Subgroups — C₄₁×#24198 #23800399

Order 10496.

1180 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#459	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#440	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2580	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2042	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2012	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1961	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1879	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1862	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1805	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
14	C₄₁×#559	—	2	4	no	—
15	C₄₁×#559	—	2	4	no	—
16	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×#124×C₄₁	—	2	1	no	—
18	C₂×#124×C₄₁	C₂²	1	2	yes	—
19	C₂×#123×C₄₁	—	2	2	no	—
20	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₄₁	—	2	1	no	—
23	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
25	C₄₁×#459	—	2	4	no	—
26	C₄₁×#440	—	2	4	no	—
27	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×#103×C₄₁	C₂²	1	2	yes	—
31	C₂×#102×C₄₁	C₂²	1	2	yes	—
32	C₂×#102×C₄₁	C₂²	1	2	yes	—
33	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
34	C₂×#102×C₄₁	C₂²	1	2	yes	—
35	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
36	C₄₁×#399	—	2	1	no	—
37	C₄₁×#393	—	2	1	no	—
38	C₄₁×#390	C₂²	1	2	yes	—
39	C₄₁×#389	C₂²	1	2	yes	—
40	C₄₁×#385	—	2	2	no	—
41	C₄₁×#382	C₂²	1	1	yes	characteristic
42	C₄₁×#340	C₂²	1	2	yes	—
43	C₄₁×#336	C₂²	1	2	yes	—
44	C₄₁×#336	C₂²	1	2	yes	—
45	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₄₁	—	2	1	no	—
48	C₂²×D₄×C₄₁	—	2	1	no	—
49	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
50	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₄₁	—	2	2	no	—
52	C₂³×C₄×C₄₁	—	2	1	no	—
53	C₂×SD₁₆×C₄₁	—	2	2	no	—
54	C₂×SD₁₆×C₄₁	—	2	2	no	—
55	C₂×SD₁₆×C₄₁	—	2	2	no	—
56	C₂×D₈×C₄₁	—	2	4	no	—
57	C₂×D₈×C₄₁	—	2	4	no	—
58	C₂×D₈×C₄₁	—	4	2	no	—
59	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
62	#127×C₄₁	—	2	2	no	—
63	#127×C₄₁	D₄	1	4	yes	—
64	#124×C₄₁	—	2	8	no	—
65	#124×C₄₁	—	2	4	no	—
66	#124×C₄₁	—	2	4	no	—
67	#123×C₄₁	—	2	16	no	—
68	C₄×Q₈×C₄₁	—	2	2	no	—
69	C₄×Q₈×C₄₁	—	2	2	no	—
70	C₄×Q₈×C₄₁	—	2	2	no	—
71	C₄×Q₈×C₄₁	D₄	1	4	yes	—
72	C₄×D₄×C₄₁	—	2	8	no	—
73	C₄×D₄×C₄₁	—	2	4	no	—
74	C₄×D₄×C₄₁	—	2	4	no	—
75	C₄×D₄×C₄₁	—	2	2	no	—
76	C₄×D₄×C₄₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
79	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
81	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
90	C₂×C₄²×C₄₁	—	2	1	no	—
91	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
92	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
93	#107×C₄₁	—	2	2	no	—
94	C₄⋊C₈×C₄₁	—	2	2	no	—
95	#103×C₄₁	—	2	2	no	—
96	#103×C₄₁	—	4	2	no	—
97	#102×C₄₁	—	2	4	no	—
98	#102×C₄₁	—	2	4	no	—
99	#102×C₄₁	—	2	2	no	—
100	#102×C₄₁	—	4	2	no	—
101	#97×C₄₁	—	2	2	no	—
102	#95×C₄₁	—	2	1	no	—
103	#95×C₄₁	—	2	1	no	—
104	#95×C₄₁	—	2	2	no	—
105	C₂⁴×C₄₁	—	2	1	no	—
106	C₂⁴×C₄₁	—	2	2	no	—
107	C₂×Q₈×C₄₁	—	2	2	no	—
108	C₂×Q₈×C₄₁	—	2	1	no	—
109	C₂×Q₈×C₄₁	—	2	1	no	—
110	C₂×Q₈×C₄₁	—	2	1	no	—
111	C₂×Q₈×C₄₁	—	2	1	no	—
112	C₂×Q₈×C₄₁	—	2	1	no	—
113	C₂×Q₈×C₄₁	—	4	2	no	—
114	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
115	C₂×D₄×C₄₁	—	2	4	no	—
116	C₂×D₄×C₄₁	—	2	4	no	—
117	C₂×D₄×C₄₁	—	2	1	no	—
118	C₂×D₄×C₄₁	—	2	1	no	—
119	C₂×D₄×C₄₁	—	2	2	no	—
120	C₂×D₄×C₄₁	—	2	2	no	—
121	C₂×D₄×C₄₁	—	2	2	no	—
122	C₂×D₄×C₄₁	—	2	1	no	—
123	C₂×D₄×C₄₁	—	2	2	no	—
124	C₂×D₄×C₄₁	—	2	1	no	—
125	C₂×D₄×C₄₁	—	2	2	no	—
126	C₂×D₄×C₄₁	—	2	1	no	—
127	C₂×D₄×C₄₁	—	4	4	no	—
128	C₂×D₄×C₄₁	—	8	2	no	—
129	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
130	C₂×D₄×C₄₁	#41	1	2	yes	—
131	C₂²×C₄×C₄₁	—	2	1	no	—
132	C₂²×C₄×C₄₁	—	2	1	no	—
133	C₂²×C₄×C₄₁	—	2	8	no	—
134	C₂²×C₄×C₄₁	—	2	2	no	—
135	C₂²×C₄×C₄₁	—	2	1	no	—
136	C₂²×C₄×C₄₁	—	4	2	no	—
137	C₂²×C₄×C₄₁	—	8	2	no	—
138	C₂²×C₄×C₄₁	—	8	2	no	—
139	C₂²×C₄×C₄₁	—	8	2	no	—
140	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
145	SD₁₆×C₄₁	—	2	8	no	—
146	D₈×C₄₁	—	4	8	no	—
147	C₂×C₈×C₄₁	—	2	1	no	—
148	C₂×C₈×C₄₁	—	2	1	no	—
149	C₂×C₈×C₄₁	—	2	1	no	—
150	C₂×C₈×C₄₁	—	2	2	no	—
151	C₂×C₈×C₄₁	—	2	2	no	—
152	C₂×C₈×C₄₁	—	2	1	no	—
153	C₂×C₈×C₄₁	—	2	2	no	—
154	C₂×C₈×C₄₁	—	2	1	no	—
155	C₂×C₈×C₄₁	#41	1	2	yes	—
156	C₄⋊C₄×C₄₁	—	2	4	no	—
157	C₄⋊C₄×C₄₁	—	2	2	no	—
158	C₄⋊C₄×C₄₁	—	2	4	no	—
159	C₄⋊C₄×C₄₁	—	2	1	no	—
160	C₄⋊C₄×C₄₁	—	2	1	no	—
161	C₄⋊C₄×C₄₁	—	2	1	no	—
162	C₄⋊C₄×C₄₁	—	2	2	no	—
163	C₄⋊C₄×C₄₁	—	2	2	no	—
164	C₄⋊C₄×C₄₁	—	8	2	no	—
165	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
166	C₂²⋊C₄×C₄₁	—	2	8	no	—
167	C₂²⋊C₄×C₄₁	—	2	8	no	—
168	C₂²⋊C₄×C₄₁	—	2	8	no	—
169	C₂²⋊C₄×C₄₁	—	4	4	no	—
170	C₂²⋊C₄×C₄₁	—	4	2	no	—
171	C₂²⋊C₄×C₄₁	—	8	2	no	—
172	C₄²×C₄₁	—	2	4	no	—
173	C₄²×C₄₁	—	2	2	no	—
174	C₄²×C₄₁	—	2	2	no	—
175	C₄²×C₄₁	C₂×D₄	1	2	yes	—
176	C₄²×C₄₁	C₂×D₄	1	2	yes	—
177	C₂³×C₄₁	—	2	4	no	—
178	C₂³×C₄₁	—	2	2	no	—
179	C₂³×C₄₁	—	4	1	no	—
180	C₂³×C₄₁	—	4	2	no	—
181	C₂³×C₄₁	—	4	2	no	—
182	C₂³×C₄₁	—	4	1	no	—
183	C₂³×C₄₁	—	4	2	no	—
184	C₂³×C₄₁	—	4	2	no	—
185	C₂³×C₄₁	—	4	2	no	—
186	C₂³×C₄₁	—	4	2	no	—
187	C₂³×C₄₁	—	4	2	no	—
188	C₂³×C₄₁	—	4	2	no	—
189	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
190	Q₈×C₄₁	—	2	2	no	—
191	Q₈×C₄₁	—	2	2	no	—
192	Q₈×C₄₁	—	4	4	no	—
193	Q₈×C₄₁	—	4	1	no	—
194	Q₈×C₄₁	—	4	1	no	—
195	D₄×C₄₁	—	2	2	no	—
196	D₄×C₄₁	—	2	2	no	—
197	D₄×C₄₁	—	2	4	no	—
198	D₄×C₄₁	—	2	4	no	—
199	D₄×C₄₁	—	4	8	no	—
200	D₄×C₄₁	—	4	2	no	—
201	D₄×C₄₁	—	4	1	no	—
202	D₄×C₄₁	—	4	2	no	—
203	D₄×C₄₁	—	4	1	no	—
204	D₄×C₄₁	—	8	4	no	—
205	C₂×C₄×C₄₁	—	2	2	no	—
206	C₂×C₄×C₄₁	—	2	2	no	—
207	C₂×C₄×C₄₁	—	2	2	no	—
208	C₂×C₄×C₄₁	—	2	2	no	—
209	C₂×C₄×C₄₁	—	2	2	no	—
210	C₂×C₄×C₄₁	—	2	2	no	—
211	C₂×C₄×C₄₁	—	2	2	no	—
212	C₂×C₄×C₄₁	—	2	2	no	—
213	C₂×C₄×C₄₁	—	2	2	no	—
214	C₂×C₄×C₄₁	—	2	2	no	—
215	C₂×C₄×C₄₁	—	4	1	no	—
216	C₂×C₄×C₄₁	—	4	1	no	—
217	C₂×C₄×C₄₁	—	4	2	no	—
218	C₂×C₄×C₄₁	—	4	1	no	—
219	C₂×C₄×C₄₁	—	4	2	no	—
220	C₂×C₄×C₄₁	—	4	2	no	—
221	C₂×C₄×C₄₁	—	4	2	no	—
222	C₂×C₄×C₄₁	—	4	2	no	—
223	C₂×C₄×C₄₁	—	8	4	no	—
224	C₂×C₄×C₄₁	—	8	8	no	—
225	C₂×C₄×C₄₁	#120	1	2	yes	—
226	C₂×C₄×C₄₁	#120	1	2	yes	—
227	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
230	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
231	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
232	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
233	C₈×C₄₁	—	2	2	no	—
234	C₈×C₄₁	—	4	1	no	—
235	C₈×C₄₁	—	4	2	no	—
236	#24198	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
237	C₂²×C₄₁	—	4	4	no	—
238	C₂²×C₄₁	—	4	2	no	—
239	C₂²×C₄₁	—	4	4	no	—
240	C₂²×C₄₁	—	4	4	no	—
241	C₂²×C₄₁	—	4	4	no	—
242	C₂²×C₄₁	—	8	2	no	—
243	C₂²×C₄₁	—	8	2	no	—
244	C₂²×C₄₁	—	8	2	no	—
245	C₂²×C₄₁	—	8	2	no	—
246	C₂²×C₄₁	—	8	2	no	—
247	C₂²×C₄₁	—	8	1	no	—
248	C₂²×C₄₁	—	8	1	no	—
249	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
250	C₂²×C₄₁	#548	1	1	yes	characteristic
251	C₂²×C₄₁	#550	1	1	yes	characteristic
252	C₂²×C₄₁	#575	1	2	yes	—
253	C₂²×C₄₁	#576	1	2	yes	—
254	C₄×C₄₁	—	2	2	no	—
255	C₄×C₄₁	—	2	2	no	—
256	C₄×C₄₁	—	4	2	no	—
257	C₄×C₄₁	—	4	2	no	—
258	C₄×C₄₁	—	4	2	no	—
259	C₄×C₄₁	—	4	2	no	—
260	C₄×C₄₁	—	8	2	no	—
261	C₄×C₄₁	—	8	1	no	—
262	C₄×C₄₁	#525	1	1	yes	characteristic
263	C₄×C₄₁	#537	1	1	yes	characteristic
264	C₄×C₄₁	#546	1	1	yes	characteristic
265	C₄×C₄₁	#547	1	1	yes	characteristic
266	C₂×#559	C₂×C₄₁	1	1	yes	characteristic, complemented
267	C₂×#459	C₂×C₄₁	1	1	yes	characteristic
268	C₂×#440	C₂×C₄₁	1	1	yes	characteristic, complemented
269	#2580	C₂×C₄₁	1	1	yes	characteristic, complemented
270	#2042	C₂×C₄₁	1	2	yes	complemented
271	#2012	C₂×C₄₁	1	2	yes	complemented
272	#1985	C₂×C₄₁	1	1	yes	characteristic, complemented
273	#1961	C₂×C₄₁	1	1	yes	characteristic, complemented
274	#1879	C₂×C₄₁	1	2	yes	complemented
275	#1862	C₂×C₄₁	1	2	yes	complemented
276	#1805	C₂×C₄₁	1	1	yes	characteristic, complemented
277	C₂×C₄₁	—	8	2	no	—
278	C₂×C₄₁	—	8	4	no	—
279	C₂×C₄₁	#2407	1	1	yes	characteristic
280	C₂×C₄₁	#3005	1	1	yes	characteristic
281	C₂×C₄₁	#3008	1	1	yes	characteristic
282	C₂×C₄₁	#3370	1	2	yes	—
283	C₂×C₄₁	#3403	1	2	yes	—
284	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
285	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
286	#559	—	2	4	no	—
287	#559	—	2	4	no	—
288	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
289	C₂×#124	—	2	1	no	—
290	C₂×#124	C₂²×C₄₁	1	2	yes	—
291	C₂×#123	—	2	2	no	—
292	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
293	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
294	C₂×C₄×D₄	—	2	1	no	—
295	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
296	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
297	#459	—	2	4	no	—
298	#440	—	2	4	no	—
299	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
300	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
301	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
302	C₂×#103	C₂²×C₄₁	1	2	yes	—
303	C₂×#102	C₂²×C₄₁	1	2	yes	—
304	C₂×#102	C₂²×C₄₁	1	2	yes	—
305	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
306	C₂×#102	C₂²×C₄₁	1	2	yes	—
307	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
308	#399	—	2	1	no	—
309	#393	—	2	1	no	—
310	#390	C₂²×C₄₁	1	2	yes	—
311	#389	C₂²×C₄₁	1	2	yes	—
312	#385	—	2	2	no	—
313	#382	C₂²×C₄₁	1	1	yes	characteristic
314	#340	C₂²×C₄₁	1	2	yes	—
315	#336	C₂²×C₄₁	1	2	yes	—
316	#336	C₂²×C₄₁	1	2	yes	—
317	C₄₁	#24198	1	1	yes	characteristic, direct product factor, Sylow
318	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
319	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
320	C₂²×D₄	—	2	1	no	—
321	C₂²×D₄	—	2	1	no	—
322	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
323	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
324	C₂³×C₄	—	2	2	no	—
325	C₂³×C₄	—	2	1	no	—
326	C₂×SD₁₆	—	2	2	no	—
327	C₂×SD₁₆	—	2	2	no	—
328	C₂×SD₁₆	—	2	2	no	—
329	C₂×D₈	—	2	4	no	—
330	C₂×D₈	—	2	4	no	—
331	C₂×D₈	—	4	2	no	—
332	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
333	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
334	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
335	#127	—	2	2	no	—
336	#127	D₄×C₄₁	1	4	yes	—
337	#124	—	2	8	no	—
338	#124	—	2	4	no	—
339	#124	—	2	4	no	—
340	#123	—	2	16	no	—
341	C₄×Q₈	—	2	2	no	—
342	C₄×Q₈	—	2	2	no	—
343	C₄×Q₈	—	2	2	no	—
344	C₄×Q₈	D₄×C₄₁	1	4	yes	—
345	C₄×D₄	—	2	8	no	—
346	C₄×D₄	—	2	4	no	—
347	C₄×D₄	—	2	4	no	—
348	C₄×D₄	—	2	2	no	—
349	C₄×D₄	—	2	2	no	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
353	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
354	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
356	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₂²⋊C₄	—	2	2	no	—
359	C₂×C₂²⋊C₄	—	2	2	no	—
360	C₂×C₂²⋊C₄	—	2	1	no	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₄²	—	2	1	no	—
364	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
365	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
366	#107	—	2	2	no	—
367	C₄⋊C₈	—	2	2	no	—
368	#103	—	2	2	no	—
369	#103	—	4	2	no	—
370	#102	—	2	4	no	—
371	#102	—	2	4	no	—
372	#102	—	2	2	no	—
373	#102	—	4	2	no	—
374	#97	—	2	2	no	—
375	#95	—	2	1	no	—
376	#95	—	2	1	no	—
377	#95	—	2	2	no	—
378	C₂⁴	—	2	1	no	—
379	C₂⁴	—	2	2	no	—
380	C₂×Q₈	—	2	2	no	—
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	—	2	1	no	—
383	C₂×Q₈	—	2	1	no	—
384	C₂×Q₈	—	2	1	no	—
385	C₂×Q₈	—	2	1	no	—
386	C₂×Q₈	—	4	2	no	—
387	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	1	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	4	4	no	—
401	C₂×D₄	—	8	2	no	—
402	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
403	C₂×D₄	#41×C₄₁	1	2	yes	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	2	8	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	—	4	2	no	—
410	C₂²×C₄	—	8	2	no	—
411	C₂²×C₄	—	8	2	no	—
412	C₂²×C₄	—	8	2	no	—
413	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
414	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
415	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
416	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
417	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
418	SD₁₆	—	2	8	no	—
419	D₈	—	4	8	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	1	no	—
428	C₂×C₈	#41×C₄₁	1	2	yes	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	2	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	8	2	no	—
438	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
439	C₂²⋊C₄	—	2	8	no	—
440	C₂²⋊C₄	—	2	8	no	—
441	C₂²⋊C₄	—	2	8	no	—
442	C₂²⋊C₄	—	4	4	no	—
443	C₂²⋊C₄	—	4	2	no	—
444	C₂²⋊C₄	—	8	2	no	—
445	C₄²	—	2	4	no	—
446	C₄²	—	2	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	C₂×D₄×C₄₁	1	2	yes	—
449	C₄²	C₂×D₄×C₄₁	1	2	yes	—
450	C₂³	—	2	4	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	4	1	no	—
453	C₂³	—	4	2	no	—
454	C₂³	—	4	2	no	—
455	C₂³	—	4	1	no	—
456	C₂³	—	4	2	no	—
457	C₂³	—	4	2	no	—
458	C₂³	—	4	2	no	—
459	C₂³	—	4	2	no	—
460	C₂³	—	4	2	no	—
461	C₂³	—	4	2	no	—
462	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
463	Q₈	—	2	2	no	—
464	Q₈	—	2	2	no	—
465	Q₈	—	4	4	no	—
466	Q₈	—	4	1	no	—
467	Q₈	—	4	1	no	—
468	D₄	—	2	2	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	4	no	—
471	D₄	—	2	4	no	—
472	D₄	—	4	8	no	—
473	D₄	—	4	2	no	—
474	D₄	—	4	1	no	—
475	D₄	—	4	2	no	—
476	D₄	—	4	1	no	—
477	D₄	—	8	4	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	8	4	no	—
497	C₂×C₄	—	8	8	no	—
498	C₂×C₄	#120×C₄₁	1	2	yes	—
499	C₂×C₄	#120×C₄₁	1	2	yes	—
500	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
501	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
502	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
503	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
504	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
505	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
506	C₈	—	2	2	no	—
507	C₈	—	4	1	no	—
508	C₈	—	4	2	no	—
509	C₂²	—	4	4	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	4	no	—
512	C₂²	—	4	4	no	—
513	C₂²	—	4	4	no	—
514	C₂²	—	8	2	no	—
515	C₂²	—	8	2	no	—
516	C₂²	—	8	2	no	—
517	C₂²	—	8	2	no	—
518	C₂²	—	8	2	no	—
519	C₂²	—	8	1	no	—
520	C₂²	—	8	1	no	—
521	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
522	C₂²	C₄₁×#548	1	1	yes	characteristic
523	C₂²	C₄₁×#550	1	1	yes	characteristic
524	C₂²	C₄₁×#575	1	2	yes	—
525	C₂²	C₄₁×#576	1	2	yes	—
526	C₄	—	2	2	no	—
527	C₄	—	2	2	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	2	no	—
532	C₄	—	8	2	no	—
533	C₄	—	8	1	no	—
534	C₄	C₄₁×#525	1	1	yes	characteristic
535	C₄	C₄₁×#537	1	1	yes	characteristic
536	C₄	C₄₁×#546	1	1	yes	characteristic
537	C₄	C₄₁×#547	1	1	yes	characteristic
538	C₂	—	8	2	no	—
539	C₂	—	8	4	no	—
540	C₂	C₄₁×#2407	1	1	yes	characteristic
541	C₂	C₄₁×#3005	1	1	yes	characteristic
542	C₂	C₄₁×#3008	1	1	yes	characteristic
543	C₂	C₄₁×#3370	1	2	yes	—
544	C₂	C₄₁×#3403	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#459	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄×Q₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#559	C₂	semidirect
3. C₂×C₄₁×#440	C₂	semidirect
4. C₄₁×#2580	C₂	semidirect
5. C₄₁×#2042	C₂	semidirect
6. C₄₁×#2012	C₂	semidirect
7. C₄₁×#1985	C₂	semidirect
8. C₄₁×#1961	C₂	semidirect
9. C₄₁×#1879	C₂	semidirect
10. C₄₁×#1862	C₂	semidirect
11. C₄₁×#1805	C₂	semidirect
236. #24198	C₄₁	direct (N × H)
266. C₂×#559	C₂×C₄₁	semidirect
268. C₂×#440	C₂×C₄₁	semidirect
269. #2580	C₂×C₄₁	semidirect
270. #2042	C₂×C₄₁	semidirect
271. #2012	C₂×C₄₁	semidirect
272. #1985	C₂×C₄₁	semidirect
273. #1961	C₂×C₄₁	semidirect
274. #1879	C₂×C₄₁	semidirect
275. #1862	C₂×C₄₁	semidirect
276. #1805	C₂×C₄₁	semidirect
317. C₄₁	#24198	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#559	2
C₂×C₄₁×#459	2
C₂×C₄₁×#440	2
C₄₁×#2580	2
C₄₁×#2042	2
C₄₁×#2012	2
C₄₁×#1985	2
C₄₁×#1961	2
C₄₁×#1879	2
C₄₁×#1862	2
C₄₁×#1805	2
#24198	41

Subgroups — C41×#24198 #23800399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24198 #23800399