Subgroups — C₄₁×#24176 #23800377

Order 10496.

1238 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×#3506	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3077	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3036	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#127×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2037	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2002	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1994	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1962	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1883	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1859	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1800	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
13	C₄₁×#559	—	2	2	no	—
14	C₄₁×#556	—	2	1	no	—
15	C₄₁×#555	—	2	1	no	—
16	C₄₁×#553	—	2	2	no	—
17	C₄₁×#550	—	2	1	no	—
18	C₄₁×#547	—	2	1	no	—
19	C₄₁×#538	—	2	2	no	—
20	C₄₁×#535	C₂²	1	2	yes	—
21	C₄₁×#533	C₂²	1	1	yes	characteristic
22	C₄₁×#532	C₂²	1	1	yes	characteristic
23	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#525	—	2	2	no	—
25	C₄₁×#520	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₄₁	—	2	2	no	—
27	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
29	C₄₁×#478	—	2	2	no	—
30	C₄₁×#476	—	2	1	no	—
31	C₄₁×#474	—	2	1	no	—
32	C₄₁×#463	—	2	2	no	—
33	C₄₁×#461	—	2	1	no	—
34	C₄₁×#459	—	2	1	no	—
35	C₄₁×#428	C₂²	1	2	yes	—
36	C₄₁×#423	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	C₄₁×#418	C₂²	1	2	yes	—
41	C₄₁×#417	C₂²	1	2	yes	—
42	C₂×#100×C₄₁	C₂²	1	2	yes	—
43	C₄₁×#390	—	2	2	no	—
44	C₄₁×#390	C₂²	1	1	yes	characteristic
45	C₄₁×#390	C₂²	1	1	yes	characteristic
46	C₄₁×#384	C₂²	1	1	yes	characteristic
47	C₄³×C₄₁	C₂²	1	1	yes	characteristic
48	C₄₁×#343	C₂²	1	2	yes	—
49	C₄₁×#339	C₂²	1	2	yes	—
50	C₄₁×#339	C₂²	1	2	yes	—
51	C₂×#41×C₄₁	—	2	2	no	—
52	C₂×#41×C₄₁	C₂³	1	2	yes	—
53	C₂²×D₄×C₄₁	—	2	2	no	—
54	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
55	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₄₁	—	2	1	no	—
57	C₂³×C₄×C₄₁	D₄	1	2	yes	—
58	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
59	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
60	C₂×SD₁₆×C₄₁	—	2	2	no	—
61	C₂×D₈×C₄₁	—	2	2	no	—
62	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
63	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic, complemented
65	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic, complemented
66	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic, complemented
67	#127×C₄₁	—	2	2	no	—
68	#127×C₄₁	—	2	2	no	—
69	#127×C₄₁	—	2	2	no	—
70	#127×C₄₁	D₄	1	1	yes	characteristic, complemented
71	#126×C₄₁	—	2	4	no	—
72	#125×C₄₁	D₄	1	2	yes	complemented
73	#124×C₄₁	—	2	4	no	—
74	#124×C₄₁	—	2	2	no	—
75	#124×C₄₁	D₄	1	2	yes	complemented
76	#123×C₄₁	—	2	4	no	—
77	#123×C₄₁	—	2	4	no	—
78	C₂²⋊Q₈×C₄₁	—	2	2	no	—
79	C₂²⋊Q₈×C₄₁	—	2	1	no	—
80	C₂²⋊Q₈×C₄₁	—	2	1	no	—
81	#121×C₄₁	—	2	2	no	—
82	#121×C₄₁	—	2	4	no	—
83	#121×C₄₁	—	2	1	no	—
84	#121×C₄₁	—	2	1	no	—
85	#121×C₄₁	—	2	2	no	—
86	#121×C₄₁	—	2	2	no	—
87	#120×C₄₁	—	2	4	no	—
88	C₄×Q₈×C₄₁	—	2	2	no	—
89	C₄×Q₈×C₄₁	—	2	1	no	—
90	C₄×Q₈×C₄₁	—	2	1	no	—
91	C₄×D₄×C₄₁	—	2	4	no	—
92	C₄×D₄×C₄₁	—	2	2	no	—
93	C₄×D₄×C₄₁	—	2	4	no	—
94	C₄×D₄×C₄₁	—	2	1	no	—
95	C₄×D₄×C₄₁	—	2	1	no	—
96	C₄×D₄×C₄₁	—	2	1	no	—
97	C₄×D₄×C₄₁	—	2	1	no	—
98	C₄×D₄×C₄₁	—	2	1	no	—
99	C₄×D₄×C₄₁	—	2	1	no	—
100	C₄×D₄×C₄₁	—	2	1	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	—	4	8	no	—
105	#117×C₄₁	—	2	2	no	—
106	#117×C₄₁	—	2	2	no	—
107	#117×C₄₁	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
109	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
110	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
112	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
116	C₂×C₄²×C₄₁	—	2	1	no	—
117	C₂×C₄²×C₄₁	—	2	1	no	—
118	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
121	#107×C₄₁	—	2	2	no	—
122	#106×C₄₁	—	2	2	no	—
123	C₄⋊C₈×C₄₁	—	2	1	no	—
124	C₄⋊C₈×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	#103×C₄₁	—	2	2	no	—
130	#102×C₄₁	—	2	2	no	—
131	#102×C₄₁	—	2	2	no	—
132	#102×C₄₁	—	2	2	no	—
133	#100×C₄₁	—	4	2	no	—
134	C₂⁴×C₄₁	—	2	1	no	—
135	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
136	#41×C₄₁	—	2	2	no	—
137	#41×C₄₁	—	2	2	no	—
138	#41×C₄₁	—	2	2	no	—
139	#41×C₄₁	—	2	2	no	—
140	#41×C₄₁	—	8	4	no	—
141	C₂×Q₈×C₄₁	—	2	1	no	—
142	C₂×Q₈×C₄₁	—	2	1	no	—
143	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
144	C₂×D₄×C₄₁	—	2	8	no	—
145	C₂×D₄×C₄₁	—	2	2	no	—
146	C₂×D₄×C₄₁	—	2	4	no	—
147	C₂×D₄×C₄₁	—	2	4	no	—
148	C₂×D₄×C₄₁	—	2	1	no	—
149	C₂×D₄×C₄₁	—	2	1	no	—
150	C₂×D₄×C₄₁	—	2	1	no	—
151	C₂×D₄×C₄₁	—	2	1	no	—
152	C₂×D₄×C₄₁	—	2	1	no	—
153	C₂×D₄×C₄₁	—	2	1	no	—
154	C₂×D₄×C₄₁	—	2	1	no	—
155	C₂×D₄×C₄₁	—	2	1	no	—
156	C₂×D₄×C₄₁	—	2	1	no	—
157	C₂×D₄×C₄₁	—	2	1	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	—	4	4	no	—
162	C₂×D₄×C₄₁	—	4	4	no	—
163	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
164	C₂×D₄×C₄₁	#41	1	2	yes	complemented
165	C₂²×C₄×C₄₁	—	2	1	no	—
166	C₂²×C₄×C₄₁	—	2	1	no	—
167	C₂²×C₄×C₄₁	—	2	4	no	—
168	C₂²×C₄×C₄₁	—	2	2	no	—
169	C₂²×C₄×C₄₁	—	2	2	no	—
170	C₂²×C₄×C₄₁	—	2	2	no	—
171	C₂²×C₄×C₄₁	—	2	1	no	—
172	C₂²×C₄×C₄₁	—	4	4	no	—
173	C₂²×C₄×C₄₁	—	4	4	no	—
174	C₂²×C₄×C₄₁	—	4	1	no	—
175	C₂²×C₄×C₄₁	—	4	1	no	—
176	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
178	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
180	SD₁₆×C₄₁	—	4	4	no	—
181	D₈×C₄₁	—	4	4	no	—
182	M₁₆×C₄₁	—	2	2	no	—
183	M₁₆×C₄₁	—	2	2	no	—
184	M₁₆×C₄₁	—	2	2	no	—
185	M₁₆×C₄₁	—	2	2	no	—
186	C₂×C₈×C₄₁	—	2	2	no	—
187	C₂×C₈×C₄₁	—	2	2	no	—
188	C₄⋊C₄×C₄₁	—	2	4	no	—
189	C₄⋊C₄×C₄₁	—	2	1	no	—
190	C₄⋊C₄×C₄₁	—	2	1	no	—
191	C₄⋊C₄×C₄₁	—	2	2	no	—
192	C₄⋊C₄×C₄₁	—	2	1	no	—
193	C₄⋊C₄×C₄₁	—	2	2	no	—
194	C₄⋊C₄×C₄₁	—	2	1	no	—
195	C₄⋊C₄×C₄₁	—	2	1	no	—
196	C₄⋊C₄×C₄₁	—	2	1	no	—
197	C₄⋊C₄×C₄₁	—	2	1	no	—
198	C₄⋊C₄×C₄₁	—	2	1	no	—
199	C₄⋊C₄×C₄₁	—	4	4	no	—
200	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
201	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
202	C₂²⋊C₄×C₄₁	—	2	4	no	—
203	C₂²⋊C₄×C₄₁	—	2	4	no	—
204	C₂²⋊C₄×C₄₁	—	2	4	no	—
205	C₂²⋊C₄×C₄₁	—	2	2	no	—
206	C₂²⋊C₄×C₄₁	—	2	2	no	—
207	C₂²⋊C₄×C₄₁	—	2	2	no	—
208	C₂²⋊C₄×C₄₁	—	2	2	no	—
209	C₂²⋊C₄×C₄₁	—	2	2	no	—
210	C₂²⋊C₄×C₄₁	—	4	4	no	—
211	C₂²⋊C₄×C₄₁	—	4	4	no	—
212	C₂²⋊C₄×C₄₁	—	4	1	no	—
213	C₂²⋊C₄×C₄₁	—	4	1	no	—
214	C₄²×C₄₁	—	2	4	no	—
215	C₄²×C₄₁	—	2	4	no	—
216	C₄²×C₄₁	—	2	2	no	—
217	C₄²×C₄₁	—	2	1	no	—
218	C₄²×C₄₁	—	2	1	no	—
219	C₄²×C₄₁	—	2	1	no	—
220	C₄²×C₄₁	—	4	2	no	—
221	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
222	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
223	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
224	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
225	C₄²×C₄₁	#41	1	2	yes	complemented
226	C₂³×C₄₁	—	2	2	no	—
227	C₂³×C₄₁	—	2	2	no	—
228	C₂³×C₄₁	—	2	2	no	—
229	C₂³×C₄₁	—	2	2	no	—
230	C₂³×C₄₁	—	2	2	no	—
231	C₂³×C₄₁	—	4	2	no	—
232	C₂³×C₄₁	—	4	2	no	—
233	C₂³×C₄₁	—	4	2	no	—
234	C₂³×C₄₁	—	4	2	no	—
235	C₂³×C₄₁	—	4	1	no	—
236	C₂³×C₄₁	—	4	1	no	—
237	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
238	Q₈×C₄₁	—	2	2	no	—
239	Q₈×C₄₁	—	2	2	no	—
240	Q₈×C₄₁	—	4	2	no	—
241	Q₈×C₄₁	—	8	1	no	—
242	D₄×C₄₁	—	2	2	no	—
243	D₄×C₄₁	—	2	2	no	—
244	D₄×C₄₁	—	2	2	no	—
245	D₄×C₄₁	—	2	2	no	—
246	D₄×C₄₁	—	4	16	no	—
247	D₄×C₄₁	—	4	2	no	—
248	D₄×C₄₁	—	4	2	no	—
249	D₄×C₄₁	—	4	1	no	—
250	D₄×C₄₁	—	4	1	no	—
251	D₄×C₄₁	—	4	1	no	—
252	D₄×C₄₁	—	4	1	no	—
253	D₄×C₄₁	—	4	1	no	—
254	D₄×C₄₁	—	4	1	no	—
255	D₄×C₄₁	—	4	2	no	—
256	D₄×C₄₁	—	8	1	no	—
257	D₄×C₄₁	—	8	4	no	—
258	D₄×C₄₁	—	8	4	no	—
259	C₂×C₄×C₄₁	—	2	2	no	—
260	C₂×C₄×C₄₁	—	2	2	no	—
261	C₂×C₄×C₄₁	—	2	2	no	—
262	C₂×C₄×C₄₁	—	2	2	no	—
263	C₂×C₄×C₄₁	—	2	2	no	—
264	C₂×C₄×C₄₁	—	2	2	no	—
265	C₂×C₄×C₄₁	—	2	2	no	—
266	C₂×C₄×C₄₁	—	2	2	no	—
267	C₂×C₄×C₄₁	—	2	2	no	—
268	C₂×C₄×C₄₁	—	4	1	no	—
269	C₂×C₄×C₄₁	—	4	1	no	—
270	C₂×C₄×C₄₁	—	4	1	no	—
271	C₂×C₄×C₄₁	—	4	2	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	1	no	—
277	C₂×C₄×C₄₁	—	8	2	no	—
278	C₂×C₄×C₄₁	—	8	2	no	—
279	C₂×C₄×C₄₁	—	8	4	no	—
280	C₂×C₄×C₄₁	—	8	2	no	—
281	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
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	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
288	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
289	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
290	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
291	C₈×C₄₁	—	4	2	no	—
292	C₈×C₄₁	—	4	2	no	—
293	#24176	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
294	C₂²×C₄₁	—	2	1	no	—
295	C₂²×C₄₁	—	2	1	no	—
296	C₂²×C₄₁	—	4	4	no	—
297	C₂²×C₄₁	—	4	4	no	—
298	C₂²×C₄₁	—	4	2	no	—
299	C₂²×C₄₁	—	4	2	no	—
300	C₂²×C₄₁	—	4	2	no	—
301	C₂²×C₄₁	—	4	2	no	—
302	C₂²×C₄₁	—	4	2	no	—
303	C₂²×C₄₁	—	4	2	no	—
304	C₂²×C₄₁	—	4	2	no	—
305	C₂²×C₄₁	—	8	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	#525	1	1	yes	characteristic
311	C₂²×C₄₁	#537	1	1	yes	characteristic
312	C₄×C₄₁	—	2	4	no	—
313	C₄×C₄₁	—	2	1	no	—
314	C₄×C₄₁	—	2	1	no	—
315	C₄×C₄₁	—	2	1	no	—
316	C₄×C₄₁	—	4	2	no	—
317	C₄×C₄₁	—	4	2	no	—
318	C₄×C₄₁	—	4	2	no	—
319	C₄×C₄₁	—	4	2	no	—
320	C₄×C₄₁	—	4	2	no	—
321	C₄×C₄₁	—	8	1	no	—
322	C₄×C₄₁	#453	1	2	yes	—
323	C₄×C₄₁	#546	1	1	yes	characteristic
324	C₄×C₄₁	#547	1	1	yes	characteristic
325	C₄×C₄₁	#548	1	1	yes	characteristic
326	C₄×C₄₁	#550	1	1	yes	characteristic
327	#3506	C₂×C₄₁	1	1	yes	characteristic, complemented
328	#3077	C₂×C₄₁	1	1	yes	characteristic, complemented
329	#3036	C₂×C₄₁	1	1	yes	characteristic, complemented
330	C₄×#127	C₂×C₄₁	1	1	yes	characteristic, complemented
331	#2037	C₂×C₄₁	1	2	yes	complemented
332	#2002	C₂×C₄₁	1	2	yes	complemented
333	#1994	C₂×C₄₁	1	1	yes	characteristic, complemented
334	#1962	C₂×C₄₁	1	1	yes	characteristic, complemented
335	#1883	C₂×C₄₁	1	2	yes	complemented
336	#1859	C₂×C₄₁	1	2	yes	complemented
337	#1800	C₂×C₄₁	1	1	yes	characteristic, complemented
338	C₂×C₄₁	—	2	1	no	—
339	C₂×C₄₁	—	2	1	no	—
340	C₂×C₄₁	—	8	2	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×#453	1	1	yes	characteristic
345	C₂×C₄₁	#3018	1	1	yes	characteristic
346	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
347	#559	—	2	2	no	—
348	#556	—	2	1	no	—
349	#555	—	2	1	no	—
350	#553	—	2	2	no	—
351	#550	—	2	1	no	—
352	#547	—	2	1	no	—
353	#538	—	2	2	no	—
354	#535	C₂²×C₄₁	1	2	yes	—
355	#533	C₂²×C₄₁	1	1	yes	characteristic
356	#532	C₂²×C₄₁	1	1	yes	characteristic
357	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
358	#525	—	2	2	no	—
359	#520	C₂²×C₄₁	1	2	yes	—
360	C₂×C₄×D₄	—	2	2	no	—
361	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
362	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
363	#478	—	2	2	no	—
364	#476	—	2	1	no	—
365	#474	—	2	1	no	—
366	#463	—	2	2	no	—
367	#461	—	2	1	no	—
368	#459	—	2	1	no	—
369	#428	C₂²×C₄₁	1	2	yes	—
370	#423	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	#418	C₂²×C₄₁	1	2	yes	—
375	#417	C₂²×C₄₁	1	2	yes	—
376	C₂×#100	C₂²×C₄₁	1	2	yes	—
377	#390	—	2	2	no	—
378	#390	C₂²×C₄₁	1	1	yes	characteristic
379	#390	C₂²×C₄₁	1	1	yes	characteristic
380	#384	C₂²×C₄₁	1	1	yes	characteristic
381	C₄³	C₂²×C₄₁	1	1	yes	characteristic
382	#343	C₂²×C₄₁	1	2	yes	—
383	#339	C₂²×C₄₁	1	2	yes	—
384	#339	C₂²×C₄₁	1	2	yes	—
385	C₄₁	#24176	1	1	yes	characteristic, direct product factor, Sylow
386	C₂×#41	—	2	2	no	—
387	C₂×#41	C₂³×C₄₁	1	2	yes	—
388	C₂²×D₄	—	2	2	no	—
389	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
390	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
391	C₂³×C₄	—	2	1	no	—
392	C₂³×C₄	D₄×C₄₁	1	2	yes	—
393	AGL(1,ℤ/8)	—	2	4	no	—
394	AGL(1,ℤ/8)	—	4	2	no	—
395	C₂×SD₁₆	—	2	2	no	—
396	C₂×D₈	—	2	2	no	—
397	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
398	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
399	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic, complemented
400	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic, complemented
401	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic, complemented
402	#127	—	2	2	no	—
403	#127	—	2	2	no	—
404	#127	—	2	2	no	—
405	#127	D₄×C₄₁	1	1	yes	characteristic, complemented
406	#126	—	2	4	no	—
407	#125	D₄×C₄₁	1	2	yes	complemented
408	#124	—	2	4	no	—
409	#124	—	2	2	no	—
410	#124	D₄×C₄₁	1	2	yes	complemented
411	#123	—	2	4	no	—
412	#123	—	2	4	no	—
413	C₂²⋊Q₈	—	2	2	no	—
414	C₂²⋊Q₈	—	2	1	no	—
415	C₂²⋊Q₈	—	2	1	no	—
416	#121	—	2	2	no	—
417	#121	—	2	4	no	—
418	#121	—	2	1	no	—
419	#121	—	2	1	no	—
420	#121	—	2	2	no	—
421	#121	—	2	2	no	—
422	#120	—	2	4	no	—
423	C₄×Q₈	—	2	2	no	—
424	C₄×Q₈	—	2	1	no	—
425	C₄×Q₈	—	2	1	no	—
426	C₄×D₄	—	2	4	no	—
427	C₄×D₄	—	2	2	no	—
428	C₄×D₄	—	2	4	no	—
429	C₄×D₄	—	2	1	no	—
430	C₄×D₄	—	2	1	no	—
431	C₄×D₄	—	2	1	no	—
432	C₄×D₄	—	2	1	no	—
433	C₄×D₄	—	2	1	no	—
434	C₄×D₄	—	2	1	no	—
435	C₄×D₄	—	2	1	no	—
436	C₄×D₄	—	2	1	no	—
437	C₄×D₄	—	2	1	no	—
438	C₄×D₄	—	2	1	no	—
439	C₄×D₄	—	4	8	no	—
440	#117	—	2	2	no	—
441	#117	—	2	2	no	—
442	#117	C₂³×C₄₁	1	2	yes	—
443	C₂×C₄⋊C₄	—	2	2	no	—
444	C₂×C₄⋊C₄	—	2	2	no	—
445	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
447	C₂×C₂²⋊C₄	—	2	2	no	—
448	C₂×C₂²⋊C₄	—	2	2	no	—
449	C₂×C₂²⋊C₄	—	2	1	no	—
450	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
451	C₂×C₄²	—	2	1	no	—
452	C₂×C₄²	—	2	1	no	—
453	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
454	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
455	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
456	#107	—	2	2	no	—
457	#106	—	2	2	no	—
458	C₄⋊C₈	—	2	1	no	—
459	C₄⋊C₈	—	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	#103	—	2	2	no	—
465	#102	—	2	2	no	—
466	#102	—	2	2	no	—
467	#102	—	2	2	no	—
468	#100	—	4	2	no	—
469	C₂⁴	—	2	1	no	—
470	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
471	#41	—	2	2	no	—
472	#41	—	2	2	no	—
473	#41	—	2	2	no	—
474	#41	—	2	2	no	—
475	#41	—	8	4	no	—
476	C₂×Q₈	—	2	1	no	—
477	C₂×Q₈	—	2	1	no	—
478	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
479	C₂×D₄	—	2	8	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	4	no	—
482	C₂×D₄	—	2	4	no	—
483	C₂×D₄	—	2	1	no	—
484	C₂×D₄	—	2	1	no	—
485	C₂×D₄	—	2	1	no	—
486	C₂×D₄	—	2	1	no	—
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	—	2	1	no	—
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	2	1	no	—
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	2	1	no	—
496	C₂×D₄	—	4	4	no	—
497	C₂×D₄	—	4	4	no	—
498	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
499	C₂×D₄	#41×C₄₁	1	2	yes	complemented
500	C₂²×C₄	—	2	1	no	—
501	C₂²×C₄	—	2	1	no	—
502	C₂²×C₄	—	2	4	no	—
503	C₂²×C₄	—	2	2	no	—
504	C₂²×C₄	—	2	2	no	—
505	C₂²×C₄	—	2	2	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	4	4	no	—
508	C₂²×C₄	—	4	4	no	—
509	C₂²×C₄	—	4	1	no	—
510	C₂²×C₄	—	4	1	no	—
511	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
512	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
513	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
514	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
515	SD₁₆	—	4	4	no	—
516	D₈	—	4	4	no	—
517	M₁₆	—	2	2	no	—
518	M₁₆	—	2	2	no	—
519	M₁₆	—	2	2	no	—
520	M₁₆	—	2	2	no	—
521	C₂×C₈	—	2	2	no	—
522	C₂×C₈	—	2	2	no	—
523	C₄⋊C₄	—	2	4	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	—	2	1	no	—
530	C₄⋊C₄	—	2	1	no	—
531	C₄⋊C₄	—	2	1	no	—
532	C₄⋊C₄	—	2	1	no	—
533	C₄⋊C₄	—	2	1	no	—
534	C₄⋊C₄	—	4	4	no	—
535	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
536	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	4	no	—
540	C₂²⋊C₄	—	2	2	no	—
541	C₂²⋊C₄	—	2	2	no	—
542	C₂²⋊C₄	—	2	2	no	—
543	C₂²⋊C₄	—	2	2	no	—
544	C₂²⋊C₄	—	2	2	no	—
545	C₂²⋊C₄	—	4	4	no	—
546	C₂²⋊C₄	—	4	4	no	—
547	C₂²⋊C₄	—	4	1	no	—
548	C₂²⋊C₄	—	4	1	no	—
549	C₄²	—	2	4	no	—
550	C₄²	—	2	4	no	—
551	C₄²	—	2	2	no	—
552	C₄²	—	2	1	no	—
553	C₄²	—	2	1	no	—
554	C₄²	—	2	1	no	—
555	C₄²	—	4	2	no	—
556	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
557	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
558	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
559	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
560	C₄²	#41×C₄₁	1	2	yes	complemented
561	C₂³	—	2	2	no	—
562	C₂³	—	2	2	no	—
563	C₂³	—	2	2	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	2	2	no	—
566	C₂³	—	4	2	no	—
567	C₂³	—	4	2	no	—
568	C₂³	—	4	2	no	—
569	C₂³	—	4	2	no	—
570	C₂³	—	4	1	no	—
571	C₂³	—	4	1	no	—
572	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
573	Q₈	—	2	2	no	—
574	Q₈	—	2	2	no	—
575	Q₈	—	4	2	no	—
576	Q₈	—	8	1	no	—
577	D₄	—	2	2	no	—
578	D₄	—	2	2	no	—
579	D₄	—	2	2	no	—
580	D₄	—	2	2	no	—
581	D₄	—	4	16	no	—
582	D₄	—	4	2	no	—
583	D₄	—	4	2	no	—
584	D₄	—	4	1	no	—
585	D₄	—	4	1	no	—
586	D₄	—	4	1	no	—
587	D₄	—	4	1	no	—
588	D₄	—	4	1	no	—
589	D₄	—	4	1	no	—
590	D₄	—	4	2	no	—
591	D₄	—	8	1	no	—
592	D₄	—	8	4	no	—
593	D₄	—	8	4	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	2	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	1	no	—
612	C₂×C₄	—	8	2	no	—
613	C₂×C₄	—	8	2	no	—
614	C₂×C₄	—	8	4	no	—
615	C₂×C₄	—	8	2	no	—
616	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
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	C₂²×D₄×C₄₁	1	1	yes	characteristic
621	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
622	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
623	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
624	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
625	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
626	C₈	—	4	2	no	—
627	C₈	—	4	2	no	—
628	C₂²	—	2	1	no	—
629	C₂²	—	2	1	no	—
630	C₂²	—	4	4	no	—
631	C₂²	—	4	4	no	—
632	C₂²	—	4	2	no	—
633	C₂²	—	4	2	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	4	2	no	—
639	C₂²	—	8	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×#525	1	1	yes	characteristic
645	C₂²	C₄₁×#537	1	1	yes	characteristic
646	C₄	—	2	4	no	—
647	C₄	—	2	1	no	—
648	C₄	—	2	1	no	—
649	C₄	—	2	1	no	—
650	C₄	—	4	2	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	2	no	—
653	C₄	—	4	2	no	—
654	C₄	—	4	2	no	—
655	C₄	—	8	1	no	—
656	C₄	C₄₁×#453	1	2	yes	—
657	C₄	C₄₁×#546	1	1	yes	characteristic
658	C₄	C₄₁×#547	1	1	yes	characteristic
659	C₄	C₄₁×#548	1	1	yes	characteristic
660	C₄	C₄₁×#550	1	1	yes	characteristic
661	C₂	—	2	1	no	—
662	C₂	—	2	1	no	—
663	C₂	—	8	2	no	—
664	C₂	—	8	2	no	—
665	C₂	—	8	2	no	—
666	C₂	C₄₁×#2407	1	1	yes	characteristic
667	C₂	C₂×C₄₁×#453	1	1	yes	characteristic
668	C₂	C₄₁×#3018	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×#3506	C₂	semidirect
2. C₄₁×#3077	C₂	semidirect
3. C₄₁×#3036	C₂	semidirect
4. C₄×#127×C₄₁	C₂	semidirect
5. C₄₁×#2037	C₂	semidirect
6. C₄₁×#2002	C₂	semidirect
7. C₄₁×#1994	C₂	semidirect
8. C₄₁×#1962	C₂	semidirect
9. C₄₁×#1883	C₂	semidirect
10. C₄₁×#1859	C₂	semidirect
11. C₄₁×#1800	C₂	semidirect
64. C₄⋊Q₈×C₄₁	D₄	semidirect
65. C₄⋊Q₈×C₄₁	D₄	semidirect
66. C₄⋊Q₈×C₄₁	D₄	semidirect
70. #127×C₄₁	D₄	semidirect
72. #125×C₄₁	D₄	semidirect
75. #124×C₄₁	D₄	semidirect
164. C₂×D₄×C₄₁	#41	semidirect
225. C₄²×C₄₁	#41	semidirect
293. #24176	C₄₁	direct (N × H)
327. #3506	C₂×C₄₁	semidirect
328. #3077	C₂×C₄₁	semidirect
329. #3036	C₂×C₄₁	semidirect
330. C₄×#127	C₂×C₄₁	semidirect
331. #2037	C₂×C₄₁	semidirect
332. #2002	C₂×C₄₁	semidirect
333. #1994	C₂×C₄₁	semidirect
334. #1962	C₂×C₄₁	semidirect
335. #1883	C₂×C₄₁	semidirect
336. #1859	C₂×C₄₁	semidirect
337. #1800	C₂×C₄₁	semidirect
385. C₄₁	#24176	direct (N × H)
399. C₄⋊Q₈	D₄×C₄₁	semidirect
400. C₄⋊Q₈	D₄×C₄₁	semidirect
401. C₄⋊Q₈	D₄×C₄₁	semidirect
405. #127	D₄×C₄₁	semidirect
407. #125	D₄×C₄₁	semidirect
410. #124	D₄×C₄₁	semidirect
499. C₂×D₄	#41×C₄₁	semidirect
560. C₄²	#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×#3506	2
C₄₁×#3077	2
C₄₁×#3036	2
C₄×#127×C₄₁	2
C₄₁×#2037	2
C₄₁×#2002	2
C₄₁×#1994	2
C₄₁×#1962	2
C₄₁×#1883	2
C₄₁×#1859	2
C₄₁×#1800	2
#24176	41

Subgroups — C41×#24176 #23800377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24176 #23800377