Subgroups — C₄₁×#22077 #23798278

Order 10496.

824 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#485	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2296	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1942	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1941	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1933	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1933	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1920	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1918	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1873	C₂	1	1	yes	maximal, characteristic
11	C₄₁×#1865	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#1863	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄₁×#1854	C₂	1	1	yes	maximal, characteristic
14	C₄₁×#1825	C₂	1	1	yes	maximal, characteristic
15	C₄×C₂²⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
16	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
21	C₄₁×#485	C₄	1	8	yes	complemented
22	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
34	C₄₁×#391	C₂²	1	1	yes	characteristic
35	C₄₁×#389	C₂²	1	1	yes	characteristic
36	C₄₁×#389	C₂²	1	1	yes	characteristic
37	C₄₁×#389	C₂²	1	1	yes	characteristic
38	C₄₁×#389	C₂²	1	1	yes	characteristic
39	C₄₁×#389	C₂²	1	1	yes	characteristic
40	C₄₁×#387	—	2	1	no	—
41	C₄₁×#385	—	2	1	no	—
42	C₄₁×#385	—	2	1	no	—
43	C₄₁×#385	—	2	1	no	—
44	C₄₁×#380	C₂²	1	1	yes	characteristic
45	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
46	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
47	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
48	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
49	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
50	C₄₁×#340	C₂²	1	1	yes	characteristic
51	C₄₁×#340	C₂²	1	1	yes	characteristic
52	C₄₁×#340	C₂²	1	1	yes	characteristic
53	C₄₁×#340	C₂²	1	1	yes	characteristic
54	C₄₁×#340	C₂²	1	1	yes	characteristic
55	C₄₁×#340	C₂²	1	1	yes	characteristic
56	C₄₁×#340	C₂²	1	1	yes	characteristic
57	C₄₁×#340	C₂²	1	1	yes	characteristic
58	C₄₁×#336	C₂²	1	1	yes	characteristic
59	C₄₁×#336	C₂²	1	1	yes	characteristic
60	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
62	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
68	C₂²⋊Q₈×C₄₁	—	2	2	no	—
69	C₂²⋊Q₈×C₄₁	C₂×C₄	1	4	yes	—
70	C₄×Q₈×C₄₁	—	4	1	no	—
71	C₄×Q₈×C₄₁	—	4	1	no	—
72	#117×C₄₁	—	2	2	no	—
73	#117×C₄₁	—	4	1	no	—
74	#117×C₄₁	—	4	1	no	—
75	#117×C₄₁	C₂×C₄	1	4	yes	—
76	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
94	C₂×C₄²×C₄₁	—	2	1	no	—
95	C₂×C₄²×C₄₁	—	2	1	no	—
96	C₂×C₄²×C₄₁	—	2	1	no	—
97	C₂×C₄²×C₄₁	—	2	1	no	—
98	C₂×C₄²×C₄₁	—	4	1	no	—
99	C₂×C₄²×C₄₁	—	4	1	no	—
100	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
104	#107×C₄₁	—	4	1	no	—
105	#107×C₄₁	C₂×C₄	1	4	yes	complemented
106	#106×C₄₁	—	4	1	no	—
107	#106×C₄₁	C₂×C₄	1	4	yes	complemented
108	#103×C₄₁	—	4	1	no	—
109	#103×C₄₁	—	4	1	no	—
110	#103×C₄₁	C₂×C₄	1	4	yes	complemented
111	#103×C₄₁	C₂×C₄	1	4	yes	complemented
112	C₂²⋊C₈×C₄₁	—	2	2	no	—
113	C₂²⋊C₈×C₄₁	C₂×C₄	1	4	yes	—
114	C₄×C₈×C₄₁	—	4	1	no	—
115	C₄×C₈×C₄₁	—	4	1	no	—
116	#95×C₄₁	—	2	1	no	—
117	#95×C₄₁	—	2	1	no	—
118	#95×C₄₁	—	2	1	no	—
119	#95×C₄₁	—	2	1	no	—
120	#95×C₄₁	—	2	1	no	—
121	#95×C₄₁	—	2	1	no	—
122	#95×C₄₁	—	2	1	no	—
123	#95×C₄₁	—	2	1	no	—
124	#95×C₄₁	—	2	1	no	—
125	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Q₈×C₄₁	—	2	1	no	—
127	C₂×Q₈×C₄₁	—	2	1	no	—
128	C₂×Q₈×C₄₁	—	2	1	no	—
129	C₂×Q₈×C₄₁	—	2	1	no	—
130	C₂×Q₈×C₄₁	—	2	1	no	—
131	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
132	C₂²×C₄×C₄₁	—	2	1	no	—
133	C₂²×C₄×C₄₁	—	2	1	no	—
134	C₂²×C₄×C₄₁	—	2	1	no	—
135	C₂²×C₄×C₄₁	—	2	1	no	—
136	C₂²×C₄×C₄₁	—	2	1	no	—
137	C₂²×C₄×C₄₁	—	2	1	no	—
138	C₂²×C₄×C₄₁	—	2	1	no	—
139	C₂²×C₄×C₄₁	—	2	1	no	—
140	C₂²×C₄×C₄₁	—	2	2	no	—
141	C₂²×C₄×C₄₁	—	2	2	no	—
142	C₂²×C₄×C₄₁	—	2	2	no	—
143	C₂²×C₄×C₄₁	—	2	1	no	—
144	C₂²×C₄×C₄₁	—	2	1	no	—
145	C₂²×C₄×C₄₁	—	2	1	no	—
146	C₂²×C₄×C₄₁	—	2	1	no	—
147	C₂²×C₄×C₄₁	—	4	1	no	—
148	C₂²×C₄×C₄₁	—	4	1	no	—
149	C₂²×C₄×C₄₁	—	4	1	no	—
150	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
151	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₄₁	#41	1	2	yes	—
157	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
158	C₂×C₈×C₄₁	—	2	1	no	—
159	C₂×C₈×C₄₁	—	2	1	no	—
160	C₂×C₈×C₄₁	—	2	1	no	—
161	C₂×C₈×C₄₁	—	2	1	no	—
162	C₂×C₈×C₄₁	—	2	1	no	—
163	C₂×C₈×C₄₁	—	2	1	no	—
164	C₂×C₈×C₄₁	—	2	1	no	—
165	C₂×C₈×C₄₁	—	2	1	no	—
166	C₂×C₈×C₄₁	—	2	1	no	—
167	C₂×C₈×C₄₁	—	2	1	no	—
168	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
169	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
170	C₄⋊C₄×C₄₁	—	2	4	no	—
171	C₄⋊C₄×C₄₁	—	2	1	no	—
172	C₄⋊C₄×C₄₁	—	2	1	no	—
173	C₄⋊C₄×C₄₁	—	2	1	no	—
174	C₄⋊C₄×C₄₁	—	2	1	no	—
175	C₄⋊C₄×C₄₁	—	2	1	no	—
176	C₄⋊C₄×C₄₁	—	2	1	no	—
177	C₄⋊C₄×C₄₁	—	2	1	no	—
178	C₄⋊C₄×C₄₁	—	2	1	no	—
179	C₄⋊C₄×C₄₁	—	2	1	no	—
180	C₄⋊C₄×C₄₁	—	2	1	no	—
181	C₄⋊C₄×C₄₁	—	2	1	no	—
182	C₄⋊C₄×C₄₁	—	4	2	no	—
183	C₄⋊C₄×C₄₁	—	4	2	no	—
184	C₄⋊C₄×C₄₁	—	4	2	no	—
185	C₄⋊C₄×C₄₁	—	4	2	no	—
186	C₄⋊C₄×C₄₁	—	4	2	no	—
187	C₄⋊C₄×C₄₁	—	4	2	no	—
188	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
189	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
190	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
191	C₂²⋊C₄×C₄₁	—	2	4	no	—
192	C₂²⋊C₄×C₄₁	—	2	4	no	—
193	C₂²⋊C₄×C₄₁	—	4	2	no	—
194	C₂²⋊C₄×C₄₁	—	4	2	no	—
195	C₄²×C₄₁	—	2	4	no	—
196	C₄²×C₄₁	—	2	1	no	—
197	C₄²×C₄₁	—	2	1	no	—
198	C₄²×C₄₁	—	2	1	no	—
199	C₄²×C₄₁	—	2	1	no	—
200	C₄²×C₄₁	—	4	1	no	—
201	C₄²×C₄₁	—	4	1	no	—
202	C₄²×C₄₁	—	4	2	no	—
203	C₄²×C₄₁	—	4	2	no	—
204	C₄²×C₄₁	—	4	2	no	—
205	C₂³×C₄₁	—	2	1	no	—
206	C₂³×C₄₁	—	2	1	no	—
207	C₂³×C₄₁	—	2	1	no	—
208	C₂³×C₄₁	—	2	1	no	—
209	C₂³×C₄₁	—	2	1	no	—
210	C₂³×C₄₁	—	2	1	no	—
211	C₂³×C₄₁	C₄×D₄	1	2	yes	—
212	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
213	Q₈×C₄₁	—	2	2	no	—
214	Q₈×C₄₁	—	2	2	no	—
215	Q₈×C₄₁	—	4	1	no	—
216	Q₈×C₄₁	—	4	1	no	—
217	C₂×C₄×C₄₁	—	2	1	no	—
218	C₂×C₄×C₄₁	—	2	1	no	—
219	C₂×C₄×C₄₁	—	2	1	no	—
220	C₂×C₄×C₄₁	—	2	1	no	—
221	C₂×C₄×C₄₁	—	2	1	no	—
222	C₂×C₄×C₄₁	—	2	1	no	—
223	C₂×C₄×C₄₁	—	2	1	no	—
224	C₂×C₄×C₄₁	—	2	1	no	—
225	C₂×C₄×C₄₁	—	2	2	no	—
226	C₂×C₄×C₄₁	—	2	2	no	—
227	C₂×C₄×C₄₁	—	2	2	no	—
228	C₂×C₄×C₄₁	—	2	2	no	—
229	C₂×C₄×C₄₁	—	2	2	no	—
230	C₂×C₄×C₄₁	—	2	2	no	—
231	C₂×C₄×C₄₁	—	2	2	no	—
232	C₂×C₄×C₄₁	—	2	2	no	—
233	C₂×C₄×C₄₁	—	2	1	no	—
234	C₂×C₄×C₄₁	—	2	1	no	—
235	C₂×C₄×C₄₁	—	2	1	no	—
236	C₂×C₄×C₄₁	—	4	4	no	—
237	C₂×C₄×C₄₁	—	4	4	no	—
238	C₂×C₄×C₄₁	—	4	1	no	—
239	C₂×C₄×C₄₁	—	4	1	no	—
240	C₂×C₄×C₄₁	—	4	1	no	—
241	C₂×C₄×C₄₁	—	4	1	no	—
242	C₂×C₄×C₄₁	—	4	1	no	—
243	C₂×C₄×C₄₁	—	4	1	no	—
244	C₂×C₄×C₄₁	—	4	1	no	—
245	C₂×C₄×C₄₁	—	4	1	no	—
246	C₂×C₄×C₄₁	—	4	1	no	—
247	C₂×C₄×C₄₁	—	4	1	no	—
248	C₂×C₄×C₄₁	—	4	1	no	—
249	C₂×C₄×C₄₁	—	4	1	no	—
250	C₂×C₄×C₄₁	—	4	1	no	—
251	C₂×C₄×C₄₁	—	4	1	no	—
252	C₂×C₄×C₄₁	—	4	1	no	—
253	C₂×C₄×C₄₁	—	4	1	no	—
254	C₂×C₄×C₄₁	—	4	1	no	—
255	C₂×C₄×C₄₁	—	4	1	no	—
256	C₂×C₄×C₄₁	—	4	1	no	—
257	C₂×C₄×C₄₁	—	4	1	no	—
258	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
259	C₂×C₄×C₄₁	#135	1	2	yes	—
260	C₂×C₄×C₄₁	#135	1	2	yes	—
261	C₂×C₄×C₄₁	#137	1	2	yes	—
262	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
263	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
264	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
265	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
266	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
267	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
268	C₈×C₄₁	—	2	2	no	—
269	C₈×C₄₁	—	2	2	no	—
270	C₈×C₄₁	—	4	1	no	—
271	C₈×C₄₁	—	4	1	no	—
272	#22077	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
273	C₂²×C₄₁	—	2	2	no	—
274	C₂²×C₄₁	—	2	2	no	—
275	C₂²×C₄₁	—	2	2	no	—
276	C₂²×C₄₁	—	4	1	no	—
277	C₂²×C₄₁	—	4	1	no	—
278	C₂²×C₄₁	—	4	1	no	—
279	C₂²×C₄₁	—	4	1	no	—
280	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
281	C₂²×C₄₁	#534	1	1	yes	characteristic
282	C₂²×C₄₁	#536	1	1	yes	characteristic
283	C₂²×C₄₁	C₂×#135	1	1	yes	characteristic
284	C₂²×C₄₁	C₂×#135	1	1	yes	characteristic
285	C₂²×C₄₁	C₂×#137	1	1	yes	characteristic
286	C₂²×C₄₁	#575	1	1	yes	characteristic
287	C₄×C₄₁	—	2	2	no	—
288	C₄×C₄₁	—	2	2	no	—
289	C₄×C₄₁	—	2	2	no	—
290	C₄×C₄₁	—	2	2	no	—
291	C₄×C₄₁	—	4	2	no	—
292	C₄×C₄₁	—	4	1	no	—
293	C₄×C₄₁	—	4	2	no	—
294	C₄×C₄₁	—	4	2	no	—
295	C₄×C₄₁	—	4	2	no	—
296	C₄×C₄₁	—	8	1	no	—
297	C₄×C₄₁	—	8	1	no	—
298	C₄×C₄₁	—	8	1	no	—
299	C₄×C₄₁	—	8	1	no	—
300	C₄×C₄₁	#520	1	1	yes	characteristic
301	C₄×C₄₁	#520	1	1	yes	characteristic
302	C₄×C₄₁	#539	1	1	yes	characteristic
303	C₄×C₄₁	#539	1	1	yes	characteristic
304	C₂×#485	C₂×C₄₁	1	1	yes	characteristic
305	C₄×C₂²⋊Q₈	C₂×C₄₁	1	1	yes	characteristic
306	#2296	C₂×C₄₁	1	1	yes	characteristic
307	#1942	C₂×C₄₁	1	1	yes	characteristic, complemented
308	#1941	C₂×C₄₁	1	1	yes	characteristic, complemented
309	#1933	C₂×C₄₁	1	1	yes	characteristic, complemented
310	#1933	C₂×C₄₁	1	1	yes	characteristic, complemented
311	#1920	C₂×C₄₁	1	1	yes	characteristic, complemented
312	#1918	C₂×C₄₁	1	1	yes	characteristic, complemented
313	#1873	C₂×C₄₁	1	1	yes	characteristic
314	#1865	C₂×C₄₁	1	1	yes	characteristic, complemented
315	#1863	C₂×C₄₁	1	1	yes	characteristic, complemented
316	#1854	C₂×C₄₁	1	1	yes	characteristic
317	#1825	C₂×C₄₁	1	1	yes	characteristic
318	C₄×C₂²⋊C₈	C₂×C₄₁	1	1	yes	characteristic
319	C₂×C₄₁	—	4	2	no	—
320	C₂×C₄₁	#2322	1	1	yes	characteristic
321	C₂×C₄₁	C₄×#135	1	1	yes	characteristic
322	C₂×C₄₁	C₄×#137	1	1	yes	characteristic
323	C₂×C₄₁	#3190	1	1	yes	characteristic
324	C₂×C₄₁	#3191	1	1	yes	characteristic
325	C₂×C₄₁	#3231	1	1	yes	characteristic
326	C₂×C₄₁	#3233	1	1	yes	characteristic
327	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
328	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
329	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
330	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
331	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
332	#485	C₄×C₄₁	1	8	yes	complemented
333	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
334	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
335	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
336	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
337	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
338	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
339	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
340	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
341	C₂×C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
342	C₂×C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
343	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
344	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
345	#391	C₂²×C₄₁	1	1	yes	characteristic
346	#389	C₂²×C₄₁	1	1	yes	characteristic
347	#389	C₂²×C₄₁	1	1	yes	characteristic
348	#389	C₂²×C₄₁	1	1	yes	characteristic
349	#389	C₂²×C₄₁	1	1	yes	characteristic
350	#389	C₂²×C₄₁	1	1	yes	characteristic
351	#387	—	2	1	no	—
352	#385	—	2	1	no	—
353	#385	—	2	1	no	—
354	#385	—	2	1	no	—
355	#380	C₂²×C₄₁	1	1	yes	characteristic
356	C₄×C₄⋊C₄	—	2	1	no	—
357	C₄×C₄⋊C₄	—	2	1	no	—
358	C₄×C₄⋊C₄	—	2	1	no	—
359	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
360	C₄×C₂²⋊C₄	—	2	1	no	—
361	#340	C₂²×C₄₁	1	1	yes	characteristic
362	#340	C₂²×C₄₁	1	1	yes	characteristic
363	#340	C₂²×C₄₁	1	1	yes	characteristic
364	#340	C₂²×C₄₁	1	1	yes	characteristic
365	#340	C₂²×C₄₁	1	1	yes	characteristic
366	#340	C₂²×C₄₁	1	1	yes	characteristic
367	#340	C₂²×C₄₁	1	1	yes	characteristic
368	#340	C₂²×C₄₁	1	1	yes	characteristic
369	#336	C₂²×C₄₁	1	1	yes	characteristic
370	#336	C₂²×C₄₁	1	1	yes	characteristic
371	C₄₁	#22077	1	1	yes	characteristic, direct product factor, Sylow
372	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
373	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
374	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
375	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
376	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
377	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
378	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
379	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
380	C₂²⋊Q₈	—	2	2	no	—
381	C₂²⋊Q₈	C₂×C₄×C₄₁	1	4	yes	—
382	C₄×Q₈	—	4	1	no	—
383	C₄×Q₈	—	4	1	no	—
384	#117	—	2	2	no	—
385	#117	—	4	1	no	—
386	#117	—	4	1	no	—
387	#117	C₂×C₄×C₄₁	1	4	yes	—
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	C₂×C₄⋊C₄	—	2	1	no	—
394	C₂×C₄⋊C₄	—	2	1	no	—
395	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
397	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
398	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
400	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
402	C₂×C₂²⋊C₄	—	2	1	no	—
403	C₂×C₂²⋊C₄	—	2	1	no	—
404	C₂×C₂²⋊C₄	—	2	1	no	—
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₄²	—	2	1	no	—
407	C₂×C₄²	—	2	1	no	—
408	C₂×C₄²	—	2	1	no	—
409	C₂×C₄²	—	2	1	no	—
410	C₂×C₄²	—	4	1	no	—
411	C₂×C₄²	—	4	1	no	—
412	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
413	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
414	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
415	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
416	#107	—	4	1	no	—
417	#107	C₂×C₄×C₄₁	1	4	yes	complemented
418	#106	—	4	1	no	—
419	#106	C₂×C₄×C₄₁	1	4	yes	complemented
420	#103	—	4	1	no	—
421	#103	—	4	1	no	—
422	#103	C₂×C₄×C₄₁	1	4	yes	complemented
423	#103	C₂×C₄×C₄₁	1	4	yes	complemented
424	C₂²⋊C₈	—	2	2	no	—
425	C₂²⋊C₈	C₂×C₄×C₄₁	1	4	yes	—
426	C₄×C₈	—	4	1	no	—
427	C₄×C₈	—	4	1	no	—
428	#95	—	2	1	no	—
429	#95	—	2	1	no	—
430	#95	—	2	1	no	—
431	#95	—	2	1	no	—
432	#95	—	2	1	no	—
433	#95	—	2	1	no	—
434	#95	—	2	1	no	—
435	#95	—	2	1	no	—
436	#95	—	2	1	no	—
437	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	4	1	no	—
460	C₂²×C₄	—	4	1	no	—
461	C₂²×C₄	—	4	1	no	—
462	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
463	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
465	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
467	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
468	C₂²×C₄	#41×C₄₁	1	2	yes	—
469	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
481	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	4	2	no	—
495	C₄⋊C₄	—	4	2	no	—
496	C₄⋊C₄	—	4	2	no	—
497	C₄⋊C₄	—	4	2	no	—
498	C₄⋊C₄	—	4	2	no	—
499	C₄⋊C₄	—	4	2	no	—
500	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
501	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
502	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	4	2	no	—
506	C₂²⋊C₄	—	4	2	no	—
507	C₄²	—	2	4	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	2	1	no	—
510	C₄²	—	2	1	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	4	1	no	—
513	C₄²	—	4	1	no	—
514	C₄²	—	4	2	no	—
515	C₄²	—	4	2	no	—
516	C₄²	—	4	2	no	—
517	C₂³	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	2	1	no	—
521	C₂³	—	2	1	no	—
522	C₂³	—	2	1	no	—
523	C₂³	C₄×D₄×C₄₁	1	2	yes	—
524	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
525	Q₈	—	2	2	no	—
526	Q₈	—	2	2	no	—
527	Q₈	—	4	1	no	—
528	Q₈	—	4	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	4	4	no	—
549	C₂×C₄	—	4	4	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
571	C₂×C₄	#135×C₄₁	1	2	yes	—
572	C₂×C₄	#135×C₄₁	1	2	yes	—
573	C₂×C₄	#137×C₄₁	1	2	yes	—
574	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
575	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
576	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
577	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
578	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
579	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
580	C₈	—	2	2	no	—
581	C₈	—	2	2	no	—
582	C₈	—	4	1	no	—
583	C₈	—	4	1	no	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	4	1	no	—
588	C₂²	—	4	1	no	—
589	C₂²	—	4	1	no	—
590	C₂²	—	4	1	no	—
591	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
592	C₂²	C₄₁×#534	1	1	yes	characteristic
593	C₂²	C₄₁×#536	1	1	yes	characteristic
594	C₂²	C₂×#135×C₄₁	1	1	yes	characteristic
595	C₂²	C₂×#135×C₄₁	1	1	yes	characteristic
596	C₂²	C₂×#137×C₄₁	1	1	yes	characteristic
597	C₂²	C₄₁×#575	1	1	yes	characteristic
598	C₄	—	2	2	no	—
599	C₄	—	2	2	no	—
600	C₄	—	2	2	no	—
601	C₄	—	2	2	no	—
602	C₄	—	4	2	no	—
603	C₄	—	4	1	no	—
604	C₄	—	4	2	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	2	no	—
607	C₄	—	8	1	no	—
608	C₄	—	8	1	no	—
609	C₄	—	8	1	no	—
610	C₄	—	8	1	no	—
611	C₄	C₄₁×#520	1	1	yes	characteristic
612	C₄	C₄₁×#520	1	1	yes	characteristic
613	C₄	C₄₁×#539	1	1	yes	characteristic
614	C₄	C₄₁×#539	1	1	yes	characteristic
615	C₂	—	4	2	no	—
616	C₂	C₄₁×#2322	1	1	yes	characteristic
617	C₂	C₄×#135×C₄₁	1	1	yes	characteristic
618	C₂	C₄×#137×C₄₁	1	1	yes	characteristic
619	C₂	C₄₁×#3190	1	1	yes	characteristic
620	C₂	C₄₁×#3191	1	1	yes	characteristic
621	C₂	C₄₁×#3231	1	1	yes	characteristic
622	C₂	C₄₁×#3233	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	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1942	C₂	semidirect
5. C₄₁×#1941	C₂	semidirect
6. C₄₁×#1933	C₂	semidirect
7. C₄₁×#1933	C₂	semidirect
8. C₄₁×#1920	C₂	semidirect
9. C₄₁×#1918	C₂	semidirect
11. C₄₁×#1865	C₂	semidirect
12. C₄₁×#1863	C₂	semidirect
21. C₄₁×#485	C₄	semidirect
105. #107×C₄₁	C₂×C₄	semidirect
107. #106×C₄₁	C₂×C₄	semidirect
110. #103×C₄₁	C₂×C₄	semidirect
111. #103×C₄₁	C₂×C₄	semidirect
272. #22077	C₄₁	direct (N × H)
307. #1942	C₂×C₄₁	semidirect
308. #1941	C₂×C₄₁	semidirect
309. #1933	C₂×C₄₁	semidirect
310. #1933	C₂×C₄₁	semidirect
311. #1920	C₂×C₄₁	semidirect
312. #1918	C₂×C₄₁	semidirect
314. #1865	C₂×C₄₁	semidirect
315. #1863	C₂×C₄₁	semidirect
332. #485	C₄×C₄₁	semidirect
371. C₄₁	#22077	direct (N × H)
417. #107	C₂×C₄×C₄₁	semidirect
419. #106	C₂×C₄×C₄₁	semidirect
422. #103	C₂×C₄×C₄₁	semidirect
423. #103	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×#485	2
C₄×C₂²⋊Q₈×C₄₁	2
C₄₁×#2296	2
C₄₁×#1942	2
C₄₁×#1941	2
C₄₁×#1933	2
C₄₁×#1920	2
C₄₁×#1918	2
C₄₁×#1873	2
C₄₁×#1865	2
C₄₁×#1863	2
C₄₁×#1854	2
C₄₁×#1825	2
C₄×C₂²⋊C₈×C₄₁	2
#22077	41

Subgroups — C41×#22077 #23798278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22077 #23798278