Subgroups — C₄₁×#23208 #23799409

Order 10496.

868 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#436	C₂	1	1	yes	maximal, characteristic
3	C₄×#124×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1954	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#1924	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1923	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1851	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1753	C₂	1	2	yes	maximal, complemented
12	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
13	C₂×#124×C₄₁	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#446	—	2	4	no	—
17	C₄₁×#436	—	2	4	no	—
18	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
26	C₂×#97×C₄₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
30	C₄₁×#391	C₄	1	1	yes	characteristic
31	C₄₁×#390	C₄	1	1	yes	characteristic
32	C₄₁×#388	C₄	1	2	yes	—
33	C₄₁×#388	C₄	1	2	yes	—
34	C₄₁×#382	C₄	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
37	C₄³×C₄₁	C₂²	1	1	yes	characteristic
38	C₄₁×#336	C₂²	1	2	yes	—
39	C₄₁×#336	C₂²	1	1	yes	characteristic
40	C₄₁×#336	C₂²	1	1	yes	characteristic
41	C₄₁×#336	C₂²	1	2	yes	—
42	C₄₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
46	C₂×M₁₆×C₄₁	—	2	4	no	—
47	C₂×M₁₆×C₄₁	—	2	4	no	—
48	C₂×M₁₆×C₄₁	—	4	2	no	—
49	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
55	#124×C₄₁	—	2	4	no	—
56	C₄×Q₈×C₄₁	—	2	2	no	—
57	C₄×Q₈×C₄₁	—	2	2	no	—
58	C₄×D₄×C₄₁	—	2	2	no	—
59	C₄×D₄×C₄₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
73	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₄₁	—	2	4	no	—
77	C₄⋊C₈×C₄₁	—	2	2	no	—
78	C₄⋊C₈×C₄₁	—	2	2	no	—
79	C₄⋊C₈×C₄₁	—	2	2	no	—
80	C₄⋊C₈×C₄₁	D₄	1	4	yes	—
81	C₂²⋊C₈×C₄₁	—	2	4	no	—
82	C₂²⋊C₈×C₄₁	—	2	4	no	—
83	C₂²⋊C₈×C₄₁	—	4	2	no	—
84	#97×C₄₁	—	2	4	no	—
85	C₄×C₈×C₄₁	—	2	2	no	—
86	C₄×C₈×C₄₁	—	2	2	no	—
87	C₄×C₈×C₄₁	—	4	2	no	—
88	#95×C₄₁	C₂×C₄	1	2	yes	—
89	#95×C₄₁	C₂×C₄	1	2	yes	—
90	#95×C₄₁	C₂×C₄	1	2	yes	—
91	#95×C₄₁	C₂×C₄	1	2	yes	—
92	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
93	C₂×Q₈×C₄₁	—	2	2	no	—
94	C₂×Q₈×C₄₁	—	2	2	no	—
95	C₂×Q₈×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	C₂²×C₄×C₄₁	—	2	2	no	—
100	C₂²×C₄×C₄₁	—	2	2	no	—
101	C₂²×C₄×C₄₁	—	2	2	no	—
102	C₂²×C₄×C₄₁	—	2	2	no	—
103	C₂²×C₄×C₄₁	—	2	2	no	—
104	C₂²×C₄×C₄₁	—	2	2	no	—
105	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
115	M₁₆×C₄₁	—	4	8	no	—
116	C₂×C₈×C₄₁	—	2	1	no	—
117	C₂×C₈×C₄₁	—	2	1	no	—
118	C₂×C₈×C₄₁	—	2	2	no	—
119	C₂×C₈×C₄₁	—	2	2	no	—
120	C₂×C₈×C₄₁	—	2	1	no	—
121	C₂×C₈×C₄₁	—	2	1	no	—
122	C₂×C₈×C₄₁	—	2	1	no	—
123	C₂×C₈×C₄₁	—	2	1	no	—
124	C₂×C₈×C₄₁	—	2	2	no	—
125	C₂×C₈×C₄₁	—	2	1	no	—
126	C₂×C₈×C₄₁	—	2	1	no	—
127	C₂×C₈×C₄₁	—	2	2	no	—
128	C₂×C₈×C₄₁	—	2	2	no	—
129	C₂×C₈×C₄₁	—	2	1	no	—
130	C₂×C₈×C₄₁	—	2	1	no	—
131	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
132	C₂×C₈×C₄₁	#41	1	2	yes	—
133	C₂×C₈×C₄₁	#41	1	2	yes	—
134	C₄⋊C₄×C₄₁	—	2	2	no	—
135	C₄⋊C₄×C₄₁	—	2	2	no	—
136	C₄⋊C₄×C₄₁	—	2	4	no	—
137	C₂²⋊C₄×C₄₁	—	2	4	no	—
138	C₂²⋊C₄×C₄₁	—	2	4	no	—
139	C₂²⋊C₄×C₄₁	—	2	4	no	—
140	C₂²⋊C₄×C₄₁	—	2	4	no	—
141	C₂²⋊C₄×C₄₁	—	4	2	no	—
142	C₄²×C₄₁	—	2	2	no	—
143	C₄²×C₄₁	—	2	2	no	—
144	C₄²×C₄₁	—	2	2	no	—
145	C₄²×C₄₁	—	2	1	no	—
146	C₄²×C₄₁	—	2	1	no	—
147	C₄²×C₄₁	—	2	2	no	—
148	C₄²×C₄₁	—	2	1	no	—
149	C₄²×C₄₁	—	2	1	no	—
150	C₄²×C₄₁	—	4	1	no	—
151	C₄²×C₄₁	M₁₆	1	4	yes	complemented
152	C₄²×C₄₁	C₂×D₄	1	2	yes	—
153	C₄²×C₄₁	#41	1	2	yes	—
154	C₂³×C₄₁	—	2	2	no	—
155	C₂³×C₄₁	—	2	2	no	—
156	C₂³×C₄₁	—	2	2	no	—
157	C₂³×C₄₁	—	2	2	no	—
158	C₂³×C₄₁	—	2	2	no	—
159	C₂³×C₄₁	—	2	2	no	—
160	C₂³×C₄₁	—	2	2	no	—
161	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
162	Q₈×C₄₁	—	4	4	no	—
163	D₄×C₄₁	—	4	4	no	—
164	C₂×C₄×C₄₁	—	2	1	no	—
165	C₂×C₄×C₄₁	—	2	1	no	—
166	C₂×C₄×C₄₁	—	2	1	no	—
167	C₂×C₄×C₄₁	—	2	1	no	—
168	C₂×C₄×C₄₁	—	2	1	no	—
169	C₂×C₄×C₄₁	—	2	1	no	—
170	C₂×C₄×C₄₁	—	2	1	no	—
171	C₂×C₄×C₄₁	—	2	1	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	1	no	—
176	C₂×C₄×C₄₁	—	2	2	no	—
177	C₂×C₄×C₄₁	—	2	2	no	—
178	C₂×C₄×C₄₁	—	2	2	no	—
179	C₂×C₄×C₄₁	—	2	1	no	—
180	C₂×C₄×C₄₁	—	2	2	no	—
181	C₂×C₄×C₄₁	—	2	2	no	—
182	C₂×C₄×C₄₁	—	2	1	no	—
183	C₂×C₄×C₄₁	—	2	2	no	—
184	C₂×C₄×C₄₁	—	4	2	no	—
185	C₂×C₄×C₄₁	—	4	2	no	—
186	C₂×C₄×C₄₁	—	4	2	no	—
187	C₂×C₄×C₄₁	—	4	2	no	—
188	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
189	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
190	C₂×C₄×C₄₁	C₂×M₁₆	1	2	yes	—
191	C₂×C₄×C₄₁	C₂×M₁₆	1	2	yes	—
192	C₂×C₄×C₄₁	C₂×M₁₆	1	2	yes	—
193	C₂×C₄×C₄₁	#131	1	2	yes	—
194	C₂×C₄×C₄₁	#131	1	2	yes	—
195	C₂×C₄×C₄₁	C₂²×D₄	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	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
201	C₈×C₄₁	—	2	4	no	—
202	C₈×C₄₁	—	2	2	no	—
203	C₈×C₄₁	—	2	2	no	—
204	C₈×C₄₁	—	4	1	no	—
205	C₈×C₄₁	—	4	1	no	—
206	C₈×C₄₁	—	4	2	no	—
207	#23208	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂²×C₄₁	—	4	2	no	—
209	C₂²×C₄₁	—	4	2	no	—
210	C₂²×C₄₁	—	4	2	no	—
211	C₂²×C₄₁	—	4	2	no	—
212	C₂²×C₄₁	—	4	2	no	—
213	C₂²×C₄₁	—	4	2	no	—
214	C₂²×C₄₁	—	4	2	no	—
215	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
216	C₂²×C₄₁	C₄×#41	1	1	yes	characteristic
217	C₂²×C₄₁	#520	1	1	yes	characteristic
218	C₂²×C₄₁	C₂²×M₁₆	1	1	yes	characteristic
219	C₂²×C₄₁	C₂×#131	1	1	yes	characteristic
220	C₂²×C₄₁	C₂×#131	1	1	yes	characteristic
221	C₂²×C₄₁	#568	1	1	yes	characteristic
222	C₄×C₄₁	—	2	2	no	—
223	C₄×C₄₁	—	2	2	no	—
224	C₄×C₄₁	—	2	2	no	—
225	C₄×C₄₁	—	2	2	no	—
226	C₄×C₄₁	—	4	2	no	—
227	C₄×C₄₁	—	4	2	no	—
228	C₄×C₄₁	—	4	2	no	—
229	C₄×C₄₁	—	4	1	no	—
230	C₄×C₄₁	#436	1	4	yes	—
231	C₄×C₄₁	#546	1	1	yes	characteristic
232	C₄×C₄₁	#548	1	1	yes	characteristic
233	C₄×C₄₁	#556	1	1	yes	characteristic
234	C₄×C₄₁	#559	1	1	yes	characteristic
235	C₂×#446	C₂×C₄₁	1	1	yes	characteristic, complemented
236	C₂×#436	C₂×C₄₁	1	1	yes	characteristic
237	C₄×#124	C₂×C₄₁	1	1	yes	characteristic
238	#1984	C₂×C₄₁	1	1	yes	characteristic, complemented
239	#1954	C₂×C₄₁	1	1	yes	characteristic
240	#1924	C₂×C₄₁	1	2	yes	complemented
241	#1923	C₂×C₄₁	1	2	yes	complemented
242	#1851	C₂×C₄₁	1	2	yes	complemented
243	C₄×C₄⋊C₈	C₂×C₄₁	1	1	yes	characteristic, complemented
244	C₄×C₄⋊C₈	C₂×C₄₁	1	1	yes	characteristic, complemented
245	#1753	C₂×C₄₁	1	2	yes	complemented
246	C₂×C₄₁	—	8	2	no	—
247	C₂×C₄₁	#2367	1	1	yes	characteristic
248	C₂×C₄₁	C₂×#436	1	1	yes	characteristic
249	C₂×C₄₁	#2935	1	1	yes	characteristic
250	C₂×C₄₁	#2965	1	1	yes	characteristic
251	C₂×C₄₁	#2969	1	1	yes	characteristic
252	C₂×C₄₁	#2989	1	1	yes	characteristic
253	C₂×C₄₁	#2994	1	1	yes	characteristic
254	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
255	C₂×#124	C₄×C₄₁	1	1	yes	characteristic
256	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
257	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
258	#446	—	2	4	no	—
259	#436	—	2	4	no	—
260	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
261	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
262	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
263	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
264	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
265	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
266	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
267	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
268	C₂×#97	C₂²×C₄₁	1	2	yes	—
269	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
270	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
271	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
272	#391	C₄×C₄₁	1	1	yes	characteristic
273	#390	C₄×C₄₁	1	1	yes	characteristic
274	#388	C₄×C₄₁	1	2	yes	—
275	#388	C₄×C₄₁	1	2	yes	—
276	#382	C₄×C₄₁	1	1	yes	characteristic
277	C₄×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
278	C₄×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
279	C₄³	C₂²×C₄₁	1	1	yes	characteristic
280	#336	C₂²×C₄₁	1	2	yes	—
281	#336	C₂²×C₄₁	1	2	yes	—
282	#336	C₂²×C₄₁	1	1	yes	characteristic
283	#336	C₂²×C₄₁	1	2	yes	—
284	#336	C₂²×C₄₁	1	1	yes	characteristic
285	C₄₁	#23208	1	1	yes	characteristic, direct product factor, Sylow
286	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
287	C₂²×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
288	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
289	C₂×M₁₆	—	2	4	no	—
290	C₂×M₁₆	—	2	4	no	—
291	C₂×M₁₆	—	4	2	no	—
292	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
293	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
294	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
295	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
296	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
298	#124	—	2	4	no	—
299	C₄×Q₈	—	2	2	no	—
300	C₄×Q₈	—	2	2	no	—
301	C₄×D₄	—	2	2	no	—
302	C₄×D₄	—	2	2	no	—
303	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
306	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
307	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
308	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
309	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
310	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
311	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
312	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
313	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
314	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
315	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
316	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
317	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
318	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
319	C₄⋊C₈	—	2	4	no	—
320	C₄⋊C₈	—	2	2	no	—
321	C₄⋊C₈	—	2	2	no	—
322	C₄⋊C₈	—	2	2	no	—
323	C₄⋊C₈	D₄×C₄₁	1	4	yes	—
324	C₂²⋊C₈	—	2	4	no	—
325	C₂²⋊C₈	—	2	4	no	—
326	C₂²⋊C₈	—	4	2	no	—
327	#97	—	2	4	no	—
328	C₄×C₈	—	2	2	no	—
329	C₄×C₈	—	2	2	no	—
330	C₄×C₈	—	4	2	no	—
331	#95	C₂×C₄×C₄₁	1	2	yes	—
332	#95	C₂×C₄×C₄₁	1	2	yes	—
333	#95	C₂×C₄×C₄₁	1	2	yes	—
334	#95	C₂×C₄×C₄₁	1	2	yes	—
335	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	2	2	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
349	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
350	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
351	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
352	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
353	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
354	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
355	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
356	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
357	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
358	M₁₆	—	4	8	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	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
375	C₂×C₈	#41×C₄₁	1	2	yes	—
376	C₂×C₈	#41×C₄₁	1	2	yes	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	4	no	—
380	C₂²⋊C₄	—	2	4	no	—
381	C₂²⋊C₄	—	2	4	no	—
382	C₂²⋊C₄	—	2	4	no	—
383	C₂²⋊C₄	—	2	4	no	—
384	C₂²⋊C₄	—	4	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	2	1	no	—
392	C₄²	—	2	1	no	—
393	C₄²	—	4	1	no	—
394	C₄²	M₁₆×C₄₁	1	4	yes	complemented
395	C₄²	C₂×D₄×C₄₁	1	2	yes	—
396	C₄²	#41×C₄₁	1	2	yes	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	2	no	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	2	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	2	no	—
404	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
405	Q₈	—	4	4	no	—
406	D₄	—	4	4	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
432	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
433	C₂×C₄	C₂×M₁₆×C₄₁	1	2	yes	—
434	C₂×C₄	C₂×M₁₆×C₄₁	1	2	yes	—
435	C₂×C₄	C₂×M₁₆×C₄₁	1	2	yes	—
436	C₂×C₄	#131×C₄₁	1	2	yes	—
437	C₂×C₄	#131×C₄₁	1	2	yes	—
438	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
439	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
440	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
441	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
442	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
443	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
444	C₈	—	2	4	no	—
445	C₈	—	2	2	no	—
446	C₈	—	2	2	no	—
447	C₈	—	4	1	no	—
448	C₈	—	4	1	no	—
449	C₈	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	2	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	2	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	2	no	—
457	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
458	C₂²	C₄×#41×C₄₁	1	1	yes	characteristic
459	C₂²	C₄₁×#520	1	1	yes	characteristic
460	C₂²	C₂²×M₁₆×C₄₁	1	1	yes	characteristic
461	C₂²	C₂×#131×C₄₁	1	1	yes	characteristic
462	C₂²	C₂×#131×C₄₁	1	1	yes	characteristic
463	C₂²	C₄₁×#568	1	1	yes	characteristic
464	C₄	—	2	2	no	—
465	C₄	—	2	2	no	—
466	C₄	—	2	2	no	—
467	C₄	—	2	2	no	—
468	C₄	—	4	2	no	—
469	C₄	—	4	2	no	—
470	C₄	—	4	2	no	—
471	C₄	—	4	1	no	—
472	C₄	C₄₁×#436	1	4	yes	—
473	C₄	C₄₁×#546	1	1	yes	characteristic
474	C₄	C₄₁×#548	1	1	yes	characteristic
475	C₄	C₄₁×#556	1	1	yes	characteristic
476	C₄	C₄₁×#559	1	1	yes	characteristic
477	C₂	—	8	2	no	—
478	C₂	C₄₁×#2367	1	1	yes	characteristic
479	C₂	C₂×C₄₁×#436	1	1	yes	characteristic
480	C₂	C₄₁×#2935	1	1	yes	characteristic
481	C₂	C₄₁×#2965	1	1	yes	characteristic
482	C₂	C₄₁×#2969	1	1	yes	characteristic
483	C₂	C₄₁×#2989	1	1	yes	characteristic
484	C₂	C₄₁×#2994	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#446	C₂	semidirect
4. C₄₁×#1984	C₂	semidirect
6. C₄₁×#1924	C₂	semidirect
7. C₄₁×#1923	C₂	semidirect
8. C₄₁×#1851	C₂	semidirect
9. C₄×C₄⋊C₈×C₄₁	C₂	semidirect
10. C₄×C₄⋊C₈×C₄₁	C₂	semidirect
11. C₄₁×#1753	C₂	semidirect
151. C₄²×C₄₁	M₁₆	semidirect
207. #23208	C₄₁	direct (N × H)
235. C₂×#446	C₂×C₄₁	semidirect
238. #1984	C₂×C₄₁	semidirect
240. #1924	C₂×C₄₁	semidirect
241. #1923	C₂×C₄₁	semidirect
242. #1851	C₂×C₄₁	semidirect
243. C₄×C₄⋊C₈	C₂×C₄₁	semidirect
244. C₄×C₄⋊C₈	C₂×C₄₁	semidirect
245. #1753	C₂×C₄₁	semidirect
285. C₄₁	#23208	direct (N × H)
394. C₄²	M₁₆×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×#446	2
C₂×C₄₁×#436	2
C₄×#124×C₄₁	2
C₄₁×#1984	2
C₄₁×#1954	2
C₄₁×#1924	2
C₄₁×#1923	2
C₄₁×#1851	2
C₄×C₄⋊C₈×C₄₁	2
C₄₁×#1753	2
#23208	41

Subgroups — C41×#23208 #23799409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23208 #23799409