Subgroups — C₄₁×#24171 #23800372

Order 10496.

1106 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3511	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3036	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×AGL(1,ℤ/8)×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2563	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2040	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2006	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1980	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1883	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1860	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1806	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
14	C₄₁×#556	—	2	1	no	—
15	C₄₁×#555	—	2	1	no	—
16	C₄₁×#554	—	2	1	no	—
17	C₄₁×#553	—	2	2	no	—
18	C₄₁×#548	—	2	2	no	—
19	C₄₁×#547	—	2	1	no	—
20	C₄₁×#542	—	2	2	no	—
21	C₄₁×#537	C₂²	1	2	yes	—
22	C₄₁×#533	—	2	1	no	—
23	C₄₁×#532	—	2	1	no	—
24	C₄₁×#532	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₄₁×#520	C₂²	1	2	yes	—
27	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
29	C₄₁×#463	—	2	2	no	—
30	C₄₁×#461	—	2	1	no	—
31	C₄₁×#459	—	2	1	no	—
32	C₄₁×#442	—	2	1	no	—
33	C₄₁×#440	—	2	1	no	—
34	C₄×SD₁₆×C₄₁	—	2	1	no	—
35	C₄×D₈×C₄₁	—	2	1	no	—
36	C₄₁×#428	C₂²	1	1	yes	characteristic
37	C₄₁×#423	C₂²	1	1	yes	characteristic
38	C₂×#104×C₄₁	C₂²	1	2	yes	—
39	C₂×#104×C₄₁	C₂²	1	2	yes	—
40	C₂×#104×C₄₁	C₂²	1	2	yes	—
41	C₄₁×#418	C₂²	1	1	yes	characteristic
42	C₄₁×#417	C₂²	1	2	yes	—
43	C₄₁×#417	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
45	C₄₁×#393	—	2	2	no	—
46	C₄₁×#390	C₂²	1	2	yes	—
47	C₄₁×#382	C₂²	1	2	yes	—
48	C₄₁×#380	—	2	2	no	—
49	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
50	C₄₁×#343	C₂²	1	2	yes	—
51	C₄₁×#343	C₂²	1	2	yes	—
52	C₄₁×#339	C₂²	1	2	yes	—
53	C₂×#41×C₄₁	—	2	2	no	—
54	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
55	C₂×#41×C₄₁	C₂³	1	2	yes	—
56	C₂²×Q₈×C₄₁	D₄	1	2	yes	—
57	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₄₁	—	2	1	no	—
59	AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
60	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
61	AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
62	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
63	C₂×SD₁₆×C₄₁	—	2	1	no	—
64	C₂×SD₁₆×C₄₁	—	2	2	no	—
65	C₂×D₈×C₄₁	—	2	1	no	—
66	C₂×D₈×C₄₁	—	2	2	no	—
67	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
68	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
69	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₄₁	—	2	1	no	—
71	C₄⋊Q₈×C₄₁	—	2	1	no	—
72	C₄⋊Q₈×C₄₁	—	2	1	no	—
73	C₄⋊Q₈×C₄₁	—	2	1	no	—
74	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
75	#127×C₄₁	D₄	1	1	yes	characteristic
76	#126×C₄₁	—	2	4	no	—
77	#125×C₄₁	—	2	2	no	—
78	#124×C₄₁	—	2	2	no	—
79	#124×C₄₁	—	2	4	no	—
80	#124×C₄₁	D₄	1	2	yes	—
81	#123×C₄₁	—	2	4	no	—
82	C₂²⋊Q₈×C₄₁	—	2	2	no	—
83	C₂²⋊Q₈×C₄₁	—	2	4	no	—
84	C₂²⋊Q₈×C₄₁	—	2	2	no	—
85	C₂²⋊Q₈×C₄₁	—	2	2	no	—
86	#121×C₄₁	—	2	2	no	—
87	#121×C₄₁	—	2	2	no	—
88	#121×C₄₁	—	2	2	no	—
89	#121×C₄₁	—	2	1	no	—
90	#121×C₄₁	—	2	1	no	—
91	#121×C₄₁	—	2	4	no	—
92	C₄×Q₈×C₄₁	—	2	2	no	—
93	C₄×Q₈×C₄₁	—	2	2	no	—
94	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
95	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
96	C₄×D₄×C₄₁	—	2	2	no	—
97	C₄×D₄×C₄₁	—	2	2	no	—
98	C₄×D₄×C₄₁	—	2	1	no	—
99	C₄×D₄×C₄₁	—	2	1	no	—
100	C₄×D₄×C₄₁	—	2	4	no	—
101	C₄×D₄×C₄₁	—	2	1	no	—
102	C₄×D₄×C₄₁	—	2	1	no	—
103	C₄×D₄×C₄₁	—	2	1	no	—
104	C₄×D₄×C₄₁	—	2	1	no	—
105	C₄×D₄×C₄₁	—	2	1	no	—
106	C₄×D₄×C₄₁	—	2	1	no	—
107	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
108	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
109	#117×C₄₁	—	2	2	no	—
110	#117×C₄₁	—	2	2	no	—
111	#117×C₄₁	—	2	1	no	—
112	#117×C₄₁	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
114	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
115	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
116	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
119	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
120	C₂×C₄²×C₄₁	—	2	1	no	—
121	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
122	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
123	#107×C₄₁	—	2	1	no	—
124	#106×C₄₁	—	2	1	no	—
125	C₄⋊C₈×C₄₁	—	2	1	no	—
126	C₄⋊C₈×C₄₁	—	2	1	no	—
127	#104×C₄₁	—	2	4	no	—
128	#104×C₄₁	—	4	2	no	—
129	#104×C₄₁	—	4	2	no	—
130	#103×C₄₁	—	2	1	no	—
131	#103×C₄₁	—	2	2	no	—
132	#102×C₄₁	—	2	1	no	—
133	#102×C₄₁	—	2	2	no	—
134	#102×C₄₁	—	2	1	no	—
135	#102×C₄₁	—	2	1	no	—
136	#97×C₄₁	—	2	1	no	—
137	C₄×C₈×C₄₁	—	2	1	no	—
138	#95×C₄₁	—	2	1	no	—
139	#95×C₄₁	—	2	4	no	—
140	C₂⁴×C₄₁	—	2	1	no	—
141	#41×C₄₁	—	2	2	no	—
142	#41×C₄₁	—	2	2	no	—
143	#41×C₄₁	—	2	1	no	—
144	#41×C₄₁	—	2	1	no	—
145	#41×C₄₁	—	2	1	no	—
146	#41×C₄₁	—	2	2	no	—
147	#41×C₄₁	—	2	2	no	—
148	#41×C₄₁	—	2	1	no	—
149	#41×C₄₁	—	8	4	no	—
150	C₂×Q₈×C₄₁	—	2	2	no	—
151	C₂×Q₈×C₄₁	—	2	2	no	—
152	C₂×Q₈×C₄₁	—	8	2	no	—
153	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
154	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
155	C₂×D₄×C₄₁	—	2	1	no	—
156	C₂×D₄×C₄₁	—	2	2	no	—
157	C₂×D₄×C₄₁	—	2	4	no	—
158	C₂×D₄×C₄₁	—	2	1	no	—
159	C₂×D₄×C₄₁	—	2	1	no	—
160	C₂×D₄×C₄₁	—	2	1	no	—
161	C₂×D₄×C₄₁	—	2	1	no	—
162	C₂×D₄×C₄₁	—	2	1	no	—
163	C₂×D₄×C₄₁	—	2	1	no	—
164	C₂×D₄×C₄₁	—	2	1	no	—
165	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
166	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
167	C₂×D₄×C₄₁	#41	1	2	yes	complemented
168	C₂²×C₄×C₄₁	—	2	4	no	—
169	C₂²×C₄×C₄₁	—	2	1	no	—
170	C₂²×C₄×C₄₁	—	2	1	no	—
171	C₂²×C₄×C₄₁	—	2	1	no	—
172	C₂²×C₄×C₄₁	—	2	2	no	—
173	C₂²×C₄×C₄₁	—	2	1	no	—
174	C₂²×C₄×C₄₁	—	2	2	no	—
175	C₂²×C₄×C₄₁	—	4	4	no	—
176	C₂²×C₄×C₄₁	—	4	1	no	—
177	C₂²×C₄×C₄₁	—	4	1	no	—
178	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
180	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
182	SD₁₆×C₄₁	—	2	4	no	—
183	SD₁₆×C₄₁	—	4	4	no	—
184	D₈×C₄₁	—	2	4	no	—
185	D₈×C₄₁	—	4	4	no	—
186	M₁₆×C₄₁	—	2	1	no	—
187	M₁₆×C₄₁	—	2	2	no	—
188	M₁₆×C₄₁	—	2	1	no	—
189	M₁₆×C₄₁	—	2	2	no	—
190	M₁₆×C₄₁	—	2	1	no	—
191	M₁₆×C₄₁	#41	1	2	yes	complemented
192	C₂×C₈×C₄₁	—	2	1	no	—
193	C₂×C₈×C₄₁	—	2	1	no	—
194	C₂×C₈×C₄₁	—	2	2	no	—
195	C₄⋊C₄×C₄₁	—	2	4	no	—
196	C₄⋊C₄×C₄₁	—	2	4	no	—
197	C₄⋊C₄×C₄₁	—	2	2	no	—
198	C₄⋊C₄×C₄₁	—	2	2	no	—
199	C₄⋊C₄×C₄₁	—	2	1	no	—
200	C₄⋊C₄×C₄₁	—	2	1	no	—
201	C₄⋊C₄×C₄₁	—	2	1	no	—
202	C₄⋊C₄×C₄₁	—	2	1	no	—
203	C₄⋊C₄×C₄₁	—	2	2	no	—
204	C₄⋊C₄×C₄₁	—	2	2	no	—
205	C₄⋊C₄×C₄₁	—	2	1	no	—
206	C₄⋊C₄×C₄₁	—	2	1	no	—
207	C₄⋊C₄×C₄₁	—	4	2	no	—
208	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
209	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
210	C₂²⋊C₄×C₄₁	—	2	4	no	—
211	C₂²⋊C₄×C₄₁	—	2	4	no	—
212	C₂²⋊C₄×C₄₁	—	2	4	no	—
213	C₂²⋊C₄×C₄₁	—	2	1	no	—
214	C₂²⋊C₄×C₄₁	—	2	1	no	—
215	C₂²⋊C₄×C₄₁	—	2	1	no	—
216	C₂²⋊C₄×C₄₁	—	2	2	no	—
217	C₂²⋊C₄×C₄₁	—	2	2	no	—
218	C₂²⋊C₄×C₄₁	—	4	1	no	—
219	C₂²⋊C₄×C₄₁	—	4	1	no	—
220	C₂²⋊C₄×C₄₁	—	8	4	no	—
221	C₄²×C₄₁	—	2	4	no	—
222	C₄²×C₄₁	—	2	1	no	—
223	C₄²×C₄₁	—	2	1	no	—
224	C₄²×C₄₁	—	2	1	no	—
225	C₄²×C₄₁	—	2	2	no	—
226	C₄²×C₄₁	—	2	2	no	—
227	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
228	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
229	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
230	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
231	C₂³×C₄₁	—	2	1	no	—
232	C₂³×C₄₁	—	2	2	no	—
233	C₂³×C₄₁	—	2	2	no	—
234	C₂³×C₄₁	—	4	2	no	—
235	C₂³×C₄₁	—	4	2	no	—
236	C₂³×C₄₁	—	4	1	no	—
237	C₂³×C₄₁	—	4	1	no	—
238	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
239	Q₈×C₄₁	—	2	1	no	—
240	Q₈×C₄₁	—	2	1	no	—
241	Q₈×C₄₁	—	2	2	no	—
242	Q₈×C₄₁	—	2	2	no	—
243	Q₈×C₄₁	—	4	4	no	—
244	Q₈×C₄₁	—	8	2	no	—
245	D₄×C₄₁	—	2	1	no	—
246	D₄×C₄₁	—	2	2	no	—
247	D₄×C₄₁	—	2	1	no	—
248	D₄×C₄₁	—	2	2	no	—
249	D₄×C₄₁	—	2	2	no	—
250	D₄×C₄₁	—	2	2	no	—
251	D₄×C₄₁	—	4	2	no	—
252	D₄×C₄₁	—	4	2	no	—
253	D₄×C₄₁	—	4	1	no	—
254	D₄×C₄₁	—	4	2	no	—
255	D₄×C₄₁	—	4	1	no	—
256	D₄×C₄₁	—	4	1	no	—
257	D₄×C₄₁	—	4	1	no	—
258	D₄×C₄₁	—	4	2	no	—
259	D₄×C₄₁	—	8	4	no	—
260	C₂×C₄×C₄₁	—	2	2	no	—
261	C₂×C₄×C₄₁	—	2	1	no	—
262	C₂×C₄×C₄₁	—	2	2	no	—
263	C₂×C₄×C₄₁	—	2	1	no	—
264	C₂×C₄×C₄₁	—	2	1	no	—
265	C₂×C₄×C₄₁	—	2	4	no	—
266	C₂×C₄×C₄₁	—	4	1	no	—
267	C₂×C₄×C₄₁	—	4	2	no	—
268	C₂×C₄×C₄₁	—	4	2	no	—
269	C₂×C₄×C₄₁	—	4	2	no	—
270	C₂×C₄×C₄₁	—	4	1	no	—
271	C₂×C₄×C₄₁	—	4	1	no	—
272	C₂×C₄×C₄₁	—	4	2	no	—
273	C₂×C₄×C₄₁	—	4	4	no	—
274	C₂×C₄×C₄₁	—	4	2	no	—
275	C₂×C₄×C₄₁	—	4	2	no	—
276	C₂×C₄×C₄₁	—	4	2	no	—
277	C₂×C₄×C₄₁	—	4	1	no	—
278	C₂×C₄×C₄₁	—	4	2	no	—
279	C₂×C₄×C₄₁	—	8	2	no	—
280	C₂×C₄×C₄₁	—	8	4	no	—
281	C₂×C₄×C₄₁	—	8	2	no	—
282	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
283	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
284	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
285	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
286	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
289	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
290	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
291	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
292	C₈×C₄₁	—	2	2	no	—
293	C₈×C₄₁	—	4	1	no	—
294	C₈×C₄₁	—	4	2	no	—
295	#24171	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
296	C₂²×C₄₁	—	2	1	no	—
297	C₂²×C₄₁	—	2	1	no	—
298	C₂²×C₄₁	—	4	4	no	—
299	C₂²×C₄₁	—	4	2	no	—
300	C₂²×C₄₁	—	4	1	no	—
301	C₂²×C₄₁	—	4	2	no	—
302	C₂²×C₄₁	—	4	1	no	—
303	C₂²×C₄₁	—	4	2	no	—
304	C₂²×C₄₁	—	4	1	no	—
305	C₂²×C₄₁	—	4	2	no	—
306	C₂²×C₄₁	—	8	2	no	—
307	C₂²×C₄₁	—	8	1	no	—
308	C₂²×C₄₁	—	8	1	no	—
309	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
310	C₂²×C₄₁	#548	1	1	yes	characteristic
311	C₂²×C₄₁	#550	1	1	yes	characteristic
312	C₄×C₄₁	—	2	1	no	—
313	C₄×C₄₁	—	2	1	no	—
314	C₄×C₄₁	—	2	1	no	—
315	C₄×C₄₁	—	4	2	no	—
316	C₄×C₄₁	—	4	1	no	—
317	C₄×C₄₁	—	4	1	no	—
318	C₄×C₄₁	—	4	1	no	—
319	C₄×C₄₁	—	4	4	no	—
320	C₄×C₄₁	—	8	2	no	—
321	C₄×C₄₁	—	8	1	no	—
322	C₄×C₄₁	#454	1	2	yes	—
323	C₄×C₄₁	#525	1	1	yes	characteristic
324	C₄×C₄₁	#537	1	1	yes	characteristic
325	C₄×C₄₁	#546	1	1	yes	characteristic
326	C₄×C₄₁	#547	1	1	yes	characteristic
327	#3511	C₂×C₄₁	1	1	yes	characteristic, complemented
328	#3036	C₂×C₄₁	1	1	yes	characteristic, complemented
329	C₄×AGL(1,ℤ/8)	C₂×C₄₁	1	1	yes	characteristic, complemented
330	#2563	C₂×C₄₁	1	1	yes	characteristic, complemented
331	#2040	C₂×C₄₁	1	2	yes	complemented
332	#2006	C₂×C₄₁	1	2	yes	complemented
333	#1997	C₂×C₄₁	1	1	yes	characteristic, complemented
334	#1980	C₂×C₄₁	1	1	yes	characteristic, complemented
335	#1883	C₂×C₄₁	1	2	yes	complemented
336	#1860	C₂×C₄₁	1	2	yes	complemented
337	#1806	C₂×C₄₁	1	1	yes	characteristic, complemented
338	C₂×C₄₁	—	2	1	no	—
339	C₂×C₄₁	—	2	1	no	—
340	C₂×C₄₁	—	8	1	no	—
341	C₂×C₄₁	—	8	2	no	—
342	C₂×C₄₁	—	8	2	no	—
343	C₂×C₄₁	#2407	1	1	yes	characteristic
344	C₂×C₄₁	C₂×#454	1	1	yes	characteristic
345	C₂×C₄₁	#3019	1	1	yes	characteristic
346	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	1	yes	characteristic
347	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
348	#556	—	2	1	no	—
349	#555	—	2	1	no	—
350	#554	—	2	1	no	—
351	#553	—	2	2	no	—
352	#548	—	2	2	no	—
353	#547	—	2	1	no	—
354	#542	—	2	2	no	—
355	#537	C₂²×C₄₁	1	2	yes	—
356	#533	—	2	1	no	—
357	#532	—	2	1	no	—
358	#532	C₂²×C₄₁	1	1	yes	characteristic
359	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
360	#520	C₂²×C₄₁	1	2	yes	—
361	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
362	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
363	#463	—	2	2	no	—
364	#461	—	2	1	no	—
365	#459	—	2	1	no	—
366	#442	—	2	1	no	—
367	#440	—	2	1	no	—
368	C₄×SD₁₆	—	2	1	no	—
369	C₄×D₈	—	2	1	no	—
370	#428	C₂²×C₄₁	1	1	yes	characteristic
371	#423	C₂²×C₄₁	1	1	yes	characteristic
372	C₂×#104	C₂²×C₄₁	1	2	yes	—
373	C₂×#104	C₂²×C₄₁	1	2	yes	—
374	C₂×#104	C₂²×C₄₁	1	2	yes	—
375	#418	C₂²×C₄₁	1	1	yes	characteristic
376	#417	C₂²×C₄₁	1	2	yes	—
377	#417	C₂²×C₄₁	1	1	yes	characteristic
378	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
379	#393	—	2	2	no	—
380	#390	C₂²×C₄₁	1	2	yes	—
381	#382	C₂²×C₄₁	1	2	yes	—
382	#380	—	2	2	no	—
383	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
384	#343	C₂²×C₄₁	1	2	yes	—
385	#343	C₂²×C₄₁	1	2	yes	—
386	#339	C₂²×C₄₁	1	2	yes	—
387	C₄₁	#24171	1	1	yes	characteristic, direct product factor, Sylow
388	C₂×#41	—	2	2	no	—
389	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
390	C₂×#41	C₂³×C₄₁	1	2	yes	—
391	C₂²×Q₈	D₄×C₄₁	1	2	yes	—
392	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
393	C₂³×C₄	—	2	1	no	—
394	AGL(1,ℤ/8)	—	2	2	no	—
395	AGL(1,ℤ/8)	—	2	4	no	—
396	AGL(1,ℤ/8)	—	2	2	no	—
397	AGL(1,ℤ/8)	—	4	2	no	—
398	C₂×SD₁₆	—	2	1	no	—
399	C₂×SD₁₆	—	2	2	no	—
400	C₂×D₈	—	2	1	no	—
401	C₂×D₈	—	2	2	no	—
402	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
403	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
404	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
405	C₄⋊Q₈	—	2	1	no	—
406	C₄⋊Q₈	—	2	1	no	—
407	C₄⋊Q₈	—	2	1	no	—
408	C₄⋊Q₈	—	2	1	no	—
409	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
410	#127	D₄×C₄₁	1	1	yes	characteristic
411	#126	—	2	4	no	—
412	#125	—	2	2	no	—
413	#124	—	2	2	no	—
414	#124	—	2	4	no	—
415	#124	D₄×C₄₁	1	2	yes	—
416	#123	—	2	4	no	—
417	C₂²⋊Q₈	—	2	2	no	—
418	C₂²⋊Q₈	—	2	4	no	—
419	C₂²⋊Q₈	—	2	2	no	—
420	C₂²⋊Q₈	—	2	2	no	—
421	#121	—	2	2	no	—
422	#121	—	2	2	no	—
423	#121	—	2	2	no	—
424	#121	—	2	1	no	—
425	#121	—	2	1	no	—
426	#121	—	2	4	no	—
427	C₄×Q₈	—	2	2	no	—
428	C₄×Q₈	—	2	2	no	—
429	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
430	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
431	C₄×D₄	—	2	2	no	—
432	C₄×D₄	—	2	2	no	—
433	C₄×D₄	—	2	1	no	—
434	C₄×D₄	—	2	1	no	—
435	C₄×D₄	—	2	4	no	—
436	C₄×D₄	—	2	1	no	—
437	C₄×D₄	—	2	1	no	—
438	C₄×D₄	—	2	1	no	—
439	C₄×D₄	—	2	1	no	—
440	C₄×D₄	—	2	1	no	—
441	C₄×D₄	—	2	1	no	—
442	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
443	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
444	#117	—	2	2	no	—
445	#117	—	2	2	no	—
446	#117	—	2	1	no	—
447	#117	C₂³×C₄₁	1	1	yes	characteristic
448	C₂×C₄⋊C₄	—	2	2	no	—
449	C₂×C₄⋊C₄	—	2	1	no	—
450	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
451	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
452	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
453	C₂×C₂²⋊C₄	—	2	1	no	—
454	C₂×C₂²⋊C₄	—	2	4	no	—
455	C₂×C₄²	—	2	1	no	—
456	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
457	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
458	#107	—	2	1	no	—
459	#106	—	2	1	no	—
460	C₄⋊C₈	—	2	1	no	—
461	C₄⋊C₈	—	2	1	no	—
462	#104	—	2	4	no	—
463	#104	—	4	2	no	—
464	#104	—	4	2	no	—
465	#103	—	2	1	no	—
466	#103	—	2	2	no	—
467	#102	—	2	1	no	—
468	#102	—	2	2	no	—
469	#102	—	2	1	no	—
470	#102	—	2	1	no	—
471	#97	—	2	1	no	—
472	C₄×C₈	—	2	1	no	—
473	#95	—	2	1	no	—
474	#95	—	2	4	no	—
475	C₂⁴	—	2	1	no	—
476	#41	—	2	2	no	—
477	#41	—	2	2	no	—
478	#41	—	2	1	no	—
479	#41	—	2	1	no	—
480	#41	—	2	1	no	—
481	#41	—	2	2	no	—
482	#41	—	2	2	no	—
483	#41	—	2	1	no	—
484	#41	—	8	4	no	—
485	C₂×Q₈	—	2	2	no	—
486	C₂×Q₈	—	2	2	no	—
487	C₂×Q₈	—	8	2	no	—
488	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
489	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	2	2	no	—
492	C₂×D₄	—	2	4	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	2	1	no	—
496	C₂×D₄	—	2	1	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	1	no	—
499	C₂×D₄	—	2	1	no	—
500	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
501	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
502	C₂×D₄	#41×C₄₁	1	2	yes	complemented
503	C₂²×C₄	—	2	4	no	—
504	C₂²×C₄	—	2	1	no	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	2	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	2	2	no	—
510	C₂²×C₄	—	4	4	no	—
511	C₂²×C₄	—	4	1	no	—
512	C₂²×C₄	—	4	1	no	—
513	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
514	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
515	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
516	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
517	SD₁₆	—	2	4	no	—
518	SD₁₆	—	4	4	no	—
519	D₈	—	2	4	no	—
520	D₈	—	4	4	no	—
521	M₁₆	—	2	1	no	—
522	M₁₆	—	2	2	no	—
523	M₁₆	—	2	1	no	—
524	M₁₆	—	2	2	no	—
525	M₁₆	—	2	1	no	—
526	M₁₆	#41×C₄₁	1	2	yes	complemented
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	2	1	no	—
529	C₂×C₈	—	2	2	no	—
530	C₄⋊C₄	—	2	4	no	—
531	C₄⋊C₄	—	2	4	no	—
532	C₄⋊C₄	—	2	2	no	—
533	C₄⋊C₄	—	2	2	no	—
534	C₄⋊C₄	—	2	1	no	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	2	1	no	—
538	C₄⋊C₄	—	2	2	no	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	1	no	—
541	C₄⋊C₄	—	2	1	no	—
542	C₄⋊C₄	—	4	2	no	—
543	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
544	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
545	C₂²⋊C₄	—	2	4	no	—
546	C₂²⋊C₄	—	2	4	no	—
547	C₂²⋊C₄	—	2	4	no	—
548	C₂²⋊C₄	—	2	1	no	—
549	C₂²⋊C₄	—	2	1	no	—
550	C₂²⋊C₄	—	2	1	no	—
551	C₂²⋊C₄	—	2	2	no	—
552	C₂²⋊C₄	—	2	2	no	—
553	C₂²⋊C₄	—	4	1	no	—
554	C₂²⋊C₄	—	4	1	no	—
555	C₂²⋊C₄	—	8	4	no	—
556	C₄²	—	2	4	no	—
557	C₄²	—	2	1	no	—
558	C₄²	—	2	1	no	—
559	C₄²	—	2	1	no	—
560	C₄²	—	2	2	no	—
561	C₄²	—	2	2	no	—
562	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
563	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
564	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
565	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
566	C₂³	—	2	1	no	—
567	C₂³	—	2	2	no	—
568	C₂³	—	2	2	no	—
569	C₂³	—	4	2	no	—
570	C₂³	—	4	2	no	—
571	C₂³	—	4	1	no	—
572	C₂³	—	4	1	no	—
573	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
574	Q₈	—	2	1	no	—
575	Q₈	—	2	1	no	—
576	Q₈	—	2	2	no	—
577	Q₈	—	2	2	no	—
578	Q₈	—	4	4	no	—
579	Q₈	—	8	2	no	—
580	D₄	—	2	1	no	—
581	D₄	—	2	2	no	—
582	D₄	—	2	1	no	—
583	D₄	—	2	2	no	—
584	D₄	—	2	2	no	—
585	D₄	—	2	2	no	—
586	D₄	—	4	2	no	—
587	D₄	—	4	2	no	—
588	D₄	—	4	1	no	—
589	D₄	—	4	2	no	—
590	D₄	—	4	1	no	—
591	D₄	—	4	1	no	—
592	D₄	—	4	1	no	—
593	D₄	—	4	2	no	—
594	D₄	—	8	4	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	2	no	—
603	C₂×C₄	—	4	2	no	—
604	C₂×C₄	—	4	2	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	4	no	—
609	C₂×C₄	—	4	2	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	2	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	2	no	—
614	C₂×C₄	—	8	2	no	—
615	C₂×C₄	—	8	4	no	—
616	C₂×C₄	—	8	2	no	—
617	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
618	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
619	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
620	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
621	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
622	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
623	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
624	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
625	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
626	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
627	C₈	—	2	2	no	—
628	C₈	—	4	1	no	—
629	C₈	—	4	2	no	—
630	C₂²	—	2	1	no	—
631	C₂²	—	2	1	no	—
632	C₂²	—	4	4	no	—
633	C₂²	—	4	2	no	—
634	C₂²	—	4	1	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	4	1	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	4	1	no	—
639	C₂²	—	4	2	no	—
640	C₂²	—	8	2	no	—
641	C₂²	—	8	1	no	—
642	C₂²	—	8	1	no	—
643	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
644	C₂²	C₄₁×#548	1	1	yes	characteristic
645	C₂²	C₄₁×#550	1	1	yes	characteristic
646	C₄	—	2	1	no	—
647	C₄	—	2	1	no	—
648	C₄	—	2	1	no	—
649	C₄	—	4	2	no	—
650	C₄	—	4	1	no	—
651	C₄	—	4	1	no	—
652	C₄	—	4	1	no	—
653	C₄	—	4	4	no	—
654	C₄	—	8	2	no	—
655	C₄	—	8	1	no	—
656	C₄	C₄₁×#454	1	2	yes	—
657	C₄	C₄₁×#525	1	1	yes	characteristic
658	C₄	C₄₁×#537	1	1	yes	characteristic
659	C₄	C₄₁×#546	1	1	yes	characteristic
660	C₄	C₄₁×#547	1	1	yes	characteristic
661	C₂	—	2	1	no	—
662	C₂	—	2	1	no	—
663	C₂	—	8	2	no	—
664	C₂	—	8	1	no	—
665	C₂	—	8	2	no	—
666	C₂	C₄₁×#2407	1	1	yes	characteristic
667	C₂	C₂×C₄₁×#454	1	1	yes	characteristic
668	C₂	C₄₁×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
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×#3511	C₂	semidirect
2. C₄₁×#3036	C₂	semidirect
3. C₄×AGL(1,ℤ/8)×C₄₁	C₂	semidirect
4. C₄₁×#2563	C₂	semidirect
5. C₄₁×#2040	C₂	semidirect
6. C₄₁×#2006	C₂	semidirect
7. C₄₁×#1997	C₂	semidirect
8. C₄₁×#1980	C₂	semidirect
9. C₄₁×#1883	C₂	semidirect
10. C₄₁×#1860	C₂	semidirect
11. C₄₁×#1806	C₂	semidirect
167. C₂×D₄×C₄₁	#41	semidirect
191. M₁₆×C₄₁	#41	semidirect
295. #24171	C₄₁	direct (N × H)
327. #3511	C₂×C₄₁	semidirect
328. #3036	C₂×C₄₁	semidirect
329. C₄×AGL(1,ℤ/8)	C₂×C₄₁	semidirect
330. #2563	C₂×C₄₁	semidirect
331. #2040	C₂×C₄₁	semidirect
332. #2006	C₂×C₄₁	semidirect
333. #1997	C₂×C₄₁	semidirect
334. #1980	C₂×C₄₁	semidirect
335. #1883	C₂×C₄₁	semidirect
336. #1860	C₂×C₄₁	semidirect
337. #1806	C₂×C₄₁	semidirect
387. C₄₁	#24171	direct (N × H)
502. C₂×D₄	#41×C₄₁	semidirect
526. M₁₆	#41×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3511	2
C₄₁×#3036	2
C₄×AGL(1,ℤ/8)×C₄₁	2
C₄₁×#2563	2
C₄₁×#2040	2
C₄₁×#2006	2
C₄₁×#1997	2
C₄₁×#1980	2
C₄₁×#1883	2
C₄₁×#1860	2
C₄₁×#1806	2
#24171	41

Subgroups — C41×#24171 #23800372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24171 #23800372