Subgroups — C₄₁×#18526 #23794727

Order 10496.

1068 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3151	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#3112	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1630	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1620	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1555	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1491	C₂	1	2	yes	maximal, complemented
8	C₂×AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
9	C₄₁×#559	—	2	2	no	—
10	C₄₁×#559	—	2	4	no	—
11	C₄₁×#559	C₂²	1	4	yes	—
12	C₄₁×#532	C₂²	1	1	yes	characteristic
13	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×#124×C₄₁	—	2	2	no	—
15	C₂×#123×C₄₁	—	2	2	no	—
16	C₂×C₄×Q₈×C₄₁	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₄₁	—	2	2	no	—
18	C₄₁×#496	—	2	2	no	—
19	C₄₁×#493	C₂²	1	2	yes	—
20	C₄₁×#492	C₂²	1	2	yes	—
21	C₄₁×#480	—	2	2	no	—
22	C₄₁×#477	C₂²	1	4	yes	complemented
23	C₄₁×#460	C₂²	1	4	yes	complemented
24	C₄₁×#448	—	2	2	no	—
25	C₄₁×#440	—	2	4	no	—
26	C₄₁×#432	C₂²	1	2	yes	—
27	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
28	C₄₁×#331	C₂²	1	4	yes	—
29	C₄₁×Q₈⋊C₈	C₂²	1	8	yes	complemented
30	C₂×#41×C₄₁	—	2	1	no	—
31	C₂²×Q₈×C₄₁	D₄	1	2	yes	—
32	C₂²×D₄×C₄₁	—	2	2	no	—
33	C₂²×D₄×C₄₁	D₄	1	2	yes	—
34	C₂³×C₄×C₄₁	—	2	2	no	—
35	AGL(1,ℤ/8)×C₄₁	—	4	8	no	—
36	C₂×SD₁₆×C₄₁	—	2	4	no	—
37	C₂×SD₁₆×C₄₁	—	2	4	no	—
38	C₂×D₈×C₄₁	—	2	4	no	—
39	C₂×M₁₆×C₄₁	—	2	1	no	—
40	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
41	#127×C₄₁	—	2	2	no	—
42	#127×C₄₁	D₄	1	2	yes	—
43	#127×C₄₁	C₂³	1	1	yes	characteristic
44	#127×C₄₁	C₂³	1	2	yes	—
45	#124×C₄₁	—	2	8	no	—
46	#124×C₄₁	—	2	8	no	—
47	#124×C₄₁	—	2	1	no	—
48	#123×C₄₁	—	2	8	no	—
49	#123×C₄₁	—	2	8	no	—
50	#121×C₄₁	—	2	2	no	—
51	C₄×Q₈×C₄₁	—	2	2	no	—
52	C₄×Q₈×C₄₁	—	2	2	no	—
53	C₄×Q₈×C₄₁	D₄	1	4	yes	—
54	C₄×Q₈×C₄₁	C₂³	1	4	yes	—
55	C₄×D₄×C₄₁	—	2	8	no	—
56	C₄×D₄×C₄₁	—	2	8	no	—
57	C₄×D₄×C₄₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
61	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
62	C₂×C₄²×C₄₁	—	2	2	no	—
63	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
64	#107×C₄₁	—	2	4	no	—
65	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
66	#103×C₄₁	—	2	4	no	—
67	#102×C₄₁	—	2	4	no	—
68	C₂²⋊C₈×C₄₁	—	2	2	no	—
69	#97×C₄₁	—	2	1	no	—
70	#97×C₄₁	—	2	2	no	—
71	C₄×C₈×C₄₁	C₂³	1	2	yes	—
72	C₂⁴×C₄₁	—	2	2	no	—
73	#41×C₄₁	—	4	4	no	—
74	C₂×Q₈×C₄₁	—	2	2	no	—
75	C₂×Q₈×C₄₁	—	2	1	no	—
76	C₂×Q₈×C₄₁	—	4	4	no	—
77	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
78	C₂×D₄×C₄₁	—	2	4	no	—
79	C₂×D₄×C₄₁	—	2	2	no	—
80	C₂×D₄×C₄₁	—	2	4	no	—
81	C₂×D₄×C₄₁	—	2	2	no	—
82	C₂×D₄×C₄₁	—	2	1	no	—
83	C₂×D₄×C₄₁	—	4	8	no	—
84	C₂×D₄×C₄₁	—	4	4	no	—
85	C₂×D₄×C₄₁	—	4	1	no	—
86	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₄₁	—	2	2	no	—
88	C₂²×C₄×C₄₁	—	2	2	no	—
89	C₂²×C₄×C₄₁	—	2	8	no	—
90	C₂²×C₄×C₄₁	—	4	1	no	—
91	C₂²×C₄×C₄₁	—	8	4	no	—
92	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
94	SD₁₆×C₄₁	—	4	8	no	—
95	SD₁₆×C₄₁	—	4	8	no	—
96	D₈×C₄₁	—	4	8	no	—
97	M₁₆×C₄₁	—	2	4	no	—
98	C₂×C₈×C₄₁	—	2	2	no	—
99	C₂×C₈×C₄₁	—	2	4	no	—
100	C₄⋊C₄×C₄₁	—	2	4	no	—
101	C₄⋊C₄×C₄₁	—	2	2	no	—
102	C₄⋊C₄×C₄₁	—	2	4	no	—
103	C₄⋊C₄×C₄₁	—	2	2	no	—
104	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
105	C₂²⋊C₄×C₄₁	—	2	16	no	—
106	C₂²⋊C₄×C₄₁	—	2	8	no	—
107	C₂²⋊C₄×C₄₁	—	4	2	no	—
108	C₄²×C₄₁	—	2	4	no	—
109	C₄²×C₄₁	—	2	4	no	—
110	C₄²×C₄₁	C₂×D₄	1	2	yes	—
111	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
113	C₂³×C₄₁	—	2	4	no	—
114	C₂³×C₄₁	—	4	8	no	—
115	C₂³×C₄₁	—	4	4	no	—
116	C₂³×C₄₁	—	4	1	no	—
117	C₂³×C₄₁	#120	1	1	yes	characteristic
118	Q₈×C₄₁	—	2	4	no	—
119	Q₈×C₄₁	—	2	4	no	—
120	Q₈×C₄₁	—	4	4	no	—
121	Q₈×C₄₁	—	4	2	no	—
122	D₄×C₄₁	—	4	4	no	—
123	D₄×C₄₁	—	4	8	no	—
124	D₄×C₄₁	—	4	2	no	—
125	D₄×C₄₁	—	4	8	no	—
126	D₄×C₄₁	—	4	4	no	—
127	D₄×C₄₁	—	8	2	no	—
128	D₄×C₄₁	—	8	2	no	—
129	C₂×C₄×C₄₁	—	2	2	no	—
130	C₂×C₄×C₄₁	—	2	4	no	—
131	C₂×C₄×C₄₁	—	2	4	no	—
132	C₂×C₄×C₄₁	—	2	4	no	—
133	C₂×C₄×C₄₁	—	4	4	no	—
134	C₂×C₄×C₄₁	—	4	4	no	—
135	C₂×C₄×C₄₁	—	4	1	no	—
136	C₂×C₄×C₄₁	—	8	2	no	—
137	C₂×C₄×C₄₁	—	8	8	no	—
138	C₂×C₄×C₄₁	#120	1	2	yes	—
139	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
140	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
142	C₈×C₄₁	—	2	4	no	—
143	C₈×C₄₁	—	4	4	no	—
144	#18526	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₄₁	—	2	2	no	—
146	C₂²×C₄₁	—	4	8	no	—
147	C₂²×C₄₁	—	4	4	no	—
148	C₂²×C₄₁	—	8	8	no	—
149	C₂²×C₄₁	—	8	2	no	—
150	C₂²×C₄₁	—	8	1	no	—
151	C₂²×C₄₁	#453	1	2	yes	—
152	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
153	C₄×C₄₁	—	2	2	no	—
154	C₄×C₄₁	—	2	1	no	—
155	C₄×C₄₁	—	2	1	no	—
156	C₄×C₄₁	—	4	4	no	—
157	C₄×C₄₁	—	4	4	no	—
158	C₄×C₄₁	—	8	1	no	—
159	C₄×C₄₁	#575	1	4	yes	—
160	C₂×#559	C₂×C₄₁	1	1	yes	characteristic, complemented
161	#3151	C₂×C₄₁	1	1	yes	characteristic
162	#3112	C₂×C₄₁	1	2	yes	complemented
163	#1630	C₂×C₄₁	1	4	yes	complemented
164	#1620	C₂×C₄₁	1	4	yes	complemented
165	#1555	C₂×C₄₁	1	1	yes	characteristic, complemented
166	#1491	C₂×C₄₁	1	2	yes	complemented
167	C₂×C₄₁	—	2	2	no	—
168	C₂×C₄₁	—	8	4	no	—
169	C₂×C₄₁	—	16	1	no	—
170	C₂×C₄₁	#3005	1	2	yes	—
171	C₂×C₄₁	C₂×#453	1	1	yes	characteristic
172	C₂×AGL(1,ℤ/8)	—	2	2	no	—
173	#559	—	2	2	no	—
174	#559	—	2	4	no	—
175	#559	C₂²×C₄₁	1	4	yes	—
176	#532	C₂²×C₄₁	1	1	yes	characteristic
177	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
178	C₂×#124	—	2	2	no	—
179	C₂×#123	—	2	2	no	—
180	C₂×C₄×Q₈	C₂²×C₄₁	1	2	yes	—
181	C₂×C₄×D₄	—	2	2	no	—
182	#496	—	2	2	no	—
183	#493	C₂²×C₄₁	1	2	yes	—
184	#492	C₂²×C₄₁	1	2	yes	—
185	#480	—	2	2	no	—
186	#477	C₂²×C₄₁	1	4	yes	complemented
187	#460	C₂²×C₄₁	1	4	yes	complemented
188	#448	—	2	2	no	—
189	#440	—	2	4	no	—
190	#432	C₂²×C₄₁	1	2	yes	—
191	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
192	#331	C₂²×C₄₁	1	4	yes	—
193	Q₈⋊C₈	C₂²×C₄₁	1	8	yes	complemented
194	C₄₁	#18526	1	1	yes	characteristic, direct product factor, Sylow
195	C₂×#41	—	2	1	no	—
196	C₂²×Q₈	D₄×C₄₁	1	2	yes	—
197	C₂²×D₄	—	2	2	no	—
198	C₂²×D₄	D₄×C₄₁	1	2	yes	—
199	C₂³×C₄	—	2	2	no	—
200	AGL(1,ℤ/8)	—	4	8	no	—
201	C₂×SD₁₆	—	2	4	no	—
202	C₂×SD₁₆	—	2	4	no	—
203	C₂×D₈	—	2	4	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
206	#127	—	2	2	no	—
207	#127	D₄×C₄₁	1	2	yes	—
208	#127	C₂³×C₄₁	1	1	yes	characteristic
209	#127	C₂³×C₄₁	1	2	yes	—
210	#124	—	2	8	no	—
211	#124	—	2	8	no	—
212	#124	—	2	1	no	—
213	#123	—	2	8	no	—
214	#123	—	2	8	no	—
215	#121	—	2	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	D₄×C₄₁	1	4	yes	—
219	C₄×Q₈	C₂³×C₄₁	1	4	yes	—
220	C₄×D₄	—	2	8	no	—
221	C₄×D₄	—	2	8	no	—
222	C₄×D₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
225	C₂×C₂²⋊C₄	—	2	4	no	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₄²	—	2	2	no	—
228	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
229	#107	—	2	4	no	—
230	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
231	#103	—	2	4	no	—
232	#102	—	2	4	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	#97	—	2	1	no	—
235	#97	—	2	2	no	—
236	C₄×C₈	C₂³×C₄₁	1	2	yes	—
237	C₂⁴	—	2	2	no	—
238	#41	—	4	4	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	4	4	no	—
242	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
243	C₂×D₄	—	2	4	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	4	8	no	—
249	C₂×D₄	—	4	4	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	8	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	8	4	no	—
257	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
258	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
259	SD₁₆	—	4	8	no	—
260	SD₁₆	—	4	8	no	—
261	D₈	—	4	8	no	—
262	M₁₆	—	2	4	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
270	C₂²⋊C₄	—	2	16	no	—
271	C₂²⋊C₄	—	2	8	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	2	4	no	—
274	C₄²	—	2	4	no	—
275	C₄²	C₂×D₄×C₄₁	1	2	yes	—
276	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
277	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
278	C₂³	—	2	4	no	—
279	C₂³	—	4	8	no	—
280	C₂³	—	4	4	no	—
281	C₂³	—	4	1	no	—
282	C₂³	#120×C₄₁	1	1	yes	characteristic
283	Q₈	—	2	4	no	—
284	Q₈	—	2	4	no	—
285	Q₈	—	4	4	no	—
286	Q₈	—	4	2	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	8	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	8	no	—
291	D₄	—	4	4	no	—
292	D₄	—	8	2	no	—
293	D₄	—	8	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	8	8	no	—
303	C₂×C₄	#120×C₄₁	1	2	yes	—
304	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
307	C₈	—	2	4	no	—
308	C₈	—	4	4	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	8	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	8	8	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	1	no	—
315	C₂²	C₄₁×#453	1	2	yes	—
316	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	4	no	—
321	C₄	—	4	4	no	—
322	C₄	—	8	1	no	—
323	C₄	C₄₁×#575	1	4	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	8	4	no	—
326	C₂	—	16	1	no	—
327	C₂	C₄₁×#3005	1	2	yes	—
328	C₂	C₂×C₄₁×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₄₁×#3151	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ 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. C₂×C₄₁×#559	C₂	semidirect
3. C₄₁×#3112	C₂	semidirect
4. C₄₁×#1630	C₂	semidirect
5. C₄₁×#1620	C₂	semidirect
6. C₄₁×#1555	C₂	semidirect
7. C₄₁×#1491	C₂	semidirect
22. C₄₁×#477	C₂²	semidirect
23. C₄₁×#460	C₂²	semidirect
29. C₄₁×Q₈⋊C₈	C₂²	semidirect
144. #18526	C₄₁	direct (N × H)
160. C₂×#559	C₂×C₄₁	semidirect
162. #3112	C₂×C₄₁	semidirect
163. #1630	C₂×C₄₁	semidirect
164. #1620	C₂×C₄₁	semidirect
165. #1555	C₂×C₄₁	semidirect
166. #1491	C₂×C₄₁	semidirect
186. #477	C₂²×C₄₁	semidirect
187. #460	C₂²×C₄₁	semidirect
193. Q₈⋊C₈	C₂²×C₄₁	semidirect
194. C₄₁	#18526	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#559	2
C₄₁×#3151	2
C₄₁×#3112	2
C₄₁×#1630	2
C₄₁×#1620	2
C₄₁×#1555	2
C₄₁×#1491	2
#18526	41

Subgroups — C41×#18526 #23794727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#18526 #23794727