Subgroups — C₄₁×#26079 #23802280

Order 10496.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#501	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#428	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2442	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2439	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#2085	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2085	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#2059	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#2057	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1943	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1942	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1854	C₂	1	1	yes	maximal, characteristic
12	C₄₁×#1852	C₂	1	1	yes	maximal, characteristic
13	C₄₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄₁×#1809	C₂	1	1	yes	maximal, characteristic
15	C₄₁×#1753	C₂	1	1	yes	maximal, characteristic
16	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×#125×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₄₁×#501	—	2	4	no	—
22	C₄₁×#428	—	2	4	no	—
23	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
34	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
36	C₄₁×#400	—	2	1	no	—
37	C₄₁×#398	—	2	1	no	—
38	C₄₁×#396	—	2	1	no	—
39	C₄₁×#393	—	2	1	no	—
40	C₄₁×#391	—	2	1	no	—
41	C₄₁×#389	C₂²	1	1	yes	characteristic
42	C₄₁×#389	C₂²	1	1	yes	characteristic
43	C₄₁×#389	C₂²	1	1	yes	characteristic
44	C₄₁×#389	C₂²	1	1	yes	characteristic
45	C₄₁×#387	—	2	1	no	—
46	C₄₁×#385	—	2	1	no	—
47	C₄₁×#382	C₂²	1	1	yes	characteristic
48	C₄₁×#382	C₂²	1	1	yes	characteristic
49	C₄₁×#380	—	2	1	no	—
50	C₄₁×#380	C₂²	1	1	yes	characteristic
51	C₄₁×#380	C₂²	1	1	yes	characteristic
52	C₄×C₂²⋊C₄×C₄₁	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×#340	C₂²	1	1	yes	characteristic
59	C₄₁×#336	C₂²	1	1	yes	characteristic
60	C₄₁×#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₄₁	—	2	1	no	—
62	C₂³×C₄×C₄₁	—	2	1	no	—
63	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
64	C₂×M₁₆×C₄₁	—	2	2	no	—
65	C₂×M₁₆×C₄₁	—	2	2	no	—
66	C₂×M₁₆×C₄₁	Q₈	1	4	yes	complemented
67	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₄₁	—	2	2	no	—
72	C₄⋊Q₈×C₄₁	—	4	1	no	—
73	#125×C₄₁	—	2	2	no	—
74	#125×C₄₁	—	4	1	no	—
75	#117×C₄₁	—	2	2	no	—
76	#117×C₄₁	—	4	1	no	—
77	C₂×C₄⋊C₄×C₄₁	—	2	2	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	—	2	1	no	—
84	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
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×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
95	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₄₁	D₄	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₄₁	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₄₁	—	2	1	no	—
101	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
103	#107×C₄₁	—	2	2	no	—
104	#107×C₄₁	—	2	2	no	—
105	#107×C₄₁	—	2	2	no	—
106	#107×C₄₁	—	4	1	no	—
107	#107×C₄₁	—	4	1	no	—
108	#107×C₄₁	D₄	1	4	yes	—
109	#106×C₄₁	—	2	2	no	—
110	#106×C₄₁	—	2	2	no	—
111	#106×C₄₁	—	2	2	no	—
112	#106×C₄₁	—	2	2	no	—
113	C₂²⋊C₈×C₄₁	Q₈	1	4	yes	complemented
114	#97×C₄₁	—	2	2	no	—
115	C₄×C₈×C₄₁	—	2	2	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	#95×C₄₁	—	2	1	no	—
126	#95×C₄₁	D₄	1	1	yes	characteristic
127	#95×C₄₁	D₄	1	1	yes	characteristic
128	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
129	C₂×Q₈×C₄₁	—	4	2	no	—
130	C₂×Q₈×C₄₁	—	4	2	no	—
131	C₂×Q₈×C₄₁	—	8	1	no	—
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	1	no	—
141	C₂²×C₄×C₄₁	—	2	1	no	—
142	C₂²×C₄×C₄₁	—	2	1	no	—
143	C₂²×C₄×C₄₁	—	2	1	no	—
144	C₂²×C₄×C₄₁	—	4	2	no	—
145	C₂²×C₄×C₄₁	—	4	2	no	—
146	C₂²×C₄×C₄₁	—	8	1	no	—
147	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
153	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
154	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
155	M₁₆×C₄₁	—	2	8	no	—
156	C₂×C₈×C₄₁	—	2	1	no	—
157	C₂×C₈×C₄₁	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
163	C₂×C₈×C₄₁	C₂×Q₈	1	2	yes	—
164	C₂×C₈×C₄₁	C₂×Q₈	1	2	yes	—
165	C₂×C₈×C₄₁	C₂×Q₈	1	2	yes	—
166	C₂×C₈×C₄₁	C₂×Q₈	1	2	yes	—
167	C₂×C₈×C₄₁	#41	1	2	yes	—
168	C₄⋊C₄×C₄₁	—	2	1	no	—
169	C₄⋊C₄×C₄₁	—	2	1	no	—
170	C₄⋊C₄×C₄₁	—	2	1	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	—	8	1	no	—
188	C₄⋊C₄×C₄₁	—	8	1	no	—
189	C₄⋊C₄×C₄₁	—	8	1	no	—
190	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
191	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
192	C₂²⋊C₄×C₄₁	—	4	1	no	—
193	C₂²⋊C₄×C₄₁	—	4	1	no	—
194	C₂²⋊C₄×C₄₁	—	4	2	no	—
195	C₂²⋊C₄×C₄₁	—	4	2	no	—
196	C₂²⋊C₄×C₄₁	—	8	1	no	—
197	C₄²×C₄₁	—	2	1	no	—
198	C₄²×C₄₁	—	2	1	no	—
199	C₄²×C₄₁	—	2	1	no	—
200	C₄²×C₄₁	—	2	1	no	—
201	C₄²×C₄₁	—	4	2	no	—
202	C₂³×C₄₁	—	2	1	no	—
203	C₂³×C₄₁	—	2	1	no	—
204	C₂³×C₄₁	—	2	1	no	—
205	C₂³×C₄₁	—	2	1	no	—
206	C₂³×C₄₁	—	2	1	no	—
207	C₂³×C₄₁	—	2	1	no	—
208	C₂³×C₄₁	C₄⋊Q₈	1	2	yes	—
209	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
210	Q₈×C₄₁	—	8	4	no	—
211	C₂×C₄×C₄₁	—	2	1	no	—
212	C₂×C₄×C₄₁	—	2	1	no	—
213	C₂×C₄×C₄₁	—	2	2	no	—
214	C₂×C₄×C₄₁	—	2	2	no	—
215	C₂×C₄×C₄₁	—	2	2	no	—
216	C₂×C₄×C₄₁	—	2	2	no	—
217	C₂×C₄×C₄₁	—	2	1	no	—
218	C₂×C₄×C₄₁	—	2	1	no	—
219	C₂×C₄×C₄₁	—	2	2	no	—
220	C₂×C₄×C₄₁	—	2	2	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	1	no	—
226	C₂×C₄×C₄₁	—	2	1	no	—
227	C₂×C₄×C₄₁	—	2	1	no	—
228	C₂×C₄×C₄₁	—	2	1	no	—
229	C₂×C₄×C₄₁	—	2	1	no	—
230	C₂×C₄×C₄₁	—	2	1	no	—
231	C₂×C₄×C₄₁	—	4	1	no	—
232	C₂×C₄×C₄₁	—	4	1	no	—
233	C₂×C₄×C₄₁	—	4	1	no	—
234	C₂×C₄×C₄₁	—	4	1	no	—
235	C₂×C₄×C₄₁	—	4	1	no	—
236	C₂×C₄×C₄₁	—	4	1	no	—
237	C₂×C₄×C₄₁	—	4	1	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	—	8	4	no	—
254	C₂×C₄×C₄₁	C₄⋊Q₈	1	2	yes	—
255	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
257	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
258	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
259	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
260	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
261	C₈×C₄₁	—	2	2	no	—
262	C₈×C₄₁	—	2	2	no	—
263	C₈×C₄₁	—	2	2	no	—
264	C₈×C₄₁	C₂²⋊Q₈	1	4	yes	—
265	#26079	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂²×C₄₁	—	2	2	no	—
267	C₂²×C₄₁	—	2	2	no	—
268	C₂²×C₄₁	—	2	2	no	—
269	C₂²×C₄₁	—	4	1	no	—
270	C₂²×C₄₁	—	4	1	no	—
271	C₂²×C₄₁	—	4	1	no	—
272	C₂²×C₄₁	—	4	1	no	—
273	C₂²×C₄₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
274	C₂²×C₄₁	D₄²	1	1	yes	characteristic
275	C₂²×C₄₁	#547	1	1	yes	characteristic
276	C₂²×C₄₁	#575	1	1	yes	characteristic
277	C₂²×C₄₁	#575	1	1	yes	characteristic
278	C₂²×C₄₁	#576	1	1	yes	characteristic
279	C₂²×C₄₁	#578	1	1	yes	characteristic
280	C₄×C₄₁	—	2	2	no	—
281	C₄×C₄₁	—	4	1	no	—
282	C₄×C₄₁	—	4	1	no	—
283	C₄×C₄₁	—	4	1	no	—
284	C₄×C₄₁	—	4	2	no	—
285	C₄×C₄₁	—	4	2	no	—
286	C₄×C₄₁	—	4	1	no	—
287	C₄×C₄₁	—	8	1	no	—
288	C₄×C₄₁	—	8	1	no	—
289	C₄×C₄₁	—	8	1	no	—
290	C₄×C₄₁	—	8	1	no	—
291	C₄×C₄₁	—	8	1	no	—
292	C₄×C₄₁	—	8	1	no	—
293	C₄×C₄₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
294	C₄×C₄₁	#544	1	1	yes	characteristic
295	C₄×C₄₁	D₄×Q₈	1	1	yes	characteristic
296	C₄×C₄₁	#554	1	1	yes	characteristic
297	C₂×#501	C₂×C₄₁	1	1	yes	characteristic, complemented
298	C₂×#428	C₂×C₄₁	1	1	yes	characteristic
299	#2442	C₂×C₄₁	1	1	yes	characteristic
300	#2439	C₂×C₄₁	1	1	yes	characteristic
301	#2085	C₂×C₄₁	1	1	yes	characteristic, complemented
302	#2085	C₂×C₄₁	1	1	yes	characteristic, complemented
303	#2059	C₂×C₄₁	1	1	yes	characteristic, complemented
304	#2057	C₂×C₄₁	1	1	yes	characteristic, complemented
305	#1943	C₂×C₄₁	1	1	yes	characteristic, complemented
306	#1942	C₂×C₄₁	1	1	yes	characteristic, complemented
307	#1854	C₂×C₄₁	1	1	yes	characteristic
308	#1852	C₂×C₄₁	1	1	yes	characteristic
309	#1846	C₂×C₄₁	1	1	yes	characteristic, complemented
310	#1809	C₂×C₄₁	1	1	yes	characteristic
311	#1753	C₂×C₄₁	1	1	yes	characteristic
312	C₂×C₄₁	—	4	2	no	—
313	C₂×C₄₁	#2440	1	1	yes	characteristic
314	C₂×C₄₁	#3164	1	1	yes	characteristic
315	C₂×C₄₁	#3166	1	1	yes	characteristic
316	C₂×C₄₁	#3273	1	1	yes	characteristic
317	C₂×C₄₁	#3277	1	1	yes	characteristic
318	C₂×C₄₁	#3347	1	1	yes	characteristic
319	C₂×C₄₁	#3349	1	1	yes	characteristic
320	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
321	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
322	C₂×#125	C₂²×C₄₁	1	1	yes	characteristic
323	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
324	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
325	#501	—	2	4	no	—
326	#428	—	2	4	no	—
327	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
328	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
329	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
330	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
331	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
332	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
333	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
334	C₂×#106	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×C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
338	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
339	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
340	#400	—	2	1	no	—
341	#398	—	2	1	no	—
342	#396	—	2	1	no	—
343	#393	—	2	1	no	—
344	#391	—	2	1	no	—
345	#389	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	#387	—	2	1	no	—
350	#385	—	2	1	no	—
351	#382	C₂²×C₄₁	1	1	yes	characteristic
352	#382	C₂²×C₄₁	1	1	yes	characteristic
353	#380	—	2	1	no	—
354	#380	C₂²×C₄₁	1	1	yes	characteristic
355	#380	C₂²×C₄₁	1	1	yes	characteristic
356	C₄×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
357	#340	C₂²×C₄₁	1	1	yes	characteristic
358	#340	C₂²×C₄₁	1	1	yes	characteristic
359	#340	C₂²×C₄₁	1	1	yes	characteristic
360	#340	C₂²×C₄₁	1	1	yes	characteristic
361	#340	C₂²×C₄₁	1	1	yes	characteristic
362	#340	C₂²×C₄₁	1	1	yes	characteristic
363	#336	C₂²×C₄₁	1	1	yes	characteristic
364	#336	C₂²×C₄₁	1	1	yes	characteristic
365	C₄₁	#26079	1	1	yes	characteristic, direct product factor, Sylow
366	C₂²×Q₈	—	2	1	no	—
367	C₂³×C₄	—	2	1	no	—
368	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
369	C₂×M₁₆	—	2	2	no	—
370	C₂×M₁₆	—	2	2	no	—
371	C₂×M₁₆	Q₈×C₄₁	1	4	yes	complemented
372	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
373	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
374	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
375	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
376	C₄⋊Q₈	—	2	2	no	—
377	C₄⋊Q₈	—	4	1	no	—
378	#125	—	2	2	no	—
379	#125	—	4	1	no	—
380	#117	—	2	2	no	—
381	#117	—	4	1	no	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
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	—	4	1	no	—
392	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
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×C₂²⋊C₄	—	2	1	no	—
401	C₂×C₂²⋊C₄	—	2	1	no	—
402	C₂×C₂²⋊C₄	—	2	1	no	—
403	C₂×C₂²⋊C₄	D₄×C₄₁	1	1	yes	characteristic
404	C₂×C₂²⋊C₄	D₄×C₄₁	1	1	yes	characteristic
405	C₂×C₄²	—	2	1	no	—
406	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
408	#107	—	2	2	no	—
409	#107	—	2	2	no	—
410	#107	—	2	2	no	—
411	#107	—	4	1	no	—
412	#107	—	4	1	no	—
413	#107	D₄×C₄₁	1	4	yes	—
414	#106	—	2	2	no	—
415	#106	—	2	2	no	—
416	#106	—	2	2	no	—
417	#106	—	2	2	no	—
418	C₂²⋊C₈	Q₈×C₄₁	1	4	yes	complemented
419	#97	—	2	2	no	—
420	C₄×C₈	—	2	2	no	—
421	#95	—	2	1	no	—
422	#95	—	2	1	no	—
423	#95	—	2	1	no	—
424	#95	—	2	1	no	—
425	#95	—	2	1	no	—
426	#95	—	2	1	no	—
427	#95	—	2	1	no	—
428	#95	—	2	1	no	—
429	#95	—	2	1	no	—
430	#95	—	2	1	no	—
431	#95	D₄×C₄₁	1	1	yes	characteristic
432	#95	D₄×C₄₁	1	1	yes	characteristic
433	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
434	C₂×Q₈	—	4	2	no	—
435	C₂×Q₈	—	4	2	no	—
436	C₂×Q₈	—	8	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
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	—	4	2	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	—	8	1	no	—
452	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
457	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
458	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
459	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
460	M₁₆	—	2	8	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
468	C₂×C₈	C₂×Q₈×C₄₁	1	2	yes	—
469	C₂×C₈	C₂×Q₈×C₄₁	1	2	yes	—
470	C₂×C₈	C₂×Q₈×C₄₁	1	2	yes	—
471	C₂×C₈	C₂×Q₈×C₄₁	1	2	yes	—
472	C₂×C₈	#41×C₄₁	1	2	yes	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	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	—	4	2	no	—
488	C₄⋊C₄	—	4	2	no	—
489	C₄⋊C₄	—	4	2	no	—
490	C₄⋊C₄	—	4	2	no	—
491	C₄⋊C₄	—	4	2	no	—
492	C₄⋊C₄	—	8	1	no	—
493	C₄⋊C₄	—	8	1	no	—
494	C₄⋊C₄	—	8	1	no	—
495	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
496	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
497	C₂²⋊C₄	—	4	1	no	—
498	C₂²⋊C₄	—	4	1	no	—
499	C₂²⋊C₄	—	4	2	no	—
500	C₂²⋊C₄	—	4	2	no	—
501	C₂²⋊C₄	—	8	1	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	4	2	no	—
507	C₂³	—	2	1	no	—
508	C₂³	—	2	1	no	—
509	C₂³	—	2	1	no	—
510	C₂³	—	2	1	no	—
511	C₂³	—	2	1	no	—
512	C₂³	—	2	1	no	—
513	C₂³	C₄⋊Q₈×C₄₁	1	2	yes	—
514	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
515	Q₈	—	8	4	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	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	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	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	—	8	4	no	—
559	C₂×C₄	C₄⋊Q₈×C₄₁	1	2	yes	—
560	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
561	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
562	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
563	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
564	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
565	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
566	C₈	—	2	2	no	—
567	C₈	—	2	2	no	—
568	C₈	—	2	2	no	—
569	C₈	C₂²⋊Q₈×C₄₁	1	4	yes	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	4	1	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	1	no	—
577	C₂²	C₂×C₄⋊Q₈×C₄₁	1	1	yes	characteristic
578	C₂²	D₄²×C₄₁	1	1	yes	characteristic
579	C₂²	C₄₁×#547	1	1	yes	characteristic
580	C₂²	C₄₁×#575	1	1	yes	characteristic
581	C₂²	C₄₁×#575	1	1	yes	characteristic
582	C₂²	C₄₁×#576	1	1	yes	characteristic
583	C₂²	C₄₁×#578	1	1	yes	characteristic
584	C₄	—	2	2	no	—
585	C₄	—	4	1	no	—
586	C₄	—	4	1	no	—
587	C₄	—	4	1	no	—
588	C₄	—	4	2	no	—
589	C₄	—	4	2	no	—
590	C₄	—	4	1	no	—
591	C₄	—	8	1	no	—
592	C₄	—	8	1	no	—
593	C₄	—	8	1	no	—
594	C₄	—	8	1	no	—
595	C₄	—	8	1	no	—
596	C₄	—	8	1	no	—
597	C₄	C₂×C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
598	C₄	C₄₁×#544	1	1	yes	characteristic
599	C₄	D₄×Q₈×C₄₁	1	1	yes	characteristic
600	C₄	C₄₁×#554	1	1	yes	characteristic
601	C₂	—	4	2	no	—
602	C₂	C₄₁×#2440	1	1	yes	characteristic
603	C₂	C₄₁×#3164	1	1	yes	characteristic
604	C₂	C₄₁×#3166	1	1	yes	characteristic
605	C₂	C₄₁×#3273	1	1	yes	characteristic
606	C₂	C₄₁×#3277	1	1	yes	characteristic
607	C₂	C₄₁×#3347	1	1	yes	characteristic
608	C₂	C₄₁×#3349	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂²×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
1. C₂×C₄₁×#501	C₂	semidirect
5. C₄₁×#2085	C₂	semidirect
6. C₄₁×#2085	C₂	semidirect
7. C₄₁×#2059	C₂	semidirect
8. C₄₁×#2057	C₂	semidirect
9. C₄₁×#1943	C₂	semidirect
10. C₄₁×#1942	C₂	semidirect
13. C₄₁×#1846	C₂	semidirect
66. C₂×M₁₆×C₄₁	Q₈	semidirect
113. C₂²⋊C₈×C₄₁	Q₈	semidirect
265. #26079	C₄₁	direct (N × H)
297. C₂×#501	C₂×C₄₁	semidirect
301. #2085	C₂×C₄₁	semidirect
302. #2085	C₂×C₄₁	semidirect
303. #2059	C₂×C₄₁	semidirect
304. #2057	C₂×C₄₁	semidirect
305. #1943	C₂×C₄₁	semidirect
306. #1942	C₂×C₄₁	semidirect
309. #1846	C₂×C₄₁	semidirect
365. C₄₁	#26079	direct (N × H)
371. C₂×M₁₆	Q₈×C₄₁	semidirect
418. C₂²⋊C₈	Q₈×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×#501	2
C₂×C₄₁×#428	2
C₄₁×#2442	2
C₄₁×#2439	2
C₄₁×#2085	2
C₄₁×#2059	2
C₄₁×#2057	2
C₄₁×#1943	2
C₄₁×#1942	2
C₄₁×#1854	2
C₄₁×#1852	2
C₄₁×#1846	2
C₄₁×#1809	2
C₄₁×#1753	2
#26079	41

Subgroups — C41×#26079 #23802280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#26079 #23802280