Subgroups — C₁₁×#11269 #14588462

Order 2816.

642 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×#341	C₂	1	2	yes	maximal
3	C₂×C₁₁×#341	C₂	1	2	yes	maximal
4	C₁₁×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₁	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
11	C₁₁×#423	—	2	4	no	—
12	C₂×C₁₁×C₄⋊C₈	—	2	2	no	—
13	C₁₁×#341	—	2	4	no	—
14	C₁₁×#341	—	2	4	no	—
15	C₁₁×#336	—	2	2	no	—
16	C₁₁×#336	C₄	1	2	yes	—
17	C₁₁×#336	C₄	1	2	yes	—
18	C₂³×C₄×C₁₁	Q₈	1	2	yes	—
19	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
20	C₂×C₁₁×M₁₆	—	2	2	no	—
21	C₂×C₁₁×M₁₆	—	2	1	no	—
22	C₂×C₁₁×M₁₆	—	2	1	no	—
23	C₂×C₁₁×M₁₆	—	2	2	no	—
24	C₂×C₁₁×M₁₆	—	2	2	no	—
25	C₂×C₁₁×M₁₆	—	2	2	no	—
26	C₂×C₁₁×M₁₆	—	2	2	no	—
27	C₂×C₁₁×M₁₆	—	2	2	no	—
28	C₂×C₁₁×M₁₆	—	2	2	no	—
29	C₂×C₁₁×M₁₆	—	2	2	no	—
30	C₂×C₁₁×M₁₆	—	2	2	no	—
31	C₂×C₁₁×M₁₆	D₄	1	2	yes	—
32	C₂×C₁₁×M₁₆	D₄	1	4	yes	—
33	C₂×C₁₁×M₁₆	Q₈	1	2	yes	—
34	C₂²×C₈×C₁₁	—	2	2	no	—
35	C₂²×C₈×C₁₁	—	2	2	no	—
36	C₂²×C₈×C₁₁	—	2	1	no	—
37	C₂²×C₈×C₁₁	—	2	1	no	—
38	C₂²×C₈×C₁₁	—	2	2	no	—
39	C₂²×C₈×C₁₁	—	2	2	no	—
40	C₂²×C₈×C₁₁	—	2	2	no	—
41	C₂²×C₈×C₁₁	—	2	2	no	—
42	C₂²×C₈×C₁₁	—	2	2	no	—
43	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₁₁	—	2	2	no	—
45	C₂×C₄²×C₁₁	—	2	2	no	—
46	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₁	D₄	1	2	yes	—
49	C₁₁×C₄⋊C₈	—	2	8	no	—
50	#11269	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂⁴×C₁₁	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₁	—	2	4	no	—
53	C₂²×C₄×C₁₁	—	2	4	no	—
54	C₂²×C₄×C₁₁	—	4	2	no	—
55	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₁	#41	1	2	yes	—
63	C₂²×C₄×C₁₁	#41	1	2	yes	—
64	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
65	C₂²×C₄×C₁₁	#41	1	2	yes	—
66	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
67	C₁₁×M₁₆	—	2	4	no	—
68	C₁₁×M₁₆	—	2	4	no	—
69	C₁₁×M₁₆	—	2	4	no	—
70	C₁₁×M₁₆	—	2	4	no	—
71	C₁₁×M₁₆	—	4	2	no	—
72	C₁₁×M₁₆	—	4	2	no	—
73	C₁₁×M₁₆	—	4	2	no	—
74	C₁₁×M₁₆	—	4	2	no	—
75	C₂×C₈×C₁₁	—	2	4	no	—
76	C₂×C₈×C₁₁	—	2	4	no	—
77	C₂×C₈×C₁₁	—	2	4	no	—
78	C₂×C₈×C₁₁	—	2	4	no	—
79	C₂×C₈×C₁₁	—	2	2	no	—
80	C₂×C₈×C₁₁	—	2	2	no	—
81	C₂×C₈×C₁₁	—	4	1	no	—
82	C₂×C₈×C₁₁	—	4	2	no	—
83	C₂×C₈×C₁₁	—	4	2	no	—
84	C₂×C₈×C₁₁	—	4	2	no	—
85	C₂×C₈×C₁₁	—	4	1	no	—
86	C₂×C₈×C₁₁	—	4	2	no	—
87	C₂×C₈×C₁₁	—	4	1	no	—
88	C₂×C₈×C₁₁	—	4	1	no	—
89	C₂×C₈×C₁₁	—	4	2	no	—
90	C₂×C₈×C₁₁	—	4	2	no	—
91	C₄²×C₁₁	—	2	2	no	—
92	C₄²×C₁₁	—	2	2	no	—
93	C₄²×C₁₁	—	2	2	no	—
94	C₄²×C₁₁	—	2	2	no	—
95	C₂×#423	C₂×C₁₁	1	1	yes	characteristic
96	C₂×#341	C₂×C₁₁	1	2	yes	—
97	C₂×#341	C₂×C₁₁	1	2	yes	—
98	#1728	C₂×C₁₁	1	1	yes	characteristic
99	C₂×#336	C₂×C₁₁	1	1	yes	characteristic
100	C₂³×C₁₁	—	2	2	no	—
101	C₂³×C₁₁	—	2	2	no	—
102	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂³×C₁₁	C₂²⋊Q₈	1	2	yes	—
104	C₂³×C₁₁	#125	1	2	yes	—
105	C₂³×C₁₁	#125	1	1	yes	characteristic
106	C₂³×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₁	—	2	2	no	—
108	C₂×C₄×C₁₁	—	2	2	no	—
109	C₂×C₄×C₁₁	—	2	2	no	—
110	C₂×C₄×C₁₁	—	2	2	no	—
111	C₂×C₄×C₁₁	—	2	2	no	—
112	C₂×C₄×C₁₁	—	4	8	no	—
113	C₂×C₄×C₁₁	#108	1	4	yes	—
114	C₂×C₄×C₁₁	#117	1	2	yes	—
115	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₄×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₁	#120	1	1	yes	characteristic
119	C₂×C₄×C₁₁	#121	1	2	yes	—
120	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₁	#123	1	2	yes	—
124	C₂×C₄×C₁₁	#123	1	2	yes	—
125	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
126	C₂×C₄×C₁₁	#124	1	2	yes	—
127	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
128	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
129	C₂×C₄×C₁₁	#125	1	2	yes	—
130	C₂×C₄×C₁₁	#126	1	2	yes	—
131	C₂×C₄×C₁₁	#126	1	2	yes	—
132	C₈×C₁₁	—	4	4	no	—
133	C₈×C₁₁	—	4	2	no	—
134	C₈×C₁₁	—	4	4	no	—
135	C₈×C₁₁	—	4	2	no	—
136	C₂²×M₁₆	C₂²×C₁₁	1	2	yes	—
137	C₂²×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
138	C₂²×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
139	C₂³×C₈	C₂²×C₁₁	1	2	yes	—
140	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic
141	#423	—	2	4	no	—
142	C₂×C₄⋊C₈	—	2	2	no	—
143	#341	—	2	4	no	—
144	#341	—	2	4	no	—
145	#336	—	2	2	no	—
146	#336	C₄×C₁₁	1	2	yes	—
147	#336	C₄×C₁₁	1	2	yes	—
148	C₂²×C₁₁	—	2	2	no	—
149	C₂²×C₁₁	—	2	2	no	—
150	C₂²×C₁₁	—	2	4	no	—
151	C₂²×C₁₁	—	2	2	no	—
152	C₂²×C₁₁	—	2	2	no	—
153	C₂²×C₁₁	#382	1	1	yes	characteristic
154	C₂²×C₁₁	#385	1	1	yes	characteristic
155	C₂²×C₁₁	#389	1	1	yes	characteristic
156	C₂²×C₁₁	#398	1	2	yes	—
157	C₂²×C₁₁	#400	1	2	yes	—
158	C₂²×C₁₁	C₂×#108	1	2	yes	—
159	C₂²×C₁₁	#430	1	2	yes	—
160	C₄×C₁₁	—	2	4	no	—
161	C₄×C₁₁	—	4	2	no	—
162	C₄×C₁₁	#387	1	2	yes	—
163	C₄×C₁₁	#388	1	2	yes	—
164	C₄×C₁₁	#393	1	1	yes	characteristic
165	C₄×C₁₁	#394	1	1	yes	characteristic
166	C₄×C₁₁	#397	1	1	yes	characteristic
167	C₄×C₁₁	#400	1	1	yes	characteristic
168	C₂³×C₄	Q₈×C₁₁	1	2	yes	—
169	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	1	no	—
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	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	D₄×C₁₁	1	2	yes	—
182	C₂×M₁₆	D₄×C₁₁	1	4	yes	—
183	C₂×M₁₆	Q₈×C₁₁	1	2	yes	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	C₂×C₄×C₁₁	1	4	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	C₂×C₄×C₁₁	1	1	yes	characteristic
197	C₂×C₄²	C₂×C₄×C₁₁	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₁₁	1	2	yes	—
199	C₄⋊C₈	—	2	8	no	—
200	C₂×C₁₁	—	2	2	no	—
201	C₂×C₁₁	—	2	2	no	—
202	C₂×C₁₁	#1442	1	1	yes	characteristic
203	C₂×C₁₁	#1837	1	1	yes	characteristic
204	C₂×C₁₁	#1838	1	1	yes	characteristic
205	C₂×C₁₁	#1855	1	2	yes	—
206	C₂×C₁₁	#1950	1	2	yes	—
207	C₂⁴	C₂×Q₈×C₁₁	1	1	yes	characteristic
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₁₁×C₄⋊C₄	1	2	yes	—
212	C₂²×C₄	C₁₁×C₄⋊C₄	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×Q₈×C₁₁	1	1	yes	characteristic
218	C₂²×C₄	C₁₁×#41	1	2	yes	—
219	C₂²×C₄	C₁₁×#41	1	2	yes	—
220	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
221	C₂²×C₄	C₁₁×#41	1	2	yes	—
222	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	—	4	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₁₁	#11269	1	1	yes	characteristic, direct product factor, Sylow
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₁₁×C₄⋊C₄	1	1	yes	characteristic
255	C₂³	C₁₁×C₂²⋊Q₈	1	2	yes	—
256	C₂³	C₁₁×#125	1	2	yes	—
257	C₂³	C₁₁×#125	1	1	yes	characteristic
258	C₂³	C₁₁×C₄⋊Q₈	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	C₁₁×#108	1	4	yes	—
266	C₂×C₄	C₁₁×#117	1	2	yes	—
267	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
268	C₂×C₄	C₄×D₄×C₁₁	1	1	yes	characteristic
269	C₂×C₄	C₄×Q₈×C₁₁	1	1	yes	characteristic
270	C₂×C₄	C₁₁×#120	1	1	yes	characteristic
271	C₂×C₄	C₁₁×#121	1	2	yes	—
272	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
273	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
274	C₂×C₄	C₁₁×C₂²⋊Q₈	1	1	yes	characteristic
275	C₂×C₄	C₁₁×#123	1	2	yes	—
276	C₂×C₄	C₁₁×#123	1	2	yes	—
277	C₂×C₄	C₁₁×#123	1	1	yes	characteristic
278	C₂×C₄	C₁₁×#124	1	2	yes	—
279	C₂×C₄	C₁₁×#124	1	1	yes	characteristic
280	C₂×C₄	C₁₁×#124	1	1	yes	characteristic
281	C₂×C₄	C₁₁×#125	1	2	yes	—
282	C₂×C₄	C₁₁×#126	1	2	yes	—
283	C₂×C₄	C₁₁×#126	1	2	yes	—
284	C₈	—	4	4	no	—
285	C₈	—	4	2	no	—
286	C₈	—	4	4	no	—
287	C₈	—	4	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	4	no	—
293	C₂²	C₁₁×#382	1	1	yes	characteristic
294	C₂²	C₁₁×#385	1	1	yes	characteristic
295	C₂²	C₁₁×#389	1	1	yes	characteristic
296	C₂²	C₁₁×#398	1	2	yes	—
297	C₂²	C₁₁×#400	1	2	yes	—
298	C₂²	C₂×C₁₁×#108	1	2	yes	—
299	C₂²	C₁₁×#430	1	2	yes	—
300	C₄	—	2	4	no	—
301	C₄	—	4	2	no	—
302	C₄	C₁₁×#387	1	2	yes	—
303	C₄	C₁₁×#388	1	2	yes	—
304	C₄	C₁₁×#393	1	1	yes	characteristic
305	C₄	C₁₁×#394	1	1	yes	characteristic
306	C₄	C₁₁×#397	1	1	yes	characteristic
307	C₄	C₁₁×#400	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	C₁₁×#1442	1	1	yes	characteristic
311	C₂	C₁₁×#1837	1	1	yes	characteristic
312	C₂	C₁₁×#1838	1	1	yes	characteristic
313	C₂	C₁₁×#1855	1	2	yes	—
314	C₂	C₁₁×#1950	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₁	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₁
socle	C₂³×C₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₁	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁ ◁ C₂²×C₄²×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
50. #11269	C₁₁	direct (N × H)
251. C₁₁	#11269	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₁×#423	2
C₂×C₁₁×#341	2
C₁₁×#1728	2
C₂×C₁₁×#336	2
#11269	11

Subgroups — C11×#11269 #14588462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#11269 #14588462