Subgroups — C₄₁×#11271 #23787472

Order 10496.

634 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#106×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#343	C₂	1	2	yes	maximal
3	C₂×C₄₁×#341	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1736	C₂	1	2	yes	maximal
5	C₄₁×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
11	C₂×#106×C₄₁	—	2	2	no	—
12	C₂×#106×C₄₁	—	2	4	no	—
13	C₄₁×#343	—	2	4	no	—
14	C₄₁×#341	C₄	1	4	yes	complemented
15	C₄₁×#340	—	2	4	no	—
16	C₄₁×#336	—	2	2	no	—
17	C₂³×C₄×C₄₁	—	2	2	no	—
18	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₄₁	—	2	2	no	—
20	C₂×M₁₆×C₄₁	—	2	1	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	—	2	2	no	—
25	C₂×M₁₆×C₄₁	—	2	2	no	—
26	C₂×M₁₆×C₄₁	—	2	2	no	—
27	C₂×M₁₆×C₄₁	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₄₁	Q₈	1	2	yes	—
29	C₂²×C₈×C₄₁	—	2	2	no	—
30	C₂²×C₈×C₄₁	—	2	2	no	—
31	C₂²×C₈×C₄₁	—	2	1	no	—
32	C₂²×C₈×C₄₁	—	2	2	no	—
33	C₂²×C₈×C₄₁	C₂×C₄	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	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₁	D₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
46	C₂×C₄²×C₄₁	—	2	1	no	—
47	#106×C₄₁	—	2	8	no	—
48	C₂⁴×C₄₁	#41	1	1	yes	characteristic
49	C₂²×C₄×C₄₁	—	2	2	no	—
50	C₂²×C₄×C₄₁	—	2	4	no	—
51	C₂²×C₄×C₄₁	—	4	2	no	—
52	C₂²×C₄×C₄₁	—	4	2	no	—
53	C₂²×C₄×C₄₁	—	8	4	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	—	2	4	no	—
67	M₁₆×C₄₁	—	4	2	no	—
68	M₁₆×C₄₁	—	4	2	no	—
69	M₁₆×C₄₁	—	4	2	no	—
70	C₂×C₈×C₄₁	—	2	4	no	—
71	C₂×C₈×C₄₁	—	2	2	no	—
72	C₂×C₈×C₄₁	—	2	4	no	—
73	C₂×C₈×C₄₁	—	2	2	no	—
74	C₂×C₈×C₄₁	—	2	4	no	—
75	C₂×C₈×C₄₁	—	2	2	no	—
76	C₂×C₈×C₄₁	—	4	1	no	—
77	C₂×C₈×C₄₁	—	4	2	no	—
78	C₂×C₈×C₄₁	—	4	2	no	—
79	C₂×C₈×C₄₁	—	4	1	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	C₄⋊C₄×C₄₁	—	2	4	no	—
84	C₄⋊C₄×C₄₁	—	4	2	no	—
85	C₄⋊C₄×C₄₁	—	4	2	no	—
86	C₂²⋊C₄×C₄₁	—	4	2	no	—
87	C₄²×C₄₁	—	4	1	no	—
88	C₄²×C₄₁	—	4	1	no	—
89	C₂³×C₄₁	—	2	4	no	—
90	C₂³×C₄₁	C₄×Q₈	1	1	yes	characteristic
91	C₂³×C₄₁	#121	1	1	yes	characteristic
92	C₂³×C₄₁	#123	1	2	yes	—
93	C₂³×C₄₁	#125	1	1	yes	characteristic
94	C₂³×C₄₁	#126	1	2	yes	—
95	C₂×C₄×C₄₁	—	4	4	no	—
96	C₂×C₄×C₄₁	—	4	2	no	—
97	C₂×C₄×C₄₁	—	4	4	no	—
98	C₂×C₄×C₄₁	—	8	4	no	—
99	C₂×C₄×C₄₁	—	8	4	no	—
100	C₂×C₄×C₄₁	—	8	4	no	—
101	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₁	#117	1	2	yes	—
103	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
106	C₂×C₄×C₄₁	#121	1	2	yes	—
107	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₁	#123	1	1	yes	characteristic
111	C₂×C₄×C₄₁	#123	1	2	yes	—
112	C₂×C₄×C₄₁	#124	1	1	yes	characteristic
113	C₂×C₄×C₄₁	#124	1	1	yes	characteristic
114	C₂×C₄×C₄₁	#124	1	2	yes	—
115	C₂×C₄×C₄₁	#125	1	2	yes	—
116	C₂×C₄×C₄₁	#125	1	2	yes	—
117	C₂×C₄×C₄₁	#126	1	2	yes	—
118	C₂×C₄×C₄₁	C₄⋊Q₈	1	1	yes	characteristic
119	C₈×C₄₁	—	2	4	no	—
120	C₈×C₄₁	—	4	4	no	—
121	C₈×C₄₁	—	4	2	no	—
122	#11271	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₄₁	—	2	4	no	—
124	C₂²×C₄₁	—	2	2	no	—
125	C₂²×C₄₁	—	2	2	no	—
126	C₂²×C₄₁	—	2	2	no	—
127	C₂²×C₄₁	—	2	2	no	—
128	C₂²×C₄₁	#387	1	2	yes	—
129	C₂²×C₄₁	#389	1	1	yes	characteristic
130	C₂²×C₄₁	#394	1	1	yes	characteristic
131	C₂²×C₄₁	#397	1	1	yes	characteristic
132	C₂²×C₄₁	#400	1	2	yes	—
133	C₂²×C₄₁	#471	1	4	yes	—
134	C₄×C₄₁	—	8	2	no	—
135	C₄×C₄₁	—	8	4	no	—
136	C₄×C₄₁	#382	1	1	yes	characteristic
137	C₄×C₄₁	#385	1	1	yes	characteristic
138	C₄×C₄₁	#388	1	2	yes	—
139	C₄×C₄₁	#393	1	1	yes	characteristic
140	C₄×C₄₁	#398	1	2	yes	—
141	C₄×C₄₁	#400	1	1	yes	characteristic
142	C₂²×#106	C₂×C₄₁	1	1	yes	characteristic
143	C₂×#343	C₂×C₄₁	1	2	yes	—
144	C₂×#341	C₂×C₄₁	1	1	yes	characteristic
145	#1736	C₂×C₄₁	1	2	yes	—
146	#1729	C₂×C₄₁	1	1	yes	characteristic
147	C₂×C₄₁	—	2	2	no	—
148	C₂×C₄₁	—	2	2	no	—
149	C₂×C₄₁	#1442	1	1	yes	characteristic
150	C₂×C₄₁	#1951	1	2	yes	—
151	C₂×C₄₁	#2046	1	2	yes	—
152	C₂×C₄₁	#2079	1	2	yes	—
153	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
154	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
155	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
156	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
157	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
158	C₂×#106	—	2	2	no	—
159	C₂×#106	—	2	4	no	—
160	#343	—	2	4	no	—
161	#341	C₄×C₄₁	1	4	yes	complemented
162	#340	—	2	4	no	—
163	#336	—	2	2	no	—
164	C₄₁	#11271	1	1	yes	characteristic, direct product factor, Sylow
165	C₂³×C₄	—	2	2	no	—
166	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	C₂×C₄×C₄₁	1	4	yes	—
176	C₂×M₁₆	Q₈×C₄₁	1	2	yes	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
182	C₂²×C₈	D₄×C₄₁	1	2	yes	—
183	C₂²×C₈	Q₈×C₄₁	1	2	yes	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	D₄×C₄₁	1	4	yes	—
192	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
193	C₂×C₂²⋊C₄	—	2	1	no	—
194	C₂×C₄²	—	2	1	no	—
195	#106	—	2	8	no	—
196	C₂⁴	#41×C₄₁	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	8	4	no	—
202	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
207	C₂²×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
208	C₂²×C₄	#41×C₄₁	1	2	yes	—
209	C₂²×C₄	#41×C₄₁	1	2	yes	—
210	C₂²×C₄	#41×C₄₁	1	2	yes	—
211	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	C₄⋊C₄×C₄₁	1	4	yes	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	4	1	no	—
237	C₂³	—	2	4	no	—
238	C₂³	C₄×Q₈×C₄₁	1	1	yes	characteristic
239	C₂³	#121×C₄₁	1	1	yes	characteristic
240	C₂³	#123×C₄₁	1	2	yes	—
241	C₂³	#125×C₄₁	1	1	yes	characteristic
242	C₂³	#126×C₄₁	1	2	yes	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
250	C₂×C₄	#117×C₄₁	1	2	yes	—
251	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
252	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
253	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
254	C₂×C₄	#121×C₄₁	1	2	yes	—
255	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
256	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
257	C₂×C₄	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
258	C₂×C₄	#123×C₄₁	1	1	yes	characteristic
259	C₂×C₄	#123×C₄₁	1	2	yes	—
260	C₂×C₄	#124×C₄₁	1	1	yes	characteristic
261	C₂×C₄	#124×C₄₁	1	1	yes	characteristic
262	C₂×C₄	#124×C₄₁	1	2	yes	—
263	C₂×C₄	#125×C₄₁	1	2	yes	—
264	C₂×C₄	#125×C₄₁	1	2	yes	—
265	C₂×C₄	#126×C₄₁	1	2	yes	—
266	C₂×C₄	C₄⋊Q₈×C₄₁	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	—	4	4	no	—
269	C₈	—	4	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	C₄₁×#387	1	2	yes	—
276	C₂²	C₄₁×#389	1	1	yes	characteristic
277	C₂²	C₄₁×#394	1	1	yes	characteristic
278	C₂²	C₄₁×#397	1	1	yes	characteristic
279	C₂²	C₄₁×#400	1	2	yes	—
280	C₂²	C₄₁×#471	1	4	yes	—
281	C₄	—	8	4	no	—
282	C₄	—	8	2	no	—
283	C₄	C₄₁×#382	1	1	yes	characteristic
284	C₄	C₄₁×#385	1	1	yes	characteristic
285	C₄	C₄₁×#388	1	2	yes	—
286	C₄	C₄₁×#393	1	1	yes	characteristic
287	C₄	C₄₁×#398	1	2	yes	—
288	C₄	C₄₁×#400	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	C₄₁×#1442	1	1	yes	characteristic
292	C₂	C₄₁×#1951	1	2	yes	—
293	C₂	C₄₁×#2046	1	2	yes	—
294	C₂	C₄₁×#2079	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	#121
commutator	C₂²×C₄	C₂²×C₄×C₄₁
Frattini, cosocle	C₂³×C₄	C₂³×C₄₁
prime radical	C₂⁴×C₄₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₄₁×#341	C₄	semidirect
122. #11271	C₄₁	direct (N × H)
161. #341	C₄×C₄₁	semidirect
164. C₄₁	#11271	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×#106×C₄₁	2
C₂×C₄₁×#343	2
C₂×C₄₁×#341	2
C₄₁×#1736	2
C₄₁×#1729	2
#11271	41

Subgroups — C41×#11271 #23787472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#11271 #23787472