Subgroups — C₁₁×#11243 #14588436

Order 2816.

670 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×#2911	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#1742	C₂	1	2	yes	maximal
4	C₂×C₁₁×#339	C₂	1	2	yes	maximal
5	C₂×C₁₁×#338	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₁×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
11	C₁₁×#428	—	2	4	no	—
12	C₁₁×#427	—	2	2	no	—
13	C₁₁×#423	—	2	2	no	—
14	C₁₁×#423	C₄	1	4	yes	complemented
15	C₂×C₁₁×C₄⋊C₈	—	2	2	no	—
16	C₁₁×#343	—	2	4	no	—
17	C₁₁×#339	—	2	2	no	—
18	C₁₁×#339	—	2	2	no	—
19	C₁₁×#338	—	2	2	no	—
20	C₂³×C₄×C₁₁	—	2	1	no	—
21	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
22	C₂×C₁₁×M₁₆	—	2	2	no	—
23	C₂×C₁₁×M₁₆	—	2	2	no	—
24	C₂×C₁₁×M₁₆	—	2	2	no	—
25	C₂×C₁₁×M₁₆	—	2	1	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	—	2	2	no	—
32	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₁×M₁₆	D₄	1	2	yes	—
35	C₂×C₁₁×M₁₆	Q₈	1	4	yes	—
36	C₂²×C₈×C₁₁	—	2	1	no	—
37	C₂²×C₈×C₁₁	—	2	2	no	—
38	C₂²×C₈×C₁₁	D₄	1	2	yes	—
39	C₁₁×#117	—	2	2	no	—
40	C₁₁×#117	—	2	2	no	—
41	C₁₁×#117	—	2	2	no	—
42	C₁₁×#117	—	2	2	no	—
43	C₂×C₁₁×C₄⋊C₄	D₄	1	4	yes	—
44	C₂×C₁₁×C₂²⋊C₄	—	2	2	no	—
45	C₂×C₄²×C₁₁	—	2	1	no	—
46	C₂×C₄²×C₁₁	—	2	1	no	—
47	C₂×C₄²×C₁₁	—	2	1	no	—
48	C₂×C₄²×C₁₁	—	2	2	no	—
49	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₁₁	Q₈	1	2	yes	—
53	C₁₁×#107	—	2	4	no	—
54	C₁₁×#106	—	2	4	no	—
55	C₁₁×C₄⋊C₈	—	2	8	no	—
56	#11243	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₁	#41	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	—	2	1	no	—
59	C₂²×C₄×C₁₁	—	2	2	no	—
60	C₂²×C₄×C₁₁	—	2	1	no	—
61	C₂²×C₄×C₁₁	—	2	2	no	—
62	C₂²×C₄×C₁₁	—	2	4	no	—
63	C₂²×C₄×C₁₁	—	4	4	no	—
64	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₁	#41	1	2	yes	—
72	C₂²×C₄×C₁₁	#41	1	2	yes	—
73	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
74	C₁₁×M₁₆	—	2	4	no	—
75	C₁₁×M₁₆	—	2	4	no	—
76	C₁₁×M₁₆	—	2	4	no	—
77	C₁₁×M₁₆	—	2	4	no	—
78	C₁₁×M₁₆	—	2	4	no	—
79	C₁₁×M₁₆	—	4	2	no	—
80	C₁₁×M₁₆	—	4	2	no	—
81	C₁₁×M₁₆	—	4	2	no	—
82	C₂×C₈×C₁₁	—	2	2	no	—
83	C₂×C₈×C₁₁	—	2	2	no	—
84	C₂×C₈×C₁₁	—	2	4	no	—
85	C₂×C₈×C₁₁	—	2	2	no	—
86	C₂×C₈×C₁₁	—	4	1	no	—
87	C₂×C₈×C₁₁	—	4	2	no	—
88	C₂×C₈×C₁₁	—	4	1	no	—
89	C₂×C₈×C₁₁	—	4	2	no	—
90	C₂×C₈×C₁₁	C₂²⋊C₄	1	4	yes	—
91	C₁₁×C₄⋊C₄	—	2	4	no	—
92	C₁₁×C₄⋊C₄	—	2	4	no	—
93	C₁₁×C₂²⋊C₄	—	4	2	no	—
94	C₁₁×C₂²⋊C₄	—	4	2	no	—
95	C₄²×C₁₁	—	2	2	no	—
96	C₄²×C₁₁	—	2	1	no	—
97	C₄²×C₁₁	—	2	1	no	—
98	C₄²×C₁₁	—	2	2	no	—
99	C₄²×C₁₁	—	4	2	no	—
100	C₄²×C₁₁	—	4	2	no	—
101	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
102	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
103	#2911	C₂×C₁₁	1	1	yes	characteristic
104	C₂×#423	C₂×C₁₁	1	1	yes	characteristic
105	#1742	C₂×C₁₁	1	2	yes	—
106	C₂×#339	C₂×C₁₁	1	2	yes	—
107	C₂×#338	C₂×C₁₁	1	1	yes	characteristic
108	C₂³×C₁₁	—	4	2	no	—
109	C₂³×C₁₁	#117	1	1	yes	characteristic
110	C₂³×C₁₁	#121	1	1	yes	characteristic
111	C₂³×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂³×C₁₁	#123	1	2	yes	—
113	C₂³×C₁₁	#125	1	2	yes	—
114	C₂×C₄×C₁₁	—	2	4	no	—
115	C₂×C₄×C₁₁	—	2	2	no	—
116	C₂×C₄×C₁₁	—	2	2	no	—
117	C₂×C₄×C₁₁	—	4	4	no	—
118	C₂×C₄×C₁₁	—	4	4	no	—
119	C₂×C₄×C₁₁	—	4	2	no	—
120	C₂×C₄×C₁₁	—	4	4	no	—
121	C₂×C₄×C₁₁	—	8	4	no	—
122	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₁₁	#120	1	1	yes	characteristic
127	C₂×C₄×C₁₁	#121	1	2	yes	—
128	C₂×C₄×C₁₁	#121	1	2	yes	—
129	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
135	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
136	C₂×C₄×C₁₁	#125	1	2	yes	—
137	C₂×C₄×C₁₁	#126	1	2	yes	—
138	C₂×C₄×C₁₁	C₄⋊Q₈	1	2	yes	—
139	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
140	C₈×C₁₁	—	2	4	no	—
141	C₈×C₁₁	—	4	2	no	—
142	C₈×C₁₁	—	4	4	no	—
143	C₂²×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
144	C₂²×M₁₆	C₂²×C₁₁	1	2	yes	—
145	C₂²×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
146	C₂×#117	C₂²×C₁₁	1	2	yes	—
147	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic
148	#428	—	2	4	no	—
149	#427	—	2	2	no	—
150	#423	—	2	2	no	—
151	#423	C₄×C₁₁	1	4	yes	complemented
152	C₂×C₄⋊C₈	—	2	2	no	—
153	#343	—	2	4	no	—
154	#339	—	2	2	no	—
155	#339	—	2	2	no	—
156	#338	—	2	2	no	—
157	C₂²×C₁₁	—	2	2	no	—
158	C₂²×C₁₁	—	2	4	no	—
159	C₂²×C₁₁	—	4	1	no	—
160	C₂²×C₁₁	—	4	1	no	—
161	C₂²×C₁₁	—	4	2	no	—
162	C₂²×C₁₁	#380	1	1	yes	characteristic
163	C₂²×C₁₁	#387	1	2	yes	—
164	C₂²×C₁₁	#394	1	1	yes	characteristic
165	C₂²×C₁₁	#398	1	2	yes	—
166	C₂²×C₁₁	#398	1	1	yes	characteristic
167	C₄×C₁₁	—	2	4	no	—
168	C₄×C₁₁	—	4	2	no	—
169	C₄×C₁₁	—	8	4	no	—
170	C₄×C₁₁	#386	1	1	yes	characteristic
171	C₄×C₁₁	#389	1	2	yes	—
172	C₄×C₁₁	#391	1	1	yes	characteristic
173	C₄×C₁₁	#395	1	1	yes	characteristic
174	C₄×C₁₁	#396	1	2	yes	—
175	C₄×C₁₁	#400	1	1	yes	characteristic
176	C₂³×C₄	—	2	1	no	—
177	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	C₂×C₄×C₁₁	1	2	yes	—
189	C₂×M₁₆	C₂×C₄×C₁₁	1	2	yes	—
190	C₂×M₁₆	D₄×C₁₁	1	2	yes	—
191	C₂×M₁₆	Q₈×C₁₁	1	4	yes	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	D₄×C₁₁	1	2	yes	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	C₂×C₄⋊C₄	D₄×C₁₁	1	4	yes	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	—	2	1	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	C₂×C₄×C₁₁	1	2	yes	—
206	C₂×C₄²	D₄×C₁₁	1	1	yes	characteristic
207	C₂×C₄²	D₄×C₁₁	1	1	yes	characteristic
208	C₂×C₄²	Q₈×C₁₁	1	2	yes	—
209	#107	—	2	4	no	—
210	#106	—	2	4	no	—
211	C₄⋊C₈	—	2	8	no	—
212	C₂×C₁₁	—	2	1	no	—
213	C₂×C₁₁	—	2	2	no	—
214	C₂×C₁₁	—	2	1	no	—
215	C₂×C₁₁	—	2	2	no	—
216	C₂×C₁₁	#1441	1	1	yes	characteristic
217	C₂×C₁₁	#2050	1	2	yes	—
218	C₂⁴	C₁₁×#41	1	1	yes	characteristic
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
230	C₂²×C₄	C₂×Q₈×C₁₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×Q₈×C₁₁	1	2	yes	—
232	C₂²×C₄	C₁₁×#41	1	2	yes	—
233	C₂²×C₄	C₁₁×#41	1	2	yes	—
234	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	4	2	no	—
241	M₁₆	—	4	2	no	—
242	M₁₆	—	4	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	C₁₁×C₂²⋊C₄	1	4	yes	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	C₁₁×C₄⋊C₄	1	2	yes	—
263	C₄²	C₁₁×C₄⋊C₄	1	2	yes	—
264	C₁₁	#11243	1	1	yes	characteristic, direct product factor, Sylow
265	C₂³	—	4	2	no	—
266	C₂³	C₁₁×#117	1	1	yes	characteristic
267	C₂³	C₁₁×#121	1	1	yes	characteristic
268	C₂³	C₁₁×C₂²⋊Q₈	1	1	yes	characteristic
269	C₂³	C₁₁×#123	1	2	yes	—
270	C₂³	C₁₁×#125	1	2	yes	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	8	4	no	—
279	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₂×C₁₁×C₄⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
282	C₂×C₄	C₄×Q₈×C₁₁	1	2	yes	—
283	C₂×C₄	C₁₁×#120	1	1	yes	characteristic
284	C₂×C₄	C₁₁×#121	1	2	yes	—
285	C₂×C₄	C₁₁×#121	1	2	yes	—
286	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
287	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
288	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
289	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
290	C₂×C₄	C₁₁×C₂²⋊Q₈	1	1	yes	characteristic
291	C₂×C₄	C₁₁×#123	1	1	yes	characteristic
292	C₂×C₄	C₁₁×#124	1	1	yes	characteristic
293	C₂×C₄	C₁₁×#125	1	2	yes	—
294	C₂×C₄	C₁₁×#126	1	2	yes	—
295	C₂×C₄	C₁₁×C₄⋊Q₈	1	2	yes	—
296	C₂×C₄	C₁₁×C₄⋊Q₈	1	1	yes	characteristic
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	C₁₁×#380	1	1	yes	characteristic
306	C₂²	C₁₁×#387	1	2	yes	—
307	C₂²	C₁₁×#394	1	1	yes	characteristic
308	C₂²	C₁₁×#398	1	2	yes	—
309	C₂²	C₁₁×#398	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	4	no	—
313	C₄	C₁₁×#386	1	1	yes	characteristic
314	C₄	C₁₁×#389	1	2	yes	—
315	C₄	C₁₁×#391	1	1	yes	characteristic
316	C₄	C₁₁×#395	1	1	yes	characteristic
317	C₄	C₁₁×#396	1	2	yes	—
318	C₄	C₁₁×#400	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	C₁₁×#1441	1	1	yes	characteristic
324	C₂	C₁₁×#2050	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₁	#124
commutator	C₂²×C₄	C₂²×C₄×C₁₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₁
socle	C₂²×C₁₁	#394
prime radical	C₂⁴×C₁₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₁ ◁ C₂³×C₄×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₁₁×#423	C₄	semidirect
56. #11243	C₁₁	direct (N × H)
151. #423	C₄×C₁₁	semidirect
264. C₁₁	#11243	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₁×#2911	2
C₂×C₁₁×#423	2
C₁₁×#1742	2
C₂×C₁₁×#339	2
C₂×C₁₁×#338	2
#11243	11

Subgroups — C11×#11243 #14588436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#11243 #14588436