Subgroups — C₄₁×#17682 #23793883

Order 10496.

876 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3515	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2971	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2923	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1535	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1534	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1531	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1529	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1529	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1528	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1526	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1524	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#557	C₄	1	1	yes	characteristic
13	C₄₁×#554	C₄	1	1	yes	characteristic
14	C₄₁×#554	C₄	1	1	yes	characteristic
15	C₄₁×#553	C₄	1	2	yes	—
16	C₄₁×#552	—	2	2	no	—
17	C₄₁×#552	—	2	1	no	—
18	C₄₁×#552	C₄	1	2	yes	—
19	C₄₁×#547	—	2	1	no	—
20	C₄₁×#547	C₄	1	1	yes	characteristic
21	C₄₁×#541	—	2	1	no	—
22	C₄₁×#539	—	2	2	no	—
23	C₄₁×#533	C₂²	1	1	yes	characteristic
24	C₄₁×#529	C₂²	1	1	yes	characteristic
25	C₄₁×#529	C₂²	1	2	yes	—
26	C₂×#125×C₄₁	C₂²	1	2	yes	—
27	C₄₁×#520	—	2	1	no	—
28	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
29	C₈×Q₈×C₄₁	—	2	1	no	—
30	C₄₁×#436	—	2	1	no	—
31	C₄₁×#436	—	2	2	no	—
32	C₄₁×#433	—	2	2	no	—
33	C₄₁×#432	C₂²	1	2	yes	—
34	C₄₁×#432	C₂²	1	1	yes	characteristic
35	C₄₁×#432	C₂²	1	2	yes	—
36	C₄₁×#432	C₂²	1	2	yes	—
37	C₄₁×#424	—	2	1	no	—
38	C₄₁×#408	—	2	1	no	—
39	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
41	C₄×M₁₆×C₄₁	C₂²	1	2	yes	—
42	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
43	C₄₁×#330	C₂²	1	1	yes	characteristic
44	C₄₁×#330	C₂²	1	1	yes	characteristic
45	C₄₁×#330	C₂²	1	2	yes	—
46	C₄₁×#330	C₂²	1	2	yes	—
47	C₄₁×#330	C₂²	1	2	yes	—
48	C₄₁×#330	C₂²	1	2	yes	—
49	C₄₁×#330	C₂²	1	2	yes	—
50	C₄₁×#329	C₂²	1	1	yes	characteristic
51	C₄₁×#329	C₂²	1	1	yes	characteristic
52	C₄₁×#329	C₂²	1	2	yes	—
53	C₂×#41×C₄₁	—	2	1	no	—
54	C₂×#41×C₄₁	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₄₁	—	2	1	no	—
56	C₂×M₁₆×C₄₁	—	2	2	no	—
57	C₂×M₁₆×C₄₁	—	2	1	no	—
58	C₂²×C₈×C₄₁	—	2	1	no	—
59	C₄⋊Q₈×C₄₁	—	2	1	no	—
60	C₄⋊Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
62	#126×C₄₁	—	2	2	no	—
63	#126×C₄₁	—	2	2	no	—
64	#126×C₄₁	—	2	2	no	—
65	#126×C₄₁	—	2	1	no	—
66	#125×C₄₁	—	2	2	no	—
67	#125×C₄₁	—	2	2	no	—
68	#125×C₄₁	—	2	1	no	—
69	#125×C₄₁	—	2	2	no	—
70	#125×C₄₁	—	2	2	no	—
71	#125×C₄₁	C₂×C₄	1	2	yes	—
72	#125×C₄₁	C₂×C₄	1	2	yes	—
73	#125×C₄₁	C₂³	1	1	yes	characteristic
74	#125×C₄₁	C₂³	1	2	yes	—
75	#124×C₄₁	C₂³	1	1	yes	characteristic
76	#123×C₄₁	—	2	2	no	—
77	#123×C₄₁	—	2	1	no	—
78	#123×C₄₁	—	2	2	no	—
79	#123×C₄₁	—	2	2	no	—
80	#123×C₄₁	—	2	2	no	—
81	C₂²⋊Q₈×C₄₁	—	2	1	no	—
82	C₂²⋊Q₈×C₄₁	—	2	1	no	—
83	C₂²⋊Q₈×C₄₁	—	2	1	no	—
84	C₂²⋊Q₈×C₄₁	—	2	2	no	—
85	C₂²⋊Q₈×C₄₁	—	2	1	no	—
86	C₂²⋊Q₈×C₄₁	—	2	2	no	—
87	C₂²⋊Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
88	C₂²⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
89	#121×C₄₁	—	2	2	no	—
90	#121×C₄₁	C₂×C₄	1	2	yes	—
91	#121×C₄₁	C₂×C₄	1	1	yes	characteristic
92	C₄×Q₈×C₄₁	—	2	1	no	—
93	C₄×Q₈×C₄₁	—	2	1	no	—
94	C₄×Q₈×C₄₁	—	2	1	no	—
95	C₄×Q₈×C₄₁	C₂×C₄	1	2	yes	—
96	C₄×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
97	C₄×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
98	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
99	C₄×D₄×C₄₁	—	2	1	no	—
100	C₄×D₄×C₄₁	—	2	2	no	—
101	C₄×D₄×C₄₁	—	2	1	no	—
102	C₄×D₄×C₄₁	—	2	2	no	—
103	C₄×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
104	C₄×D₄×C₄₁	C₂×C₄	1	2	yes	—
105	C₄×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
106	C₄×D₄×C₄₁	C₂×C₄	1	2	yes	—
107	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
108	C₄×D₄×C₄₁	D₄	1	2	yes	—
109	#117×C₄₁	—	2	2	no	—
110	#117×C₄₁	—	2	1	no	—
111	#117×C₄₁	—	2	2	no	—
112	#117×C₄₁	C₂×C₄	1	2	yes	—
113	#117×C₄₁	D₄	1	1	yes	characteristic
114	#117×C₄₁	D₄	1	2	yes	—
115	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
116	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
117	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
118	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
119	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
120	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
121	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
122	C₄⋊C₈×C₄₁	—	2	2	no	—
123	C₄⋊C₈×C₄₁	—	2	2	no	—
124	C₄⋊C₈×C₄₁	—	2	1	no	—
125	C₄⋊C₈×C₄₁	—	2	2	no	—
126	C₂²⋊C₈×C₄₁	—	2	2	no	—
127	C₂²⋊C₈×C₄₁	—	2	2	no	—
128	C₂²⋊C₈×C₄₁	—	2	2	no	—
129	C₂²⋊C₈×C₄₁	—	2	1	no	—
130	#97×C₄₁	—	2	1	no	—
131	#97×C₄₁	C₂³	1	1	yes	characteristic
132	#97×C₄₁	C₂³	1	1	yes	characteristic
133	#97×C₄₁	C₂³	1	1	yes	characteristic
134	#97×C₄₁	C₂³	1	1	yes	characteristic
135	#97×C₄₁	C₂³	1	2	yes	—
136	#97×C₄₁	C₂³	1	2	yes	—
137	C₄×C₈×C₄₁	—	2	1	no	—
138	C₄×C₈×C₄₁	—	2	1	no	—
139	C₄×C₈×C₄₁	—	2	2	no	—
140	#41×C₄₁	—	4	2	no	—
141	#41×C₄₁	—	4	4	no	—
142	C₂×Q₈×C₄₁	—	2	1	no	—
143	C₂×Q₈×C₄₁	—	2	1	no	—
144	C₂×Q₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
145	C₂×Q₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
146	C₂×D₄×C₄₁	—	2	2	no	—
147	C₂×D₄×C₄₁	—	2	1	no	—
148	C₂×D₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
149	C₂×D₄×C₄₁	C₂²⋊C₄	1	2	yes	—
150	C₂×D₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₄₁	—	2	2	no	—
152	C₂²×C₄×C₄₁	—	2	2	no	—
153	C₂²×C₄×C₄₁	—	2	1	no	—
154	C₂²×C₄×C₄₁	—	2	2	no	—
155	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
156	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
161	M₁₆×C₄₁	—	4	4	no	—
162	M₁₆×C₄₁	—	4	2	no	—
163	M₁₆×C₄₁	—	4	2	no	—
164	C₂×C₈×C₄₁	—	2	1	no	—
165	C₂×C₈×C₄₁	—	2	2	no	—
166	C₂×C₈×C₄₁	—	2	1	no	—
167	C₂×C₈×C₄₁	—	2	2	no	—
168	C₂×C₈×C₄₁	—	2	1	no	—
169	C₂×C₈×C₄₁	—	2	1	no	—
170	C₂×C₈×C₄₁	—	4	2	no	—
171	C₄⋊C₄×C₄₁	—	2	2	no	—
172	C₄⋊C₄×C₄₁	—	2	1	no	—
173	C₄⋊C₄×C₄₁	—	2	2	no	—
174	C₄⋊C₄×C₄₁	—	2	2	no	—
175	C₄⋊C₄×C₄₁	—	2	2	no	—
176	C₄⋊C₄×C₄₁	—	2	2	no	—
177	C₄⋊C₄×C₄₁	—	2	2	no	—
178	C₄⋊C₄×C₄₁	—	2	1	no	—
179	C₄⋊C₄×C₄₁	—	2	2	no	—
180	C₄⋊C₄×C₄₁	—	2	1	no	—
181	C₄⋊C₄×C₄₁	—	2	1	no	—
182	C₄⋊C₄×C₄₁	—	2	2	no	—
183	C₄⋊C₄×C₄₁	—	2	1	no	—
184	C₄⋊C₄×C₄₁	—	2	1	no	—
185	C₄⋊C₄×C₄₁	—	2	2	no	—
186	C₄⋊C₄×C₄₁	—	2	2	no	—
187	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
188	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
189	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
190	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
191	C₄⋊C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
193	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
194	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
195	C₂²⋊C₄×C₄₁	—	2	2	no	—
196	C₂²⋊C₄×C₄₁	—	2	2	no	—
197	C₂²⋊C₄×C₄₁	—	2	1	no	—
198	C₂²⋊C₄×C₄₁	—	2	1	no	—
199	C₂²⋊C₄×C₄₁	—	2	2	no	—
200	C₂²⋊C₄×C₄₁	—	2	2	no	—
201	C₂²⋊C₄×C₄₁	—	2	1	no	—
202	C₂²⋊C₄×C₄₁	—	2	1	no	—
203	C₂²⋊C₄×C₄₁	—	2	1	no	—
204	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
205	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
206	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
207	C₄²×C₄₁	—	2	2	no	—
208	C₄²×C₄₁	—	2	2	no	—
209	C₄²×C₄₁	—	2	1	no	—
210	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
211	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
212	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
213	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₄₁	C₂×D₄	1	2	yes	—
215	C₄²×C₄₁	C₂×D₄	1	2	yes	—
216	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
217	C₂³×C₄₁	—	2	1	no	—
218	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
220	Q₈×C₄₁	—	2	2	no	—
221	Q₈×C₄₁	—	4	1	no	—
222	Q₈×C₄₁	—	4	2	no	—
223	Q₈×C₄₁	—	8	1	no	—
224	D₄×C₄₁	—	2	2	no	—
225	D₄×C₄₁	—	4	2	no	—
226	D₄×C₄₁	—	4	1	no	—
227	D₄×C₄₁	—	4	4	no	—
228	D₄×C₄₁	—	8	1	no	—
229	C₂×C₄×C₄₁	—	2	2	no	—
230	C₂×C₄×C₄₁	—	2	1	no	—
231	C₂×C₄×C₄₁	—	2	2	no	—
232	C₂×C₄×C₄₁	—	2	2	no	—
233	C₂×C₄×C₄₁	—	2	1	no	—
234	C₂×C₄×C₄₁	—	2	1	no	—
235	C₂×C₄×C₄₁	—	4	2	no	—
236	C₂×C₄×C₄₁	—	4	1	no	—
237	C₂×C₄×C₄₁	—	4	2	no	—
238	C₂×C₄×C₄₁	—	4	1	no	—
239	C₂×C₄×C₄₁	—	4	2	no	—
240	C₂×C₄×C₄₁	—	4	4	no	—
241	C₂×C₄×C₄₁	—	4	2	no	—
242	C₂×C₄×C₄₁	—	8	2	no	—
243	C₂×C₄×C₄₁	—	8	2	no	—
244	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
246	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
247	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
248	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
249	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
251	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
253	C₈×C₄₁	—	2	2	no	—
254	C₈×C₄₁	—	4	1	no	—
255	C₈×C₄₁	—	4	1	no	—
256	C₈×C₄₁	—	4	2	no	—
257	C₈×C₄₁	—	4	1	no	—
258	C₈×C₄₁	—	4	2	no	—
259	#17682	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
260	C₂²×C₄₁	—	2	1	no	—
261	C₂²×C₄₁	—	2	2	no	—
262	C₂²×C₄₁	—	2	1	no	—
263	C₂²×C₄₁	—	2	2	no	—
264	C₂²×C₄₁	—	2	2	no	—
265	C₂²×C₄₁	—	2	1	no	—
266	C₂²×C₄₁	—	4	1	no	—
267	C₂²×C₄₁	—	4	1	no	—
268	C₂²×C₄₁	—	4	1	no	—
269	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
270	C₄×C₄₁	—	2	1	no	—
271	C₄×C₄₁	—	2	1	no	—
272	C₄×C₄₁	—	2	2	no	—
273	C₄×C₄₁	—	2	1	no	—
274	C₄×C₄₁	—	2	2	no	—
275	C₄×C₄₁	—	2	1	no	—
276	C₄×C₄₁	—	2	2	no	—
277	C₄×C₄₁	—	2	1	no	—
278	C₄×C₄₁	—	4	1	no	—
279	C₄×C₄₁	—	4	1	no	—
280	C₄×C₄₁	—	4	2	no	—
281	C₄×C₄₁	—	4	1	no	—
282	C₄×C₄₁	—	4	2	no	—
283	C₄×C₄₁	#421	1	2	yes	—
284	#3515	C₂×C₄₁	1	1	yes	characteristic
285	#2971	C₂×C₄₁	1	1	yes	characteristic, complemented
286	#2923	C₂×C₄₁	1	1	yes	characteristic, complemented
287	#1535	C₂×C₄₁	1	2	yes	complemented
288	#1534	C₂×C₄₁	1	2	yes	complemented
289	#1531	C₂×C₄₁	1	1	yes	characteristic, complemented
290	#1529	C₂×C₄₁	1	1	yes	characteristic, complemented
291	#1529	C₂×C₄₁	1	2	yes	complemented
292	#1528	C₂×C₄₁	1	2	yes	complemented
293	#1526	C₂×C₄₁	1	1	yes	characteristic, complemented
294	#1524	C₂×C₄₁	1	1	yes	characteristic, complemented
295	C₂×C₄₁	—	4	1	no	—
296	C₂×C₄₁	—	4	2	no	—
297	C₂×C₄₁	—	8	1	no	—
298	C₂×C₄₁	#2889	1	1	yes	characteristic
299	C₂×C₄₁	C₂×#421	1	1	yes	characteristic
300	C₂×C₄₁	#2901	1	1	yes	characteristic
301	#557	C₄×C₄₁	1	1	yes	characteristic
302	#554	C₄×C₄₁	1	1	yes	characteristic
303	#554	C₄×C₄₁	1	1	yes	characteristic
304	#553	C₄×C₄₁	1	2	yes	—
305	#552	—	2	2	no	—
306	#552	—	2	1	no	—
307	#552	C₄×C₄₁	1	2	yes	—
308	#547	—	2	1	no	—
309	#547	C₄×C₄₁	1	1	yes	characteristic
310	#541	—	2	1	no	—
311	#539	—	2	2	no	—
312	#533	C₂²×C₄₁	1	1	yes	characteristic
313	#529	C₂²×C₄₁	1	1	yes	characteristic
314	#529	C₂²×C₄₁	1	2	yes	—
315	C₂×#125	C₂²×C₄₁	1	2	yes	—
316	#520	—	2	1	no	—
317	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
318	C₈×Q₈	—	2	1	no	—
319	#436	—	2	1	no	—
320	#436	—	2	2	no	—
321	#433	—	2	2	no	—
322	#432	C₂²×C₄₁	1	2	yes	—
323	#432	C₂²×C₄₁	1	2	yes	—
324	#432	C₂²×C₄₁	1	1	yes	characteristic
325	#432	C₂²×C₄₁	1	2	yes	—
326	#424	—	2	1	no	—
327	#408	—	2	1	no	—
328	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
329	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
330	C₄×M₁₆	C₂²×C₄₁	1	2	yes	—
331	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
332	#330	C₂²×C₄₁	1	1	yes	characteristic
333	#330	C₂²×C₄₁	1	1	yes	characteristic
334	#330	C₂²×C₄₁	1	2	yes	—
335	#330	C₂²×C₄₁	1	2	yes	—
336	#330	C₂²×C₄₁	1	2	yes	—
337	#330	C₂²×C₄₁	1	2	yes	—
338	#330	C₂²×C₄₁	1	2	yes	—
339	#329	C₂²×C₄₁	1	1	yes	characteristic
340	#329	C₂²×C₄₁	1	1	yes	characteristic
341	#329	C₂²×C₄₁	1	2	yes	—
342	C₄₁	#17682	1	1	yes	characteristic, direct product factor, Sylow
343	C₂×#41	—	2	1	no	—
344	C₂×#41	D₄×C₄₁	1	1	yes	characteristic
345	C₂×M₁₆	—	2	1	no	—
346	C₂×M₁₆	—	2	2	no	—
347	C₂×M₁₆	—	2	1	no	—
348	C₂²×C₈	—	2	1	no	—
349	C₄⋊Q₈	—	2	1	no	—
350	C₄⋊Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
351	C₄⋊Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
352	#126	—	2	2	no	—
353	#126	—	2	2	no	—
354	#126	—	2	2	no	—
355	#126	—	2	1	no	—
356	#125	—	2	2	no	—
357	#125	—	2	2	no	—
358	#125	—	2	1	no	—
359	#125	—	2	2	no	—
360	#125	—	2	2	no	—
361	#125	C₂×C₄×C₄₁	1	2	yes	—
362	#125	C₂×C₄×C₄₁	1	2	yes	—
363	#125	C₂³×C₄₁	1	1	yes	characteristic
364	#125	C₂³×C₄₁	1	2	yes	—
365	#124	C₂³×C₄₁	1	1	yes	characteristic
366	#123	—	2	2	no	—
367	#123	—	2	1	no	—
368	#123	—	2	2	no	—
369	#123	—	2	2	no	—
370	#123	—	2	2	no	—
371	C₂²⋊Q₈	—	2	1	no	—
372	C₂²⋊Q₈	—	2	1	no	—
373	C₂²⋊Q₈	—	2	1	no	—
374	C₂²⋊Q₈	—	2	2	no	—
375	C₂²⋊Q₈	—	2	1	no	—
376	C₂²⋊Q₈	—	2	2	no	—
377	C₂²⋊Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
378	C₂²⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
379	#121	—	2	2	no	—
380	#121	C₂×C₄×C₄₁	1	2	yes	—
381	#121	C₂×C₄×C₄₁	1	1	yes	characteristic
382	C₄×Q₈	—	2	1	no	—
383	C₄×Q₈	—	2	1	no	—
384	C₄×Q₈	—	2	1	no	—
385	C₄×Q₈	C₂×C₄×C₄₁	1	2	yes	—
386	C₄×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
387	C₄×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
388	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
389	C₄×D₄	—	2	1	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	—	2	1	no	—
392	C₄×D₄	—	2	2	no	—
393	C₄×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
394	C₄×D₄	C₂×C₄×C₄₁	1	2	yes	—
395	C₄×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
396	C₄×D₄	C₂×C₄×C₄₁	1	2	yes	—
397	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
398	C₄×D₄	D₄×C₄₁	1	2	yes	—
399	#117	—	2	2	no	—
400	#117	—	2	1	no	—
401	#117	—	2	2	no	—
402	#117	C₂×C₄×C₄₁	1	2	yes	—
403	#117	D₄×C₄₁	1	1	yes	characteristic
404	#117	D₄×C₄₁	1	2	yes	—
405	C₂×C₄⋊C₄	—	2	2	no	—
406	C₂×C₄⋊C₄	—	2	2	no	—
407	C₂×C₄⋊C₄	—	2	1	no	—
408	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
409	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
410	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
411	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
412	C₄⋊C₈	—	2	2	no	—
413	C₄⋊C₈	—	2	2	no	—
414	C₄⋊C₈	—	2	1	no	—
415	C₄⋊C₈	—	2	2	no	—
416	C₂²⋊C₈	—	2	2	no	—
417	C₂²⋊C₈	—	2	2	no	—
418	C₂²⋊C₈	—	2	2	no	—
419	C₂²⋊C₈	—	2	1	no	—
420	#97	—	2	1	no	—
421	#97	C₂³×C₄₁	1	1	yes	characteristic
422	#97	C₂³×C₄₁	1	1	yes	characteristic
423	#97	C₂³×C₄₁	1	1	yes	characteristic
424	#97	C₂³×C₄₁	1	1	yes	characteristic
425	#97	C₂³×C₄₁	1	2	yes	—
426	#97	C₂³×C₄₁	1	2	yes	—
427	C₄×C₈	—	2	1	no	—
428	C₄×C₈	—	2	1	no	—
429	C₄×C₈	—	2	2	no	—
430	#41	—	4	2	no	—
431	#41	—	4	4	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
435	C₂×Q₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
439	C₂×D₄	C₂²⋊C₄×C₄₁	1	2	yes	—
440	C₂×D₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
446	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
447	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
449	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
450	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
451	M₁₆	—	4	4	no	—
452	M₁₆	—	4	2	no	—
453	M₁₆	—	4	2	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	2	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	2	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	4	2	no	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	—	2	2	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	2	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
478	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
479	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
480	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
481	C₄⋊C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
482	C₄⋊C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
483	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
484	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
485	C₂²⋊C₄	—	2	2	no	—
486	C₂²⋊C₄	—	2	2	no	—
487	C₂²⋊C₄	—	2	1	no	—
488	C₂²⋊C₄	—	2	1	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	2	2	no	—
491	C₂²⋊C₄	—	2	1	no	—
492	C₂²⋊C₄	—	2	1	no	—
493	C₂²⋊C₄	—	2	1	no	—
494	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
495	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
496	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
497	C₄²	—	2	2	no	—
498	C₄²	—	2	2	no	—
499	C₄²	—	2	1	no	—
500	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
501	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
502	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
503	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
504	C₄²	C₂×D₄×C₄₁	1	2	yes	—
505	C₄²	C₂×D₄×C₄₁	1	2	yes	—
506	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
507	C₂³	—	2	1	no	—
508	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
509	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
510	Q₈	—	2	2	no	—
511	Q₈	—	4	1	no	—
512	Q₈	—	4	2	no	—
513	Q₈	—	8	1	no	—
514	D₄	—	2	2	no	—
515	D₄	—	4	2	no	—
516	D₄	—	4	1	no	—
517	D₄	—	4	4	no	—
518	D₄	—	8	1	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	4	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	8	2	no	—
533	C₂×C₄	—	8	2	no	—
534	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
535	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
536	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
537	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
538	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
539	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
540	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
543	C₈	—	2	2	no	—
544	C₈	—	4	1	no	—
545	C₈	—	4	1	no	—
546	C₈	—	4	2	no	—
547	C₈	—	4	1	no	—
548	C₈	—	4	2	no	—
549	C₂²	—	2	1	no	—
550	C₂²	—	2	2	no	—
551	C₂²	—	2	1	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	1	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	4	1	no	—
557	C₂²	—	4	1	no	—
558	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
559	C₄	—	2	1	no	—
560	C₄	—	2	1	no	—
561	C₄	—	2	2	no	—
562	C₄	—	2	1	no	—
563	C₄	—	2	2	no	—
564	C₄	—	2	1	no	—
565	C₄	—	2	2	no	—
566	C₄	—	2	1	no	—
567	C₄	—	4	1	no	—
568	C₄	—	4	1	no	—
569	C₄	—	4	2	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	2	no	—
572	C₄	C₄₁×#421	1	2	yes	—
573	C₂	—	4	1	no	—
574	C₂	—	4	2	no	—
575	C₂	—	8	1	no	—
576	C₂	C₄₁×#2889	1	1	yes	characteristic
577	C₂	C₂×C₄₁×#421	1	1	yes	characteristic
578	C₂	C₄₁×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₄₁×#3515	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄×#41×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2971	C₂	semidirect
3. C₄₁×#2923	C₂	semidirect
4. C₄₁×#1535	C₂	semidirect
5. C₄₁×#1534	C₂	semidirect
6. C₄₁×#1531	C₂	semidirect
7. C₄₁×#1529	C₂	semidirect
8. C₄₁×#1529	C₂	semidirect
9. C₄₁×#1528	C₂	semidirect
10. C₄₁×#1526	C₂	semidirect
11. C₄₁×#1524	C₂	semidirect
259. #17682	C₄₁	direct (N × H)
285. #2971	C₂×C₄₁	semidirect
286. #2923	C₂×C₄₁	semidirect
287. #1535	C₂×C₄₁	semidirect
288. #1534	C₂×C₄₁	semidirect
289. #1531	C₂×C₄₁	semidirect
290. #1529	C₂×C₄₁	semidirect
291. #1529	C₂×C₄₁	semidirect
292. #1528	C₂×C₄₁	semidirect
293. #1526	C₂×C₄₁	semidirect
294. #1524	C₂×C₄₁	semidirect
342. C₄₁	#17682	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×#3515	2
C₄₁×#2971	2
C₄₁×#2923	2
C₄₁×#1535	2
C₄₁×#1534	2
C₄₁×#1531	2
C₄₁×#1529	2
C₄₁×#1528	2
C₄₁×#1526	2
C₄₁×#1524	2
#17682	41

Subgroups — C41×#17682 #23793883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#17682 #23793883