Subgroups — C₄₁×#21093 #23797294

Order 10496.

1198 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#417	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#417	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2297	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1936	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1935	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1934	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1933	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1736	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1736	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₄₁	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
20	C₄₁×#417	—	2	4	no	—
21	C₄₁×#417	C₄	1	8	yes	complemented
22	C₂×#103×C₄₁	C₂²	1	2	yes	—
23	C₂×#103×C₄₁	C₂²	1	2	yes	—
24	C₂×#102×C₄₁	C₂²	1	2	yes	—
25	C₂×#102×C₄₁	C₂²	1	2	yes	—
26	C₄×M₁₆×C₄₁	—	2	4	no	—
27	C₂×#97×C₄₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
29	C₄₁×#391	C₂²	1	1	yes	characteristic
30	C₄₁×#391	C₂²	1	1	yes	characteristic
31	C₄₁×#390	C₂²	1	1	yes	characteristic
32	C₄₁×#390	C₂²	1	1	yes	characteristic
33	C₄₁×#389	—	2	1	no	—
34	C₄₁×#389	—	2	1	no	—
35	C₄₁×#389	C₂²	1	2	yes	—
36	C₄₁×#389	C₂²	1	2	yes	—
37	C₄₁×#386	—	2	1	no	—
38	C₄₁×#386	—	2	1	no	—
39	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
40	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
41	C₂×#95×C₄₁	—	2	1	no	—
42	C₄₁×#340	C₂²	1	2	yes	—
43	C₄₁×#340	C₂²	1	2	yes	—
44	C₄₁×#340	C₂²	1	2	yes	—
45	C₄₁×#340	C₂²	1	2	yes	—
46	C₂×#41×C₄₁	—	2	2	no	—
47	C₂×#41×C₄₁	—	2	2	no	—
48	C₂×#41×C₄₁	—	2	2	no	—
49	C₂×#41×C₄₁	—	2	2	no	—
50	C₂×#41×C₄₁	—	2	2	no	—
51	C₂×#41×C₄₁	C₂×C₄	1	4	yes	—
52	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₄₁	—	2	1	no	—
54	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₄₁	—	2	1	no	—
56	C₂³×C₄×C₄₁	—	2	1	no	—
57	C₂³×C₄×C₄₁	—	2	1	no	—
58	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₄₁	—	2	2	no	—
62	C₂×M₁₆×C₄₁	—	2	2	no	—
63	C₂×M₁₆×C₄₁	—	2	2	no	—
64	C₂×M₁₆×C₄₁	—	2	2	no	—
65	C₂×M₁₆×C₄₁	C₂×C₄	1	4	yes	—
66	C₂×M₁₆×C₄₁	D₄	1	4	yes	—
67	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
69	#117×C₄₁	—	2	2	no	—
70	#117×C₄₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₄₁	—	4	4	no	—
79	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
80	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
84	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
89	C₂×C₄²×C₄₁	—	2	2	no	—
90	C₂×C₄²×C₄₁	—	2	2	no	—
91	C₂×C₄²×C₄₁	—	2	1	no	—
92	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
96	#103×C₄₁	—	2	4	no	—
97	#103×C₄₁	C₂×C₄	1	8	yes	complemented
98	#102×C₄₁	—	2	4	no	—
99	#102×C₄₁	C₂×C₄	1	8	yes	complemented
100	#97×C₄₁	—	2	4	no	—
101	C₄×C₈×C₄₁	—	2	4	no	—
102	#95×C₄₁	—	2	2	no	—
103	#95×C₄₁	—	2	2	no	—
104	#95×C₄₁	—	2	1	no	—
105	#95×C₄₁	—	2	1	no	—
106	#95×C₄₁	—	2	1	no	—
107	#95×C₄₁	—	2	1	no	—
108	C₂⁴×C₄₁	—	2	1	no	—
109	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
110	#41×C₄₁	—	2	8	no	—
111	#41×C₄₁	—	2	8	no	—
112	#41×C₄₁	—	4	4	no	—
113	#41×C₄₁	—	4	4	no	—
114	C₂×Q₈×C₄₁	—	2	1	no	—
115	C₂×Q₈×C₄₁	—	2	1	no	—
116	C₂×Q₈×C₄₁	—	2	1	no	—
117	C₂×Q₈×C₄₁	—	2	1	no	—
118	C₂×Q₈×C₄₁	—	2	1	no	—
119	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
120	C₂×D₄×C₄₁	—	2	4	no	—
121	C₂×D₄×C₄₁	—	2	1	no	—
122	C₂×D₄×C₄₁	—	2	1	no	—
123	C₂×D₄×C₄₁	—	2	1	no	—
124	C₂×D₄×C₄₁	—	2	1	no	—
125	C₂×D₄×C₄₁	—	2	1	no	—
126	C₂×D₄×C₄₁	—	4	2	no	—
127	C₂×D₄×C₄₁	—	4	2	no	—
128	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₄₁	—	2	4	no	—
130	C₂²×C₄×C₄₁	—	2	2	no	—
131	C₂²×C₄×C₄₁	—	2	2	no	—
132	C₂²×C₄×C₄₁	—	2	1	no	—
133	C₂²×C₄×C₄₁	—	2	2	no	—
134	C₂²×C₄×C₄₁	—	2	4	no	—
135	C₂²×C₄×C₄₁	—	2	2	no	—
136	C₂²×C₄×C₄₁	—	2	2	no	—
137	C₂²×C₄×C₄₁	—	2	1	no	—
138	C₂²×C₄×C₄₁	—	2	1	no	—
139	C₂²×C₄×C₄₁	—	2	2	no	—
140	C₂²×C₄×C₄₁	—	4	1	no	—
141	C₂²×C₄×C₄₁	—	4	1	no	—
142	C₂²×C₄×C₄₁	—	4	2	no	—
143	C₂²×C₄×C₄₁	—	4	2	no	—
144	C₂²×C₄×C₄₁	—	4	2	no	—
145	C₂²×C₄×C₄₁	—	4	2	no	—
146	C₂²×C₄×C₄₁	—	4	2	no	—
147	C₂²×C₄×C₄₁	—	4	2	no	—
148	C₂²×C₄×C₄₁	—	4	2	no	—
149	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
150	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₄₁	#41	1	2	yes	—
157	C₂²×C₄×C₄₁	#41	1	2	yes	—
158	C₂²×C₄×C₄₁	#41	1	2	yes	—
159	C₂²×C₄×C₄₁	#41	1	2	yes	—
160	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
161	M₁₆×C₄₁	—	2	8	no	—
162	M₁₆×C₄₁	—	2	8	no	—
163	C₂×C₈×C₄₁	—	2	2	no	—
164	C₂×C₈×C₄₁	—	2	2	no	—
165	C₂×C₈×C₄₁	—	2	2	no	—
166	C₂×C₈×C₄₁	—	2	2	no	—
167	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
168	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
169	C₄⋊C₄×C₄₁	—	2	4	no	—
170	C₄⋊C₄×C₄₁	—	2	2	no	—
171	C₄⋊C₄×C₄₁	—	2	2	no	—
172	C₄⋊C₄×C₄₁	—	2	2	no	—
173	C₄⋊C₄×C₄₁	—	2	1	no	—
174	C₄⋊C₄×C₄₁	—	2	1	no	—
175	C₄⋊C₄×C₄₁	—	2	1	no	—
176	C₄⋊C₄×C₄₁	—	2	1	no	—
177	C₄⋊C₄×C₄₁	—	4	4	no	—
178	C₄⋊C₄×C₄₁	—	4	2	no	—
179	C₄⋊C₄×C₄₁	—	4	4	no	—
180	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
181	C₂²⋊C₄×C₄₁	—	4	2	no	—
182	C₂²⋊C₄×C₄₁	—	4	2	no	—
183	C₂²⋊C₄×C₄₁	—	4	2	no	—
184	C₂²⋊C₄×C₄₁	—	4	2	no	—
185	C₂²⋊C₄×C₄₁	—	4	2	no	—
186	C₄²×C₄₁	—	2	2	no	—
187	C₄²×C₄₁	—	2	2	no	—
188	C₄²×C₄₁	—	2	1	no	—
189	C₄²×C₄₁	—	2	1	no	—
190	C₄²×C₄₁	—	2	1	no	—
191	C₄²×C₄₁	—	2	1	no	—
192	C₄²×C₄₁	—	4	2	no	—
193	C₂³×C₄₁	—	2	1	no	—
194	C₂³×C₄₁	—	2	1	no	—
195	C₂³×C₄₁	—	2	1	no	—
196	C₂³×C₄₁	—	2	1	no	—
197	C₂³×C₄₁	—	2	2	no	—
198	C₂³×C₄₁	—	4	1	no	—
199	C₂³×C₄₁	—	4	1	no	—
200	C₂³×C₄₁	—	4	1	no	—
201	C₂³×C₄₁	—	4	1	no	—
202	C₂³×C₄₁	—	4	1	no	—
203	C₂³×C₄₁	—	4	1	no	—
204	C₂³×C₄₁	C₄×D₄	1	2	yes	—
205	C₂³×C₄₁	#121	1	2	yes	—
206	C₂³×C₄₁	#124	1	2	yes	—
207	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
208	Q₈×C₄₁	—	2	2	no	—
209	Q₈×C₄₁	—	2	2	no	—
210	Q₈×C₄₁	—	4	1	no	—
211	Q₈×C₄₁	—	4	1	no	—
212	D₄×C₄₁	—	2	2	no	—
213	D₄×C₄₁	—	2	2	no	—
214	D₄×C₄₁	—	4	8	no	—
215	D₄×C₄₁	—	4	1	no	—
216	D₄×C₄₁	—	4	1	no	—
217	C₂×C₄×C₄₁	—	2	1	no	—
218	C₂×C₄×C₄₁	—	2	1	no	—
219	C₂×C₄×C₄₁	—	2	1	no	—
220	C₂×C₄×C₄₁	—	2	1	no	—
221	C₂×C₄×C₄₁	—	2	4	no	—
222	C₂×C₄×C₄₁	—	2	2	no	—
223	C₂×C₄×C₄₁	—	2	1	no	—
224	C₂×C₄×C₄₁	—	2	1	no	—
225	C₂×C₄×C₄₁	—	2	1	no	—
226	C₂×C₄×C₄₁	—	2	1	no	—
227	C₂×C₄×C₄₁	—	2	1	no	—
228	C₂×C₄×C₄₁	—	2	1	no	—
229	C₂×C₄×C₄₁	—	4	4	no	—
230	C₂×C₄×C₄₁	—	4	4	no	—
231	C₂×C₄×C₄₁	—	4	8	no	—
232	C₂×C₄×C₄₁	—	4	1	no	—
233	C₂×C₄×C₄₁	—	4	1	no	—
234	C₂×C₄×C₄₁	—	4	2	no	—
235	C₂×C₄×C₄₁	—	4	2	no	—
236	C₂×C₄×C₄₁	—	4	8	no	—
237	C₂×C₄×C₄₁	—	4	2	no	—
238	C₂×C₄×C₄₁	—	4	1	no	—
239	C₂×C₄×C₄₁	—	4	1	no	—
240	C₂×C₄×C₄₁	—	4	1	no	—
241	C₂×C₄×C₄₁	—	4	2	no	—
242	C₂×C₄×C₄₁	—	4	1	no	—
243	C₂×C₄×C₄₁	—	4	1	no	—
244	C₂×C₄×C₄₁	—	4	1	no	—
245	C₂×C₄×C₄₁	—	4	2	no	—
246	C₂×C₄×C₄₁	—	8	4	no	—
247	C₂×C₄×C₄₁	#117	1	4	yes	—
248	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
249	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
250	C₂×C₄×C₄₁	#121	1	2	yes	—
251	C₂×C₄×C₄₁	#123	1	4	yes	—
252	C₂×C₄×C₄₁	#124	1	2	yes	—
253	C₂×C₄×C₄₁	#126	1	4	yes	—
254	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
255	C₂×C₄×C₄₁	#137	1	2	yes	—
256	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
257	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
259	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
261	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
262	C₈×C₄₁	—	2	4	no	—
263	C₈×C₄₁	—	2	4	no	—
264	#21093	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
265	C₂²×C₄₁	—	2	2	no	—
266	C₂²×C₄₁	—	2	2	no	—
267	C₂²×C₄₁	—	2	2	no	—
268	C₂²×C₄₁	—	2	2	no	—
269	C₂²×C₄₁	—	2	2	no	—
270	C₂²×C₄₁	—	2	2	no	—
271	C₂²×C₄₁	—	4	2	no	—
272	C₂²×C₄₁	—	4	2	no	—
273	C₂²×C₄₁	—	4	2	no	—
274	C₂²×C₄₁	—	8	1	no	—
275	C₂²×C₄₁	—	8	1	no	—
276	C₂²×C₄₁	—	8	1	no	—
277	C₂²×C₄₁	—	8	1	no	—
278	C₂²×C₄₁	#388	1	4	yes	—
279	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
280	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
281	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
282	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
283	C₂²×C₄₁	C₂×#137	1	1	yes	characteristic
284	C₂²×C₄₁	#575	1	1	yes	characteristic
285	C₂²×C₄₁	#575	1	1	yes	characteristic
286	C₄×C₄₁	—	2	2	no	—
287	C₄×C₄₁	—	2	2	no	—
288	C₄×C₄₁	—	4	4	no	—
289	C₄×C₄₁	—	4	1	no	—
290	C₄×C₄₁	—	4	1	no	—
291	C₄×C₄₁	—	4	2	no	—
292	C₄×C₄₁	—	8	1	no	—
293	C₄×C₄₁	—	8	1	no	—
294	C₄×C₄₁	—	8	2	no	—
295	C₄×C₄₁	#388	1	4	yes	—
296	C₄×C₄₁	C₂×#117	1	1	yes	characteristic
297	C₄×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
298	C₄×C₄₁	C₂×#123	1	1	yes	characteristic
299	C₄×C₄₁	C₂×#126	1	1	yes	characteristic
300	C₂×#417	C₂×C₄₁	1	1	yes	characteristic
301	C₂×#417	C₂×C₄₁	1	1	yes	characteristic
302	C₂×C₄×M₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
303	#2297	C₂×C₄₁	1	1	yes	characteristic
304	C₂×#389	C₂×C₄₁	1	1	yes	characteristic, complemented
305	#1936	C₂×C₄₁	1	2	yes	complemented
306	#1935	C₂×C₄₁	1	2	yes	complemented
307	#1934	C₂×C₄₁	1	2	yes	complemented
308	#1933	C₂×C₄₁	1	2	yes	complemented
309	#1736	C₂×C₄₁	1	1	yes	characteristic, complemented
310	#1736	C₂×C₄₁	1	1	yes	characteristic, complemented
311	C₂×C₄₁	—	2	4	no	—
312	C₂×C₄₁	—	8	2	no	—
313	C₂×C₄₁	C₂×#388	1	1	yes	characteristic
314	C₂×C₄₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
315	C₂×C₄₁	C₄×#137	1	1	yes	characteristic
316	C₂×C₄₁	#3055	1	1	yes	characteristic
317	C₂×C₄₁	#3056	1	1	yes	characteristic
318	C₂×C₄₁	#3141	1	1	yes	characteristic
319	C₂×C₄₁	#3142	1	1	yes	characteristic
320	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
321	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
322	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
323	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
324	C₂²×C₄⋊C₄	—	2	1	no	—
325	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
326	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
327	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
328	#417	—	2	4	no	—
329	#417	C₄×C₄₁	1	8	yes	complemented
330	C₂×#103	C₂²×C₄₁	1	2	yes	—
331	C₂×#103	C₂²×C₄₁	1	2	yes	—
332	C₂×#102	C₂²×C₄₁	1	2	yes	—
333	C₂×#102	C₂²×C₄₁	1	2	yes	—
334	C₄×M₁₆	—	2	4	no	—
335	C₂×#97	C₂²×C₄₁	1	2	yes	—
336	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
337	#391	C₂²×C₄₁	1	1	yes	characteristic
338	#391	C₂²×C₄₁	1	1	yes	characteristic
339	#390	C₂²×C₄₁	1	1	yes	characteristic
340	#390	C₂²×C₄₁	1	1	yes	characteristic
341	#389	—	2	1	no	—
342	#389	—	2	1	no	—
343	#389	C₂²×C₄₁	1	2	yes	—
344	#389	C₂²×C₄₁	1	2	yes	—
345	#386	—	2	1	no	—
346	#386	—	2	1	no	—
347	C₄×C₂²⋊C₄	—	2	1	no	—
348	C₄×C₂²⋊C₄	—	2	1	no	—
349	C₂×#95	—	2	1	no	—
350	#340	C₂²×C₄₁	1	2	yes	—
351	#340	C₂²×C₄₁	1	2	yes	—
352	#340	C₂²×C₄₁	1	2	yes	—
353	#340	C₂²×C₄₁	1	2	yes	—
354	C₄₁	#21093	1	1	yes	characteristic, direct product factor, Sylow
355	C₂×#41	—	2	2	no	—
356	C₂×#41	—	2	2	no	—
357	C₂×#41	—	2	2	no	—
358	C₂×#41	—	2	2	no	—
359	C₂×#41	—	2	2	no	—
360	C₂×#41	C₂×C₄×C₄₁	1	4	yes	—
361	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
362	C₂²×D₄	—	2	1	no	—
363	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
364	C₂³×C₄	—	2	1	no	—
365	C₂³×C₄	—	2	1	no	—
366	C₂³×C₄	—	2	1	no	—
367	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
368	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
369	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
370	C₂×M₁₆	—	2	2	no	—
371	C₂×M₁₆	—	2	2	no	—
372	C₂×M₁₆	—	2	2	no	—
373	C₂×M₁₆	—	2	2	no	—
374	C₂×M₁₆	C₂×C₄×C₄₁	1	4	yes	—
375	C₂×M₁₆	D₄×C₄₁	1	4	yes	—
376	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
377	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
378	#117	—	2	2	no	—
379	#117	—	2	2	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	2	no	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	2	no	—
384	C₂×C₄⋊C₄	—	2	2	no	—
385	C₂×C₄⋊C₄	—	2	2	no	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	4	4	no	—
388	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
390	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
393	C₂×C₂²⋊C₄	—	2	1	no	—
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₂²⋊C₄	—	2	1	no	—
396	C₂×C₂²⋊C₄	—	2	1	no	—
397	C₂×C₂²⋊C₄	—	2	1	no	—
398	C₂×C₄²	—	2	2	no	—
399	C₂×C₄²	—	2	2	no	—
400	C₂×C₄²	—	2	1	no	—
401	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
402	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
403	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
405	#103	—	2	4	no	—
406	#103	C₂×C₄×C₄₁	1	8	yes	complemented
407	#102	—	2	4	no	—
408	#102	C₂×C₄×C₄₁	1	8	yes	complemented
409	#97	—	2	4	no	—
410	C₄×C₈	—	2	4	no	—
411	#95	—	2	2	no	—
412	#95	—	2	2	no	—
413	#95	—	2	1	no	—
414	#95	—	2	1	no	—
415	#95	—	2	1	no	—
416	#95	—	2	1	no	—
417	C₂⁴	—	2	1	no	—
418	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
419	#41	—	2	8	no	—
420	#41	—	2	8	no	—
421	#41	—	4	4	no	—
422	#41	—	4	4	no	—
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	—	2	1	no	—
425	C₂×Q₈	—	2	1	no	—
426	C₂×Q₈	—	2	1	no	—
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	4	2	no	—
436	C₂×D₄	—	4	2	no	—
437	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
438	C₂²×C₄	—	2	4	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	4	1	no	—
450	C₂²×C₄	—	4	1	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	4	2	no	—
453	C₂²×C₄	—	4	2	no	—
454	C₂²×C₄	—	4	2	no	—
455	C₂²×C₄	—	4	2	no	—
456	C₂²×C₄	—	4	2	no	—
457	C₂²×C₄	—	4	2	no	—
458	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
459	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
465	C₂²×C₄	#41×C₄₁	1	2	yes	—
466	C₂²×C₄	#41×C₄₁	1	2	yes	—
467	C₂²×C₄	#41×C₄₁	1	2	yes	—
468	C₂²×C₄	#41×C₄₁	1	2	yes	—
469	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
470	M₁₆	—	2	8	no	—
471	M₁₆	—	2	8	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	—	2	2	no	—
474	C₂×C₈	—	2	2	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
477	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	2	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	4	4	no	—
487	C₄⋊C₄	—	4	2	no	—
488	C₄⋊C₄	—	4	4	no	—
489	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
490	C₂²⋊C₄	—	4	2	no	—
491	C₂²⋊C₄	—	4	2	no	—
492	C₂²⋊C₄	—	4	2	no	—
493	C₂²⋊C₄	—	4	2	no	—
494	C₂²⋊C₄	—	4	2	no	—
495	C₄²	—	2	2	no	—
496	C₄²	—	2	2	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	2	1	no	—
500	C₄²	—	2	1	no	—
501	C₄²	—	4	2	no	—
502	C₂³	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	2	1	no	—
505	C₂³	—	2	1	no	—
506	C₂³	—	2	2	no	—
507	C₂³	—	4	1	no	—
508	C₂³	—	4	1	no	—
509	C₂³	—	4	1	no	—
510	C₂³	—	4	1	no	—
511	C₂³	—	4	1	no	—
512	C₂³	—	4	1	no	—
513	C₂³	C₄×D₄×C₄₁	1	2	yes	—
514	C₂³	#121×C₄₁	1	2	yes	—
515	C₂³	#124×C₄₁	1	2	yes	—
516	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
517	Q₈	—	2	2	no	—
518	Q₈	—	2	2	no	—
519	Q₈	—	4	1	no	—
520	Q₈	—	4	1	no	—
521	D₄	—	2	2	no	—
522	D₄	—	2	2	no	—
523	D₄	—	4	8	no	—
524	D₄	—	4	1	no	—
525	D₄	—	4	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	4	4	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	4	8	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	8	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	2	no	—
555	C₂×C₄	—	8	4	no	—
556	C₂×C₄	#117×C₄₁	1	4	yes	—
557	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
558	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
559	C₂×C₄	#121×C₄₁	1	2	yes	—
560	C₂×C₄	#123×C₄₁	1	4	yes	—
561	C₂×C₄	#124×C₄₁	1	2	yes	—
562	C₂×C₄	#126×C₄₁	1	4	yes	—
563	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
564	C₂×C₄	#137×C₄₁	1	2	yes	—
565	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
566	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
567	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
568	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
569	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
570	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
571	C₈	—	2	4	no	—
572	C₈	—	2	4	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	4	2	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	8	1	no	—
583	C₂²	—	8	1	no	—
584	C₂²	—	8	1	no	—
585	C₂²	—	8	1	no	—
586	C₂²	C₄₁×#388	1	4	yes	—
587	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
588	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
589	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
590	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
591	C₂²	C₂×#137×C₄₁	1	1	yes	characteristic
592	C₂²	C₄₁×#575	1	1	yes	characteristic
593	C₂²	C₄₁×#575	1	1	yes	characteristic
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	4	1	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	2	no	—
599	C₄	—	4	4	no	—
600	C₄	—	8	2	no	—
601	C₄	—	8	1	no	—
602	C₄	—	8	1	no	—
603	C₄	C₄₁×#388	1	4	yes	—
604	C₄	C₂×#117×C₄₁	1	1	yes	characteristic
605	C₄	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
606	C₄	C₂×#123×C₄₁	1	1	yes	characteristic
607	C₄	C₂×#126×C₄₁	1	1	yes	characteristic
608	C₂	—	2	4	no	—
609	C₂	—	8	2	no	—
610	C₂	C₂×C₄₁×#388	1	1	yes	characteristic
611	C₂	C₄×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
612	C₂	C₄×#137×C₄₁	1	1	yes	characteristic
613	C₂	C₄₁×#3055	1	1	yes	characteristic
614	C₂	C₄₁×#3056	1	1	yes	characteristic
615	C₂	C₄₁×#3141	1	1	yes	characteristic
616	C₂	C₄₁×#3142	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×M₁₆×C₄₁	C₂	semidirect
5. C₂×C₄₁×#389	C₂	semidirect
6. C₄₁×#1936	C₂	semidirect
7. C₄₁×#1935	C₂	semidirect
8. C₄₁×#1934	C₂	semidirect
9. C₄₁×#1933	C₂	semidirect
10. C₄₁×#1736	C₂	semidirect
11. C₄₁×#1736	C₂	semidirect
21. C₄₁×#417	C₄	semidirect
97. #103×C₄₁	C₂×C₄	semidirect
99. #102×C₄₁	C₂×C₄	semidirect
264. #21093	C₄₁	direct (N × H)
302. C₂×C₄×M₁₆	C₂×C₄₁	semidirect
304. C₂×#389	C₂×C₄₁	semidirect
305. #1936	C₂×C₄₁	semidirect
306. #1935	C₂×C₄₁	semidirect
307. #1934	C₂×C₄₁	semidirect
308. #1933	C₂×C₄₁	semidirect
309. #1736	C₂×C₄₁	semidirect
310. #1736	C₂×C₄₁	semidirect
329. #417	C₄×C₄₁	semidirect
354. C₄₁	#21093	direct (N × H)
406. #103	C₂×C₄×C₄₁	semidirect
408. #102	C₂×C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#417	2
C₂×C₄×M₁₆×C₄₁	2
C₄₁×#2297	2
C₂×C₄₁×#389	2
C₄₁×#1936	2
C₄₁×#1935	2
C₄₁×#1934	2
C₄₁×#1933	2
C₄₁×#1736	2
#21093	41

Subgroups — C41×#21093 #23797294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21093 #23797294