁	—	4	1	no	—
261	C₂×C₄×C₄₁	—	4	1	no	—
262	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
263	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
264	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
265	C₂×C₄×C₄₁	#137	1	2	yes	—
266	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
267	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
268	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
269	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
270	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
271	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
272	C₈×C₄₁	—	2	2	no	—
273	C₈×C₄₁	—	2	2	no	—
274	C₈×C₄₁	—	4	1	no	—
275	C₈×C₄₁	—	4	1	no	—
276	#21121	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
277	C₂²×C₄₁	—	2	4	no	—
278	C₂²×C₄₁	—	4	2	no	—
279	C₂²×C₄₁	—	4	2	no	—
280	C₂²×C₄₁	—	4	2	no	—
281	C₂²×C₄₁	—	4	2	no	—
282	C₂²×C₄₁	—	4	2	no	—
283	C₂²×C₄₁	—	4	2	no	—
284	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
285	C₂²×C₄₁	#525	1	1	yes	characteristic
286	C₂²×C₄₁	#525	1	1	yes	characteristic
287	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
288	C₂²×C₄₁	C₂×#137	1	1	yes	characteristic
289	C₂²×C₄₁	#575	1	1	yes	characteristic
290	C₂²×C₄₁	#575	1	1	yes	characteristic
291	C₄×C₄₁	—	2	2	no	—
292	C₄×C₄₁	—	2	2	no	—
293	C₄×C₄₁	—	2	2	no	—
294	C₄×C₄₁	—	2	2	no	—
295	C₄×C₄₁	—	4	2	no	—
296	C₄×C₄₁	—	4	1	no	—
297	C₄×C₄₁	—	4	2	no	—
298	C₄×C₄₁	—	4	1	no	—
299	C₄×C₄₁	—	4	2	no	—
300	C₄×C₄₁	—	4	2	no	—
301	C₄×C₄₁	—	4	2	no	—
302	C₄×C₄₁	—	8	1	no	—
303	C₄×C₄₁	—	8	1	no	—
304	C₄×C₄₁	C₂×#117	1	1	yes	characteristic
305	C₄×C₄₁	C₄×#41	1	1	yes	characteristic
306	C₄×C₄₁	#529	1	1	yes	characteristic
307	C₄×C₄₁	#529	1	1	yes	characteristic
308	C₂×#440	C₂×C₄₁	1	1	yes	characteristic
309	#2295	C₂×C₄₁	1	1	yes	characteristic, complemented
310	C₄²×D₄	C₂×C₄₁	1	1	yes	characteristic
311	#1928	C₂×C₄₁	1	1	yes	characteristic, complemented
312	#1927	C₂×C₄₁	1	1	yes	characteristic, complemented
313	#1921	C₂×C₄₁	1	1	yes	characteristic, complemented
314	#1918	C₂×C₄₁	1	1	yes	characteristic, complemented
315	#1877	C₂×C₄₁	1	1	yes	characteristic
316	#1877	C₂×C₄₁	1	1	yes	characteristic
317	#1864	C₂×C₄₁	1	1	yes	characteristic
318	#1862	C₂×C₄₁	1	1	yes	characteristic
319	#1776	C₂×C₄₁	1	1	yes	characteristic, complemented
320	#1763	C₂×C₄₁	1	1	yes	characteristic, complemented
321	#1762	C₂×C₄₁	1	1	yes	characteristic
322	#1745	C₂×C₄₁	1	1	yes	characteristic, complemented
323	C₂×C₄₁	—	4	4	no	—
324	C₂×C₄₁	#2290	1	1	yes	characteristic
325	C₂×C₄₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
326	C₂×C₄₁	C₄×#137	1	1	yes	characteristic
327	C₂×C₄₁	#3110	1	1	yes	characteristic
328	C₂×C₄₁	#3111	1	1	yes	characteristic
329	C₂×C₄₁	#3114	1	1	yes	characteristic
330	C₂×C₄₁	#3116	1	1	yes	characteristic
331	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
332	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
333	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
334	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
335	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
336	C₂²×C₄²	—	2	1	no	—
337	#440	C₄×C₄₁	1	8	yes	complemented
338	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
339	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
340	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
341	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
342	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
343	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
344	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
345	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
346	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
347	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
348	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
349	#391	C₂²×C₄₁	1	1	yes	characteristic
350	#391	C₂²×C₄₁	1	1	yes	characteristic
351	#389	C₂²×C₄₁	1	1	yes	characteristic
352	#389	C₂²×C₄₁	1	1	yes	characteristic
353	#387	—	2	1	no	—
354	#387	—	2	1	no	—
355	C₄×C₄⋊C₄	—	2	1	no	—
356	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
357	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
358	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
359	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
360	C₄×C₂²⋊C₄	—	2	1	no	—
361	C₄×C₂²⋊C₄	—	2	1	no	—
362	C₄×C₂²⋊C₄	—	2	1	no	—
363	#376	—	2	1	no	—
364	C₄³	C₂²×C₄₁	1	1	yes	characteristic
365	#340	C₂²×C₄₁	1	1	yes	characteristic
366	#340	C₂²×C₄₁	1	1	yes	characteristic
367	#340	C₂²×C₄₁	1	1	yes	characteristic
368	#340	C₂²×C₄₁	1	1	yes	characteristic
369	#340	C₂²×C₄₁	1	1	yes	characteristic
370	#340	C₂²×C₄₁	1	1	yes	characteristic
371	#336	C₂²×C₄₁	1	1	yes	characteristic
372	#336	C₂²×C₄₁	1	1	yes	characteristic
373	#336	C₂²×C₄₁	1	1	yes	characteristic
374	#336	C₂²×C₄₁	1	1	yes	characteristic
375	C₄₁	#21121	1	1	yes	characteristic, direct product factor, Sylow
376	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
377	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
378	C₂³×C₄	—	2	1	no	—
379	C₂³×C₄	—	2	1	no	—
380	C₂³×C₄	—	2	1	no	—
381	C₂×SD₁₆	—	2	2	no	—
382	C₂×SD₁₆	—	2	2	no	—
383	C₂×SD₁₆	C₂×C₄×C₄₁	1	4	yes	—
384	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
385	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
386	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
387	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
388	C₄×Q₈	—	2	2	no	—
389	C₄×Q₈	C₂×C₄×C₄₁	1	4	yes	complemented
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	—	2	2	no	—
392	C₄×D₄	—	2	2	no	—
393	C₄×D₄	—	2	2	no	—
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	C₂×C₄×C₄₁	1	4	yes	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	—	2	1	no	—
398	C₂×C₄⋊C₄	—	2	1	no	—
399	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
400	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
403	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
404	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₂²⋊C₄	—	2	1	no	—
408	C₂×C₄²	—	2	1	no	—
409	C₂×C₄²	—	2	1	no	—
410	C₂×C₄²	—	2	1	no	—
411	C₂×C₄²	—	2	1	no	—
412	C₂×C₄²	—	2	1	no	—
413	C₂×C₄²	—	4	4	no	—
414	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
415	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
416	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
417	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
421	#107	—	2	2	no	—
422	#107	C₂×C₄×C₄₁	1	4	yes	complemented
423	#106	—	2	2	no	—
424	#103	—	2	2	no	—
425	#103	—	2	2	no	—
426	#103	C₂×C₄×C₄₁	1	4	yes	complemented
427	#102	—	2	2	no	—
428	#102	—	2	2	no	—
429	#102	C₂×C₄×C₄₁	1	4	yes	—
430	#97	—	2	2	no	—
431	#97	C₂×C₄×C₄₁	1	4	yes	complemented
432	#95	—	2	1	no	—
433	#95	—	2	1	no	—
434	#95	—	2	1	no	—
435	#95	—	2	1	no	—
436	#95	—	2	1	no	—
437	#95	—	2	1	no	—
438	#95	—	2	1	no	—
439	#95	—	2	1	no	—
440	#95	—	2	1	no	—
441	#95	—	2	1	no	—
442	C₂⁴	—	2	1	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
453	C₂×D₄	#41×C₄₁	1	2	yes	—
454	C₂×D₄	#41×C₄₁	1	2	yes	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	4	2	no	—
466	C₂²×C₄	—	4	2	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	—	4	2	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
477	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
478	SD₁₆	—	2	8	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	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
490	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	4	2	no	—
501	C₄⋊C₄	—	4	2	no	—
502	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
503	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
504	C₄⋊C₄	#41×C₄₁	1	2	yes	—
505	C₄⋊C₄	#41×C₄₁	1	2	yes	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₄²	—	2	4	no	—
510	C₄²	—	2	2	no	—
511	C₄²	—	2	2	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	—	2	1	no	—
516	C₄²	—	4	1	no	—
517	C₄²	—	4	1	no	—
518	C₄²	—	4	2	no	—
519	C₄²	—	4	2	no	—
520	C₄²	—	4	2	no	—
521	C₄²	—	4	2	no	—
522	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
523	C₄²	#41×C₄₁	1	2	yes	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	4	1	no	—
528	C₂³	—	4	1	no	—
529	C₂³	—	4	1	no	—
530	C₂³	—	4	1	no	—
531	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
532	Q₈	—	2	2	no	—
533	Q₈	—	2	2	no	—
534	Q₈	—	4	1	no	—
535	Q₈	—	4	1	no	—
536	D₄	—	2	2	no	—
537	D₄	—	2	2	no	—
538	D₄	—	2	2	no	—
539	D₄	#117×C₄₁	1	4	yes	—
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	—	2	1	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	4	8	no	—
560	C₂×C₄	—	4	8	no	—
561	C₂×C₄	—	4	8	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
579	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
580	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
581	C₂×C₄	#137×C₄₁	1	2	yes	—
582	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
583	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
584	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
585	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
586	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
587	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
588	C₈	—	2	2	no	—
589	C₈	—	2	2	no	—
590	C₈	—	4	1	no	—
591	C₈	—	4	1	no	—
592	C₂²	—	2	4	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	4	2	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	4	2	no	—
599	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
600	C₂²	C₄₁×#525	1	1	yes	characteristic
601	C₂²	C₄₁×#525	1	1	yes	characteristic
602	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
603	C₂²	C₂×#137×C₄₁	1	1	yes	characteristic
604	C₂²	C₄₁×#575	1	1	yes	characteristic
605	C₂²	C₄₁×#575	1	1	yes	characteristic
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	2	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	2	no	—
612	C₄	—	4	1	no	—
613	C₄	—	4	2	no	—
614	C₄	—	4	1	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	2	no	—
617	C₄	—	8	1	no	—
618	C₄	—	8	1	no	—
619	C₄	C₂×#117×C₄₁	1	1	yes	characteristic
620	C₄	C₄×#41×C₄₁	1	1	yes	characteristic
621	C₄	C₄₁×#529	1	1	yes	characteristic
622	C₄	C₄₁×#529	1	1	yes	characteristic
623	C₂	—	4	4	no	—
624	C₂	C₄₁×#2290	1	1	yes	characteristic
625	C₂	C₄×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
626	C₂	C₄×#137×C₄₁	1	1	yes	characteristic
627	C₂	C₄₁×#3110	1	1	yes	characteristic
628	C₂	C₄₁×#3111	1	1	yes	characteristic
629	C₂	C₄₁×#3114	1	1	yes	characteristic
630	C₂	C₄₁×#3116	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	C₂²×D₄×C₄₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2295	C₂	semidirect
4. C₄₁×#1928	C₂	semidirect
5. C₄₁×#1927	C₂	semidirect
6. C₄₁×#1921	C₂	semidirect
7. C₄₁×#1918	C₂	semidirect
12. C₄₁×#1776	C₂	semidirect
13. C₄₁×#1763	C₂	semidirect
15. C₄₁×#1745	C₂	semidirect
22. C₄₁×#440	C₄	semidirect
73. C₄×Q₈×C₄₁	C₂×C₄	semidirect
106. #107×C₄₁	C₂×C₄	semidirect
110. #103×C₄₁	C₂×C₄	semidirect
115. #97×C₄₁	C₂×C₄	semidirect
276. #21121	C₄₁	direct (N × H)
309. #2295	C₂×C₄₁	semidirect
311. #1928	C₂×C₄₁	semidirect
312. #1927	C₂×C₄₁	semidirect
313. #1921	C₂×C₄₁	semidirect
314. #1918	C₂×C₄₁	semidirect
319. #1776	C₂×C₄₁	semidirect
320. #1763	C₂×C₄₁	semidirect
322. #1745	C₂×C₄₁	semidirect
337. #440	C₄×C₄₁	semidirect
375. C₄₁	#21121	direct (N × H)
389. C₄×Q₈	C₂×C₄×C₄₁	semidirect
422. #107	C₂×C₄×C₄₁	semidirect
426. #103	C₂×C₄×C₄₁	semidirect
431. #97	C₂×C₄×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×#440	2
C₄₁×#2295	2
C₄²×D₄×C₄₁	2
C₄₁×#1928	2
C₄₁×#1927	2
C₄₁×#1921	2
C₄₁×#1918	2
C₄₁×#1877	2
C₄₁×#1864	2
C₄₁×#1862	2
C₄₁×#1776	2
C₄₁×#1763	2
C₄₁×#1762	2
C₄₁×#1745	2
#21121	41

Subgroups — C41×#21121 #23797322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21121 #23797322