Subgroups — C₁₁×#11223 #14588416

Order 2816.

706 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×#2905	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#2296	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₁×#344	C₂	1	2	yes	maximal
4	C₁₁×#1742	C₂	1	1	yes	maximal, characteristic
5	C₁₁×#1728	C₂	1	2	yes	maximal
6	C₂²×C₁₁×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₁×M₁₆	C₂²	1	2	yes	—
8	C₂×C₁₁×#117	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
11	C₁₁×#424	—	2	4	no	—
12	C₁₁×#423	—	2	2	no	—
13	C₁₁×#389	—	2	2	no	—
14	C₁₁×#385	C₄	1	4	yes	—
15	C₄×C₁₁×C₄⋊C₄	—	2	2	no	—
16	C₁₁×#344	—	2	4	no	—
17	C₁₁×#343	—	2	2	no	—
18	C₁₁×#336	—	2	4	no	—
19	C₂³×C₄×C₁₁	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
21	C₂×C₁₁×M₁₆	—	2	2	no	—
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	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	—	2	2	no	—
32	C₂×C₁₁×M₁₆	Q₈	1	4	yes	—
33	C₂²×C₈×C₁₁	—	2	2	no	—
34	C₂²×C₈×C₁₁	D₄	1	4	yes	—
35	C₁₁×#117	—	2	2	no	—
36	C₁₁×#117	—	2	1	no	—
37	C₁₁×#117	—	2	1	no	—
38	C₁₁×#117	—	2	2	no	—
39	C₁₁×#117	D₄	1	2	yes	—
40	C₁₁×#117	Q₈	1	2	yes	—
41	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
42	C₂×C₁₁×C₄⋊C₄	—	2	4	no	—
43	C₂×C₁₁×C₄⋊C₄	D₄	1	2	yes	—
44	C₂×C₁₁×C₂²⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₁₁×C₂²⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₁₁	—	2	1	no	—
47	C₂×C₄²×C₁₁	—	2	2	no	—
48	C₂×C₄²×C₁₁	—	2	1	no	—
49	C₂×C₄²×C₁₁	—	2	1	no	—
50	C₂×C₄²×C₁₁	—	2	2	no	—
51	C₂×C₄²×C₁₁	D₄	1	2	yes	—
52	C₁₁×C₄⋊C₈	—	2	8	no	—
53	C₁₁×#95	—	2	4	no	—
54	#11223	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁴×C₁₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	—	2	2	no	—
57	C₂²×C₄×C₁₁	—	2	2	no	—
58	C₂²×C₄×C₁₁	—	2	4	no	—
59	C₂²×C₄×C₁₁	—	4	4	no	—
60	C₂²×C₄×C₁₁	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
69	C₂²×C₄×C₁₁	#41	1	2	yes	—
70	C₂²×C₄×C₁₁	#41	1	2	yes	—
71	C₁₁×M₁₆	—	2	4	no	—
72	C₁₁×M₁₆	—	2	4	no	—
73	C₁₁×M₁₆	—	4	2	no	—
74	C₁₁×M₁₆	—	4	2	no	—
75	C₁₁×M₁₆	—	4	2	no	—
76	C₁₁×M₁₆	—	4	2	no	—
77	C₁₁×M₁₆	—	4	2	no	—
78	C₁₁×M₁₆	—	4	2	no	—
79	C₂×C₈×C₁₁	—	2	4	no	—
80	C₂×C₈×C₁₁	—	2	4	no	—
81	C₂×C₈×C₁₁	—	2	4	no	—
82	C₂×C₈×C₁₁	—	2	4	no	—
83	C₂×C₈×C₁₁	—	4	2	no	—
84	C₂×C₈×C₁₁	—	4	2	no	—
85	C₁₁×C₄⋊C₄	—	2	2	no	—
86	C₁₁×C₄⋊C₄	—	2	4	no	—
87	C₁₁×C₄⋊C₄	—	2	2	no	—
88	C₁₁×C₄⋊C₄	—	4	8	no	—
89	C₁₁×C₂²⋊C₄	—	2	2	no	—
90	C₁₁×C₂²⋊C₄	—	2	4	no	—
91	C₁₁×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
92	C₄²×C₁₁	—	2	2	no	—
93	C₄²×C₁₁	—	2	2	no	—
94	C₄²×C₁₁	—	2	2	no	—
95	C₄²×C₁₁	—	2	2	no	—
96	C₄²×C₁₁	—	4	1	no	—
97	C₄²×C₁₁	—	4	2	no	—
98	C₄²×C₁₁	—	4	1	no	—
99	#2905	C₂×C₁₁	1	1	yes	characteristic
100	#2296	C₂×C₁₁	1	1	yes	characteristic
101	C₂×#344	C₂×C₁₁	1	2	yes	—
102	#1742	C₂×C₁₁	1	1	yes	characteristic
103	#1728	C₂×C₁₁	1	2	yes	—
104	C₂³×C₁₁	—	4	2	no	—
105	C₂³×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₁₁	#117	1	1	yes	characteristic
108	C₂³×C₁₁	C₄×D₄	1	2	yes	—
109	C₂³×C₁₁	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₁₁	—	2	4	no	—
111	C₂×C₄×C₁₁	—	2	8	no	—
112	C₂×C₄×C₁₁	—	2	4	no	—
113	C₂×C₄×C₁₁	—	2	4	no	—
114	C₂×C₄×C₁₁	—	4	4	no	—
115	C₂×C₄×C₁₁	—	4	2	no	—
116	C₂×C₄×C₁₁	—	4	4	no	—
117	C₂×C₄×C₁₁	—	4	2	no	—
118	C₂×C₄×C₁₁	—	4	4	no	—
119	C₂×C₄×C₁₁	#120	1	1	yes	characteristic
120	C₂×C₄×C₁₁	#121	1	2	yes	—
121	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
123	C₂×C₄×C₁₁	#121	1	2	yes	—
124	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₁₁	#123	1	2	yes	—
130	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
131	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
132	C₂×C₄×C₁₁	#125	1	2	yes	—
133	C₂×C₄×C₁₁	#125	1	2	yes	—
134	C₂×C₄×C₁₁	#126	1	2	yes	—
135	C₂×C₄×C₁₁	C₄⋊Q₈	1	2	yes	—
136	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
137	C₈×C₁₁	—	4	4	no	—
138	C₈×C₁₁	—	4	4	no	—
139	C₂²×M₁₆	C₂²×C₁₁	1	2	yes	—
140	C₂²×M₁₆	C₂²×C₁₁	1	2	yes	—
141	C₂×#117	C₂²×C₁₁	1	1	yes	characteristic
142	C₂×#117	C₂²×C₁₁	1	1	yes	characteristic
143	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic
144	#424	—	2	4	no	—
145	#423	—	2	2	no	—
146	#389	—	2	2	no	—
147	#385	C₄×C₁₁	1	4	yes	—
148	C₄×C₄⋊C₄	—	2	2	no	—
149	#344	—	2	4	no	—
150	#343	—	2	2	no	—
151	#336	—	2	4	no	—
152	C₂²×C₁₁	—	2	2	no	—
153	C₂²×C₁₁	—	2	4	no	—
154	C₂²×C₁₁	—	4	1	no	—
155	C₂²×C₁₁	—	4	1	no	—
156	C₂²×C₁₁	—	4	2	no	—
157	C₂²×C₁₁	#380	1	1	yes	characteristic
158	C₂²×C₁₁	#386	1	1	yes	characteristic
159	C₂²×C₁₁	#387	1	2	yes	—
160	C₂²×C₁₁	#389	1	2	yes	—
161	C₂²×C₁₁	#391	1	1	yes	characteristic
162	C₄×C₁₁	—	4	4	no	—
163	C₄×C₁₁	—	4	4	no	—
164	C₄×C₁₁	—	4	4	no	—
165	C₄×C₁₁	#394	1	1	yes	characteristic
166	C₄×C₁₁	#395	1	1	yes	characteristic
167	C₄×C₁₁	#396	1	2	yes	—
168	C₄×C₁₁	#398	1	1	yes	characteristic
169	C₄×C₁₁	#398	1	2	yes	—
170	C₄×C₁₁	#400	1	1	yes	characteristic
171	C₂³×C₄	C₂×C₄×C₁₁	1	2	yes	—
172	C₂³×C₄	C₂³×C₁₁	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	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	Q₈×C₁₁	1	4	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	D₄×C₁₁	1	4	yes	—
187	#117	—	2	2	no	—
188	#117	—	2	1	no	—
189	#117	—	2	1	no	—
190	#117	—	2	2	no	—
191	#117	D₄×C₁₁	1	2	yes	—
192	#117	Q₈×C₁₁	1	2	yes	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	D₄×C₁₁	1	2	yes	—
196	C₂×C₂²⋊C₄	C₂×C₄×C₁₁	1	2	yes	—
197	C₂×C₂²⋊C₄	C₂×C₄×C₁₁	1	2	yes	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	D₄×C₁₁	1	2	yes	—
204	C₄⋊C₈	—	2	8	no	—
205	#95	—	2	4	no	—
206	C₂×C₁₁	—	2	1	no	—
207	C₂×C₁₁	—	2	2	no	—
208	C₂×C₁₁	—	2	2	no	—
209	C₂×C₁₁	—	2	1	no	—
210	C₂×C₁₁	#1441	1	1	yes	characteristic
211	C₂×C₁₁	#1891	1	2	yes	—
212	C₂⁴	C₂²×C₄×C₁₁	1	1	yes	characteristic
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	C₁₁×C₂²⋊C₄	1	4	yes	—
218	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×Q₈×C₁₁	1	2	yes	—
223	C₂²×C₄	C₂×Q₈×C₁₁	1	1	yes	characteristic
224	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
225	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
226	C₂²×C₄	C₁₁×#41	1	2	yes	—
227	C₂²×C₄	C₁₁×#41	1	2	yes	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	—	4	2	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	4	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	4	8	no	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	C₁₁×C₄⋊C₄	1	4	yes	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	1	no	—
254	C₄²	—	4	2	no	—
255	C₄²	—	4	1	no	—
256	C₁₁	#11223	1	1	yes	characteristic, direct product factor, Sylow
257	C₂³	—	4	2	no	—
258	C₂³	C₂×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
259	C₂³	C₂×C₁₁×C₄⋊C₄	1	1	yes	characteristic
260	C₂³	C₁₁×#117	1	1	yes	characteristic
261	C₂³	C₄×D₄×C₁₁	1	2	yes	—
262	C₂³	C₄×Q₈×C₁₁	1	2	yes	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₁₁×#120	1	1	yes	characteristic
273	C₂×C₄	C₁₁×#121	1	2	yes	—
274	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
275	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
276	C₂×C₄	C₁₁×#121	1	2	yes	—
277	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
278	C₂×C₄	C₁₁×C₂²⋊Q₈	1	1	yes	characteristic
279	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
280	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
281	C₂×C₄	C₁₁×C₂²⋊Q₈	1	1	yes	characteristic
282	C₂×C₄	C₁₁×#123	1	2	yes	—
283	C₂×C₄	C₁₁×#123	1	1	yes	characteristic
284	C₂×C₄	C₁₁×#124	1	1	yes	characteristic
285	C₂×C₄	C₁₁×#125	1	2	yes	—
286	C₂×C₄	C₁₁×#125	1	2	yes	—
287	C₂×C₄	C₁₁×#126	1	2	yes	—
288	C₂×C₄	C₁₁×C₄⋊Q₈	1	2	yes	—
289	C₂×C₄	C₁₁×C₄⋊Q₈	1	1	yes	characteristic
290	C₈	—	4	4	no	—
291	C₈	—	4	4	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	2	no	—
297	C₂²	C₁₁×#380	1	1	yes	characteristic
298	C₂²	C₁₁×#386	1	1	yes	characteristic
299	C₂²	C₁₁×#387	1	2	yes	—
300	C₂²	C₁₁×#389	1	2	yes	—
301	C₂²	C₁₁×#391	1	1	yes	characteristic
302	C₄	—	4	4	no	—
303	C₄	—	4	4	no	—
304	C₄	—	4	4	no	—
305	C₄	C₁₁×#394	1	1	yes	characteristic
306	C₄	C₁₁×#395	1	1	yes	characteristic
307	C₄	C₁₁×#396	1	2	yes	—
308	C₄	C₁₁×#398	1	1	yes	characteristic
309	C₄	C₁₁×#398	1	2	yes	—
310	C₄	C₁₁×#400	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	—	2	2	no	—
314	C₂	—	2	1	no	—
315	C₂	C₁₁×#1441	1	1	yes	characteristic
316	C₂	C₁₁×#1891	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₁	#120
commutator	C₂⁴	C₂²×C₄×C₁₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₁
socle	C₂²×C₁₁	#386
prime radical	C₂⁴×C₁₁	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ 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
54. #11223	C₁₁	direct (N × H)
256. C₁₁	#11223	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×#2905	2
C₁₁×#2296	2
C₂×C₁₁×#344	2
C₁₁×#1742	2
C₁₁×#1728	2
#11223	11

Subgroups — C11×#11223 #14588416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#11223 #14588416