Subgroups — C₄₁×#23171 #23799372

Order 10496.

626 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3160	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2962	C₂	1	2	yes	maximal
3	C₄₁×#1986	C₂	1	2	yes	maximal
4	C₄₁×#1946	C₂	1	2	yes	maximal
5	C₄₁×#1922	C₂	1	4	yes	maximal
6	C₄×#108×C₄₁	C₂	1	1	yes	maximal, characteristic
7	C₄₁×#1773	C₂	1	2	yes	maximal
8	C₄₁×#1770	C₂	1	1	yes	maximal, characteristic
9	C₄₁×#533	C₂²	1	2	yes	—
10	C₄₁×#500	C₄	1	1	yes	characteristic, complemented
11	C₄₁×#500	C₄	1	1	yes	characteristic, complemented
12	C₄₁×#499	C₄	1	2	yes	complemented
13	C₄₁×#499	C₄	1	2	yes	complemented
14	C₄₁×#498	C₄	1	2	yes	complemented
15	C₄₁×#446	—	2	4	no	—
16	C₄₁×#446	—	2	2	no	—
17	C₈×Q₈×C₄₁	—	2	2	no	—
18	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
20	C₄₁×#427	C₂²	1	2	yes	—
21	C₄₁×#427	C₂²	1	2	yes	—
22	C₄₁×#424	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#405	C₂²	1	4	yes	—
25	C₄×M₁₆×C₄₁	C₂²	1	2	yes	—
26	C₂×#97×C₄₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
29	C₄₁×#341	C₂²	1	2	yes	—
30	C₄₁×#339	C₂²	1	4	yes	—
31	C₄₁×#339	C₂²	1	4	yes	—
32	C₄₁×#336	C₂²	1	2	yes	—
33	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
34	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
39	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
40	#125×C₄₁	C₂×C₄	1	4	yes	—
41	C₂²⋊Q₈×C₄₁	—	2	2	no	—
42	C₂²⋊Q₈×C₄₁	—	2	2	no	—
43	C₄×Q₈×C₄₁	—	2	2	no	—
44	C₄×Q₈×C₄₁	—	2	2	no	—
45	#117×C₄₁	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
47	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
48	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
49	#108×C₄₁	—	2	2	no	—
50	#108×C₄₁	Q₈	1	4	yes	complemented
51	#107×C₄₁	C₂×C₄	1	2	yes	—
52	#107×C₄₁	C₂×C₄	1	2	yes	—
53	#107×C₄₁	C₂×C₄	1	2	yes	—
54	#107×C₄₁	C₂×C₄	1	2	yes	—
55	#106×C₄₁	C₂×C₄	1	4	yes	—
56	#106×C₄₁	C₂×C₄	1	4	yes	—
57	C₄⋊C₈×C₄₁	—	2	4	no	—
58	C₄⋊C₈×C₄₁	—	2	4	no	—
59	C₄⋊C₈×C₄₁	D₄	1	2	yes	—
60	C₄⋊C₈×C₄₁	D₄	1	2	yes	—
61	#97×C₄₁	—	2	4	no	—
62	#97×C₄₁	—	2	4	no	—
63	#97×C₄₁	—	2	2	no	—
64	C₄×C₈×C₄₁	—	2	4	no	—
65	C₄×C₈×C₄₁	—	2	2	no	—
66	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
70	#95×C₄₁	—	2	1	no	—
71	C₂×Q₈×C₄₁	—	2	2	no	—
72	C₂×Q₈×C₄₁	—	2	2	no	—
73	C₂²×C₄×C₄₁	—	2	2	no	—
74	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
75	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
76	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
77	M₁₆×C₄₁	—	2	2	no	—
78	M₁₆×C₄₁	—	2	2	no	—
79	M₁₆×C₄₁	—	2	2	no	—
80	M₁₆×C₄₁	C₂×Q₈	1	4	yes	—
81	C₂×C₈×C₄₁	—	2	2	no	—
82	C₂×C₈×C₄₁	—	2	2	no	—
83	C₂×C₈×C₄₁	—	2	1	no	—
84	C₂×C₈×C₄₁	—	2	1	no	—
85	C₂×C₈×C₄₁	—	2	2	no	—
86	C₂×C₈×C₄₁	—	2	2	no	—
87	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₄₁	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₄₁	#41	1	2	yes	—
94	C₄⋊C₄×C₄₁	—	2	4	no	—
95	C₄⋊C₄×C₄₁	—	2	4	no	—
96	C₄⋊C₄×C₄₁	—	2	2	no	—
97	C₄⋊C₄×C₄₁	—	2	2	no	—
98	C₄⋊C₄×C₄₁	—	4	2	no	—
99	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
100	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
101	C₂²⋊C₄×C₄₁	—	2	4	no	—
102	C₄²×C₄₁	—	2	4	no	—
103	C₄²×C₄₁	—	2	2	no	—
104	C₄²×C₄₁	—	2	2	no	—
105	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
110	Q₈×C₄₁	—	4	4	no	—
111	Q₈×C₄₁	—	8	2	no	—
112	C₂×C₄×C₄₁	—	2	2	no	—
113	C₂×C₄×C₄₁	—	2	4	no	—
114	C₂×C₄×C₄₁	—	2	4	no	—
115	C₂×C₄×C₄₁	—	4	1	no	—
116	C₂×C₄×C₄₁	—	4	2	no	—
117	C₂×C₄×C₄₁	—	4	2	no	—
118	C₂×C₄×C₄₁	—	4	2	no	—
119	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
129	C₈×C₄₁	—	2	1	no	—
130	C₈×C₄₁	—	2	1	no	—
131	C₈×C₄₁	—	2	4	no	—
132	C₈×C₄₁	—	2	4	no	—
133	C₈×C₄₁	—	4	2	no	—
134	C₈×C₄₁	C₄×Q₈	1	2	yes	—
135	C₈×C₄₁	C₄×Q₈	1	2	yes	—
136	#23171	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₄₁	—	2	1	no	—
138	C₂²×C₄₁	—	2	1	no	—
139	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₄₁	#556	1	1	yes	characteristic
141	C₂²×C₄₁	#557	1	1	yes	characteristic
142	C₄×C₄₁	—	2	1	no	—
143	C₄×C₄₁	—	2	1	no	—
144	C₄×C₄₁	—	2	1	no	—
145	C₄×C₄₁	—	4	4	no	—
146	C₄×C₄₁	—	4	4	no	—
147	C₄×C₄₁	—	8	2	no	—
148	C₄×C₄₁	#444	1	2	yes	—
149	C₄×C₄₁	C₂×C₄×Q₈	1	1	yes	characteristic
150	C₄×C₄₁	#519	1	1	yes	characteristic
151	C₄×C₄₁	#547	1	1	yes	characteristic
152	C₄×C₄₁	D₄×Q₈	1	1	yes	characteristic
153	#3160	C₂×C₄₁	1	1	yes	characteristic
154	#2962	C₂×C₄₁	1	2	yes	—
155	#1986	C₂×C₄₁	1	2	yes	—
156	#1946	C₂×C₄₁	1	2	yes	—
157	#1922	C₂×C₄₁	1	4	yes	—
158	C₄×#108	C₂×C₄₁	1	1	yes	characteristic
159	#1773	C₂×C₄₁	1	2	yes	—
160	#1770	C₂×C₄₁	1	1	yes	characteristic
161	C₂×C₄₁	—	2	1	no	—
162	C₂×C₄₁	—	2	1	no	—
163	C₂×C₄₁	#2365	1	1	yes	characteristic
164	C₂×C₄₁	C₂×#444	1	1	yes	characteristic
165	C₂×C₄₁	#2957	1	1	yes	characteristic
166	#533	C₂²×C₄₁	1	2	yes	—
167	#500	C₄×C₄₁	1	1	yes	characteristic, complemented
168	#500	C₄×C₄₁	1	1	yes	characteristic, complemented
169	#499	C₄×C₄₁	1	2	yes	complemented
170	#499	C₄×C₄₁	1	2	yes	complemented
171	#498	C₄×C₄₁	1	2	yes	complemented
172	#446	—	2	4	no	—
173	#446	—	2	2	no	—
174	C₈×Q₈	—	2	2	no	—
175	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
176	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
177	#427	C₂²×C₄₁	1	2	yes	—
178	#427	C₂²×C₄₁	1	2	yes	—
179	#424	C₂²×C₄₁	1	4	yes	—
180	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
181	#405	C₂²×C₄₁	1	4	yes	—
182	C₄×M₁₆	C₂²×C₄₁	1	2	yes	—
183	C₂×#97	C₂²×C₄₁	1	2	yes	—
184	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
185	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
186	#341	C₂²×C₄₁	1	2	yes	—
187	#339	C₂²×C₄₁	1	4	yes	—
188	#339	C₂²×C₄₁	1	4	yes	—
189	#336	C₂²×C₄₁	1	2	yes	—
190	C₄₁	#23171	1	1	yes	characteristic, direct product factor, Sylow
191	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
192	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
193	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
195	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
196	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
197	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
198	#125	C₂×C₄×C₄₁	1	4	yes	—
199	C₂²⋊Q₈	—	2	2	no	—
200	C₂²⋊Q₈	—	2	2	no	—
201	C₄×Q₈	—	2	2	no	—
202	C₄×Q₈	—	2	2	no	—
203	#117	C₂³×C₄₁	1	4	yes	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
206	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
207	#108	—	2	2	no	—
208	#108	Q₈×C₄₁	1	4	yes	complemented
209	#107	C₂×C₄×C₄₁	1	2	yes	—
210	#107	C₂×C₄×C₄₁	1	2	yes	—
211	#107	C₂×C₄×C₄₁	1	2	yes	—
212	#107	C₂×C₄×C₄₁	1	2	yes	—
213	#106	C₂×C₄×C₄₁	1	4	yes	—
214	#106	C₂×C₄×C₄₁	1	4	yes	—
215	C₄⋊C₈	—	2	4	no	—
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	D₄×C₄₁	1	2	yes	—
218	C₄⋊C₈	D₄×C₄₁	1	2	yes	—
219	#97	—	2	4	no	—
220	#97	—	2	4	no	—
221	#97	—	2	2	no	—
222	C₄×C₈	—	2	4	no	—
223	C₄×C₈	—	2	2	no	—
224	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
225	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
226	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
227	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
228	#95	—	2	1	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
233	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
234	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	2	2	no	—
238	M₁₆	C₂×Q₈×C₄₁	1	4	yes	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
248	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
250	C₂×C₈	C₂×Q₈×C₄₁	1	2	yes	—
251	C₂×C₈	#41×C₄₁	1	2	yes	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
258	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
264	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
265	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
266	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
267	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
268	Q₈	—	4	4	no	—
269	Q₈	—	8	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
278	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
279	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
280	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
281	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
283	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
285	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
286	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	—	2	4	no	—
290	C₈	—	2	4	no	—
291	C₈	—	4	2	no	—
292	C₈	C₄×Q₈×C₄₁	1	2	yes	—
293	C₈	C₄×Q₈×C₄₁	1	2	yes	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
297	C₂²	C₄₁×#556	1	1	yes	characteristic
298	C₂²	C₄₁×#557	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	4	no	—
303	C₄	—	4	4	no	—
304	C₄	—	8	2	no	—
305	C₄	C₄₁×#444	1	2	yes	—
306	C₄	C₂×C₄×Q₈×C₄₁	1	1	yes	characteristic
307	C₄	C₄₁×#519	1	1	yes	characteristic
308	C₄	C₄₁×#547	1	1	yes	characteristic
309	C₄	D₄×Q₈×C₄₁	1	1	yes	characteristic
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	C₄₁×#2365	1	1	yes	characteristic
313	C₂	C₂×C₄₁×#444	1	1	yes	characteristic
314	C₂	C₄₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂×C₄×D₄
prime radical	C₂³×C₄₁	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂×C₄⋊C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄₁×#500	C₄	semidirect
11. C₄₁×#500	C₄	semidirect
12. C₄₁×#499	C₄	semidirect
13. C₄₁×#499	C₄	semidirect
14. C₄₁×#498	C₄	semidirect
50. #108×C₄₁	Q₈	semidirect
136. #23171	C₄₁	direct (N × H)
167. #500	C₄×C₄₁	semidirect
168. #500	C₄×C₄₁	semidirect
169. #499	C₄×C₄₁	semidirect
170. #499	C₄×C₄₁	semidirect
171. #498	C₄×C₄₁	semidirect
190. C₄₁	#23171	direct (N × H)
208. #108	Q₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3160	2
C₄₁×#2962	2
C₄₁×#1986	2
C₄₁×#1946	2
C₄₁×#1922	2
C₄×#108×C₄₁	2
C₄₁×#1773	2
C₄₁×#1770	2
#23171	41

Subgroups — C41×#23171 #23799372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23171 #23799372