Subgroups — C₄₁×#23453 #23799654

Order 10496.

1282 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#417	C₂	1	2	yes	maximal
3	C₄₁×#2297	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2026	C₂	1	4	yes	maximal, complemented
5	C₄₁×#2025	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1736	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1728	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
10	C₂×#125×C₄₁	—	2	8	no	—
11	C₂×#125×C₄₁	—	2	2	no	—
12	C₂×#125×C₄₁	C₂²	1	4	yes	—
13	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₄₁	—	2	2	no	—
15	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#417	—	2	8	no	—
18	C₂×#103×C₄₁	C₂²	1	4	yes	—
19	C₂×#102×C₄₁	C₂²	1	4	yes	—
20	C₄₁×#391	C₂²	1	2	yes	—
21	C₄₁×#390	C₂²	1	2	yes	—
22	C₄₁×#386	—	2	2	no	—
23	C₄×C₂²⋊C₄×C₄₁	—	2	2	no	—
24	C₄₁×#340	C₂²	1	8	yes	—
25	C₄₁×#336	C₂²	1	4	yes	—
26	C₂×#41×C₄₁	—	2	2	no	—
27	C₂×#41×C₄₁	—	2	4	no	—
28	C₂×#41×C₄₁	—	2	4	no	—
29	C₂×#41×C₄₁	Q₈	1	4	yes	—
30	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
31	C₂²×D₄×C₄₁	—	2	1	no	—
32	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₄₁	—	2	2	no	—
34	C₂³×C₄×C₄₁	—	2	1	no	—
35	C₂³×C₄×C₄₁	D₄	1	2	yes	—
36	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₄₁	—	2	4	no	—
38	C₂×M₁₆×C₄₁	—	2	4	no	—
39	C₂×M₁₆×C₄₁	—	2	4	no	—
40	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
41	#125×C₄₁	—	2	16	no	—
42	#125×C₄₁	—	2	16	no	—
43	#117×C₄₁	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₁	—	2	16	no	—
45	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
46	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₁	D₄	1	8	yes	—
49	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
51	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
52	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
53	C₂×C₄²×C₄₁	—	2	4	no	—
54	C₂×C₄²×C₄₁	—	2	1	no	—
55	C₂×C₄²×C₄₁	D₄	1	2	yes	—
56	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
57	#103×C₄₁	—	2	8	no	—
58	#102×C₄₁	—	2	8	no	—
59	#95×C₄₁	—	2	4	no	—
60	C₂⁴×C₄₁	—	2	1	no	—
61	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
62	#41×C₄₁	—	2	16	no	—
63	#41×C₄₁	—	4	8	no	—
64	C₂×Q₈×C₄₁	—	2	1	no	—
65	C₂×Q₈×C₄₁	—	2	2	no	—
66	C₂×Q₈×C₄₁	—	2	2	no	—
67	C₂×Q₈×C₄₁	C₂×Q₈	1	2	yes	—
68	C₂×D₄×C₄₁	—	2	4	no	—
69	C₂×D₄×C₄₁	—	2	2	no	—
70	C₂×D₄×C₄₁	—	2	2	no	—
71	C₂×D₄×C₄₁	—	2	1	no	—
72	C₂×D₄×C₄₁	—	4	2	no	—
73	C₂×D₄×C₄₁	—	4	2	no	—
74	C₂×D₄×C₄₁	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₄₁	—	2	4	no	—
76	C₂²×C₄×C₄₁	—	2	8	no	—
77	C₂²×C₄×C₄₁	—	2	1	no	—
78	C₂²×C₄×C₄₁	—	2	4	no	—
79	C₂²×C₄×C₄₁	—	2	4	no	—
80	C₂²×C₄×C₄₁	—	2	2	no	—
81	C₂²×C₄×C₄₁	—	4	4	no	—
82	C₂²×C₄×C₄₁	—	4	8	no	—
83	C₂²×C₄×C₄₁	—	4	2	no	—
84	C₂²×C₄×C₄₁	—	4	2	no	—
85	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₄₁	#41	1	4	yes	—
90	C₂²×C₄×C₄₁	#41	1	2	yes	—
91	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
92	M₁₆×C₄₁	—	4	8	no	—
93	C₂×C₈×C₄₁	—	2	4	no	—
94	C₂×C₈×C₄₁	—	2	4	no	—
95	C₂×C₈×C₄₁	—	2	4	no	—
96	C₄⋊C₄×C₄₁	—	2	32	no	—
97	C₄⋊C₄×C₄₁	—	2	8	no	—
98	C₄⋊C₄×C₄₁	—	2	4	no	—
99	C₄⋊C₄×C₄₁	—	2	4	no	—
100	C₄⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
101	C₂²⋊C₄×C₄₁	—	4	8	no	—
102	C₂²⋊C₄×C₄₁	—	4	2	no	—
103	C₄²×C₄₁	—	2	4	no	—
104	C₄²×C₄₁	—	2	4	no	—
105	C₄²×C₄₁	—	4	2	no	—
106	C₂³×C₄₁	—	2	4	no	—
107	C₂³×C₄₁	—	2	2	no	—
108	C₂³×C₄₁	—	4	1	no	—
109	C₂³×C₄₁	—	4	1	no	—
110	C₂³×C₄₁	—	4	4	no	—
111	C₂³×C₄₁	#120	1	2	yes	—
112	C₂³×C₄₁	C₂²⋊Q₈	1	2	yes	—
113	C₂³×C₄₁	#124	1	2	yes	—
114	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₄₁	—	2	4	no	—
116	Q₈×C₄₁	—	4	2	no	—
117	D₄×C₄₁	—	2	4	no	—
118	D₄×C₄₁	—	4	8	no	—
119	D₄×C₄₁	—	4	2	no	—
120	C₂×C₄×C₄₁	—	2	4	no	—
121	C₂×C₄×C₄₁	—	2	8	no	—
122	C₂×C₄×C₄₁	—	2	2	no	—
123	C₂×C₄×C₄₁	—	2	4	no	—
124	C₂×C₄×C₄₁	—	2	4	no	—
125	C₂×C₄×C₄₁	—	4	8	no	—
126	C₂×C₄×C₄₁	—	4	16	no	—
127	C₂×C₄×C₄₁	—	4	1	no	—
128	C₂×C₄×C₄₁	—	4	1	no	—
129	C₂×C₄×C₄₁	—	4	8	no	—
130	C₂×C₄×C₄₁	—	4	8	no	—
131	C₂×C₄×C₄₁	—	4	2	no	—
132	C₂×C₄×C₄₁	—	4	4	no	—
133	C₂×C₄×C₄₁	#120	1	2	yes	—
134	C₂×C₄×C₄₁	C₂²⋊Q₈	1	8	yes	—
135	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₄₁	#124	1	8	yes	—
137	C₂×C₄×C₄₁	#124	1	2	yes	—
138	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
139	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
141	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
142	C₈×C₄₁	—	4	4	no	—
143	#23453	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₄₁	—	2	4	no	—
145	C₂²×C₄₁	—	2	4	no	—
146	C₂²×C₄₁	—	2	4	no	—
147	C₂²×C₄₁	—	4	4	no	—
148	C₂²×C₄₁	—	4	2	no	—
149	C₂²×C₄₁	—	8	2	no	—
150	C₂²×C₄₁	—	8	2	no	—
151	C₂²×C₄₁	#393	1	4	yes	—
152	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
153	C₂²×C₄₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
154	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
155	C₂²×C₄₁	#575	1	4	yes	—
156	C₄×C₄₁	—	4	2	no	—
157	C₄×C₄₁	—	4	4	no	—
158	C₄×C₄₁	—	4	8	no	—
159	C₄×C₄₁	—	8	2	no	—
160	C₄×C₄₁	#393	1	4	yes	—
161	C₄×C₄₁	C₂×C₂²⋊Q₈	1	2	yes	—
162	C₄×C₄₁	C₂×#124	1	2	yes	—
163	C₂²×#125	C₂×C₄₁	1	1	yes	characteristic, complemented
164	C₂×#417	C₂×C₄₁	1	2	yes	—
165	#2297	C₂×C₄₁	1	1	yes	characteristic
166	#2026	C₂×C₄₁	1	4	yes	complemented
167	#2025	C₂×C₄₁	1	4	yes	complemented
168	#1736	C₂×C₄₁	1	2	yes	complemented
169	#1728	C₂×C₄₁	1	1	yes	characteristic, complemented
170	C₂×C₄₁	—	2	4	no	—
171	C₂×C₄₁	—	8	2	no	—
172	C₂×C₄₁	C₂×#393	1	1	yes	characteristic
173	C₂×C₄₁	#3005	1	2	yes	—
174	C₂×C₄₁	#3080	1	2	yes	—
175	C₂×C₄₁	#3142	1	2	yes	—
176	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
177	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
178	C₂×#125	—	2	8	no	—
179	C₂×#125	—	2	2	no	—
180	C₂×#125	C₂²×C₄₁	1	4	yes	—
181	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
182	C₂²×C₄⋊C₄	—	2	2	no	—
183	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
184	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
185	#417	—	2	8	no	—
186	C₂×#103	C₂²×C₄₁	1	4	yes	—
187	C₂×#102	C₂²×C₄₁	1	4	yes	—
188	#391	C₂²×C₄₁	1	2	yes	—
189	#390	C₂²×C₄₁	1	2	yes	—
190	#386	—	2	2	no	—
191	C₄×C₂²⋊C₄	—	2	2	no	—
192	#340	C₂²×C₄₁	1	8	yes	—
193	#336	C₂²×C₄₁	1	4	yes	—
194	C₄₁	#23453	1	1	yes	characteristic, direct product factor, Sylow
195	C₂×#41	—	2	2	no	—
196	C₂×#41	—	2	4	no	—
197	C₂×#41	—	2	4	no	—
198	C₂×#41	Q₈×C₄₁	1	4	yes	—
199	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
202	C₂³×C₄	—	2	2	no	—
203	C₂³×C₄	—	2	1	no	—
204	C₂³×C₄	D₄×C₄₁	1	2	yes	—
205	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
206	C₂×M₁₆	—	2	4	no	—
207	C₂×M₁₆	—	2	4	no	—
208	C₂×M₁₆	—	2	4	no	—
209	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
210	#125	—	2	16	no	—
211	#125	—	2	16	no	—
212	#117	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	16	no	—
214	C₂×C₄⋊C₄	—	2	8	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	D₄×C₄₁	1	8	yes	—
218	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	2	4	no	—
222	C₂×C₄²	—	2	4	no	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	D₄×C₄₁	1	2	yes	—
225	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
226	#103	—	2	8	no	—
227	#102	—	2	8	no	—
228	#95	—	2	4	no	—
229	C₂⁴	—	2	1	no	—
230	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
231	#41	—	2	16	no	—
232	#41	—	4	8	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	C₂×Q₈×C₄₁	1	2	yes	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	C₂×Q₈×C₄₁	1	2	yes	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	4	4	no	—
251	C₂²×C₄	—	4	8	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
255	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
257	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
258	C₂²×C₄	#41×C₄₁	1	4	yes	—
259	C₂²×C₄	#41×C₄₁	1	2	yes	—
260	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
261	M₁₆	—	4	8	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	2	4	no	—
265	C₄⋊C₄	—	2	32	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
270	C₂²⋊C₄	—	4	8	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₄²	—	2	4	no	—
273	C₄²	—	2	4	no	—
274	C₄²	—	4	2	no	—
275	C₂³	—	2	4	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	4	no	—
280	C₂³	#120×C₄₁	1	2	yes	—
281	C₂³	C₂²⋊Q₈×C₄₁	1	2	yes	—
282	C₂³	#124×C₄₁	1	2	yes	—
283	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
284	Q₈	—	2	4	no	—
285	Q₈	—	4	2	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	8	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	8	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	8	no	—
295	C₂×C₄	—	4	16	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	8	no	—
299	C₂×C₄	—	4	8	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	#120×C₄₁	1	2	yes	—
303	C₂×C₄	C₂²⋊Q₈×C₄₁	1	8	yes	—
304	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
305	C₂×C₄	#124×C₄₁	1	8	yes	—
306	C₂×C₄	#124×C₄₁	1	2	yes	—
307	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
308	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
310	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
311	C₈	—	4	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	C₄₁×#393	1	4	yes	—
320	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
321	C₂²	C₂×C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
322	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
323	C₂²	C₄₁×#575	1	4	yes	—
324	C₄	—	4	2	no	—
325	C₄	—	4	4	no	—
326	C₄	—	4	8	no	—
327	C₄	—	8	2	no	—
328	C₄	C₄₁×#393	1	4	yes	—
329	C₄	C₂×C₂²⋊Q₈×C₄₁	1	2	yes	—
330	C₄	C₂×#124×C₄₁	1	2	yes	—
331	C₂	—	2	4	no	—
332	C₂	—	8	2	no	—
333	C₂	C₂×C₄₁×#393	1	1	yes	characteristic
334	C₂	C₄₁×#3005	1	2	yes	—
335	C₂	C₄₁×#3080	1	2	yes	—
336	C₂	C₄₁×#3142	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂²×#41×C₄₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#125×C₄₁	C₂	semidirect
4. C₄₁×#2026	C₂	semidirect
5. C₄₁×#2025	C₂	semidirect
6. C₄₁×#1736	C₂	semidirect
7. C₄₁×#1728	C₂	semidirect
143. #23453	C₄₁	direct (N × H)
163. C₂²×#125	C₂×C₄₁	semidirect
166. #2026	C₂×C₄₁	semidirect
167. #2025	C₂×C₄₁	semidirect
168. #1736	C₂×C₄₁	semidirect
169. #1728	C₂×C₄₁	semidirect
194. C₄₁	#23453	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#125×C₄₁	2
C₂×C₄₁×#417	2
C₄₁×#2297	2
C₄₁×#2026	2
C₄₁×#2025	2
C₄₁×#1736	2
C₄₁×#1728	2
#23453	41

Subgroups — C41×#23453 #23799654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23453 #23799654