Subgroups — C₄₁×#18570 #23794771

Order 10496.

1028 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3519	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#3239	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3141	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1646	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1643	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1642	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1639	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1551	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1483	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₄₁	—	2	1	no	—
13	C₄₁×#555	C₂²	1	1	yes	characteristic
14	C₄₁×#554	C₂²	1	1	yes	characteristic
15	C₄₁×#553	—	2	1	no	—
16	C₄₁×#553	—	2	1	no	—
17	C₄₁×#550	C₂²	1	1	yes	characteristic
18	C₄₁×#546	—	2	2	no	—
19	C₄₁×#546	—	2	1	no	—
20	D₄²×C₄₁	—	2	1	no	—
21	D₄²×C₄₁	C₂²	1	1	yes	characteristic
22	C₄₁×#540	—	2	1	no	—
23	C₄₁×#538	—	2	2	no	—
24	C₄₁×#532	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
26	C₄₁×#529	C₂²	1	1	yes	characteristic
27	C₄₁×#525	—	2	1	no	—
28	C₂×#121×C₄₁	—	2	1	no	—
29	C₄₁×#518	—	2	1	no	—
30	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
31	C₄₁×#501	—	2	1	no	—
32	C₄₁×#489	—	2	2	no	—
33	C₄₁×#489	—	2	1	no	—
34	C₄₁×#487	C₂²	1	2	yes	complemented
35	C₄₁×#486	C₂²	1	2	yes	complemented
36	C₄₁×#484	—	2	1	no	—
37	C₄₁×#481	—	2	1	no	—
38	C₄₁×#475	C₂²	1	2	yes	complemented
39	C₄₁×#474	C₂²	1	2	yes	complemented
40	C₄₁×#461	—	2	1	no	—
41	C₄₁×#461	—	2	1	no	—
42	C₄₁×#461	C₂²	1	2	yes	—
43	C₄₁×#461	C₂²	1	2	yes	complemented
44	C₄₁×#432	C₂²	1	1	yes	characteristic, complemented
45	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
46	C₄₁×#417	—	2	2	no	—
47	C₂×#103×C₄₁	—	2	1	no	—
48	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic, complemented
49	C₄₁×#333	C₂²	1	2	yes	complemented
50	C₄₁×#331	C₂²	1	2	yes	complemented
51	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
52	C₄₁×D₄⋊C₈	C₂²	1	2	yes	—
53	C₄₁×D₄⋊C₈	C₂²	1	2	yes	complemented
54	C₄₁×D₄⋊C₈	C₂²	1	2	yes	complemented
55	C₂×#41×C₄₁	—	2	1	no	—
56	C₂×#41×C₄₁	D₄	1	2	yes	—
57	C₂²×Q₈×C₄₁	—	2	1	no	—
58	C₂²×D₄×C₄₁	—	2	2	no	—
59	C₂²×D₄×C₄₁	—	2	1	no	—
60	C₂²×D₄×C₄₁	—	2	1	no	—
61	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
62	C₂³×C₄×C₄₁	—	2	1	no	—
63	C₂×SD₁₆×C₄₁	—	2	2	no	—
64	C₂×SD₁₆×C₄₁	—	2	2	no	—
65	C₂×SD₁₆×C₄₁	—	4	4	no	—
66	C₂×M₁₆×C₄₁	—	2	1	no	—
67	C₂²×C₈×C₄₁	—	2	1	no	—
68	C₄⋊Q₈×C₄₁	—	2	1	no	—
69	C₄⋊Q₈×C₄₁	—	2	1	no	—
70	C₄⋊Q₈×C₄₁	—	2	1	no	—
71	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
73	#127×C₄₁	—	2	1	no	—
74	#127×C₄₁	C₂³	1	1	yes	characteristic
75	#126×C₄₁	—	2	2	no	—
76	#126×C₄₁	—	2	1	no	—
77	#125×C₄₁	—	2	1	no	—
78	#125×C₄₁	D₄	1	1	yes	characteristic
79	#124×C₄₁	—	2	2	no	—
80	#124×C₄₁	D₄	1	1	yes	characteristic
81	#124×C₄₁	D₄	1	2	yes	—
82	#123×C₄₁	—	2	2	no	—
83	#123×C₄₁	—	2	2	no	—
84	#123×C₄₁	—	2	2	no	—
85	#123×C₄₁	—	2	1	no	—
86	C₂²⋊Q₈×C₄₁	—	2	1	no	—
87	C₂²⋊Q₈×C₄₁	—	2	2	no	—
88	C₂²⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
89	#121×C₄₁	—	2	2	no	—
90	#121×C₄₁	—	2	2	no	—
91	#121×C₄₁	—	2	2	no	—
92	#121×C₄₁	—	2	2	no	—
93	#121×C₄₁	—	2	1	no	—
94	#121×C₄₁	—	2	2	no	—
95	#121×C₄₁	—	2	1	no	—
96	#121×C₄₁	—	2	1	no	—
97	#121×C₄₁	—	2	1	no	—
98	#121×C₄₁	—	2	1	no	—
99	#121×C₄₁	—	2	1	no	—
100	#121×C₄₁	—	2	1	no	—
101	#121×C₄₁	D₄	1	1	yes	characteristic
102	#120×C₄₁	—	2	2	no	—
103	#120×C₄₁	—	2	2	no	—
104	#120×C₄₁	—	2	2	no	—
105	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
106	C₄×D₄×C₄₁	—	2	2	no	—
107	C₄×D₄×C₄₁	—	2	2	no	—
108	C₄×D₄×C₄₁	—	2	2	no	—
109	C₄×D₄×C₄₁	—	2	1	no	—
110	C₄×D₄×C₄₁	—	2	1	no	—
111	C₄×D₄×C₄₁	—	2	1	no	—
112	C₄×D₄×C₄₁	—	2	1	no	—
113	C₄×D₄×C₄₁	—	2	1	no	—
114	C₄×D₄×C₄₁	—	2	1	no	—
115	C₄×D₄×C₄₁	—	2	1	no	—
116	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
117	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
118	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
119	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
120	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
121	#117×C₄₁	—	2	1	no	—
122	#117×C₄₁	—	2	1	no	—
123	#117×C₄₁	—	2	1	no	—
124	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
125	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
126	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
127	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
128	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
129	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
130	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
131	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
132	#107×C₄₁	—	2	2	no	—
133	#106×C₄₁	—	2	2	no	—
134	C₄⋊C₈×C₄₁	—	2	1	no	—
135	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
136	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
137	#103×C₄₁	—	2	4	no	—
138	#103×C₄₁	—	2	2	no	—
139	#103×C₄₁	—	2	2	no	—
140	#103×C₄₁	—	2	2	no	—
141	#102×C₄₁	—	2	4	no	—
142	#102×C₄₁	—	2	2	no	—
143	C₂²⋊C₈×C₄₁	—	2	1	no	—
144	#97×C₄₁	—	2	1	no	—
145	#97×C₄₁	—	2	1	no	—
146	C₄×C₈×C₄₁	C₂³	1	2	yes	—
147	C₂⁴×C₄₁	—	2	2	no	—
148	C₂⁴×C₄₁	—	2	1	no	—
149	#41×C₄₁	—	4	4	no	—
150	#41×C₄₁	—	8	2	no	—
151	C₂×Q₈×C₄₁	—	2	2	no	—
152	C₂×Q₈×C₄₁	—	4	2	no	—
153	C₂×Q₈×C₄₁	—	4	2	no	—
154	C₂×Q₈×C₄₁	—	4	1	no	—
155	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
156	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
157	C₂×D₄×C₄₁	—	2	2	no	—
158	C₂×D₄×C₄₁	—	2	2	no	—
159	C₂×D₄×C₄₁	—	2	2	no	—
160	C₂×D₄×C₄₁	—	2	1	no	—
161	C₂×D₄×C₄₁	—	2	2	no	—
162	C₂×D₄×C₄₁	—	2	2	no	—
163	C₂×D₄×C₄₁	—	2	2	no	—
164	C₂×D₄×C₄₁	—	2	2	no	—
165	C₂×D₄×C₄₁	—	2	1	no	—
166	C₂×D₄×C₄₁	—	2	1	no	—
167	C₂×D₄×C₄₁	—	2	1	no	—
168	C₂×D₄×C₄₁	—	4	1	no	—
169	C₂×D₄×C₄₁	—	4	2	no	—
170	C₂×D₄×C₄₁	—	4	2	no	—
171	C₂×D₄×C₄₁	—	8	2	no	—
172	C₂×D₄×C₄₁	—	8	4	no	—
173	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
174	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
175	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
176	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₄₁	—	2	2	no	—
178	C₂²×C₄×C₄₁	—	2	2	no	—
179	C₂²×C₄×C₄₁	—	2	1	no	—
180	C₂²×C₄×C₄₁	—	2	1	no	—
181	C₂²×C₄×C₄₁	—	2	1	no	—
182	C₂²×C₄×C₄₁	—	2	1	no	—
183	C₂²×C₄×C₄₁	—	2	1	no	—
184	C₂²×C₄×C₄₁	—	2	2	no	—
185	C₂²×C₄×C₄₁	—	2	1	no	—
186	C₂²×C₄×C₄₁	—	4	1	no	—
187	C₂²×C₄×C₄₁	—	8	2	no	—
188	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
191	SD₁₆×C₄₁	—	4	4	no	—
192	SD₁₆×C₄₁	—	4	4	no	—
193	M₁₆×C₄₁	—	4	2	no	—
194	C₂×C₈×C₄₁	—	2	2	no	—
195	C₂×C₈×C₄₁	—	2	2	no	—
196	C₂×C₈×C₄₁	—	2	2	no	—
197	C₂×C₈×C₄₁	—	4	2	no	—
198	C₄⋊C₄×C₄₁	—	2	2	no	—
199	C₄⋊C₄×C₄₁	—	2	2	no	—
200	C₄⋊C₄×C₄₁	—	2	1	no	—
201	C₄⋊C₄×C₄₁	—	2	1	no	—
202	C₄⋊C₄×C₄₁	—	2	1	no	—
203	C₄⋊C₄×C₄₁	—	2	1	no	—
204	C₄⋊C₄×C₄₁	—	2	1	no	—
205	C₄⋊C₄×C₄₁	—	2	1	no	—
206	C₄⋊C₄×C₄₁	—	2	1	no	—
207	C₄⋊C₄×C₄₁	—	2	2	no	—
208	C₄⋊C₄×C₄₁	—	2	2	no	—
209	C₄⋊C₄×C₄₁	—	4	1	no	—
210	C₄⋊C₄×C₄₁	—	4	1	no	—
211	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
212	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
213	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
214	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
215	C₂²⋊C₄×C₄₁	—	2	2	no	—
216	C₂²⋊C₄×C₄₁	—	2	2	no	—
217	C₂²⋊C₄×C₄₁	—	2	2	no	—
218	C₂²⋊C₄×C₄₁	—	2	2	no	—
219	C₂²⋊C₄×C₄₁	—	2	2	no	—
220	C₂²⋊C₄×C₄₁	—	2	1	no	—
221	C₂²⋊C₄×C₄₁	—	2	1	no	—
222	C₂²⋊C₄×C₄₁	—	2	1	no	—
223	C₂²⋊C₄×C₄₁	—	2	1	no	—
224	C₂²⋊C₄×C₄₁	—	2	2	no	—
225	C₂²⋊C₄×C₄₁	—	2	1	no	—
226	C₂²⋊C₄×C₄₁	—	2	1	no	—
227	C₂²⋊C₄×C₄₁	—	2	2	no	—
228	C₂²⋊C₄×C₄₁	—	2	1	no	—
229	C₂²⋊C₄×C₄₁	—	2	1	no	—
230	C₂²⋊C₄×C₄₁	—	2	1	no	—
231	C₄²×C₄₁	—	2	2	no	—
232	C₄²×C₄₁	—	2	1	no	—
233	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
234	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
235	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
236	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
237	C₂³×C₄₁	—	2	1	no	—
238	C₂³×C₄₁	—	2	1	no	—
239	C₂³×C₄₁	—	2	1	no	—
240	C₂³×C₄₁	—	2	2	no	—
241	C₂³×C₄₁	—	4	2	no	—
242	C₂³×C₄₁	—	4	4	no	—
243	C₂³×C₄₁	—	4	2	no	—
244	C₂³×C₄₁	—	8	1	no	—
245	C₂³×C₄₁	—	8	2	no	—
246	C₂³×C₄₁	—	8	2	no	—
247	C₂³×C₄₁	#120	1	1	yes	characteristic
248	Q₈×C₄₁	—	2	2	no	—
249	Q₈×C₄₁	—	4	2	no	—
250	Q₈×C₄₁	—	4	2	no	—
251	Q₈×C₄₁	—	4	2	no	—
252	Q₈×C₄₁	—	8	2	no	—
253	D₄×C₄₁	—	2	1	no	—
254	D₄×C₄₁	—	2	1	no	—
255	D₄×C₄₁	—	2	1	no	—
256	D₄×C₄₁	—	2	2	no	—
257	D₄×C₄₁	—	4	2	no	—
258	D₄×C₄₁	—	4	2	no	—
259	D₄×C₄₁	—	4	4	no	—
260	D₄×C₄₁	—	4	2	no	—
261	D₄×C₄₁	—	4	4	no	—
262	D₄×C₄₁	—	4	1	no	—
263	D₄×C₄₁	—	4	1	no	—
264	D₄×C₄₁	—	4	2	no	—
265	D₄×C₄₁	—	8	2	no	—
266	D₄×C₄₁	—	8	2	no	—
267	D₄×C₄₁	—	8	2	no	—
268	D₄×C₄₁	—	8	2	no	—
269	D₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
270	C₂×C₄×C₄₁	—	2	2	no	—
271	C₂×C₄×C₄₁	—	2	1	no	—
272	C₂×C₄×C₄₁	—	2	1	no	—
273	C₂×C₄×C₄₁	—	2	2	no	—
274	C₂×C₄×C₄₁	—	2	1	no	—
275	C₂×C₄×C₄₁	—	2	1	no	—
276	C₂×C₄×C₄₁	—	2	1	no	—
277	C₂×C₄×C₄₁	—	2	2	no	—
278	C₂×C₄×C₄₁	—	2	1	no	—
279	C₂×C₄×C₄₁	—	2	1	no	—
280	C₂×C₄×C₄₁	—	4	1	no	—
281	C₂×C₄×C₄₁	—	4	1	no	—
282	C₂×C₄×C₄₁	—	4	2	no	—
283	C₂×C₄×C₄₁	—	4	2	no	—
284	C₂×C₄×C₄₁	—	4	2	no	—
285	C₂×C₄×C₄₁	—	8	1	no	—
286	C₂×C₄×C₄₁	—	8	1	no	—
287	C₂×C₄×C₄₁	—	8	2	no	—
288	C₂×C₄×C₄₁	—	8	1	no	—
289	C₂×C₄×C₄₁	—	8	2	no	—
290	C₂×C₄×C₄₁	—	8	2	no	—
291	C₂×C₄×C₄₁	—	8	2	no	—
292	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
293	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
294	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
295	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
297	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
298	C₈×C₄₁	—	4	2	no	—
299	C₈×C₄₁	—	4	2	no	—
300	C₈×C₄₁	—	4	2	no	—
301	#18570	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
302	C₂²×C₄₁	—	2	1	no	—
303	C₂²×C₄₁	—	2	1	no	—
304	C₂²×C₄₁	—	2	2	no	—
305	C₂²×C₄₁	—	4	1	no	—
306	C₂²×C₄₁	—	4	1	no	—
307	C₂²×C₄₁	—	4	1	no	—
308	C₂²×C₄₁	—	4	1	no	—
309	C₂²×C₄₁	—	4	2	no	—
310	C₂²×C₄₁	—	4	1	no	—
311	C₂²×C₄₁	—	4	2	no	—
312	C₂²×C₄₁	—	4	1	no	—
313	C₂²×C₄₁	—	4	2	no	—
314	C₂²×C₄₁	—	4	1	no	—
315	C₂²×C₄₁	—	4	1	no	—
316	C₂²×C₄₁	—	8	4	no	—
317	C₂²×C₄₁	—	16	4	no	—
318	C₂²×C₄₁	#455	1	2	yes	—
319	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
320	C₄×C₄₁	—	2	1	no	—
321	C₄×C₄₁	—	2	1	no	—
322	C₄×C₄₁	—	2	1	no	—
323	C₄×C₄₁	—	2	1	no	—
324	C₄×C₄₁	—	4	1	no	—
325	C₄×C₄₁	—	4	1	no	—
326	C₄×C₄₁	—	4	1	no	—
327	C₄×C₄₁	—	4	1	no	—
328	C₄×C₄₁	—	4	1	no	—
329	C₄×C₄₁	—	4	2	no	—
330	C₄×C₄₁	—	8	2	no	—
331	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
332	C₄×C₄₁	#575	1	1	yes	characteristic
333	C₄×C₄₁	#577	1	2	yes	—
334	#3519	C₂×C₄₁	1	1	yes	characteristic
335	#3239	C₂×C₄₁	1	1	yes	characteristic, complemented
336	#3141	C₂×C₄₁	1	1	yes	characteristic, complemented
337	C₂×#461	C₂×C₄₁	1	1	yes	characteristic, complemented
338	#1646	C₂×C₄₁	1	2	yes	complemented
339	#1643	C₂×C₄₁	1	2	yes	complemented
340	#1642	C₂×C₄₁	1	2	yes	complemented
341	#1639	C₂×C₄₁	1	2	yes	complemented
342	#1551	C₂×C₄₁	1	1	yes	characteristic, complemented
343	#1490	C₂×C₄₁	1	1	yes	characteristic, complemented
344	#1483	C₂×C₄₁	1	1	yes	characteristic, complemented
345	C₂×C₄₁	—	2	2	no	—
346	C₂×C₄₁	—	4	2	no	—
347	C₂×C₄₁	—	8	1	no	—
348	C₂×C₄₁	—	8	2	no	—
349	C₂×C₄₁	—	8	1	no	—
350	C₂×C₄₁	#3006	1	1	yes	characteristic
351	C₂×C₄₁	#3009	1	1	yes	characteristic
352	C₂×C₄₁	C₂×#455	1	1	yes	characteristic
353	C₂²×SD₁₆	—	2	1	no	—
354	#555	C₂²×C₄₁	1	1	yes	characteristic
355	#554	C₂²×C₄₁	1	1	yes	characteristic
356	#553	—	2	1	no	—
357	#553	—	2	1	no	—
358	#550	C₂²×C₄₁	1	1	yes	characteristic
359	#546	—	2	2	no	—
360	#546	—	2	1	no	—
361	D₄²	—	2	1	no	—
362	D₄²	C₂²×C₄₁	1	1	yes	characteristic
363	#540	—	2	1	no	—
364	#538	—	2	2	no	—
365	#532	C₂²×C₄₁	1	1	yes	characteristic
366	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
367	#529	C₂²×C₄₁	1	1	yes	characteristic
368	#525	—	2	1	no	—
369	C₂×#121	—	2	1	no	—
370	#518	—	2	1	no	—
371	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
372	#501	—	2	1	no	—
373	#489	—	2	2	no	—
374	#489	—	2	1	no	—
375	#487	C₂²×C₄₁	1	2	yes	complemented
376	#486	C₂²×C₄₁	1	2	yes	complemented
377	#484	—	2	1	no	—
378	#481	—	2	1	no	—
379	#475	C₂²×C₄₁	1	2	yes	complemented
380	#474	C₂²×C₄₁	1	2	yes	complemented
381	#461	—	2	1	no	—
382	#461	—	2	1	no	—
383	#461	C₂²×C₄₁	1	2	yes	—
384	#461	C₂²×C₄₁	1	2	yes	complemented
385	#432	C₂²×C₄₁	1	1	yes	characteristic, complemented
386	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
387	#417	—	2	2	no	—
388	C₂×#103	—	2	1	no	—
389	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic, complemented
390	#333	C₂²×C₄₁	1	2	yes	complemented
391	#331	C₂²×C₄₁	1	2	yes	complemented
392	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	complemented
393	D₄⋊C₈	C₂²×C₄₁	1	2	yes	—
394	D₄⋊C₈	C₂²×C₄₁	1	2	yes	complemented
395	D₄⋊C₈	C₂²×C₄₁	1	2	yes	complemented
396	C₄₁	#18570	1	1	yes	characteristic, direct product factor, Sylow
397	C₂×#41	—	2	1	no	—
398	C₂×#41	D₄×C₄₁	1	2	yes	—
399	C₂²×Q₈	—	2	1	no	—
400	C₂²×D₄	—	2	2	no	—
401	C₂²×D₄	—	2	1	no	—
402	C₂²×D₄	—	2	1	no	—
403	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
404	C₂³×C₄	—	2	1	no	—
405	C₂×SD₁₆	—	2	2	no	—
406	C₂×SD₁₆	—	2	2	no	—
407	C₂×SD₁₆	—	4	4	no	—
408	C₂×M₁₆	—	2	1	no	—
409	C₂²×C₈	—	2	1	no	—
410	C₄⋊Q₈	—	2	1	no	—
411	C₄⋊Q₈	—	2	1	no	—
412	C₄⋊Q₈	—	2	1	no	—
413	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
414	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
415	#127	—	2	1	no	—
416	#127	C₂³×C₄₁	1	1	yes	characteristic
417	#126	—	2	2	no	—
418	#126	—	2	1	no	—
419	#125	—	2	1	no	—
420	#125	D₄×C₄₁	1	1	yes	characteristic
421	#124	—	2	2	no	—
422	#124	D₄×C₄₁	1	1	yes	characteristic
423	#124	D₄×C₄₁	1	2	yes	—
424	#123	—	2	2	no	—
425	#123	—	2	2	no	—
426	#123	—	2	2	no	—
427	#123	—	2	1	no	—
428	C₂²⋊Q₈	—	2	1	no	—
429	C₂²⋊Q₈	—	2	2	no	—
430	C₂²⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
431	#121	—	2	2	no	—
432	#121	—	2	2	no	—
433	#121	—	2	2	no	—
434	#121	—	2	2	no	—
435	#121	—	2	1	no	—
436	#121	—	2	2	no	—
437	#121	—	2	1	no	—
438	#121	—	2	1	no	—
439	#121	—	2	1	no	—
440	#121	—	2	1	no	—
441	#121	—	2	1	no	—
442	#121	—	2	1	no	—
443	#121	D₄×C₄₁	1	1	yes	characteristic
444	#120	—	2	2	no	—
445	#120	—	2	2	no	—
446	#120	—	2	2	no	—
447	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
448	C₄×D₄	—	2	2	no	—
449	C₄×D₄	—	2	2	no	—
450	C₄×D₄	—	2	2	no	—
451	C₄×D₄	—	2	1	no	—
452	C₄×D₄	—	2	1	no	—
453	C₄×D₄	—	2	1	no	—
454	C₄×D₄	—	2	1	no	—
455	C₄×D₄	—	2	1	no	—
456	C₄×D₄	—	2	1	no	—
457	C₄×D₄	—	2	1	no	—
458	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
459	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
460	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
461	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
462	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
463	#117	—	2	1	no	—
464	#117	—	2	1	no	—
465	#117	—	2	1	no	—
466	C₂×C₄⋊C₄	—	2	1	no	—
467	C₂×C₄⋊C₄	—	2	1	no	—
468	C₂×C₄⋊C₄	—	2	1	no	—
469	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
470	C₂×C₂²⋊C₄	—	2	2	no	—
471	C₂×C₂²⋊C₄	—	2	2	no	—
472	C₂×C₂²⋊C₄	—	2	1	no	—
473	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
474	#107	—	2	2	no	—
475	#106	—	2	2	no	—
476	C₄⋊C₈	—	2	1	no	—
477	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
478	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
479	#103	—	2	4	no	—
480	#103	—	2	2	no	—
481	#103	—	2	2	no	—
482	#103	—	2	2	no	—
483	#102	—	2	4	no	—
484	#102	—	2	2	no	—
485	C₂²⋊C₈	—	2	1	no	—
486	#97	—	2	1	no	—
487	#97	—	2	1	no	—
488	C₄×C₈	C₂³×C₄₁	1	2	yes	—
489	C₂⁴	—	2	2	no	—
490	C₂⁴	—	2	1	no	—
491	#41	—	4	4	no	—
492	#41	—	8	2	no	—
493	C₂×Q₈	—	2	2	no	—
494	C₂×Q₈	—	4	2	no	—
495	C₂×Q₈	—	4	2	no	—
496	C₂×Q₈	—	4	1	no	—
497	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
498	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
499	C₂×D₄	—	2	2	no	—
500	C₂×D₄	—	2	2	no	—
501	C₂×D₄	—	2	2	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	—	2	2	no	—
505	C₂×D₄	—	2	2	no	—
506	C₂×D₄	—	2	2	no	—
507	C₂×D₄	—	2	1	no	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	—	2	1	no	—
510	C₂×D₄	—	4	1	no	—
511	C₂×D₄	—	4	2	no	—
512	C₂×D₄	—	4	2	no	—
513	C₂×D₄	—	8	2	no	—
514	C₂×D₄	—	8	4	no	—
515	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
516	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
517	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
518	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
519	C₂²×C₄	—	2	2	no	—
520	C₂²×C₄	—	2	2	no	—
521	C₂²×C₄	—	2	1	no	—
522	C₂²×C₄	—	2	1	no	—
523	C₂²×C₄	—	2	1	no	—
524	C₂²×C₄	—	2	1	no	—
525	C₂²×C₄	—	2	1	no	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	—	4	1	no	—
529	C₂²×C₄	—	8	2	no	—
530	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
531	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
532	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
533	SD₁₆	—	4	4	no	—
534	SD₁₆	—	4	4	no	—
535	M₁₆	—	4	2	no	—
536	C₂×C₈	—	2	2	no	—
537	C₂×C₈	—	2	2	no	—
538	C₂×C₈	—	2	2	no	—
539	C₂×C₈	—	4	2	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	—	2	1	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	1	no	—
548	C₄⋊C₄	—	2	1	no	—
549	C₄⋊C₄	—	2	2	no	—
550	C₄⋊C₄	—	2	2	no	—
551	C₄⋊C₄	—	4	1	no	—
552	C₄⋊C₄	—	4	1	no	—
553	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
554	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
555	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
556	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
557	C₂²⋊C₄	—	2	2	no	—
558	C₂²⋊C₄	—	2	2	no	—
559	C₂²⋊C₄	—	2	2	no	—
560	C₂²⋊C₄	—	2	2	no	—
561	C₂²⋊C₄	—	2	2	no	—
562	C₂²⋊C₄	—	2	1	no	—
563	C₂²⋊C₄	—	2	1	no	—
564	C₂²⋊C₄	—	2	1	no	—
565	C₂²⋊C₄	—	2	1	no	—
566	C₂²⋊C₄	—	2	2	no	—
567	C₂²⋊C₄	—	2	1	no	—
568	C₂²⋊C₄	—	2	1	no	—
569	C₂²⋊C₄	—	2	2	no	—
570	C₂²⋊C₄	—	2	1	no	—
571	C₂²⋊C₄	—	2	1	no	—
572	C₂²⋊C₄	—	2	1	no	—
573	C₄²	—	2	2	no	—
574	C₄²	—	2	1	no	—
575	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
576	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
577	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
578	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
579	C₂³	—	2	1	no	—
580	C₂³	—	2	1	no	—
581	C₂³	—	2	1	no	—
582	C₂³	—	2	2	no	—
583	C₂³	—	4	2	no	—
584	C₂³	—	4	4	no	—
585	C₂³	—	4	2	no	—
586	C₂³	—	8	1	no	—
587	C₂³	—	8	2	no	—
588	C₂³	—	8	2	no	—
589	C₂³	#120×C₄₁	1	1	yes	characteristic
590	Q₈	—	2	2	no	—
591	Q₈	—	4	2	no	—
592	Q₈	—	4	2	no	—
593	Q₈	—	4	2	no	—
594	Q₈	—	8	2	no	—
595	D₄	—	2	1	no	—
596	D₄	—	2	1	no	—
597	D₄	—	2	1	no	—
598	D₄	—	2	2	no	—
599	D₄	—	4	2	no	—
600	D₄	—	4	2	no	—
601	D₄	—	4	4	no	—
602	D₄	—	4	2	no	—
603	D₄	—	4	4	no	—
604	D₄	—	4	1	no	—
605	D₄	—	4	1	no	—
606	D₄	—	4	2	no	—
607	D₄	—	8	2	no	—
608	D₄	—	8	2	no	—
609	D₄	—	8	2	no	—
610	D₄	—	8	2	no	—
611	D₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	4	1	no	—
623	C₂×C₄	—	4	1	no	—
624	C₂×C₄	—	4	2	no	—
625	C₂×C₄	—	4	2	no	—
626	C₂×C₄	—	4	2	no	—
627	C₂×C₄	—	8	1	no	—
628	C₂×C₄	—	8	1	no	—
629	C₂×C₄	—	8	2	no	—
630	C₂×C₄	—	8	1	no	—
631	C₂×C₄	—	8	2	no	—
632	C₂×C₄	—	8	2	no	—
633	C₂×C₄	—	8	2	no	—
634	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
635	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
636	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
637	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
638	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
639	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
640	C₈	—	4	2	no	—
641	C₈	—	4	2	no	—
642	C₈	—	4	2	no	—
643	C₂²	—	2	1	no	—
644	C₂²	—	2	1	no	—
645	C₂²	—	2	2	no	—
646	C₂²	—	4	1	no	—
647	C₂²	—	4	1	no	—
648	C₂²	—	4	1	no	—
649	C₂²	—	4	1	no	—
650	C₂²	—	4	2	no	—
651	C₂²	—	4	1	no	—
652	C₂²	—	4	2	no	—
653	C₂²	—	4	1	no	—
654	C₂²	—	4	2	no	—
655	C₂²	—	4	1	no	—
656	C₂²	—	4	1	no	—
657	C₂²	—	8	4	no	—
658	C₂²	—	16	4	no	—
659	C₂²	C₄₁×#455	1	2	yes	—
660	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
661	C₄	—	2	1	no	—
662	C₄	—	2	1	no	—
663	C₄	—	2	1	no	—
664	C₄	—	2	1	no	—
665	C₄	—	4	1	no	—
666	C₄	—	4	1	no	—
667	C₄	—	4	1	no	—
668	C₄	—	4	1	no	—
669	C₄	—	4	1	no	—
670	C₄	—	4	2	no	—
671	C₄	—	8	2	no	—
672	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
673	C₄	C₄₁×#575	1	1	yes	characteristic
674	C₄	C₄₁×#577	1	2	yes	—
675	C₂	—	2	2	no	—
676	C₂	—	4	2	no	—
677	C₂	—	8	1	no	—
678	C₂	—	8	1	no	—
679	C₂	—	8	2	no	—
680	C₂	C₄₁×#3006	1	1	yes	characteristic
681	C₂	C₄₁×#3009	1	1	yes	characteristic
682	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×#3519	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×#3239	C₂	semidirect
3. C₄₁×#3141	C₂	semidirect
4. C₂×C₄₁×#461	C₂	semidirect
5. C₄₁×#1646	C₂	semidirect
6. C₄₁×#1643	C₂	semidirect
7. C₄₁×#1642	C₂	semidirect
8. C₄₁×#1639	C₂	semidirect
9. C₄₁×#1551	C₂	semidirect
10. C₄₁×#1490	C₂	semidirect
11. C₄₁×#1483	C₂	semidirect
25. C₂×C₄⋊Q₈×C₄₁	C₂²	semidirect
34. C₄₁×#487	C₂²	semidirect
35. C₄₁×#486	C₂²	semidirect
38. C₄₁×#475	C₂²	semidirect
39. C₄₁×#474	C₂²	semidirect
43. C₄₁×#461	C₂²	semidirect
44. C₄₁×#432	C₂²	semidirect
45. C₂×C₄⋊C₈×C₄₁	C₂²	semidirect
48. C₄×M₁₆×C₄₁	C₂²	semidirect
49. C₄₁×#333	C₂²	semidirect
50. C₄₁×#331	C₂²	semidirect
51. C₄₁×Q₈⋊C₈	C₂²	semidirect
53. C₄₁×D₄⋊C₈	C₂²	semidirect
54. C₄₁×D₄⋊C₈	C₂²	semidirect
301. #18570	C₄₁	direct (N × H)
335. #3239	C₂×C₄₁	semidirect
336. #3141	C₂×C₄₁	semidirect
337. C₂×#461	C₂×C₄₁	semidirect
338. #1646	C₂×C₄₁	semidirect
339. #1643	C₂×C₄₁	semidirect
340. #1642	C₂×C₄₁	semidirect
341. #1639	C₂×C₄₁	semidirect
342. #1551	C₂×C₄₁	semidirect
343. #1490	C₂×C₄₁	semidirect
344. #1483	C₂×C₄₁	semidirect
366. C₂×C₄⋊Q₈	C₂²×C₄₁	semidirect
375. #487	C₂²×C₄₁	semidirect
376. #486	C₂²×C₄₁	semidirect
379. #475	C₂²×C₄₁	semidirect
380. #474	C₂²×C₄₁	semidirect
384. #461	C₂²×C₄₁	semidirect
385. #432	C₂²×C₄₁	semidirect
386. C₂×C₄⋊C₈	C₂²×C₄₁	semidirect
389. C₄×M₁₆	C₂²×C₄₁	semidirect
390. #333	C₂²×C₄₁	semidirect
391. #331	C₂²×C₄₁	semidirect
392. Q₈⋊C₈	C₂²×C₄₁	semidirect
394. D₄⋊C₈	C₂²×C₄₁	semidirect
395. D₄⋊C₈	C₂²×C₄₁	semidirect
396. C₄₁	#18570	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×#3519	2
C₄₁×#3239	2
C₄₁×#3141	2
C₂×C₄₁×#461	2
C₄₁×#1646	2
C₄₁×#1643	2
C₄₁×#1642	2
C₄₁×#1639	2
C₄₁×#1551	2
C₄₁×#1490	2
C₄₁×#1483	2
#18570	41

Subgroups — C41×#18570 #23794771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#18570 #23794771