Subgroups — C₄₁×#11258 #23787459

Order 10496.

654 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1735	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#339	C₂	1	2	yes	maximal
4	C₄₁×#1729	C₂	1	2	yes	maximal
5	C₂×C₄₁×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
7	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×#117×C₄₁	C₂²	1	2	yes	—
10	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×#108×C₄₁	—	2	4	no	—
12	C₂×#108×C₄₁	—	2	2	no	—
13	C₄₁×#340	C₄	1	4	yes	complemented
14	C₄₁×#339	—	2	4	no	—
15	C₄₁×#336	—	2	2	no	—
16	C₄₁×#336	—	2	4	no	—
17	C₂³×C₄×C₄₁	—	2	1	no	—
18	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₄₁	—	2	2	no	—
20	C₂×M₁₆×C₄₁	—	2	2	no	—
21	C₂×M₁₆×C₄₁	—	2	2	no	—
22	C₂×M₁₆×C₄₁	—	2	2	no	—
23	C₂×M₁₆×C₄₁	—	2	2	no	—
24	C₂×M₁₆×C₄₁	D₄	1	4	yes	—
25	C₂²×C₈×C₄₁	—	2	2	no	—
26	C₂²×C₈×C₄₁	—	2	2	no	—
27	C₂²×C₈×C₄₁	—	2	1	no	—
28	C₂²×C₈×C₄₁	—	2	2	no	—
29	C₂²×C₈×C₄₁	—	2	1	no	—
30	C₂²×C₈×C₄₁	—	2	2	no	—
31	C₂²×C₈×C₄₁	—	2	2	no	—
32	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₄₁	D₄	1	2	yes	—
34	C₂²×C₈×C₄₁	D₄	1	2	yes	—
35	C₂²×C₈×C₄₁	Q₈	1	2	yes	—
36	#117×C₄₁	—	2	2	no	—
37	#117×C₄₁	—	2	2	no	—
38	#117×C₄₁	—	2	2	no	—
39	#117×C₄₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
42	C₂×C₄²×C₄₁	—	2	2	no	—
43	C₂×C₄²×C₄₁	—	2	2	no	—
44	C₂×C₄²×C₄₁	—	2	1	no	—
45	C₂×C₄²×C₄₁	—	2	2	no	—
46	C₂×C₄²×C₄₁	Q₈	1	2	yes	—
47	#108×C₄₁	—	2	8	no	—
48	C₂⁴×C₄₁	#41	1	1	yes	characteristic
49	C₂²×C₄×C₄₁	—	2	4	no	—
50	C₂²×C₄×C₄₁	—	2	2	no	—
51	C₂²×C₄×C₄₁	—	2	4	no	—
52	C₂²×C₄×C₄₁	—	4	2	no	—
53	C₂²×C₄×C₄₁	—	4	2	no	—
54	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₁	#41	1	2	yes	—
61	C₂²×C₄×C₄₁	#41	1	2	yes	—
62	C₂²×C₄×C₄₁	#41	1	2	yes	—
63	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
64	M₁₆×C₄₁	—	2	4	no	—
65	M₁₆×C₄₁	—	2	4	no	—
66	M₁₆×C₄₁	—	4	2	no	—
67	M₁₆×C₄₁	—	4	2	no	—
68	C₂×C₈×C₄₁	—	2	4	no	—
69	C₂×C₈×C₄₁	—	2	2	no	—
70	C₂×C₈×C₄₁	—	2	4	no	—
71	C₂×C₈×C₄₁	—	2	2	no	—
72	C₂×C₈×C₄₁	—	2	2	no	—
73	C₂×C₈×C₄₁	—	2	4	no	—
74	C₂×C₈×C₄₁	—	2	4	no	—
75	C₂×C₈×C₄₁	—	4	2	no	—
76	C₂×C₈×C₄₁	—	4	2	no	—
77	C₂×C₈×C₄₁	—	4	1	no	—
78	C₂×C₈×C₄₁	—	4	1	no	—
79	C₂×C₈×C₄₁	—	4	2	no	—
80	C₂×C₈×C₄₁	C₄⋊C₄	1	4	yes	—
81	C₄⋊C₄×C₄₁	—	2	4	no	—
82	C₄⋊C₄×C₄₁	—	2	4	no	—
83	C₂²⋊C₄×C₄₁	—	4	2	no	—
84	C₂²⋊C₄×C₄₁	—	4	2	no	—
85	C₄²×C₄₁	—	2	4	no	—
86	C₄²×C₄₁	—	4	1	no	—
87	C₄²×C₄₁	—	4	2	no	—
88	C₄²×C₄₁	—	4	2	no	—
89	C₄²×C₄₁	—	4	1	no	—
90	C₂³×C₄₁	—	2	2	no	—
91	C₂³×C₄₁	—	2	2	no	—
92	C₂³×C₄₁	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂³×C₄₁	#124	1	2	yes	—
95	C₂³×C₄₁	#125	1	1	yes	characteristic
96	C₂³×C₄₁	#126	1	2	yes	—
97	C₂×C₄×C₄₁	—	2	4	no	—
98	C₂×C₄×C₄₁	—	4	8	no	—
99	C₂×C₄×C₄₁	—	4	2	no	—
100	C₂×C₄×C₄₁	—	4	2	no	—
101	C₂×C₄×C₄₁	—	4	4	no	—
102	C₂×C₄×C₄₁	—	4	4	no	—
103	C₂×C₄×C₄₁	—	4	4	no	—
104	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₁	#117	1	2	yes	—
106	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₄₁	C₄×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
109	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
110	C₂×C₄×C₄₁	#121	1	2	yes	—
111	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₄₁	#123	1	1	yes	characteristic
114	C₂×C₄×C₄₁	#123	1	2	yes	—
115	C₂×C₄×C₄₁	#123	1	2	yes	—
116	C₂×C₄×C₄₁	#124	1	1	yes	characteristic
117	C₂×C₄×C₄₁	#124	1	1	yes	characteristic
118	C₂×C₄×C₄₁	#125	1	2	yes	—
119	C₂×C₄×C₄₁	#125	1	2	yes	—
120	C₂×C₄×C₄₁	#126	1	2	yes	—
121	C₂×C₄×C₄₁	C₄⋊Q₈	1	1	yes	characteristic
122	C₈×C₄₁	—	2	4	no	—
123	C₈×C₄₁	—	4	4	no	—
124	C₈×C₄₁	—	4	2	no	—
125	#11258	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₄₁	—	2	4	no	—
127	C₂²×C₄₁	—	2	4	no	—
128	C₂²×C₄₁	—	2	1	no	—
129	C₂²×C₄₁	—	2	1	no	—
130	C₂²×C₄₁	—	2	1	no	—
131	C₂²×C₄₁	—	2	1	no	—
132	C₂²×C₄₁	#388	1	2	yes	—
133	C₂²×C₄₁	#389	1	1	yes	characteristic
134	C₂²×C₄₁	#393	1	1	yes	characteristic
135	C₂²×C₄₁	#400	1	2	yes	—
136	C₂²×C₄₁	#400	1	1	yes	characteristic
137	C₂²×C₄₁	#443	1	2	yes	—
138	C₂²×C₄₁	#444	1	2	yes	—
139	C₄×C₄₁	—	4	4	no	—
140	C₄×C₄₁	—	8	4	no	—
141	C₄×C₄₁	#382	1	1	yes	characteristic
142	C₄×C₄₁	#385	1	1	yes	characteristic
143	C₄×C₄₁	#387	1	2	yes	—
144	C₄×C₄₁	#394	1	1	yes	characteristic
145	C₄×C₄₁	#397	1	1	yes	characteristic
146	C₄×C₄₁	#398	1	2	yes	—
147	C₂²×#108	C₂×C₄₁	1	1	yes	characteristic
148	#1735	C₂×C₄₁	1	1	yes	characteristic
149	C₂×#339	C₂×C₄₁	1	2	yes	—
150	#1729	C₂×C₄₁	1	2	yes	—
151	C₂×#336	C₂×C₄₁	1	1	yes	characteristic
152	C₂×C₄₁	—	2	1	no	—
153	C₂×C₄₁	—	2	2	no	—
154	C₂×C₄₁	—	2	1	no	—
155	C₂×C₄₁	#1442	1	1	yes	characteristic
156	C₂×C₄₁	#1929	1	4	yes	—
157	C₂×C₄₁	#1945	1	1	yes	characteristic
158	C₂×C₄₁	#1946	1	1	yes	characteristic
159	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
160	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
161	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
162	C₂×#117	C₂²×C₄₁	1	2	yes	—
163	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
164	C₂×#108	—	2	4	no	—
165	C₂×#108	—	2	2	no	—
166	#340	C₄×C₄₁	1	4	yes	complemented
167	#339	—	2	4	no	—
168	#336	—	2	2	no	—
169	#336	—	2	4	no	—
170	C₄₁	#11258	1	1	yes	characteristic, direct product factor, Sylow
171	C₂³×C₄	—	2	1	no	—
172	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	D₄×C₄₁	1	4	yes	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
187	C₂²×C₈	D₄×C₄₁	1	2	yes	—
188	C₂²×C₈	D₄×C₄₁	1	2	yes	—
189	C₂²×C₈	Q₈×C₄₁	1	2	yes	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	Q₈×C₄₁	1	2	yes	—
201	#108	—	2	8	no	—
202	C₂⁴	#41×C₄₁	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
214	C₂²×C₄	#41×C₄₁	1	2	yes	—
215	C₂²×C₄	#41×C₄₁	1	2	yes	—
216	C₂²×C₄	#41×C₄₁	1	2	yes	—
217	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	C₄⋊C₄×C₄₁	1	4	yes	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	4	1	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	4	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	C₄×D₄×C₄₁	1	1	yes	characteristic
247	C₂³	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
248	C₂³	#124×C₄₁	1	2	yes	—
249	C₂³	#125×C₄₁	1	1	yes	characteristic
250	C₂³	#126×C₄₁	1	2	yes	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
259	C₂×C₄	#117×C₄₁	1	2	yes	—
260	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
261	C₂×C₄	C₄×Q₈×C₄₁	1	1	yes	characteristic
262	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
263	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
264	C₂×C₄	#121×C₄₁	1	2	yes	—
265	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
266	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
267	C₂×C₄	#123×C₄₁	1	1	yes	characteristic
268	C₂×C₄	#123×C₄₁	1	2	yes	—
269	C₂×C₄	#123×C₄₁	1	2	yes	—
270	C₂×C₄	#124×C₄₁	1	1	yes	characteristic
271	C₂×C₄	#124×C₄₁	1	1	yes	characteristic
272	C₂×C₄	#125×C₄₁	1	2	yes	—
273	C₂×C₄	#125×C₄₁	1	2	yes	—
274	C₂×C₄	#126×C₄₁	1	2	yes	—
275	C₂×C₄	C₄⋊Q₈×C₄₁	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	4	4	no	—
278	C₈	—	4	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	C₄₁×#388	1	2	yes	—
286	C₂²	C₄₁×#389	1	1	yes	characteristic
287	C₂²	C₄₁×#393	1	1	yes	characteristic
288	C₂²	C₄₁×#400	1	2	yes	—
289	C₂²	C₄₁×#400	1	1	yes	characteristic
290	C₂²	C₄₁×#443	1	2	yes	—
291	C₂²	C₄₁×#444	1	2	yes	—
292	C₄	—	4	4	no	—
293	C₄	—	8	4	no	—
294	C₄	C₄₁×#382	1	1	yes	characteristic
295	C₄	C₄₁×#385	1	1	yes	characteristic
296	C₄	C₄₁×#387	1	2	yes	—
297	C₄	C₄₁×#394	1	1	yes	characteristic
298	C₄	C₄₁×#397	1	1	yes	characteristic
299	C₄	C₄₁×#398	1	2	yes	—
300	C₂	—	2	1	no	—
301	C₂	—	2	2	no	—
302	C₂	—	2	1	no	—
303	C₂	C₄₁×#1442	1	1	yes	characteristic
304	C₂	C₄₁×#1929	1	4	yes	—
305	C₂	C₄₁×#1945	1	1	yes	characteristic
306	C₂	C₄₁×#1946	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₄₁
Frattini, cosocle	C₂³×C₄	C₂³×C₄₁
socle	C₂³×C₄₁	C₄×D₄
prime radical	C₂⁴×C₄₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ C₂³×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄₁×#340	C₄	semidirect
125. #11258	C₄₁	direct (N × H)
166. #340	C₄×C₄₁	semidirect
170. C₄₁	#11258	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×#108×C₄₁	2
C₄₁×#1735	2
C₂×C₄₁×#339	2
C₄₁×#1729	2
C₂×C₄₁×#336	2
#11258	41

Subgroups — C41×#11258 #23787459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#11258 #23787459