Subgroups — C₄₁×#23220 #23799421

Order 10496.

950 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#486	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2366	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2365	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1982	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#1936	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1926	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1864	C₂	1	2	yes	maximal, complemented
9	C₄×#106×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1762	C₂	1	2	yes	maximal, complemented
12	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₄₁	—	2	1	no	—
15	C₂×C₄×D₄×C₄₁	—	2	1	no	—
16	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
17	C₄₁×#486	C₄	1	8	yes	complemented
18	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×#102×C₄₁	C₂²	1	2	yes	—
22	C₂×#102×C₄₁	C₂²	1	2	yes	—
23	C₂×#102×C₄₁	C₂²	1	2	yes	—
24	C₂×#102×C₄₁	C₂²	1	2	yes	—
25	C₂×#97×C₄₁	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#391	—	2	1	no	—
28	C₄₁×#390	—	2	1	no	—
29	C₄₁×#390	C₂²	1	2	yes	—
30	C₄₁×#389	C₂²	1	2	yes	—
31	C₄₁×#387	—	2	2	no	—
32	C₄₁×#386	—	2	2	no	—
33	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
37	C₄₁×#340	C₂²	1	1	yes	characteristic
38	C₄₁×#340	C₂²	1	2	yes	—
39	C₄₁×#340	C₂²	1	1	yes	characteristic
40	C₄₁×#340	C₂²	1	2	yes	—
41	C₄₁×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₄₁	—	2	1	no	—
43	C₂²×D₄×C₄₁	—	2	1	no	—
44	C₂²×D₄×C₄₁	—	2	1	no	—
45	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
46	C₂³×C₄×C₄₁	—	2	2	no	—
47	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
50	C₄⋊Q₈×C₄₁	—	2	2	no	—
51	C₄⋊Q₈×C₄₁	C₂×C₄	1	4	yes	complemented
52	#127×C₄₁	—	2	2	no	—
53	#127×C₄₁	C₂×C₄	1	4	yes	—
54	C₄×Q₈×C₄₁	—	4	4	no	—
55	C₄×D₄×C₄₁	—	2	4	no	—
56	C₄×D₄×C₄₁	—	4	2	no	—
57	C₄×D₄×C₄₁	—	4	4	no	—
58	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
70	C₂×C₄²×C₄₁	—	2	1	no	—
71	C₂×C₄²×C₄₁	—	2	2	no	—
72	C₂×C₄²×C₄₁	—	2	1	no	—
73	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
74	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
75	#106×C₄₁	—	2	2	no	—
76	#106×C₄₁	—	2	2	no	—
77	C₄⋊C₈×C₄₁	—	2	2	no	—
78	#102×C₄₁	—	2	4	no	—
79	#102×C₄₁	—	4	2	no	—
80	#102×C₄₁	C₂×C₄	1	8	yes	complemented
81	#102×C₄₁	C₂×C₄	1	8	yes	complemented
82	#97×C₄₁	—	4	2	no	—
83	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	complemented
84	#95×C₄₁	—	2	1	no	—
85	#95×C₄₁	—	2	1	no	—
86	#95×C₄₁	—	2	2	no	—
87	#95×C₄₁	—	2	2	no	—
88	#95×C₄₁	D₄	1	2	yes	—
89	C₂⁴×C₄₁	—	2	2	no	—
90	C₂×Q₈×C₄₁	—	2	4	no	—
91	C₂×Q₈×C₄₁	—	4	2	no	—
92	C₂×Q₈×C₄₁	—	4	2	no	—
93	C₂×D₄×C₄₁	—	2	4	no	—
94	C₂×D₄×C₄₁	—	2	4	no	—
95	C₂×D₄×C₄₁	—	2	2	no	—
96	C₂×D₄×C₄₁	—	2	2	no	—
97	C₂×D₄×C₄₁	—	2	2	no	—
98	C₂×D₄×C₄₁	—	2	2	no	—
99	C₂×D₄×C₄₁	—	2	2	no	—
100	C₂×D₄×C₄₁	—	4	2	no	—
101	C₂×D₄×C₄₁	—	4	2	no	—
102	C₂×D₄×C₄₁	—	8	1	no	—
103	C₂×D₄×C₄₁	—	8	1	no	—
104	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
105	C₂²×C₄×C₄₁	—	2	1	no	—
106	C₂²×C₄×C₄₁	—	2	2	no	—
107	C₂²×C₄×C₄₁	—	2	1	no	—
108	C₂²×C₄×C₄₁	—	2	1	no	—
109	C₂²×C₄×C₄₁	—	2	1	no	—
110	C₂²×C₄×C₄₁	—	4	4	no	—
111	C₂²×C₄×C₄₁	—	4	4	no	—
112	C₂²×C₄×C₄₁	—	8	2	no	—
113	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
118	C₂×C₈×C₄₁	—	2	1	no	—
119	C₂×C₈×C₄₁	—	2	1	no	—
120	C₂×C₈×C₄₁	—	2	1	no	—
121	C₂×C₈×C₄₁	—	2	1	no	—
122	C₂×C₈×C₄₁	—	2	2	no	—
123	C₂×C₈×C₄₁	—	2	2	no	—
124	C₂×C₈×C₄₁	—	2	2	no	—
125	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₄₁	—	2	4	no	—
128	C₄⋊C₄×C₄₁	—	2	2	no	—
129	C₄⋊C₄×C₄₁	—	2	2	no	—
130	C₄⋊C₄×C₄₁	—	2	2	no	—
131	C₄⋊C₄×C₄₁	—	2	1	no	—
132	C₄⋊C₄×C₄₁	—	2	1	no	—
133	C₄⋊C₄×C₄₁	—	2	1	no	—
134	C₄⋊C₄×C₄₁	—	2	1	no	—
135	C₄⋊C₄×C₄₁	—	2	1	no	—
136	C₄⋊C₄×C₄₁	—	2	1	no	—
137	C₄⋊C₄×C₄₁	—	4	4	no	—
138	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
139	C₄⋊C₄×C₄₁	#41	1	2	yes	—
140	C₄⋊C₄×C₄₁	#41	1	2	yes	—
141	C₂²⋊C₄×C₄₁	—	4	4	no	—
142	C₂²⋊C₄×C₄₁	—	4	4	no	—
143	C₂²⋊C₄×C₄₁	—	8	2	no	—
144	C₄²×C₄₁	—	2	2	no	—
145	C₄²×C₄₁	—	2	2	no	—
146	C₄²×C₄₁	—	4	4	no	—
147	C₄²×C₄₁	—	4	2	no	—
148	C₄²×C₄₁	—	4	2	no	—
149	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
150	C₄²×C₄₁	#41	1	2	yes	—
151	C₂³×C₄₁	—	2	4	no	—
152	C₂³×C₄₁	—	4	2	no	—
153	C₂³×C₄₁	—	4	2	no	—
154	C₂³×C₄₁	—	4	2	no	—
155	C₂³×C₄₁	—	4	2	no	—
156	C₂³×C₄₁	—	4	2	no	—
157	C₂³×C₄₁	—	4	2	no	—
158	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₄₁	—	4	8	no	—
160	D₄×C₄₁	—	2	4	no	—
161	D₄×C₄₁	—	2	4	no	—
162	D₄×C₄₁	—	4	2	no	—
163	D₄×C₄₁	—	4	2	no	—
164	D₄×C₄₁	—	4	8	no	—
165	D₄×C₄₁	—	8	4	no	—
166	C₂×C₄×C₄₁	—	2	1	no	—
167	C₂×C₄×C₄₁	—	2	1	no	—
168	C₂×C₄×C₄₁	—	2	4	no	—
169	C₂×C₄×C₄₁	—	2	2	no	—
170	C₂×C₄×C₄₁	—	2	2	no	—
171	C₂×C₄×C₄₁	—	2	2	no	—
172	C₂×C₄×C₄₁	—	2	2	no	—
173	C₂×C₄×C₄₁	—	2	2	no	—
174	C₂×C₄×C₄₁	—	2	2	no	—
175	C₂×C₄×C₄₁	—	2	2	no	—
176	C₂×C₄×C₄₁	—	2	2	no	—
177	C₂×C₄×C₄₁	—	4	1	no	—
178	C₂×C₄×C₄₁	—	4	2	no	—
179	C₂×C₄×C₄₁	—	4	1	no	—
180	C₂×C₄×C₄₁	—	4	2	no	—
181	C₂×C₄×C₄₁	—	4	1	no	—
182	C₂×C₄×C₄₁	—	4	1	no	—
183	C₂×C₄×C₄₁	—	4	1	no	—
184	C₂×C₄×C₄₁	—	4	1	no	—
185	C₂×C₄×C₄₁	—	4	1	no	—
186	C₂×C₄×C₄₁	—	4	1	no	—
187	C₂×C₄×C₄₁	—	4	1	no	—
188	C₂×C₄×C₄₁	—	4	1	no	—
189	C₂×C₄×C₄₁	—	8	8	no	—
190	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
191	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
192	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
193	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
194	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
199	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
200	C₈×C₄₁	—	2	2	no	—
201	C₈×C₄₁	—	2	2	no	—
202	C₈×C₄₁	—	4	2	no	—
203	#23220	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂²×C₄₁	—	4	4	no	—
205	C₂²×C₄₁	—	4	4	no	—
206	C₂²×C₄₁	—	4	4	no	—
207	C₂²×C₄₁	—	8	2	no	—
208	C₂²×C₄₁	—	8	2	no	—
209	C₂²×C₄₁	—	8	2	no	—
210	C₂²×C₄₁	—	8	2	no	—
211	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
212	C₂²×C₄₁	#546	1	1	yes	characteristic
213	C₂²×C₄₁	#548	1	1	yes	characteristic
214	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
215	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₂²×C₄₁	#577	1	1	yes	characteristic
217	C₂²×C₄₁	#577	1	1	yes	characteristic
218	C₄×C₄₁	—	2	2	no	—
219	C₄×C₄₁	—	4	2	no	—
220	C₄×C₄₁	—	4	2	no	—
221	C₄×C₄₁	—	4	4	no	—
222	C₄×C₄₁	—	4	2	no	—
223	C₄×C₄₁	—	4	2	no	—
224	C₄×C₄₁	—	8	1	no	—
225	C₄×C₄₁	—	8	1	no	—
226	C₄×C₄₁	#442	1	4	yes	—
227	C₄×C₄₁	C₄×#41	1	1	yes	characteristic
228	C₄×C₄₁	#520	1	1	yes	characteristic
229	C₄×C₄₁	#556	1	1	yes	characteristic
230	C₄×C₄₁	#559	1	1	yes	characteristic
231	C₂×#486	C₂×C₄₁	1	1	yes	characteristic
232	#2366	C₂×C₄₁	1	1	yes	characteristic
233	#2365	C₂×C₄₁	1	1	yes	characteristic, complemented
234	#1988	C₂×C₄₁	1	1	yes	characteristic, complemented
235	#1982	C₂×C₄₁	1	1	yes	characteristic
236	#1936	C₂×C₄₁	1	2	yes	complemented
237	#1926	C₂×C₄₁	1	2	yes	complemented
238	#1864	C₂×C₄₁	1	2	yes	complemented
239	C₄×#106	C₂×C₄₁	1	1	yes	characteristic, complemented
240	#1770	C₂×C₄₁	1	1	yes	characteristic, complemented
241	#1762	C₂×C₄₁	1	2	yes	complemented
242	C₂×C₄₁	—	8	4	no	—
243	C₂×C₄₁	#2367	1	1	yes	characteristic
244	C₂×C₄₁	C₂×#442	1	1	yes	characteristic
245	C₂×C₄₁	#2949	1	1	yes	characteristic
246	C₂×C₄₁	#3324	1	1	yes	characteristic
247	C₂×C₄₁	#3332	1	1	yes	characteristic
248	C₂×C₄₁	#3368	1	1	yes	characteristic
249	C₂×C₄₁	#3373	1	1	yes	characteristic
250	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
251	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
252	C₂×C₄×Q₈	—	2	1	no	—
253	C₂×C₄×D₄	—	2	1	no	—
254	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
255	#486	C₄×C₄₁	1	8	yes	complemented
256	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
257	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
258	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
259	C₂×#102	C₂²×C₄₁	1	2	yes	—
260	C₂×#102	C₂²×C₄₁	1	2	yes	—
261	C₂×#102	C₂²×C₄₁	1	2	yes	—
262	C₂×#102	C₂²×C₄₁	1	2	yes	—
263	C₂×#97	C₂²×C₄₁	1	2	yes	—
264	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
265	#391	—	2	1	no	—
266	#390	—	2	1	no	—
267	#390	C₂²×C₄₁	1	2	yes	—
268	#389	C₂²×C₄₁	1	2	yes	—
269	#387	—	2	2	no	—
270	#386	—	2	2	no	—
271	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
272	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
273	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
274	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
275	#340	C₂²×C₄₁	1	2	yes	—
276	#340	C₂²×C₄₁	1	2	yes	—
277	#340	C₂²×C₄₁	1	1	yes	characteristic
278	#340	C₂²×C₄₁	1	1	yes	characteristic
279	#336	C₂²×C₄₁	1	2	yes	—
280	C₄₁	#23220	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²×Q₈	—	2	1	no	—
282	C₂²×D₄	—	2	1	no	—
283	C₂²×D₄	—	2	1	no	—
284	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
285	C₂³×C₄	—	2	2	no	—
286	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
287	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
288	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
289	C₄⋊Q₈	—	2	2	no	—
290	C₄⋊Q₈	C₂×C₄×C₄₁	1	4	yes	complemented
291	#127	—	2	2	no	—
292	#127	C₂×C₄×C₄₁	1	4	yes	—
293	C₄×Q₈	—	4	4	no	—
294	C₄×D₄	—	2	4	no	—
295	C₄×D₄	—	4	2	no	—
296	C₄×D₄	—	4	4	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
300	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
304	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
306	C₂×C₂²⋊C₄	—	2	2	no	—
307	C₂×C₂²⋊C₄	—	2	2	no	—
308	C₂×C₂²⋊C₄	—	2	2	no	—
309	C₂×C₄²	—	2	1	no	—
310	C₂×C₄²	—	2	2	no	—
311	C₂×C₄²	—	2	1	no	—
312	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
313	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
314	#106	—	2	2	no	—
315	#106	—	2	2	no	—
316	C₄⋊C₈	—	2	2	no	—
317	#102	—	2	4	no	—
318	#102	—	4	2	no	—
319	#102	C₂×C₄×C₄₁	1	8	yes	complemented
320	#102	C₂×C₄×C₄₁	1	8	yes	complemented
321	#97	—	4	2	no	—
322	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	complemented
323	#95	—	2	1	no	—
324	#95	—	2	1	no	—
325	#95	—	2	2	no	—
326	#95	—	2	2	no	—
327	#95	D₄×C₄₁	1	2	yes	—
328	C₂⁴	—	2	2	no	—
329	C₂×Q₈	—	2	4	no	—
330	C₂×Q₈	—	4	2	no	—
331	C₂×Q₈	—	4	2	no	—
332	C₂×D₄	—	2	4	no	—
333	C₂×D₄	—	2	4	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	4	2	no	—
340	C₂×D₄	—	4	2	no	—
341	C₂×D₄	—	8	1	no	—
342	C₂×D₄	—	8	1	no	—
343	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	4	4	no	—
350	C₂²×C₄	—	4	4	no	—
351	C₂²×C₄	—	8	2	no	—
352	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
354	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
356	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
365	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
366	C₄⋊C₄	—	2	4	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	4	4	no	—
377	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
378	C₄⋊C₄	#41×C₄₁	1	2	yes	—
379	C₄⋊C₄	#41×C₄₁	1	2	yes	—
380	C₂²⋊C₄	—	4	4	no	—
381	C₂²⋊C₄	—	4	4	no	—
382	C₂²⋊C₄	—	8	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	2	2	no	—
385	C₄²	—	4	4	no	—
386	C₄²	—	4	2	no	—
387	C₄²	—	4	2	no	—
388	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
389	C₄²	#41×C₄₁	1	2	yes	—
390	C₂³	—	2	4	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	4	2	no	—
394	C₂³	—	4	2	no	—
395	C₂³	—	4	2	no	—
396	C₂³	—	4	2	no	—
397	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
398	Q₈	—	4	8	no	—
399	D₄	—	2	4	no	—
400	D₄	—	2	4	no	—
401	D₄	—	4	2	no	—
402	D₄	—	4	2	no	—
403	D₄	—	4	8	no	—
404	D₄	—	8	4	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	8	8	no	—
429	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
430	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
431	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
432	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
433	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
434	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
435	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
437	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
438	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
439	C₈	—	2	2	no	—
440	C₈	—	2	2	no	—
441	C₈	—	4	2	no	—
442	C₂²	—	4	4	no	—
443	C₂²	—	4	4	no	—
444	C₂²	—	4	4	no	—
445	C₂²	—	8	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	—	8	2	no	—
448	C₂²	—	8	2	no	—
449	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
450	C₂²	C₄₁×#546	1	1	yes	characteristic
451	C₂²	C₄₁×#548	1	1	yes	characteristic
452	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
453	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
454	C₂²	C₄₁×#577	1	1	yes	characteristic
455	C₂²	C₄₁×#577	1	1	yes	characteristic
456	C₄	—	2	2	no	—
457	C₄	—	4	2	no	—
458	C₄	—	4	2	no	—
459	C₄	—	4	2	no	—
460	C₄	—	4	2	no	—
461	C₄	—	4	4	no	—
462	C₄	—	8	1	no	—
463	C₄	—	8	1	no	—
464	C₄	C₄₁×#442	1	4	yes	—
465	C₄	C₄×#41×C₄₁	1	1	yes	characteristic
466	C₄	C₄₁×#520	1	1	yes	characteristic
467	C₄	C₄₁×#556	1	1	yes	characteristic
468	C₄	C₄₁×#559	1	1	yes	characteristic
469	C₂	—	8	4	no	—
470	C₂	C₄₁×#2367	1	1	yes	characteristic
471	C₂	C₂×C₄₁×#442	1	1	yes	characteristic
472	C₂	C₄₁×#2949	1	1	yes	characteristic
473	C₂	C₄₁×#3324	1	1	yes	characteristic
474	C₂	C₄₁×#3332	1	1	yes	characteristic
475	C₂	C₄₁×#3368	1	1	yes	characteristic
476	C₂	C₄₁×#3373	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	C₂×#127×C₄₁	C₂²
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
3. C₄₁×#2365	C₂	semidirect
4. C₄₁×#1988	C₂	semidirect
6. C₄₁×#1936	C₂	semidirect
7. C₄₁×#1926	C₂	semidirect
8. C₄₁×#1864	C₂	semidirect
9. C₄×#106×C₄₁	C₂	semidirect
10. C₄₁×#1770	C₂	semidirect
11. C₄₁×#1762	C₂	semidirect
17. C₄₁×#486	C₄	semidirect
51. C₄⋊Q₈×C₄₁	C₂×C₄	semidirect
80. #102×C₄₁	C₂×C₄	semidirect
81. #102×C₄₁	C₂×C₄	semidirect
83. C₄×C₈×C₄₁	C₂×C₄	semidirect
203. #23220	C₄₁	direct (N × H)
233. #2365	C₂×C₄₁	semidirect
234. #1988	C₂×C₄₁	semidirect
236. #1936	C₂×C₄₁	semidirect
237. #1926	C₂×C₄₁	semidirect
238. #1864	C₂×C₄₁	semidirect
239. C₄×#106	C₂×C₄₁	semidirect
240. #1770	C₂×C₄₁	semidirect
241. #1762	C₂×C₄₁	semidirect
255. #486	C₄×C₄₁	semidirect
280. C₄₁	#23220	direct (N × H)
290. C₄⋊Q₈	C₂×C₄×C₄₁	semidirect
319. #102	C₂×C₄×C₄₁	semidirect
320. #102	C₂×C₄×C₄₁	semidirect
322. C₄×C₈	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×#486	2
C₄₁×#2366	2
C₄₁×#2365	2
C₄₁×#1988	2
C₄₁×#1982	2
C₄₁×#1936	2
C₄₁×#1926	2
C₄₁×#1864	2
C₄×#106×C₄₁	2
C₄₁×#1770	2
C₄₁×#1762	2
#23220	41

Subgroups — C41×#23220 #23799421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23220 #23799421