Subgroups — C₄₁×#33693 #23809894

Order 10496.

512 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#2955	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2239	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2231	C₂	1	2	yes	maximal, complemented
5	C₄₁×#2214	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2191	C₂	1	2	yes	maximal
7	C₄₁×#2180	C₂	1	4	yes	maximal, complemented
8	C₄₁×#2166	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#575	C₂²	1	1	yes	characteristic
10	C₂×#131×C₄₁	—	2	1	no	—
11	C₄₁×#541	—	2	1	no	—
12	C₄₁×#510	C₂²	1	4	yes	—
13	C₄₁×#499	C₂²	1	1	yes	characteristic, complemented
14	C₄₁×#494	C₂²	1	1	yes	characteristic, complemented
15	C₄₁×#491	—	2	1	no	—
16	C₄₁×#470	C₂²	1	2	yes	—
17	C₄₁×#467	—	2	1	no	—
18	C₄₁×#444	—	2	1	no	—
19	C₄₁×#444	C₂²	1	2	yes	—
20	C₄₁×#443	—	2	1	no	—
21	C₄₁×#443	C₂²	1	2	yes	—
22	C₄₁×#441	C₂²	1	2	yes	complemented
23	C₄×Q₁₆×C₄₁	—	2	1	no	—
24	C₄₁×#430	C₂²	1	1	yes	characteristic
25	C₄₁×#421	—	2	1	no	—
26	C₄₁×#405	C₂²	1	1	yes	characteristic
27	C₄₁×#365	C₂²	1	4	yes	complemented
28	C₄₁×#364	C₂²	1	2	yes	complemented
29	C₄₁×#362	C₂²	1	4	yes	complemented
30	C₄₁×#360	C₂²	1	4	yes	—
31	C₄₁×#349	C₂²	1	2	yes	—
32	C₄₁×#347	C₂²	1	2	yes	complemented
33	C₂×#41×C₄₁	—	2	1	no	—
34	#137×C₄₁	—	2	2	no	—
35	AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
36	#135×C₄₁	—	2	1	no	—
37	#135×C₄₁	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₄₁	—	2	1	no	—
39	C₂×Q₁₆×C₄₁	C₂³	1	2	yes	—
40	#131×C₄₁	—	2	2	no	—
41	#131×C₄₁	—	2	2	no	—
42	#131×C₄₁	—	4	1	no	—
43	#131×C₄₁	—	4	1	no	—
44	C₂×M₁₆×C₄₁	—	2	1	no	—
45	C₂×M₁₆×C₄₁	—	4	1	no	—
46	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₁	—	4	1	no	—
48	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
49	C₄⋊Q₈×C₄₁	—	2	1	no	—
50	#126×C₄₁	—	4	2	no	—
51	#125×C₄₁	—	2	1	no	—
52	#125×C₄₁	—	4	2	no	—
53	C₂²⋊Q₈×C₄₁	—	2	2	no	—
54	C₄×Q₈×C₄₁	—	2	2	no	—
55	#117×C₄₁	D₄	1	1	yes	characteristic
56	Q₃₂×C₄₁	—	2	4	no	—
57	SD₃₂×C₄₁	—	2	4	no	—
58	M₃₂×C₄₁	C₂³	1	4	yes	—
59	#108×C₄₁	—	2	1	no	—
60	#108×C₄₁	C₂³	1	2	yes	—
61	#107×C₄₁	—	2	1	no	—
62	#106×C₄₁	C₂³	1	2	yes	—
63	#104×C₄₁	—	2	2	no	—
64	#104×C₄₁	—	2	2	no	—
65	#103×C₄₁	—	2	2	no	—
66	#103×C₄₁	—	2	2	no	—
67	#97×C₄₁	D₄	1	1	yes	characteristic
68	#97×C₄₁	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
71	#41×C₄₁	—	2	2	no	—
72	#41×C₄₁	—	2	2	no	—
73	#41×C₄₁	—	4	1	no	—
74	#41×C₄₁	—	4	1	no	—
75	C₂×Q₈×C₄₁	—	2	1	no	—
76	C₂×Q₈×C₄₁	—	2	2	no	—
77	C₂×D₄×C₄₁	—	2	1	no	—
78	C₂×D₄×C₄₁	—	4	1	no	—
79	C₂²×C₄×C₄₁	—	4	1	no	—
80	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₄₁	—	2	1	no	—
82	Q₁₆×C₄₁	—	2	4	no	—
83	Q₁₆×C₄₁	—	4	1	no	—
84	Q₁₆×C₄₁	—	4	1	no	—
85	Q₁₆×C₄₁	#41	1	2	yes	—
86	SD₁₆×C₄₁	—	2	2	no	—
87	SD₁₆×C₄₁	—	2	2	no	—
88	D₈×C₄₁	—	2	1	no	—
89	D₈×C₄₁	#41	1	2	yes	—
90	M₁₆×C₄₁	—	2	2	no	—
91	M₁₆×C₄₁	—	2	2	no	—
92	M₁₆×C₄₁	—	4	2	no	—
93	M₁₆×C₄₁	—	4	1	no	—
94	M₁₆×C₄₁	—	4	2	no	—
95	C₂×C₈×C₄₁	—	2	1	no	—
96	C₂×C₈×C₄₁	—	2	1	no	—
97	C₂×C₈×C₄₁	—	4	2	no	—
98	C₂×C₈×C₄₁	—	4	1	no	—
99	C₂×C₈×C₄₁	—	4	2	no	—
100	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₄₁	—	2	1	no	—
105	C₄⋊C₄×C₄₁	—	2	2	no	—
106	C₄⋊C₄×C₄₁	—	4	2	no	—
107	C₄⋊C₄×C₄₁	—	4	2	no	—
108	C₄⋊C₄×C₄₁	—	4	2	no	—
109	C₄⋊C₄×C₄₁	—	4	2	no	—
110	C₂²⋊C₄×C₄₁	—	2	1	no	—
111	C₂²⋊C₄×C₄₁	—	4	2	no	—
112	C₄²×C₄₁	—	4	2	no	—
113	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₄₁	—	2	4	no	—
116	C₂³×C₄₁	—	4	1	no	—
117	C₂³×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
118	Q₈×C₄₁	—	2	2	no	—
119	Q₈×C₄₁	—	4	1	no	—
120	Q₈×C₄₁	—	4	2	no	—
121	Q₈×C₄₁	—	4	2	no	—
122	D₄×C₄₁	—	2	2	no	—
123	D₄×C₄₁	—	4	1	no	—
124	D₄×C₄₁	—	4	2	no	—
125	D₄×C₄₁	—	4	2	no	—
126	C₂×C₄×C₄₁	—	2	1	no	—
127	C₂×C₄×C₄₁	—	2	1	no	—
128	C₂×C₄×C₄₁	—	2	1	no	—
129	C₂×C₄×C₄₁	—	2	1	no	—
130	C₂×C₄×C₄₁	—	4	2	no	—
131	C₂×C₄×C₄₁	—	4	1	no	—
132	C₂×C₄×C₄₁	—	4	2	no	—
133	C₂×C₄×C₄₁	—	4	2	no	—
134	C₂×C₄×C₄₁	—	4	2	no	—
135	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
137	C₈×C₄₁	—	2	1	no	—
138	C₈×C₄₁	—	2	1	no	—
139	C₈×C₄₁	—	4	2	no	—
140	C₈×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₈×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₈×C₄₁	C₂×#41	1	2	yes	—
143	#33693	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₄₁	—	2	1	no	—
145	C₂²×C₄₁	—	4	1	no	—
146	C₂²×C₄₁	—	4	2	no	—
147	C₂²×C₄₁	—	8	2	no	—
148	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₄×C₄₁	—	2	1	no	—
150	C₄×C₄₁	—	4	1	no	—
151	C₄×C₄₁	—	4	1	no	—
152	C₄×C₄₁	—	4	2	no	—
153	C₄×C₄₁	—	8	2	no	—
154	C₄×C₄₁	—	8	2	no	—
155	C₄×C₄₁	#525	1	1	yes	characteristic
156	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	#3177	C₂×C₄₁	1	1	yes	characteristic, complemented
158	#2955	C₂×C₄₁	1	1	yes	characteristic
159	#2239	C₂×C₄₁	1	2	yes	complemented
160	#2231	C₂×C₄₁	1	2	yes	complemented
161	#2214	C₂×C₄₁	1	2	yes	complemented
162	#2191	C₂×C₄₁	1	2	yes	—
163	#2180	C₂×C₄₁	1	4	yes	complemented
164	#2166	C₂×C₄₁	1	1	yes	characteristic, complemented
165	C₂×C₄₁	—	2	1	no	—
166	C₂×C₄₁	—	4	1	no	—
167	C₂×C₄₁	—	8	2	no	—
168	C₂×C₄₁	#3109	1	1	yes	characteristic
169	#575	C₂²×C₄₁	1	1	yes	characteristic
170	C₂×#131	—	2	1	no	—
171	#541	—	2	1	no	—
172	#510	C₂²×C₄₁	1	4	yes	—
173	#499	C₂²×C₄₁	1	1	yes	characteristic, complemented
174	#494	C₂²×C₄₁	1	1	yes	characteristic, complemented
175	#491	—	2	1	no	—
176	#470	C₂²×C₄₁	1	2	yes	—
177	#467	—	2	1	no	—
178	#444	—	2	1	no	—
179	#444	C₂²×C₄₁	1	2	yes	—
180	#443	—	2	1	no	—
181	#443	C₂²×C₄₁	1	2	yes	—
182	#441	C₂²×C₄₁	1	2	yes	complemented
183	C₄×Q₁₆	—	2	1	no	—
184	#430	C₂²×C₄₁	1	1	yes	characteristic
185	#421	—	2	1	no	—
186	#405	C₂²×C₄₁	1	1	yes	characteristic
187	#365	C₂²×C₄₁	1	4	yes	complemented
188	#364	C₂²×C₄₁	1	2	yes	complemented
189	#362	C₂²×C₄₁	1	4	yes	complemented
190	#360	C₂²×C₄₁	1	4	yes	—
191	#349	C₂²×C₄₁	1	2	yes	—
192	#347	C₂²×C₄₁	1	2	yes	complemented
193	C₄₁	#33693	1	1	yes	characteristic, direct product factor, Sylow
194	C₂×#41	—	2	1	no	—
195	#137	—	2	2	no	—
196	AGL(1,ℤ/8)	—	2	2	no	—
197	#135	—	2	1	no	—
198	#135	C₂³×C₄₁	1	2	yes	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	C₂³×C₄₁	1	2	yes	—
201	#131	—	2	2	no	—
202	#131	—	2	2	no	—
203	#131	—	4	1	no	—
204	#131	—	4	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	4	1	no	—
207	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
208	C₂²×C₈	—	4	1	no	—
209	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	1	no	—
211	#126	—	4	2	no	—
212	#125	—	2	1	no	—
213	#125	—	4	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₄×Q₈	—	2	2	no	—
216	#117	D₄×C₄₁	1	1	yes	characteristic
217	Q₃₂	—	2	4	no	—
218	SD₃₂	—	2	4	no	—
219	M₃₂	C₂³×C₄₁	1	4	yes	—
220	#108	—	2	1	no	—
221	#108	C₂³×C₄₁	1	2	yes	—
222	#107	—	2	1	no	—
223	#106	C₂³×C₄₁	1	2	yes	—
224	#104	—	2	2	no	—
225	#104	—	2	2	no	—
226	#103	—	2	2	no	—
227	#103	—	2	2	no	—
228	#97	D₄×C₄₁	1	1	yes	characteristic
229	#97	C₂³×C₄₁	1	1	yes	characteristic
230	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	#41	—	4	1	no	—
235	#41	—	4	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
242	Q₁₆	—	2	1	no	—
243	Q₁₆	—	2	4	no	—
244	Q₁₆	—	4	1	no	—
245	Q₁₆	—	4	1	no	—
246	Q₁₆	#41×C₄₁	1	2	yes	—
247	SD₁₆	—	2	2	no	—
248	SD₁₆	—	2	2	no	—
249	D₈	—	2	1	no	—
250	D₈	#41×C₄₁	1	2	yes	—
251	M₁₆	—	2	2	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	4	2	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	4	2	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
264	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
276	C₁₆	—	2	4	no	—
277	C₂³	—	4	1	no	—
278	C₂³	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
279	Q₈	—	2	2	no	—
280	Q₈	—	4	1	no	—
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	D₄	—	2	2	no	—
284	D₄	—	4	1	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	4	2	no	—
301	C₈	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
302	C₈	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
303	C₈	C₂×#41×C₄₁	1	2	yes	—
304	C₂²	—	2	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	2	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	C₄₁×#525	1	1	yes	characteristic
316	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
317	C₂	—	2	1	no	—
318	C₂	—	4	1	no	—
319	C₂	—	8	2	no	—
320	C₂	C₄₁×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
prime radical	C₄₁×#575	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁ ◁ C₂×C₄×C₄₁ ◁ C₄₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3177	C₂	semidirect
3. C₄₁×#2239	C₂	semidirect
4. C₄₁×#2231	C₂	semidirect
5. C₄₁×#2214	C₂	semidirect
7. C₄₁×#2180	C₂	semidirect
8. C₄₁×#2166	C₂	semidirect
13. C₄₁×#499	C₂²	semidirect
14. C₄₁×#494	C₂²	semidirect
22. C₄₁×#441	C₂²	semidirect
27. C₄₁×#365	C₂²	semidirect
28. C₄₁×#364	C₂²	semidirect
29. C₄₁×#362	C₂²	semidirect
32. C₄₁×#347	C₂²	semidirect
143. #33693	C₄₁	direct (N × H)
157. #3177	C₂×C₄₁	semidirect
159. #2239	C₂×C₄₁	semidirect
160. #2231	C₂×C₄₁	semidirect
161. #2214	C₂×C₄₁	semidirect
163. #2180	C₂×C₄₁	semidirect
164. #2166	C₂×C₄₁	semidirect
173. #499	C₂²×C₄₁	semidirect
174. #494	C₂²×C₄₁	semidirect
182. #441	C₂²×C₄₁	semidirect
187. #365	C₂²×C₄₁	semidirect
188. #364	C₂²×C₄₁	semidirect
189. #362	C₂²×C₄₁	semidirect
192. #347	C₂²×C₄₁	semidirect
193. C₄₁	#33693	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#3177	2
C₄₁×#2955	2
C₄₁×#2239	2
C₄₁×#2231	2
C₄₁×#2214	2
C₄₁×#2191	2
C₄₁×#2180	2
C₄₁×#2166	2
#33693	41

Subgroups — C41×#33693 #23809894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33693 #23809894