Subgroups — C₄₁×#25425 #23801626

Order 10496.

1048 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#495	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#470	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#469	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₁×#442	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2427	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2042	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2000	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1971	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1941	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1877	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1753	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
14	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×#124×C₄₁	C₂²	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
18	C₂×#121×C₄₁	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
20	C₄₁×#495	—	2	4	no	—
21	C₄₁×#470	—	2	4	no	—
22	C₄₁×#469	—	2	4	no	—
23	C₄₁×#442	—	2	4	no	—
24	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#103×C₄₁	C₂²	1	2	yes	—
27	C₂×#103×C₄₁	C₂²	1	2	yes	—
28	C₂×#102×C₄₁	C₂²	1	2	yes	—
29	C₂×#102×C₄₁	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
33	C₄₁×#397	—	2	2	no	—
34	C₄₁×#394	—	2	2	no	—
35	C₄₁×#391	C₂²	1	2	yes	—
36	C₄₁×#389	C₂²	1	2	yes	—
37	C₄₁×#386	—	2	2	no	—
38	C₄₁×#380	—	2	2	no	—
39	C₄×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
40	C₄₁×#340	C₂²	1	2	yes	—
41	C₄₁×#336	C₂²	1	1	yes	characteristic
42	C₄₁×#336	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
44	C₂²×D₄×C₄₁	—	2	2	no	—
45	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
46	C₂³×C₄×C₄₁	—	2	2	no	—
47	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
48	C₂×Q₁₆×C₄₁	—	2	2	no	—
49	C₂×Q₁₆×C₄₁	—	2	2	no	—
50	C₂×Q₁₆×C₄₁	—	2	2	no	—
51	C₂×SD₁₆×C₄₁	—	2	4	no	—
52	C₂×SD₁₆×C₄₁	—	2	4	no	—
53	C₂×SD₁₆×C₄₁	—	4	2	no	—
54	C₂×D₈×C₄₁	—	2	2	no	—
55	C₂×D₈×C₄₁	—	2	2	no	—
56	C₂×D₈×C₄₁	D₄	1	4	yes	—
57	C₂×M₁₆×C₄₁	—	2	2	no	—
58	C₂×M₁₆×C₄₁	—	2	2	no	—
59	C₂×M₁₆×C₄₁	—	2	2	no	—
60	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
64	#124×C₄₁	—	2	4	no	—
65	#124×C₄₁	—	4	2	no	—
66	C₂²⋊Q₈×C₄₁	—	2	4	no	—
67	C₂²⋊Q₈×C₄₁	—	4	2	no	—
68	#121×C₄₁	—	2	4	no	—
69	#121×C₄₁	—	4	2	no	—
70	C₄×D₄×C₄₁	—	2	4	no	—
71	C₄×D₄×C₄₁	—	4	2	no	—
72	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
76	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
82	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
84	#106×C₄₁	—	2	2	no	—
85	#106×C₄₁	D₄	1	4	yes	complemented
86	#103×C₄₁	—	2	4	no	—
87	#103×C₄₁	—	4	2	no	—
88	#102×C₄₁	—	2	4	no	—
89	#102×C₄₁	—	2	4	no	—
90	C₂²⋊C₈×C₄₁	—	2	2	no	—
91	#97×C₄₁	—	2	2	no	—
92	C₄×C₈×C₄₁	—	2	2	no	—
93	#95×C₄₁	—	2	2	no	—
94	#95×C₄₁	—	2	2	no	—
95	#95×C₄₁	—	2	2	no	—
96	#95×C₄₁	D₄	1	2	yes	—
97	C₂⁴×C₄₁	—	2	2	no	—
98	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
99	C₂×Q₈×C₄₁	—	2	2	no	—
100	C₂×Q₈×C₄₁	—	2	2	no	—
101	C₂×Q₈×C₄₁	—	2	2	no	—
102	C₂×Q₈×C₄₁	—	2	2	no	—
103	C₂×Q₈×C₄₁	—	2	2	no	—
104	C₂×Q₈×C₄₁	—	2	2	no	—
105	C₂×D₄×C₄₁	—	2	2	no	—
106	C₂×D₄×C₄₁	—	2	2	no	—
107	C₂×D₄×C₄₁	—	2	2	no	—
108	C₂×D₄×C₄₁	—	2	2	no	—
109	C₂×D₄×C₄₁	—	2	2	no	—
110	C₂×D₄×C₄₁	—	4	4	no	—
111	C₂×D₄×C₄₁	—	4	4	no	—
112	C₂×D₄×C₄₁	—	8	2	no	—
113	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₄₁	—	2	2	no	—
115	C₂²×C₄×C₄₁	—	2	1	no	—
116	C₂²×C₄×C₄₁	—	2	1	no	—
117	C₂²×C₄×C₄₁	—	2	1	no	—
118	C₂²×C₄×C₄₁	—	2	1	no	—
119	C₂²×C₄×C₄₁	—	2	1	no	—
120	C₂²×C₄×C₄₁	—	2	1	no	—
121	C₂²×C₄×C₄₁	—	2	2	no	—
122	C₂²×C₄×C₄₁	—	2	2	no	—
123	C₂²×C₄×C₄₁	—	4	4	no	—
124	C₂²×C₄×C₄₁	—	4	4	no	—
125	C₂²×C₄×C₄₁	—	8	2	no	—
126	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
130	Q₁₆×C₄₁	—	4	4	no	—
131	SD₁₆×C₄₁	—	4	8	no	—
132	D₈×C₄₁	—	2	8	no	—
133	M₁₆×C₄₁	—	4	4	no	—
134	C₂×C₈×C₄₁	—	2	1	no	—
135	C₂×C₈×C₄₁	—	2	1	no	—
136	C₂×C₈×C₄₁	—	2	1	no	—
137	C₂×C₈×C₄₁	—	2	1	no	—
138	C₂×C₈×C₄₁	—	2	1	no	—
139	C₂×C₈×C₄₁	—	2	1	no	—
140	C₂×C₈×C₄₁	—	2	1	no	—
141	C₂×C₈×C₄₁	—	2	1	no	—
142	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₄₁	#41	1	2	yes	—
145	C₂×C₈×C₄₁	#41	1	2	yes	—
146	C₄⋊C₄×C₄₁	—	2	2	no	—
147	C₄⋊C₄×C₄₁	—	2	2	no	—
148	C₄⋊C₄×C₄₁	—	2	2	no	—
149	C₄⋊C₄×C₄₁	—	4	4	no	—
150	C₄⋊C₄×C₄₁	—	8	2	no	—
151	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
152	C₂²⋊C₄×C₄₁	—	4	2	no	—
153	C₂²⋊C₄×C₄₁	—	4	4	no	—
154	C₂²⋊C₄×C₄₁	—	4	4	no	—
155	C₂²⋊C₄×C₄₁	—	4	4	no	—
156	C₂²⋊C₄×C₄₁	—	4	4	no	—
157	C₂²⋊C₄×C₄₁	—	4	4	no	—
158	C₄²×C₄₁	—	2	1	no	—
159	C₄²×C₄₁	—	2	1	no	—
160	C₄²×C₄₁	—	2	1	no	—
161	C₄²×C₄₁	—	2	1	no	—
162	C₂³×C₄₁	—	2	1	no	—
163	C₂³×C₄₁	—	2	1	no	—
164	C₂³×C₄₁	—	2	4	no	—
165	C₂³×C₄₁	—	2	1	no	—
166	C₂³×C₄₁	—	2	2	no	—
167	C₂³×C₄₁	—	2	2	no	—
168	C₂³×C₄₁	—	4	2	no	—
169	C₂³×C₄₁	—	4	2	no	—
170	C₂³×C₄₁	—	4	2	no	—
171	C₂³×C₄₁	—	4	2	no	—
172	C₂³×C₄₁	—	4	2	no	—
173	C₂³×C₄₁	—	4	2	no	—
174	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
175	Q₈×C₄₁	—	4	2	no	—
176	Q₈×C₄₁	—	4	2	no	—
177	Q₈×C₄₁	—	4	2	no	—
178	Q₈×C₄₁	—	4	2	no	—
179	D₄×C₄₁	—	2	4	no	—
180	D₄×C₄₁	—	2	4	no	—
181	D₄×C₄₁	—	4	2	no	—
182	D₄×C₄₁	—	4	2	no	—
183	D₄×C₄₁	—	8	8	no	—
184	C₂×C₄×C₄₁	—	2	1	no	—
185	C₂×C₄×C₄₁	—	2	2	no	—
186	C₂×C₄×C₄₁	—	2	2	no	—
187	C₂×C₄×C₄₁	—	2	1	no	—
188	C₂×C₄×C₄₁	—	2	4	no	—
189	C₂×C₄×C₄₁	—	2	1	no	—
190	C₂×C₄×C₄₁	—	2	2	no	—
191	C₂×C₄×C₄₁	—	2	2	no	—
192	C₂×C₄×C₄₁	—	2	2	no	—
193	C₂×C₄×C₄₁	—	2	2	no	—
194	C₂×C₄×C₄₁	—	4	2	no	—
195	C₂×C₄×C₄₁	—	4	2	no	—
196	C₂×C₄×C₄₁	—	4	1	no	—
197	C₂×C₄×C₄₁	—	4	1	no	—
198	C₂×C₄×C₄₁	—	4	1	no	—
199	C₂×C₄×C₄₁	—	4	1	no	—
200	C₂×C₄×C₄₁	—	4	2	no	—
201	C₂×C₄×C₄₁	—	4	2	no	—
202	C₂×C₄×C₄₁	—	4	2	no	—
203	C₂×C₄×C₄₁	—	4	2	no	—
204	C₂×C₄×C₄₁	—	4	2	no	—
205	C₂×C₄×C₄₁	—	4	2	no	—
206	C₂×C₄×C₄₁	—	8	8	no	—
207	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
213	C₈×C₄₁	—	2	2	no	—
214	C₈×C₄₁	—	4	1	no	—
215	C₈×C₄₁	—	4	1	no	—
216	C₈×C₄₁	#121	1	4	yes	—
217	#25425	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂²×C₄₁	—	4	1	no	—
219	C₂²×C₄₁	—	4	1	no	—
220	C₂²×C₄₁	—	4	1	no	—
221	C₂²×C₄₁	—	4	4	no	—
222	C₂²×C₄₁	—	4	2	no	—
223	C₂²×C₄₁	—	4	4	no	—
224	C₂²×C₄₁	—	4	4	no	—
225	C₂²×C₄₁	—	4	2	no	—
226	C₂²×C₄₁	—	8	2	no	—
227	C₂²×C₄₁	—	8	2	no	—
228	C₂²×C₄₁	—	8	2	no	—
229	C₂²×C₄₁	—	8	2	no	—
230	C₂²×C₄₁	#532	1	1	yes	characteristic
231	C₂²×C₄₁	D₄²	1	1	yes	characteristic
232	C₂²×C₄₁	D₄²	1	1	yes	characteristic
233	C₂²×C₄₁	#575	1	2	yes	—
234	C₂²×C₄₁	#577	1	2	yes	—
235	C₄×C₄₁	—	4	1	no	—
236	C₄×C₄₁	—	4	2	no	—
237	C₄×C₄₁	—	4	2	no	—
238	C₄×C₄₁	—	4	4	no	—
239	C₄×C₄₁	—	8	2	no	—
240	C₄×C₄₁	—	8	2	no	—
241	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
242	C₄×C₄₁	#540	1	1	yes	characteristic
243	C₄×C₄₁	#546	1	1	yes	characteristic
244	C₄×C₄₁	#550	1	1	yes	characteristic
245	C₂×#495	C₂×C₄₁	1	1	yes	characteristic, complemented
246	C₂×#470	C₂×C₄₁	1	1	yes	characteristic, complemented
247	C₂×#469	C₂×C₄₁	1	1	yes	characteristic
248	C₂×#442	C₂×C₄₁	1	1	yes	characteristic, complemented
249	#2427	C₂×C₄₁	1	2	yes	complemented
250	#2042	C₂×C₄₁	1	2	yes	complemented
251	#2000	C₂×C₄₁	1	2	yes	complemented
252	#1971	C₂×C₄₁	1	1	yes	characteristic, complemented
253	#1941	C₂×C₄₁	1	1	yes	characteristic, complemented
254	#1877	C₂×C₄₁	1	2	yes	complemented
255	#1753	C₂×C₄₁	1	1	yes	characteristic, complemented
256	C₂×C₄₁	—	8	1	no	—
257	C₂×C₄₁	—	8	4	no	—
258	C₂×C₄₁	#2428	1	1	yes	characteristic
259	C₂×C₄₁	#3173	1	1	yes	characteristic
260	C₂×C₄₁	#3183	1	1	yes	characteristic
261	C₂×C₄₁	#3267	1	2	yes	—
262	C₂×C₄₁	#3275	1	2	yes	—
263	C₂²×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
264	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
265	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
266	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
267	C₂×#124	C₂²×C₄₁	1	2	yes	—
268	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
269	C₂×#121	C₂²×C₄₁	1	2	yes	—
270	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
271	#495	—	2	4	no	—
272	#470	—	2	4	no	—
273	#469	—	2	4	no	—
274	#442	—	2	4	no	—
275	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
276	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
277	C₂×#103	C₂²×C₄₁	1	2	yes	—
278	C₂×#103	C₂²×C₄₁	1	2	yes	—
279	C₂×#102	C₂²×C₄₁	1	2	yes	—
280	C₂×#102	C₂²×C₄₁	1	2	yes	—
281	C₂×C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
282	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
283	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
284	#397	—	2	2	no	—
285	#394	—	2	2	no	—
286	#391	C₂²×C₄₁	1	2	yes	—
287	#389	C₂²×C₄₁	1	2	yes	—
288	#386	—	2	2	no	—
289	#380	—	2	2	no	—
290	C₄×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
291	#340	C₂²×C₄₁	1	2	yes	—
292	#336	C₂²×C₄₁	1	1	yes	characteristic
293	#336	C₂²×C₄₁	1	1	yes	characteristic
294	C₄₁	#25425	1	1	yes	characteristic, direct product factor, Sylow
295	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
296	C₂²×D₄	—	2	2	no	—
297	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
298	C₂³×C₄	—	2	2	no	—
299	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
300	C₂×Q₁₆	—	2	2	no	—
301	C₂×Q₁₆	—	2	2	no	—
302	C₂×Q₁₆	—	2	2	no	—
303	C₂×SD₁₆	—	2	4	no	—
304	C₂×SD₁₆	—	2	4	no	—
305	C₂×SD₁₆	—	4	2	no	—
306	C₂×D₈	—	2	2	no	—
307	C₂×D₈	—	2	2	no	—
308	C₂×D₈	D₄×C₄₁	1	4	yes	—
309	C₂×M₁₆	—	2	2	no	—
310	C₂×M₁₆	—	2	2	no	—
311	C₂×M₁₆	—	2	2	no	—
312	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
313	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
314	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
315	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
316	#124	—	2	4	no	—
317	#124	—	4	2	no	—
318	C₂²⋊Q₈	—	2	4	no	—
319	C₂²⋊Q₈	—	4	2	no	—
320	#121	—	2	4	no	—
321	#121	—	4	2	no	—
322	C₄×D₄	—	2	4	no	—
323	C₄×D₄	—	4	2	no	—
324	C₂×C₄⋊C₄	—	2	2	no	—
325	C₂×C₄⋊C₄	—	2	2	no	—
326	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₂²⋊C₄	—	2	2	no	—
333	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
334	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
336	#106	—	2	2	no	—
337	#106	D₄×C₄₁	1	4	yes	complemented
338	#103	—	2	4	no	—
339	#103	—	4	2	no	—
340	#102	—	2	4	no	—
341	#102	—	2	4	no	—
342	C₂²⋊C₈	—	2	2	no	—
343	#97	—	2	2	no	—
344	C₄×C₈	—	2	2	no	—
345	#95	—	2	2	no	—
346	#95	—	2	2	no	—
347	#95	—	2	2	no	—
348	#95	D₄×C₄₁	1	2	yes	—
349	C₂⁴	—	2	2	no	—
350	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
351	C₂×Q₈	—	2	2	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	2	no	—
354	C₂×Q₈	—	2	2	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	4	4	no	—
363	C₂×D₄	—	4	4	no	—
364	C₂×D₄	—	8	2	no	—
365	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	4	4	no	—
376	C₂²×C₄	—	4	4	no	—
377	C₂²×C₄	—	8	2	no	—
378	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
379	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
380	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
381	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
382	Q₁₆	—	4	4	no	—
383	SD₁₆	—	4	8	no	—
384	D₈	—	2	8	no	—
385	M₁₆	—	4	4	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
395	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
396	C₂×C₈	#41×C₄₁	1	2	yes	—
397	C₂×C₈	#41×C₄₁	1	2	yes	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	4	4	no	—
402	C₄⋊C₄	—	8	2	no	—
403	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
404	C₂²⋊C₄	—	4	2	no	—
405	C₂²⋊C₄	—	4	4	no	—
406	C₂²⋊C₄	—	4	4	no	—
407	C₂²⋊C₄	—	4	4	no	—
408	C₂²⋊C₄	—	4	4	no	—
409	C₂²⋊C₄	—	4	4	no	—
410	C₄²	—	2	1	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	2	1	no	—
413	C₄²	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	—	2	1	no	—
416	C₂³	—	2	4	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	2	no	—
426	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
427	Q₈	—	4	2	no	—
428	Q₈	—	4	2	no	—
429	Q₈	—	4	2	no	—
430	Q₈	—	4	2	no	—
431	D₄	—	2	4	no	—
432	D₄	—	2	4	no	—
433	D₄	—	4	2	no	—
434	D₄	—	4	2	no	—
435	D₄	—	8	8	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	8	8	no	—
459	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
462	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
463	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
464	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
465	C₈	—	2	2	no	—
466	C₈	—	4	1	no	—
467	C₈	—	4	1	no	—
468	C₈	#121×C₄₁	1	4	yes	—
469	C₂²	—	4	1	no	—
470	C₂²	—	4	1	no	—
471	C₂²	—	4	1	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	4	4	no	—
475	C₂²	—	4	4	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	8	2	no	—
478	C₂²	—	8	2	no	—
479	C₂²	—	8	2	no	—
480	C₂²	—	8	2	no	—
481	C₂²	C₄₁×#532	1	1	yes	characteristic
482	C₂²	D₄²×C₄₁	1	1	yes	characteristic
483	C₂²	D₄²×C₄₁	1	1	yes	characteristic
484	C₂²	C₄₁×#575	1	2	yes	—
485	C₂²	C₄₁×#577	1	2	yes	—
486	C₄	—	4	1	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	4	no	—
490	C₄	—	8	2	no	—
491	C₄	—	8	2	no	—
492	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
493	C₄	C₄₁×#540	1	1	yes	characteristic
494	C₄	C₄₁×#546	1	1	yes	characteristic
495	C₄	C₄₁×#550	1	1	yes	characteristic
496	C₂	—	8	1	no	—
497	C₂	—	8	4	no	—
498	C₂	C₄₁×#2428	1	1	yes	characteristic
499	C₂	C₄₁×#3173	1	1	yes	characteristic
500	C₂	C₄₁×#3183	1	1	yes	characteristic
501	C₂	C₄₁×#3267	1	2	yes	—
502	C₂	C₄₁×#3275	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#469	C₂
LCS	G ▷ C₂²×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
1. C₂×C₄₁×#495	C₂	semidirect
2. C₂×C₄₁×#470	C₂	semidirect
4. C₂×C₄₁×#442	C₂	semidirect
5. C₄₁×#2427	C₂	semidirect
6. C₄₁×#2042	C₂	semidirect
7. C₄₁×#2000	C₂	semidirect
8. C₄₁×#1971	C₂	semidirect
9. C₄₁×#1941	C₂	semidirect
10. C₄₁×#1877	C₂	semidirect
11. C₄₁×#1753	C₂	semidirect
85. #106×C₄₁	D₄	semidirect
217. #25425	C₄₁	direct (N × H)
245. C₂×#495	C₂×C₄₁	semidirect
246. C₂×#470	C₂×C₄₁	semidirect
248. C₂×#442	C₂×C₄₁	semidirect
249. #2427	C₂×C₄₁	semidirect
250. #2042	C₂×C₄₁	semidirect
251. #2000	C₂×C₄₁	semidirect
252. #1971	C₂×C₄₁	semidirect
253. #1941	C₂×C₄₁	semidirect
254. #1877	C₂×C₄₁	semidirect
255. #1753	C₂×C₄₁	semidirect
294. C₄₁	#25425	direct (N × H)
337. #106	D₄×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×#495	2
C₂×C₄₁×#470	2
C₂×C₄₁×#469	2
C₂×C₄₁×#442	2
C₄₁×#2427	2
C₄₁×#2042	2
C₄₁×#2000	2
C₄₁×#1971	2
C₄₁×#1941	2
C₄₁×#1877	2
C₄₁×#1753	2
#25425	41

Subgroups — C41×#25425 #23801626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#25425 #23801626