Subgroups — C₄₁×#33125 #23809326

Order 10496.

794 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3094	C₂	1	2	yes	maximal, complemented
2	C₂×C₄₁×#469	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2873	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1978	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1951	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1931	C₂	1	4	yes	maximal, complemented
7	C₄₁×#1900	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1811	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×#121×C₄₁	C₂²	1	2	yes	—
13	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
14	C₄₁×#482	—	2	4	no	—
15	C₄₁×#481	—	2	2	no	—
16	C₄₁×#480	—	2	2	no	—
17	C₄₁×#469	—	2	2	no	—
18	C₄₁×#469	—	2	2	no	—
19	C₂×#108×C₄₁	C₂²	1	2	yes	—
20	C₄₁×#428	C₂²	1	4	yes	—
21	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
22	C₄₁×#418	C₂²	1	4	yes	—
23	C₂×#102×C₄₁	C₂²	1	2	yes	—
24	C₂×#100×C₄₁	C₂²	1	4	yes	—
25	C₄₁×#407	C₂²	1	2	yes	—
26	C₄₁×#405	—	2	2	no	—
27	C₄₁×#405	C₂²	1	2	yes	—
28	C₄₁×#405	C₂²	1	2	yes	—
29	C₄×M₁₆×C₄₁	—	2	2	no	—
30	C₄₁×#343	C₂²	1	4	yes	—
31	C₄₁×#341	C₂²	1	2	yes	—
32	C₂²×D₄×C₄₁	—	2	2	no	—
33	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
34	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
35	C₂×D₈×C₄₁	—	2	2	no	—
36	C₂×D₈×C₄₁	—	2	4	no	—
37	C₂×M₁₆×C₄₁	—	2	1	no	—
38	C₂×M₁₆×C₄₁	—	2	4	no	—
39	C₂×M₁₆×C₄₁	—	2	4	no	—
40	C₂×M₁₆×C₄₁	D₄	1	2	yes	—
41	C₂×M₁₆×C₄₁	D₄	1	2	yes	—
42	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
43	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₄₁	—	2	1	no	—
45	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
47	#121×C₄₁	—	2	2	no	—
48	#121×C₄₁	—	2	2	no	—
49	#121×C₄₁	—	2	2	no	—
50	#121×C₄₁	—	2	2	no	—
51	#117×C₄₁	—	2	1	no	—
52	#117×C₄₁	—	2	1	no	—
53	#117×C₄₁	D₄	1	2	yes	—
54	#117×C₄₁	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
58	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
59	C₂×C₄²×C₄₁	—	2	1	no	—
60	#108×C₄₁	—	2	4	no	—
61	#107×C₄₁	—	2	4	no	—
62	#106×C₄₁	—	2	2	no	—
63	#106×C₄₁	—	2	4	no	—
64	#102×C₄₁	—	2	4	no	—
65	#102×C₄₁	—	2	4	no	—
66	#102×C₄₁	—	2	4	no	—
67	#100×C₄₁	—	2	8	no	—
68	C₂²⋊C₈×C₄₁	—	2	2	no	—
69	C₂²⋊C₈×C₄₁	—	2	2	no	—
70	#97×C₄₁	—	2	2	no	—
71	#97×C₄₁	—	2	2	no	—
72	C₄×C₈×C₄₁	—	2	2	no	—
73	C₄×C₈×C₄₁	—	2	2	no	—
74	C₂⁴×C₄₁	—	2	2	no	—
75	C₂⁴×C₄₁	#41	1	1	yes	characteristic
76	C₂×D₄×C₄₁	—	2	2	no	—
77	C₂×D₄×C₄₁	—	2	2	no	—
78	C₂×D₄×C₄₁	—	2	4	no	—
79	C₂×D₄×C₄₁	—	2	4	no	—
80	C₂×D₄×C₄₁	—	4	2	no	—
81	C₂×D₄×C₄₁	—	4	2	no	—
82	C₂×D₄×C₄₁	—	8	4	no	—
83	C₂²×C₄×C₄₁	—	2	2	no	—
84	C₂²×C₄×C₄₁	—	2	2	no	—
85	C₂²×C₄×C₄₁	—	2	2	no	—
86	C₂²×C₄×C₄₁	—	2	2	no	—
87	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
92	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
93	D₈×C₄₁	—	4	4	no	—
94	M₁₆×C₄₁	—	2	4	no	—
95	M₁₆×C₄₁	—	2	2	no	—
96	M₁₆×C₄₁	—	2	4	no	—
97	M₁₆×C₄₁	—	2	4	no	—
98	M₁₆×C₄₁	—	2	4	no	—
99	M₁₆×C₄₁	—	2	4	no	—
100	M₁₆×C₄₁	#41	1	4	yes	—
101	C₂×C₈×C₄₁	—	2	4	no	—
102	C₂×C₈×C₄₁	—	2	2	no	—
103	C₂×C₈×C₄₁	—	2	2	no	—
104	C₂×C₈×C₄₁	—	2	4	no	—
105	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₄₁	#41	1	4	yes	—
108	C₄⋊C₄×C₄₁	—	2	2	no	—
109	C₄⋊C₄×C₄₁	—	2	2	no	—
110	C₄⋊C₄×C₄₁	—	2	2	no	—
111	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
112	C₂²⋊C₄×C₄₁	—	2	2	no	—
113	C₂²⋊C₄×C₄₁	—	2	2	no	—
114	C₂²⋊C₄×C₄₁	—	4	2	no	—
115	C₂²⋊C₄×C₄₁	—	4	2	no	—
116	C₄²×C₄₁	—	2	2	no	—
117	C₄²×C₄₁	—	2	2	no	—
118	C₂³×C₄₁	—	2	2	no	—
119	C₂³×C₄₁	—	4	2	no	—
120	C₂³×C₄₁	—	4	2	no	—
121	C₂³×C₄₁	—	4	4	no	—
122	C₂³×C₄₁	—	8	2	no	—
123	C₂³×C₄₁	—	8	2	no	—
124	C₂³×C₄₁	#121	1	1	yes	characteristic
125	C₂³×C₄₁	#121	1	1	yes	characteristic
126	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
127	D₄×C₄₁	—	4	2	no	—
128	D₄×C₄₁	—	4	2	no	—
129	D₄×C₄₁	—	4	4	no	—
130	D₄×C₄₁	—	8	4	no	—
131	D₄×C₄₁	—	8	4	no	—
132	C₂×C₄×C₄₁	—	2	2	no	—
133	C₂×C₄×C₄₁	—	2	1	no	—
134	C₂×C₄×C₄₁	—	2	1	no	—
135	C₂×C₄×C₄₁	—	2	1	no	—
136	C₂×C₄×C₄₁	—	2	1	no	—
137	C₂×C₄×C₄₁	—	2	2	no	—
138	C₂×C₄×C₄₁	—	2	2	no	—
139	C₂×C₄×C₄₁	—	2	4	no	—
140	C₂×C₄×C₄₁	—	4	2	no	—
141	C₂×C₄×C₄₁	—	8	2	no	—
142	C₂×C₄×C₄₁	—	8	2	no	—
143	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
144	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
148	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
149	C₈×C₄₁	—	2	2	no	—
150	C₈×C₄₁	—	2	4	no	—
151	C₈×C₄₁	—	4	4	no	—
152	C₈×C₄₁	#123	1	4	yes	—
153	#33125	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂²×C₄₁	—	2	2	no	—
155	C₂²×C₄₁	—	2	1	no	—
156	C₂²×C₄₁	—	2	1	no	—
157	C₂²×C₄₁	—	2	2	no	—
158	C₂²×C₄₁	—	2	2	no	—
159	C₂²×C₄₁	—	8	2	no	—
160	C₂²×C₄₁	—	8	2	no	—
161	C₂²×C₄₁	—	8	2	no	—
162	C₂²×C₄₁	—	8	2	no	—
163	C₂²×C₄₁	—	16	4	no	—
164	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
165	C₂²×C₄₁	#529	1	2	yes	—
166	C₄×C₄₁	—	2	1	no	—
167	C₄×C₄₁	—	2	1	no	—
168	C₄×C₄₁	—	4	4	no	—
169	C₄×C₄₁	—	8	2	no	—
170	C₄×C₄₁	C₂×#123	1	1	yes	characteristic
171	C₄×C₄₁	#525	1	1	yes	characteristic
172	C₄×C₄₁	#532	1	2	yes	—
173	#3094	C₂×C₄₁	1	2	yes	complemented
174	C₂×#469	C₂×C₄₁	1	1	yes	characteristic
175	#2873	C₂×C₄₁	1	1	yes	characteristic, complemented
176	#1978	C₂×C₄₁	1	2	yes	complemented
177	#1951	C₂×C₄₁	1	2	yes	complemented
178	#1931	C₂×C₄₁	1	4	yes	complemented
179	#1900	C₂×C₄₁	1	2	yes	complemented
180	#1811	C₂×C₄₁	1	1	yes	characteristic, complemented
181	C₂×C₄₁	—	2	2	no	—
182	C₂×C₄₁	—	4	1	no	—
183	C₂×C₄₁	—	4	1	no	—
184	C₂×C₄₁	—	16	2	no	—
185	C₂×C₄₁	#2854	1	1	yes	characteristic
186	C₂×C₄₁	#3068	1	2	yes	—
187	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
188	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
189	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
190	C₂×#121	C₂²×C₄₁	1	2	yes	—
191	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
192	#482	—	2	4	no	—
193	#481	—	2	2	no	—
194	#480	—	2	2	no	—
195	#469	—	2	2	no	—
196	#469	—	2	2	no	—
197	C₂×#108	C₂²×C₄₁	1	2	yes	—
198	#428	C₂²×C₄₁	1	4	yes	—
199	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
200	#418	C₂²×C₄₁	1	4	yes	—
201	C₂×#102	C₂²×C₄₁	1	2	yes	—
202	C₂×#100	C₂²×C₄₁	1	4	yes	—
203	#407	C₂²×C₄₁	1	2	yes	—
204	#405	—	2	2	no	—
205	#405	C₂²×C₄₁	1	2	yes	—
206	#405	C₂²×C₄₁	1	2	yes	—
207	C₄×M₁₆	—	2	2	no	—
208	#343	C₂²×C₄₁	1	4	yes	—
209	#341	C₂²×C₄₁	1	2	yes	—
210	C₄₁	#33125	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²×D₄	—	2	2	no	—
212	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
213	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
214	C₂×D₈	—	2	2	no	—
215	C₂×D₈	—	2	4	no	—
216	C₂×M₁₆	—	2	1	no	—
217	C₂×M₁₆	—	2	4	no	—
218	C₂×M₁₆	—	2	4	no	—
219	C₂×M₁₆	D₄×C₄₁	1	2	yes	—
220	C₂×M₁₆	D₄×C₄₁	1	2	yes	—
221	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
222	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
226	#121	—	2	2	no	—
227	#121	—	2	2	no	—
228	#121	—	2	2	no	—
229	#121	—	2	2	no	—
230	#117	—	2	1	no	—
231	#117	—	2	1	no	—
232	#117	D₄×C₄₁	1	2	yes	—
233	#117	C₂³×C₄₁	1	2	yes	—
234	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
235	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
236	C₂×C₂²⋊C₄	—	2	1	no	—
237	C₂×C₂²⋊C₄	—	2	2	no	—
238	C₂×C₄²	—	2	1	no	—
239	#108	—	2	4	no	—
240	#107	—	2	4	no	—
241	#106	—	2	2	no	—
242	#106	—	2	4	no	—
243	#102	—	2	4	no	—
244	#102	—	2	4	no	—
245	#102	—	2	4	no	—
246	#100	—	2	8	no	—
247	C₂²⋊C₈	—	2	2	no	—
248	C₂²⋊C₈	—	2	2	no	—
249	#97	—	2	2	no	—
250	#97	—	2	2	no	—
251	C₄×C₈	—	2	2	no	—
252	C₄×C₈	—	2	2	no	—
253	C₂⁴	—	2	2	no	—
254	C₂⁴	#41×C₄₁	1	1	yes	characteristic
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	4	no	—
258	C₂×D₄	—	2	4	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	8	4	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
270	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
271	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
272	D₈	—	4	4	no	—
273	M₁₆	—	2	4	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	4	no	—
276	M₁₆	—	2	4	no	—
277	M₁₆	—	2	4	no	—
278	M₁₆	—	2	4	no	—
279	M₁₆	#41×C₄₁	1	4	yes	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
286	C₂×C₈	#41×C₄₁	1	4	yes	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
291	C₂²⋊C₄	—	2	2	no	—
292	C₂²⋊C₄	—	2	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₂³	—	2	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	4	no	—
301	C₂³	—	8	2	no	—
302	C₂³	—	8	2	no	—
303	C₂³	#121×C₄₁	1	1	yes	characteristic
304	C₂³	#121×C₄₁	1	1	yes	characteristic
305	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	4	no	—
309	D₄	—	8	4	no	—
310	D₄	—	8	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	2	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
323	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
324	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
326	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
327	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
328	C₈	—	2	2	no	—
329	C₈	—	2	4	no	—
330	C₈	—	4	4	no	—
331	C₈	#123×C₄₁	1	4	yes	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	16	4	no	—
342	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
343	C₂²	C₄₁×#529	1	2	yes	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	4	4	no	—
347	C₄	—	8	2	no	—
348	C₄	C₂×#123×C₄₁	1	1	yes	characteristic
349	C₄	C₄₁×#525	1	1	yes	characteristic
350	C₄	C₄₁×#532	1	2	yes	—
351	C₂	—	2	2	no	—
352	C₂	—	4	1	no	—
353	C₂	—	4	1	no	—
354	C₂	—	16	2	no	—
355	C₂	C₄₁×#2854	1	1	yes	characteristic
356	C₂	C₄₁×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#469	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3094	C₂	semidirect
3. C₄₁×#2873	C₂	semidirect
4. C₄₁×#1978	C₂	semidirect
5. C₄₁×#1951	C₂	semidirect
6. C₄₁×#1931	C₂	semidirect
7. C₄₁×#1900	C₂	semidirect
8. C₄₁×#1811	C₂	semidirect
153. #33125	C₄₁	direct (N × H)
173. #3094	C₂×C₄₁	semidirect
175. #2873	C₂×C₄₁	semidirect
176. #1978	C₂×C₄₁	semidirect
177. #1951	C₂×C₄₁	semidirect
178. #1931	C₂×C₄₁	semidirect
179. #1900	C₂×C₄₁	semidirect
180. #1811	C₂×C₄₁	semidirect
210. C₄₁	#33125	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×#3094	2
C₂×C₄₁×#469	2
C₄₁×#2873	2
C₄₁×#1978	2
C₄₁×#1951	2
C₄₁×#1931	2
C₄₁×#1900	2
C₄₁×#1811	2
#33125	41

Subgroups — C41×#33125 #23809326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33125 #23809326