Subgroups — C₄₁×#22778 #23798979

Order 10496.

894 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#440	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2356	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1933	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1921	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1918	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1877	C₂	1	1	yes	maximal, characteristic
9	C₄₁×#1877	C₂	1	1	yes	maximal, characteristic
10	C₄₁×#1836	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1804	C₂	1	1	yes	maximal, characteristic
12	C₄₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄₁×#1771	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄₁×#1762	C₂	1	1	yes	maximal, characteristic
15	C₄×#102×C₄₁	C₂	1	1	yes	maximal, characteristic
16	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₄₁	—	2	1	no	—
22	C₄₁×#440	C₄	1	8	yes	complemented
23	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
34	C₄₁×#391	C₂²	1	1	yes	characteristic
35	C₄₁×#391	C₂²	1	1	yes	characteristic
36	C₄₁×#389	C₂²	1	1	yes	characteristic
37	C₄₁×#389	C₂²	1	1	yes	characteristic
38	C₄₁×#387	—	2	1	no	—
39	C₄₁×#387	—	2	1	no	—
40	C₄₁×#385	—	2	1	no	—
41	C₄₁×#385	—	2	1	no	—
42	C₄₁×#384	C₂²	1	1	yes	characteristic
43	C₄₁×#383	—	2	1	no	—
44	C₄₁×#382	C₂²	1	1	yes	characteristic
45	C₄₁×#380	—	2	1	no	—
46	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
48	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
49	C₄₁×#376	—	2	1	no	—
50	C₄₁×#340	C₂²	1	1	yes	characteristic
51	C₄₁×#340	C₂²	1	1	yes	characteristic
52	C₄₁×#340	C₂²	1	1	yes	characteristic
53	C₄₁×#340	C₂²	1	1	yes	characteristic
54	C₄₁×#340	C₂²	1	1	yes	characteristic
55	C₄₁×#340	C₂²	1	1	yes	characteristic
56	C₄₁×#340	C₂²	1	1	yes	characteristic
57	C₄₁×#340	C₂²	1	1	yes	characteristic
58	C₄₁×#336	C₂²	1	1	yes	characteristic
59	C₄₁×#336	C₂²	1	1	yes	characteristic
60	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
61	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₄₁	—	2	1	no	—
63	C₂³×C₄×C₄₁	—	2	1	no	—
64	C₂³×C₄×C₄₁	—	2	1	no	—
65	C₂×SD₁₆×C₄₁	—	2	2	no	—
66	C₂×SD₁₆×C₄₁	—	2	2	no	—
67	C₂×SD₁₆×C₄₁	C₂×C₄	1	4	yes	—
68	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₄₁	—	2	2	no	—
73	C₄×Q₈×C₄₁	C₂×C₄	1	4	yes	complemented
74	C₄×D₄×C₄₁	—	2	2	no	—
75	C₄×D₄×C₄₁	—	2	2	no	—
76	C₄×D₄×C₄₁	—	2	2	no	—
77	C₄×D₄×C₄₁	—	4	1	no	—
78	C₄×D₄×C₄₁	—	4	1	no	—
79	C₄×D₄×C₄₁	C₂×C₄	1	4	yes	—
80	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₄₁	—	4	4	no	—
86	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
97	C₂×C₄²×C₄₁	—	2	1	no	—
98	C₂×C₄²×C₄₁	—	2	1	no	—
99	C₂×C₄²×C₄₁	—	2	1	no	—
100	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
103	#107×C₄₁	C₂×C₄	1	4	yes	complemented
104	C₄⋊C₈×C₄₁	—	2	2	no	—
105	#103×C₄₁	—	4	1	no	—
106	#103×C₄₁	—	4	1	no	—
107	#103×C₄₁	C₂×C₄	1	4	yes	complemented
108	#102×C₄₁	—	2	2	no	—
109	#102×C₄₁	—	2	2	no	—
110	#102×C₄₁	C₂×C₄	1	4	yes	—
111	#97×C₄₁	—	4	1	no	—
112	#97×C₄₁	C₂×C₄	1	4	yes	complemented
113	C₄×C₈×C₄₁	—	4	1	no	—
114	#95×C₄₁	—	2	1	no	—
115	#95×C₄₁	—	2	1	no	—
116	#95×C₄₁	—	2	1	no	—
117	#95×C₄₁	—	2	1	no	—
118	#95×C₄₁	—	2	1	no	—
119	#95×C₄₁	—	2	1	no	—
120	#95×C₄₁	—	2	1	no	—
121	#95×C₄₁	—	2	1	no	—
122	#95×C₄₁	—	2	1	no	—
123	#95×C₄₁	—	2	1	no	—
124	#95×C₄₁	D₄	1	1	yes	characteristic
125	#95×C₄₁	D₄	1	1	yes	characteristic
126	C₂⁴×C₄₁	—	2	1	no	—
127	C₂×Q₈×C₄₁	—	2	1	no	—
128	C₂×Q₈×C₄₁	—	2	1	no	—
129	C₂×Q₈×C₄₁	—	2	1	no	—
130	C₂×Q₈×C₄₁	—	2	1	no	—
131	C₂×Q₈×C₄₁	—	2	1	no	—
132	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
133	C₂×D₄×C₄₁	—	2	1	no	—
134	C₂×D₄×C₄₁	—	2	1	no	—
135	C₂×D₄×C₄₁	—	2	1	no	—
136	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
137	C₂×D₄×C₄₁	#41	1	2	yes	—
138	C₂×D₄×C₄₁	#41	1	2	yes	—
139	C₂²×C₄×C₄₁	—	2	1	no	—
140	C₂²×C₄×C₄₁	—	2	1	no	—
141	C₂²×C₄×C₄₁	—	2	1	no	—
142	C₂²×C₄×C₄₁	—	2	4	no	—
143	C₂²×C₄×C₄₁	—	2	1	no	—
144	C₂²×C₄×C₄₁	—	2	1	no	—
145	C₂²×C₄×C₄₁	—	2	1	no	—
146	C₂²×C₄×C₄₁	—	2	1	no	—
147	C₂²×C₄×C₄₁	—	4	2	no	—
148	C₂²×C₄×C₄₁	—	4	2	no	—
149	C₂²×C₄×C₄₁	—	4	2	no	—
150	C₂²×C₄×C₄₁	—	4	2	no	—
151	C₂²×C₄×C₄₁	—	4	2	no	—
152	C₂²×C₄×C₄₁	—	4	2	no	—
153	C₂²×C₄×C₄₁	—	8	1	no	—
154	C₂²×C₄×C₄₁	—	8	1	no	—
155	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
162	SD₁₆×C₄₁	—	2	8	no	—
163	C₂×C₈×C₄₁	—	2	1	no	—
164	C₂×C₈×C₄₁	—	2	1	no	—
165	C₂×C₈×C₄₁	—	2	1	no	—
166	C₂×C₈×C₄₁	—	2	1	no	—
167	C₂×C₈×C₄₁	—	2	1	no	—
168	C₂×C₈×C₄₁	—	2	1	no	—
169	C₂×C₈×C₄₁	—	2	1	no	—
170	C₂×C₈×C₄₁	—	2	1	no	—
171	C₂×C₈×C₄₁	—	2	1	no	—
172	C₂×C₈×C₄₁	—	2	1	no	—
173	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
174	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
175	C₄⋊C₄×C₄₁	—	2	4	no	—
176	C₄⋊C₄×C₄₁	—	2	2	no	—
177	C₄⋊C₄×C₄₁	—	2	1	no	—
178	C₄⋊C₄×C₄₁	—	2	1	no	—
179	C₄⋊C₄×C₄₁	—	2	1	no	—
180	C₄⋊C₄×C₄₁	—	2	1	no	—
181	C₄⋊C₄×C₄₁	—	2	1	no	—
182	C₄⋊C₄×C₄₁	—	2	1	no	—
183	C₄⋊C₄×C₄₁	—	2	1	no	—
184	C₄⋊C₄×C₄₁	—	2	1	no	—
185	C₄⋊C₄×C₄₁	—	2	1	no	—
186	C₄⋊C₄×C₄₁	—	2	1	no	—
187	C₄⋊C₄×C₄₁	—	4	1	no	—
188	C₄⋊C₄×C₄₁	—	4	2	no	—
189	C₄⋊C₄×C₄₁	—	4	2	no	—
190	C₄⋊C₄×C₄₁	—	4	2	no	—
191	C₄⋊C₄×C₄₁	—	8	1	no	—
192	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
193	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
194	C₂²⋊C₄×C₄₁	—	2	4	no	—
195	C₂²⋊C₄×C₄₁	—	4	2	no	—
196	C₂²⋊C₄×C₄₁	—	4	2	no	—
197	C₄²×C₄₁	—	2	4	no	—
198	C₄²×C₄₁	—	2	1	no	—
199	C₄²×C₄₁	—	2	1	no	—
200	C₄²×C₄₁	—	2	1	no	—
201	C₄²×C₄₁	—	2	1	no	—
202	C₄²×C₄₁	—	4	2	no	—
203	C₄²×C₄₁	—	4	2	no	—
204	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
205	C₄²×C₄₁	#41	1	2	yes	—
206	C₂³×C₄₁	—	2	2	no	—
207	C₂³×C₄₁	—	2	2	no	—
208	C₂³×C₄₁	—	2	2	no	—
209	C₂³×C₄₁	—	4	1	no	—
210	C₂³×C₄₁	—	4	1	no	—
211	C₂³×C₄₁	—	4	1	no	—
212	C₂³×C₄₁	—	4	1	no	—
213	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
214	Q₈×C₄₁	—	2	2	no	—
215	Q₈×C₄₁	—	2	2	no	—
216	Q₈×C₄₁	—	4	1	no	—
217	Q₈×C₄₁	—	4	1	no	—
218	D₄×C₄₁	—	2	2	no	—
219	D₄×C₄₁	—	2	2	no	—
220	D₄×C₄₁	—	2	2	no	—
221	D₄×C₄₁	#117	1	4	yes	—
222	C₂×C₄×C₄₁	—	2	1	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	2	no	—
229	C₂×C₄×C₄₁	—	2	2	no	—
230	C₂×C₄×C₄₁	—	2	2	no	—
231	C₂×C₄×C₄₁	—	2	1	no	—
232	C₂×C₄×C₄₁	—	2	1	no	—
233	C₂×C₄×C₄₁	—	2	1	no	—
234	C₂×C₄×C₄₁	—	2	1	no	—
235	C₂×C₄×C₄₁	—	2	1	no	—
236	C₂×C₄×C₄₁	—	2	1	no	—
237	C₂×C₄×C₄₁	—	2	1	no	—
238	C₂×C₄×C₄₁	—	2	1	no	—
239	C₂×C₄×C₄₁	—	2	1	no	—
240	C₂×C₄×C₄₁	—	4	8	no	—
241	C₂×C₄×C₄₁	—	4	1	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	1	no	—
246	C₂×C₄×C₄₁	—	4	1	no	—
247	C₂×C₄×C₄₁	—	4	1	no	—
248	C₂×C₄×C₄₁	—	4	1	no	—
249	C₂×C₄×C₄₁	—	4	1	no	—
250	C₂×C₄×C₄₁	—	4	1	no	—
251	C₂×C₄×C₄₁	—	4	1	no	—
252	C₂×C₄×C₄₁	—	4	1	no	—
253	C₂×C₄×C₄₁	—	4	1	no	—
254	C₂×C₄×C₄₁	—	4	1	no	—
255	C₂×C₄×C₄₁	—	4	1	no	—
256	C₂×C₄×C₄₁	—	4	1	no	—
257	C₂×C₄×C₄₁	—	4	1	no	—
258	C₂×C₄×C₄₁	—	4	1	no	—
259	C₂×C₄×C₄₁	—	8	4	no	—
260	C₂×C₄×C₄₁	—	8	4	no	—
261	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
262	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
263	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
264	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
265	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
266	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
267	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
268	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
269	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
270	C₈×C₄₁	—	2	2	no	—
271	C₈×C₄₁	—	2	2	no	—
272	C₈×C₄₁	—	4	1	no	—
273	C₈×C₄₁	—	4	1	no	—
274	#22778	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
275	C₂²×C₄₁	—	2	4	no	—
276	C₂²×C₄₁	—	4	2	no	—
277	C₂²×C₄₁	—	4	2	no	—
278	C₂²×C₄₁	—	4	2	no	—
279	C₂²×C₄₁	—	4	2	no	—
280	C₂²×C₄₁	—	4	2	no	—
281	C₂²×C₄₁	—	4	2	no	—
282	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
283	C₂²×C₄₁	#546	1	1	yes	characteristic
284	C₂²×C₄₁	#548	1	1	yes	characteristic
285	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
286	C₂²×C₄₁	#575	1	1	yes	characteristic
287	C₂²×C₄₁	#577	1	1	yes	characteristic
288	C₂²×C₄₁	#578	1	1	yes	characteristic
289	C₄×C₄₁	—	2	2	no	—
290	C₄×C₄₁	—	2	2	no	—
291	C₄×C₄₁	—	2	2	no	—
292	C₄×C₄₁	—	4	2	no	—
293	C₄×C₄₁	—	4	2	no	—
294	C₄×C₄₁	—	4	2	no	—
295	C₄×C₄₁	—	4	1	no	—
296	C₄×C₄₁	—	4	1	no	—
297	C₄×C₄₁	—	4	1	no	—
298	C₄×C₄₁	—	8	1	no	—
299	C₄×C₄₁	—	8	1	no	—
300	C₄×C₄₁	—	8	1	no	—
301	C₄×C₄₁	—	8	1	no	—
302	C₄×C₄₁	C₂×#117	1	1	yes	characteristic
303	C₄×C₄₁	#520	1	1	yes	characteristic
304	C₄×C₄₁	#552	1	1	yes	characteristic
305	C₄×C₄₁	#553	1	1	yes	characteristic
306	C₂×#440	C₂×C₄₁	1	1	yes	characteristic
307	#2356	C₂×C₄₁	1	1	yes	characteristic, complemented
308	D₄×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
309	#1935	C₂×C₄₁	1	1	yes	characteristic, complemented
310	#1933	C₂×C₄₁	1	1	yes	characteristic, complemented
311	#1921	C₂×C₄₁	1	1	yes	characteristic, complemented
312	#1918	C₂×C₄₁	1	1	yes	characteristic, complemented
313	#1877	C₂×C₄₁	1	1	yes	characteristic
314	#1877	C₂×C₄₁	1	1	yes	characteristic
315	#1836	C₂×C₄₁	1	1	yes	characteristic, complemented
316	#1804	C₂×C₄₁	1	1	yes	characteristic
317	#1803	C₂×C₄₁	1	1	yes	characteristic, complemented
318	#1771	C₂×C₄₁	1	1	yes	characteristic, complemented
319	#1762	C₂×C₄₁	1	1	yes	characteristic
320	C₄×#102	C₂×C₄₁	1	1	yes	characteristic
321	C₂×C₄₁	—	4	4	no	—
322	C₂×C₄₁	#2352	1	1	yes	characteristic
323	C₂×C₄₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
324	C₂×C₄₁	#2947	1	1	yes	characteristic
325	C₂×C₄₁	#3310	1	1	yes	characteristic
326	C₂×C₄₁	#3315	1	1	yes	characteristic
327	C₂×C₄₁	#3367	1	1	yes	characteristic
328	C₂×C₄₁	#3376	1	1	yes	characteristic
329	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
330	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
331	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
332	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
333	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
334	C₂²×C₄⋊C₄	—	2	1	no	—
335	#440	C₄×C₄₁	1	8	yes	complemented
336	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
337	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
338	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
339	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
340	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
341	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
342	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
343	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
344	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
345	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
346	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
347	#391	C₂²×C₄₁	1	1	yes	characteristic
348	#391	C₂²×C₄₁	1	1	yes	characteristic
349	#389	C₂²×C₄₁	1	1	yes	characteristic
350	#389	C₂²×C₄₁	1	1	yes	characteristic
351	#387	—	2	1	no	—
352	#387	—	2	1	no	—
353	#385	—	2	1	no	—
354	#385	—	2	1	no	—
355	#384	C₂²×C₄₁	1	1	yes	characteristic
356	#383	—	2	1	no	—
357	#382	C₂²×C₄₁	1	1	yes	characteristic
358	#380	—	2	1	no	—
359	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
360	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
361	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
362	#376	—	2	1	no	—
363	#340	C₂²×C₄₁	1	1	yes	characteristic
364	#340	C₂²×C₄₁	1	1	yes	characteristic
365	#340	C₂²×C₄₁	1	1	yes	characteristic
366	#340	C₂²×C₄₁	1	1	yes	characteristic
367	#340	C₂²×C₄₁	1	1	yes	characteristic
368	#340	C₂²×C₄₁	1	1	yes	characteristic
369	#340	C₂²×C₄₁	1	1	yes	characteristic
370	#340	C₂²×C₄₁	1	1	yes	characteristic
371	#336	C₂²×C₄₁	1	1	yes	characteristic
372	#336	C₂²×C₄₁	1	1	yes	characteristic
373	C₄₁	#22778	1	1	yes	characteristic, direct product factor, Sylow
374	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
375	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
376	C₂³×C₄	—	2	1	no	—
377	C₂³×C₄	—	2	1	no	—
378	C₂³×C₄	—	2	1	no	—
379	C₂×SD₁₆	—	2	2	no	—
380	C₂×SD₁₆	—	2	2	no	—
381	C₂×SD₁₆	C₂×C₄×C₄₁	1	4	yes	—
382	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
383	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
384	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
385	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
386	C₄×Q₈	—	2	2	no	—
387	C₄×Q₈	C₂×C₄×C₄₁	1	4	yes	complemented
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	—	4	1	no	—
392	C₄×D₄	—	4	1	no	—
393	C₄×D₄	C₂×C₄×C₄₁	1	4	yes	—
394	C₂×C₄⋊C₄	—	2	1	no	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	—	2	1	no	—
398	C₂×C₄⋊C₄	—	2	1	no	—
399	C₂×C₄⋊C₄	—	4	4	no	—
400	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
401	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
403	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
404	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
408	C₂×C₂²⋊C₄	—	2	1	no	—
409	C₂×C₂²⋊C₄	—	2	1	no	—
410	C₂×C₂²⋊C₄	—	2	1	no	—
411	C₂×C₄²	—	2	1	no	—
412	C₂×C₄²	—	2	1	no	—
413	C₂×C₄²	—	2	1	no	—
414	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
415	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
417	#107	C₂×C₄×C₄₁	1	4	yes	complemented
418	C₄⋊C₈	—	2	2	no	—
419	#103	—	4	1	no	—
420	#103	—	4	1	no	—
421	#103	C₂×C₄×C₄₁	1	4	yes	complemented
422	#102	—	2	2	no	—
423	#102	—	2	2	no	—
424	#102	C₂×C₄×C₄₁	1	4	yes	—
425	#97	—	4	1	no	—
426	#97	C₂×C₄×C₄₁	1	4	yes	complemented
427	C₄×C₈	—	4	1	no	—
428	#95	—	2	1	no	—
429	#95	—	2	1	no	—
430	#95	—	2	1	no	—
431	#95	—	2	1	no	—
432	#95	—	2	1	no	—
433	#95	—	2	1	no	—
434	#95	—	2	1	no	—
435	#95	—	2	1	no	—
436	#95	—	2	1	no	—
437	#95	—	2	1	no	—
438	#95	D₄×C₄₁	1	1	yes	characteristic
439	#95	D₄×C₄₁	1	1	yes	characteristic
440	C₂⁴	—	2	1	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
451	C₂×D₄	#41×C₄₁	1	2	yes	—
452	C₂×D₄	#41×C₄₁	1	2	yes	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	4	2	no	—
462	C₂²×C₄	—	4	2	no	—
463	C₂²×C₄	—	4	2	no	—
464	C₂²×C₄	—	4	2	no	—
465	C₂²×C₄	—	4	2	no	—
466	C₂²×C₄	—	4	2	no	—
467	C₂²×C₄	—	8	1	no	—
468	C₂²×C₄	—	8	1	no	—
469	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
475	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
476	SD₁₆	—	2	8	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
488	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	4	1	no	—
502	C₄⋊C₄	—	4	2	no	—
503	C₄⋊C₄	—	4	2	no	—
504	C₄⋊C₄	—	4	2	no	—
505	C₄⋊C₄	—	8	1	no	—
506	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
507	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	—	4	2	no	—
510	C₂²⋊C₄	—	4	2	no	—
511	C₄²	—	2	4	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	—	2	1	no	—
516	C₄²	—	4	2	no	—
517	C₄²	—	4	2	no	—
518	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
519	C₄²	#41×C₄₁	1	2	yes	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	2	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	4	1	no	—
524	C₂³	—	4	1	no	—
525	C₂³	—	4	1	no	—
526	C₂³	—	4	1	no	—
527	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
528	Q₈	—	2	2	no	—
529	Q₈	—	2	2	no	—
530	Q₈	—	4	1	no	—
531	Q₈	—	4	1	no	—
532	D₄	—	2	2	no	—
533	D₄	—	2	2	no	—
534	D₄	—	2	2	no	—
535	D₄	#117×C₄₁	1	4	yes	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	4	8	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	8	4	no	—
574	C₂×C₄	—	8	4	no	—
575	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
576	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
577	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
578	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
579	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
580	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
581	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
582	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
583	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
584	C₈	—	2	2	no	—
585	C₈	—	2	2	no	—
586	C₈	—	4	1	no	—
587	C₈	—	4	1	no	—
588	C₂²	—	2	4	no	—
589	C₂²	—	4	2	no	—
590	C₂²	—	4	2	no	—
591	C₂²	—	4	2	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	4	2	no	—
595	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
596	C₂²	C₄₁×#546	1	1	yes	characteristic
597	C₂²	C₄₁×#548	1	1	yes	characteristic
598	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
599	C₂²	C₄₁×#575	1	1	yes	characteristic
600	C₂²	C₄₁×#577	1	1	yes	characteristic
601	C₂²	C₄₁×#578	1	1	yes	characteristic
602	C₄	—	2	2	no	—
603	C₄	—	2	2	no	—
604	C₄	—	2	2	no	—
605	C₄	—	4	1	no	—
606	C₄	—	4	1	no	—
607	C₄	—	4	1	no	—
608	C₄	—	4	2	no	—
609	C₄	—	4	2	no	—
610	C₄	—	4	2	no	—
611	C₄	—	8	1	no	—
612	C₄	—	8	1	no	—
613	C₄	—	8	1	no	—
614	C₄	—	8	1	no	—
615	C₄	C₂×#117×C₄₁	1	1	yes	characteristic
616	C₄	C₄₁×#520	1	1	yes	characteristic
617	C₄	C₄₁×#552	1	1	yes	characteristic
618	C₄	C₄₁×#553	1	1	yes	characteristic
619	C₂	—	4	4	no	—
620	C₂	C₄₁×#2352	1	1	yes	characteristic
621	C₂	C₄×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
622	C₂	C₄₁×#2947	1	1	yes	characteristic
623	C₂	C₄₁×#3310	1	1	yes	characteristic
624	C₂	C₄₁×#3315	1	1	yes	characteristic
625	C₂	C₄₁×#3367	1	1	yes	characteristic
626	C₂	C₄₁×#3376	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×D₄×C₄₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄×C₄⋊C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2356	C₂	semidirect
4. C₄₁×#1935	C₂	semidirect
5. C₄₁×#1933	C₂	semidirect
6. C₄₁×#1921	C₂	semidirect
7. C₄₁×#1918	C₂	semidirect
10. C₄₁×#1836	C₂	semidirect
12. C₄₁×#1803	C₂	semidirect
13. C₄₁×#1771	C₂	semidirect
22. C₄₁×#440	C₄	semidirect
73. C₄×Q₈×C₄₁	C₂×C₄	semidirect
103. #107×C₄₁	C₂×C₄	semidirect
107. #103×C₄₁	C₂×C₄	semidirect
112. #97×C₄₁	C₂×C₄	semidirect
274. #22778	C₄₁	direct (N × H)
307. #2356	C₂×C₄₁	semidirect
309. #1935	C₂×C₄₁	semidirect
310. #1933	C₂×C₄₁	semidirect
311. #1921	C₂×C₄₁	semidirect
312. #1918	C₂×C₄₁	semidirect
315. #1836	C₂×C₄₁	semidirect
317. #1803	C₂×C₄₁	semidirect
318. #1771	C₂×C₄₁	semidirect
335. #440	C₄×C₄₁	semidirect
373. C₄₁	#22778	direct (N × H)
387. C₄×Q₈	C₂×C₄×C₄₁	semidirect
417. #107	C₂×C₄×C₄₁	semidirect
421. #103	C₂×C₄×C₄₁	semidirect
426. #97	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×#440	2
C₄₁×#2356	2
D₄×C₄⋊C₄×C₄₁	2
C₄₁×#1935	2
C₄₁×#1933	2
C₄₁×#1921	2
C₄₁×#1918	2
C₄₁×#1877	2
C₄₁×#1836	2
C₄₁×#1804	2
C₄₁×#1803	2
C₄₁×#1771	2
C₄₁×#1762	2
C₄×#102×C₄₁	2
#22778	41

Subgroups — C41×#22778 #23798979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22778 #23798979