Subgroups — C₄₁×#17514 #23793715

Order 10496.

926 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2965	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2925	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1497	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1491	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1485	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1481	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1479	C₂	1	2	yes	maximal, complemented
11	C₂×C₄₁×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈²×C₄₁	C₄	1	1	yes	characteristic
13	C₄₁×#557	—	2	1	no	—
14	C₄₁×#557	C₄	1	1	yes	characteristic
15	C₄₁×#554	—	2	2	no	—
16	C₄₁×#554	—	2	1	no	—
17	C₄₁×#554	C₄	1	2	yes	—
18	C₄₁×#554	C₄	1	1	yes	characteristic
19	D₄×Q₈×C₄₁	C₄	1	1	yes	characteristic
20	D₄×Q₈×C₄₁	C₄	1	2	yes	—
21	C₄₁×#533	—	2	1	no	—
22	C₄₁×#533	—	2	2	no	—
23	C₄₁×#533	C₂²	1	1	yes	characteristic
24	C₄₁×#533	C₂²	1	2	yes	—
25	C₄×#41×C₄₁	—	2	1	no	—
26	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₄₁	C₂²	1	2	yes	—
29	C₄₁×#446	—	2	1	no	—
30	C₄₁×#436	—	2	1	no	—
31	C₄₁×#435	—	2	2	no	—
32	C₄₁×#433	—	2	2	no	—
33	C₄₁×#432	C₂²	1	2	yes	—
34	C₄₁×#431	C₂²	1	2	yes	—
35	C₄₁×#431	C₂²	1	2	yes	—
36	C₄₁×#431	C₂²	1	1	yes	characteristic
37	C₄₁×#423	C₂²	1	1	yes	characteristic
38	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
39	C₄₁×#408	—	2	1	no	—
40	C₄₁×#405	—	2	1	no	—
41	C₄×M₁₆×C₄₁	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
43	C₄₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
44	C₄₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
45	C₄₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
46	C₄₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
47	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
51	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
52	C₄₁×Q₈⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₄₁	D₄	1	1	yes	characteristic
54	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×Q₈×C₄₁	C₂×C₄	1	2	yes	—
56	C₂×M₁₆×C₄₁	—	2	2	no	—
57	C₂×M₁₆×C₄₁	—	2	1	no	—
58	C₂²×C₈×C₄₁	—	2	1	no	—
59	C₂²×C₈×C₄₁	—	2	1	no	—
60	C₄⋊Q₈×C₄₁	—	2	2	no	—
61	C₄⋊Q₈×C₄₁	—	2	1	no	—
62	C₄⋊Q₈×C₄₁	—	2	2	no	—
63	C₄⋊Q₈×C₄₁	—	2	1	no	—
64	C₄⋊Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
65	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
66	C₄⋊Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
67	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
68	#125×C₄₁	—	2	2	no	—
69	#125×C₄₁	—	2	1	no	—
70	#125×C₄₁	—	2	2	no	—
71	#125×C₄₁	—	2	1	no	—
72	#125×C₄₁	—	2	1	no	—
73	#125×C₄₁	—	2	2	no	—
74	C₂²⋊Q₈×C₄₁	—	2	1	no	—
75	C₂²⋊Q₈×C₄₁	—	2	2	no	—
76	C₂²⋊Q₈×C₄₁	—	2	2	no	—
77	C₂²⋊Q₈×C₄₁	—	2	2	no	—
78	C₂²⋊Q₈×C₄₁	—	2	2	no	—
79	C₂²⋊Q₈×C₄₁	—	2	2	no	—
80	C₂²⋊Q₈×C₄₁	—	2	2	no	—
81	C₂²⋊Q₈×C₄₁	—	2	2	no	—
82	C₂²⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
83	C₂²⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
84	C₂²⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
85	C₄×Q₈×C₄₁	—	2	1	no	—
86	C₄×Q₈×C₄₁	—	2	1	no	—
87	C₄×Q₈×C₄₁	—	2	2	no	—
88	C₄×Q₈×C₄₁	—	2	1	no	—
89	C₄×Q₈×C₄₁	—	2	1	no	—
90	C₄×Q₈×C₄₁	—	2	2	no	—
91	C₄×Q₈×C₄₁	—	2	2	no	—
92	C₄×Q₈×C₄₁	—	2	1	no	—
93	C₄×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
94	C₄×Q₈×C₄₁	C₂×C₄	1	2	yes	—
95	C₄×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
96	C₄×Q₈×C₄₁	C₂×C₄	1	2	yes	—
97	C₄×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
98	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
99	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
100	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
101	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
102	C₄×D₄×C₄₁	—	2	2	no	—
103	C₄×D₄×C₄₁	—	2	1	no	—
104	C₄×D₄×C₄₁	—	2	2	no	—
105	C₄×D₄×C₄₁	—	2	1	no	—
106	C₄×D₄×C₄₁	C₂×C₄	1	2	yes	—
107	C₄×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
108	C₄×D₄×C₄₁	D₄	1	2	yes	—
109	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
110	#117×C₄₁	—	2	2	no	—
111	#117×C₄₁	—	2	1	no	—
112	#117×C₄₁	D₄	1	1	yes	characteristic
113	#117×C₄₁	D₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
115	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
116	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
118	C₂×C₄²×C₄₁	—	2	2	no	—
119	C₂×C₄²×C₄₁	—	2	1	no	—
120	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
122	C₄⋊C₈×C₄₁	—	2	1	no	—
123	C₄⋊C₈×C₄₁	—	2	1	no	—
124	C₄⋊C₈×C₄₁	—	2	2	no	—
125	C₄⋊C₈×C₄₁	—	2	1	no	—
126	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
127	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
128	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
129	C₂²⋊C₈×C₄₁	—	2	2	no	—
130	C₂²⋊C₈×C₄₁	—	2	2	no	—
131	C₂²⋊C₈×C₄₁	—	2	1	no	—
132	C₂²⋊C₈×C₄₁	—	2	2	no	—
133	#97×C₄₁	—	2	2	no	—
134	#97×C₄₁	—	2	2	no	—
135	C₄×C₈×C₄₁	—	2	1	no	—
136	C₄×C₈×C₄₁	—	2	2	no	—
137	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₄₁	C₂³	1	2	yes	—
140	#41×C₄₁	—	2	2	no	—
141	#41×C₄₁	—	2	2	no	—
142	C₂×Q₈×C₄₁	—	2	1	no	—
143	C₂×Q₈×C₄₁	—	2	2	no	—
144	C₂×Q₈×C₄₁	—	2	2	no	—
145	C₂×Q₈×C₄₁	—	2	1	no	—
146	C₂×Q₈×C₄₁	—	2	2	no	—
147	C₂×Q₈×C₄₁	—	2	4	no	—
148	C₂×Q₈×C₄₁	—	4	1	no	—
149	C₂×Q₈×C₄₁	—	4	2	no	—
150	C₂×Q₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
151	C₂×Q₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
152	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
153	C₂×Q₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
154	C₂×D₄×C₄₁	C₂²⋊C₄	1	2	yes	—
155	C₂×D₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₄₁	—	2	2	no	—
157	C₂²×C₄×C₄₁	—	2	1	no	—
158	C₂²×C₄×C₄₁	—	2	2	no	—
159	C₂²×C₄×C₄₁	—	2	1	no	—
160	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
161	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
166	M₁₆×C₄₁	—	4	4	no	—
167	M₁₆×C₄₁	—	4	2	no	—
168	C₂×C₈×C₄₁	—	2	1	no	—
169	C₂×C₈×C₄₁	—	2	1	no	—
170	C₂×C₈×C₄₁	—	2	2	no	—
171	C₂×C₈×C₄₁	—	2	1	no	—
172	C₂×C₈×C₄₁	—	2	2	no	—
173	C₂×C₈×C₄₁	—	2	1	no	—
174	C₂×C₈×C₄₁	—	4	2	no	—
175	C₂×C₈×C₄₁	—	4	2	no	—
176	C₄⋊C₄×C₄₁	—	2	2	no	—
177	C₄⋊C₄×C₄₁	—	2	1	no	—
178	C₄⋊C₄×C₄₁	—	2	2	no	—
179	C₄⋊C₄×C₄₁	—	2	1	no	—
180	C₄⋊C₄×C₄₁	—	2	1	no	—
181	C₄⋊C₄×C₄₁	—	2	2	no	—
182	C₄⋊C₄×C₄₁	—	2	2	no	—
183	C₄⋊C₄×C₄₁	—	2	1	no	—
184	C₄⋊C₄×C₄₁	—	2	2	no	—
185	C₄⋊C₄×C₄₁	—	2	1	no	—
186	C₄⋊C₄×C₄₁	—	2	2	no	—
187	C₄⋊C₄×C₄₁	—	2	1	no	—
188	C₄⋊C₄×C₄₁	—	2	2	no	—
189	C₄⋊C₄×C₄₁	—	2	2	no	—
190	C₄⋊C₄×C₄₁	—	2	2	no	—
191	C₄⋊C₄×C₄₁	—	2	1	no	—
192	C₄⋊C₄×C₄₁	—	2	1	no	—
193	C₄⋊C₄×C₄₁	—	2	2	no	—
194	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
195	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
197	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
199	C₄⋊C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
200	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
201	C₂²⋊C₄×C₄₁	—	2	2	no	—
202	C₂²⋊C₄×C₄₁	—	2	2	no	—
203	C₂²⋊C₄×C₄₁	—	2	2	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	1	no	—
209	C₄²×C₄₁	—	2	2	no	—
210	C₄²×C₄₁	—	2	1	no	—
211	C₄²×C₄₁	—	2	1	no	—
212	C₄²×C₄₁	—	2	2	no	—
213	C₄²×C₄₁	—	2	1	no	—
214	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
215	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
216	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₄₁	C₂×D₄	1	2	yes	—
218	C₄²×C₄₁	C₂×D₄	1	2	yes	—
219	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
221	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
222	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	Q₈×C₄₁	—	2	1	no	—
224	Q₈×C₄₁	—	2	2	no	—
225	Q₈×C₄₁	—	2	1	no	—
226	Q₈×C₄₁	—	2	2	no	—
227	Q₈×C₄₁	—	2	1	no	—
228	Q₈×C₄₁	—	2	2	no	—
229	Q₈×C₄₁	—	4	2	no	—
230	Q₈×C₄₁	—	4	4	no	—
231	Q₈×C₄₁	—	8	1	no	—
232	Q₈×C₄₁	—	8	2	no	—
233	Q₈×C₄₁	#131	1	2	yes	—
234	D₄×C₄₁	—	2	4	no	—
235	D₄×C₄₁	—	2	2	no	—
236	C₂×C₄×C₄₁	—	2	1	no	—
237	C₂×C₄×C₄₁	—	2	1	no	—
238	C₂×C₄×C₄₁	—	2	2	no	—
239	C₂×C₄×C₄₁	—	2	1	no	—
240	C₂×C₄×C₄₁	—	2	2	no	—
241	C₂×C₄×C₄₁	—	2	2	no	—
242	C₂×C₄×C₄₁	—	2	2	no	—
243	C₂×C₄×C₄₁	—	2	1	no	—
244	C₂×C₄×C₄₁	—	2	2	no	—
245	C₂×C₄×C₄₁	—	2	1	no	—
246	C₂×C₄×C₄₁	—	2	2	no	—
247	C₂×C₄×C₄₁	—	2	1	no	—
248	C₂×C₄×C₄₁	—	4	1	no	—
249	C₂×C₄×C₄₁	—	4	2	no	—
250	C₂×C₄×C₄₁	—	4	2	no	—
251	C₂×C₄×C₄₁	—	4	4	no	—
252	C₂×C₄×C₄₁	—	8	1	no	—
253	C₂×C₄×C₄₁	—	8	2	no	—
254	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
255	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
256	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
259	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄×C₄₁	#137	1	2	yes	—
261	C₂×C₄×C₄₁	#137	1	2	yes	—
262	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
263	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
265	C₈×C₄₁	—	2	2	no	—
266	C₈×C₄₁	—	4	1	no	—
267	C₈×C₄₁	—	4	2	no	—
268	C₈×C₄₁	—	4	1	no	—
269	C₈×C₄₁	—	4	1	no	—
270	C₈×C₄₁	—	4	2	no	—
271	#17514	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂²×C₄₁	—	2	1	no	—
273	C₂²×C₄₁	—	2	1	no	—
274	C₂²×C₄₁	—	2	2	no	—
275	C₂²×C₄₁	—	2	2	no	—
276	C₂²×C₄₁	—	2	2	no	—
277	C₂²×C₄₁	#419	1	2	yes	—
278	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
279	C₄×C₄₁	—	2	1	no	—
280	C₄×C₄₁	—	2	2	no	—
281	C₄×C₄₁	—	2	2	no	—
282	C₄×C₄₁	—	2	2	no	—
283	C₄×C₄₁	—	2	1	no	—
284	C₄×C₄₁	—	2	1	no	—
285	C₄×C₄₁	—	2	2	no	—
286	C₄×C₄₁	—	4	1	no	—
287	C₄×C₄₁	—	4	2	no	—
288	C₄×C₄₁	—	4	1	no	—
289	C₄×C₄₁	—	4	1	no	—
290	C₄×C₄₁	—	4	2	no	—
291	C₄×C₄₁	#419	1	2	yes	—
292	C₄×C₄₁	C₂×#131	1	1	yes	characteristic
293	C₄×C₄₁	#568	1	1	yes	characteristic
294	C₄×C₄₁	C₂×#137	1	1	yes	characteristic
295	C₄×C₄₁	C₂×#137	1	1	yes	characteristic
296	Q₈×#41	C₂×C₄₁	1	1	yes	characteristic
297	#2965	C₂×C₄₁	1	1	yes	characteristic
298	#2925	C₂×C₄₁	1	1	yes	characteristic
299	#1497	C₂×C₄₁	1	2	yes	complemented
300	#1493	C₂×C₄₁	1	1	yes	characteristic, complemented
301	#1491	C₂×C₄₁	1	2	yes	complemented
302	#1487	C₂×C₄₁	1	1	yes	characteristic, complemented
303	#1485	C₂×C₄₁	1	2	yes	complemented
304	#1481	C₂×C₄₁	1	1	yes	characteristic, complemented
305	#1479	C₂×C₄₁	1	2	yes	complemented
306	C₂×Q₈⋊C₈	C₂×C₄₁	1	1	yes	characteristic, complemented
307	C₂×C₄₁	—	2	2	no	—
308	C₂×C₄₁	—	4	2	no	—
309	C₂×C₄₁	#2880	1	1	yes	characteristic
310	C₂×C₄₁	C₂×#419	1	1	yes	characteristic
311	C₂×C₄₁	#2901	1	1	yes	characteristic
312	Q₈²	C₄×C₄₁	1	1	yes	characteristic
313	#557	—	2	1	no	—
314	#557	C₄×C₄₁	1	1	yes	characteristic
315	#554	—	2	2	no	—
316	#554	—	2	1	no	—
317	#554	C₄×C₄₁	1	2	yes	—
318	#554	C₄×C₄₁	1	1	yes	characteristic
319	D₄×Q₈	C₄×C₄₁	1	1	yes	characteristic
320	D₄×Q₈	C₄×C₄₁	1	2	yes	—
321	#533	—	2	1	no	—
322	#533	—	2	2	no	—
323	#533	C₂²×C₄₁	1	1	yes	characteristic
324	#533	C₂²×C₄₁	1	2	yes	—
325	C₄×#41	—	2	1	no	—
326	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
327	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
328	C₂×C₄×Q₈	C₂²×C₄₁	1	2	yes	—
329	#446	—	2	1	no	—
330	#436	—	2	1	no	—
331	#435	—	2	2	no	—
332	#433	—	2	2	no	—
333	#432	C₂²×C₄₁	1	2	yes	—
334	#431	C₂²×C₄₁	1	2	yes	—
335	#431	C₂²×C₄₁	1	2	yes	—
336	#431	C₂²×C₄₁	1	1	yes	characteristic
337	#423	C₂²×C₄₁	1	1	yes	characteristic
338	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
339	#408	—	2	1	no	—
340	#405	—	2	1	no	—
341	C₄×M₁₆	C₂²×C₄₁	1	2	yes	—
342	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
343	Q₈⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
344	Q₈⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
345	Q₈⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
346	Q₈⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
347	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
348	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
349	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
350	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
351	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
352	Q₈⋊C₈	C₂²×C₄₁	1	2	yes	—
353	C₄₁	#17514	1	1	yes	characteristic, direct product factor, Sylow
354	C₂×#41	D₄×C₄₁	1	1	yes	characteristic
355	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
356	C₂²×Q₈	C₂×C₄×C₄₁	1	2	yes	—
357	C₂×M₁₆	—	2	2	no	—
358	C₂×M₁₆	—	2	1	no	—
359	C₂²×C₈	—	2	1	no	—
360	C₂²×C₈	—	2	1	no	—
361	C₄⋊Q₈	—	2	2	no	—
362	C₄⋊Q₈	—	2	1	no	—
363	C₄⋊Q₈	—	2	2	no	—
364	C₄⋊Q₈	—	2	1	no	—
365	C₄⋊Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
366	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
367	C₄⋊Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
368	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
369	#125	—	2	2	no	—
370	#125	—	2	1	no	—
371	#125	—	2	2	no	—
372	#125	—	2	1	no	—
373	#125	—	2	1	no	—
374	#125	—	2	2	no	—
375	C₂²⋊Q₈	—	2	1	no	—
376	C₂²⋊Q₈	—	2	2	no	—
377	C₂²⋊Q₈	—	2	2	no	—
378	C₂²⋊Q₈	—	2	2	no	—
379	C₂²⋊Q₈	—	2	2	no	—
380	C₂²⋊Q₈	—	2	2	no	—
381	C₂²⋊Q₈	—	2	2	no	—
382	C₂²⋊Q₈	—	2	2	no	—
383	C₂²⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
384	C₂²⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
385	C₂²⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
386	C₄×Q₈	—	2	1	no	—
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	—	2	2	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	—	2	1	no	—
391	C₄×Q₈	—	2	2	no	—
392	C₄×Q₈	—	2	2	no	—
393	C₄×Q₈	—	2	1	no	—
394	C₄×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
395	C₄×Q₈	C₂×C₄×C₄₁	1	2	yes	—
396	C₄×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
397	C₄×Q₈	C₂×C₄×C₄₁	1	2	yes	—
398	C₄×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
399	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
400	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
401	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
402	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	—	2	2	no	—
406	C₄×D₄	—	2	1	no	—
407	C₄×D₄	C₂×C₄×C₄₁	1	2	yes	—
408	C₄×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
409	C₄×D₄	D₄×C₄₁	1	2	yes	—
410	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
411	#117	—	2	2	no	—
412	#117	—	2	1	no	—
413	#117	D₄×C₄₁	1	1	yes	characteristic
414	#117	D₄×C₄₁	1	2	yes	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	2	no	—
417	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
419	C₂×C₄²	—	2	2	no	—
420	C₂×C₄²	—	2	1	no	—
421	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
423	C₄⋊C₈	—	2	1	no	—
424	C₄⋊C₈	—	2	1	no	—
425	C₄⋊C₈	—	2	2	no	—
426	C₄⋊C₈	—	2	1	no	—
427	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
428	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
429	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
430	C₂²⋊C₈	—	2	2	no	—
431	C₂²⋊C₈	—	2	2	no	—
432	C₂²⋊C₈	—	2	1	no	—
433	C₂²⋊C₈	—	2	2	no	—
434	#97	—	2	2	no	—
435	#97	—	2	2	no	—
436	C₄×C₈	—	2	1	no	—
437	C₄×C₈	—	2	2	no	—
438	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
439	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
440	C₄×C₈	C₂³×C₄₁	1	2	yes	—
441	#41	—	2	2	no	—
442	#41	—	2	2	no	—
443	C₂×Q₈	—	2	2	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	—	2	4	no	—
449	C₂×Q₈	—	4	1	no	—
450	C₂×Q₈	—	4	2	no	—
451	C₂×Q₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
452	C₂×Q₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
453	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
454	C₂×Q₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
455	C₂×D₄	C₂²⋊C₄×C₄₁	1	2	yes	—
456	C₂×D₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
462	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
464	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
465	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
466	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
467	M₁₆	—	4	4	no	—
468	M₁₆	—	4	2	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	—	2	2	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	4	2	no	—
476	C₂×C₈	—	4	2	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	2	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
496	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
497	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
498	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
499	C₄⋊C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
500	C₄⋊C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
501	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
506	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
507	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
508	C₄²	—	2	2	no	—
509	C₄²	—	2	1	no	—
510	C₄²	—	2	2	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	1	no	—
515	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
516	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
517	C₄²	C₂×D₄×C₄₁	1	2	yes	—
518	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
519	C₄²	C₂×D₄×C₄₁	1	2	yes	—
520	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
521	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
522	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
523	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
524	Q₈	—	2	1	no	—
525	Q₈	—	2	2	no	—
526	Q₈	—	2	1	no	—
527	Q₈	—	2	2	no	—
528	Q₈	—	2	1	no	—
529	Q₈	—	2	2	no	—
530	Q₈	—	4	2	no	—
531	Q₈	—	4	4	no	—
532	Q₈	—	8	1	no	—
533	Q₈	—	8	2	no	—
534	Q₈	#131×C₄₁	1	2	yes	—
535	D₄	—	2	4	no	—
536	D₄	—	2	2	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	2	no	—
552	C₂×C₄	—	4	4	no	—
553	C₂×C₄	—	8	1	no	—
554	C₂×C₄	—	8	2	no	—
555	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
556	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
557	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
558	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
559	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
560	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
561	C₂×C₄	#137×C₄₁	1	2	yes	—
562	C₂×C₄	#137×C₄₁	1	2	yes	—
563	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
564	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
565	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
566	C₈	—	2	2	no	—
567	C₈	—	4	1	no	—
568	C₈	—	4	2	no	—
569	C₈	—	4	1	no	—
570	C₈	—	4	1	no	—
571	C₈	—	4	2	no	—
572	C₂²	—	2	1	no	—
573	C₂²	—	2	1	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	C₄₁×#419	1	2	yes	—
578	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
579	C₄	—	2	1	no	—
580	C₄	—	2	2	no	—
581	C₄	—	2	2	no	—
582	C₄	—	2	2	no	—
583	C₄	—	2	1	no	—
584	C₄	—	2	1	no	—
585	C₄	—	2	2	no	—
586	C₄	—	4	1	no	—
587	C₄	—	4	2	no	—
588	C₄	—	4	1	no	—
589	C₄	—	4	1	no	—
590	C₄	—	4	2	no	—
591	C₄	C₄₁×#419	1	2	yes	—
592	C₄	C₂×#131×C₄₁	1	1	yes	characteristic
593	C₄	C₄₁×#568	1	1	yes	characteristic
594	C₄	C₂×#137×C₄₁	1	1	yes	characteristic
595	C₄	C₂×#137×C₄₁	1	1	yes	characteristic
596	C₂	—	2	2	no	—
597	C₂	—	4	2	no	—
598	C₂	C₄₁×#2880	1	1	yes	characteristic
599	C₂	C₂×C₄₁×#419	1	1	yes	characteristic
600	C₂	C₄₁×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₄²	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂²×C₂²⋊C₄
prime radical	C₂×#41×C₄₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₄×#41×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1497	C₂	semidirect
5. C₄₁×#1493	C₂	semidirect
6. C₄₁×#1491	C₂	semidirect
7. C₄₁×#1487	C₂	semidirect
8. C₄₁×#1485	C₂	semidirect
9. C₄₁×#1481	C₂	semidirect
10. C₄₁×#1479	C₂	semidirect
11. C₂×C₄₁×Q₈⋊C₈	C₂	semidirect
271. #17514	C₄₁	direct (N × H)
299. #1497	C₂×C₄₁	semidirect
300. #1493	C₂×C₄₁	semidirect
301. #1491	C₂×C₄₁	semidirect
302. #1487	C₂×C₄₁	semidirect
303. #1485	C₂×C₄₁	semidirect
304. #1481	C₂×C₄₁	semidirect
305. #1479	C₂×C₄₁	semidirect
306. C₂×Q₈⋊C₈	C₂×C₄₁	semidirect
353. C₄₁	#17514	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#41×C₄₁	2
C₄₁×#2965	2
C₄₁×#2925	2
C₄₁×#1497	2
C₄₁×#1493	2
C₄₁×#1491	2
C₄₁×#1487	2
C₄₁×#1485	2
C₄₁×#1481	2
C₄₁×#1479	2
C₂×C₄₁×Q₈⋊C₈	2
#17514	41

Subgroups — C41×#17514 #23793715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#17514 #23793715