Subgroups — C₄₁×#24118 #23800319

Order 10496.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×SD₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2537	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#2035	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2000	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1954	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1874	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1853	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#554	—	2	4	no	—
14	C₄₁×#554	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×#125×C₄₁	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₄₁	—	2	2	no	—
18	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
19	C₂×#121×C₄₁	—	2	1	no	—
20	C₂×#121×C₄₁	C₂²	1	2	yes	—
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	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
26	C₄₁×#476	—	2	4	no	—
27	C₄×SD₁₆×C₄₁	—	2	4	no	—
28	C₂×#106×C₄₁	C₂²	1	2	yes	—
29	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×#103×C₄₁	C₂²	1	2	yes	—
33	C₂×#102×C₄₁	C₂²	1	2	yes	—
34	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
38	C₄₁×#394	—	2	2	no	—
39	C₄₁×#384	C₂²	1	1	yes	characteristic
40	C₄₁×#380	—	2	1	no	—
41	C₄₁×#380	C₂²	1	2	yes	—
42	C₄₁×#340	C₂²	1	2	yes	—
43	C₄₁×#340	C₂²	1	2	yes	—
44	C₄₁×#340	C₂²	1	2	yes	—
45	C₂²×Q₈×C₄₁	—	2	1	no	—
46	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₄₁	—	2	2	no	—
48	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
49	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₄₁	—	2	2	no	—
51	C₂³×C₄×C₄₁	—	2	1	no	—
52	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
53	C₂×SD₁₆×C₄₁	—	2	2	no	—
54	C₂×SD₁₆×C₄₁	—	2	2	no	—
55	C₂×SD₁₆×C₄₁	—	2	2	no	—
56	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
58	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₄₁	—	2	2	no	—
60	C₄⋊Q₈×C₄₁	D₄	1	4	yes	complemented
61	#125×C₄₁	—	2	8	no	—
62	C₂²⋊Q₈×C₄₁	—	2	16	no	—
63	C₂²⋊Q₈×C₄₁	—	2	4	no	—
64	C₂²⋊Q₈×C₄₁	—	2	4	no	—
65	#121×C₄₁	—	2	4	no	—
66	#121×C₄₁	—	4	2	no	—
67	#121×C₄₁	—	4	4	no	—
68	C₄×Q₈×C₄₁	—	2	2	no	—
69	C₄×Q₈×C₄₁	D₄	1	4	yes	complemented
70	C₄×D₄×C₄₁	—	2	8	no	—
71	C₄×D₄×C₄₁	—	2	2	no	—
72	C₄×D₄×C₄₁	—	2	2	no	—
73	C₄×D₄×C₄₁	—	2	2	no	—
74	C₄×D₄×C₄₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
79	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
80	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
81	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
88	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
93	C₂×C₄²×C₄₁	—	2	1	no	—
94	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
95	#106×C₄₁	—	2	2	no	—
96	#106×C₄₁	—	2	4	no	—
97	C₄⋊C₈×C₄₁	—	2	2	no	—
98	#103×C₄₁	—	2	2	no	—
99	#103×C₄₁	—	4	2	no	—
100	#102×C₄₁	—	2	2	no	—
101	#102×C₄₁	—	2	4	no	—
102	C₂²⋊C₈×C₄₁	—	2	4	no	—
103	C₂²⋊C₈×C₄₁	—	4	2	no	—
104	C₄×C₈×C₄₁	—	2	2	no	—
105	#95×C₄₁	—	2	2	no	—
106	C₂⁴×C₄₁	—	2	1	no	—
107	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
108	C₂×Q₈×C₄₁	—	2	4	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	—	2	1	no	—
114	C₂×Q₈×C₄₁	—	8	1	no	—
115	C₂×Q₈×C₄₁	—	8	1	no	—
116	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
117	C₂×D₄×C₄₁	—	2	2	no	—
118	C₂×D₄×C₄₁	—	2	2	no	—
119	C₂×D₄×C₄₁	—	2	1	no	—
120	C₂×D₄×C₄₁	—	2	1	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	—	4	4	no	—
125	C₂×D₄×C₄₁	—	4	4	no	—
126	C₂×D₄×C₄₁	—	8	2	no	—
127	C₂×D₄×C₄₁	SD₁₆	1	4	yes	complemented
128	C₂×D₄×C₄₁	#41	1	2	yes	—
129	C₂²×C₄×C₄₁	—	2	1	no	—
130	C₂²×C₄×C₄₁	—	2	1	no	—
131	C₂²×C₄×C₄₁	—	2	2	no	—
132	C₂²×C₄×C₄₁	—	2	8	no	—
133	C₂²×C₄×C₄₁	—	2	2	no	—
134	C₂²×C₄×C₄₁	—	2	2	no	—
135	C₂²×C₄×C₄₁	—	2	2	no	—
136	C₂²×C₄×C₄₁	—	2	2	no	—
137	C₂²×C₄×C₄₁	—	2	2	no	—
138	C₂²×C₄×C₄₁	—	2	2	no	—
139	C₂²×C₄×C₄₁	—	2	1	no	—
140	C₂²×C₄×C₄₁	—	2	2	no	—
141	C₂²×C₄×C₄₁	—	4	2	no	—
142	C₂²×C₄×C₄₁	—	4	2	no	—
143	C₂²×C₄×C₄₁	—	4	2	no	—
144	C₂²×C₄×C₄₁	—	8	2	no	—
145	C₂²×C₄×C₄₁	—	8	2	no	—
146	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
150	SD₁₆×C₄₁	—	2	8	no	—
151	C₂×C₈×C₄₁	—	2	1	no	—
152	C₂×C₈×C₄₁	—	2	1	no	—
153	C₂×C₈×C₄₁	—	2	1	no	—
154	C₂×C₈×C₄₁	—	2	2	no	—
155	C₂×C₈×C₄₁	—	2	2	no	—
156	C₂×C₈×C₄₁	—	2	1	no	—
157	C₂×C₈×C₄₁	—	2	2	no	—
158	C₂×C₈×C₄₁	—	2	1	no	—
159	C₂×C₈×C₄₁	#41	1	2	yes	—
160	C₄⋊C₄×C₄₁	—	2	8	no	—
161	C₄⋊C₄×C₄₁	—	2	8	no	—
162	C₄⋊C₄×C₄₁	—	2	4	no	—
163	C₄⋊C₄×C₄₁	—	2	4	no	—
164	C₄⋊C₄×C₄₁	—	2	2	no	—
165	C₄⋊C₄×C₄₁	—	2	1	no	—
166	C₄⋊C₄×C₄₁	—	2	2	no	—
167	C₄⋊C₄×C₄₁	—	2	1	no	—
168	C₄⋊C₄×C₄₁	—	2	1	no	—
169	C₄⋊C₄×C₄₁	—	2	2	no	—
170	C₄⋊C₄×C₄₁	—	2	2	no	—
171	C₄⋊C₄×C₄₁	—	4	2	no	—
172	C₄⋊C₄×C₄₁	—	8	1	no	—
173	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
175	C₂²⋊C₄×C₄₁	—	2	8	no	—
176	C₂²⋊C₄×C₄₁	—	2	4	no	—
177	C₂²⋊C₄×C₄₁	—	4	4	no	—
178	C₂²⋊C₄×C₄₁	—	4	4	no	—
179	C₂²⋊C₄×C₄₁	—	4	2	no	—
180	C₄²×C₄₁	—	2	4	no	—
181	C₄²×C₄₁	C₂×D₄	1	2	yes	—
182	C₄²×C₄₁	C₂×D₄	1	2	yes	—
183	C₂³×C₄₁	—	2	2	no	—
184	C₂³×C₄₁	—	2	2	no	—
185	C₂³×C₄₁	—	2	2	no	—
186	C₂³×C₄₁	—	2	2	no	—
187	C₂³×C₄₁	—	2	2	no	—
188	C₂³×C₄₁	—	2	2	no	—
189	C₂³×C₄₁	—	2	2	no	—
190	C₂³×C₄₁	—	4	1	no	—
191	C₂³×C₄₁	—	4	1	no	—
192	C₂³×C₄₁	—	4	1	no	—
193	C₂³×C₄₁	—	4	1	no	—
194	C₂³×C₄₁	—	4	1	no	—
195	C₂³×C₄₁	—	4	1	no	—
196	C₂³×C₄₁	C₂×SD₁₆	1	4	yes	—
197	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
198	Q₈×C₄₁	—	2	2	no	—
199	Q₈×C₄₁	—	2	2	no	—
200	Q₈×C₄₁	—	4	1	no	—
201	Q₈×C₄₁	—	4	1	no	—
202	Q₈×C₄₁	—	8	4	no	—
203	D₄×C₄₁	—	2	2	no	—
204	D₄×C₄₁	—	2	2	no	—
205	D₄×C₄₁	—	2	8	no	—
206	D₄×C₄₁	—	4	1	no	—
207	D₄×C₄₁	—	4	1	no	—
208	D₄×C₄₁	—	8	8	no	—
209	C₂×C₄×C₄₁	—	2	1	no	—
210	C₂×C₄×C₄₁	—	2	1	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	4	no	—
215	C₂×C₄×C₄₁	—	2	2	no	—
216	C₂×C₄×C₄₁	—	2	2	no	—
217	C₂×C₄×C₄₁	—	4	1	no	—
218	C₂×C₄×C₄₁	—	4	1	no	—
219	C₂×C₄×C₄₁	—	4	1	no	—
220	C₂×C₄×C₄₁	—	4	2	no	—
221	C₂×C₄×C₄₁	—	4	1	no	—
222	C₂×C₄×C₄₁	—	4	2	no	—
223	C₂×C₄×C₄₁	—	4	2	no	—
224	C₂×C₄×C₄₁	—	4	2	no	—
225	C₂×C₄×C₄₁	—	4	2	no	—
226	C₂×C₄×C₄₁	—	4	2	no	—
227	C₂×C₄×C₄₁	—	4	1	no	—
228	C₂×C₄×C₄₁	—	4	2	no	—
229	C₂×C₄×C₄₁	—	4	1	no	—
230	C₂×C₄×C₄₁	—	4	2	no	—
231	C₂×C₄×C₄₁	—	4	1	no	—
232	C₂×C₄×C₄₁	—	4	2	no	—
233	C₂×C₄×C₄₁	—	8	8	no	—
234	C₂×C₄×C₄₁	—	8	4	no	—
235	C₂×C₄×C₄₁	#120	1	2	yes	—
236	C₂×C₄×C₄₁	#120	1	2	yes	—
237	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
238	C₂×C₄×C₄₁	#137	1	2	yes	—
239	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
242	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
243	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
244	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
245	C₈×C₄₁	—	2	2	no	—
246	C₈×C₄₁	—	4	1	no	—
247	C₈×C₄₁	—	4	2	no	—
248	#24118	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
249	C₂²×C₄₁	—	2	4	no	—
250	C₂²×C₄₁	—	2	4	no	—
251	C₂²×C₄₁	—	2	4	no	—
252	C₂²×C₄₁	—	4	2	no	—
253	C₂²×C₄₁	—	4	2	no	—
254	C₂²×C₄₁	—	4	2	no	—
255	C₂²×C₄₁	—	4	2	no	—
256	C₂²×C₄₁	—	4	2	no	—
257	C₂²×C₄₁	—	4	2	no	—
258	C₂²×C₄₁	—	4	2	no	—
259	C₂²×C₄₁	—	8	1	no	—
260	C₂²×C₄₁	—	8	1	no	—
261	C₂²×C₄₁	—	8	1	no	—
262	C₂²×C₄₁	—	8	1	no	—
263	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
264	C₂²×C₄₁	#546	1	1	yes	characteristic
265	C₂²×C₄₁	#547	1	1	yes	characteristic
266	C₂²×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
267	C₂²×C₄₁	C₂×#137	1	1	yes	characteristic
268	C₂²×C₄₁	#576	1	1	yes	characteristic
269	C₂²×C₄₁	#577	1	1	yes	characteristic
270	C₄×C₄₁	—	2	2	no	—
271	C₄×C₄₁	—	4	2	no	—
272	C₄×C₄₁	—	4	2	no	—
273	C₄×C₄₁	—	4	2	no	—
274	C₄×C₄₁	—	4	4	no	—
275	C₄×C₄₁	—	8	1	no	—
276	C₄×C₄₁	—	8	2	no	—
277	C₄×C₄₁	#448	1	4	yes	—
278	C₄×C₄₁	#525	1	1	yes	characteristic
279	C₄×C₄₁	#537	1	1	yes	characteristic
280	C₄×C₄₁	#548	1	1	yes	characteristic
281	C₄×C₄₁	#550	1	1	yes	characteristic
282	C₂×#554	C₂×C₄₁	1	1	yes	characteristic, complemented
283	C₂×#476	C₂×C₄₁	1	1	yes	characteristic, complemented
284	C₂×C₄×SD₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
285	#2537	C₂×C₄₁	1	1	yes	characteristic
286	#2035	C₂×C₄₁	1	2	yes	complemented
287	#2000	C₂×C₄₁	1	2	yes	complemented
288	#1990	C₂×C₄₁	1	1	yes	characteristic, complemented
289	#1954	C₂×C₄₁	1	1	yes	characteristic, complemented
290	#1874	C₂×C₄₁	1	2	yes	complemented
291	#1853	C₂×C₄₁	1	2	yes	complemented
292	#1803	C₂×C₄₁	1	1	yes	characteristic, complemented
293	C₂×C₄₁	—	4	4	no	—
294	C₂×C₄₁	—	8	2	no	—
295	C₂×C₄₁	#2407	1	1	yes	characteristic
296	C₂×C₄₁	C₂×#448	1	1	yes	characteristic
297	C₂×C₄₁	#3012	1	1	yes	characteristic
298	C₂×C₄₁	#3295	1	1	yes	characteristic
299	C₂×C₄₁	#3331	1	1	yes	characteristic
300	C₂×C₄₁	#3335	1	1	yes	characteristic
301	C₂×C₄₁	#3357	1	1	yes	characteristic
302	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
303	#554	—	2	4	no	—
304	#554	—	2	4	no	—
305	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
306	C₂×#125	—	2	1	no	—
307	C₂×C₂²⋊Q₈	—	2	2	no	—
308	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
309	C₂×#121	—	2	1	no	—
310	C₂×#121	C₂²×C₄₁	1	2	yes	—
311	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
312	C₂×C₄×D₄	—	2	1	no	—
313	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
314	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
315	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
316	#476	—	2	4	no	—
317	C₄×SD₁₆	—	2	4	no	—
318	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
319	C₂×#106	C₂²×C₄₁	1	2	yes	—
320	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
321	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
322	C₂×#103	C₂²×C₄₁	1	2	yes	—
323	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
324	C₂×#102	C₂²×C₄₁	1	2	yes	—
325	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
326	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
327	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
328	#394	—	2	2	no	—
329	#384	C₂²×C₄₁	1	1	yes	characteristic
330	#380	—	2	1	no	—
331	#380	C₂²×C₄₁	1	2	yes	—
332	#340	C₂²×C₄₁	1	2	yes	—
333	#340	C₂²×C₄₁	1	2	yes	—
334	#340	C₂²×C₄₁	1	2	yes	—
335	C₄₁	#24118	1	1	yes	characteristic, direct product factor, Sylow
336	C₂²×Q₈	—	2	1	no	—
337	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
338	C₂²×D₄	—	2	2	no	—
339	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
340	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
341	C₂³×C₄	—	2	2	no	—
342	C₂³×C₄	—	2	1	no	—
343	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
344	C₂×SD₁₆	—	2	2	no	—
345	C₂×SD₁₆	—	2	2	no	—
346	C₂×SD₁₆	—	2	2	no	—
347	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
348	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
349	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
350	C₄⋊Q₈	—	2	2	no	—
351	C₄⋊Q₈	D₄×C₄₁	1	4	yes	complemented
352	#125	—	2	8	no	—
353	C₂²⋊Q₈	—	2	16	no	—
354	C₂²⋊Q₈	—	2	4	no	—
355	C₂²⋊Q₈	—	2	4	no	—
356	#121	—	2	4	no	—
357	#121	—	4	2	no	—
358	#121	—	4	4	no	—
359	C₄×Q₈	—	2	2	no	—
360	C₄×Q₈	D₄×C₄₁	1	4	yes	complemented
361	C₄×D₄	—	2	8	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	—	2	2	no	—
366	C₂×C₄⋊C₄	—	2	2	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	1	no	—
369	C₂×C₄⋊C₄	—	2	2	no	—
370	C₂×C₄⋊C₄	—	2	4	no	—
371	C₂×C₄⋊C₄	—	2	4	no	—
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄⋊C₄	—	2	1	no	—
374	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
379	C₂×C₂²⋊C₄	—	2	2	no	—
380	C₂×C₂²⋊C₄	—	2	2	no	—
381	C₂×C₂²⋊C₄	—	2	1	no	—
382	C₂×C₂²⋊C₄	—	2	2	no	—
383	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
384	C₂×C₄²	—	2	1	no	—
385	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
386	#106	—	2	2	no	—
387	#106	—	2	4	no	—
388	C₄⋊C₈	—	2	2	no	—
389	#103	—	2	2	no	—
390	#103	—	4	2	no	—
391	#102	—	2	2	no	—
392	#102	—	2	4	no	—
393	C₂²⋊C₈	—	2	4	no	—
394	C₂²⋊C₈	—	4	2	no	—
395	C₄×C₈	—	2	2	no	—
396	#95	—	2	2	no	—
397	C₂⁴	—	2	1	no	—
398	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
399	C₂×Q₈	—	2	4	no	—
400	C₂×Q₈	—	2	1	no	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	2	1	no	—
403	C₂×Q₈	—	2	1	no	—
404	C₂×Q₈	—	2	1	no	—
405	C₂×Q₈	—	8	1	no	—
406	C₂×Q₈	—	8	1	no	—
407	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	1	no	—
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	4	4	no	—
416	C₂×D₄	—	4	4	no	—
417	C₂×D₄	—	8	2	no	—
418	C₂×D₄	SD₁₆×C₄₁	1	4	yes	complemented
419	C₂×D₄	#41×C₄₁	1	2	yes	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	8	no	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	—	4	2	no	—
434	C₂²×C₄	—	4	2	no	—
435	C₂²×C₄	—	8	2	no	—
436	C₂²×C₄	—	8	2	no	—
437	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
438	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
440	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
441	SD₁₆	—	2	8	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	2	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	#41×C₄₁	1	2	yes	—
451	C₄⋊C₄	—	2	8	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	4	2	no	—
463	C₄⋊C₄	—	8	1	no	—
464	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
465	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
466	C₂²⋊C₄	—	2	8	no	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	4	4	no	—
469	C₂²⋊C₄	—	4	4	no	—
470	C₂²⋊C₄	—	4	2	no	—
471	C₄²	—	2	4	no	—
472	C₄²	C₂×D₄×C₄₁	1	2	yes	—
473	C₄²	C₂×D₄×C₄₁	1	2	yes	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	4	1	no	—
482	C₂³	—	4	1	no	—
483	C₂³	—	4	1	no	—
484	C₂³	—	4	1	no	—
485	C₂³	—	4	1	no	—
486	C₂³	—	4	1	no	—
487	C₂³	C₂×SD₁₆×C₄₁	1	4	yes	—
488	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
489	Q₈	—	2	2	no	—
490	Q₈	—	2	2	no	—
491	Q₈	—	4	1	no	—
492	Q₈	—	4	1	no	—
493	Q₈	—	8	4	no	—
494	D₄	—	2	2	no	—
495	D₄	—	2	2	no	—
496	D₄	—	2	8	no	—
497	D₄	—	4	1	no	—
498	D₄	—	4	1	no	—
499	D₄	—	8	8	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	2	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	2	no	—
524	C₂×C₄	—	8	8	no	—
525	C₂×C₄	—	8	4	no	—
526	C₂×C₄	#120×C₄₁	1	2	yes	—
527	C₂×C₄	#120×C₄₁	1	2	yes	—
528	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
529	C₂×C₄	#137×C₄₁	1	2	yes	—
530	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
531	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
532	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
533	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
534	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
535	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
536	C₈	—	2	2	no	—
537	C₈	—	4	1	no	—
538	C₈	—	4	2	no	—
539	C₂²	—	2	4	no	—
540	C₂²	—	2	4	no	—
541	C₂²	—	2	4	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	4	2	no	—
544	C₂²	—	4	2	no	—
545	C₂²	—	4	2	no	—
546	C₂²	—	4	2	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	8	1	no	—
550	C₂²	—	8	1	no	—
551	C₂²	—	8	1	no	—
552	C₂²	—	8	1	no	—
553	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
554	C₂²	C₄₁×#546	1	1	yes	characteristic
555	C₂²	C₄₁×#547	1	1	yes	characteristic
556	C₂²	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
557	C₂²	C₂×#137×C₄₁	1	1	yes	characteristic
558	C₂²	C₄₁×#576	1	1	yes	characteristic
559	C₂²	C₄₁×#577	1	1	yes	characteristic
560	C₄	—	2	2	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	4	no	—
565	C₄	—	8	2	no	—
566	C₄	—	8	1	no	—
567	C₄	C₄₁×#448	1	4	yes	—
568	C₄	C₄₁×#525	1	1	yes	characteristic
569	C₄	C₄₁×#537	1	1	yes	characteristic
570	C₄	C₄₁×#548	1	1	yes	characteristic
571	C₄	C₄₁×#550	1	1	yes	characteristic
572	C₂	—	4	4	no	—
573	C₂	—	8	2	no	—
574	C₂	C₄₁×#2407	1	1	yes	characteristic
575	C₂	C₂×C₄₁×#448	1	1	yes	characteristic
576	C₂	C₄₁×#3012	1	1	yes	characteristic
577	C₂	C₄₁×#3295	1	1	yes	characteristic
578	C₂	C₄₁×#3331	1	1	yes	characteristic
579	C₂	C₄₁×#3335	1	1	yes	characteristic
580	C₂	C₄₁×#3357	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#2537	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄×D₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#554	C₂	semidirect
2. C₂×C₄₁×#476	C₂	semidirect
3. C₂×C₄×SD₁₆×C₄₁	C₂	semidirect
5. C₄₁×#2035	C₂	semidirect
6. C₄₁×#2000	C₂	semidirect
7. C₄₁×#1990	C₂	semidirect
8. C₄₁×#1954	C₂	semidirect
9. C₄₁×#1874	C₂	semidirect
10. C₄₁×#1853	C₂	semidirect
11. C₄₁×#1803	C₂	semidirect
60. C₄⋊Q₈×C₄₁	D₄	semidirect
69. C₄×Q₈×C₄₁	D₄	semidirect
127. C₂×D₄×C₄₁	SD₁₆	semidirect
248. #24118	C₄₁	direct (N × H)
282. C₂×#554	C₂×C₄₁	semidirect
283. C₂×#476	C₂×C₄₁	semidirect
284. C₂×C₄×SD₁₆	C₂×C₄₁	semidirect
286. #2035	C₂×C₄₁	semidirect
287. #2000	C₂×C₄₁	semidirect
288. #1990	C₂×C₄₁	semidirect
289. #1954	C₂×C₄₁	semidirect
290. #1874	C₂×C₄₁	semidirect
291. #1853	C₂×C₄₁	semidirect
292. #1803	C₂×C₄₁	semidirect
335. C₄₁	#24118	direct (N × H)
351. C₄⋊Q₈	D₄×C₄₁	semidirect
360. C₄×Q₈	D₄×C₄₁	semidirect
418. C₂×D₄	SD₁₆×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×#554	2
C₂×C₄₁×#476	2
C₂×C₄×SD₁₆×C₄₁	2
C₄₁×#2537	2
C₄₁×#2035	2
C₄₁×#2000	2
C₄₁×#1990	2
C₄₁×#1954	2
C₄₁×#1874	2
C₄₁×#1853	2
C₄₁×#1803	2
#24118	41

Subgroups — C41×#24118 #23800319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24118 #23800319