Subgroups — C₄₁×#29112 #23805313

Order 10496.

862 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2851	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#391	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2074	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2026	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1935	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1824	C₂	1	2	yes	maximal
7	C₄₁×#1787	C₂	1	2	yes	maximal
8	C₄₁×#1749	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×#125×C₄₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×#103×C₄₁	C₂²	1	4	yes	—
14	C₂×#103×C₄₁	C₂²	1	4	yes	—
15	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
16	C₂×#97×C₄₁	C₂²	1	2	yes	—
17	C₄₁×#391	—	2	2	no	—
18	C₄₁×#391	C₂²	1	2	yes	—
19	C₄₁×#391	C₂²	1	2	yes	—
20	C₄₁×#389	C₂²	1	2	yes	—
21	C₄₁×#387	—	2	2	no	—
22	C₄₁×#385	—	2	2	no	—
23	C₄₁×#382	C₂²	1	1	yes	characteristic
24	C₄₁×#380	C₂²	1	2	yes	—
25	C₂×#95×C₄₁	—	2	2	no	—
26	C₄₁×#340	C₂²	1	4	yes	—
27	C₄₁×#340	C₂²	1	4	yes	—
28	C₄₁×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₄₁	—	2	4	no	—
30	C₂²×Q₈×C₄₁	—	2	1	no	—
31	C₂²×Q₈×C₄₁	—	2	4	no	—
32	C₂²×Q₈×C₄₁	—	4	2	no	—
33	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
34	C₂³×C₄×C₄₁	—	2	1	no	—
35	C₂³×C₄×C₄₁	—	2	1	no	—
36	C₂³×C₄×C₄₁	D₄	1	2	yes	—
37	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
39	#125×C₄₁	—	4	2	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
47	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
48	C₂×C₄²×C₄₁	—	4	2	no	—
49	C₂×C₄²×C₄₁	D₄	1	2	yes	—
50	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
52	#103×C₄₁	—	2	8	no	—
53	#103×C₄₁	—	4	4	no	—
54	C₂²⋊C₈×C₄₁	—	2	4	no	—
55	#97×C₄₁	—	4	2	no	—
56	#95×C₄₁	—	2	4	no	—
57	#95×C₄₁	—	2	4	no	—
58	#95×C₄₁	—	2	2	no	—
59	#95×C₄₁	—	2	2	no	—
60	#95×C₄₁	—	2	2	no	—
61	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
62	C₂×Q₈×C₄₁	—	2	4	no	—
63	C₂×Q₈×C₄₁	—	2	2	no	—
64	C₂×Q₈×C₄₁	—	2	4	no	—
65	C₂×Q₈×C₄₁	—	2	4	no	—
66	C₂×Q₈×C₄₁	—	4	16	no	—
67	C₂×Q₈×C₄₁	—	4	2	no	—
68	C₂×Q₈×C₄₁	—	4	2	no	—
69	C₂×Q₈×C₄₁	#41	1	4	yes	—
70	C₂²×C₄×C₄₁	—	2	2	no	—
71	C₂²×C₄×C₄₁	—	2	2	no	—
72	C₂²×C₄×C₄₁	—	2	4	no	—
73	C₂²×C₄×C₄₁	—	2	4	no	—
74	C₂²×C₄×C₄₁	—	2	2	no	—
75	C₂²×C₄×C₄₁	—	2	2	no	—
76	C₂²×C₄×C₄₁	—	2	2	no	—
77	C₂²×C₄×C₄₁	—	2	2	no	—
78	C₂²×C₄×C₄₁	—	4	2	no	—
79	C₂²×C₄×C₄₁	—	4	2	no	—
80	C₂²×C₄×C₄₁	—	4	2	no	—
81	C₂²×C₄×C₄₁	—	4	2	no	—
82	C₂²×C₄×C₄₁	—	4	4	no	—
83	C₂²×C₄×C₄₁	—	4	2	no	—
84	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₁	#41	1	4	yes	—
88	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₄₁	—	2	4	no	—
90	C₂×C₈×C₄₁	—	2	4	no	—
91	C₂×C₈×C₄₁	—	2	4	no	—
92	C₄⋊C₄×C₄₁	—	2	4	no	—
93	C₄⋊C₄×C₄₁	—	2	4	no	—
94	C₄⋊C₄×C₄₁	—	2	4	no	—
95	C₄⋊C₄×C₄₁	—	4	4	no	—
96	C₄⋊C₄×C₄₁	—	4	4	no	—
97	C₄⋊C₄×C₄₁	—	4	2	no	—
98	C₂²⋊C₄×C₄₁	—	4	4	no	—
99	C₄²×C₄₁	—	2	2	no	—
100	C₄²×C₄₁	—	2	2	no	—
101	C₄²×C₄₁	—	4	2	no	—
102	C₂³×C₄₁	—	2	1	no	—
103	C₂³×C₄₁	—	2	1	no	—
104	C₂³×C₄₁	—	2	2	no	—
105	C₂³×C₄₁	#124	1	2	yes	—
106	C₂³×C₄₁	#137	1	4	yes	—
107	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₄₁	—	2	8	no	—
109	Q₈×C₄₁	—	4	8	no	—
110	Q₈×C₄₁	—	4	4	no	—
111	C₂×C₄×C₄₁	—	2	1	no	—
112	C₂×C₄×C₄₁	—	2	1	no	—
113	C₂×C₄×C₄₁	—	2	8	no	—
114	C₂×C₄×C₄₁	—	2	4	no	—
115	C₂×C₄×C₄₁	—	2	2	no	—
116	C₂×C₄×C₄₁	—	2	2	no	—
117	C₂×C₄×C₄₁	—	2	2	no	—
118	C₂×C₄×C₄₁	—	2	2	no	—
119	C₂×C₄×C₄₁	—	2	2	no	—
120	C₂×C₄×C₄₁	—	2	2	no	—
121	C₂×C₄×C₄₁	—	4	8	no	—
122	C₂×C₄×C₄₁	—	4	8	no	—
123	C₂×C₄×C₄₁	—	4	2	no	—
124	C₂×C₄×C₄₁	—	4	2	no	—
125	C₂×C₄×C₄₁	—	4	2	no	—
126	C₂×C₄×C₄₁	—	4	2	no	—
127	C₂×C₄×C₄₁	—	4	2	no	—
128	C₂×C₄×C₄₁	—	4	2	no	—
129	C₂×C₄×C₄₁	—	4	2	no	—
130	C₂×C₄×C₄₁	—	4	2	no	—
131	C₂×C₄×C₄₁	—	4	2	no	—
132	C₂×C₄×C₄₁	—	4	4	no	—
133	C₂×C₄×C₄₁	—	8	4	no	—
134	C₂×C₄×C₄₁	#124	1	2	yes	—
135	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
136	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
137	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
138	C₈×C₄₁	—	4	4	no	—
139	#29112	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₄₁	—	2	2	no	—
141	C₂²×C₄₁	—	2	4	no	—
142	C₂²×C₄₁	—	2	2	no	—
143	C₂²×C₄₁	—	2	4	no	—
144	C₂²×C₄₁	#491	1	4	yes	—
145	C₂²×C₄₁	#525	1	1	yes	characteristic
146	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
147	C₂²×C₄₁	#534	1	1	yes	characteristic
148	C₂²×C₄₁	C₂×#137	1	2	yes	—
149	C₂²×C₄₁	#575	1	2	yes	—
150	C₄×C₄₁	—	2	2	no	—
151	C₄×C₄₁	—	4	4	no	—
152	C₄×C₄₁	—	4	2	no	—
153	C₄×C₄₁	—	4	2	no	—
154	C₄×C₄₁	—	8	2	no	—
155	C₄×C₄₁	—	8	2	no	—
156	C₄×C₄₁	—	8	2	no	—
157	C₄×C₄₁	#551	1	4	yes	—
158	#2851	C₂×C₄₁	1	1	yes	characteristic
159	C₂×#391	C₂×C₄₁	1	1	yes	characteristic
160	#2074	C₂×C₄₁	1	2	yes	complemented
161	#2026	C₂×C₄₁	1	2	yes	complemented
162	#1935	C₂×C₄₁	1	4	yes	complemented
163	#1824	C₂×C₄₁	1	2	yes	—
164	#1787	C₂×C₄₁	1	2	yes	—
165	#1749	C₂×C₄₁	1	1	yes	characteristic
166	C₂×C₄₁	—	2	4	no	—
167	C₂×C₄₁	#2561	1	1	yes	characteristic
168	C₂×C₄₁	#3117	1	2	yes	—
169	C₂×C₄₁	C₂×#491	1	1	yes	characteristic
170	C₂×C₄₁	#3140	1	1	yes	characteristic
171	C₂×C₄₁	#3196	1	2	yes	—
172	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
173	C₂×#125	C₂²×C₄₁	1	1	yes	characteristic
174	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
175	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
176	C₂×#103	C₂²×C₄₁	1	4	yes	—
177	C₂×#103	C₂²×C₄₁	1	4	yes	—
178	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
179	C₂×#97	C₂²×C₄₁	1	2	yes	—
180	#391	—	2	2	no	—
181	#391	C₂²×C₄₁	1	2	yes	—
182	#391	C₂²×C₄₁	1	2	yes	—
183	#389	C₂²×C₄₁	1	2	yes	—
184	#387	—	2	2	no	—
185	#385	—	2	2	no	—
186	#382	C₂²×C₄₁	1	1	yes	characteristic
187	#380	C₂²×C₄₁	1	2	yes	—
188	C₂×#95	—	2	2	no	—
189	#340	C₂²×C₄₁	1	4	yes	—
190	#340	C₂²×C₄₁	1	4	yes	—
191	#336	C₂²×C₄₁	1	2	yes	—
192	C₄₁	#29112	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²×Q₈	—	2	4	no	—
194	C₂²×Q₈	—	2	1	no	—
195	C₂²×Q₈	—	2	4	no	—
196	C₂²×Q₈	—	4	2	no	—
197	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
198	C₂³×C₄	—	2	1	no	—
199	C₂³×C₄	—	2	1	no	—
200	C₂³×C₄	D₄×C₄₁	1	2	yes	—
201	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
203	#125	—	4	2	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
210	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₄²	—	4	2	no	—
213	C₂×C₄²	D₄×C₄₁	1	2	yes	—
214	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
216	#103	—	2	8	no	—
217	#103	—	4	4	no	—
218	C₂²⋊C₈	—	2	4	no	—
219	#97	—	4	2	no	—
220	#95	—	2	4	no	—
221	#95	—	2	4	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	2	4	no	—
230	C₂×Q₈	—	4	16	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂×Q₈	—	4	2	no	—
233	C₂×Q₈	#41×C₄₁	1	4	yes	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
251	C₂²×C₄	#41×C₄₁	1	4	yes	—
252	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	4	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	4	2	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	2	no	—
269	C₂³	#124×C₄₁	1	2	yes	—
270	C₂³	#137×C₄₁	1	4	yes	—
271	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
272	Q₈	—	2	8	no	—
273	Q₈	—	4	8	no	—
274	Q₈	—	4	4	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	8	no	—
286	C₂×C₄	—	4	8	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	#124×C₄₁	1	2	yes	—
299	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
300	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
301	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
302	C₈	—	4	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	4	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	4	no	—
307	C₂²	C₄₁×#491	1	4	yes	—
308	C₂²	C₄₁×#525	1	1	yes	characteristic
309	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
310	C₂²	C₄₁×#534	1	1	yes	characteristic
311	C₂²	C₂×#137×C₄₁	1	2	yes	—
312	C₂²	C₄₁×#575	1	2	yes	—
313	C₄	—	2	2	no	—
314	C₄	—	4	4	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₄₁×#551	1	4	yes	—
321	C₂	—	2	4	no	—
322	C₂	C₄₁×#2561	1	1	yes	characteristic
323	C₂	C₄₁×#3117	1	2	yes	—
324	C₂	C₂×C₄₁×#491	1	1	yes	characteristic
325	C₂	C₄₁×#3140	1	1	yes	characteristic
326	C₂	C₄₁×#3196	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂⁴×C₄₁	C₂×D₄
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
3. C₄₁×#2074	C₂	semidirect
4. C₄₁×#2026	C₂	semidirect
5. C₄₁×#1935	C₂	semidirect
139. #29112	C₄₁	direct (N × H)
160. #2074	C₂×C₄₁	semidirect
161. #2026	C₂×C₄₁	semidirect
162. #1935	C₂×C₄₁	semidirect
192. C₄₁	#29112	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×#2851	2
C₂×C₄₁×#391	2
C₄₁×#2074	2
C₄₁×#2026	2
C₄₁×#1935	2
C₄₁×#1824	2
C₄₁×#1787	2
C₄₁×#1749	2
#29112	41

Subgroups — C41×#29112 #23805313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#29112 #23805313