Subgroups — C₄₁×#18553 #23794754

Order 10496.

1008 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3483	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#3141	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3079	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#3039	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1646	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1645	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1642	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1641	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1560	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1489	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1489	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₄₁	—	2	1	no	—
13	Q₈²×C₄₁	C₂²	1	1	yes	characteristic
14	C₄₁×#555	—	2	2	no	—
15	C₄₁×#555	—	2	1	no	—
16	C₄₁×#555	—	2	1	no	—
17	C₄₁×#555	C₂²	1	1	yes	characteristic
18	C₄₁×#550	—	2	1	no	—
19	C₄₁×#550	—	2	1	no	—
20	C₄₁×#550	C₂²	1	1	yes	characteristic
21	C₄₁×#550	C₂²	1	1	yes	characteristic
22	C₄₁×#532	—	2	2	no	—
23	C₄₁×#532	—	2	1	no	—
24	C₄₁×#532	—	2	1	no	—
25	C₄₁×#532	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
27	C₄×#41×C₄₁	—	2	1	no	—
28	C₄×#41×C₄₁	—	2	1	no	—
29	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
30	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
31	C₄₁×#489	—	2	2	no	—
32	C₄₁×#487	C₂²	1	2	yes	—
33	C₄₁×#486	C₂²	1	2	yes	—
34	C₄₁×#477	—	2	1	no	—
35	C₄₁×#476	—	2	1	no	—
36	C₄₁×#475	C₂²	1	2	yes	—
37	C₄₁×#474	C₂²	1	2	yes	—
38	C₄₁×#462	—	2	1	no	—
39	C₄₁×#462	C₂²	1	2	yes	—
40	C₄₁×#461	—	2	1	no	—
41	C₄₁×#461	C₂²	1	2	yes	—
42	C₄₁×#428	—	2	1	no	—
43	C₄₁×#423	C₂²	1	1	yes	characteristic
44	C₄₁×#423	C₂²	1	1	yes	characteristic
45	C₄₁×#419	—	2	1	no	—
46	C₄₁×#417	—	2	1	no	—
47	C₄₁×#417	—	2	2	no	—
48	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
49	C₄₁×#333	C₂²	1	2	yes	—
50	C₄₁×#331	C₂²	1	2	yes	—
51	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
52	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
53	C₄₁×D₄⋊C₈	C₂²	1	2	yes	—
54	C₄₁×D₄⋊C₈	C₂²	1	2	yes	—
55	C₂×#41×C₄₁	—	2	2	no	—
56	C₂×#41×C₄₁	—	2	1	no	—
57	C₂×#41×C₄₁	—	2	1	no	—
58	C₂×#41×C₄₁	D₄	1	1	yes	characteristic
59	C₂×#41×C₄₁	D₄	1	1	yes	characteristic
60	C₂×#41×C₄₁	D₄	1	2	yes	—
61	C₂²×Q₈×C₄₁	—	2	1	no	—
62	#137×C₄₁	—	4	4	no	—
63	C₂×Q₁₆×C₄₁	—	2	2	no	—
64	C₂×SD₁₆×C₄₁	—	2	2	no	—
65	C₂×M₁₆×C₄₁	—	2	1	no	—
66	C₂×M₁₆×C₄₁	—	2	1	no	—
67	C₂×M₁₆×C₄₁	—	2	1	no	—
68	C₄⋊Q₈×C₄₁	—	2	2	no	—
69	C₄⋊Q₈×C₄₁	—	2	1	no	—
70	C₄⋊Q₈×C₄₁	—	2	1	no	—
71	C₄⋊Q₈×C₄₁	—	2	1	no	—
72	C₄⋊Q₈×C₄₁	—	2	1	no	—
73	C₄⋊Q₈×C₄₁	—	2	1	no	—
74	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
76	#127×C₄₁	—	2	2	no	—
77	#127×C₄₁	—	2	1	no	—
78	#127×C₄₁	—	2	1	no	—
79	#127×C₄₁	C₂³	1	1	yes	characteristic
80	#124×C₄₁	—	2	2	no	—
81	#124×C₄₁	—	2	2	no	—
82	#124×C₄₁	—	2	2	no	—
83	#124×C₄₁	—	2	2	no	—
84	#124×C₄₁	D₄	1	2	yes	—
85	#121×C₄₁	—	2	2	no	—
86	#121×C₄₁	—	2	2	no	—
87	#121×C₄₁	—	2	2	no	—
88	#121×C₄₁	—	2	2	no	—
89	#121×C₄₁	—	2	1	no	—
90	#121×C₄₁	—	2	1	no	—
91	#121×C₄₁	—	2	2	no	—
92	#121×C₄₁	—	2	2	no	—
93	#121×C₄₁	—	2	1	no	—
94	#121×C₄₁	—	2	1	no	—
95	#121×C₄₁	—	2	1	no	—
96	#121×C₄₁	—	2	1	no	—
97	C₄×Q₈×C₄₁	—	2	1	no	—
98	C₄×Q₈×C₄₁	—	2	1	no	—
99	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
100	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
101	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
102	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
103	C₄×D₄×C₄₁	—	2	2	no	—
104	C₄×D₄×C₄₁	—	2	2	no	—
105	C₄×D₄×C₄₁	—	2	2	no	—
106	C₄×D₄×C₄₁	—	2	2	no	—
107	C₄×D₄×C₄₁	—	2	1	no	—
108	C₄×D₄×C₄₁	—	2	1	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	D₄	1	1	yes	characteristic
116	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
117	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
118	C₄×D₄×C₄₁	C₂³	1	1	yes	characteristic
119	#117×C₄₁	—	2	2	no	—
120	#117×C₄₁	—	2	2	no	—
121	#117×C₄₁	—	2	1	no	—
122	#117×C₄₁	—	2	1	no	—
123	#117×C₄₁	—	2	1	no	—
124	#117×C₄₁	—	2	1	no	—
125	#117×C₄₁	D₄	1	1	yes	characteristic
126	#117×C₄₁	D₄	1	1	yes	characteristic
127	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
128	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
129	C₂×C₄²×C₄₁	—	2	2	no	—
130	C₂×C₄²×C₄₁	—	2	1	no	—
131	C₂×C₄²×C₄₁	—	2	1	no	—
132	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
133	#107×C₄₁	—	2	2	no	—
134	#106×C₄₁	—	2	2	no	—
135	C₄⋊C₈×C₄₁	—	2	1	no	—
136	C₄⋊C₈×C₄₁	—	2	1	no	—
137	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
138	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
139	#103×C₄₁	—	2	2	no	—
140	#103×C₄₁	—	2	4	no	—
141	#103×C₄₁	—	2	2	no	—
142	#103×C₄₁	—	2	2	no	—
143	#102×C₄₁	—	2	2	no	—
144	#102×C₄₁	—	2	4	no	—
145	#97×C₄₁	—	2	1	no	—
146	C₄×C₈×C₄₁	C₂³	1	2	yes	—
147	#41×C₄₁	—	2	2	no	—
148	#41×C₄₁	—	2	2	no	—
149	#41×C₄₁	—	4	1	no	—
150	#41×C₄₁	—	4	1	no	—
151	#41×C₄₁	—	4	4	no	—
152	#41×C₄₁	—	4	2	no	—
153	#41×C₄₁	—	4	2	no	—
154	#41×C₄₁	—	4	2	no	—
155	#41×C₄₁	—	4	2	no	—
156	#41×C₄₁	—	8	4	no	—
157	C₂×Q₈×C₄₁	—	2	2	no	—
158	C₂×Q₈×C₄₁	—	2	2	no	—
159	C₂×Q₈×C₄₁	—	4	2	no	—
160	C₂×Q₈×C₄₁	—	4	2	no	—
161	C₂×Q₈×C₄₁	—	4	1	no	—
162	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
163	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
164	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
165	C₂×D₄×C₄₁	—	2	2	no	—
166	C₂×D₄×C₄₁	—	2	2	no	—
167	C₂×D₄×C₄₁	—	2	1	no	—
168	C₂×D₄×C₄₁	—	2	1	no	—
169	C₂×D₄×C₄₁	—	2	2	no	—
170	C₂×D₄×C₄₁	—	2	2	no	—
171	C₂×D₄×C₄₁	—	2	2	no	—
172	C₂×D₄×C₄₁	—	2	2	no	—
173	C₂×D₄×C₄₁	—	2	1	no	—
174	C₂×D₄×C₄₁	—	2	1	no	—
175	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
176	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
177	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
178	C₂²×C₄×C₄₁	—	2	2	no	—
179	C₂²×C₄×C₄₁	—	2	2	no	—
180	C₂²×C₄×C₄₁	—	2	1	no	—
181	C₂²×C₄×C₄₁	—	2	1	no	—
182	C₂²×C₄×C₄₁	—	2	2	no	—
183	C₂²×C₄×C₄₁	—	2	1	no	—
184	C₂²×C₄×C₄₁	—	2	1	no	—
185	C₂²×C₄×C₄₁	—	2	1	no	—
186	C₂²×C₄×C₄₁	—	2	1	no	—
187	C₂²×C₄×C₄₁	—	4	1	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	Q₁₆×C₄₁	—	4	4	no	—
192	SD₁₆×C₄₁	—	4	4	no	—
193	M₁₆×C₄₁	—	4	2	no	—
194	M₁₆×C₄₁	—	4	2	no	—
195	M₁₆×C₄₁	—	4	2	no	—
196	C₂×C₈×C₄₁	—	2	2	no	—
197	C₂×C₈×C₄₁	—	2	2	no	—
198	C₂×C₈×C₄₁	—	2	2	no	—
199	C₄⋊C₄×C₄₁	—	2	2	no	—
200	C₄⋊C₄×C₄₁	—	2	2	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	1	no	—
220	C₂²⋊C₄×C₄₁	—	2	1	no	—
221	C₂²⋊C₄×C₄₁	—	2	2	no	—
222	C₂²⋊C₄×C₄₁	—	2	2	no	—
223	C₂²⋊C₄×C₄₁	—	2	1	no	—
224	C₂²⋊C₄×C₄₁	—	2	1	no	—
225	C₂²⋊C₄×C₄₁	—	2	1	no	—
226	C₂²⋊C₄×C₄₁	—	2	1	no	—
227	C₄²×C₄₁	—	2	2	no	—
228	C₄²×C₄₁	—	2	2	no	—
229	C₄²×C₄₁	—	2	1	no	—
230	C₄²×C₄₁	—	2	1	no	—
231	C₄²×C₄₁	—	2	1	no	—
232	C₄²×C₄₁	—	2	1	no	—
233	C₄²×C₄₁	—	2	1	no	—
234	C₄²×C₄₁	—	2	1	no	—
235	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
236	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
237	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
238	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
239	C₂³×C₄₁	—	2	1	no	—
240	C₂³×C₄₁	—	2	1	no	—
241	C₂³×C₄₁	—	2	2	no	—
242	C₂³×C₄₁	#120	1	1	yes	characteristic
243	Q₈×C₄₁	—	2	1	no	—
244	Q₈×C₄₁	—	2	1	no	—
245	Q₈×C₄₁	—	4	2	no	—
246	Q₈×C₄₁	—	4	2	no	—
247	Q₈×C₄₁	—	4	2	no	—
248	Q₈×C₄₁	—	8	2	no	—
249	Q₈×C₄₁	—	8	2	no	—
250	Q₈×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
251	Q₈×C₄₁	#137	1	2	yes	complemented
252	D₄×C₄₁	—	2	1	no	—
253	D₄×C₄₁	—	2	1	no	—
254	D₄×C₄₁	—	2	1	no	—
255	D₄×C₄₁	—	2	1	no	—
256	D₄×C₄₁	—	4	2	no	—
257	D₄×C₄₁	—	4	2	no	—
258	D₄×C₄₁	—	4	2	no	—
259	D₄×C₄₁	—	4	4	no	—
260	D₄×C₄₁	—	4	4	no	—
261	D₄×C₄₁	—	4	1	no	—
262	D₄×C₄₁	—	4	1	no	—
263	D₄×C₄₁	—	4	1	no	—
264	D₄×C₄₁	—	4	1	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	C₂×C₄×C₄₁	—	2	1	no	—
270	C₂×C₄×C₄₁	—	2	1	no	—
271	C₂×C₄×C₄₁	—	2	1	no	—
272	C₂×C₄×C₄₁	—	2	1	no	—
273	C₂×C₄×C₄₁	—	2	1	no	—
274	C₂×C₄×C₄₁	—	2	1	no	—
275	C₂×C₄×C₄₁	—	2	2	no	—
276	C₂×C₄×C₄₁	—	2	1	no	—
277	C₂×C₄×C₄₁	—	2	1	no	—
278	C₂×C₄×C₄₁	—	2	2	no	—
279	C₂×C₄×C₄₁	—	2	1	no	—
280	C₂×C₄×C₄₁	—	2	1	no	—
281	C₂×C₄×C₄₁	—	4	2	no	—
282	C₂×C₄×C₄₁	—	4	2	no	—
283	C₂×C₄×C₄₁	—	4	4	no	—
284	C₂×C₄×C₄₁	—	4	4	no	—
285	C₂×C₄×C₄₁	—	4	1	no	—
286	C₂×C₄×C₄₁	—	4	1	no	—
287	C₂×C₄×C₄₁	—	4	1	no	—
288	C₂×C₄×C₄₁	—	4	1	no	—
289	C₂×C₄×C₄₁	—	4	4	no	—
290	C₂×C₄×C₄₁	—	4	2	no	—
291	C₂×C₄×C₄₁	—	4	2	no	—
292	C₂×C₄×C₄₁	—	4	2	no	—
293	C₂×C₄×C₄₁	—	8	2	no	—
294	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
295	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
296	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
297	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
298	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
300	C₈×C₄₁	—	4	2	no	—
301	C₈×C₄₁	—	4	2	no	—
302	C₈×C₄₁	—	4	2	no	—
303	#18553	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
304	C₂²×C₄₁	—	2	1	no	—
305	C₂²×C₄₁	—	2	1	no	—
306	C₂²×C₄₁	—	2	1	no	—
307	C₂²×C₄₁	—	4	1	no	—
308	C₂²×C₄₁	—	4	2	no	—
309	C₂²×C₄₁	—	4	1	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	C₂×#120	1	1	yes	characteristic
317	C₄×C₄₁	—	2	1	no	—
318	C₄×C₄₁	—	2	2	no	—
319	C₄×C₄₁	—	2	1	no	—
320	C₄×C₄₁	—	2	2	no	—
321	C₄×C₄₁	—	2	1	no	—
322	C₄×C₄₁	—	4	1	no	—
323	C₄×C₄₁	—	4	1	no	—
324	C₄×C₄₁	—	4	2	no	—
325	C₄×C₄₁	—	4	1	no	—
326	C₄×C₄₁	—	4	1	no	—
327	C₄×C₄₁	—	8	2	no	—
328	C₄×C₄₁	#455	1	2	yes	—
329	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
330	C₄×C₄₁	C₂×#137	1	1	yes	characteristic
331	C₄×C₄₁	#575	1	1	yes	characteristic
332	C₄×C₄₁	#575	1	1	yes	characteristic
333	#3483	C₂×C₄₁	1	1	yes	characteristic
334	#3141	C₂×C₄₁	1	1	yes	characteristic, complemented
335	#3079	C₂×C₄₁	1	1	yes	characteristic, complemented
336	#3039	C₂×C₄₁	1	1	yes	characteristic, complemented
337	#1646	C₂×C₄₁	1	2	yes	complemented
338	#1645	C₂×C₄₁	1	2	yes	complemented
339	#1642	C₂×C₄₁	1	2	yes	complemented
340	#1641	C₂×C₄₁	1	2	yes	complemented
341	#1560	C₂×C₄₁	1	1	yes	characteristic, complemented
342	#1489	C₂×C₄₁	1	1	yes	characteristic, complemented
343	#1489	C₂×C₄₁	1	1	yes	characteristic, complemented
344	C₂×C₄₁	—	4	1	no	—
345	C₂×C₄₁	—	8	1	no	—
346	C₂×C₄₁	—	8	1	no	—
347	C₂×C₄₁	—	8	2	no	—
348	C₂×C₄₁	#3006	1	1	yes	characteristic
349	C₂×C₄₁	#3007	1	1	yes	characteristic
350	C₂×C₄₁	C₂×#455	1	1	yes	characteristic
351	C₂×#137	—	2	1	no	—
352	Q₈²	C₂²×C₄₁	1	1	yes	characteristic
353	#555	—	2	2	no	—
354	#555	—	2	1	no	—
355	#555	—	2	1	no	—
356	#555	C₂²×C₄₁	1	1	yes	characteristic
357	#550	—	2	1	no	—
358	#550	—	2	1	no	—
359	#550	C₂²×C₄₁	1	1	yes	characteristic
360	#550	C₂²×C₄₁	1	1	yes	characteristic
361	#532	—	2	2	no	—
362	#532	—	2	1	no	—
363	#532	—	2	1	no	—
364	#532	C₂²×C₄₁	1	1	yes	characteristic
365	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
366	C₄×#41	—	2	1	no	—
367	C₄×#41	—	2	1	no	—
368	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
369	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
370	#489	—	2	2	no	—
371	#487	C₂²×C₄₁	1	2	yes	—
372	#486	C₂²×C₄₁	1	2	yes	—
373	#477	—	2	1	no	—
374	#476	—	2	1	no	—
375	#475	C₂²×C₄₁	1	2	yes	—
376	#474	C₂²×C₄₁	1	2	yes	—
377	#462	—	2	1	no	—
378	#462	C₂²×C₄₁	1	2	yes	—
379	#461	—	2	1	no	—
380	#461	C₂²×C₄₁	1	2	yes	—
381	#428	—	2	1	no	—
382	#423	C₂²×C₄₁	1	1	yes	characteristic
383	#423	C₂²×C₄₁	1	1	yes	characteristic
384	#419	—	2	1	no	—
385	#417	—	2	1	no	—
386	#417	—	2	2	no	—
387	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
388	#333	C₂²×C₄₁	1	2	yes	—
389	#331	C₂²×C₄₁	1	2	yes	—
390	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
391	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
392	D₄⋊C₈	C₂²×C₄₁	1	2	yes	—
393	D₄⋊C₈	C₂²×C₄₁	1	2	yes	—
394	C₄₁	#18553	1	1	yes	characteristic, direct product factor, Sylow
395	C₂×#41	—	2	2	no	—
396	C₂×#41	—	2	1	no	—
397	C₂×#41	—	2	1	no	—
398	C₂×#41	D₄×C₄₁	1	1	yes	characteristic
399	C₂×#41	D₄×C₄₁	1	1	yes	characteristic
400	C₂×#41	D₄×C₄₁	1	2	yes	—
401	C₂²×Q₈	—	2	1	no	—
402	#137	—	4	4	no	—
403	C₂×Q₁₆	—	2	2	no	—
404	C₂×SD₁₆	—	2	2	no	—
405	C₂×M₁₆	—	2	1	no	—
406	C₂×M₁₆	—	2	1	no	—
407	C₂×M₁₆	—	2	1	no	—
408	C₄⋊Q₈	—	2	2	no	—
409	C₄⋊Q₈	—	2	1	no	—
410	C₄⋊Q₈	—	2	1	no	—
411	C₄⋊Q₈	—	2	1	no	—
412	C₄⋊Q₈	—	2	1	no	—
413	C₄⋊Q₈	—	2	1	no	—
414	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
415	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
416	#127	—	2	2	no	—
417	#127	—	2	1	no	—
418	#127	—	2	1	no	—
419	#127	C₂³×C₄₁	1	1	yes	characteristic
420	#124	—	2	2	no	—
421	#124	—	2	2	no	—
422	#124	—	2	2	no	—
423	#124	—	2	2	no	—
424	#124	D₄×C₄₁	1	2	yes	—
425	#121	—	2	2	no	—
426	#121	—	2	2	no	—
427	#121	—	2	2	no	—
428	#121	—	2	2	no	—
429	#121	—	2	1	no	—
430	#121	—	2	1	no	—
431	#121	—	2	2	no	—
432	#121	—	2	2	no	—
433	#121	—	2	1	no	—
434	#121	—	2	1	no	—
435	#121	—	2	1	no	—
436	#121	—	2	1	no	—
437	C₄×Q₈	—	2	1	no	—
438	C₄×Q₈	—	2	1	no	—
439	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
440	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
441	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
442	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
443	C₄×D₄	—	2	2	no	—
444	C₄×D₄	—	2	2	no	—
445	C₄×D₄	—	2	2	no	—
446	C₄×D₄	—	2	2	no	—
447	C₄×D₄	—	2	1	no	—
448	C₄×D₄	—	2	1	no	—
449	C₄×D₄	—	2	1	no	—
450	C₄×D₄	—	2	1	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	D₄×C₄₁	1	1	yes	characteristic
456	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
457	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
458	C₄×D₄	C₂³×C₄₁	1	1	yes	characteristic
459	#117	—	2	2	no	—
460	#117	—	2	2	no	—
461	#117	—	2	1	no	—
462	#117	—	2	1	no	—
463	#117	—	2	1	no	—
464	#117	—	2	1	no	—
465	#117	D₄×C₄₁	1	1	yes	characteristic
466	#117	D₄×C₄₁	1	1	yes	characteristic
467	C₂×C₄⋊C₄	—	2	1	no	—
468	C₂×C₄⋊C₄	—	2	1	no	—
469	C₂×C₄²	—	2	2	no	—
470	C₂×C₄²	—	2	1	no	—
471	C₂×C₄²	—	2	1	no	—
472	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
473	#107	—	2	2	no	—
474	#106	—	2	2	no	—
475	C₄⋊C₈	—	2	1	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	2	no	—
480	#103	—	2	4	no	—
481	#103	—	2	2	no	—
482	#103	—	2	2	no	—
483	#102	—	2	2	no	—
484	#102	—	2	4	no	—
485	#97	—	2	1	no	—
486	C₄×C₈	C₂³×C₄₁	1	2	yes	—
487	#41	—	2	2	no	—
488	#41	—	2	2	no	—
489	#41	—	4	1	no	—
490	#41	—	4	1	no	—
491	#41	—	4	4	no	—
492	#41	—	4	2	no	—
493	#41	—	4	2	no	—
494	#41	—	4	2	no	—
495	#41	—	4	2	no	—
496	#41	—	8	4	no	—
497	C₂×Q₈	—	2	2	no	—
498	C₂×Q₈	—	2	2	no	—
499	C₂×Q₈	—	4	2	no	—
500	C₂×Q₈	—	4	2	no	—
501	C₂×Q₈	—	4	1	no	—
502	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
503	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
504	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
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	2	no	—
510	C₂×D₄	—	2	2	no	—
511	C₂×D₄	—	2	2	no	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	2	1	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	2	yes	—
518	C₂²×C₄	—	2	2	no	—
519	C₂²×C₄	—	2	2	no	—
520	C₂²×C₄	—	2	1	no	—
521	C₂²×C₄	—	2	1	no	—
522	C₂²×C₄	—	2	2	no	—
523	C₂²×C₄	—	2	1	no	—
524	C₂²×C₄	—	2	1	no	—
525	C₂²×C₄	—	2	1	no	—
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	4	1	no	—
528	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
529	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
530	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
531	Q₁₆	—	4	4	no	—
532	SD₁₆	—	4	4	no	—
533	M₁₆	—	4	2	no	—
534	M₁₆	—	4	2	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	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	1	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	2	no	—
548	C₄⋊C₄	—	2	2	no	—
549	C₄⋊C₄	—	4	1	no	—
550	C₄⋊C₄	—	4	1	no	—
551	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
552	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
553	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
554	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
555	C₂²⋊C₄	—	2	2	no	—
556	C₂²⋊C₄	—	2	2	no	—
557	C₂²⋊C₄	—	2	2	no	—
558	C₂²⋊C₄	—	2	2	no	—
559	C₂²⋊C₄	—	2	1	no	—
560	C₂²⋊C₄	—	2	1	no	—
561	C₂²⋊C₄	—	2	2	no	—
562	C₂²⋊C₄	—	2	2	no	—
563	C₂²⋊C₄	—	2	1	no	—
564	C₂²⋊C₄	—	2	1	no	—
565	C₂²⋊C₄	—	2	1	no	—
566	C₂²⋊C₄	—	2	1	no	—
567	C₄²	—	2	2	no	—
568	C₄²	—	2	2	no	—
569	C₄²	—	2	1	no	—
570	C₄²	—	2	1	no	—
571	C₄²	—	2	1	no	—
572	C₄²	—	2	1	no	—
573	C₄²	—	2	1	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	2	no	—
582	C₂³	#120×C₄₁	1	1	yes	characteristic
583	Q₈	—	2	1	no	—
584	Q₈	—	2	1	no	—
585	Q₈	—	4	2	no	—
586	Q₈	—	4	2	no	—
587	Q₈	—	4	2	no	—
588	Q₈	—	8	2	no	—
589	Q₈	—	8	2	no	—
590	Q₈	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
591	Q₈	#137×C₄₁	1	2	yes	complemented
592	D₄	—	2	1	no	—
593	D₄	—	2	1	no	—
594	D₄	—	2	1	no	—
595	D₄	—	2	1	no	—
596	D₄	—	4	2	no	—
597	D₄	—	4	2	no	—
598	D₄	—	4	2	no	—
599	D₄	—	4	4	no	—
600	D₄	—	4	4	no	—
601	D₄	—	4	1	no	—
602	D₄	—	4	1	no	—
603	D₄	—	4	1	no	—
604	D₄	—	4	1	no	—
605	D₄	—	8	2	no	—
606	D₄	—	8	2	no	—
607	D₄	—	8	2	no	—
608	D₄	—	8	2	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	1	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	2	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	4	2	no	—
622	C₂×C₄	—	4	2	no	—
623	C₂×C₄	—	4	4	no	—
624	C₂×C₄	—	4	4	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	1	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	—	4	4	no	—
630	C₂×C₄	—	4	2	no	—
631	C₂×C₄	—	4	2	no	—
632	C₂×C₄	—	4	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	1	no	—
646	C₂²	—	4	1	no	—
647	C₂²	—	4	2	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	1	no	—
655	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
656	C₄	—	2	1	no	—
657	C₄	—	2	2	no	—
658	C₄	—	2	1	no	—
659	C₄	—	2	2	no	—
660	C₄	—	2	1	no	—
661	C₄	—	4	1	no	—
662	C₄	—	4	1	no	—
663	C₄	—	4	2	no	—
664	C₄	—	4	1	no	—
665	C₄	—	4	1	no	—
666	C₄	—	8	2	no	—
667	C₄	C₄₁×#455	1	2	yes	—
668	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
669	C₄	C₂×#137×C₄₁	1	1	yes	characteristic
670	C₄	C₄₁×#575	1	1	yes	characteristic
671	C₄	C₄₁×#575	1	1	yes	characteristic
672	C₂	—	4	1	no	—
673	C₂	—	8	1	no	—
674	C₂	—	8	1	no	—
675	C₂	—	8	2	no	—
676	C₂	C₄₁×#3006	1	1	yes	characteristic
677	C₂	C₄₁×#3007	1	1	yes	characteristic
678	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×#3483	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×#3141	C₂	semidirect
3. C₄₁×#3079	C₂	semidirect
4. C₄₁×#3039	C₂	semidirect
5. C₄₁×#1646	C₂	semidirect
6. C₄₁×#1645	C₂	semidirect
7. C₄₁×#1642	C₂	semidirect
8. C₄₁×#1641	C₂	semidirect
9. C₄₁×#1560	C₂	semidirect
10. C₄₁×#1489	C₂	semidirect
11. C₄₁×#1489	C₂	semidirect
251. Q₈×C₄₁	#137	semidirect
303. #18553	C₄₁	direct (N × H)
334. #3141	C₂×C₄₁	semidirect
335. #3079	C₂×C₄₁	semidirect
336. #3039	C₂×C₄₁	semidirect
337. #1646	C₂×C₄₁	semidirect
338. #1645	C₂×C₄₁	semidirect
339. #1642	C₂×C₄₁	semidirect
340. #1641	C₂×C₄₁	semidirect
341. #1560	C₂×C₄₁	semidirect
342. #1489	C₂×C₄₁	semidirect
343. #1489	C₂×C₄₁	semidirect
394. C₄₁	#18553	direct (N × H)
591. Q₈	#137×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3483	2
C₄₁×#3141	2
C₄₁×#3079	2
C₄₁×#3039	2
C₄₁×#1646	2
C₄₁×#1645	2
C₄₁×#1642	2
C₄₁×#1641	2
C₄₁×#1560	2
C₄₁×#1489	2
#18553	41

Subgroups — C41×#18553 #23794754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#18553 #23794754