Subgroups — C₄₁×#25668 #23801869

Order 10496.

984 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3512	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#3142	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3080	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2040	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2039	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1998	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1962	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1860	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1859	C₂	1	2	yes	maximal, complemented
11	C₄×#104×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#553	—	2	2	no	—
13	C₄₁×#553	—	2	4	no	—
14	C₄₁×#552	—	2	2	no	—
15	C₄₁×#541	—	2	1	no	—
16	C₄₁×#540	—	2	1	no	—
17	C₄₁×#537	C₂²	1	2	yes	—
18	C₄₁×#532	C₂²	1	1	yes	characteristic
19	C₄₁×#529	—	2	2	no	—
20	C₂×#125×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×#125×C₄₁	C₂²	1	1	yes	characteristic
22	C₄₁×#520	C₂²	1	2	yes	—
23	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#491	—	2	1	no	—
25	C₄₁×#490	—	2	1	no	—
26	C₄₁×#488	—	2	2	no	—
27	C₄₁×#479	—	2	2	no	—
28	C₄₁×#478	—	2	2	no	—
29	C₄₁×#428	C₂²	1	2	yes	—
30	C₄₁×#423	C₂²	1	1	yes	characteristic
31	C₂×#104×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×#104×C₄₁	C₂²	1	1	yes	characteristic
33	C₂×#104×C₄₁	C₂²	1	2	yes	—
34	C₂×#104×C₄₁	C₂²	1	2	yes	—
35	C₄₁×#417	C₂²	1	2	yes	—
36	C₄₁×#417	C₂²	1	2	yes	—
37	C₄₁×#417	C₂²	1	2	yes	—
38	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
39	C₄₁×#389	—	2	4	no	—
40	C₄₁×#384	C₂²	1	2	yes	—
41	C₄₁×#382	C₂²	1	2	yes	—
42	C₄³×C₄₁	C₂²	1	1	yes	characteristic
43	C₄₁×#343	C₂²	1	2	yes	—
44	C₄₁×#343	C₂²	1	2	yes	—
45	C₄₁×#339	C₂²	1	1	yes	characteristic
46	C₄₁×#339	C₂²	1	1	yes	characteristic
47	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
48	C₂×#41×C₄₁	C₂³	1	2	yes	—
49	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
52	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
53	#127×C₄₁	D₄	1	1	yes	characteristic
54	#126×C₄₁	—	2	4	no	—
55	#126×C₄₁	—	2	4	no	—
56	#125×C₄₁	—	2	4	no	—
57	#125×C₄₁	—	2	1	no	—
58	#125×C₄₁	—	2	1	no	—
59	#125×C₄₁	—	2	2	no	—
60	#125×C₄₁	—	2	2	no	—
61	#125×C₄₁	—	2	1	no	—
62	#125×C₄₁	—	2	1	no	—
63	#124×C₄₁	D₄	1	2	yes	—
64	#123×C₄₁	—	2	4	no	—
65	#123×C₄₁	—	2	4	no	—
66	C₂²⋊Q₈×C₄₁	—	2	2	no	—
67	C₂²⋊Q₈×C₄₁	—	2	2	no	—
68	#121×C₄₁	—	2	4	no	—
69	#121×C₄₁	—	2	2	no	—
70	#121×C₄₁	—	2	2	no	—
71	#121×C₄₁	—	2	2	no	—
72	#121×C₄₁	—	2	4	no	—
73	C₄×Q₈×C₄₁	—	2	2	no	—
74	C₄×Q₈×C₄₁	D₄	1	2	yes	—
75	C₄×D₄×C₄₁	—	2	4	no	—
76	C₄×D₄×C₄₁	—	2	2	no	—
77	C₄×D₄×C₄₁	—	2	2	no	—
78	C₄×D₄×C₄₁	—	2	4	no	—
79	C₄×D₄×C₄₁	—	2	2	no	—
80	C₄×D₄×C₄₁	D₄	1	2	yes	—
81	#117×C₄₁	—	2	4	no	—
82	#117×C₄₁	—	2	2	no	—
83	#117×C₄₁	—	2	1	no	—
84	#117×C₄₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
86	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
87	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
88	C₂×C₄⋊C₄×C₄₁	D₄	1	4	yes	—
89	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
91	C₂×C₄²×C₄₁	—	2	1	no	—
92	C₂×C₄²×C₄₁	—	2	2	no	—
93	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
96	#107×C₄₁	—	2	2	no	—
97	#106×C₄₁	—	2	2	no	—
98	C₄⋊C₈×C₄₁	—	2	2	no	—
99	#104×C₄₁	—	2	2	no	—
100	#104×C₄₁	—	2	2	no	—
101	#104×C₄₁	—	2	4	no	—
102	#104×C₄₁	—	2	4	no	—
103	#103×C₄₁	—	2	2	no	—
104	#103×C₄₁	—	2	2	no	—
105	#103×C₄₁	—	2	2	no	—
106	#102×C₄₁	—	2	2	no	—
107	#102×C₄₁	—	2	2	no	—
108	#102×C₄₁	—	2	2	no	—
109	#97×C₄₁	—	2	1	no	—
110	C₄×C₈×C₄₁	—	2	1	no	—
111	#95×C₄₁	—	2	4	no	—
112	#41×C₄₁	—	2	2	no	—
113	#41×C₄₁	—	2	2	no	—
114	#41×C₄₁	—	2	1	no	—
115	#41×C₄₁	—	2	1	no	—
116	#41×C₄₁	—	2	2	no	—
117	#41×C₄₁	—	2	2	no	—
118	#41×C₄₁	—	2	1	no	—
119	#41×C₄₁	—	2	1	no	—
120	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
122	C₂×D₄×C₄₁	—	2	1	no	—
123	C₂×D₄×C₄₁	—	2	2	no	—
124	C₂×D₄×C₄₁	—	2	4	no	—
125	C₂×D₄×C₄₁	—	2	2	no	—
126	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
127	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₄₁	—	2	1	no	—
129	C₂²×C₄×C₄₁	—	2	1	no	—
130	C₂²×C₄×C₄₁	—	2	4	no	—
131	C₂²×C₄×C₄₁	—	2	1	no	—
132	C₂²×C₄×C₄₁	—	2	2	no	—
133	C₂²×C₄×C₄₁	—	2	2	no	—
134	C₂²×C₄×C₄₁	—	2	2	no	—
135	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
136	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
138	M₁₆×C₄₁	—	2	2	no	—
139	M₁₆×C₄₁	—	2	2	no	—
140	M₁₆×C₄₁	—	2	1	no	—
141	M₁₆×C₄₁	—	2	1	no	—
142	M₁₆×C₄₁	—	2	1	no	—
143	M₁₆×C₄₁	—	2	1	no	—
144	C₂×C₈×C₄₁	—	2	1	no	—
145	C₂×C₈×C₄₁	—	2	1	no	—
146	C₂×C₈×C₄₁	—	2	2	no	—
147	C₄⋊C₄×C₄₁	—	2	4	no	—
148	C₄⋊C₄×C₄₁	—	2	4	no	—
149	C₄⋊C₄×C₄₁	—	2	4	no	—
150	C₄⋊C₄×C₄₁	—	2	4	no	—
151	C₄⋊C₄×C₄₁	—	2	2	no	—
152	C₄⋊C₄×C₄₁	—	2	2	no	—
153	C₄⋊C₄×C₄₁	—	2	2	no	—
154	C₄⋊C₄×C₄₁	—	2	2	no	—
155	C₄⋊C₄×C₄₁	—	2	2	no	—
156	C₄⋊C₄×C₄₁	—	2	2	no	—
157	C₄⋊C₄×C₄₁	—	4	8	no	—
158	C₄⋊C₄×C₄₁	—	4	2	no	—
159	C₄⋊C₄×C₄₁	—	4	2	no	—
160	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
161	C₂²⋊C₄×C₄₁	—	2	4	no	—
162	C₂²⋊C₄×C₄₁	—	2	4	no	—
163	C₂²⋊C₄×C₄₁	—	2	4	no	—
164	C₂²⋊C₄×C₄₁	—	2	1	no	—
165	C₂²⋊C₄×C₄₁	—	2	2	no	—
166	C₂²⋊C₄×C₄₁	—	2	2	no	—
167	C₂²⋊C₄×C₄₁	—	2	2	no	—
168	C₄²×C₄₁	—	2	4	no	—
169	C₄²×C₄₁	—	2	2	no	—
170	C₄²×C₄₁	—	2	1	no	—
171	C₄²×C₄₁	—	2	1	no	—
172	C₄²×C₄₁	—	2	1	no	—
173	C₄²×C₄₁	—	4	4	no	—
174	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₄₁	C₂×D₄	1	2	yes	—
177	C₄²×C₄₁	#41	1	2	yes	—
178	C₄²×C₄₁	#41	1	2	yes	—
179	C₂³×C₄₁	—	2	2	no	—
180	C₂³×C₄₁	—	2	1	no	—
181	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₄₁	—	2	1	no	—
183	Q₈×C₄₁	—	2	1	no	—
184	Q₈×C₄₁	—	2	2	no	—
185	Q₈×C₄₁	—	2	2	no	—
186	D₄×C₄₁	—	2	1	no	—
187	D₄×C₄₁	—	2	1	no	—
188	D₄×C₄₁	—	2	2	no	—
189	D₄×C₄₁	—	2	2	no	—
190	D₄×C₄₁	—	4	2	no	—
191	D₄×C₄₁	—	4	2	no	—
192	D₄×C₄₁	—	4	1	no	—
193	D₄×C₄₁	—	4	1	no	—
194	D₄×C₄₁	—	8	2	no	—
195	D₄×C₄₁	—	8	4	no	—
196	C₂×C₄×C₄₁	—	2	4	no	—
197	C₂×C₄×C₄₁	—	2	2	no	—
198	C₂×C₄×C₄₁	—	2	2	no	—
199	C₂×C₄×C₄₁	—	2	2	no	—
200	C₂×C₄×C₄₁	—	2	1	no	—
201	C₂×C₄×C₄₁	—	2	4	no	—
202	C₂×C₄×C₄₁	—	2	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	—	4	2	no	—
207	C₂×C₄×C₄₁	—	4	1	no	—
208	C₂×C₄×C₄₁	—	4	1	no	—
209	C₂×C₄×C₄₁	—	4	4	no	—
210	C₂×C₄×C₄₁	—	4	4	no	—
211	C₂×C₄×C₄₁	—	4	2	no	—
212	C₂×C₄×C₄₁	—	8	2	no	—
213	C₂×C₄×C₄₁	—	8	4	no	—
214	C₂×C₄×C₄₁	#120	1	2	yes	—
215	C₂×C₄×C₄₁	#120	1	2	yes	—
216	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
221	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
222	C₈×C₄₁	—	4	2	no	—
223	C₈×C₄₁	—	4	1	no	—
224	C₈×C₄₁	—	4	1	no	—
225	#25668	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂²×C₄₁	—	2	2	no	—
227	C₂²×C₄₁	—	4	2	no	—
228	C₂²×C₄₁	—	4	1	no	—
229	C₂²×C₄₁	—	4	2	no	—
230	C₂²×C₄₁	—	4	2	no	—
231	C₂²×C₄₁	—	4	2	no	—
232	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
233	C₂²×C₄₁	#548	1	1	yes	characteristic
234	C₂²×C₄₁	#548	1	1	yes	characteristic
235	C₄×C₄₁	—	2	4	no	—
236	C₄×C₄₁	—	2	2	no	—
237	C₄×C₄₁	—	2	2	no	—
238	C₄×C₄₁	—	4	2	no	—
239	C₄×C₄₁	—	4	2	no	—
240	C₄×C₄₁	—	4	1	no	—
241	C₄×C₄₁	—	4	4	no	—
242	C₄×C₄₁	—	4	2	no	—
243	C₄×C₄₁	—	8	2	no	—
244	C₄×C₄₁	#532	1	1	yes	characteristic
245	C₄×C₄₁	#536	1	1	yes	characteristic
246	C₄×C₄₁	#547	1	1	yes	characteristic
247	C₄×C₄₁	#547	1	1	yes	characteristic
248	#3512	C₂×C₄₁	1	1	yes	characteristic
249	#3142	C₂×C₄₁	1	1	yes	characteristic, complemented
250	#3080	C₂×C₄₁	1	1	yes	characteristic, complemented
251	#2859	C₂×C₄₁	1	1	yes	characteristic, complemented
252	#2040	C₂×C₄₁	1	2	yes	complemented
253	#2039	C₂×C₄₁	1	2	yes	complemented
254	#1998	C₂×C₄₁	1	1	yes	characteristic, complemented
255	#1962	C₂×C₄₁	1	1	yes	characteristic, complemented
256	#1860	C₂×C₄₁	1	2	yes	complemented
257	#1859	C₂×C₄₁	1	2	yes	complemented
258	C₄×#104	C₂×C₄₁	1	1	yes	characteristic, complemented
259	C₂×C₄₁	—	2	1	no	—
260	C₂×C₄₁	—	2	1	no	—
261	C₂×C₄₁	—	8	2	no	—
262	C₂×C₄₁	—	8	1	no	—
263	C₂×C₄₁	#2432	1	1	yes	characteristic
264	C₂×C₄₁	#3018	1	2	yes	—
265	#553	—	2	2	no	—
266	#553	—	2	4	no	—
267	#552	—	2	2	no	—
268	#541	—	2	1	no	—
269	#540	—	2	1	no	—
270	#537	C₂²×C₄₁	1	2	yes	—
271	#532	C₂²×C₄₁	1	1	yes	characteristic
272	#529	—	2	2	no	—
273	C₂×#125	C₂²×C₄₁	1	1	yes	characteristic
274	C₂×#125	C₂²×C₄₁	1	1	yes	characteristic
275	#520	C₂²×C₄₁	1	2	yes	—
276	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
277	#491	—	2	1	no	—
278	#490	—	2	1	no	—
279	#488	—	2	2	no	—
280	#479	—	2	2	no	—
281	#478	—	2	2	no	—
282	#428	C₂²×C₄₁	1	2	yes	—
283	#423	C₂²×C₄₁	1	1	yes	characteristic
284	C₂×#104	C₂²×C₄₁	1	2	yes	—
285	C₂×#104	C₂²×C₄₁	1	2	yes	—
286	C₂×#104	C₂²×C₄₁	1	1	yes	characteristic
287	C₂×#104	C₂²×C₄₁	1	1	yes	characteristic
288	#417	C₂²×C₄₁	1	2	yes	—
289	#417	C₂²×C₄₁	1	2	yes	—
290	#417	C₂²×C₄₁	1	2	yes	—
291	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
292	#389	—	2	4	no	—
293	#384	C₂²×C₄₁	1	2	yes	—
294	#382	C₂²×C₄₁	1	2	yes	—
295	C₄³	C₂²×C₄₁	1	1	yes	characteristic
296	#343	C₂²×C₄₁	1	2	yes	—
297	#343	C₂²×C₄₁	1	2	yes	—
298	#339	C₂²×C₄₁	1	1	yes	characteristic
299	#339	C₂²×C₄₁	1	1	yes	characteristic
300	C₄₁	#25668	1	1	yes	characteristic, direct product factor, Sylow
301	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
302	C₂×#41	C₂³×C₄₁	1	2	yes	—
303	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
304	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
305	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
306	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
307	#127	D₄×C₄₁	1	1	yes	characteristic
308	#126	—	2	4	no	—
309	#126	—	2	4	no	—
310	#125	—	2	4	no	—
311	#125	—	2	1	no	—
312	#125	—	2	1	no	—
313	#125	—	2	2	no	—
314	#125	—	2	2	no	—
315	#125	—	2	1	no	—
316	#125	—	2	1	no	—
317	#124	D₄×C₄₁	1	2	yes	—
318	#123	—	2	4	no	—
319	#123	—	2	4	no	—
320	C₂²⋊Q₈	—	2	2	no	—
321	C₂²⋊Q₈	—	2	2	no	—
322	#121	—	2	4	no	—
323	#121	—	2	2	no	—
324	#121	—	2	2	no	—
325	#121	—	2	2	no	—
326	#121	—	2	4	no	—
327	C₄×Q₈	—	2	2	no	—
328	C₄×Q₈	D₄×C₄₁	1	2	yes	—
329	C₄×D₄	—	2	4	no	—
330	C₄×D₄	—	2	2	no	—
331	C₄×D₄	—	2	2	no	—
332	C₄×D₄	—	2	4	no	—
333	C₄×D₄	—	2	2	no	—
334	C₄×D₄	D₄×C₄₁	1	2	yes	—
335	#117	—	2	4	no	—
336	#117	—	2	2	no	—
337	#117	—	2	1	no	—
338	#117	C₂³×C₄₁	1	1	yes	characteristic
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	—	2	2	no	—
341	C₂×C₄⋊C₄	—	2	4	no	—
342	C₂×C₄⋊C₄	D₄×C₄₁	1	4	yes	—
343	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
344	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
345	C₂×C₄²	—	2	1	no	—
346	C₂×C₄²	—	2	2	no	—
347	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
348	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
350	#107	—	2	2	no	—
351	#106	—	2	2	no	—
352	C₄⋊C₈	—	2	2	no	—
353	#104	—	2	4	no	—
354	#104	—	2	4	no	—
355	#104	—	2	2	no	—
356	#104	—	2	2	no	—
357	#103	—	2	2	no	—
358	#103	—	2	2	no	—
359	#103	—	2	2	no	—
360	#102	—	2	2	no	—
361	#102	—	2	2	no	—
362	#102	—	2	2	no	—
363	#97	—	2	1	no	—
364	C₄×C₈	—	2	1	no	—
365	#95	—	2	4	no	—
366	#41	—	2	2	no	—
367	#41	—	2	2	no	—
368	#41	—	2	1	no	—
369	#41	—	2	1	no	—
370	#41	—	2	2	no	—
371	#41	—	2	2	no	—
372	#41	—	2	1	no	—
373	#41	—	2	1	no	—
374	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
375	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
381	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
390	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
391	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
392	M₁₆	—	2	2	no	—
393	M₁₆	—	2	2	no	—
394	M₁₆	—	2	1	no	—
395	M₁₆	—	2	1	no	—
396	M₁₆	—	2	1	no	—
397	M₁₆	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	2	no	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	4	8	no	—
412	C₄⋊C₄	—	4	2	no	—
413	C₄⋊C₄	—	4	2	no	—
414	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
415	C₂²⋊C₄	—	2	4	no	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	2	1	no	—
419	C₂²⋊C₄	—	2	2	no	—
420	C₂²⋊C₄	—	2	2	no	—
421	C₂²⋊C₄	—	2	2	no	—
422	C₄²	—	2	4	no	—
423	C₄²	—	2	2	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	4	4	no	—
428	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
429	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
430	C₄²	C₂×D₄×C₄₁	1	2	yes	—
431	C₄²	#41×C₄₁	1	2	yes	—
432	C₄²	#41×C₄₁	1	2	yes	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	1	no	—
435	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
436	Q₈	—	2	1	no	—
437	Q₈	—	2	1	no	—
438	Q₈	—	2	2	no	—
439	Q₈	—	2	2	no	—
440	D₄	—	2	1	no	—
441	D₄	—	2	1	no	—
442	D₄	—	2	2	no	—
443	D₄	—	2	2	no	—
444	D₄	—	4	2	no	—
445	D₄	—	4	2	no	—
446	D₄	—	4	1	no	—
447	D₄	—	4	1	no	—
448	D₄	—	8	2	no	—
449	D₄	—	8	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	4	no	—
464	C₂×C₄	—	4	4	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	8	2	no	—
467	C₂×C₄	—	8	4	no	—
468	C₂×C₄	#120×C₄₁	1	2	yes	—
469	C₂×C₄	#120×C₄₁	1	2	yes	—
470	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
471	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
472	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
473	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
474	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
475	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
476	C₈	—	4	2	no	—
477	C₈	—	4	1	no	—
478	C₈	—	4	1	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	1	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
486	C₂²	C₄₁×#548	1	1	yes	characteristic
487	C₂²	C₄₁×#548	1	1	yes	characteristic
488	C₄	—	2	4	no	—
489	C₄	—	2	2	no	—
490	C₄	—	2	2	no	—
491	C₄	—	4	2	no	—
492	C₄	—	4	2	no	—
493	C₄	—	4	1	no	—
494	C₄	—	4	4	no	—
495	C₄	—	4	2	no	—
496	C₄	—	8	2	no	—
497	C₄	C₄₁×#532	1	1	yes	characteristic
498	C₄	C₄₁×#536	1	1	yes	characteristic
499	C₄	C₄₁×#547	1	1	yes	characteristic
500	C₄	C₄₁×#547	1	1	yes	characteristic
501	C₂	—	2	1	no	—
502	C₂	—	2	1	no	—
503	C₂	—	8	2	no	—
504	C₂	—	8	1	no	—
505	C₂	C₄₁×#2432	1	1	yes	characteristic
506	C₂	C₄₁×#3018	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#3512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#3142	C₂	semidirect
3. C₄₁×#3080	C₂	semidirect
4. C₄₁×#2859	C₂	semidirect
5. C₄₁×#2040	C₂	semidirect
6. C₄₁×#2039	C₂	semidirect
7. C₄₁×#1998	C₂	semidirect
8. C₄₁×#1962	C₂	semidirect
9. C₄₁×#1860	C₂	semidirect
10. C₄₁×#1859	C₂	semidirect
11. C₄×#104×C₄₁	C₂	semidirect
225. #25668	C₄₁	direct (N × H)
249. #3142	C₂×C₄₁	semidirect
250. #3080	C₂×C₄₁	semidirect
251. #2859	C₂×C₄₁	semidirect
252. #2040	C₂×C₄₁	semidirect
253. #2039	C₂×C₄₁	semidirect
254. #1998	C₂×C₄₁	semidirect
255. #1962	C₂×C₄₁	semidirect
256. #1860	C₂×C₄₁	semidirect
257. #1859	C₂×C₄₁	semidirect
258. C₄×#104	C₂×C₄₁	semidirect
300. C₄₁	#25668	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#3512	2
C₄₁×#3142	2
C₄₁×#3080	2
C₄₁×#2859	2
C₄₁×#2040	2
C₄₁×#2039	2
C₄₁×#1998	2
C₄₁×#1962	2
C₄₁×#1860	2
C₄₁×#1859	2
C₄×#104×C₄₁	2
#25668	41

Subgroups — C41×#25668 #23801869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#25668 #23801869