Subgroups — C₄₁×#25405 #23801606

Order 10496.

916 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3152	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#471	C₂	1	2	yes	maximal, complemented
3	C₄₁×#3036	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2935	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2052	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2019	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1966	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1908	C₂	1	2	yes	maximal, complemented
11	C₄×#100×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₄₁	C₂²	1	2	yes	—
13	C₂×#137×C₄₁	C₂²	1	2	yes	—
14	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
16	C₂×#131×C₄₁	C₂²	1	2	yes	—
17	C₄₁×#532	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
20	C₄₁×#497	—	2	2	no	—
21	C₄₁×#494	—	2	1	no	—
22	C₄₁×#492	—	2	1	no	—
23	C₄₁×#471	—	2	8	no	—
24	C₄₁×#463	—	2	2	no	—
25	C₄₁×#461	—	2	1	no	—
26	C₄₁×#461	—	2	1	no	—
27	C₄₁×#461	—	2	2	no	—
28	C₄₁×#461	—	2	1	no	—
29	C₄₁×#459	—	2	1	no	—
30	C₄₁×#436	—	2	1	no	—
31	C₄₁×#435	—	2	2	no	—
32	C₈×D₄×C₄₁	—	2	1	no	—
33	C₂×#108×C₄₁	C₂²	1	2	yes	—
34	C₄₁×#423	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
36	C₄₁×#417	C₂²	1	2	yes	—
37	C₂×#103×C₄₁	C₂²	1	2	yes	—
38	C₂×#101×C₄₁	C₂²	1	2	yes	—
39	C₂×#101×C₄₁	C₂²	1	2	yes	—
40	C₂×#100×C₄₁	C₂²	1	1	yes	characteristic
41	C₂×#100×C₄₁	C₂²	1	1	yes	characteristic
42	C₄₁×#408	C₂²	1	2	yes	—
43	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
45	C₄₁×#343	C₂²	1	1	yes	characteristic
46	C₄₁×#343	C₂²	1	1	yes	characteristic
47	C₄₁×#341	C₂²	1	2	yes	—
48	C₂×#41×C₄₁	C₂³	1	2	yes	—
49	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
50	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₄₁	—	2	1	no	—
52	#137×C₄₁	—	2	4	no	—
53	#137×C₄₁	—	2	4	no	—
54	#137×C₄₁	—	4	2	no	—
55	#137×C₄₁	—	4	2	no	—
56	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
57	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
58	C₂×Q₁₆×C₄₁	—	2	2	no	—
59	C₂×Q₁₆×C₄₁	—	2	2	no	—
60	C₂×SD₁₆×C₄₁	—	2	2	no	—
61	C₂×SD₁₆×C₄₁	—	2	4	no	—
62	C₂×SD₁₆×C₄₁	—	2	2	no	—
63	C₂×SD₁₆×C₄₁	—	2	2	no	—
64	C₂×SD₁₆×C₄₁	—	2	2	no	—
65	C₂×SD₁₆×C₄₁	—	2	2	no	—
66	C₂×SD₁₆×C₄₁	—	4	2	no	—
67	C₂×D₈×C₄₁	—	2	2	no	—
68	#131×C₄₁	—	2	4	no	—
69	#131×C₄₁	—	4	2	no	—
70	C₂×M₁₆×C₄₁	—	2	2	no	—
71	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
72	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₄₁	—	2	2	no	—
77	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
78	C₄⋊Q₈×C₄₁	—	2	1	no	—
79	C₄⋊Q₈×C₄₁	—	2	1	no	—
80	C₄⋊Q₈×C₄₁	—	2	1	no	—
81	C₄⋊Q₈×C₄₁	—	2	1	no	—
82	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
83	#127×C₄₁	D₄	1	1	yes	characteristic
84	#124×C₄₁	D₄	1	2	yes	—
85	#121×C₄₁	—	2	1	no	—
86	#121×C₄₁	—	2	1	no	—
87	C₄×D₄×C₄₁	—	2	1	no	—
88	C₄×D₄×C₄₁	—	2	1	no	—
89	C₄×D₄×C₄₁	—	2	1	no	—
90	C₄×D₄×C₄₁	—	2	1	no	—
91	C₄×D₄×C₄₁	—	2	1	no	—
92	C₄×D₄×C₄₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
94	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
95	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
97	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
98	#108×C₄₁	—	2	4	no	—
99	C₄⋊C₈×C₄₁	—	2	1	no	—
100	C₄⋊C₈×C₄₁	—	2	1	no	—
101	C₄⋊C₈×C₄₁	D₄	1	1	yes	characteristic
102	C₄⋊C₈×C₄₁	D₄	1	2	yes	—
103	C₄⋊C₈×C₄₁	D₄	1	1	yes	characteristic
104	#103×C₄₁	—	2	2	no	—
105	#103×C₄₁	—	2	2	no	—
106	#103×C₄₁	—	2	2	no	—
107	#102×C₄₁	—	2	2	no	—
108	#101×C₄₁	—	2	4	no	—
109	#101×C₄₁	—	4	2	no	—
110	#100×C₄₁	—	2	2	no	—
111	#100×C₄₁	D₄	1	4	yes	complemented
112	C₂²⋊C₈×C₄₁	—	2	2	no	—
113	C₂²⋊C₈×C₄₁	—	2	2	no	—
114	#97×C₄₁	—	2	1	no	—
115	#97×C₄₁	—	2	1	no	—
116	C₄×C₈×C₄₁	—	2	1	no	—
117	C₄×C₈×C₄₁	—	2	1	no	—
118	C₂⁴×C₄₁	—	2	1	no	—
119	#41×C₄₁	—	2	2	no	—
120	#41×C₄₁	—	2	2	no	—
121	#41×C₄₁	—	2	2	no	—
122	#41×C₄₁	—	2	2	no	—
123	C₂×Q₈×C₄₁	—	2	2	no	—
124	C₂×Q₈×C₄₁	—	2	2	no	—
125	C₂×Q₈×C₄₁	—	2	2	no	—
126	C₂×Q₈×C₄₁	—	2	2	no	—
127	C₂×Q₈×C₄₁	—	2	2	no	—
128	C₂×Q₈×C₄₁	—	2	2	no	—
129	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
130	C₂×D₄×C₄₁	—	2	2	no	—
131	C₂×D₄×C₄₁	—	2	1	no	—
132	C₂×D₄×C₄₁	—	2	1	no	—
133	C₂×D₄×C₄₁	—	2	1	no	—
134	C₂×D₄×C₄₁	—	2	1	no	—
135	C₂×D₄×C₄₁	—	2	1	no	—
136	C₂×D₄×C₄₁	—	2	1	no	—
137	C₂×D₄×C₄₁	—	2	1	no	—
138	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
139	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₄₁	—	2	2	no	—
141	C₂²×C₄×C₄₁	—	2	2	no	—
142	C₂²×C₄×C₄₁	—	2	1	no	—
143	C₂²×C₄×C₄₁	—	4	1	no	—
144	C₂²×C₄×C₄₁	—	4	1	no	—
145	C₂²×C₄×C₄₁	—	8	2	no	—
146	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
147	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
148	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
149	Q₁₆×C₄₁	—	4	4	no	—
150	Q₁₆×C₄₁	—	4	4	no	—
151	SD₁₆×C₄₁	—	4	4	no	—
152	SD₁₆×C₄₁	—	4	4	no	—
153	SD₁₆×C₄₁	—	4	4	no	—
154	SD₁₆×C₄₁	—	4	4	no	—
155	SD₁₆×C₄₁	—	4	4	no	—
156	D₈×C₄₁	—	4	4	no	—
157	M₁₆×C₄₁	—	2	1	no	—
158	M₁₆×C₄₁	—	2	1	no	—
159	M₁₆×C₄₁	—	2	1	no	—
160	M₁₆×C₄₁	—	2	2	no	—
161	M₁₆×C₄₁	—	2	1	no	—
162	M₁₆×C₄₁	—	2	1	no	—
163	M₁₆×C₄₁	—	2	2	no	—
164	M₁₆×C₄₁	—	4	4	no	—
165	M₁₆×C₄₁	C₂×D₄	1	2	yes	—
166	M₁₆×C₄₁	C₂×D₄	1	2	yes	—
167	M₁₆×C₄₁	#41	1	2	yes	—
168	C₂×C₈×C₄₁	—	2	1	no	—
169	C₂×C₈×C₄₁	—	2	1	no	—
170	C₂×C₈×C₄₁	—	2	1	no	—
171	C₂×C₈×C₄₁	—	2	2	no	—
172	C₂×C₈×C₄₁	—	2	2	no	—
173	C₂×C₈×C₄₁	—	2	1	no	—
174	C₂×C₈×C₄₁	—	2	2	no	—
175	C₂×C₈×C₄₁	—	4	4	no	—
176	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
177	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
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	1	no	—
183	C₄⋊C₄×C₄₁	—	4	1	no	—
184	C₄⋊C₄×C₄₁	—	4	1	no	—
185	C₄⋊C₄×C₄₁	—	4	1	no	—
186	C₂²⋊C₄×C₄₁	—	2	2	no	—
187	C₂²⋊C₄×C₄₁	—	2	2	no	—
188	C₂²⋊C₄×C₄₁	—	4	1	no	—
189	C₂²⋊C₄×C₄₁	—	4	1	no	—
190	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
194	C₂³×C₄₁	—	2	2	no	—
195	C₂³×C₄₁	—	2	2	no	—
196	C₂³×C₄₁	—	4	1	no	—
197	C₂³×C₄₁	—	4	1	no	—
198	C₂³×C₄₁	—	8	1	no	—
199	C₂³×C₄₁	—	8	1	no	—
200	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
201	Q₈×C₄₁	—	2	2	no	—
202	Q₈×C₄₁	—	2	2	no	—
203	Q₈×C₄₁	—	4	2	no	—
204	Q₈×C₄₁	—	4	2	no	—
205	Q₈×C₄₁	—	4	2	no	—
206	Q₈×C₄₁	—	4	2	no	—
207	D₄×C₄₁	—	2	2	no	—
208	D₄×C₄₁	—	2	2	no	—
209	D₄×C₄₁	—	2	2	no	—
210	D₄×C₄₁	—	2	2	no	—
211	D₄×C₄₁	—	4	2	no	—
212	D₄×C₄₁	—	4	2	no	—
213	D₄×C₄₁	—	4	1	no	—
214	D₄×C₄₁	—	4	1	no	—
215	D₄×C₄₁	—	4	2	no	—
216	D₄×C₄₁	—	8	1	no	—
217	C₂×C₄×C₄₁	—	2	2	no	—
218	C₂×C₄×C₄₁	—	2	2	no	—
219	C₂×C₄×C₄₁	—	4	1	no	—
220	C₂×C₄×C₄₁	—	4	2	no	—
221	C₂×C₄×C₄₁	—	4	2	no	—
222	C₂×C₄×C₄₁	—	4	1	no	—
223	C₂×C₄×C₄₁	—	4	2	no	—
224	C₂×C₄×C₄₁	—	4	2	no	—
225	C₂×C₄×C₄₁	—	4	2	no	—
226	C₂×C₄×C₄₁	—	8	2	no	—
227	C₂×C₄×C₄₁	—	8	2	no	—
228	C₂×C₄×C₄₁	—	8	1	no	—
229	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
230	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
231	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
232	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
233	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
237	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
238	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
239	C₈×C₄₁	—	2	2	no	—
240	C₈×C₄₁	—	2	2	no	—
241	C₈×C₄₁	—	2	2	no	—
242	C₈×C₄₁	—	2	2	no	—
243	C₈×C₄₁	—	2	2	no	—
244	C₈×C₄₁	—	4	2	no	—
245	C₈×C₄₁	—	4	1	no	—
246	#25405	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂²×C₄₁	—	2	1	no	—
248	C₂²×C₄₁	—	2	1	no	—
249	C₂²×C₄₁	—	4	2	no	—
250	C₂²×C₄₁	—	4	2	no	—
251	C₂²×C₄₁	—	4	2	no	—
252	C₂²×C₄₁	—	4	2	no	—
253	C₂²×C₄₁	—	8	2	no	—
254	C₂²×C₄₁	—	8	2	no	—
255	C₂²×C₄₁	—	8	1	no	—
256	C₂²×C₄₁	—	8	1	no	—
257	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
258	C₂²×C₄₁	#532	1	1	yes	characteristic
259	C₂²×C₄₁	#540	1	1	yes	characteristic
260	C₄×C₄₁	—	2	1	no	—
261	C₄×C₄₁	—	2	1	no	—
262	C₄×C₄₁	—	2	1	no	—
263	C₄×C₄₁	—	4	2	no	—
264	C₄×C₄₁	—	8	2	no	—
265	C₄×C₄₁	—	8	1	no	—
266	C₄×C₄₁	#471	1	2	yes	complemented
267	C₄×C₄₁	D₄²	1	1	yes	characteristic
268	C₄×C₄₁	D₄²	1	1	yes	characteristic
269	C₄×C₄₁	#546	1	1	yes	characteristic
270	C₄×C₄₁	#550	1	1	yes	characteristic
271	#3152	C₂×C₄₁	1	1	yes	characteristic, complemented
272	C₂×#471	C₂×C₄₁	1	2	yes	complemented
273	#3036	C₂×C₄₁	1	1	yes	characteristic
274	C₂×#461	C₂×C₄₁	1	1	yes	characteristic, complemented
275	#2935	C₂×C₄₁	1	1	yes	characteristic, complemented
276	#2052	C₂×C₄₁	1	2	yes	complemented
277	#2019	C₂×C₄₁	1	2	yes	complemented
278	#1966	C₂×C₄₁	1	1	yes	characteristic, complemented
279	#1950	C₂×C₄₁	1	1	yes	characteristic, complemented
280	#1908	C₂×C₄₁	1	2	yes	complemented
281	C₄×#100	C₂×C₄₁	1	1	yes	characteristic, complemented
282	C₂×C₄₁	—	2	1	no	—
283	C₂×C₄₁	—	2	1	no	—
284	C₂×C₄₁	—	8	2	no	—
285	C₂×C₄₁	—	8	2	no	—
286	C₂×C₄₁	#2428	1	1	yes	characteristic
287	C₂×C₄₁	C₂×#471	1	1	yes	characteristic
288	C₂×C₄₁	#3069	1	1	yes	characteristic
289	C₂×#137	C₂²×C₄₁	1	2	yes	—
290	C₂×#137	C₂²×C₄₁	1	2	yes	—
291	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
292	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
293	C₂×#131	C₂²×C₄₁	1	2	yes	—
294	#532	C₂²×C₄₁	1	1	yes	characteristic
295	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
296	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
297	#497	—	2	2	no	—
298	#494	—	2	1	no	—
299	#492	—	2	1	no	—
300	#471	—	2	8	no	—
301	#463	—	2	2	no	—
302	#461	—	2	1	no	—
303	#461	—	2	1	no	—
304	#461	—	2	2	no	—
305	#461	—	2	1	no	—
306	#459	—	2	1	no	—
307	#436	—	2	1	no	—
308	#435	—	2	2	no	—
309	C₈×D₄	—	2	1	no	—
310	C₂×#108	C₂²×C₄₁	1	2	yes	—
311	#423	C₂²×C₄₁	1	1	yes	characteristic
312	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
313	#417	C₂²×C₄₁	1	2	yes	—
314	C₂×#103	C₂²×C₄₁	1	2	yes	—
315	C₂×#101	C₂²×C₄₁	1	2	yes	—
316	C₂×#101	C₂²×C₄₁	1	2	yes	—
317	C₂×#100	C₂²×C₄₁	1	1	yes	characteristic
318	C₂×#100	C₂²×C₄₁	1	1	yes	characteristic
319	#408	C₂²×C₄₁	1	2	yes	—
320	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
321	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
322	#343	C₂²×C₄₁	1	1	yes	characteristic
323	#343	C₂²×C₄₁	1	1	yes	characteristic
324	#341	C₂²×C₄₁	1	2	yes	—
325	C₄₁	#25405	1	1	yes	characteristic, direct product factor, Sylow
326	C₂×#41	C₂³×C₄₁	1	2	yes	—
327	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
328	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
329	C₂³×C₄	—	2	1	no	—
330	#137	—	2	4	no	—
331	#137	—	2	4	no	—
332	#137	—	4	2	no	—
333	#137	—	4	2	no	—
334	AGL(1,ℤ/8)	—	2	4	no	—
335	AGL(1,ℤ/8)	—	4	2	no	—
336	C₂×Q₁₆	—	2	2	no	—
337	C₂×Q₁₆	—	2	2	no	—
338	C₂×SD₁₆	—	2	2	no	—
339	C₂×SD₁₆	—	2	4	no	—
340	C₂×SD₁₆	—	2	2	no	—
341	C₂×SD₁₆	—	2	2	no	—
342	C₂×SD₁₆	—	2	2	no	—
343	C₂×SD₁₆	—	2	2	no	—
344	C₂×SD₁₆	—	4	2	no	—
345	C₂×D₈	—	2	2	no	—
346	#131	—	2	4	no	—
347	#131	—	4	2	no	—
348	C₂×M₁₆	—	2	2	no	—
349	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
350	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
351	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
352	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
353	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
354	C₂²×C₈	—	2	2	no	—
355	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
356	C₄⋊Q₈	—	2	1	no	—
357	C₄⋊Q₈	—	2	1	no	—
358	C₄⋊Q₈	—	2	1	no	—
359	C₄⋊Q₈	—	2	1	no	—
360	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
361	#127	D₄×C₄₁	1	1	yes	characteristic
362	#124	D₄×C₄₁	1	2	yes	—
363	#121	—	2	1	no	—
364	#121	—	2	1	no	—
365	C₄×D₄	—	2	1	no	—
366	C₄×D₄	—	2	1	no	—
367	C₄×D₄	—	2	1	no	—
368	C₄×D₄	—	2	1	no	—
369	C₄×D₄	—	2	1	no	—
370	C₄×D₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
374	C₂×C₂²⋊C₄	—	2	1	no	—
375	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
376	#108	—	2	4	no	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	—	2	1	no	—
379	C₄⋊C₈	D₄×C₄₁	1	1	yes	characteristic
380	C₄⋊C₈	D₄×C₄₁	1	2	yes	—
381	C₄⋊C₈	D₄×C₄₁	1	1	yes	characteristic
382	#103	—	2	2	no	—
383	#103	—	2	2	no	—
384	#103	—	2	2	no	—
385	#102	—	2	2	no	—
386	#101	—	2	4	no	—
387	#101	—	4	2	no	—
388	#100	—	2	2	no	—
389	#100	D₄×C₄₁	1	4	yes	complemented
390	C₂²⋊C₈	—	2	2	no	—
391	C₂²⋊C₈	—	2	2	no	—
392	#97	—	2	1	no	—
393	#97	—	2	1	no	—
394	C₄×C₈	—	2	1	no	—
395	C₄×C₈	—	2	1	no	—
396	C₂⁴	—	2	1	no	—
397	#41	—	2	2	no	—
398	#41	—	2	2	no	—
399	#41	—	2	2	no	—
400	#41	—	2	2	no	—
401	C₂×Q₈	—	2	2	no	—
402	C₂×Q₈	—	2	2	no	—
403	C₂×Q₈	—	2	2	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	1	no	—
410	C₂×D₄	—	2	1	no	—
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
417	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	4	1	no	—
422	C₂²×C₄	—	4	1	no	—
423	C₂²×C₄	—	8	2	no	—
424	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
425	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
426	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
427	Q₁₆	—	4	4	no	—
428	Q₁₆	—	4	4	no	—
429	SD₁₆	—	4	4	no	—
430	SD₁₆	—	4	4	no	—
431	SD₁₆	—	4	4	no	—
432	SD₁₆	—	4	4	no	—
433	SD₁₆	—	4	4	no	—
434	D₈	—	4	4	no	—
435	M₁₆	—	2	1	no	—
436	M₁₆	—	2	1	no	—
437	M₁₆	—	2	1	no	—
438	M₁₆	—	2	2	no	—
439	M₁₆	—	2	1	no	—
440	M₁₆	—	2	1	no	—
441	M₁₆	—	2	2	no	—
442	M₁₆	—	4	4	no	—
443	M₁₆	C₂×D₄×C₄₁	1	2	yes	—
444	M₁₆	C₂×D₄×C₄₁	1	2	yes	—
445	M₁₆	#41×C₄₁	1	2	yes	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	2	no	—
453	C₂×C₈	—	4	4	no	—
454	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
455	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	4	1	no	—
461	C₄⋊C₄	—	4	1	no	—
462	C₄⋊C₄	—	4	1	no	—
463	C₄⋊C₄	—	4	1	no	—
464	C₂²⋊C₄	—	2	2	no	—
465	C₂²⋊C₄	—	2	2	no	—
466	C₂²⋊C₄	—	4	1	no	—
467	C₂²⋊C₄	—	4	1	no	—
468	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
469	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
470	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
471	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
472	C₂³	—	2	2	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	4	1	no	—
475	C₂³	—	4	1	no	—
476	C₂³	—	8	1	no	—
477	C₂³	—	8	1	no	—
478	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
479	Q₈	—	2	2	no	—
480	Q₈	—	2	2	no	—
481	Q₈	—	4	2	no	—
482	Q₈	—	4	2	no	—
483	Q₈	—	4	2	no	—
484	Q₈	—	4	2	no	—
485	D₄	—	2	2	no	—
486	D₄	—	2	2	no	—
487	D₄	—	2	2	no	—
488	D₄	—	2	2	no	—
489	D₄	—	4	2	no	—
490	D₄	—	4	2	no	—
491	D₄	—	4	1	no	—
492	D₄	—	4	1	no	—
493	D₄	—	4	2	no	—
494	D₄	—	8	1	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	8	2	no	—
505	C₂×C₄	—	8	2	no	—
506	C₂×C₄	—	8	1	no	—
507	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
508	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
509	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
510	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
512	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
513	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
514	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
515	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
516	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
517	C₈	—	2	2	no	—
518	C₈	—	2	2	no	—
519	C₈	—	2	2	no	—
520	C₈	—	2	2	no	—
521	C₈	—	2	2	no	—
522	C₈	—	4	2	no	—
523	C₈	—	4	1	no	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	8	2	no	—
531	C₂²	—	8	2	no	—
532	C₂²	—	8	1	no	—
533	C₂²	—	8	1	no	—
534	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
535	C₂²	C₄₁×#532	1	1	yes	characteristic
536	C₂²	C₄₁×#540	1	1	yes	characteristic
537	C₄	—	2	1	no	—
538	C₄	—	2	1	no	—
539	C₄	—	2	1	no	—
540	C₄	—	4	2	no	—
541	C₄	—	8	1	no	—
542	C₄	—	8	2	no	—
543	C₄	C₄₁×#471	1	2	yes	complemented
544	C₄	D₄²×C₄₁	1	1	yes	characteristic
545	C₄	D₄²×C₄₁	1	1	yes	characteristic
546	C₄	C₄₁×#546	1	1	yes	characteristic
547	C₄	C₄₁×#550	1	1	yes	characteristic
548	C₂	—	2	1	no	—
549	C₂	—	2	1	no	—
550	C₂	—	8	2	no	—
551	C₂	—	8	2	no	—
552	C₂	C₄₁×#2428	1	1	yes	characteristic
553	C₂	C₂×C₄₁×#471	1	1	yes	characteristic
554	C₂	C₄₁×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#3036	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3152	C₂	semidirect
2. C₂×C₄₁×#471	C₂	semidirect
4. C₂×C₄₁×#461	C₂	semidirect
5. C₄₁×#2935	C₂	semidirect
6. C₄₁×#2052	C₂	semidirect
7. C₄₁×#2019	C₂	semidirect
8. C₄₁×#1966	C₂	semidirect
9. C₄₁×#1950	C₂	semidirect
10. C₄₁×#1908	C₂	semidirect
11. C₄×#100×C₄₁	C₂	semidirect
111. #100×C₄₁	D₄	semidirect
246. #25405	C₄₁	direct (N × H)
266. C₄×C₄₁	#471	semidirect
271. #3152	C₂×C₄₁	semidirect
272. C₂×#471	C₂×C₄₁	semidirect
274. C₂×#461	C₂×C₄₁	semidirect
275. #2935	C₂×C₄₁	semidirect
276. #2052	C₂×C₄₁	semidirect
277. #2019	C₂×C₄₁	semidirect
278. #1966	C₂×C₄₁	semidirect
279. #1950	C₂×C₄₁	semidirect
280. #1908	C₂×C₄₁	semidirect
281. C₄×#100	C₂×C₄₁	semidirect
325. C₄₁	#25405	direct (N × H)
389. #100	D₄×C₄₁	semidirect
543. C₄	C₄₁×#471	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3152	2
C₂×C₄₁×#471	2
C₄₁×#3036	2
C₂×C₄₁×#461	2
C₄₁×#2935	2
C₄₁×#2052	2
C₄₁×#2019	2
C₄₁×#1966	2
C₄₁×#1950	2
C₄₁×#1908	2
C₄×#100×C₄₁	2
#25405	41

Subgroups — C41×#25405 #23801606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#25405 #23801606