Subgroups — C₄₁×#23170 #23799371

Order 10496.

638 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×#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×#1945	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×#1772	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₄₁	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×C₄×C₈×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	1	yes	characteristic
60	C₄⋊C₈×C₄₁	D₄	1	1	yes	characteristic
61	C₄⋊C₈×C₄₁	D₄	1	2	yes	—
62	#97×C₄₁	—	2	4	no	—
63	#97×C₄₁	—	2	2	no	—
64	C₄×C₈×C₄₁	—	2	4	no	—
65	C₄×C₈×C₄₁	—	2	4	no	—
66	C₄×C₈×C₄₁	—	2	2	no	—
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	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
71	#95×C₄₁	—	2	1	no	—
72	C₂×Q₈×C₄₁	—	2	2	no	—
73	C₂×Q₈×C₄₁	—	2	2	no	—
74	C₂²×C₄×C₄₁	—	2	2	no	—
75	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
76	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
77	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
78	M₁₆×C₄₁	—	2	2	no	—
79	M₁₆×C₄₁	—	2	2	no	—
80	M₁₆×C₄₁	—	2	2	no	—
81	M₁₆×C₄₁	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₄₁	—	2	2	no	—
83	C₂×C₈×C₄₁	—	2	2	no	—
84	C₂×C₈×C₄₁	—	2	1	no	—
85	C₂×C₈×C₄₁	—	2	1	no	—
86	C₂×C₈×C₄₁	—	2	2	no	—
87	C₂×C₈×C₄₁	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₄₁	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₄₁	#41	1	2	yes	—
96	C₄⋊C₄×C₄₁	—	2	4	no	—
97	C₄⋊C₄×C₄₁	—	2	4	no	—
98	C₄⋊C₄×C₄₁	—	2	4	no	—
99	C₄⋊C₄×C₄₁	—	2	2	no	—
100	C₄⋊C₄×C₄₁	—	2	2	no	—
101	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
102	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
103	C₂²⋊C₄×C₄₁	—	2	4	no	—
104	C₄²×C₄₁	—	2	4	no	—
105	C₄²×C₄₁	—	2	2	no	—
106	C₄²×C₄₁	—	2	2	no	—
107	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
111	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
112	Q₈×C₄₁	—	4	4	no	—
113	Q₈×C₄₁	—	8	2	no	—
114	C₂×C₄×C₄₁	—	2	2	no	—
115	C₂×C₄×C₄₁	—	2	4	no	—
116	C₂×C₄×C₄₁	—	2	4	no	—
117	C₂×C₄×C₄₁	—	2	4	no	—
118	C₂×C₄×C₄₁	—	4	1	no	—
119	C₂×C₄×C₄₁	—	4	2	no	—
120	C₂×C₄×C₄₁	—	4	2	no	—
121	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
131	C₈×C₄₁	—	2	1	no	—
132	C₈×C₄₁	—	2	1	no	—
133	C₈×C₄₁	—	2	2	no	—
134	C₈×C₄₁	—	2	2	no	—
135	C₈×C₄₁	—	2	4	no	—
136	C₈×C₄₁	—	4	2	no	—
137	C₈×C₄₁	C₄×Q₈	1	2	yes	—
138	C₈×C₄₁	C₄×Q₈	1	2	yes	—
139	#23170	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₄₁	—	2	1	no	—
141	C₂²×C₄₁	—	2	1	no	—
142	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₄₁	#556	1	1	yes	characteristic
144	C₂²×C₄₁	#557	1	1	yes	characteristic
145	C₄×C₄₁	—	2	1	no	—
146	C₄×C₄₁	—	2	1	no	—
147	C₄×C₄₁	—	2	1	no	—
148	C₄×C₄₁	—	4	4	no	—
149	C₄×C₄₁	—	4	4	no	—
150	C₄×C₄₁	—	4	4	no	—
151	C₄×C₄₁	#443	1	2	yes	—
152	C₄×C₄₁	C₂×C₄×Q₈	1	1	yes	characteristic
153	C₄×C₄₁	#519	1	1	yes	characteristic
154	C₄×C₄₁	#547	1	1	yes	characteristic
155	C₄×C₄₁	D₄×Q₈	1	1	yes	characteristic
156	#3160	C₂×C₄₁	1	1	yes	characteristic
157	#2962	C₂×C₄₁	1	2	yes	—
158	#1986	C₂×C₄₁	1	2	yes	—
159	#1945	C₂×C₄₁	1	2	yes	—
160	#1922	C₂×C₄₁	1	4	yes	—
161	C₄×#108	C₂×C₄₁	1	1	yes	characteristic
162	#1772	C₂×C₄₁	1	2	yes	—
163	C₈×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
164	C₂×C₄₁	—	2	1	no	—
165	C₂×C₄₁	—	2	1	no	—
166	C₂×C₄₁	#2365	1	1	yes	characteristic
167	C₂×C₄₁	C₂×#443	1	1	yes	characteristic
168	C₂×C₄₁	#2957	1	1	yes	characteristic
169	#533	C₂²×C₄₁	1	2	yes	—
170	#500	C₄×C₄₁	1	1	yes	characteristic, complemented
171	#500	C₄×C₄₁	1	1	yes	characteristic, complemented
172	#499	C₄×C₄₁	1	2	yes	complemented
173	#499	C₄×C₄₁	1	2	yes	complemented
174	#498	C₄×C₄₁	1	2	yes	complemented
175	#446	—	2	4	no	—
176	#446	—	2	2	no	—
177	C₈×Q₈	—	2	2	no	—
178	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
179	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
180	#427	C₂²×C₄₁	1	2	yes	—
181	#427	C₂²×C₄₁	1	2	yes	—
182	#424	C₂²×C₄₁	1	4	yes	—
183	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
184	#405	C₂²×C₄₁	1	4	yes	—
185	C₄×M₁₆	C₂²×C₄₁	1	2	yes	—
186	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
187	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
188	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
189	#341	C₂²×C₄₁	1	2	yes	—
190	#339	C₂²×C₄₁	1	4	yes	—
191	#339	C₂²×C₄₁	1	4	yes	—
192	#336	C₂²×C₄₁	1	2	yes	—
193	C₄₁	#23170	1	1	yes	characteristic, direct product factor, Sylow
194	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
195	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
196	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
198	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
200	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
201	#125	C₂×C₄×C₄₁	1	4	yes	—
202	C₂²⋊Q₈	—	2	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	#117	C₂³×C₄₁	1	4	yes	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
209	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
210	#108	—	2	2	no	—
211	#108	Q₈×C₄₁	1	4	yes	complemented
212	#107	C₂×C₄×C₄₁	1	2	yes	—
213	#107	C₂×C₄×C₄₁	1	2	yes	—
214	#107	C₂×C₄×C₄₁	1	2	yes	—
215	#107	C₂×C₄×C₄₁	1	2	yes	—
216	#106	C₂×C₄×C₄₁	1	4	yes	—
217	#106	C₂×C₄×C₄₁	1	4	yes	—
218	C₄⋊C₈	—	2	4	no	—
219	C₄⋊C₈	—	2	4	no	—
220	C₄⋊C₈	D₄×C₄₁	1	1	yes	characteristic
221	C₄⋊C₈	D₄×C₄₁	1	1	yes	characteristic
222	C₄⋊C₈	D₄×C₄₁	1	2	yes	—
223	#97	—	2	4	no	—
224	#97	—	2	2	no	—
225	C₄×C₈	—	2	4	no	—
226	C₄×C₈	—	2	4	no	—
227	C₄×C₈	—	2	2	no	—
228	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
229	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
230	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
231	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
232	#95	—	2	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
237	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
238	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
239	M₁₆	—	2	2	no	—
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	2	no	—
242	M₁₆	C₂×Q₈×C₄₁	1	4	yes	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
250	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
252	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
253	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
254	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
255	C₂×C₈	C₂×Q₈×C₄₁	1	2	yes	—
256	C₂×C₈	#41×C₄₁	1	2	yes	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
263	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
264	C₂²⋊C₄	—	2	4	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
269	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
270	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
271	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
272	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
273	Q₈	—	4	4	no	—
274	Q₈	—	8	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
283	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
284	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
285	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
286	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
288	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
289	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
290	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
291	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	2	no	—
295	C₈	—	2	2	no	—
296	C₈	—	2	4	no	—
297	C₈	—	4	2	no	—
298	C₈	C₄×Q₈×C₄₁	1	2	yes	—
299	C₈	C₄×Q₈×C₄₁	1	2	yes	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
303	C₂²	C₄₁×#556	1	1	yes	characteristic
304	C₂²	C₄₁×#557	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	4	4	no	—
309	C₄	—	4	4	no	—
310	C₄	—	4	4	no	—
311	C₄	C₄₁×#443	1	2	yes	—
312	C₄	C₂×C₄×Q₈×C₄₁	1	1	yes	characteristic
313	C₄	C₄₁×#519	1	1	yes	characteristic
314	C₄	C₄₁×#547	1	1	yes	characteristic
315	C₄	D₄×Q₈×C₄₁	1	1	yes	characteristic
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	C₄₁×#2365	1	1	yes	characteristic
319	C₂	C₂×C₄₁×#443	1	1	yes	characteristic
320	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
139. #23170	C₄₁	direct (N × H)
170. #500	C₄×C₄₁	semidirect
171. #500	C₄×C₄₁	semidirect
172. #499	C₄×C₄₁	semidirect
173. #499	C₄×C₄₁	semidirect
174. #498	C₄×C₄₁	semidirect
193. C₄₁	#23170	direct (N × H)
211. #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×#1945	2
C₄₁×#1922	2
C₄×#108×C₄₁	2
C₄₁×#1772	2
C₈×C₄⋊C₄×C₄₁	2
#23170	41

Subgroups — C41×#23170 #23799371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23170 #23799371