Subgroups — C₄₁×#24854 #23801055

Order 10496.

788 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#479	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#424	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2629	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2323	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#2088	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2087	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#2066	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#2066	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1989	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1894	C₂	1	1	yes	maximal, characteristic
12	C₄₁×#1892	C₂	1	1	yes	maximal, characteristic
13	C₄₁×#1809	C₂	1	1	yes	maximal, characteristic
14	C₄₁×#1807	C₂	1	1	yes	maximal, characteristic
15	C₄₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×#125×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
22	C₄₁×#479	—	2	4	no	—
23	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#424	—	2	4	no	—
28	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×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	C₄₁×#398	—	2	1	no	—
37	C₄₁×#398	—	2	1	no	—
38	C₄₁×#396	—	2	1	no	—
39	C₄₁×#396	—	2	1	no	—
40	C₄₁×#391	C₂²	1	1	yes	characteristic
41	C₄₁×#389	—	2	1	no	—
42	C₄₁×#389	C₂²	1	1	yes	characteristic
43	C₄₁×#389	C₂²	1	1	yes	characteristic
44	C₄₁×#389	C₂²	1	1	yes	characteristic
45	C₄₁×#388	—	2	1	no	—
46	C₄₁×#388	—	2	1	no	—
47	C₄₁×#387	—	2	1	no	—
48	C₄₁×#384	C₂²	1	1	yes	characteristic
49	C₄₁×#380	C₂²	1	1	yes	characteristic
50	C₄₁×#380	C₂²	1	1	yes	characteristic
51	C₄₁×#380	C₂²	1	1	yes	characteristic
52	C₄×C₄⋊C₄×C₄₁	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×#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×#340	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
63	C₂×M₁₆×C₄₁	—	2	2	no	—
64	C₂×M₁₆×C₄₁	—	2	2	no	—
65	C₂×M₁₆×C₄₁	—	4	1	no	—
66	C₂²×C₈×C₄₁	—	2	2	no	—
67	C₂²×C₈×C₄₁	—	2	2	no	—
68	C₂²×C₈×C₄₁	—	4	1	no	—
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	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
73	#125×C₄₁	—	2	2	no	—
74	#125×C₄₁	—	2	2	no	—
75	C₂²⋊Q₈×C₄₁	—	2	2	no	—
76	C₂²⋊Q₈×C₄₁	Q₈	1	4	yes	—
77	C₄×Q₈×C₄₁	—	2	2	no	—
78	C₄×Q₈×C₄₁	—	2	2	no	—
79	#117×C₄₁	—	2	2	no	—
80	#117×C₄₁	—	2	2	no	—
81	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
88	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
89	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
90	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₄₁	D₄	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₄₁	D₄	1	1	yes	characteristic
105	C₂×C₄²×C₄₁	—	2	1	no	—
106	C₂×C₄²×C₄₁	—	2	1	no	—
107	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
109	#107×C₄₁	—	2	2	no	—
110	#106×C₄₁	—	2	2	no	—
111	#106×C₄₁	—	4	1	no	—
112	#106×C₄₁	—	4	1	no	—
113	C₄⋊C₈×C₄₁	—	2	2	no	—
114	C₄⋊C₈×C₄₁	—	2	2	no	—
115	C₄⋊C₈×C₄₁	—	2	2	no	—
116	C₄⋊C₈×C₄₁	D₄	1	4	yes	—
117	#103×C₄₁	—	2	2	no	—
118	#103×C₄₁	—	2	2	no	—
119	#103×C₄₁	—	4	1	no	—
120	#103×C₄₁	—	4	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	#95×C₄₁	—	2	1	no	—
127	#95×C₄₁	—	2	1	no	—
128	#95×C₄₁	—	2	1	no	—
129	#95×C₄₁	—	2	1	no	—
130	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
131	C₂×Q₈×C₄₁	—	2	1	no	—
132	C₂×Q₈×C₄₁	—	2	1	no	—
133	C₂×Q₈×C₄₁	—	2	1	no	—
134	C₂×Q₈×C₄₁	—	2	1	no	—
135	C₂×Q₈×C₄₁	—	2	1	no	—
136	C₂×Q₈×C₄₁	C₂×Q₈	1	2	yes	—
137	C₂²×C₄×C₄₁	—	2	1	no	—
138	C₂²×C₄×C₄₁	—	2	1	no	—
139	C₂²×C₄×C₄₁	—	2	1	no	—
140	C₂²×C₄×C₄₁	—	2	1	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	1	no	—
147	C₂²×C₄×C₄₁	—	2	1	no	—
148	C₂²×C₄×C₄₁	—	2	1	no	—
149	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
155	C₂²×C₄×C₄₁	#41	1	2	yes	—
156	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
157	M₁₆×C₄₁	—	4	4	no	—
158	C₂×C₈×C₄₁	—	2	1	no	—
159	C₂×C₈×C₄₁	—	2	1	no	—
160	C₂×C₈×C₄₁	—	2	1	no	—
161	C₂×C₈×C₄₁	—	2	1	no	—
162	C₂×C₈×C₄₁	—	2	1	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	—	4	4	no	—
169	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
170	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
171	C₄⋊C₄×C₄₁	—	2	2	no	—
172	C₄⋊C₄×C₄₁	—	2	2	no	—
173	C₄⋊C₄×C₄₁	—	2	4	no	—
174	C₄⋊C₄×C₄₁	—	2	1	no	—
175	C₄⋊C₄×C₄₁	—	2	1	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	—	2	1	no	—
185	C₄⋊C₄×C₄₁	—	2	1	no	—
186	C₄⋊C₄×C₄₁	—	2	1	no	—
187	C₄⋊C₄×C₄₁	—	4	2	no	—
188	C₄⋊C₄×C₄₁	—	4	2	no	—
189	C₄⋊C₄×C₄₁	—	4	2	no	—
190	C₄⋊C₄×C₄₁	—	8	1	no	—
191	C₄⋊C₄×C₄₁	—	8	1	no	—
192	C₄⋊C₄×C₄₁	—	8	1	no	—
193	C₄⋊C₄×C₄₁	—	8	1	no	—
194	C₄⋊C₄×C₄₁	—	8	1	no	—
195	C₄⋊C₄×C₄₁	C₂×Q₈	1	2	yes	—
196	C₂²⋊C₄×C₄₁	—	2	4	no	—
197	C₂²⋊C₄×C₄₁	—	2	2	no	—
198	C₂²⋊C₄×C₄₁	—	2	2	no	—
199	C₂²⋊C₄×C₄₁	—	8	1	no	—
200	C₂²⋊C₄×C₄₁	—	8	1	no	—
201	C₂²⋊C₄×C₄₁	—	8	1	no	—
202	C₄²×C₄₁	—	2	1	no	—
203	C₄²×C₄₁	—	2	1	no	—
204	C₄²×C₄₁	—	4	2	no	—
205	C₄²×C₄₁	—	8	1	no	—
206	C₄²×C₄₁	C₂×D₄	1	2	yes	—
207	C₄²×C₄₁	#41	1	2	yes	—
208	C₂³×C₄₁	—	2	1	no	—
209	C₂³×C₄₁	—	2	1	no	—
210	C₂³×C₄₁	—	2	1	no	—
211	C₂³×C₄₁	—	2	1	no	—
212	C₂³×C₄₁	—	2	1	no	—
213	C₂³×C₄₁	—	2	1	no	—
214	C₂³×C₄₁	C₂²⋊Q₈	1	2	yes	—
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	C₂×C₄×C₄₁	—	2	1	no	—
221	C₂×C₄×C₄₁	—	2	1	no	—
222	C₂×C₄×C₄₁	—	2	1	no	—
223	C₂×C₄×C₄₁	—	2	1	no	—
224	C₂×C₄×C₄₁	—	2	1	no	—
225	C₂×C₄×C₄₁	—	2	1	no	—
226	C₂×C₄×C₄₁	—	2	2	no	—
227	C₂×C₄×C₄₁	—	2	2	no	—
228	C₂×C₄×C₄₁	—	2	2	no	—
229	C₂×C₄×C₄₁	—	2	2	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	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	—	4	1	no	—
239	C₂×C₄×C₄₁	—	4	1	no	—
240	C₂×C₄×C₄₁	—	4	1	no	—
241	C₂×C₄×C₄₁	—	4	1	no	—
242	C₂×C₄×C₄₁	—	4	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	—	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	#121	1	2	yes	—
261	C₂×C₄×C₄₁	#121	1	2	yes	—
262	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
263	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
265	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
266	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
267	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
268	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
269	C₈×C₄₁	—	2	2	no	—
270	C₈×C₄₁	—	2	2	no	—
271	C₈×C₄₁	—	4	1	no	—
272	C₈×C₄₁	—	4	1	no	—
273	#24854	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂²×C₄₁	—	2	2	no	—
275	C₂²×C₄₁	—	2	2	no	—
276	C₂²×C₄₁	—	2	2	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	C₂×#121	1	1	yes	characteristic
282	C₂²×C₄₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
283	C₂²×C₄₁	#537	1	1	yes	characteristic
284	C₂²×C₄₁	#577	1	1	yes	characteristic
285	C₂²×C₄₁	#577	1	1	yes	characteristic
286	C₂²×C₄₁	#577	1	1	yes	characteristic
287	C₂²×C₄₁	#578	1	1	yes	characteristic
288	C₄×C₄₁	—	2	2	no	—
289	C₄×C₄₁	—	2	2	no	—
290	C₄×C₄₁	—	2	2	no	—
291	C₄×C₄₁	—	4	2	no	—
292	C₄×C₄₁	—	4	2	no	—
293	C₄×C₄₁	—	4	1	no	—
294	C₄×C₄₁	—	4	1	no	—
295	C₄×C₄₁	—	8	1	no	—
296	C₄×C₄₁	—	8	1	no	—
297	C₄×C₄₁	—	8	1	no	—
298	C₄×C₄₁	—	8	1	no	—
299	C₄×C₄₁	—	8	1	no	—
300	C₄×C₄₁	—	8	1	no	—
301	C₄×C₄₁	#546	1	1	yes	characteristic
302	C₄×C₄₁	D₄×Q₈	1	1	yes	characteristic
303	C₄×C₄₁	#552	1	1	yes	characteristic
304	C₄×C₄₁	#554	1	1	yes	characteristic
305	C₂×#479	C₂×C₄₁	1	1	yes	characteristic, complemented
306	C₂×#424	C₂×C₄₁	1	1	yes	characteristic
307	#2629	C₂×C₄₁	1	1	yes	characteristic
308	#2323	C₂×C₄₁	1	1	yes	characteristic
309	#2088	C₂×C₄₁	1	1	yes	characteristic, complemented
310	#2087	C₂×C₄₁	1	1	yes	characteristic, complemented
311	#2066	C₂×C₄₁	1	1	yes	characteristic, complemented
312	#2066	C₂×C₄₁	1	1	yes	characteristic, complemented
313	#1989	C₂×C₄₁	1	1	yes	characteristic, complemented
314	#1948	C₂×C₄₁	1	1	yes	characteristic, complemented
315	#1894	C₂×C₄₁	1	1	yes	characteristic
316	#1892	C₂×C₄₁	1	1	yes	characteristic
317	#1809	C₂×C₄₁	1	1	yes	characteristic
318	#1807	C₂×C₄₁	1	1	yes	characteristic
319	#1803	C₂×C₄₁	1	1	yes	characteristic, complemented
320	C₂×C₄₁	—	4	2	no	—
321	C₂×C₄₁	#2417	1	1	yes	characteristic
322	C₂×C₄₁	#3060	1	1	yes	characteristic
323	C₂×C₄₁	#3063	1	1	yes	characteristic
324	C₂×C₄₁	#3082	1	1	yes	characteristic
325	C₂×C₄₁	#3084	1	1	yes	characteristic
326	C₂×C₄₁	#3258	1	1	yes	characteristic
327	C₂×C₄₁	#3260	1	1	yes	characteristic
328	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
329	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
330	C₂×#125	C₂²×C₄₁	1	1	yes	characteristic
331	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
332	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
333	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
334	#479	—	2	4	no	—
335	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
336	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
337	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
338	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
339	#424	—	2	4	no	—
340	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
341	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
342	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
343	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
344	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
345	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
346	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
347	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
348	#398	—	2	1	no	—
349	#398	—	2	1	no	—
350	#396	—	2	1	no	—
351	#396	—	2	1	no	—
352	#391	C₂²×C₄₁	1	1	yes	characteristic
353	#389	—	2	1	no	—
354	#389	C₂²×C₄₁	1	1	yes	characteristic
355	#389	C₂²×C₄₁	1	1	yes	characteristic
356	#389	C₂²×C₄₁	1	1	yes	characteristic
357	#388	—	2	1	no	—
358	#388	—	2	1	no	—
359	#387	—	2	1	no	—
360	#384	C₂²×C₄₁	1	1	yes	characteristic
361	#380	C₂²×C₄₁	1	1	yes	characteristic
362	#380	C₂²×C₄₁	1	1	yes	characteristic
363	#380	C₂²×C₄₁	1	1	yes	characteristic
364	C₄×C₄⋊C₄	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	#340	C₂²×C₄₁	1	1	yes	characteristic
372	#340	C₂²×C₄₁	1	1	yes	characteristic
373	C₄₁	#24854	1	1	yes	characteristic, direct product factor, Sylow
374	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
375	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
376	C₂×M₁₆	—	2	2	no	—
377	C₂×M₁₆	—	2	2	no	—
378	C₂×M₁₆	—	4	1	no	—
379	C₂²×C₈	—	2	2	no	—
380	C₂²×C₈	—	2	2	no	—
381	C₂²×C₈	—	4	1	no	—
382	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
383	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
384	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
385	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
386	#125	—	2	2	no	—
387	#125	—	2	2	no	—
388	C₂²⋊Q₈	—	2	2	no	—
389	C₂²⋊Q₈	Q₈×C₄₁	1	4	yes	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×Q₈	—	2	2	no	—
392	#117	—	2	2	no	—
393	#117	—	2	2	no	—
394	C₂×C₄⋊C₄	—	2	1	no	—
395	C₂×C₄⋊C₄	—	2	1	no	—
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	—	2	1	no	—
400	C₂×C₄⋊C₄	—	2	1	no	—
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	—	2	1	no	—
403	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
404	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
412	C₂×C₂²⋊C₄	—	2	1	no	—
413	C₂×C₂²⋊C₄	—	2	1	no	—
414	C₂×C₂²⋊C₄	—	2	1	no	—
415	C₂×C₂²⋊C₄	—	2	1	no	—
416	C₂×C₂²⋊C₄	D₄×C₄₁	1	1	yes	characteristic
417	C₂×C₂²⋊C₄	D₄×C₄₁	1	1	yes	characteristic
418	C₂×C₄²	—	2	1	no	—
419	C₂×C₄²	—	2	1	no	—
420	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
422	#107	—	2	2	no	—
423	#106	—	2	2	no	—
424	#106	—	4	1	no	—
425	#106	—	4	1	no	—
426	C₄⋊C₈	—	2	2	no	—
427	C₄⋊C₈	—	2	2	no	—
428	C₄⋊C₈	—	2	2	no	—
429	C₄⋊C₈	D₄×C₄₁	1	4	yes	—
430	#103	—	2	2	no	—
431	#103	—	2	2	no	—
432	#103	—	4	1	no	—
433	#103	—	4	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	#95	—	2	1	no	—
443	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
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	—	2	1	no	—
449	C₂×Q₈	C₂×Q₈×C₄₁	1	2	yes	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	1	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	C₂×D₄×C₄₁	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
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	1	yes	characteristic
467	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
468	C₂²×C₄	#41×C₄₁	1	2	yes	—
469	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
470	M₁₆	—	4	4	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	4	4	no	—
482	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
483	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	4	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	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	—	4	2	no	—
503	C₄⋊C₄	—	8	1	no	—
504	C₄⋊C₄	—	8	1	no	—
505	C₄⋊C₄	—	8	1	no	—
506	C₄⋊C₄	—	8	1	no	—
507	C₄⋊C₄	—	8	1	no	—
508	C₄⋊C₄	C₂×Q₈×C₄₁	1	2	yes	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	8	1	no	—
513	C₂²⋊C₄	—	8	1	no	—
514	C₂²⋊C₄	—	8	1	no	—
515	C₄²	—	2	1	no	—
516	C₄²	—	2	1	no	—
517	C₄²	—	4	2	no	—
518	C₄²	—	8	1	no	—
519	C₄²	C₂×D₄×C₄₁	1	2	yes	—
520	C₄²	#41×C₄₁	1	2	yes	—
521	C₂³	—	2	1	no	—
522	C₂³	—	2	1	no	—
523	C₂³	—	2	1	no	—
524	C₂³	—	2	1	no	—
525	C₂³	—	2	1	no	—
526	C₂³	—	2	1	no	—
527	C₂³	C₂²⋊Q₈×C₄₁	1	2	yes	—
528	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
529	Q₈	—	2	2	no	—
530	Q₈	—	2	2	no	—
531	Q₈	—	4	1	no	—
532	Q₈	—	4	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	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	#121×C₄₁	1	2	yes	—
574	C₂×C₄	#121×C₄₁	1	2	yes	—
575	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
576	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
577	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
578	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
579	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
580	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
581	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
582	C₈	—	2	2	no	—
583	C₈	—	2	2	no	—
584	C₈	—	4	1	no	—
585	C₈	—	4	1	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	4	1	no	—
590	C₂²	—	4	1	no	—
591	C₂²	—	4	1	no	—
592	C₂²	—	4	1	no	—
593	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
594	C₂²	C₂×C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
595	C₂²	C₄₁×#537	1	1	yes	characteristic
596	C₂²	C₄₁×#577	1	1	yes	characteristic
597	C₂²	C₄₁×#577	1	1	yes	characteristic
598	C₂²	C₄₁×#577	1	1	yes	characteristic
599	C₂²	C₄₁×#578	1	1	yes	characteristic
600	C₄	—	2	2	no	—
601	C₄	—	2	2	no	—
602	C₄	—	2	2	no	—
603	C₄	—	4	1	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	2	no	—
607	C₄	—	8	1	no	—
608	C₄	—	8	1	no	—
609	C₄	—	8	1	no	—
610	C₄	—	8	1	no	—
611	C₄	—	8	1	no	—
612	C₄	—	8	1	no	—
613	C₄	C₄₁×#546	1	1	yes	characteristic
614	C₄	D₄×Q₈×C₄₁	1	1	yes	characteristic
615	C₄	C₄₁×#552	1	1	yes	characteristic
616	C₄	C₄₁×#554	1	1	yes	characteristic
617	C₂	—	4	2	no	—
618	C₂	C₄₁×#2417	1	1	yes	characteristic
619	C₂	C₄₁×#3060	1	1	yes	characteristic
620	C₂	C₄₁×#3063	1	1	yes	characteristic
621	C₂	C₄₁×#3082	1	1	yes	characteristic
622	C₂	C₄₁×#3084	1	1	yes	characteristic
623	C₂	C₄₁×#3258	1	1	yes	characteristic
624	C₂	C₄₁×#3260	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×#117×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#479	C₂	semidirect
5. C₄₁×#2088	C₂	semidirect
6. C₄₁×#2087	C₂	semidirect
7. C₄₁×#2066	C₂	semidirect
8. C₄₁×#2066	C₂	semidirect
9. C₄₁×#1989	C₂	semidirect
10. C₄₁×#1948	C₂	semidirect
15. C₄₁×#1803	C₂	semidirect
273. #24854	C₄₁	direct (N × H)
305. C₂×#479	C₂×C₄₁	semidirect
309. #2088	C₂×C₄₁	semidirect
310. #2087	C₂×C₄₁	semidirect
311. #2066	C₂×C₄₁	semidirect
312. #2066	C₂×C₄₁	semidirect
313. #1989	C₂×C₄₁	semidirect
314. #1948	C₂×C₄₁	semidirect
319. #1803	C₂×C₄₁	semidirect
373. C₄₁	#24854	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#479	2
C₂×C₄₁×#424	2
C₄₁×#2629	2
C₄₁×#2323	2
C₄₁×#2088	2
C₄₁×#2087	2
C₄₁×#2066	2
C₄₁×#1989	2
C₄₁×#1948	2
C₄₁×#1894	2
C₄₁×#1892	2
C₄₁×#1809	2
C₄₁×#1807	2
C₄₁×#1803	2
#24854	41

Subgroups — C41×#24854 #23801055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24854 #23801055