Subgroups — C₄₁×#32998 #23809199

Order 10496.

1284 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3428	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3094	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3006	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2500	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1883	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1883	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1860	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1859	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1826	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1822	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1789	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
14	C₂×#121×C₄₁	C₂²	1	1	yes	characteristic
15	C₄₁×#520	—	2	1	no	—
16	C₄₁×#520	—	2	1	no	—
17	C₄₁×#520	C₂²	1	2	yes	—
18	C₄₁×#520	C₂²	1	2	yes	—
19	C₄₁×#519	—	2	1	no	—
20	C₄₁×#518	—	2	1	no	—
21	C₄₁×#518	—	2	2	no	—
22	C₄×#41×C₄₁	—	2	2	no	—
23	C₄×#41×C₄₁	—	2	2	no	—
24	C₂×#117×C₄₁	—	2	2	no	—
25	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#482	—	2	2	no	—
28	C₄₁×#481	—	2	1	no	—
29	C₄₁×#480	—	2	1	no	—
30	C₄₁×#449	—	2	2	no	—
31	C₄₁×#448	—	2	1	no	—
32	C₄₁×#447	—	2	1	no	—
33	C₄₁×#428	C₂²	1	2	yes	complemented
34	C₂×#104×C₄₁	C₂²	1	2	yes	—
35	C₂×#104×C₄₁	C₂²	1	2	yes	—
36	C₄₁×#418	C₂²	1	2	yes	complemented
37	C₄₁×#417	C₂²	1	2	yes	—
38	C₄₁×#407	C₂²	1	1	yes	characteristic
39	C₄₁×#407	C₂²	1	1	yes	characteristic
40	C₄₁×#394	—	2	2	no	—
41	C₄₁×#390	C₂²	1	1	yes	characteristic, complemented
42	C₄₁×#390	C₂²	1	1	yes	characteristic, complemented
43	C₄₁×#388	—	2	2	no	—
44	C₄₁×#384	C₂²	1	1	yes	characteristic, complemented
45	C₄₁×#382	C₂²	1	1	yes	characteristic, complemented
46	C₄×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
47	C₄₁×#343	C₂²	1	2	yes	complemented
48	C₄₁×#343	C₂²	1	2	yes	complemented
49	C₄₁×#339	C₂²	1	2	yes	complemented
50	C₄₁×#339	C₂²	1	2	yes	complemented
51	C₂×#41×C₄₁	—	2	4	no	—
52	C₂×#41×C₄₁	—	2	4	no	—
53	C₂×#41×C₄₁	—	2	1	no	—
54	C₂×#41×C₄₁	—	2	1	no	—
55	C₂×#41×C₄₁	D₄	1	2	yes	—
56	C₂×#41×C₄₁	C₂³	1	2	yes	—
57	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
58	C₂²×D₄×C₄₁	—	2	1	no	—
59	C₂²×D₄×C₄₁	—	2	1	no	—
60	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
61	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₄₁	—	2	1	no	—
63	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
64	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
65	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
66	C₂×SD₁₆×C₄₁	—	2	2	no	—
67	C₂×D₈×C₄₁	—	2	2	no	—
68	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
69	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
70	#121×C₄₁	—	2	1	no	—
71	#121×C₄₁	—	2	1	no	—
72	#121×C₄₁	—	2	1	no	—
73	#121×C₄₁	—	2	1	no	—
74	C₄×Q₈×C₄₁	—	2	2	no	—
75	C₄×Q₈×C₄₁	—	2	2	no	—
76	C₄×Q₈×C₄₁	—	2	2	no	—
77	C₄×Q₈×C₄₁	—	2	2	no	—
78	C₄×D₄×C₄₁	—	2	4	no	—
79	C₄×D₄×C₄₁	—	2	4	no	—
80	C₄×D₄×C₄₁	—	2	2	no	—
81	C₄×D₄×C₄₁	—	2	2	no	—
82	C₄×D₄×C₄₁	—	2	2	no	—
83	C₄×D₄×C₄₁	—	2	2	no	—
84	C₄×D₄×C₄₁	—	2	4	no	—
85	C₄×D₄×C₄₁	—	2	4	no	—
86	#117×C₄₁	—	2	4	no	—
87	#117×C₄₁	—	2	4	no	—
88	#117×C₄₁	—	2	1	no	—
89	#117×C₄₁	—	2	1	no	—
90	#117×C₄₁	—	2	1	no	—
91	#117×C₄₁	—	2	1	no	—
92	#117×C₄₁	—	2	1	no	—
93	#117×C₄₁	—	2	1	no	—
94	#117×C₄₁	—	2	2	no	—
95	#117×C₄₁	—	2	2	no	—
96	#117×C₄₁	D₄	1	2	yes	—
97	#117×C₄₁	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
99	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
100	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
101	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
106	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
107	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
108	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
110	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
113	C₂×C₄²×C₄₁	—	2	4	no	—
114	C₂×C₄²×C₄₁	—	2	1	no	—
115	C₂×C₄²×C₄₁	D₄	1	2	yes	—
116	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
118	#107×C₄₁	—	2	2	no	—
119	#106×C₄₁	—	2	2	no	—
120	#104×C₄₁	—	2	4	no	—
121	#104×C₄₁	—	2	4	no	—
122	#103×C₄₁	—	2	2	no	—
123	#102×C₄₁	—	2	2	no	—
124	#102×C₄₁	—	2	2	no	—
125	#102×C₄₁	—	2	2	no	—
126	C₂²⋊C₈×C₄₁	—	2	1	no	—
127	C₂²⋊C₈×C₄₁	—	2	1	no	—
128	C₂²⋊C₈×C₄₁	—	2	1	no	—
129	C₂²⋊C₈×C₄₁	—	2	1	no	—
130	#95×C₄₁	—	2	1	no	—
131	#95×C₄₁	—	2	2	no	—
132	C₂⁴×C₄₁	—	2	1	no	—
133	C₂⁴×C₄₁	—	2	1	no	—
134	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
135	#41×C₄₁	—	2	4	no	—
136	#41×C₄₁	—	2	4	no	—
137	#41×C₄₁	—	2	8	no	—
138	#41×C₄₁	—	2	4	no	—
139	#41×C₄₁	—	2	4	no	—
140	#41×C₄₁	—	2	2	no	—
141	#41×C₄₁	—	2	2	no	—
142	#41×C₄₁	#41	1	4	yes	—
143	#41×C₄₁	#41	1	4	yes	—
144	C₂×Q₈×C₄₁	—	2	4	no	—
145	C₂×Q₈×C₄₁	—	2	1	no	—
146	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
147	C₂×D₄×C₄₁	—	2	4	no	—
148	C₂×D₄×C₄₁	—	2	4	no	—
149	C₂×D₄×C₄₁	—	2	2	no	—
150	C₂×D₄×C₄₁	—	2	2	no	—
151	C₂×D₄×C₄₁	—	2	4	no	—
152	C₂×D₄×C₄₁	—	2	1	no	—
153	C₂×D₄×C₄₁	—	2	1	no	—
154	C₂×D₄×C₄₁	—	2	1	no	—
155	C₂×D₄×C₄₁	—	2	2	no	—
156	C₂×D₄×C₄₁	—	2	2	no	—
157	C₂×D₄×C₄₁	—	4	1	no	—
158	C₂×D₄×C₄₁	—	4	1	no	—
159	C₂×D₄×C₄₁	—	8	2	no	—
160	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₄₁	—	2	8	no	—
162	C₂²×C₄×C₄₁	—	2	1	no	—
163	C₂²×C₄×C₄₁	—	2	2	no	—
164	C₂²×C₄×C₄₁	—	2	2	no	—
165	C₂²×C₄×C₄₁	—	2	2	no	—
166	C₂²×C₄×C₄₁	—	2	2	no	—
167	C₂²×C₄×C₄₁	—	2	1	no	—
168	C₂²×C₄×C₄₁	—	2	4	no	—
169	C₂²×C₄×C₄₁	—	2	4	no	—
170	C₂²×C₄×C₄₁	—	2	1	no	—
171	C₂²×C₄×C₄₁	—	2	1	no	—
172	C₂²×C₄×C₄₁	—	2	1	no	—
173	C₂²×C₄×C₄₁	—	2	1	no	—
174	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
180	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
182	SD₁₆×C₄₁	—	4	4	no	—
183	D₈×C₄₁	—	4	4	no	—
184	M₁₆×C₄₁	—	2	2	no	—
185	M₁₆×C₄₁	—	2	2	no	—
186	M₁₆×C₄₁	—	2	2	no	—
187	M₁₆×C₄₁	—	2	2	no	—
188	C₂×C₈×C₄₁	—	2	2	no	—
189	C₂×C₈×C₄₁	—	2	2	no	—
190	C₄⋊C₄×C₄₁	—	2	4	no	—
191	C₄⋊C₄×C₄₁	—	2	4	no	—
192	C₄⋊C₄×C₄₁	—	2	1	no	—
193	C₄⋊C₄×C₄₁	—	2	1	no	—
194	C₄⋊C₄×C₄₁	—	2	1	no	—
195	C₄⋊C₄×C₄₁	—	2	1	no	—
196	C₄⋊C₄×C₄₁	—	2	1	no	—
197	C₄⋊C₄×C₄₁	—	2	1	no	—
198	C₄⋊C₄×C₄₁	—	2	1	no	—
199	C₄⋊C₄×C₄₁	—	2	1	no	—
200	C₄⋊C₄×C₄₁	—	8	2	no	—
201	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
202	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
203	C₂²⋊C₄×C₄₁	—	2	4	no	—
204	C₂²⋊C₄×C₄₁	—	2	4	no	—
205	C₂²⋊C₄×C₄₁	—	2	2	no	—
206	C₂²⋊C₄×C₄₁	—	2	2	no	—
207	C₂²⋊C₄×C₄₁	—	2	2	no	—
208	C₂²⋊C₄×C₄₁	—	2	2	no	—
209	C₂²⋊C₄×C₄₁	—	4	4	no	—
210	C₂²⋊C₄×C₄₁	—	4	2	no	—
211	C₂²⋊C₄×C₄₁	—	4	1	no	—
212	C₂²⋊C₄×C₄₁	—	4	1	no	—
213	C₂²⋊C₄×C₄₁	—	8	2	no	—
214	C₄²×C₄₁	—	2	4	no	—
215	C₄²×C₄₁	—	2	4	no	—
216	C₄²×C₄₁	—	2	2	no	—
217	C₄²×C₄₁	—	2	2	no	—
218	C₄²×C₄₁	—	2	2	no	—
219	C₄²×C₄₁	—	2	2	no	—
220	C₄²×C₄₁	—	2	2	no	—
221	C₄²×C₄₁	—	2	2	no	—
222	C₂³×C₄₁	—	2	4	no	—
223	C₂³×C₄₁	—	2	2	no	—
224	C₂³×C₄₁	—	2	1	no	—
225	C₂³×C₄₁	—	2	1	no	—
226	C₂³×C₄₁	—	4	1	no	—
227	C₂³×C₄₁	—	4	1	no	—
228	C₂³×C₄₁	—	4	2	no	—
229	C₂³×C₄₁	—	4	2	no	—
230	C₂³×C₄₁	—	4	1	no	—
231	C₂³×C₄₁	—	4	1	no	—
232	C₂³×C₄₁	—	4	1	no	—
233	C₂³×C₄₁	—	8	1	no	—
234	C₂³×C₄₁	—	8	1	no	—
235	C₂³×C₄₁	#120	1	1	yes	characteristic
236	C₂³×C₄₁	#120	1	1	yes	characteristic
237	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
238	Q₈×C₄₁	—	2	4	no	—
239	Q₈×C₄₁	—	2	2	no	—
240	Q₈×C₄₁	#123	1	4	yes	—
241	D₄×C₄₁	—	2	8	no	—
242	D₄×C₄₁	—	2	4	no	—
243	D₄×C₄₁	—	2	4	no	—
244	D₄×C₄₁	—	2	4	no	—
245	D₄×C₄₁	—	2	2	no	—
246	D₄×C₄₁	—	4	2	no	—
247	D₄×C₄₁	—	4	2	no	—
248	D₄×C₄₁	—	8	2	no	—
249	D₄×C₄₁	—	8	2	no	—
250	D₄×C₄₁	#123	1	4	yes	—
251	C₂×C₄×C₄₁	—	2	4	no	—
252	C₂×C₄×C₄₁	—	2	8	no	—
253	C₂×C₄×C₄₁	—	2	2	no	—
254	C₂×C₄×C₄₁	—	2	1	no	—
255	C₂×C₄×C₄₁	—	2	1	no	—
256	C₂×C₄×C₄₁	—	2	2	no	—
257	C₂×C₄×C₄₁	—	2	2	no	—
258	C₂×C₄×C₄₁	—	2	2	no	—
259	C₂×C₄×C₄₁	—	2	4	no	—
260	C₂×C₄×C₄₁	—	2	4	no	—
261	C₂×C₄×C₄₁	—	2	2	no	—
262	C₂×C₄×C₄₁	—	2	2	no	—
263	C₂×C₄×C₄₁	—	2	2	no	—
264	C₂×C₄×C₄₁	—	2	2	no	—
265	C₂×C₄×C₄₁	—	2	2	no	—
266	C₂×C₄×C₄₁	—	2	2	no	—
267	C₂×C₄×C₄₁	—	4	1	no	—
268	C₂×C₄×C₄₁	—	4	1	no	—
269	C₂×C₄×C₄₁	—	4	2	no	—
270	C₂×C₄×C₄₁	—	4	2	no	—
271	C₂×C₄×C₄₁	—	4	1	no	—
272	C₂×C₄×C₄₁	—	4	1	no	—
273	C₂×C₄×C₄₁	—	4	2	no	—
274	C₂×C₄×C₄₁	—	4	1	no	—
275	C₂×C₄×C₄₁	—	4	1	no	—
276	C₂×C₄×C₄₁	—	8	1	no	—
277	C₂×C₄×C₄₁	—	8	1	no	—
278	C₂×C₄×C₄₁	—	8	4	no	—
279	C₂×C₄×C₄₁	—	8	4	no	—
280	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
281	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
282	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
285	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
286	C₈×C₄₁	—	4	2	no	—
287	C₈×C₄₁	—	4	2	no	—
288	#32998	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
289	C₂²×C₄₁	—	2	1	no	—
290	C₂²×C₄₁	—	2	1	no	—
291	C₂²×C₄₁	—	2	1	no	—
292	C₂²×C₄₁	—	2	1	no	—
293	C₂²×C₄₁	—	2	4	no	—
294	C₂²×C₄₁	—	2	1	no	—
295	C₂²×C₄₁	—	2	1	no	—
296	C₂²×C₄₁	—	2	1	no	—
297	C₂²×C₄₁	—	2	1	no	—
298	C₂²×C₄₁	—	4	2	no	—
299	C₂²×C₄₁	—	4	2	no	—
300	C₂²×C₄₁	—	8	1	no	—
301	C₂²×C₄₁	—	8	1	no	—
302	C₂²×C₄₁	—	8	2	no	—
303	C₂²×C₄₁	—	8	2	no	—
304	C₂²×C₄₁	—	8	4	no	—
305	C₂²×C₄₁	—	8	1	no	—
306	C₂²×C₄₁	—	8	1	no	—
307	C₂²×C₄₁	—	8	1	no	—
308	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
309	C₂²×C₄₁	#525	1	1	yes	characteristic
310	C₂²×C₄₁	#525	1	1	yes	characteristic
311	C₄×C₄₁	—	2	1	no	—
312	C₄×C₄₁	—	2	4	no	—
313	C₄×C₄₁	—	2	1	no	—
314	C₄×C₄₁	—	4	2	no	—
315	C₄×C₄₁	—	4	2	no	—
316	C₄×C₄₁	—	4	2	no	—
317	C₄×C₄₁	—	4	2	no	—
318	C₄×C₄₁	—	4	2	no	—
319	C₄×C₄₁	—	8	1	no	—
320	C₄×C₄₁	—	8	1	no	—
321	C₄×C₄₁	C₂×#123	1	2	yes	—
322	C₄×C₄₁	#525	1	2	yes	—
323	#3428	C₂×C₄₁	1	1	yes	characteristic, complemented
324	#3094	C₂×C₄₁	1	1	yes	characteristic, complemented
325	#3006	C₂×C₄₁	1	1	yes	characteristic
326	#2500	C₂×C₄₁	1	1	yes	characteristic, complemented
327	#1883	C₂×C₄₁	1	2	yes	complemented
328	#1883	C₂×C₄₁	1	2	yes	complemented
329	#1860	C₂×C₄₁	1	2	yes	complemented
330	#1859	C₂×C₄₁	1	2	yes	complemented
331	#1826	C₂×C₄₁	1	1	yes	characteristic, complemented
332	#1822	C₂×C₄₁	1	1	yes	characteristic, complemented
333	#1789	C₂×C₄₁	1	1	yes	characteristic, complemented
334	C₂×C₄₁	—	2	1	no	—
335	C₂×C₄₁	—	2	1	no	—
336	C₂×C₄₁	—	4	1	no	—
337	C₂×C₄₁	—	4	1	no	—
338	C₂×C₄₁	—	4	4	no	—
339	C₂×C₄₁	—	16	1	no	—
340	C₂×C₄₁	#2849	1	1	yes	characteristic
341	C₂×C₄₁	#3018	1	1	yes	characteristic
342	C₂×C₄₁	#3021	1	1	yes	characteristic
343	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
344	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
345	C₂×#121	C₂²×C₄₁	1	1	yes	characteristic
346	#520	—	2	1	no	—
347	#520	—	2	1	no	—
348	#520	C₂²×C₄₁	1	2	yes	—
349	#520	C₂²×C₄₁	1	2	yes	—
350	#519	—	2	1	no	—
351	#518	—	2	1	no	—
352	#518	—	2	2	no	—
353	C₄×#41	—	2	2	no	—
354	C₄×#41	—	2	2	no	—
355	C₂×#117	—	2	2	no	—
356	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
357	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
358	#482	—	2	2	no	—
359	#481	—	2	1	no	—
360	#480	—	2	1	no	—
361	#449	—	2	2	no	—
362	#448	—	2	1	no	—
363	#447	—	2	1	no	—
364	#428	C₂²×C₄₁	1	2	yes	complemented
365	C₂×#104	C₂²×C₄₁	1	2	yes	—
366	C₂×#104	C₂²×C₄₁	1	2	yes	—
367	#418	C₂²×C₄₁	1	2	yes	complemented
368	#417	C₂²×C₄₁	1	2	yes	—
369	#407	C₂²×C₄₁	1	1	yes	characteristic
370	#407	C₂²×C₄₁	1	1	yes	characteristic
371	#394	—	2	2	no	—
372	#390	C₂²×C₄₁	1	1	yes	characteristic, complemented
373	#390	C₂²×C₄₁	1	1	yes	characteristic, complemented
374	#388	—	2	2	no	—
375	#384	C₂²×C₄₁	1	1	yes	characteristic, complemented
376	#382	C₂²×C₄₁	1	1	yes	characteristic, complemented
377	C₄×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
378	#343	C₂²×C₄₁	1	2	yes	complemented
379	#343	C₂²×C₄₁	1	2	yes	complemented
380	#339	C₂²×C₄₁	1	2	yes	complemented
381	#339	C₂²×C₄₁	1	2	yes	complemented
382	C₄₁	#32998	1	1	yes	characteristic, direct product factor, Sylow
383	C₂×#41	—	2	4	no	—
384	C₂×#41	—	2	4	no	—
385	C₂×#41	—	2	1	no	—
386	C₂×#41	—	2	1	no	—
387	C₂×#41	D₄×C₄₁	1	2	yes	—
388	C₂×#41	C₂³×C₄₁	1	2	yes	—
389	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
390	C₂²×D₄	—	2	1	no	—
391	C₂²×D₄	—	2	1	no	—
392	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
393	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
394	C₂³×C₄	—	2	1	no	—
395	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
396	AGL(1,ℤ/8)	—	2	4	no	—
397	AGL(1,ℤ/8)	—	2	4	no	—
398	C₂×SD₁₆	—	2	2	no	—
399	C₂×D₈	—	2	2	no	—
400	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
401	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
402	#121	—	2	1	no	—
403	#121	—	2	1	no	—
404	#121	—	2	1	no	—
405	#121	—	2	1	no	—
406	C₄×Q₈	—	2	2	no	—
407	C₄×Q₈	—	2	2	no	—
408	C₄×Q₈	—	2	2	no	—
409	C₄×Q₈	—	2	2	no	—
410	C₄×D₄	—	2	4	no	—
411	C₄×D₄	—	2	4	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	2	no	—
415	C₄×D₄	—	2	2	no	—
416	C₄×D₄	—	2	4	no	—
417	C₄×D₄	—	2	4	no	—
418	#117	—	2	4	no	—
419	#117	—	2	4	no	—
420	#117	—	2	1	no	—
421	#117	—	2	1	no	—
422	#117	—	2	1	no	—
423	#117	—	2	1	no	—
424	#117	—	2	1	no	—
425	#117	—	2	1	no	—
426	#117	—	2	2	no	—
427	#117	—	2	2	no	—
428	#117	D₄×C₄₁	1	2	yes	—
429	#117	C₂³×C₄₁	1	2	yes	—
430	C₂×C₄⋊C₄	—	2	2	no	—
431	C₂×C₄⋊C₄	—	2	2	no	—
432	C₂×C₄⋊C₄	—	2	2	no	—
433	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
434	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
435	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
436	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
437	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
438	C₂×C₂²⋊C₄	—	2	4	no	—
439	C₂×C₂²⋊C₄	—	2	1	no	—
440	C₂×C₂²⋊C₄	—	2	1	no	—
441	C₂×C₂²⋊C₄	—	2	1	no	—
442	C₂×C₂²⋊C₄	—	2	2	no	—
443	C₂×C₂²⋊C₄	—	2	2	no	—
444	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
445	C₂×C₄²	—	2	4	no	—
446	C₂×C₄²	—	2	1	no	—
447	C₂×C₄²	D₄×C₄₁	1	2	yes	—
448	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
449	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
450	#107	—	2	2	no	—
451	#106	—	2	2	no	—
452	#104	—	2	4	no	—
453	#104	—	2	4	no	—
454	#103	—	2	2	no	—
455	#102	—	2	2	no	—
456	#102	—	2	2	no	—
457	#102	—	2	2	no	—
458	C₂²⋊C₈	—	2	1	no	—
459	C₂²⋊C₈	—	2	1	no	—
460	C₂²⋊C₈	—	2	1	no	—
461	C₂²⋊C₈	—	2	1	no	—
462	#95	—	2	1	no	—
463	#95	—	2	2	no	—
464	C₂⁴	—	2	1	no	—
465	C₂⁴	—	2	1	no	—
466	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
467	#41	—	2	4	no	—
468	#41	—	2	4	no	—
469	#41	—	2	8	no	—
470	#41	—	2	4	no	—
471	#41	—	2	4	no	—
472	#41	—	2	2	no	—
473	#41	—	2	2	no	—
474	#41	#41×C₄₁	1	4	yes	—
475	#41	#41×C₄₁	1	4	yes	—
476	C₂×Q₈	—	2	4	no	—
477	C₂×Q₈	—	2	1	no	—
478	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
479	C₂×D₄	—	2	4	no	—
480	C₂×D₄	—	2	4	no	—
481	C₂×D₄	—	2	2	no	—
482	C₂×D₄	—	2	2	no	—
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	2	1	no	—
485	C₂×D₄	—	2	1	no	—
486	C₂×D₄	—	2	1	no	—
487	C₂×D₄	—	2	2	no	—
488	C₂×D₄	—	2	2	no	—
489	C₂×D₄	—	4	1	no	—
490	C₂×D₄	—	4	1	no	—
491	C₂×D₄	—	8	2	no	—
492	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
493	C₂²×C₄	—	2	8	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	2	no	—
496	C₂²×C₄	—	2	2	no	—
497	C₂²×C₄	—	2	2	no	—
498	C₂²×C₄	—	2	2	no	—
499	C₂²×C₄	—	2	1	no	—
500	C₂²×C₄	—	2	4	no	—
501	C₂²×C₄	—	2	4	no	—
502	C₂²×C₄	—	2	1	no	—
503	C₂²×C₄	—	2	1	no	—
504	C₂²×C₄	—	2	1	no	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
507	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
508	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
509	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
510	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
511	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
512	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
513	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
514	SD₁₆	—	4	4	no	—
515	D₈	—	4	4	no	—
516	M₁₆	—	2	2	no	—
517	M₁₆	—	2	2	no	—
518	M₁₆	—	2	2	no	—
519	M₁₆	—	2	2	no	—
520	C₂×C₈	—	2	2	no	—
521	C₂×C₈	—	2	2	no	—
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	4	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	2	1	no	—
530	C₄⋊C₄	—	2	1	no	—
531	C₄⋊C₄	—	2	1	no	—
532	C₄⋊C₄	—	8	2	no	—
533	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
534	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₂²⋊C₄	—	2	2	no	—
538	C₂²⋊C₄	—	2	2	no	—
539	C₂²⋊C₄	—	2	2	no	—
540	C₂²⋊C₄	—	2	2	no	—
541	C₂²⋊C₄	—	4	4	no	—
542	C₂²⋊C₄	—	4	2	no	—
543	C₂²⋊C₄	—	4	1	no	—
544	C₂²⋊C₄	—	4	1	no	—
545	C₂²⋊C₄	—	8	2	no	—
546	C₄²	—	2	4	no	—
547	C₄²	—	2	4	no	—
548	C₄²	—	2	2	no	—
549	C₄²	—	2	2	no	—
550	C₄²	—	2	2	no	—
551	C₄²	—	2	2	no	—
552	C₄²	—	2	2	no	—
553	C₄²	—	2	2	no	—
554	C₂³	—	2	4	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	1	no	—
557	C₂³	—	2	1	no	—
558	C₂³	—	4	1	no	—
559	C₂³	—	4	1	no	—
560	C₂³	—	4	2	no	—
561	C₂³	—	4	2	no	—
562	C₂³	—	4	1	no	—
563	C₂³	—	4	1	no	—
564	C₂³	—	4	1	no	—
565	C₂³	—	8	1	no	—
566	C₂³	—	8	1	no	—
567	C₂³	#120×C₄₁	1	1	yes	characteristic
568	C₂³	#120×C₄₁	1	1	yes	characteristic
569	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
570	Q₈	—	2	4	no	—
571	Q₈	—	2	2	no	—
572	Q₈	#123×C₄₁	1	4	yes	—
573	D₄	—	2	8	no	—
574	D₄	—	2	4	no	—
575	D₄	—	2	4	no	—
576	D₄	—	2	4	no	—
577	D₄	—	2	2	no	—
578	D₄	—	4	2	no	—
579	D₄	—	4	2	no	—
580	D₄	—	8	2	no	—
581	D₄	—	8	2	no	—
582	D₄	#123×C₄₁	1	4	yes	—
583	C₂×C₄	—	2	4	no	—
584	C₂×C₄	—	2	8	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	4	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	2	no	—
602	C₂×C₄	—	4	2	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	2	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	8	1	no	—
609	C₂×C₄	—	8	1	no	—
610	C₂×C₄	—	8	4	no	—
611	C₂×C₄	—	8	4	no	—
612	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
613	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
614	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
615	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
616	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
617	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
618	C₈	—	4	2	no	—
619	C₈	—	4	2	no	—
620	C₂²	—	2	1	no	—
621	C₂²	—	2	1	no	—
622	C₂²	—	2	1	no	—
623	C₂²	—	2	1	no	—
624	C₂²	—	2	4	no	—
625	C₂²	—	2	1	no	—
626	C₂²	—	2	1	no	—
627	C₂²	—	2	1	no	—
628	C₂²	—	2	1	no	—
629	C₂²	—	4	2	no	—
630	C₂²	—	4	2	no	—
631	C₂²	—	8	1	no	—
632	C₂²	—	8	1	no	—
633	C₂²	—	8	2	no	—
634	C₂²	—	8	2	no	—
635	C₂²	—	8	4	no	—
636	C₂²	—	8	1	no	—
637	C₂²	—	8	1	no	—
638	C₂²	—	8	1	no	—
639	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
640	C₂²	C₄₁×#525	1	1	yes	characteristic
641	C₂²	C₄₁×#525	1	1	yes	characteristic
642	C₄	—	2	1	no	—
643	C₄	—	2	4	no	—
644	C₄	—	2	1	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	2	no	—
647	C₄	—	4	2	no	—
648	C₄	—	4	2	no	—
649	C₄	—	4	2	no	—
650	C₄	—	8	1	no	—
651	C₄	—	8	1	no	—
652	C₄	C₂×#123×C₄₁	1	2	yes	—
653	C₄	C₄₁×#525	1	2	yes	—
654	C₂	—	2	1	no	—
655	C₂	—	2	1	no	—
656	C₂	—	4	1	no	—
657	C₂	—	4	1	no	—
658	C₂	—	4	4	no	—
659	C₂	—	16	1	no	—
660	C₂	C₄₁×#2849	1	1	yes	characteristic
661	C₂	C₄₁×#3018	1	1	yes	characteristic
662	C₂	C₄₁×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#3006	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3428	C₂	semidirect
2. C₄₁×#3094	C₂	semidirect
4. C₄₁×#2500	C₂	semidirect
5. C₄₁×#1883	C₂	semidirect
6. C₄₁×#1883	C₂	semidirect
7. C₄₁×#1860	C₂	semidirect
8. C₄₁×#1859	C₂	semidirect
9. C₄₁×#1826	C₂	semidirect
10. C₄₁×#1822	C₂	semidirect
11. C₄₁×#1789	C₂	semidirect
33. C₄₁×#428	C₂²	semidirect
36. C₄₁×#418	C₂²	semidirect
41. C₄₁×#390	C₂²	semidirect
42. C₄₁×#390	C₂²	semidirect
44. C₄₁×#384	C₂²	semidirect
45. C₄₁×#382	C₂²	semidirect
47. C₄₁×#343	C₂²	semidirect
48. C₄₁×#343	C₂²	semidirect
49. C₄₁×#339	C₂²	semidirect
50. C₄₁×#339	C₂²	semidirect
288. #32998	C₄₁	direct (N × H)
323. #3428	C₂×C₄₁	semidirect
324. #3094	C₂×C₄₁	semidirect
326. #2500	C₂×C₄₁	semidirect
327. #1883	C₂×C₄₁	semidirect
328. #1883	C₂×C₄₁	semidirect
329. #1860	C₂×C₄₁	semidirect
330. #1859	C₂×C₄₁	semidirect
331. #1826	C₂×C₄₁	semidirect
332. #1822	C₂×C₄₁	semidirect
333. #1789	C₂×C₄₁	semidirect
364. #428	C₂²×C₄₁	semidirect
367. #418	C₂²×C₄₁	semidirect
372. #390	C₂²×C₄₁	semidirect
373. #390	C₂²×C₄₁	semidirect
375. #384	C₂²×C₄₁	semidirect
376. #382	C₂²×C₄₁	semidirect
378. #343	C₂²×C₄₁	semidirect
379. #343	C₂²×C₄₁	semidirect
380. #339	C₂²×C₄₁	semidirect
381. #339	C₂²×C₄₁	semidirect
382. C₄₁	#32998	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×#3428	2
C₄₁×#3094	2
C₄₁×#3006	2
C₄₁×#2500	2
C₄₁×#1883	2
C₄₁×#1860	2
C₄₁×#1859	2
C₄₁×#1826	2
C₄₁×#1822	2
C₄₁×#1789	2
#32998	41

Subgroups — C41×#32998 #23809199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#32998 #23809199