Subgroups — #14670231

Order 2816.

462 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×(C₂×D₁₁)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₈×(C₂×D₁₁)⋊C₄	C₂	1	1	yes	maximal, characteristic
5	#13066694	C₂	1	1	yes	maximal, characteristic, complemented
6	#13066693	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₁	C₂²	1	1	yes	characteristic
9	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₁₁)⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₄×(C₂×D₁₁)⋊C₄	C₄	1	1	yes	characteristic
17	C₂×C₁₆×D₁₁	—	2	2	no	—
18	C₂×C₁₆×D₁₁	—	2	2	no	—
19	C₂×C₁₆×D₁₁	—	2	2	no	—
20	C₁₁×C₄⋊C₁₆	—	2	2	no	—
21	#10622188	C₄	1	1	yes	characteristic
22	#10622162	C₄	1	1	yes	characteristic
23	(C₂×D₁₁)⋊C₁₆	—	2	2	no	—
24	Dic₁₁⋊C₁₆	—	2	2	no	—
25	C₂²×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂×(C₂×D₁₁)⋊C₄	C₈	1	1	yes	characteristic
30	C₂×(C₂×D₁₁)⋊C₄	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₁₁	—	2	1	no	—
35	C₂×C₈×D₁₁	—	2	1	no	—
36	C₂×C₈×D₁₁	—	2	1	no	—
37	C₂×C₈×D₁₁	—	2	1	no	—
38	C₂×C₈×D₁₁	D₄	1	2	yes	—
39	C₂×C₈×D₁₁	Q₈	1	2	yes	—
40	C₂×C₁₁×C₁₆	—	2	1	no	—
41	C₂×C₁₁×C₁₆	—	2	1	no	—
42	C₂×C₁₁×C₁₆	—	2	1	no	—
43	C₂×C₁₁×C₁₆	—	2	1	no	—
44	C₂×C₁₁×C₁₆	—	2	1	no	—
45	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
46	C₄×C₈×C₁₁	—	2	1	no	—
47	C₄×C₈×C₁₁	—	2	1	no	—
48	C₁₁⋊#95	C₈	1	1	yes	characteristic
49	C₁₁⋊#95	C₈	1	1	yes	characteristic
50	(C₂×D₁₁)⋊C₈	—	2	2	no	—
51	C₈×Dic₁₁	—	2	1	no	—
52	C₈×Dic₁₁	—	2	1	no	—
53	C₂×C₁₁⋊C₁₆	—	2	1	no	—
54	C₂×C₁₁⋊C₁₆	—	2	1	no	—
55	C₂×C₁₁⋊C₁₆	—	2	1	no	—
56	C₂×C₁₁⋊C₁₆	—	2	1	no	—
57	C₂×C₁₁⋊C₁₆	—	2	1	no	—
58	C₂×C₁₁⋊C₁₆	D₄	1	2	yes	—
59	C₁₆×D₁₁	—	2	8	no	—
60	#12912	—	11	1	no	maximal, Sylow
61	C₂³×D₁₁	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
66	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
67	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
68	C₂×C₄×D₁₁	—	2	1	no	—
69	C₂×C₄×D₁₁	—	2	1	no	—
70	C₂×C₄×D₁₁	—	2	1	no	—
71	C₂×C₄×D₁₁	—	2	1	no	—
72	C₂×C₄×D₁₁	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₁₁	M₁₆	1	2	yes	—
74	C₂×C₈×C₁₁	—	2	1	no	—
75	C₂×C₈×C₁₁	—	2	1	no	—
76	C₂×C₈×C₁₁	—	2	1	no	—
77	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
83	C₄²×C₁₁	—	2	1	no	—
84	C₄²×C₁₁	—	2	1	no	—
85	(C₂×D₁₁)⋊C₄	—	2	2	no	—
86	(C₂×D₁₁)⋊C₄	C₁₆	1	4	yes	complemented
87	C₄×Dic₁₁	—	2	1	no	—
88	C₄×Dic₁₁	—	2	1	no	—
89	C₂×C₁₁⋊C₈	—	2	1	no	—
90	C₂×C₁₁⋊C₈	—	2	1	no	—
91	C₂×C₁₁⋊C₈	—	2	1	no	—
92	C₂×C₁₁⋊C₈	—	2	1	no	—
93	C₂×C₁₁⋊C₈	—	2	1	no	—
94	C₂×C₁₁⋊C₈	C₄⋊C₄	1	2	yes	—
95	C₈×D₁₁	—	2	2	no	—
96	C₈×D₁₁	—	2	2	no	—
97	C₈×D₁₁	—	2	2	no	—
98	C₈×D₁₁	—	2	2	no	—
99	C₁₁×C₁₆	—	2	2	no	—
100	C₁₁×C₁₆	—	4	1	no	—
101	C₁₁⋊C₁₆	—	2	2	no	—
102	C₁₁⋊C₁₆	—	4	1	no	—
103	C₂³×C₁₆	—	11	1	no	—
104	C₂×C₄⋊C₁₆	—	11	1	no	—
105	C₂×C₄⋊C₁₆	D₁₁	1	1	yes	characteristic, complemented
106	C₂×C₂²⋊C₁₆	—	11	1	no	—
107	C₈×C₂²⋊C₄	—	11	1	no	—
108	#1374	—	11	1	no	—
109	#1374	—	11	1	no	—
110	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
111	C₂²×D₁₁	—	2	1	no	—
112	C₂²×D₁₁	—	2	1	no	—
113	C₂²×D₁₁	—	2	1	no	—
114	C₂²×D₁₁	—	2	1	no	—
115	C₂²×D₁₁	C₄⋊C₈	1	2	yes	—
116	C₂²×D₁₁	C₂×C₁₆	1	2	yes	—
117	C₂²×D₁₁	M₃₂	1	2	yes	—
118	C₂×C₄×C₁₁	—	2	1	no	—
119	C₂×C₄×C₁₁	—	2	1	no	—
120	C₂×C₄×C₁₁	—	2	1	no	—
121	C₂×C₄×C₁₁	—	2	1	no	—
122	C₂×C₄×C₁₁	—	2	1	no	—
123	C₂×C₄×C₁₁	C₂×C₁₆	1	2	yes	complemented
124	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₁	#131	1	1	yes	characteristic
129	C₂×C₄×C₁₁	#131	1	1	yes	characteristic
130	C₂×Dic₁₁	—	2	1	no	—
131	C₂×Dic₁₁	—	2	1	no	—
132	C₂×Dic₁₁	—	2	1	no	—
133	C₂×Dic₁₁	—	2	1	no	—
134	C₂×Dic₁₁	—	2	1	no	—
135	C₂×Dic₁₁	—	2	1	no	—
136	C₂×Dic₁₁	—	2	1	no	—
137	C₂×Dic₁₁	C₄⋊C₈	1	2	yes	—
138	C₂×Dic₁₁	C₂×C₁₆	1	2	yes	complemented
139	C₄×D₁₁	—	2	2	no	—
140	C₄×D₁₁	—	2	2	no	—
141	C₄×D₁₁	—	2	2	no	—
142	C₄×D₁₁	—	2	2	no	—
143	C₈×C₁₁	—	4	1	no	—
144	C₈×C₁₁	#120	1	1	yes	characteristic
145	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
146	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
147	C₈×C₁₁	#123	1	1	yes	characteristic
148	C₁₁⋊C₈	—	2	2	no	—
149	C₁₁⋊C₈	—	4	1	no	—
150	C₂³×C₈	—	11	1	no	—
151	C₂²×C₁₆	—	11	1	no	—
152	C₂²×C₁₆	—	11	1	no	—
153	C₂²×C₁₆	—	22	2	no	—
154	C₂²×C₁₆	—	22	2	no	—
155	C₂²×C₁₆	—	22	2	no	—
156	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
157	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
158	C₂×C₂²⋊C₈	—	11	1	no	—
159	C₂×C₄×C₈	—	11	1	no	—
160	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
161	C₄×C₂²⋊C₄	—	11	1	no	—
162	C₄⋊C₁₆	—	2	2	no	—
163	C₄⋊C₁₆	—	22	2	no	—
164	C₂²⋊C₁₆	—	22	2	no	—
165	#336	—	11	1	no	—
166	#336	—	11	1	no	—
167	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
168	C₂²×C₁₁	C₈×D₄	1	1	yes	characteristic
169	C₂²×C₁₁	C₈×D₄	1	1	yes	characteristic
170	C₂²×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
171	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
172	C₂²×C₁₁	#504	1	1	yes	characteristic
173	C₂²×C₁₁	#504	1	1	yes	characteristic
174	C₂×D₁₁	—	2	2	no	—
175	C₂×D₁₁	—	2	2	no	—
176	C₂×D₁₁	—	2	2	no	—
177	C₂×D₁₁	—	2	2	no	—
178	C₂×D₁₁	—	2	2	no	—
179	C₂×D₁₁	—	2	2	no	—
180	C₂×D₁₁	C₄⋊C₁₆	1	4	yes	complemented
181	C₄×C₁₁	—	2	2	no	—
182	C₄×C₁₁	—	4	1	no	—
183	C₄×C₁₁	#385	1	1	yes	characteristic
184	C₄×C₁₁	#424	1	1	yes	characteristic
185	C₄×C₁₁	#435	1	1	yes	characteristic
186	C₄×C₁₁	#435	1	1	yes	characteristic
187	Dic₁₁	—	2	2	no	—
188	Dic₁₁	—	2	2	no	—
189	Dic₁₁	—	4	1	no	—
190	C₂³×C₄	—	11	1	no	—
191	C₂²×C₈	—	11	1	no	—
192	C₂²×C₈	—	11	2	no	—
193	C₂²×C₈	—	11	2	no	—
194	C₂²×C₈	—	11	1	no	—
195	C₂²×C₈	—	22	1	no	—
196	C₂²×C₈	—	22	1	no	—
197	C₂²×C₈	—	22	1	no	—
198	C₂²×C₈	—	22	1	no	—
199	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	—	11	1	no	—
202	C₂×C₂²⋊C₄	—	11	1	no	—
203	C₂×C₄²	—	11	1	no	—
204	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	2	1	no	—
209	C₂×C₁₆	—	2	1	no	—
210	C₂×C₁₆	—	11	2	no	—
211	C₂×C₁₆	—	22	8	no	—
212	C₂×C₁₆	—	22	1	no	—
213	C₂×C₁₆	—	22	1	no	—
214	C₂×C₁₆	—	22	1	no	—
215	C₂×C₁₆	—	22	1	no	—
216	C₂×C₁₆	—	22	1	no	—
217	C₂×C₁₆	C₂²⋊D₁₁	1	2	yes	—
218	C₂²⋊C₈	—	22	2	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	22	1	no	—
222	C₄×C₈	—	22	1	no	—
223	#95	—	11	1	no	—
224	#95	—	11	1	no	—
225	C₂×C₁₁	#1850	1	1	yes	characteristic
226	C₂×C₁₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
227	C₂×C₁₁	#2151	1	1	yes	characteristic
228	C₂×C₁₁	D₄×C₁₆	1	1	yes	characteristic
229	C₂×C₁₁	D₄×C₁₆	1	1	yes	characteristic
230	C₂×C₁₁	#2168	1	1	yes	characteristic
231	C₂×C₁₁	#2168	1	1	yes	characteristic
232	D₁₁	—	2	4	no	—
233	C₂⁴	—	11	1	no	—
234	C₂²×C₄	—	11	1	no	—
235	C₂²×C₄	—	11	1	no	—
236	C₂²×C₄	—	11	2	no	—
237	C₂²×C₄	—	11	2	no	—
238	C₂²×C₄	—	11	1	no	—
239	C₂²×C₄	—	22	1	no	—
240	C₂²×C₄	—	22	1	no	—
241	C₂²×C₄	—	22	1	no	—
242	C₂²×C₄	—	22	1	no	—
243	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
244	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	11	2	no	—
250	C₂×C₈	—	22	1	no	—
251	C₂×C₈	—	22	1	no	—
252	C₂×C₈	—	22	2	no	—
253	C₂×C₈	—	22	2	no	—
254	C₂×C₈	—	22	2	no	—
255	C₂×C₈	—	22	2	no	—
256	C₂×C₈	—	22	1	no	—
257	C₂×C₈	—	22	1	no	—
258	C₂×C₈	—	22	1	no	—
259	C₂×C₈	#4339	1	1	yes	characteristic
260	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
261	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
262	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
263	C₂×C₈	#4343	1	1	yes	characteristic
264	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	11	4	no	—
266	C₂²⋊C₄	—	22	2	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	22	1	no	—
270	C₄²	—	22	1	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	4	1	no	—
273	C₁₆	—	22	2	no	—
274	C₁₆	—	44	1	no	—
275	C₁₁	#12912	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	11	2	no	—
277	C₂³	—	11	2	no	—
278	C₂³	—	11	2	no	—
279	C₂³	—	22	1	no	—
280	C₂³	—	22	1	no	—
281	C₂³	—	22	1	no	—
282	C₂³	—	22	1	no	—
283	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	11	2	no	—
290	C₂×C₄	—	11	2	no	—
291	C₂×C₄	—	22	1	no	—
292	C₂×C₄	—	22	1	no	—
293	C₂×C₄	—	22	1	no	—
294	C₂×C₄	—	22	1	no	—
295	C₂×C₄	—	22	2	no	—
296	C₂×C₄	—	22	2	no	—
297	C₂×C₄	—	22	2	no	—
298	C₂×C₄	—	22	2	no	—
299	C₂×C₄	—	22	1	no	—
300	C₂×C₄	—	22	1	no	—
301	C₂×C₄	—	22	1	no	—
302	C₂×C₄	C₁₆×D₁₁	1	2	yes	complemented
303	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
304	C₂×C₄	#66946	1	1	yes	characteristic
305	C₂×C₄	#66954	1	1	yes	characteristic
306	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
307	C₂×C₄	#66960	1	1	yes	characteristic
308	C₂×C₄	C₄×C₂²⋊D₁₁	1	1	yes	characteristic
309	C₈	—	4	1	no	—
310	C₈	—	22	2	no	—
311	C₈	—	44	1	no	—
312	C₈	#66947	1	1	yes	characteristic
313	C₈	#66949	1	1	yes	characteristic
314	C₈	#66990	1	1	yes	characteristic
315	C₈	#66999	1	1	yes	characteristic
316	C₂²	—	11	4	no	—
317	C₂²	—	22	2	no	—
318	C₂²	—	22	2	no	—
319	C₂²	—	22	2	no	—
320	C₂²	—	22	2	no	—
321	C₂²	—	22	2	no	—
322	C₂²	—	22	2	no	—
323	C₂²	D₁₁×C₄⋊C₈	1	1	yes	characteristic
324	C₂²	#10622447	1	1	yes	characteristic
325	C₂²	C₂×C₁₆×D₁₁	1	1	yes	characteristic
326	C₂²	#10622514	1	1	yes	characteristic
327	C₂²	D₁₁×M₃₂	1	1	yes	characteristic
328	C₂²	#10622520	1	1	yes	characteristic
329	C₂²	C₈×C₂²⋊D₁₁	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	4	1	no	—
332	C₄	—	22	2	no	—
333	C₄	—	22	2	no	—
334	C₄	—	44	1	no	—
335	C₄	#10622449	1	1	yes	characteristic
336	C₄	#10622452	1	1	yes	characteristic
337	C₄	#10622602	1	1	yes	characteristic
338	C₄	#10622740	1	1	yes	characteristic
339	C₂	—	22	4	no	—
340	C₂	#13068792	1	1	yes	characteristic
341	C₂	D₁₁×C₄⋊C₁₆	1	1	yes	characteristic
342	C₂	#13070373	1	1	yes	characteristic
343	C₂	C₁₆×C₂²⋊D₁₁	1	1	yes	characteristic
344	C₂	#13070435	1	1	yes	characteristic
345	C₂	#13070439	1	1	yes	characteristic
346	C₂	#13070440	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂²×C₁₆
Fitting	C₂×C₁₁×C₄⋊C₁₆	C₂
socle	C₂³×C₁₁	C₂²×C₈
prime radical	C₂³×D₁₁	C₂×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₁
hypocenter	C₁₁	#12912
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₁⋊C₁₆	C₂	semidirect
5. #13066694	C₂	semidirect
6. #13066693	C₂	semidirect
7. C₂×C₁₁×C₄⋊C₁₆	C₂	semidirect
86. (C₂×D₁₁)⋊C₄	C₁₆	semidirect
105. C₂×C₄⋊C₁₆	D₁₁	semidirect
123. C₂×C₄×C₁₁	C₂×C₁₆	semidirect
138. C₂×Dic₁₁	C₂×C₁₆	semidirect
180. C₂×D₁₁	C₄⋊C₁₆	semidirect
275. C₁₁	#12912	semidirect
302. C₂×C₄	C₁₆×D₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₁₁	2
C₂×Dic₁₁⋊C₁₆	2
C₂×(C₂×D₁₁)⋊C₁₆	2
C₈×(C₂×D₁₁)⋊C₄	2
#13066694	2
#13066693	2
C₂×C₁₁×C₄⋊C₁₆	2
#12912	11