Subgroups — C₄₁×#20971 #23797172

Order 10496.

1986 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#525	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#417	C₂	1	2	yes	maximal, complemented
4	C₂×C₄₁×#380	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1895	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1894	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1893	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1892	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1736	C₂	1	2	yes	maximal, complemented
10	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₄₁	—	2	4	no	—
13	C₂²×M₁₆×C₄₁	—	2	8	no	—
14	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
16	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
17	C₄₁×#525	—	2	4	no	—
18	C₄₁×#525	C₄	1	8	yes	complemented
19	C₂×#123×C₄₁	—	2	1	no	—
20	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×#121×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×#121×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#120×C₄₁	—	2	1	no	—
25	C₂×C₄×D₄×C₄₁	—	2	1	no	—
26	C₂×C₄×D₄×C₄₁	—	2	1	no	—
27	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
28	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₄₁	—	2	1	no	—
30	C₄₁×#417	—	2	8	no	—
31	C₂×#103×C₄₁	C₂²	1	4	yes	complemented
32	C₂×#102×C₄₁	C₂²	1	4	yes	complemented
33	C₄₁×#380	—	2	1	no	—
34	C₄₁×#380	—	2	1	no	—
35	C₄₁×#380	C₂²	1	2	yes	—
36	C₄₁×#380	C₂²	1	2	yes	—
37	C₂×#95×C₄₁	—	2	1	no	—
38	C₄₁×#340	C₂²	1	4	yes	complemented
39	C₄₁×#340	C₂²	1	4	yes	complemented
40	C₂⁵×C₄₁	D₄	1	1	yes	characteristic
41	C₂×#41×C₄₁	—	2	2	no	—
42	C₂×#41×C₄₁	—	2	4	no	—
43	C₂×#41×C₄₁	—	2	4	no	—
44	C₂×#41×C₄₁	C₂×C₄	1	4	yes	—
45	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₄₁	—	2	1	no	—
47	C₂²×D₄×C₄₁	—	2	1	no	—
48	C₂²×D₄×C₄₁	—	2	1	no	—
49	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₄₁	—	2	4	no	—
51	C₂³×C₄×C₄₁	—	2	4	no	—
52	C₂³×C₄×C₄₁	—	2	1	no	—
53	C₂³×C₄×C₄₁	—	2	2	no	—
54	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
55	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
57	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₄₁	D₄	1	4	yes	—
59	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
62	C₂×M₁₆×C₄₁	—	2	32	no	—
63	C₂×M₁₆×C₄₁	—	2	4	no	—
64	C₂×M₁₆×C₄₁	—	2	4	no	—
65	C₂×M₁₆×C₄₁	—	2	4	no	—
66	C₂×M₁₆×C₄₁	—	2	8	no	—
67	C₂×M₁₆×C₄₁	D₄	1	8	yes	complemented
68	C₂²×C₈×C₄₁	—	2	4	no	—
69	C₂²×C₈×C₄₁	—	2	8	no	—
70	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
71	#123×C₄₁	—	2	8	no	—
72	C₂²⋊Q₈×C₄₁	—	2	2	no	—
73	C₂²⋊Q₈×C₄₁	—	2	2	no	—
74	C₂²⋊Q₈×C₄₁	C₂×C₄	1	4	yes	complemented
75	C₂²⋊Q₈×C₄₁	C₂×C₄	1	4	yes	complemented
76	#121×C₄₁	—	2	2	no	—
77	#121×C₄₁	—	2	2	no	—
78	#121×C₄₁	C₂×C₄	1	4	yes	complemented
79	#121×C₄₁	C₂×C₄	1	4	yes	complemented
80	#120×C₄₁	—	2	8	no	—
81	C₄×D₄×C₄₁	—	2	8	no	—
82	C₄×D₄×C₄₁	—	2	8	no	—
83	#117×C₄₁	—	2	2	no	—
84	#117×C₄₁	C₂×C₄	1	4	yes	—
85	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
88	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
89	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
90	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
100	C₂×C₂²⋊C₄×C₄₁	—	4	4	no	—
101	C₂×C₄²×C₄₁	—	2	1	no	—
102	#103×C₄₁	—	2	8	no	—
103	#102×C₄₁	—	2	8	no	—
104	#95×C₄₁	—	2	2	no	—
105	#95×C₄₁	—	2	2	no	—
106	C₂⁴×C₄₁	—	2	4	no	—
107	C₂⁴×C₄₁	—	2	2	no	—
108	C₂⁴×C₄₁	—	2	2	no	—
109	C₂⁴×C₄₁	—	2	1	no	—
110	C₂⁴×C₄₁	—	4	2	no	—
111	C₂⁴×C₄₁	C₂²⋊C₄	1	4	yes	—
112	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
113	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
114	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
115	#41×C₄₁	—	2	16	no	—
116	#41×C₄₁	—	4	4	no	—
117	#41×C₄₁	—	4	4	no	—
118	C₂×Q₈×C₄₁	—	2	1	no	—
119	C₂×Q₈×C₄₁	—	2	2	no	—
120	C₂×Q₈×C₄₁	—	2	2	no	—
121	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
122	C₂×D₄×C₄₁	—	2	4	no	—
123	C₂×D₄×C₄₁	—	2	4	no	—
124	C₂×D₄×C₄₁	—	2	4	no	—
125	C₂×D₄×C₄₁	—	2	2	no	—
126	C₂×D₄×C₄₁	—	2	1	no	—
127	C₂×D₄×C₄₁	—	2	2	no	—
128	C₂×D₄×C₄₁	—	4	4	no	—
129	C₂×D₄×C₄₁	—	4	4	no	—
130	C₂×D₄×C₄₁	—	4	2	no	—
131	C₂×D₄×C₄₁	—	4	2	no	—
132	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
133	C₂²×C₄×C₄₁	—	2	4	no	—
134	C₂²×C₄×C₄₁	—	2	16	no	—
135	C₂²×C₄×C₄₁	—	2	8	no	—
136	C₂²×C₄×C₄₁	—	2	4	no	—
137	C₂²×C₄×C₄₁	—	2	4	no	—
138	C₂²×C₄×C₄₁	—	2	8	no	—
139	C₂²×C₄×C₄₁	—	2	2	no	—
140	C₂²×C₄×C₄₁	—	2	2	no	—
141	C₂²×C₄×C₄₁	—	2	2	no	—
142	C₂²×C₄×C₄₁	—	2	2	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	4	no	—
148	C₂²×C₄×C₄₁	—	4	4	no	—
149	C₂²×C₄×C₄₁	—	4	4	no	—
150	C₂²×C₄×C₄₁	—	4	4	no	—
151	C₂²×C₄×C₄₁	—	4	4	no	—
152	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
153	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
154	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
155	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
156	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
157	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
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	2	yes	—
161	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
165	C₂²×C₄×C₄₁	#41	1	4	yes	—
166	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
167	M₁₆×C₄₁	—	2	16	no	—
168	M₁₆×C₄₁	—	2	16	no	—
169	C₂×C₈×C₄₁	—	2	16	no	—
170	C₂×C₈×C₄₁	—	2	4	no	—
171	C₂×C₈×C₄₁	—	2	4	no	—
172	C₂×C₈×C₄₁	C₂×D₄	1	8	yes	—
173	C₄⋊C₄×C₄₁	—	2	4	no	—
174	C₄⋊C₄×C₄₁	—	2	4	no	—
175	C₄⋊C₄×C₄₁	—	2	1	no	—
176	C₄⋊C₄×C₄₁	—	2	1	no	—
177	C₄⋊C₄×C₄₁	—	2	4	no	—
178	C₄⋊C₄×C₄₁	—	2	4	no	—
179	C₄⋊C₄×C₄₁	—	4	4	no	—
180	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
181	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
182	C₂²⋊C₄×C₄₁	—	2	4	no	—
183	C₂²⋊C₄×C₄₁	—	2	4	no	—
184	C₂²⋊C₄×C₄₁	—	2	4	no	—
185	C₂²⋊C₄×C₄₁	—	2	4	no	—
186	C₂²⋊C₄×C₄₁	—	2	4	no	—
187	C₂²⋊C₄×C₄₁	—	4	4	no	—
188	C₂²⋊C₄×C₄₁	—	4	4	no	—
189	C₄²×C₄₁	—	2	4	no	—
190	C₂³×C₄₁	—	2	2	no	—
191	C₂³×C₄₁	—	2	1	no	—
192	C₂³×C₄₁	—	2	1	no	—
193	C₂³×C₄₁	—	2	2	no	—
194	C₂³×C₄₁	—	2	2	no	—
195	C₂³×C₄₁	—	2	2	no	—
196	C₂³×C₄₁	—	2	1	no	—
197	C₂³×C₄₁	—	2	1	no	—
198	C₂³×C₄₁	—	2	2	no	—
199	C₂³×C₄₁	—	2	8	no	—
200	C₂³×C₄₁	—	2	8	no	—
201	C₂³×C₄₁	—	2	2	no	—
202	C₂³×C₄₁	—	4	4	no	—
203	C₂³×C₄₁	—	4	8	no	—
204	C₂³×C₄₁	—	4	8	no	—
205	C₂³×C₄₁	—	4	2	no	—
206	C₂³×C₄₁	—	4	1	no	—
207	C₂³×C₄₁	—	4	1	no	—
208	C₂³×C₄₁	—	4	2	no	—
209	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
210	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
211	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
212	C₂³×C₄₁	#121	1	4	yes	—
213	C₂³×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
214	C₂³×C₄₁	#137	1	2	yes	—
215	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
216	Q₈×C₄₁	—	2	4	no	—
217	Q₈×C₄₁	—	4	1	no	—
218	Q₈×C₄₁	—	4	1	no	—
219	D₄×C₄₁	—	2	4	no	—
220	D₄×C₄₁	—	4	8	no	—
221	D₄×C₄₁	—	4	8	no	—
222	D₄×C₄₁	—	4	8	no	—
223	D₄×C₄₁	—	4	1	no	—
224	D₄×C₄₁	—	4	1	no	—
225	C₂×C₄×C₄₁	—	2	2	no	—
226	C₂×C₄×C₄₁	—	2	2	no	—
227	C₂×C₄×C₄₁	—	2	8	no	—
228	C₂×C₄×C₄₁	—	2	4	no	—
229	C₂×C₄×C₄₁	—	2	4	no	—
230	C₂×C₄×C₄₁	—	2	2	no	—
231	C₂×C₄×C₄₁	—	2	2	no	—
232	C₂×C₄×C₄₁	—	2	2	no	—
233	C₂×C₄×C₄₁	—	2	2	no	—
234	C₂×C₄×C₄₁	—	2	8	no	—
235	C₂×C₄×C₄₁	—	2	8	no	—
236	C₂×C₄×C₄₁	—	2	2	no	—
237	C₂×C₄×C₄₁	—	2	2	no	—
238	C₂×C₄×C₄₁	—	2	2	no	—
239	C₂×C₄×C₄₁	—	4	2	no	—
240	C₂×C₄×C₄₁	—	4	1	no	—
241	C₂×C₄×C₄₁	—	4	1	no	—
242	C₂×C₄×C₄₁	—	4	2	no	—
243	C₂×C₄×C₄₁	—	4	8	no	—
244	C₂×C₄×C₄₁	—	4	4	no	—
245	C₂×C₄×C₄₁	—	4	4	no	—
246	C₂×C₄×C₄₁	—	4	4	no	—
247	C₂×C₄×C₄₁	—	8	8	no	—
248	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
249	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
250	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
251	C₂×C₄×C₄₁	C₄×D₄	1	8	yes	—
252	C₂×C₄×C₄₁	#120	1	4	yes	—
253	C₂×C₄×C₄₁	#121	1	4	yes	—
254	C₂×C₄×C₄₁	#123	1	4	yes	—
255	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
256	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
259	C₈×C₄₁	—	2	8	no	—
260	#20971	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
261	C₂²×C₄₁	—	2	2	no	—
262	C₂²×C₄₁	—	2	2	no	—
263	C₂²×C₄₁	—	2	2	no	—
264	C₂²×C₄₁	—	2	2	no	—
265	C₂²×C₄₁	—	2	4	no	—
266	C₂²×C₄₁	—	2	4	no	—
267	C₂²×C₄₁	—	2	4	no	—
268	C₂²×C₄₁	—	2	4	no	—
269	C₂²×C₄₁	—	2	2	no	—
270	C₂²×C₄₁	—	2	2	no	—
271	C₂²×C₄₁	—	2	2	no	—
272	C₂²×C₄₁	—	4	16	no	—
273	C₂²×C₄₁	—	4	2	no	—
274	C₂²×C₄₁	—	4	2	no	—
275	C₂²×C₄₁	—	4	2	no	—
276	C₂²×C₄₁	—	4	2	no	—
277	C₂²×C₄₁	—	4	2	no	—
278	C₂²×C₄₁	—	8	2	no	—
279	C₂²×C₄₁	—	8	2	no	—
280	C₂²×C₄₁	#386	1	4	yes	—
281	C₂²×C₄₁	#417	1	4	yes	complemented
282	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
283	C₂²×C₄₁	C₂×#121	1	2	yes	—
284	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
285	C₂²×C₄₁	C₂×#137	1	1	yes	characteristic
286	C₂²×C₄₁	#575	1	2	yes	—
287	C₄×C₄₁	—	2	2	no	—
288	C₄×C₄₁	—	2	2	no	—
289	C₄×C₄₁	—	4	2	no	—
290	C₄×C₄₁	—	4	2	no	—
291	C₄×C₄₁	—	4	2	no	—
292	C₄×C₄₁	—	8	4	no	—
293	C₄×C₄₁	#386	1	4	yes	—
294	C₄×C₄₁	C₂×C₄×D₄	1	2	yes	—
295	C₄×C₄₁	C₂×#120	1	1	yes	characteristic
296	C₄×C₄₁	C₂×#123	1	1	yes	characteristic
297	C₂³×M₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
298	C₂×#525	C₂×C₄₁	1	1	yes	characteristic
299	C₂×#417	C₂×C₄₁	1	2	yes	complemented
300	C₂×#380	C₂×C₄₁	1	1	yes	characteristic, complemented
301	#1895	C₂×C₄₁	1	2	yes	complemented
302	#1894	C₂×C₄₁	1	2	yes	complemented
303	#1893	C₂×C₄₁	1	2	yes	complemented
304	#1892	C₂×C₄₁	1	2	yes	complemented
305	#1736	C₂×C₄₁	1	2	yes	complemented
306	C₂×C₄₁	—	2	4	no	—
307	C₂×C₄₁	—	2	4	no	—
308	C₂×C₄₁	—	4	2	no	—
309	C₂×C₄₁	—	8	2	no	—
310	C₂×C₄₁	C₂×#386	1	1	yes	characteristic
311	C₂×C₄₁	C₂×#417	1	1	yes	characteristic
312	C₂×C₄₁	#2896	1	1	yes	characteristic
313	C₂×C₄₁	#3055	1	2	yes	—
314	C₂×C₄₁	#3056	1	2	yes	—
315	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
316	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
317	C₂²×M₁₆	—	2	4	no	—
318	C₂²×M₁₆	—	2	8	no	—
319	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
320	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
321	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
322	#525	—	2	4	no	—
323	#525	C₄×C₄₁	1	8	yes	complemented
324	C₂×#123	—	2	1	no	—
325	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
326	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
327	C₂×#121	C₂²×C₄₁	1	1	yes	characteristic
328	C₂×#121	C₂²×C₄₁	1	1	yes	characteristic
329	C₂×#120	—	2	1	no	—
330	C₂×C₄×D₄	—	2	1	no	—
331	C₂×C₄×D₄	—	2	1	no	—
332	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
333	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
334	C₂²×C₂²⋊C₄	—	2	1	no	—
335	#417	—	2	8	no	—
336	C₂×#103	C₂²×C₄₁	1	4	yes	complemented
337	C₂×#102	C₂²×C₄₁	1	4	yes	complemented
338	#380	—	2	1	no	—
339	#380	—	2	1	no	—
340	#380	C₂²×C₄₁	1	2	yes	—
341	#380	C₂²×C₄₁	1	2	yes	—
342	C₂×#95	—	2	1	no	—
343	#340	C₂²×C₄₁	1	4	yes	complemented
344	#340	C₂²×C₄₁	1	4	yes	complemented
345	C₄₁	#20971	1	1	yes	characteristic, direct product factor, Sylow
346	C₂⁵	D₄×C₄₁	1	1	yes	characteristic
347	C₂×#41	—	2	2	no	—
348	C₂×#41	—	2	4	no	—
349	C₂×#41	—	2	4	no	—
350	C₂×#41	C₂×C₄×C₄₁	1	4	yes	—
351	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
352	C₂²×D₄	—	2	1	no	—
353	C₂²×D₄	—	2	1	no	—
354	C₂²×D₄	—	2	1	no	—
355	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
356	C₂³×C₄	—	2	4	no	—
357	C₂³×C₄	—	2	4	no	—
358	C₂³×C₄	—	2	1	no	—
359	C₂³×C₄	—	2	2	no	—
360	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
361	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
362	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
363	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
364	C₂³×C₄	D₄×C₄₁	1	4	yes	—
365	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
366	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
367	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
368	C₂×M₁₆	—	2	32	no	—
369	C₂×M₁₆	—	2	4	no	—
370	C₂×M₁₆	—	2	4	no	—
371	C₂×M₁₆	—	2	4	no	—
372	C₂×M₁₆	—	2	8	no	—
373	C₂×M₁₆	D₄×C₄₁	1	8	yes	complemented
374	C₂²×C₈	—	2	4	no	—
375	C₂²×C₈	—	2	8	no	—
376	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
377	#123	—	2	8	no	—
378	C₂²⋊Q₈	—	2	2	no	—
379	C₂²⋊Q₈	—	2	2	no	—
380	C₂²⋊Q₈	C₂×C₄×C₄₁	1	4	yes	complemented
381	C₂²⋊Q₈	C₂×C₄×C₄₁	1	4	yes	complemented
382	#121	—	2	2	no	—
383	#121	—	2	2	no	—
384	#121	C₂×C₄×C₄₁	1	4	yes	complemented
385	#121	C₂×C₄×C₄₁	1	4	yes	complemented
386	#120	—	2	8	no	—
387	C₄×D₄	—	2	8	no	—
388	C₄×D₄	—	2	8	no	—
389	#117	—	2	2	no	—
390	#117	C₂×C₄×C₄₁	1	4	yes	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	C₂×C₄⋊C₄	—	2	4	no	—
394	C₂×C₄⋊C₄	—	2	4	no	—
395	C₂×C₄⋊C₄	—	2	2	no	—
396	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
397	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
398	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
399	C₂×C₂²⋊C₄	—	2	1	no	—
400	C₂×C₂²⋊C₄	—	2	1	no	—
401	C₂×C₂²⋊C₄	—	2	1	no	—
402	C₂×C₂²⋊C₄	—	2	1	no	—
403	C₂×C₂²⋊C₄	—	2	1	no	—
404	C₂×C₂²⋊C₄	—	2	2	no	—
405	C₂×C₂²⋊C₄	—	2	2	no	—
406	C₂×C₂²⋊C₄	—	4	4	no	—
407	C₂×C₄²	—	2	1	no	—
408	#103	—	2	8	no	—
409	#102	—	2	8	no	—
410	#95	—	2	2	no	—
411	#95	—	2	2	no	—
412	C₂⁴	—	2	4	no	—
413	C₂⁴	—	2	2	no	—
414	C₂⁴	—	2	2	no	—
415	C₂⁴	—	2	1	no	—
416	C₂⁴	—	4	2	no	—
417	C₂⁴	C₂²⋊C₄×C₄₁	1	4	yes	—
418	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
419	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
420	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
421	#41	—	2	16	no	—
422	#41	—	4	4	no	—
423	#41	—	4	4	no	—
424	C₂×Q₈	—	2	1	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	2	no	—
427	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
428	C₂×D₄	—	2	4	no	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	4	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	4	4	no	—
435	C₂×D₄	—	4	4	no	—
436	C₂×D₄	—	4	2	no	—
437	C₂×D₄	—	4	2	no	—
438	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
439	C₂²×C₄	—	2	4	no	—
440	C₂²×C₄	—	2	16	no	—
441	C₂²×C₄	—	2	8	no	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	8	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	4	no	—
454	C₂²×C₄	—	4	4	no	—
455	C₂²×C₄	—	4	4	no	—
456	C₂²×C₄	—	4	4	no	—
457	C₂²×C₄	—	4	4	no	—
458	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
459	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
460	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
461	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
462	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
463	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
464	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
465	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
468	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
470	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
471	C₂²×C₄	#41×C₄₁	1	4	yes	—
472	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
473	M₁₆	—	2	16	no	—
474	M₁₆	—	2	16	no	—
475	C₂×C₈	—	2	16	no	—
476	C₂×C₈	—	2	4	no	—
477	C₂×C₈	—	2	4	no	—
478	C₂×C₈	C₂×D₄×C₄₁	1	8	yes	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	4	4	no	—
486	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
487	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
488	C₂²⋊C₄	—	2	4	no	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	2	4	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	4	4	no	—
494	C₂²⋊C₄	—	4	4	no	—
495	C₄²	—	2	4	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	1	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	2	2	no	—
505	C₂³	—	2	8	no	—
506	C₂³	—	2	8	no	—
507	C₂³	—	2	2	no	—
508	C₂³	—	4	4	no	—
509	C₂³	—	4	8	no	—
510	C₂³	—	4	8	no	—
511	C₂³	—	4	2	no	—
512	C₂³	—	4	1	no	—
513	C₂³	—	4	1	no	—
514	C₂³	—	4	2	no	—
515	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
516	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
517	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
518	C₂³	#121×C₄₁	1	4	yes	—
519	C₂³	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
520	C₂³	#137×C₄₁	1	2	yes	—
521	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
522	Q₈	—	2	4	no	—
523	Q₈	—	4	1	no	—
524	Q₈	—	4	1	no	—
525	D₄	—	2	4	no	—
526	D₄	—	4	8	no	—
527	D₄	—	4	8	no	—
528	D₄	—	4	8	no	—
529	D₄	—	4	1	no	—
530	D₄	—	4	1	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	8	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	8	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	8	no	—
550	C₂×C₄	—	4	4	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	4	4	no	—
553	C₂×C₄	—	8	8	no	—
554	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
555	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
556	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
557	C₂×C₄	C₄×D₄×C₄₁	1	8	yes	—
558	C₂×C₄	#120×C₄₁	1	4	yes	—
559	C₂×C₄	#121×C₄₁	1	4	yes	—
560	C₂×C₄	#123×C₄₁	1	4	yes	—
561	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
562	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
563	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
564	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
565	C₈	—	2	8	no	—
566	C₂²	—	2	2	no	—
567	C₂²	—	2	4	no	—
568	C₂²	—	2	4	no	—
569	C₂²	—	2	4	no	—
570	C₂²	—	2	4	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	4	16	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	4	2	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	8	2	no	—
584	C₂²	—	8	2	no	—
585	C₂²	C₄₁×#386	1	4	yes	—
586	C₂²	C₄₁×#417	1	4	yes	complemented
587	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
588	C₂²	C₂×#121×C₄₁	1	2	yes	—
589	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
590	C₂²	C₂×#137×C₄₁	1	1	yes	characteristic
591	C₂²	C₄₁×#575	1	2	yes	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	2	no	—
597	C₄	—	8	4	no	—
598	C₄	C₄₁×#386	1	4	yes	—
599	C₄	C₂×C₄×D₄×C₄₁	1	2	yes	—
600	C₄	C₂×#120×C₄₁	1	1	yes	characteristic
601	C₄	C₂×#123×C₄₁	1	1	yes	characteristic
602	C₂	—	2	4	no	—
603	C₂	—	2	4	no	—
604	C₂	—	4	2	no	—
605	C₂	—	8	2	no	—
606	C₂	C₂×C₄₁×#386	1	1	yes	characteristic
607	C₂	C₂×C₄₁×#417	1	1	yes	characteristic
608	C₂	C₄₁×#2896	1	1	yes	characteristic
609	C₂	C₄₁×#3055	1	2	yes	—
610	C₂	C₄₁×#3056	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂⁴×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×M₁₆×C₄₁	C₂	semidirect
3. C₂×C₄₁×#417	C₂	semidirect
4. C₂×C₄₁×#380	C₂	semidirect
5. C₄₁×#1895	C₂	semidirect
6. C₄₁×#1894	C₂	semidirect
7. C₄₁×#1893	C₂	semidirect
8. C₄₁×#1892	C₂	semidirect
9. C₄₁×#1736	C₂	semidirect
18. C₄₁×#525	C₄	semidirect
31. C₂×#103×C₄₁	C₂²	semidirect
32. C₂×#102×C₄₁	C₂²	semidirect
38. C₄₁×#340	C₂²	semidirect
39. C₄₁×#340	C₂²	semidirect
67. C₂×M₁₆×C₄₁	D₄	semidirect
74. C₂²⋊Q₈×C₄₁	C₂×C₄	semidirect
75. C₂²⋊Q₈×C₄₁	C₂×C₄	semidirect
78. #121×C₄₁	C₂×C₄	semidirect
79. #121×C₄₁	C₂×C₄	semidirect
260. #20971	C₄₁	direct (N × H)
281. C₂²×C₄₁	#417	semidirect
297. C₂³×M₁₆	C₂×C₄₁	semidirect
299. C₂×#417	C₂×C₄₁	semidirect
300. C₂×#380	C₂×C₄₁	semidirect
301. #1895	C₂×C₄₁	semidirect
302. #1894	C₂×C₄₁	semidirect
303. #1893	C₂×C₄₁	semidirect
304. #1892	C₂×C₄₁	semidirect
305. #1736	C₂×C₄₁	semidirect
323. #525	C₄×C₄₁	semidirect
336. C₂×#103	C₂²×C₄₁	semidirect
337. C₂×#102	C₂²×C₄₁	semidirect
343. #340	C₂²×C₄₁	semidirect
344. #340	C₂²×C₄₁	semidirect
345. C₄₁	#20971	direct (N × H)
373. C₂×M₁₆	D₄×C₄₁	semidirect
380. C₂²⋊Q₈	C₂×C₄×C₄₁	semidirect
381. C₂²⋊Q₈	C₂×C₄×C₄₁	semidirect
384. #121	C₂×C₄×C₄₁	semidirect
385. #121	C₂×C₄×C₄₁	semidirect
586. C₂²	C₄₁×#417	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₄₁	2
C₂×C₄₁×#525	2
C₂×C₄₁×#417	2
C₂×C₄₁×#380	2
C₄₁×#1895	2
C₄₁×#1894	2
C₄₁×#1893	2
C₄₁×#1892	2
C₄₁×#1736	2
#20971	41

Subgroups — C41×#20971 #23797172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20971 #23797172