Subgroups — C₄₁×#22570 #23798771

Order 10496.

1218 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#102×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2344	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₂²⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1934	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1925	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1862	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1823	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1786	C₂	1	1	yes	maximal, characteristic
10	C₄×#102×C₄₁	C₂	1	2	yes	maximal, complemented
11	C₈×C₂²⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
15	C₂×#117×C₄₁	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
17	C₂²×C₂²⋊C₄×C₄₁	—	2	1	no	—
18	C₂×#107×C₄₁	C₂²	1	2	yes	complemented
19	C₂×#102×C₄₁	C₄	1	8	yes	complemented
20	C₂×#102×C₄₁	C₂²	1	2	yes	—
21	C₂×#102×C₄₁	C₂²	1	2	yes	—
22	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	complemented
28	C₄₁×#390	C₂²	1	2	yes	—
29	C₄₁×#389	C₂²	1	2	yes	complemented
30	C₄₁×#387	—	2	2	no	—
31	C₄₁×#386	—	2	2	no	—
32	C₄₁×#381	—	2	1	no	—
33	C₄₁×#380	C₂²	1	1	yes	characteristic
34	C₄₁×#380	C₂²	1	1	yes	characteristic
35	C₄₁×#379	—	2	1	no	—
36	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	complemented
37	C₄×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
38	C₄₁×#340	C₂²	1	2	yes	complemented
39	C₄₁×#340	C₂²	1	2	yes	complemented
40	C₄₁×#340	C₂²	1	2	yes	complemented
41	C₄₁×#336	C₂²	1	2	yes	complemented
42	C₂⁵×C₄₁	—	2	1	no	—
43	C₂²×D₄×C₄₁	—	2	4	no	—
44	C₂²×D₄×C₄₁	—	2	1	no	—
45	C₂²×D₄×C₄₁	—	2	2	no	—
46	C₂²×D₄×C₄₁	—	4	2	no	—
47	C₂²×D₄×C₄₁	C₂×C₄	1	4	yes	—
48	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₄₁	—	2	1	no	—
51	C₂³×C₄×C₄₁	—	2	2	no	—
52	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₁	—	4	2	no	—
54	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	complemented
55	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
58	C₄×D₄×C₄₁	—	2	4	no	—
59	C₄×D₄×C₄₁	—	4	2	no	—
60	#117×C₄₁	—	4	4	no	—
61	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	complemented
65	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₄₁	—	4	4	no	—
75	C₂×C₂²⋊C₄×C₄₁	D₄	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₄₁	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₄₁	—	2	2	no	—
78	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
79	#107×C₄₁	—	4	2	no	—
80	#102×C₄₁	—	2	4	no	—
81	#102×C₄₁	—	4	2	no	—
82	#102×C₄₁	C₂×C₄	1	4	yes	—
83	#102×C₄₁	C₂×C₄	1	4	yes	—
84	#102×C₄₁	C₂×C₄	1	4	yes	complemented
85	#102×C₄₁	C₂×C₄	1	4	yes	complemented
86	C₂²⋊C₈×C₄₁	—	2	2	no	—
87	C₄×C₈×C₄₁	—	4	2	no	—
88	#95×C₄₁	—	2	2	no	—
89	#95×C₄₁	—	2	1	no	—
90	#95×C₄₁	—	2	1	no	—
91	#95×C₄₁	—	2	2	no	—
92	#95×C₄₁	D₄	1	2	yes	—
93	C₂⁴×C₄₁	—	2	1	no	—
94	C₂⁴×C₄₁	—	2	1	no	—
95	C₂⁴×C₄₁	—	2	4	no	—
96	C₂⁴×C₄₁	—	4	4	no	—
97	C₂⁴×C₄₁	—	4	2	no	—
98	C₂⁴×C₄₁	—	4	2	no	—
99	C₂⁴×C₄₁	—	8	2	no	—
100	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
101	C₂×D₄×C₄₁	—	2	8	no	—
102	C₂×D₄×C₄₁	—	2	4	no	—
103	C₂×D₄×C₄₁	—	2	1	no	—
104	C₂×D₄×C₄₁	—	2	2	no	—
105	C₂×D₄×C₄₁	—	2	1	no	—
106	C₂×D₄×C₄₁	—	2	2	no	—
107	C₂×D₄×C₄₁	—	2	2	no	—
108	C₂×D₄×C₄₁	—	4	4	no	—
109	C₂×D₄×C₄₁	—	4	8	no	—
110	C₂×D₄×C₄₁	—	4	4	no	—
111	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
112	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
113	C₂×D₄×C₄₁	#41	1	4	yes	—
114	C₂²×C₄×C₄₁	—	2	2	no	—
115	C₂²×C₄×C₄₁	—	2	2	no	—
116	C₂²×C₄×C₄₁	—	2	4	no	—
117	C₂²×C₄×C₄₁	—	2	1	no	—
118	C₂²×C₄×C₄₁	—	2	1	no	—
119	C₂²×C₄×C₄₁	—	2	2	no	—
120	C₂²×C₄×C₄₁	—	2	2	no	—
121	C₂²×C₄×C₄₁	—	4	4	no	—
122	C₂²×C₄×C₄₁	—	4	2	no	—
123	C₂²×C₄×C₄₁	—	4	2	no	—
124	C₂²×C₄×C₄₁	—	4	4	no	—
125	C₂²×C₄×C₄₁	—	8	2	no	—
126	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
127	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₄₁	#41	1	2	yes	—
131	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
132	C₂×C₈×C₄₁	—	2	2	no	—
133	C₂×C₈×C₄₁	—	2	2	no	—
134	C₂×C₈×C₄₁	—	2	2	no	—
135	C₂×C₈×C₄₁	—	2	2	no	—
136	C₂×C₈×C₄₁	—	2	2	no	—
137	C₂×C₈×C₄₁	—	4	4	no	—
138	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
139	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
140	C₄⋊C₄×C₄₁	—	2	4	no	—
141	C₄⋊C₄×C₄₁	—	2	1	no	—
142	C₄⋊C₄×C₄₁	—	2	1	no	—
143	C₄⋊C₄×C₄₁	—	2	2	no	—
144	C₄⋊C₄×C₄₁	—	2	2	no	—
145	C₄⋊C₄×C₄₁	—	2	2	no	—
146	C₄⋊C₄×C₄₁	—	2	2	no	—
147	C₄⋊C₄×C₄₁	—	4	4	no	—
148	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
149	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
150	C₂²⋊C₄×C₄₁	—	2	2	no	—
151	C₂²⋊C₄×C₄₁	—	4	4	no	—
152	C₂²⋊C₄×C₄₁	—	4	2	no	—
153	C₂²⋊C₄×C₄₁	—	4	2	no	—
154	C₂²⋊C₄×C₄₁	—	4	4	no	—
155	C₂²⋊C₄×C₄₁	—	8	2	no	—
156	C₂²⋊C₄×C₄₁	D₈	1	4	yes	—
157	C₄²×C₄₁	—	2	2	no	—
158	C₄²×C₄₁	—	2	2	no	—
159	C₄²×C₄₁	—	4	4	no	—
160	C₂³×C₄₁	—	2	2	no	—
161	C₂³×C₄₁	—	2	2	no	—
162	C₂³×C₄₁	—	2	2	no	—
163	C₂³×C₄₁	—	2	4	no	—
164	C₂³×C₄₁	—	2	2	no	—
165	C₂³×C₄₁	—	4	8	no	—
166	C₂³×C₄₁	—	4	1	no	—
167	C₂³×C₄₁	—	4	1	no	—
168	C₂³×C₄₁	—	4	2	no	—
169	C₂³×C₄₁	—	4	2	no	—
170	C₂³×C₄₁	—	4	1	no	—
171	C₂³×C₄₁	—	4	1	no	—
172	C₂³×C₄₁	—	4	2	no	—
173	C₂³×C₄₁	—	4	8	no	—
174	C₂³×C₄₁	—	8	4	no	—
175	C₂³×C₄₁	—	8	8	no	—
176	C₂³×C₄₁	—	8	8	no	—
177	C₂³×C₄₁	C₄×D₄	1	2	yes	—
178	C₂³×C₄₁	C₂×D₈	1	2	yes	—
179	C₂³×C₄₁	#135	1	2	yes	—
180	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
181	D₄×C₄₁	—	2	2	no	—
182	D₄×C₄₁	—	2	2	no	—
183	D₄×C₄₁	—	2	2	no	—
184	D₄×C₄₁	—	2	4	no	—
185	D₄×C₄₁	—	4	8	no	—
186	D₄×C₄₁	—	4	2	no	—
187	D₄×C₄₁	#117	1	4	yes	complemented
188	C₂×C₄×C₄₁	—	2	2	no	—
189	C₂×C₄×C₄₁	—	2	2	no	—
190	C₂×C₄×C₄₁	—	2	2	no	—
191	C₂×C₄×C₄₁	—	2	2	no	—
192	C₂×C₄×C₄₁	—	2	2	no	—
193	C₂×C₄×C₄₁	—	2	4	no	—
194	C₂×C₄×C₄₁	—	2	2	no	—
195	C₂×C₄×C₄₁	—	2	2	no	—
196	C₂×C₄×C₄₁	—	2	2	no	—
197	C₂×C₄×C₄₁	—	2	2	no	—
198	C₂×C₄×C₄₁	—	2	2	no	—
199	C₂×C₄×C₄₁	—	4	1	no	—
200	C₂×C₄×C₄₁	—	4	1	no	—
201	C₂×C₄×C₄₁	—	4	1	no	—
202	C₂×C₄×C₄₁	—	4	1	no	—
203	C₂×C₄×C₄₁	—	4	8	no	—
204	C₂×C₄×C₄₁	—	4	2	no	—
205	C₂×C₄×C₄₁	—	4	2	no	—
206	C₂×C₄×C₄₁	—	4	2	no	—
207	C₂×C₄×C₄₁	—	4	2	no	—
208	C₂×C₄×C₄₁	—	4	2	no	—
209	C₂×C₄×C₄₁	—	4	2	no	—
210	C₂×C₄×C₄₁	—	8	8	no	—
211	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
212	C₂×C₄×C₄₁	C₂×D₈	1	4	yes	—
213	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
215	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
217	C₈×C₄₁	—	2	2	no	—
218	C₈×C₄₁	—	2	2	no	—
219	C₈×C₄₁	—	4	2	no	—
220	#22570	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂²×C₄₁	—	2	2	no	—
222	C₂²×C₄₁	—	2	2	no	—
223	C₂²×C₄₁	—	2	4	no	—
224	C₂²×C₄₁	—	2	4	no	—
225	C₂²×C₄₁	—	2	4	no	—
226	C₂²×C₄₁	—	4	2	no	—
227	C₂²×C₄₁	—	4	2	no	—
228	C₂²×C₄₁	—	4	4	no	—
229	C₂²×C₄₁	—	4	4	no	—
230	C₂²×C₄₁	—	4	2	no	—
231	C₂²×C₄₁	—	4	2	no	—
232	C₂²×C₄₁	—	4	2	no	—
233	C₂²×C₄₁	—	8	16	no	—
234	C₂²×C₄₁	—	8	2	no	—
235	C₂²×C₄₁	—	8	2	no	—
236	C₂²×C₄₁	C₄×D₈	1	4	yes	—
237	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
238	C₂²×C₄₁	#546	1	2	yes	—
239	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
240	C₂²×C₄₁	C₂×#135	1	1	yes	characteristic
241	C₂²×C₄₁	#577	1	2	yes	—
242	C₄×C₄₁	—	2	2	no	—
243	C₄×C₄₁	—	2	4	no	—
244	C₄×C₄₁	—	4	2	no	—
245	C₄×C₄₁	—	4	2	no	—
246	C₄×C₄₁	—	4	2	no	—
247	C₄×C₄₁	—	8	2	no	—
248	C₄×C₄₁	—	8	2	no	—
249	C₄×C₄₁	C₂×#117	1	1	yes	characteristic
250	C₄×C₄₁	#518	1	1	yes	characteristic
251	C₄×C₄₁	#551	1	2	yes	—
252	C₂²×#102	C₂×C₄₁	1	1	yes	characteristic
253	#2344	C₂×C₄₁	1	1	yes	characteristic, complemented
254	D₄×C₂²⋊C₄	C₂×C₄₁	1	1	yes	characteristic
255	#1934	C₂×C₄₁	1	2	yes	complemented
256	#1925	C₂×C₄₁	1	2	yes	complemented
257	#1862	C₂×C₄₁	1	2	yes	complemented
258	#1823	C₂×C₄₁	1	1	yes	characteristic, complemented
259	#1808	C₂×C₄₁	1	1	yes	characteristic, complemented
260	#1786	C₂×C₄₁	1	1	yes	characteristic
261	C₄×#102	C₂×C₄₁	1	2	yes	complemented
262	C₈×C₂²⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
263	C₂×C₄₁	—	2	4	no	—
264	C₂×C₄₁	—	4	4	no	—
265	C₂×C₄₁	—	8	2	no	—
266	C₂×C₄₁	#2342	1	1	yes	characteristic
267	C₂×C₄₁	C₂×C₄×D₈	1	1	yes	characteristic
268	C₂×C₄₁	#2949	1	1	yes	characteristic
269	C₂×C₄₁	#3294	1	2	yes	—
270	C₂×C₄₁	#3296	1	2	yes	—
271	C₂³×D₄	C₂²×C₄₁	1	1	yes	characteristic
272	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
273	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
274	C₂×#117	—	2	1	no	—
275	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
276	C₂²×C₂²⋊C₄	—	2	1	no	—
277	C₂×#107	C₂²×C₄₁	1	2	yes	complemented
278	C₂×#102	C₄×C₄₁	1	8	yes	complemented
279	C₂×#102	C₂²×C₄₁	1	2	yes	—
280	C₂×#102	C₂²×C₄₁	1	2	yes	—
281	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
282	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
283	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
284	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
285	C₂×C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
286	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	complemented
287	#390	C₂²×C₄₁	1	2	yes	—
288	#389	C₂²×C₄₁	1	2	yes	complemented
289	#387	—	2	2	no	—
290	#386	—	2	2	no	—
291	#381	—	2	1	no	—
292	#380	C₂²×C₄₁	1	1	yes	characteristic
293	#380	C₂²×C₄₁	1	1	yes	characteristic
294	#379	—	2	1	no	—
295	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	complemented
296	C₄×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
297	#340	C₂²×C₄₁	1	2	yes	complemented
298	#340	C₂²×C₄₁	1	2	yes	complemented
299	#340	C₂²×C₄₁	1	2	yes	complemented
300	#336	C₂²×C₄₁	1	2	yes	complemented
301	C₄₁	#22570	1	1	yes	characteristic, direct product factor, Sylow
302	C₂⁵	—	2	1	no	—
303	C₂²×D₄	—	2	4	no	—
304	C₂²×D₄	—	2	1	no	—
305	C₂²×D₄	—	2	2	no	—
306	C₂²×D₄	—	4	2	no	—
307	C₂²×D₄	C₂×C₄×C₄₁	1	4	yes	—
308	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
309	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
310	C₂³×C₄	—	2	1	no	—
311	C₂³×C₄	—	2	2	no	—
312	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
313	C₂²×C₈	—	4	2	no	—
314	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	complemented
315	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
316	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
317	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
318	C₄×D₄	—	2	4	no	—
319	C₄×D₄	—	4	2	no	—
320	#117	—	4	4	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	2	no	—
323	C₂×C₄⋊C₄	—	2	2	no	—
324	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	complemented
325	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
326	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
329	C₂×C₂²⋊C₄	—	2	1	no	—
330	C₂×C₂²⋊C₄	—	2	1	no	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₂²⋊C₄	—	2	2	no	—
333	C₂×C₂²⋊C₄	—	2	2	no	—
334	C₂×C₂²⋊C₄	—	4	4	no	—
335	C₂×C₂²⋊C₄	D₄×C₄₁	1	1	yes	characteristic
336	C₂×C₂²⋊C₄	D₄×C₄₁	1	1	yes	characteristic
337	C₂×C₄²	—	2	2	no	—
338	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
339	#107	—	4	2	no	—
340	#102	—	2	4	no	—
341	#102	—	4	2	no	—
342	#102	C₂×C₄×C₄₁	1	4	yes	—
343	#102	C₂×C₄×C₄₁	1	4	yes	—
344	#102	C₂×C₄×C₄₁	1	4	yes	complemented
345	#102	C₂×C₄×C₄₁	1	4	yes	complemented
346	C₂²⋊C₈	—	2	2	no	—
347	C₄×C₈	—	4	2	no	—
348	#95	—	2	2	no	—
349	#95	—	2	1	no	—
350	#95	—	2	1	no	—
351	#95	—	2	2	no	—
352	#95	D₄×C₄₁	1	2	yes	—
353	C₂⁴	—	2	1	no	—
354	C₂⁴	—	2	1	no	—
355	C₂⁴	—	2	4	no	—
356	C₂⁴	—	4	4	no	—
357	C₂⁴	—	4	2	no	—
358	C₂⁴	—	4	2	no	—
359	C₂⁴	—	8	2	no	—
360	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
361	C₂×D₄	—	2	8	no	—
362	C₂×D₄	—	2	4	no	—
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	4	4	no	—
369	C₂×D₄	—	4	8	no	—
370	C₂×D₄	—	4	4	no	—
371	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
372	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
373	C₂×D₄	#41×C₄₁	1	4	yes	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	4	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	4	4	no	—
382	C₂²×C₄	—	4	2	no	—
383	C₂²×C₄	—	4	2	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	8	2	no	—
386	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
387	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
390	C₂²×C₄	#41×C₄₁	1	2	yes	—
391	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	4	4	no	—
398	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
399	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	4	4	no	—
408	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
409	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
410	C₂²⋊C₄	—	2	2	no	—
411	C₂²⋊C₄	—	4	4	no	—
412	C₂²⋊C₄	—	4	2	no	—
413	C₂²⋊C₄	—	4	2	no	—
414	C₂²⋊C₄	—	4	4	no	—
415	C₂²⋊C₄	—	8	2	no	—
416	C₂²⋊C₄	D₈×C₄₁	1	4	yes	—
417	C₄²	—	2	2	no	—
418	C₄²	—	2	2	no	—
419	C₄²	—	4	4	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	4	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	4	8	no	—
426	C₂³	—	4	1	no	—
427	C₂³	—	4	1	no	—
428	C₂³	—	4	2	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	1	no	—
431	C₂³	—	4	1	no	—
432	C₂³	—	4	2	no	—
433	C₂³	—	4	8	no	—
434	C₂³	—	8	4	no	—
435	C₂³	—	8	8	no	—
436	C₂³	—	8	8	no	—
437	C₂³	C₄×D₄×C₄₁	1	2	yes	—
438	C₂³	C₂×D₈×C₄₁	1	2	yes	—
439	C₂³	#135×C₄₁	1	2	yes	—
440	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
441	D₄	—	2	2	no	—
442	D₄	—	2	2	no	—
443	D₄	—	2	2	no	—
444	D₄	—	2	4	no	—
445	D₄	—	4	8	no	—
446	D₄	—	4	2	no	—
447	D₄	#117×C₄₁	1	4	yes	complemented
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	8	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	8	8	no	—
471	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
472	C₂×C₄	C₂×D₈×C₄₁	1	4	yes	—
473	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
474	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
475	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
476	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
477	C₈	—	2	2	no	—
478	C₈	—	2	2	no	—
479	C₈	—	4	2	no	—
480	C₂²	—	2	4	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	2	2	no	—
484	C₂²	—	2	4	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	8	16	no	—
493	C₂²	—	8	2	no	—
494	C₂²	—	8	2	no	—
495	C₂²	C₄×D₈×C₄₁	1	4	yes	—
496	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
497	C₂²	C₄₁×#546	1	2	yes	—
498	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
499	C₂²	C₂×#135×C₄₁	1	1	yes	characteristic
500	C₂²	C₄₁×#577	1	2	yes	—
501	C₄	—	2	4	no	—
502	C₄	—	2	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	2	no	—
506	C₄	—	8	2	no	—
507	C₄	—	8	2	no	—
508	C₄	C₂×#117×C₄₁	1	1	yes	characteristic
509	C₄	C₄₁×#518	1	1	yes	characteristic
510	C₄	C₄₁×#551	1	2	yes	—
511	C₂	—	2	4	no	—
512	C₂	—	4	4	no	—
513	C₂	—	8	2	no	—
514	C₂	C₄₁×#2342	1	1	yes	characteristic
515	C₂	C₂×C₄×D₈×C₄₁	1	1	yes	characteristic
516	C₂	C₄₁×#2949	1	1	yes	characteristic
517	C₂	C₄₁×#3294	1	2	yes	—
518	C₂	C₄₁×#3296	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2344	C₂	semidirect
4. C₄₁×#1934	C₂	semidirect
5. C₄₁×#1925	C₂	semidirect
6. C₄₁×#1862	C₂	semidirect
7. C₄₁×#1823	C₂	semidirect
8. C₄₁×#1808	C₂	semidirect
10. C₄×#102×C₄₁	C₂	semidirect
11. C₈×C₂²⋊C₄×C₄₁	C₂	semidirect
18. C₂×#107×C₄₁	C₂²	semidirect
19. C₂×#102×C₄₁	C₄	semidirect
27. C₂×C₄×C₈×C₄₁	C₂²	semidirect
29. C₄₁×#389	C₂²	semidirect
36. C₄×C₄⋊C₄×C₄₁	C₂²	semidirect
38. C₄₁×#340	C₂²	semidirect
39. C₄₁×#340	C₂²	semidirect
40. C₄₁×#340	C₂²	semidirect
41. C₄₁×#336	C₂²	semidirect
54. C₂²×C₈×C₄₁	C₂×C₄	semidirect
64. C₂×C₄⋊C₄×C₄₁	C₂×C₄	semidirect
84. #102×C₄₁	C₂×C₄	semidirect
85. #102×C₄₁	C₂×C₄	semidirect
187. D₄×C₄₁	#117	semidirect
220. #22570	C₄₁	direct (N × H)
253. #2344	C₂×C₄₁	semidirect
255. #1934	C₂×C₄₁	semidirect
256. #1925	C₂×C₄₁	semidirect
257. #1862	C₂×C₄₁	semidirect
258. #1823	C₂×C₄₁	semidirect
259. #1808	C₂×C₄₁	semidirect
261. C₄×#102	C₂×C₄₁	semidirect
262. C₈×C₂²⋊C₄	C₂×C₄₁	semidirect
277. C₂×#107	C₂²×C₄₁	semidirect
278. C₂×#102	C₄×C₄₁	semidirect
286. C₂×C₄×C₈	C₂²×C₄₁	semidirect
288. #389	C₂²×C₄₁	semidirect
295. C₄×C₄⋊C₄	C₂²×C₄₁	semidirect
297. #340	C₂²×C₄₁	semidirect
298. #340	C₂²×C₄₁	semidirect
299. #340	C₂²×C₄₁	semidirect
300. #336	C₂²×C₄₁	semidirect
301. C₄₁	#22570	direct (N × H)
314. C₂²×C₈	C₂×C₄×C₄₁	semidirect
324. C₂×C₄⋊C₄	C₂×C₄×C₄₁	semidirect
344. #102	C₂×C₄×C₄₁	semidirect
345. #102	C₂×C₄×C₄₁	semidirect
447. D₄	#117×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#102×C₄₁	2
C₄₁×#2344	2
D₄×C₂²⋊C₄×C₄₁	2
C₄₁×#1934	2
C₄₁×#1925	2
C₄₁×#1862	2
C₄₁×#1823	2
C₄₁×#1808	2
C₄₁×#1786	2
C₄×#102×C₄₁	2
C₈×C₂²⋊C₄×C₄₁	2
#22570	41

Subgroups — C41×#22570 #23798771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22570 #23798771