Subgroups — C₄₁×#18634 #23794835

Order 10496.

1024 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3508	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#3229	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#3039	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1642	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1640	C₂	1	4	yes	maximal, complemented
7	C₄₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1493	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1476	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₄₁	—	2	1	no	—
11	C₄₁×#559	—	2	1	no	—
12	C₄₁×#559	C₂²	1	2	yes	—
13	C₄₁×#550	—	2	1	no	—
14	C₄₁×#550	C₂²	1	2	yes	—
15	C₄₁×#546	—	2	2	no	—
16	C₄₁×#546	—	2	2	no	—
17	C₄₁×#540	—	2	2	no	—
18	C₄₁×#535	—	2	1	no	—
19	C₄₁×#533	C₂²	1	1	yes	characteristic
20	C₄₁×#532	—	2	1	no	—
21	C₄₁×#532	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#518	—	2	1	no	—
25	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
26	C₄₁×#498	—	2	1	no	—
27	C₄₁×#487	—	2	1	no	—
28	C₄₁×#486	—	2	2	no	—
29	C₄₁×#486	C₂²	1	4	yes	—
30	C₄₁×#483	—	2	2	no	—
31	C₄₁×#475	C₂²	1	4	yes	complemented
32	C₄₁×#462	—	2	1	no	—
33	C₄₁×#462	C₂²	1	2	yes	complemented
34	C₄₁×#461	—	2	1	no	—
35	C₄₁×#461	C₂²	1	2	yes	complemented
36	C₄₁×#431	C₂²	1	1	yes	characteristic
37	C₄₁×#423	C₂²	1	1	yes	characteristic
38	C₄₁×#419	—	2	1	no	—
39	C₂×#102×C₄₁	—	2	2	no	—
40	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
41	C₄₁×#331	C₂²	1	4	yes	—
42	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
43	C₄₁×Q₈⋊C₈	C₂²	1	4	yes	complemented
44	C₄₁×D₄⋊C₈	C₂²	1	2	yes	complemented
45	C₂×#41×C₄₁	—	2	2	no	—
46	C₂×#41×C₄₁	—	2	2	no	—
47	C₂×#41×C₄₁	D₄	1	1	yes	characteristic
48	C₂²×Q₈×C₄₁	—	2	1	no	—
49	C₂²×D₄×C₄₁	—	2	1	no	—
50	C₂²×D₄×C₄₁	—	2	1	no	—
51	C₂²×D₄×C₄₁	D₄	1	2	yes	—
52	#137×C₄₁	—	4	4	no	—
53	C₂×Q₁₆×C₄₁	—	2	2	no	—
54	C₂×SD₁₆×C₄₁	—	2	2	no	—
55	C₂×M₁₆×C₄₁	—	2	1	no	—
56	C₂²×C₈×C₄₁	—	2	1	no	—
57	C₄⋊Q₈×C₄₁	—	2	1	no	—
58	C₄⋊Q₈×C₄₁	—	2	1	no	—
59	C₄⋊Q₈×C₄₁	—	2	1	no	—
60	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
62	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
63	#127×C₄₁	—	2	2	no	—
64	#127×C₄₁	—	2	1	no	—
65	#127×C₄₁	—	2	1	no	—
66	#127×C₄₁	D₄	1	2	yes	—
67	#127×C₄₁	C₂³	1	2	yes	—
68	#125×C₄₁	—	2	1	no	—
69	#124×C₄₁	—	2	2	no	—
70	#124×C₄₁	—	2	2	no	—
71	#124×C₄₁	—	2	2	no	—
72	#123×C₄₁	—	2	4	no	—
73	#123×C₄₁	—	2	4	no	—
74	C₂²⋊Q₈×C₄₁	—	2	1	no	—
75	C₂²⋊Q₈×C₄₁	D₄	1	2	yes	—
76	#121×C₄₁	—	2	2	no	—
77	#121×C₄₁	—	2	2	no	—
78	#121×C₄₁	—	2	2	no	—
79	#121×C₄₁	—	2	4	no	—
80	#121×C₄₁	—	2	4	no	—
81	#121×C₄₁	—	2	2	no	—
82	#120×C₄₁	—	2	4	no	—
83	C₄×Q₈×C₄₁	—	2	1	no	—
84	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
85	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
86	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
87	C₄×D₄×C₄₁	—	2	2	no	—
88	C₄×D₄×C₄₁	—	2	2	no	—
89	C₄×D₄×C₄₁	—	2	2	no	—
90	C₄×D₄×C₄₁	—	2	1	no	—
91	C₄×D₄×C₄₁	—	2	1	no	—
92	C₄×D₄×C₄₁	—	2	2	no	—
93	C₄×D₄×C₄₁	—	2	2	no	—
94	C₄×D₄×C₄₁	—	2	2	no	—
95	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
96	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
97	#117×C₄₁	—	2	1	no	—
98	#117×C₄₁	—	2	1	no	—
99	#117×C₄₁	D₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
101	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
103	C₂×C₄²×C₄₁	—	2	1	no	—
104	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
105	#106×C₄₁	—	2	4	no	—
106	C₄⋊C₈×C₄₁	—	2	1	no	—
107	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
108	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
109	#103×C₄₁	—	2	4	no	—
110	#103×C₄₁	—	2	2	no	—
111	#103×C₄₁	—	2	2	no	—
112	#102×C₄₁	—	2	8	no	—
113	C₂²⋊C₈×C₄₁	—	2	1	no	—
114	C₄×C₈×C₄₁	—	2	1	no	—
115	C₄×C₈×C₄₁	—	2	1	no	—
116	C₄×C₈×C₄₁	C₂³	1	2	yes	—
117	C₂⁴×C₄₁	—	2	2	no	—
118	#41×C₄₁	—	2	2	no	—
119	#41×C₄₁	—	2	2	no	—
120	#41×C₄₁	—	8	4	no	—
121	#41×C₄₁	—	8	4	no	—
122	C₂×Q₈×C₄₁	—	2	1	no	—
123	C₂×Q₈×C₄₁	—	2	2	no	—
124	C₂×Q₈×C₄₁	—	4	1	no	—
125	C₂×Q₈×C₄₁	—	4	2	no	—
126	C₂×Q₈×C₄₁	—	4	2	no	—
127	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
128	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
129	C₂×D₄×C₄₁	—	2	1	no	—
130	C₂×D₄×C₄₁	—	2	4	no	—
131	C₂×D₄×C₄₁	—	2	4	no	—
132	C₂×D₄×C₄₁	—	2	2	no	—
133	C₂×D₄×C₄₁	—	2	2	no	—
134	C₂×D₄×C₄₁	—	2	2	no	—
135	C₂×D₄×C₄₁	—	2	2	no	—
136	C₂×D₄×C₄₁	—	2	2	no	—
137	C₂×D₄×C₄₁	—	4	4	no	—
138	C₂×D₄×C₄₁	—	4	2	no	—
139	C₂×D₄×C₄₁	—	4	2	no	—
140	C₂×D₄×C₄₁	—	8	2	no	—
141	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
142	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
143	C₂²×C₄×C₄₁	—	2	4	no	—
144	C₂²×C₄×C₄₁	—	2	4	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	—	4	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	Q₁₆×C₄₁	—	4	4	no	—
153	SD₁₆×C₄₁	—	4	4	no	—
154	M₁₆×C₄₁	—	4	2	no	—
155	C₂×C₈×C₄₁	—	2	2	no	—
156	C₂×C₈×C₄₁	—	2	2	no	—
157	C₂×C₈×C₄₁	—	2	2	no	—
158	C₂×C₈×C₄₁	—	4	2	no	—
159	C₄⋊C₄×C₄₁	—	2	1	no	—
160	C₄⋊C₄×C₄₁	—	2	1	no	—
161	C₄⋊C₄×C₄₁	—	2	1	no	—
162	C₄⋊C₄×C₄₁	—	2	1	no	—
163	C₄⋊C₄×C₄₁	—	2	2	no	—
164	C₄⋊C₄×C₄₁	—	2	1	no	—
165	C₄⋊C₄×C₄₁	—	2	2	no	—
166	C₄⋊C₄×C₄₁	—	2	2	no	—
167	C₄⋊C₄×C₄₁	—	2	2	no	—
168	C₄⋊C₄×C₄₁	—	4	1	no	—
169	C₄⋊C₄×C₄₁	—	4	1	no	—
170	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
171	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
172	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
173	C₂²⋊C₄×C₄₁	—	2	4	no	—
174	C₂²⋊C₄×C₄₁	—	2	4	no	—
175	C₂²⋊C₄×C₄₁	—	2	4	no	—
176	C₂²⋊C₄×C₄₁	—	2	4	no	—
177	C₂²⋊C₄×C₄₁	—	2	1	no	—
178	C₂²⋊C₄×C₄₁	—	2	2	no	—
179	C₂²⋊C₄×C₄₁	—	2	1	no	—
180	C₂²⋊C₄×C₄₁	—	2	1	no	—
181	C₄²×C₄₁	—	2	1	no	—
182	C₄²×C₄₁	—	2	1	no	—
183	C₄²×C₄₁	—	2	1	no	—
184	C₄²×C₄₁	—	2	2	no	—
185	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
189	C₂³×C₄₁	—	2	4	no	—
190	C₂³×C₄₁	—	2	1	no	—
191	C₂³×C₄₁	—	4	4	no	—
192	C₂³×C₄₁	—	8	2	no	—
193	C₂³×C₄₁	—	8	2	no	—
194	C₂³×C₄₁	#120	1	1	yes	characteristic
195	Q₈×C₄₁	—	2	1	no	—
196	Q₈×C₄₁	—	2	1	no	—
197	Q₈×C₄₁	—	2	4	no	—
198	Q₈×C₄₁	—	4	2	no	—
199	Q₈×C₄₁	—	4	2	no	—
200	Q₈×C₄₁	—	4	2	no	—
201	Q₈×C₄₁	—	8	2	no	—
202	D₄×C₄₁	—	2	1	no	—
203	D₄×C₄₁	—	2	1	no	—
204	D₄×C₄₁	—	4	4	no	—
205	D₄×C₄₁	—	4	2	no	—
206	D₄×C₄₁	—	4	2	no	—
207	D₄×C₄₁	—	4	2	no	—
208	D₄×C₄₁	—	4	4	no	—
209	D₄×C₄₁	—	4	4	no	—
210	D₄×C₄₁	—	4	4	no	—
211	D₄×C₄₁	—	8	2	no	—
212	D₄×C₄₁	—	8	4	no	—
213	D₄×C₄₁	—	8	4	no	—
214	C₂×C₄×C₄₁	—	2	2	no	—
215	C₂×C₄×C₄₁	—	2	2	no	—
216	C₂×C₄×C₄₁	—	2	1	no	—
217	C₂×C₄×C₄₁	—	2	1	no	—
218	C₂×C₄×C₄₁	—	2	1	no	—
219	C₂×C₄×C₄₁	—	2	2	no	—
220	C₂×C₄×C₄₁	—	2	2	no	—
221	C₂×C₄×C₄₁	—	4	2	no	—
222	C₂×C₄×C₄₁	—	4	2	no	—
223	C₂×C₄×C₄₁	—	4	2	no	—
224	C₂×C₄×C₄₁	—	4	2	no	—
225	C₂×C₄×C₄₁	—	8	4	no	—
226	C₂×C₄×C₄₁	—	8	4	no	—
227	C₂×C₄×C₄₁	—	8	2	no	—
228	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
229	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
230	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
231	C₂×C₄×C₄₁	#135	1	2	yes	—
232	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
233	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
236	C₈×C₄₁	—	4	2	no	—
237	C₈×C₄₁	—	4	2	no	—
238	C₈×C₄₁	—	4	2	no	—
239	#18634	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂²×C₄₁	—	2	1	no	—
241	C₂²×C₄₁	—	2	1	no	—
242	C₂²×C₄₁	—	4	1	no	—
243	C₂²×C₄₁	—	4	4	no	—
244	C₂²×C₄₁	—	4	1	no	—
245	C₂²×C₄₁	—	4	4	no	—
246	C₂²×C₄₁	—	4	1	no	—
247	C₂²×C₄₁	—	4	4	no	—
248	C₂²×C₄₁	—	16	4	no	—
249	C₂²×C₄₁	#449	1	2	yes	—
250	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
251	C₄×C₄₁	—	2	1	no	—
252	C₄×C₄₁	—	2	1	no	—
253	C₄×C₄₁	—	4	1	no	—
254	C₄×C₄₁	—	4	1	no	—
255	C₄×C₄₁	—	4	2	no	—
256	C₄×C₄₁	—	4	1	no	—
257	C₄×C₄₁	—	4	2	no	—
258	C₄×C₄₁	—	8	2	no	—
259	C₄×C₄₁	#449	1	2	yes	—
260	C₄×C₄₁	#455	1	2	yes	—
261	C₄×C₄₁	C₂×#135	1	1	yes	characteristic
262	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
263	C₄×C₄₁	#577	1	2	yes	—
264	#3508	C₂×C₄₁	1	1	yes	characteristic
265	#3229	C₂×C₄₁	1	1	yes	characteristic, complemented
266	C₂×#486	C₂×C₄₁	1	1	yes	characteristic, complemented
267	#3039	C₂×C₄₁	1	1	yes	characteristic, complemented
268	#1642	C₂×C₄₁	1	4	yes	complemented
269	#1640	C₂×C₄₁	1	4	yes	complemented
270	#1545	C₂×C₄₁	1	1	yes	characteristic, complemented
271	#1493	C₂×C₄₁	1	1	yes	characteristic, complemented
272	#1476	C₂×C₄₁	1	1	yes	characteristic, complemented
273	C₂×C₄₁	—	2	2	no	—
274	C₂×C₄₁	—	8	1	no	—
275	C₂×C₄₁	—	8	4	no	—
276	C₂×C₄₁	C₂×#449	1	1	yes	characteristic
277	C₂×C₄₁	#3009	1	1	yes	characteristic
278	C₂×C₄₁	C₂×#455	1	1	yes	characteristic
279	C₂×#137	—	2	1	no	—
280	#559	—	2	1	no	—
281	#559	C₂²×C₄₁	1	2	yes	—
282	#550	—	2	1	no	—
283	#550	C₂²×C₄₁	1	2	yes	—
284	#546	—	2	2	no	—
285	#546	—	2	2	no	—
286	#540	—	2	2	no	—
287	#535	—	2	1	no	—
288	#533	C₂²×C₄₁	1	1	yes	characteristic
289	#532	—	2	1	no	—
290	#532	—	2	1	no	—
291	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
292	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
293	#518	—	2	1	no	—
294	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
295	#498	—	2	1	no	—
296	#487	—	2	1	no	—
297	#486	—	2	2	no	—
298	#486	C₂²×C₄₁	1	4	yes	—
299	#483	—	2	2	no	—
300	#475	C₂²×C₄₁	1	4	yes	complemented
301	#462	—	2	1	no	—
302	#462	C₂²×C₄₁	1	2	yes	complemented
303	#461	—	2	1	no	—
304	#461	C₂²×C₄₁	1	2	yes	complemented
305	#431	C₂²×C₄₁	1	1	yes	characteristic
306	#423	C₂²×C₄₁	1	1	yes	characteristic
307	#419	—	2	1	no	—
308	C₂×#102	—	2	2	no	—
309	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
310	#331	C₂²×C₄₁	1	4	yes	—
311	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	complemented
312	Q₈⋊C₈	C₂²×C₄₁	1	4	yes	complemented
313	D₄⋊C₈	C₂²×C₄₁	1	2	yes	complemented
314	C₄₁	#18634	1	1	yes	characteristic, direct product factor, Sylow
315	C₂×#41	—	2	2	no	—
316	C₂×#41	—	2	2	no	—
317	C₂×#41	D₄×C₄₁	1	1	yes	characteristic
318	C₂²×Q₈	—	2	1	no	—
319	C₂²×D₄	—	2	1	no	—
320	C₂²×D₄	—	2	1	no	—
321	C₂²×D₄	D₄×C₄₁	1	2	yes	—
322	#137	—	4	4	no	—
323	C₂×Q₁₆	—	2	2	no	—
324	C₂×SD₁₆	—	2	2	no	—
325	C₂×M₁₆	—	2	1	no	—
326	C₂²×C₈	—	2	1	no	—
327	C₄⋊Q₈	—	2	1	no	—
328	C₄⋊Q₈	—	2	1	no	—
329	C₄⋊Q₈	—	2	1	no	—
330	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
331	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
332	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
333	#127	—	2	2	no	—
334	#127	—	2	1	no	—
335	#127	—	2	1	no	—
336	#127	D₄×C₄₁	1	2	yes	—
337	#127	C₂³×C₄₁	1	2	yes	—
338	#125	—	2	1	no	—
339	#124	—	2	2	no	—
340	#124	—	2	2	no	—
341	#124	—	2	2	no	—
342	#123	—	2	4	no	—
343	#123	—	2	4	no	—
344	C₂²⋊Q₈	—	2	1	no	—
345	C₂²⋊Q₈	D₄×C₄₁	1	2	yes	—
346	#121	—	2	2	no	—
347	#121	—	2	2	no	—
348	#121	—	2	2	no	—
349	#121	—	2	4	no	—
350	#121	—	2	4	no	—
351	#121	—	2	2	no	—
352	#120	—	2	4	no	—
353	C₄×Q₈	—	2	1	no	—
354	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
355	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
356	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	1	no	—
361	C₄×D₄	—	2	1	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
366	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
367	#117	—	2	1	no	—
368	#117	—	2	1	no	—
369	#117	D₄×C₄₁	1	1	yes	characteristic
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₂²⋊C₄	—	2	4	no	—
373	C₂×C₄²	—	2	1	no	—
374	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
375	#106	—	2	4	no	—
376	C₄⋊C₈	—	2	1	no	—
377	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
378	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
379	#103	—	2	4	no	—
380	#103	—	2	2	no	—
381	#103	—	2	2	no	—
382	#102	—	2	8	no	—
383	C₂²⋊C₈	—	2	1	no	—
384	C₄×C₈	—	2	1	no	—
385	C₄×C₈	—	2	1	no	—
386	C₄×C₈	C₂³×C₄₁	1	2	yes	—
387	C₂⁴	—	2	2	no	—
388	#41	—	2	2	no	—
389	#41	—	2	2	no	—
390	#41	—	8	4	no	—
391	#41	—	8	4	no	—
392	C₂×Q₈	—	2	1	no	—
393	C₂×Q₈	—	2	2	no	—
394	C₂×Q₈	—	4	1	no	—
395	C₂×Q₈	—	4	2	no	—
396	C₂×Q₈	—	4	2	no	—
397	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
398	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	4	4	no	—
408	C₂×D₄	—	4	2	no	—
409	C₂×D₄	—	4	2	no	—
410	C₂×D₄	—	8	2	no	—
411	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
412	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	4	1	no	—
419	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
421	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
422	Q₁₆	—	4	4	no	—
423	SD₁₆	—	4	4	no	—
424	M₁₆	—	4	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	4	2	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	2	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	4	1	no	—
439	C₄⋊C₄	—	4	1	no	—
440	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
441	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
442	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
443	C₂²⋊C₄	—	2	4	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	1	no	—
448	C₂²⋊C₄	—	2	2	no	—
449	C₂²⋊C₄	—	2	1	no	—
450	C₂²⋊C₄	—	2	1	no	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	2	1	no	—
454	C₄²	—	2	2	no	—
455	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
456	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
457	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
458	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
459	C₂³	—	2	4	no	—
460	C₂³	—	2	1	no	—
461	C₂³	—	4	4	no	—
462	C₂³	—	8	2	no	—
463	C₂³	—	8	2	no	—
464	C₂³	#120×C₄₁	1	1	yes	characteristic
465	Q₈	—	2	1	no	—
466	Q₈	—	2	1	no	—
467	Q₈	—	2	4	no	—
468	Q₈	—	4	2	no	—
469	Q₈	—	4	2	no	—
470	Q₈	—	4	2	no	—
471	Q₈	—	8	2	no	—
472	D₄	—	2	1	no	—
473	D₄	—	2	1	no	—
474	D₄	—	4	4	no	—
475	D₄	—	4	2	no	—
476	D₄	—	4	2	no	—
477	D₄	—	4	2	no	—
478	D₄	—	4	4	no	—
479	D₄	—	4	4	no	—
480	D₄	—	4	4	no	—
481	D₄	—	8	2	no	—
482	D₄	—	8	4	no	—
483	D₄	—	8	4	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	8	4	no	—
496	C₂×C₄	—	8	4	no	—
497	C₂×C₄	—	8	2	no	—
498	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
499	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
500	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
501	C₂×C₄	#135×C₄₁	1	2	yes	—
502	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
503	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
504	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
506	C₈	—	4	2	no	—
507	C₈	—	4	2	no	—
508	C₈	—	4	2	no	—
509	C₂²	—	2	1	no	—
510	C₂²	—	2	1	no	—
511	C₂²	—	4	1	no	—
512	C₂²	—	4	4	no	—
513	C₂²	—	4	1	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	1	no	—
516	C₂²	—	4	4	no	—
517	C₂²	—	16	4	no	—
518	C₂²	C₄₁×#449	1	2	yes	—
519	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
520	C₄	—	2	1	no	—
521	C₄	—	2	1	no	—
522	C₄	—	4	1	no	—
523	C₄	—	4	1	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	1	no	—
526	C₄	—	4	2	no	—
527	C₄	—	8	2	no	—
528	C₄	C₄₁×#449	1	2	yes	—
529	C₄	C₄₁×#455	1	2	yes	—
530	C₄	C₂×#135×C₄₁	1	1	yes	characteristic
531	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
532	C₄	C₄₁×#577	1	2	yes	—
533	C₂	—	2	2	no	—
534	C₂	—	8	4	no	—
535	C₂	—	8	1	no	—
536	C₂	C₂×C₄₁×#449	1	1	yes	characteristic
537	C₂	C₄₁×#3009	1	1	yes	characteristic
538	C₂	C₂×C₄₁×#455	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₄₁×#3508	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ 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×#3229	C₂	semidirect
3. C₂×C₄₁×#486	C₂	semidirect
4. C₄₁×#3039	C₂	semidirect
5. C₄₁×#1642	C₂	semidirect
6. C₄₁×#1640	C₂	semidirect
7. C₄₁×#1545	C₂	semidirect
8. C₄₁×#1493	C₂	semidirect
9. C₄₁×#1476	C₂	semidirect
31. C₄₁×#475	C₂²	semidirect
33. C₄₁×#462	C₂²	semidirect
35. C₄₁×#461	C₂²	semidirect
42. C₄₁×Q₈⋊C₈	C₂²	semidirect
43. C₄₁×Q₈⋊C₈	C₂²	semidirect
44. C₄₁×D₄⋊C₈	C₂²	semidirect
239. #18634	C₄₁	direct (N × H)
265. #3229	C₂×C₄₁	semidirect
266. C₂×#486	C₂×C₄₁	semidirect
267. #3039	C₂×C₄₁	semidirect
268. #1642	C₂×C₄₁	semidirect
269. #1640	C₂×C₄₁	semidirect
270. #1545	C₂×C₄₁	semidirect
271. #1493	C₂×C₄₁	semidirect
272. #1476	C₂×C₄₁	semidirect
300. #475	C₂²×C₄₁	semidirect
302. #462	C₂²×C₄₁	semidirect
304. #461	C₂²×C₄₁	semidirect
311. Q₈⋊C₈	C₂²×C₄₁	semidirect
312. Q₈⋊C₈	C₂²×C₄₁	semidirect
313. D₄⋊C₈	C₂²×C₄₁	semidirect
314. C₄₁	#18634	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×#3508	2
C₄₁×#3229	2
C₂×C₄₁×#486	2
C₄₁×#3039	2
C₄₁×#1642	2
C₄₁×#1640	2
C₄₁×#1545	2
C₄₁×#1493	2
C₄₁×#1476	2
#18634	41

Subgroups — C41×#18634 #23794835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#18634 #23794835