Subgroups — C₁₁×#11225 #14588418

Order 2816.

686 conjugacy classes of subgroups.

The subgroup lattice has 344 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×#2317	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#1740	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#1736	C₂	1	2	yes	maximal
5	C₂×C₁₁×H(3,ℤ/4)	C₂	1	2	yes	maximal
6	C₂²×C₁₁×M₁₆	C₂²	1	2	yes	—
7	C₂×C₁₁×#117	C₂²	1	2	yes	—
8	C₂×C₁₁×#117	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₁₁×C₄⋊C₄	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	—	2	2	no	—
15	C₁₁×#384	C₄	1	2	yes	complemented
16	C₁₁×#382	C₄	1	2	yes	complemented
17	C₁₁×#342	—	2	2	no	—
18	C₁₁×#340	—	2	4	no	—
19	C₁₁×H(3,ℤ/4)	—	2	4	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	2	no	—
26	C₂×C₁₁×M₁₆	—	2	2	no	—
27	C₂×C₁₁×M₁₆	—	2	2	no	—
28	C₂²×C₈×C₁₁	D₄	1	4	yes	—
29	C₁₁×#117	—	2	2	no	—
30	C₁₁×#117	—	2	1	no	—
31	C₁₁×#117	—	2	1	no	—
32	C₁₁×#117	—	2	2	no	—
33	C₁₁×#117	—	2	2	no	—
34	C₁₁×#117	—	2	2	no	—
35	C₁₁×#117	—	2	2	no	—
36	C₁₁×#117	D₄	1	2	yes	—
37	C₁₁×#117	Q₈	1	4	yes	—
38	C₁₁×#117	Q₈	1	2	yes	—
39	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
40	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₁×C₄⋊C₄	—	2	4	no	—
45	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₁₁×C₂²⋊C₄	—	2	1	no	—
49	C₂×C₁₁×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₁₁×C₂²⋊C₄	D₄	1	2	yes	—
51	C₂×C₄²×C₁₁	—	2	2	no	—
52	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₁₁	D₄	1	2	yes	—
54	C₁₁×C₄⋊C₈	—	2	8	no	—
55	C₁₁×#95	—	2	4	no	—
56	#11225	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	—	2	2	no	—
59	C₂²×C₄×C₁₁	—	2	2	no	—
60	C₂²×C₄×C₁₁	—	4	2	no	—
61	C₂²×C₄×C₁₁	—	4	2	no	—
62	C₂²×C₄×C₁₁	—	8	2	no	—
63	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
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	1	yes	characteristic
72	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
73	C₂²×C₄×C₁₁	#41	1	2	yes	—
74	C₂²×C₄×C₁₁	#41	1	2	yes	—
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	C₁₁×C₄⋊C₄	—	2	2	no	—
83	C₁₁×C₄⋊C₄	—	2	2	no	—
84	C₁₁×C₄⋊C₄	—	2	2	no	—
85	C₁₁×C₄⋊C₄	—	2	2	no	—
86	C₁₁×C₄⋊C₄	—	2	4	no	—
87	C₁₁×C₄⋊C₄	—	2	4	no	—
88	C₁₁×C₄⋊C₄	—	4	1	no	—
89	C₁₁×C₄⋊C₄	—	4	2	no	—
90	C₁₁×C₄⋊C₄	—	4	1	no	—
91	C₁₁×C₄⋊C₄	—	8	4	no	—
92	C₁₁×C₂²⋊C₄	—	2	2	no	—
93	C₁₁×C₂²⋊C₄	—	2	4	no	—
94	C₁₁×C₂²⋊C₄	—	2	2	no	—
95	C₁₁×C₂²⋊C₄	—	2	4	no	—
96	C₁₁×C₂²⋊C₄	—	4	2	no	—
97	C₄²×C₁₁	—	2	2	no	—
98	C₄²×C₁₁	—	2	2	no	—
99	C₄²×C₁₁	—	2	2	no	—
100	C₄²×C₁₁	—	2	4	no	—
101	C₄²×C₁₁	—	4	2	no	—
102	C₄²×C₁₁	C₄⋊C₄	1	4	yes	—
103	#2905	C₂×C₁₁	1	1	yes	characteristic
104	#2317	C₂×C₁₁	1	1	yes	characteristic
105	#1740	C₂×C₁₁	1	1	yes	characteristic
106	#1736	C₂×C₁₁	1	2	yes	—
107	C₂×H(3,ℤ/4)	C₂×C₁₁	1	2	yes	—
108	C₂³×C₁₁	—	4	1	no	—
109	C₂³×C₁₁	—	4	1	no	—
110	C₂³×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂³×C₁₁	#121	1	2	yes	—
112	C₂³×C₁₁	#121	1	1	yes	characteristic
113	C₂³×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂³×C₁₁	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₁	—	2	4	no	—
116	C₂×C₄×C₁₁	—	2	2	no	—
117	C₂×C₄×C₁₁	—	2	2	no	—
118	C₂×C₄×C₁₁	—	4	2	no	—
119	C₂×C₄×C₁₁	—	4	2	no	—
120	C₂×C₄×C₁₁	—	4	2	no	—
121	C₂×C₄×C₁₁	—	4	2	no	—
122	C₂×C₄×C₁₁	—	4	4	no	—
123	C₂×C₄×C₁₁	—	4	2	no	—
124	C₂×C₄×C₁₁	—	4	4	no	—
125	C₂×C₄×C₁₁	—	8	2	no	—
126	C₂×C₄×C₁₁	—	8	2	no	—
127	C₂×C₄×C₁₁	—	8	4	no	—
128	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₁₁	#120	1	1	yes	characteristic
133	C₂×C₄×C₁₁	#121	1	2	yes	—
134	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₁	#123	1	2	yes	—
139	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
140	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
141	C₂×C₄×C₁₁	#125	1	2	yes	—
142	C₂×C₄×C₁₁	#125	1	2	yes	—
143	C₂×C₄×C₁₁	#126	1	2	yes	—
144	C₂×C₄×C₁₁	C₄⋊Q₈	1	2	yes	—
145	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
146	C₈×C₁₁	—	4	4	no	—
147	C₂²×M₁₆	C₂²×C₁₁	1	2	yes	—
148	C₂×#117	C₂²×C₁₁	1	2	yes	—
149	C₂×#117	C₂²×C₁₁	1	1	yes	characteristic
150	C₂×#117	C₂²×C₁₁	1	1	yes	characteristic
151	C₂²×C₄⋊C₄	C₂²×C₁₁	1	1	yes	characteristic
152	#424	—	2	4	no	—
153	#423	—	2	2	no	—
154	#389	—	2	2	no	—
155	#385	—	2	2	no	—
156	#384	C₄×C₁₁	1	2	yes	complemented
157	#382	C₄×C₁₁	1	2	yes	complemented
158	#342	—	2	2	no	—
159	#340	—	2	4	no	—
160	H(3,ℤ/4)	—	2	4	no	—
161	C₂²×C₁₁	—	2	1	no	—
162	C₂²×C₁₁	—	2	2	no	—
163	C₂²×C₁₁	—	2	1	no	—
164	C₂²×C₁₁	—	2	2	no	—
165	C₂²×C₁₁	—	4	1	no	—
166	C₂²×C₁₁	—	4	1	no	—
167	C₂²×C₁₁	—	4	2	no	—
168	C₂²×C₁₁	#380	1	1	yes	characteristic
169	C₂²×C₁₁	#386	1	1	yes	characteristic
170	C₂²×C₁₁	#391	1	1	yes	characteristic
171	C₂²×C₁₁	#396	1	2	yes	—
172	C₂²×C₁₁	#398	1	2	yes	—
173	C₄×C₁₁	—	4	4	no	—
174	C₄×C₁₁	—	4	2	no	—
175	C₄×C₁₁	—	4	2	no	—
176	C₄×C₁₁	—	8	4	no	—
177	C₄×C₁₁	—	8	2	no	—
178	C₄×C₁₁	#387	1	2	yes	—
179	C₄×C₁₁	#389	1	2	yes	—
180	C₄×C₁₁	#394	1	1	yes	characteristic
181	C₄×C₁₁	#395	1	1	yes	characteristic
182	C₄×C₁₁	#398	1	1	yes	characteristic
183	C₄×C₁₁	#400	1	1	yes	characteristic
184	C₂³×C₄	—	2	1	no	—
185	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂²×C₈	D₄×C₁₁	1	4	yes	—
193	#117	—	2	2	no	—
194	#117	—	2	1	no	—
195	#117	—	2	1	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	D₄×C₁₁	1	2	yes	—
201	#117	Q₈×C₁₁	1	4	yes	—
202	#117	Q₈×C₁₁	1	2	yes	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	2	yes	—
210	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	1	yes	characteristic
211	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	—	2	1	no	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₂²⋊C₄	D₄×C₁₁	1	2	yes	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	C₂×C₄×C₁₁	1	2	yes	—
217	C₂×C₄²	D₄×C₁₁	1	2	yes	—
218	C₄⋊C₈	—	2	8	no	—
219	#95	—	2	4	no	—
220	C₂×C₁₁	—	2	1	no	—
221	C₂×C₁₁	—	2	2	no	—
222	C₂×C₁₁	—	2	1	no	—
223	C₂×C₁₁	—	2	2	no	—
224	C₂×C₁₁	#1441	1	1	yes	characteristic
225	C₂×C₁₁	#1909	1	1	yes	characteristic
226	C₂×C₁₁	#1910	1	1	yes	characteristic
227	C₂⁴	C₂×D₄×C₁₁	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	8	2	no	—
233	C₂²×C₄	C₁₁×C₂²⋊C₄	1	2	yes	—
234	C₂²×C₄	C₁₁×C₂²⋊C₄	1	2	yes	—
235	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
239	C₂²×C₄	C₂×Q₈×C₁₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×Q₈×C₁₁	1	2	yes	—
241	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
242	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
243	C₂²×C₄	C₁₁×#41	1	2	yes	—
244	C₂²×C₄	C₁₁×#41	1	2	yes	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	4	2	no	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	8	4	no	—
262	C₂²⋊C₄	—	2	2	no	—
263	C₂²⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	2	4	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	4	2	no	—
272	C₄²	C₁₁×C₄⋊C₄	1	4	yes	—
273	C₁₁	#11225	1	1	yes	characteristic, direct product factor, Sylow
274	C₂³	—	4	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	C₂×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
277	C₂³	C₁₁×#121	1	2	yes	—
278	C₂³	C₁₁×#121	1	1	yes	characteristic
279	C₂³	C₁₁×C₂²⋊Q₈	1	1	yes	characteristic
280	C₂³	C₁₁×C₂²⋊Q₈	1	2	yes	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	—	8	4	no	—
294	C₂×C₄	C₂×C₁₁×C₄⋊C₄	1	1	yes	characteristic
295	C₂×C₄	C₁₁×#117	1	1	yes	characteristic
296	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
297	C₂×C₄	C₄×Q₈×C₁₁	1	2	yes	—
298	C₂×C₄	C₁₁×#120	1	1	yes	characteristic
299	C₂×C₄	C₁₁×#121	1	2	yes	—
300	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
301	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
302	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
303	C₂×C₄	C₁₁×C₂²⋊Q₈	1	1	yes	characteristic
304	C₂×C₄	C₁₁×#123	1	2	yes	—
305	C₂×C₄	C₁₁×#123	1	1	yes	characteristic
306	C₂×C₄	C₁₁×#124	1	1	yes	characteristic
307	C₂×C₄	C₁₁×#125	1	2	yes	—
308	C₂×C₄	C₁₁×#125	1	2	yes	—
309	C₂×C₄	C₁₁×#126	1	2	yes	—
310	C₂×C₄	C₁₁×C₄⋊Q₈	1	2	yes	—
311	C₂×C₄	C₁₁×C₄⋊Q₈	1	1	yes	characteristic
312	C₈	—	4	4	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	2	no	—
320	C₂²	C₁₁×#380	1	1	yes	characteristic
321	C₂²	C₁₁×#386	1	1	yes	characteristic
322	C₂²	C₁₁×#391	1	1	yes	characteristic
323	C₂²	C₁₁×#396	1	2	yes	—
324	C₂²	C₁₁×#398	1	2	yes	—
325	C₄	—	4	4	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	4	no	—
329	C₄	—	8	2	no	—
330	C₄	C₁₁×#387	1	2	yes	—
331	C₄	C₁₁×#389	1	2	yes	—
332	C₄	C₁₁×#394	1	1	yes	characteristic
333	C₄	C₁₁×#395	1	1	yes	characteristic
334	C₄	C₁₁×#398	1	1	yes	characteristic
335	C₄	C₁₁×#400	1	1	yes	characteristic
336	C₂	—	2	1	no	—
337	C₂	—	2	2	no	—
338	C₂	—	2	1	no	—
339	C₂	—	2	2	no	—
340	C₂	C₁₁×#1441	1	1	yes	characteristic
341	C₂	C₁₁×#1909	1	1	yes	characteristic
342	C₂	C₁₁×#1910	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₁	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₁
prime radical	C₂⁴×C₁₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₂³×C₄×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₁₁×#384	C₄	semidirect
16. C₁₁×#382	C₄	semidirect
56. #11225	C₁₁	direct (N × H)
156. #384	C₄×C₁₁	semidirect
157. #382	C₄×C₁₁	semidirect
273. C₁₁	#11225	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×#2317	2
C₁₁×#1740	2
C₁₁×#1736	2
C₂×C₁₁×H(3,ℤ/4)	2
#11225	11

Subgroups — C11×#11225 #14588418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#11225 #14588418