Subgroups — #14669693

Order 2816.

566 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	C₂×C₄×C₈×D₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×(C₂×D₁₁)⋊C₁₆	C₂	1	2	yes	maximal
4	#13066694	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₁₁	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₁₁	C₄	1	1	yes	characteristic
9	C₂²×C₁₁×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
13	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
14	C₂×C₄×C₁₁⋊C₈	C₄	1	1	yes	characteristic
15	C₄×C₈×D₁₁	—	2	4	no	—
16	(C₂×D₁₁)⋊C₁₆	—	2	4	no	—
17	Dic₁₁⋊C₁₆	—	2	4	no	—
18	C₂²×C₄×D₁₁	C₈	1	1	yes	characteristic
19	C₂²×C₄×D₁₁	C₈	1	1	yes	characteristic
20	C₂²×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₁	C₈	1	1	yes	characteristic
25	C₂×C₄×Dic₁₁	C₈	1	1	yes	characteristic
26	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₁₁	—	2	2	no	—
30	C₂×C₈×D₁₁	—	2	2	no	—
31	C₂×C₈×D₁₁	—	2	2	no	—
32	C₂×C₈×D₁₁	—	2	2	no	—
33	C₂×C₈×D₁₁	—	2	2	no	—
34	C₂×C₈×D₁₁	D₄	1	2	yes	—
35	C₂×C₈×D₁₁	Q₈	1	2	yes	—
36	C₄²×D₁₁	—	2	2	no	—
37	C₄²×D₁₁	—	2	2	no	—
38	C₂×C₁₁×C₁₆	—	2	2	no	—
39	C₂×C₁₁×C₁₆	—	2	2	no	—
40	C₂×C₁₁×C₁₆	—	2	2	no	—
41	C₄×C₈×C₁₁	—	2	2	no	—
42	C₈×Dic₁₁	—	2	2	no	—
43	C₈×Dic₁₁	D₄	1	4	yes	—
44	C₂×C₁₁⋊C₁₆	—	2	2	no	—
45	C₂×C₁₁⋊C₁₆	—	2	2	no	—
46	C₂×C₁₁⋊C₁₆	—	2	2	no	—
47	C₄×C₁₁⋊C₈	—	2	2	no	—
48	#12783	—	11	1	no	maximal, Sylow
49	C₂³×D₁₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
54	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
56	C₂×C₄×D₁₁	—	2	2	no	—
57	C₂×C₄×D₁₁	—	2	2	no	—
58	C₂×C₄×D₁₁	—	2	2	no	—
59	C₂×C₄×D₁₁	—	2	2	no	—
60	C₂×C₄×D₁₁	—	2	2	no	—
61	C₂×C₄×D₁₁	—	2	1	no	—
62	C₂×C₄×D₁₁	—	2	1	no	—
63	C₂×C₄×D₁₁	—	2	1	no	—
64	C₂×C₄×D₁₁	—	2	1	no	—
65	C₂×C₄×D₁₁	C₁₆	1	4	yes	complemented
66	C₂×C₄×D₁₁	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×D₁₁	M₁₆	1	2	yes	—
68	C₂×C₈×C₁₁	—	2	1	no	—
69	C₂×C₈×C₁₁	—	2	1	no	—
70	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
71	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₁₁	#41	1	2	yes	—
75	C₄²×C₁₁	—	2	1	no	—
76	C₄²×C₁₁	—	2	1	no	—
77	C₄×Dic₁₁	—	2	2	no	—
78	C₄×Dic₁₁	—	2	2	no	—
79	C₄×Dic₁₁	C₂²⋊C₄	1	2	yes	—
80	C₄×Dic₁₁	M₁₆	1	2	yes	—
81	C₂×C₁₁⋊C₈	—	2	1	no	—
82	C₂×C₁₁⋊C₈	—	2	1	no	—
83	C₂×C₁₁⋊C₈	—	2	1	no	—
84	C₂×C₁₁⋊C₈	—	2	1	no	—
85	C₂×C₁₁⋊C₈	C₂²⋊C₄	1	2	yes	—
86	C₂×C₁₁⋊C₈	C₄⋊C₄	1	2	yes	—
87	C₈×D₁₁	—	2	8	no	—
88	C₈×D₁₁	—	2	4	no	—
89	C₈×D₁₁	—	2	4	no	—
90	C₁₁×C₁₆	—	4	2	no	—
91	C₁₁⋊C₁₆	—	4	2	no	—
92	C₂²×C₄×C₈	—	11	1	no	—
93	C₂×C₄⋊C₁₆	—	11	2	no	—
94	C₂×C₂²⋊C₁₆	—	11	2	no	—
95	#1374	—	11	1	no	—
96	#1374	D₁₁	1	1	yes	characteristic, complemented
97	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
98	C₂²×D₁₁	—	2	1	no	—
99	C₂²×D₁₁	—	2	1	no	—
100	C₂²×D₁₁	—	2	1	no	—
101	C₂²×D₁₁	—	2	1	no	—
102	C₂²×D₁₁	C₄⋊C₈	1	2	yes	—
103	C₂²×D₁₁	C₂×C₁₆	1	2	yes	—
104	C₂²×D₁₁	M₃₂	1	2	yes	—
105	C₂×C₄×C₁₁	—	2	1	no	—
106	C₂×C₄×C₁₁	—	2	1	no	—
107	C₂×C₄×C₁₁	—	2	1	no	—
108	C₂×C₄×C₁₁	—	2	1	no	—
109	C₂×C₄×C₁₁	C₂×C₁₆	1	2	yes	—
110	C₂×C₄×C₁₁	M₃₂	1	2	yes	—
111	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
117	C₂×Dic₁₁	—	2	1	no	—
118	C₂×Dic₁₁	—	2	1	no	—
119	C₂×Dic₁₁	—	2	1	no	—
120	C₂×Dic₁₁	—	2	1	no	—
121	C₂×Dic₁₁	C₂²⋊C₈	1	2	yes	—
122	C₂×Dic₁₁	C₂²⋊C₈	1	2	yes	—
123	C₂×Dic₁₁	C₄⋊C₈	1	2	yes	—
124	C₂×Dic₁₁	C₂×C₁₆	1	2	yes	—
125	C₂×Dic₁₁	M₃₂	1	2	yes	—
126	C₄×D₁₁	—	2	8	no	—
127	C₄×D₁₁	—	2	2	no	—
128	C₄×D₁₁	—	2	2	no	—
129	C₄×D₁₁	—	2	2	no	—
130	C₄×D₁₁	—	2	2	no	—
131	C₄×D₁₁	—	2	8	no	—
132	C₈×C₁₁	—	2	2	no	—
133	C₈×C₁₁	#121	1	2	yes	—
134	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
135	C₁₁⋊C₈	—	2	2	no	—
136	C₁₁⋊C₈	—	2	2	no	—
137	C₂³×C₈	—	11	1	no	—
138	C₂³×C₈	—	11	1	no	—
139	C₂²×C₄²	—	11	1	no	—
140	C₂²×C₁₆	—	11	2	no	—
141	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
142	C₂×C₄×C₈	—	11	1	no	—
143	C₂×C₄×C₈	—	11	1	no	—
144	C₂×C₄×C₈	—	11	1	no	—
145	C₂×C₄×C₈	—	22	4	no	—
146	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
147	C₄⋊C₁₆	—	22	4	no	—
148	C₂²⋊C₁₆	—	22	4	no	—
149	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
151	C₂²×C₁₁	#431	1	1	yes	characteristic
152	C₂²×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
153	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
154	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
155	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
156	C₂×D₁₁	—	2	2	no	—
157	C₂×D₁₁	—	2	2	no	—
158	C₂×D₁₁	—	2	2	no	—
159	C₂×D₁₁	—	2	2	no	—
160	C₂×D₁₁	—	2	2	no	—
161	C₂×D₁₁	—	2	2	no	—
162	C₂×D₁₁	C₄⋊C₁₆	1	4	yes	complemented
163	C₄×C₁₁	—	2	2	no	—
164	C₄×C₁₁	—	2	2	no	—
165	C₄×C₁₁	#380	1	1	yes	characteristic
166	C₄×C₁₁	#407	1	1	yes	characteristic
167	C₄×C₁₁	#423	1	1	yes	characteristic
168	C₄×C₁₁	#432	1	1	yes	characteristic
169	Dic₁₁	—	2	2	no	—
170	Dic₁₁	—	2	2	no	—
171	Dic₁₁	C₂²⋊C₁₆	1	4	yes	complemented
172	C₂³×C₄	—	11	1	no	—
173	C₂³×C₄	—	11	1	no	—
174	C₂³×C₄	—	11	1	no	—
175	C₂²×C₈	—	11	1	no	—
176	C₂²×C₈	—	11	1	no	—
177	C₂²×C₈	—	11	2	no	—
178	C₂²×C₈	—	11	2	no	—
179	C₂²×C₈	—	22	2	no	—
180	C₂²×C₈	—	22	2	no	—
181	C₂²×C₈	—	22	2	no	—
182	C₂²×C₈	—	22	2	no	—
183	C₂²×C₈	—	22	2	no	—
184	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
186	C₂×C₄²	—	11	1	no	—
187	C₂×C₄²	—	11	1	no	—
188	C₂×C₄²	—	11	1	no	—
189	C₂×C₄²	—	22	2	no	—
190	C₂×C₄²	—	22	2	no	—
191	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
192	C₂×C₁₆	—	2	2	no	—
193	C₂×C₁₆	—	2	2	no	—
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	22	2	no	—
196	C₂×C₁₆	—	22	2	no	—
197	C₂×C₁₆	—	22	2	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	—	11	4	no	—
200	C₄×C₈	—	22	2	no	—
201	C₄×C₈	—	22	2	no	—
202	C₂×C₁₁	#1797	1	1	yes	characteristic
203	C₂×C₁₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
204	C₂×C₁₁	#2112	1	1	yes	characteristic
205	C₂×C₁₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
206	C₂×C₁₁	#2150	1	1	yes	characteristic
207	C₂×C₁₁	#2162	1	1	yes	characteristic
208	C₂×C₁₁	#2163	1	1	yes	characteristic
209	D₁₁	—	2	4	no	—
210	C₂⁴	—	11	1	no	—
211	C₂²×C₄	—	11	1	no	—
212	C₂²×C₄	—	11	1	no	—
213	C₂²×C₄	—	11	1	no	—
214	C₂²×C₄	—	11	4	no	—
215	C₂²×C₄	—	11	2	no	—
216	C₂²×C₄	—	11	2	no	—
217	C₂²×C₄	—	22	2	no	—
218	C₂²×C₄	—	22	2	no	—
219	C₂²×C₄	—	22	2	no	—
220	C₂²×C₄	—	22	2	no	—
221	C₂²×C₄	—	22	2	no	—
222	C₂²×C₄	—	22	1	no	—
223	C₂²×C₄	—	22	1	no	—
224	C₂²×C₄	—	22	1	no	—
225	C₂²×C₄	—	22	1	no	—
226	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
227	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	11	2	no	—
232	C₂×C₈	—	11	2	no	—
233	C₂×C₈	—	22	8	no	—
234	C₂×C₈	—	22	1	no	—
235	C₂×C₈	—	22	1	no	—
236	C₂×C₈	—	22	1	no	—
237	C₂×C₈	—	22	1	no	—
238	C₂×C₈	—	22	4	no	—
239	C₂×C₈	—	22	4	no	—
240	C₂×C₈	#4313	1	2	yes	—
241	C₂×C₈	#4339	1	2	yes	—
242	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
243	C₂×C₈	D₄×D₁₁	1	2	yes	—
244	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
245	C₄²	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	11	2	no	—
248	C₄²	—	11	2	no	—
249	C₄²	—	22	2	no	—
250	C₄²	—	22	2	no	—
251	C₁₆	—	4	2	no	—
252	C₁₆	—	44	2	no	—
253	C₁₁	#12783	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	11	2	no	—
255	C₂³	—	11	2	no	—
256	C₂³	—	11	2	no	—
257	C₂³	—	22	1	no	—
258	C₂³	—	22	1	no	—
259	C₂³	—	22	1	no	—
260	C₂³	—	22	1	no	—
261	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	11	2	no	—
267	C₂×C₄	—	11	2	no	—
268	C₂×C₄	—	11	2	no	—
269	C₂×C₄	—	11	2	no	—
270	C₂×C₄	—	11	2	no	—
271	C₂×C₄	—	22	8	no	—
272	C₂×C₄	—	22	8	no	—
273	C₂×C₄	—	22	1	no	—
274	C₂×C₄	—	22	1	no	—
275	C₂×C₄	—	22	1	no	—
276	C₂×C₄	—	22	1	no	—
277	C₂×C₄	—	22	2	no	—
278	C₂×C₄	—	22	2	no	—
279	C₂×C₄	—	22	2	no	—
280	C₂×C₄	—	22	2	no	—
281	C₂×C₄	C₁₆×D₁₁	1	2	yes	—
282	C₂×C₄	#66862	1	2	yes	—
283	C₂×C₄	#66925	1	1	yes	characteristic
284	C₂×C₄	C₂²⋊C₄×D₁₁	1	1	yes	characteristic
285	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
286	C₂×C₄	C₂×#4313	1	1	yes	characteristic
287	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	22	2	no	—
291	C₈	—	22	2	no	—
292	C₈	#66937	1	2	yes	—
293	C₈	#66949	1	2	yes	—
294	C₂²	—	11	4	no	—
295	C₂²	—	22	2	no	—
296	C₂²	—	22	2	no	—
297	C₂²	—	22	2	no	—
298	C₂²	—	22	2	no	—
299	C₂²	—	22	2	no	—
300	C₂²	—	22	2	no	—
301	C₂²	#10622298	1	1	yes	characteristic
302	C₂²	D₁₁×C₂²⋊C₈	1	1	yes	characteristic
303	C₂²	D₁₁×C₄⋊C₈	1	1	yes	characteristic
304	C₂²	C₂×C₁₆×D₁₁	1	1	yes	characteristic
305	C₂²	C₂×#66862	1	1	yes	characteristic
306	C₂²	D₁₁×M₃₂	1	1	yes	characteristic
307	C₂²	D₁₁×M₃₂	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	11	4	no	—
311	C₄	—	22	2	no	—
312	C₄	—	22	2	no	—
313	C₄	#10622280	1	1	yes	characteristic
314	C₄	#10622318	1	1	yes	characteristic
315	C₄	#10622339	1	1	yes	characteristic
316	C₄	#10622450	1	1	yes	characteristic
317	C₂	—	22	4	no	—
318	C₂	#13068595	1	1	yes	characteristic
319	C₂	D₁₁×C₂²⋊C₁₆	1	1	yes	characteristic
320	C₂	#13070225	1	1	yes	characteristic
321	C₂	D₁₁×C₄⋊C₁₆	1	1	yes	characteristic
322	C₂	#13070370	1	1	yes	characteristic
323	C₂	#13070413	1	1	yes	characteristic
324	C₂	#13070417	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂²×C₁₆
Fitting	C₁₁×#1374	C₂
socle	C₂³×C₁₁	C₂²×C₈
prime radical	C₂³×D₁₁	C₂×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1374	D₁₁
hypocenter	C₁₁	#12783
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₁⋊C₁₆	C₂	semidirect
4. #13066694	C₂	semidirect
5. C₁₁×#1374	C₂	semidirect
65. C₂×C₄×D₁₁	C₁₆	semidirect
96. #1374	D₁₁	semidirect
162. C₂×D₁₁	C₄⋊C₁₆	semidirect
171. Dic₁₁	C₂²⋊C₁₆	semidirect
253. C₁₁	#12783	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₁₁	2
C₂×Dic₁₁⋊C₁₆	2
C₂×(C₂×D₁₁)⋊C₁₆	2
#13066694	2
C₁₁×#1374	2
#12783	11