Subgroups — C₄₁×#29133 #23805334

Order 10496.

754 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2647	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#2328	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2101	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2041	C₂	1	2	yes	maximal
5	C₄₁×#1936	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1934	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1804	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1802	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1768	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
13	C₂×#102×C₄₁	C₂²	1	2	yes	—
14	C₂×#102×C₄₁	C₂²	1	2	yes	—
15	C₂×#102×C₄₁	C₂²	1	2	yes	—
16	C₂×#102×C₄₁	C₂²	1	2	yes	—
17	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
19	C₄₁×#400	—	2	2	no	—
20	C₄₁×#398	—	2	2	no	—
21	C₄₁×#390	C₂²	1	2	yes	—
22	C₄₁×#389	C₂²	1	2	yes	—
23	C₄₁×#389	C₂²	1	2	yes	—
24	C₄₁×#386	—	2	2	no	—
25	C₄₁×#386	—	2	2	no	—
26	C₄₁×#384	C₂²	1	1	yes	characteristic
27	C₄₁×#382	C₂²	1	1	yes	characteristic
28	C₄₁×#382	C₂²	1	1	yes	characteristic
29	C₄₁×#376	C₂²	1	1	yes	characteristic
30	C₄₁×#340	C₂²	1	2	yes	—
31	C₄₁×#340	C₂²	1	2	yes	—
32	C₄₁×#340	C₂²	1	2	yes	—
33	C₄₁×#340	C₂²	1	2	yes	—
34	C₄₁×#336	C₂²	1	2	yes	—
35	C₂²×D₄×C₄₁	—	2	1	no	—
36	C₂²×D₄×C₄₁	—	2	1	no	—
37	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
38	C₂³×C₄×C₄₁	—	2	2	no	—
39	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
41	#127×C₄₁	—	2	4	no	—
42	C₄×D₄×C₄₁	—	2	4	no	—
43	C₄×D₄×C₄₁	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
50	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
53	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₁	—	2	4	no	—
57	#102×C₄₁	—	2	4	no	—
58	#102×C₄₁	—	2	4	no	—
59	#102×C₄₁	—	4	2	no	—
60	#102×C₄₁	—	4	2	no	—
61	#97×C₄₁	—	4	1	no	—
62	C₄×C₈×C₄₁	—	4	1	no	—
63	#95×C₄₁	—	2	1	no	—
64	#95×C₄₁	—	2	1	no	—
65	#95×C₄₁	—	2	2	no	—
66	#95×C₄₁	—	2	2	no	—
67	#95×C₄₁	—	2	2	no	—
68	#95×C₄₁	D₄	1	2	yes	—
69	#95×C₄₁	D₄	1	2	yes	—
70	C₂⁴×C₄₁	—	2	2	no	—
71	C₂×D₄×C₄₁	—	2	4	no	—
72	C₂×D₄×C₄₁	—	2	4	no	—
73	C₂×D₄×C₄₁	—	2	2	no	—
74	C₂×D₄×C₄₁	—	2	2	no	—
75	C₂×D₄×C₄₁	—	2	2	no	—
76	C₂×D₄×C₄₁	—	2	2	no	—
77	C₂×D₄×C₄₁	—	2	2	no	—
78	C₂×D₄×C₄₁	—	8	2	no	—
79	C₂×D₄×C₄₁	—	8	2	no	—
80	C₂×D₄×C₄₁	#41	1	4	yes	—
81	C₂²×C₄×C₄₁	—	2	2	no	—
82	C₂²×C₄×C₄₁	—	2	2	no	—
83	C₂²×C₄×C₄₁	—	2	2	no	—
84	C₂²×C₄×C₄₁	—	4	4	no	—
85	C₂²×C₄×C₄₁	—	8	2	no	—
86	C₂²×C₄×C₄₁	—	8	2	no	—
87	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
92	C₂×C₈×C₄₁	—	2	2	no	—
93	C₂×C₈×C₄₁	—	2	2	no	—
94	C₂×C₈×C₄₁	—	2	2	no	—
95	C₂×C₈×C₄₁	—	2	2	no	—
96	C₂×C₈×C₄₁	—	2	2	no	—
97	C₂×C₈×C₄₁	—	2	2	no	—
98	C₄⋊C₄×C₄₁	—	2	2	no	—
99	C₄⋊C₄×C₄₁	—	2	2	no	—
100	C₄⋊C₄×C₄₁	—	2	2	no	—
101	C₄⋊C₄×C₄₁	—	2	2	no	—
102	C₄⋊C₄×C₄₁	—	2	2	no	—
103	C₄⋊C₄×C₄₁	—	2	2	no	—
104	C₄⋊C₄×C₄₁	—	8	2	no	—
105	C₄⋊C₄×C₄₁	—	8	2	no	—
106	C₄⋊C₄×C₄₁	—	8	2	no	—
107	C₂²⋊C₄×C₄₁	—	4	4	no	—
108	C₂²⋊C₄×C₄₁	—	8	2	no	—
109	C₂²⋊C₄×C₄₁	—	8	2	no	—
110	C₄²×C₄₁	—	2	2	no	—
111	C₄²×C₄₁	—	2	1	no	—
112	C₄²×C₄₁	—	2	1	no	—
113	C₄²×C₄₁	#41	1	4	yes	—
114	C₂³×C₄₁	—	2	4	no	—
115	C₂³×C₄₁	—	4	2	no	—
116	C₂³×C₄₁	—	4	2	no	—
117	C₂³×C₄₁	—	4	2	no	—
118	C₂³×C₄₁	—	4	2	no	—
119	C₂³×C₄₁	—	4	2	no	—
120	C₂³×C₄₁	—	4	2	no	—
121	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
122	D₄×C₄₁	—	2	4	no	—
123	D₄×C₄₁	—	2	4	no	—
124	D₄×C₄₁	—	4	2	no	—
125	D₄×C₄₁	—	4	2	no	—
126	D₄×C₄₁	—	8	4	no	—
127	D₄×C₄₁	—	8	4	no	—
128	C₂×C₄×C₄₁	—	2	2	no	—
129	C₂×C₄×C₄₁	—	2	2	no	—
130	C₂×C₄×C₄₁	—	2	1	no	—
131	C₂×C₄×C₄₁	—	2	1	no	—
132	C₂×C₄×C₄₁	—	2	2	no	—
133	C₂×C₄×C₄₁	—	2	2	no	—
134	C₂×C₄×C₄₁	—	2	2	no	—
135	C₂×C₄×C₄₁	—	2	2	no	—
136	C₂×C₄×C₄₁	—	2	2	no	—
137	C₂×C₄×C₄₁	—	4	2	no	—
138	C₂×C₄×C₄₁	—	4	2	no	—
139	C₂×C₄×C₄₁	—	4	2	no	—
140	C₂×C₄×C₄₁	—	4	2	no	—
141	C₂×C₄×C₄₁	—	4	2	no	—
142	C₂×C₄×C₄₁	—	4	2	no	—
143	C₂×C₄×C₄₁	—	4	2	no	—
144	C₂×C₄×C₄₁	—	4	2	no	—
145	C₂×C₄×C₄₁	—	4	2	no	—
146	C₂×C₄×C₄₁	—	8	8	no	—
147	C₂×C₄×C₄₁	#124	1	4	yes	—
148	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
149	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
150	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
152	C₈×C₄₁	—	4	2	no	—
153	C₈×C₄₁	—	4	2	no	—
154	#29133	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂²×C₄₁	—	4	4	no	—
156	C₂²×C₄₁	—	4	4	no	—
157	C₂²×C₄₁	—	4	4	no	—
158	C₂²×C₄₁	—	8	2	no	—
159	C₂²×C₄₁	—	8	2	no	—
160	C₂²×C₄₁	—	8	2	no	—
161	C₂²×C₄₁	—	8	2	no	—
162	C₂²×C₄₁	#525	1	1	yes	characteristic
163	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
164	C₂²×C₄₁	#537	1	1	yes	characteristic
165	C₂²×C₄₁	#576	1	2	yes	—
166	C₂²×C₄₁	#577	1	2	yes	—
167	C₄×C₄₁	—	2	4	no	—
168	C₄×C₄₁	—	4	1	no	—
169	C₄×C₄₁	—	4	1	no	—
170	C₄×C₄₁	—	4	2	no	—
171	C₄×C₄₁	—	8	2	no	—
172	C₄×C₄₁	—	8	2	no	—
173	C₄×C₄₁	—	8	2	no	—
174	C₄×C₄₁	#552	1	2	yes	—
175	C₄×C₄₁	#553	1	2	yes	—
176	#2647	C₂×C₄₁	1	1	yes	characteristic, complemented
177	#2328	C₂×C₄₁	1	1	yes	characteristic
178	#2101	C₂×C₄₁	1	2	yes	complemented
179	#2041	C₂×C₄₁	1	2	yes	—
180	#1936	C₂×C₄₁	1	2	yes	complemented
181	#1934	C₂×C₄₁	1	2	yes	complemented
182	#1804	C₂×C₄₁	1	2	yes	complemented
183	#1802	C₂×C₄₁	1	2	yes	complemented
184	#1768	C₂×C₄₁	1	1	yes	characteristic, complemented
185	C₂×C₄₁	—	8	4	no	—
186	C₂×C₄₁	#2562	1	1	yes	characteristic
187	C₂×C₄₁	#3122	1	1	yes	characteristic
188	C₂×C₄₁	#3123	1	1	yes	characteristic
189	C₂×C₄₁	#3136	1	2	yes	—
190	C₂×C₄₁	#3257	1	2	yes	—
191	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
192	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
193	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
194	C₂×#102	C₂²×C₄₁	1	2	yes	—
195	C₂×#102	C₂²×C₄₁	1	2	yes	—
196	C₂×#102	C₂²×C₄₁	1	2	yes	—
197	C₂×#102	C₂²×C₄₁	1	2	yes	—
198	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
199	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
200	#400	—	2	2	no	—
201	#398	—	2	2	no	—
202	#390	C₂²×C₄₁	1	2	yes	—
203	#389	C₂²×C₄₁	1	2	yes	—
204	#389	C₂²×C₄₁	1	2	yes	—
205	#386	—	2	2	no	—
206	#386	—	2	2	no	—
207	#384	C₂²×C₄₁	1	1	yes	characteristic
208	#382	C₂²×C₄₁	1	1	yes	characteristic
209	#382	C₂²×C₄₁	1	1	yes	characteristic
210	#376	C₂²×C₄₁	1	1	yes	characteristic
211	#340	C₂²×C₄₁	1	2	yes	—
212	#340	C₂²×C₄₁	1	2	yes	—
213	#340	C₂²×C₄₁	1	2	yes	—
214	#340	C₂²×C₄₁	1	2	yes	—
215	#336	C₂²×C₄₁	1	2	yes	—
216	C₄₁	#29133	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²×D₄	—	2	1	no	—
218	C₂²×D₄	—	2	1	no	—
219	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
220	C₂³×C₄	—	2	2	no	—
221	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
222	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
223	#127	—	2	4	no	—
224	C₄×D₄	—	2	4	no	—
225	C₄×D₄	—	2	4	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
230	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
231	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
232	C₂×C₂²⋊C₄	—	2	2	no	—
233	C₂×C₂²⋊C₄	—	2	2	no	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
236	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
237	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
238	C₄⋊C₈	—	2	4	no	—
239	#102	—	2	4	no	—
240	#102	—	2	4	no	—
241	#102	—	4	2	no	—
242	#102	—	4	2	no	—
243	#97	—	4	1	no	—
244	C₄×C₈	—	4	1	no	—
245	#95	—	2	1	no	—
246	#95	—	2	1	no	—
247	#95	—	2	2	no	—
248	#95	—	2	2	no	—
249	#95	—	2	2	no	—
250	#95	D₄×C₄₁	1	2	yes	—
251	#95	D₄×C₄₁	1	2	yes	—
252	C₂⁴	—	2	2	no	—
253	C₂×D₄	—	2	4	no	—
254	C₂×D₄	—	2	4	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	8	2	no	—
261	C₂×D₄	—	8	2	no	—
262	C₂×D₄	#41×C₄₁	1	4	yes	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	4	4	no	—
267	C₂²×C₄	—	8	2	no	—
268	C₂²×C₄	—	8	2	no	—
269	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
271	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
273	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	8	2	no	—
287	C₄⋊C₄	—	8	2	no	—
288	C₄⋊C₄	—	8	2	no	—
289	C₂²⋊C₄	—	4	4	no	—
290	C₂²⋊C₄	—	8	2	no	—
291	C₂²⋊C₄	—	8	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	#41×C₄₁	1	4	yes	—
296	C₂³	—	2	4	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
304	D₄	—	2	4	no	—
305	D₄	—	2	4	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	4	no	—
309	D₄	—	8	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	8	no	—
329	C₂×C₄	#124×C₄₁	1	4	yes	—
330	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
331	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
332	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
333	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
334	C₈	—	4	2	no	—
335	C₈	—	4	2	no	—
336	C₂²	—	4	4	no	—
337	C₂²	—	4	4	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	C₄₁×#525	1	1	yes	characteristic
344	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
345	C₂²	C₄₁×#537	1	1	yes	characteristic
346	C₂²	C₄₁×#576	1	2	yes	—
347	C₂²	C₄₁×#577	1	2	yes	—
348	C₄	—	2	4	no	—
349	C₄	—	4	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	8	2	no	—
353	C₄	—	8	2	no	—
354	C₄	—	8	2	no	—
355	C₄	C₄₁×#552	1	2	yes	—
356	C₄	C₄₁×#553	1	2	yes	—
357	C₂	—	8	4	no	—
358	C₂	C₄₁×#2562	1	1	yes	characteristic
359	C₂	C₄₁×#3122	1	1	yes	characteristic
360	C₂	C₄₁×#3123	1	1	yes	characteristic
361	C₂	C₄₁×#3136	1	2	yes	—
362	C₂	C₄₁×#3257	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×#127×C₄₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#2647	C₂	semidirect
3. C₄₁×#2101	C₂	semidirect
5. C₄₁×#1936	C₂	semidirect
6. C₄₁×#1934	C₂	semidirect
7. C₄₁×#1804	C₂	semidirect
8. C₄₁×#1802	C₂	semidirect
9. C₄₁×#1768	C₂	semidirect
154. #29133	C₄₁	direct (N × H)
176. #2647	C₂×C₄₁	semidirect
178. #2101	C₂×C₄₁	semidirect
180. #1936	C₂×C₄₁	semidirect
181. #1934	C₂×C₄₁	semidirect
182. #1804	C₂×C₄₁	semidirect
183. #1802	C₂×C₄₁	semidirect
184. #1768	C₂×C₄₁	semidirect
216. C₄₁	#29133	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×#2647	2
C₄₁×#2328	2
C₄₁×#2101	2
C₄₁×#2041	2
C₄₁×#1936	2
C₄₁×#1934	2
C₄₁×#1804	2
C₄₁×#1802	2
C₄₁×#1768	2
#29133	41

Subgroups — C41×#29133 #23805334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#29133 #23805334