Subgroups — #14669691

Order 2816.

566 conjugacy classes of subgroups.

The subgroup lattice has 350 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×C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×(C₂×D₁₁)⋊C₁₆	C₂	1	2	yes	maximal
4	#13066693	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₁×C₄⋊C₁₆	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	2	no	—
16	C₄×C₈×D₁₁	—	2	2	no	—
17	C₁₁×C₄⋊C₁₆	—	2	2	no	—
18	(C₂×D₁₁)⋊C₁₆	—	2	4	no	—
19	C₄⋊(C₁₁⋊C₁₆)	—	2	2	no	—
20	C₂²×C₄×D₁₁	C₈	1	1	yes	characteristic
21	C₂²×C₄×D₁₁	C₈	1	1	yes	characteristic
22	C₂²×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₁₁	C₈	1	1	yes	characteristic
27	C₂×C₄×Dic₁₁	C₈	1	1	yes	characteristic
28	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₁₁	—	2	2	no	—
32	C₂×C₈×D₁₁	—	2	2	no	—
33	C₂×C₈×D₁₁	—	2	1	no	—
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	2	no	—
38	C₂×C₈×D₁₁	D₄	1	2	yes	—
39	C₂×C₈×D₁₁	Q₈	1	2	yes	—
40	C₄²×D₁₁	—	2	2	no	—
41	C₄²×D₁₁	—	2	2	no	—
42	C₂×C₁₁×C₁₆	—	2	2	no	—
43	C₂×C₁₁×C₁₆	—	2	2	no	—
44	C₂×C₁₁×C₁₆	—	2	2	no	—
45	C₄×C₈×C₁₁	D₄	1	2	yes	—
46	C₄×C₈×C₁₁	D₄	1	2	yes	—
47	C₈×Dic₁₁	—	2	2	no	—
48	C₈×Dic₁₁	—	2	1	no	—
49	C₈×Dic₁₁	—	2	1	no	—
50	C₂×C₁₁⋊C₁₆	—	2	2	no	—
51	C₂×C₁₁⋊C₁₆	—	2	2	no	—
52	C₂×C₁₁⋊C₁₆	—	2	2	no	—
53	C₄×C₁₁⋊C₈	—	2	2	no	—
54	#12783	—	11	1	no	maximal, Sylow
55	C₂³×D₁₁	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
60	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
62	C₂×C₄×D₁₁	—	2	2	no	—
63	C₂×C₄×D₁₁	—	2	2	no	—
64	C₂×C₄×D₁₁	—	2	2	no	—
65	C₂×C₄×D₁₁	—	2	2	no	—
66	C₂×C₄×D₁₁	—	2	2	no	—
67	C₂×C₄×D₁₁	—	2	1	no	—
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	C₁₆	1	4	yes	complemented
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	C₂²⋊C₄	1	2	yes	—
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	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₁₁	M₁₆	1	2	yes	—
85	C₄×Dic₁₁	—	2	2	no	—
86	C₄×Dic₁₁	—	2	2	no	—
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	C₄⋊C₄	1	2	yes	—
94	C₂×C₁₁⋊C₈	M₁₆	1	2	yes	—
95	C₈×D₁₁	—	2	8	no	—
96	C₈×D₁₁	—	2	2	no	—
97	C₈×D₁₁	—	2	2	no	—
98	C₈×D₁₁	—	2	2	no	—
99	C₈×D₁₁	—	2	2	no	—
100	C₁₁×C₁₆	—	4	2	no	—
101	C₁₁⋊C₁₆	—	4	2	no	—
102	C₂²×C₄×C₈	—	11	1	no	—
103	C₂×C₄⋊C₁₆	—	11	1	no	—
104	C₂×C₄⋊C₁₆	D₁₁	1	1	yes	characteristic, complemented
105	C₂×C₂²⋊C₁₆	—	11	2	no	—
106	#1374	—	11	2	no	—
107	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
108	C₂²×D₁₁	—	2	1	no	—
109	C₂²×D₁₁	—	2	1	no	—
110	C₂²×D₁₁	—	2	1	no	—
111	C₂²×D₁₁	—	2	1	no	—
112	C₂²×D₁₁	C₄⋊C₈	1	2	yes	—
113	C₂²×D₁₁	C₂×C₁₆	1	2	yes	—
114	C₂²×D₁₁	M₃₂	1	2	yes	—
115	C₂×C₄×C₁₁	—	2	1	no	—
116	C₂×C₄×C₁₁	—	2	1	no	—
117	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
118	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
119	C₂×C₄×C₁₁	C₂×C₁₆	1	2	yes	—
120	C₂×C₄×C₁₁	M₃₂	1	2	yes	—
121	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
127	C₂×Dic₁₁	—	2	1	no	—
128	C₂×Dic₁₁	—	2	1	no	—
129	C₂×Dic₁₁	—	2	1	no	—
130	C₂×Dic₁₁	—	2	1	no	—
131	C₂×Dic₁₁	—	2	1	no	—
132	C₂×Dic₁₁	—	2	1	no	—
133	C₂×Dic₁₁	C₄⋊C₈	1	2	yes	—
134	C₂×Dic₁₁	C₂×C₁₆	1	2	yes	—
135	C₂×Dic₁₁	M₃₂	1	2	yes	—
136	C₄×D₁₁	—	2	8	no	—
137	C₄×D₁₁	—	2	8	no	—
138	C₄×D₁₁	—	2	2	no	—
139	C₄×D₁₁	—	2	2	no	—
140	C₄×D₁₁	—	2	2	no	—
141	C₄×D₁₁	—	2	2	no	—
142	C₈×C₁₁	—	2	2	no	—
143	C₈×C₁₁	#121	1	1	yes	characteristic
144	C₈×C₁₁	#121	1	1	yes	characteristic
145	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
146	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
147	C₁₁⋊C₈	—	2	2	no	—
148	C₁₁⋊C₈	—	2	2	no	—
149	C₂³×C₈	—	11	1	no	—
150	C₂³×C₈	—	11	1	no	—
151	C₂²×C₄²	—	11	1	no	—
152	C₂²×C₁₆	—	11	2	no	—
153	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
154	C₂×C₄×C₈	—	11	1	no	—
155	C₂×C₄×C₈	—	11	1	no	—
156	C₂×C₄×C₈	—	11	1	no	—
157	C₂×C₄×C₈	—	22	2	no	—
158	C₂×C₄×C₈	—	22	2	no	—
159	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
160	C₄⋊C₁₆	—	2	2	no	—
161	C₄⋊C₁₆	—	22	2	no	—
162	C₂²⋊C₁₆	—	22	4	no	—
163	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
164	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
165	C₂²×C₁₁	#431	1	1	yes	characteristic
166	C₂²×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
167	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
168	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
169	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
170	C₂×D₁₁	—	2	2	no	—
171	C₂×D₁₁	—	2	2	no	—
172	C₂×D₁₁	—	2	2	no	—
173	C₂×D₁₁	—	2	2	no	—
174	C₂×D₁₁	—	2	2	no	—
175	C₂×D₁₁	—	2	2	no	—
176	C₂×D₁₁	C₄⋊C₁₆	1	4	yes	complemented
177	C₄×C₁₁	—	2	2	no	—
178	C₄×C₁₁	C₂²⋊C₁₆	1	4	yes	complemented
179	C₄×C₁₁	#380	1	1	yes	characteristic
180	C₄×C₁₁	#407	1	1	yes	characteristic
181	C₄×C₁₁	#423	1	1	yes	characteristic
182	C₄×C₁₁	#432	1	1	yes	characteristic
183	Dic₁₁	—	2	2	no	—
184	Dic₁₁	—	2	2	no	—
185	Dic₁₁	—	2	2	no	—
186	C₂³×C₄	—	11	1	no	—
187	C₂³×C₄	—	11	1	no	—
188	C₂³×C₄	—	11	1	no	—
189	C₂²×C₈	—	11	1	no	—
190	C₂²×C₈	—	11	1	no	—
191	C₂²×C₈	—	11	2	no	—
192	C₂²×C₈	—	11	2	no	—
193	C₂²×C₈	—	22	2	no	—
194	C₂²×C₈	—	22	2	no	—
195	C₂²×C₈	—	22	2	no	—
196	C₂²×C₈	—	22	1	no	—
197	C₂²×C₈	—	22	1	no	—
198	C₂²×C₈	—	22	1	no	—
199	C₂²×C₈	—	22	1	no	—
200	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
201	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
202	C₂×C₄²	—	11	1	no	—
203	C₂×C₄²	—	11	1	no	—
204	C₂×C₄²	—	11	1	no	—
205	C₂×C₄²	—	22	2	no	—
206	C₂×C₄²	—	22	2	no	—
207	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
208	C₂×C₁₆	—	2	2	no	—
209	C₂×C₁₆	—	2	2	no	—
210	C₂×C₁₆	—	2	2	no	—
211	C₂×C₁₆	—	22	2	no	—
212	C₂×C₁₆	—	22	2	no	—
213	C₂×C₁₆	—	22	2	no	—
214	C₄×C₈	—	22	2	no	—
215	C₄×C₈	—	22	1	no	—
216	C₄×C₈	—	22	1	no	—
217	C₄×C₈	—	22	2	no	—
218	C₄×C₈	D₄₄	1	2	yes	—
219	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
220	C₂×C₁₁	#1797	1	1	yes	characteristic
221	C₂×C₁₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
222	C₂×C₁₁	#2112	1	1	yes	characteristic
223	C₂×C₁₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
224	C₂×C₁₁	#2150	1	1	yes	characteristic
225	C₂×C₁₁	#2162	1	1	yes	characteristic
226	C₂×C₁₁	#2163	1	1	yes	characteristic
227	D₁₁	—	2	4	no	—
228	C₂⁴	—	11	1	no	—
229	C₂²×C₄	—	11	1	no	—
230	C₂²×C₄	—	11	1	no	—
231	C₂²×C₄	—	11	1	no	—
232	C₂²×C₄	—	11	4	no	—
233	C₂²×C₄	—	11	2	no	—
234	C₂²×C₄	—	11	2	no	—
235	C₂²×C₄	—	22	2	no	—
236	C₂²×C₄	—	22	2	no	—
237	C₂²×C₄	—	22	2	no	—
238	C₂²×C₄	—	22	2	no	—
239	C₂²×C₄	—	22	2	no	—
240	C₂²×C₄	—	22	1	no	—
241	C₂²×C₄	—	22	1	no	—
242	C₂²×C₄	—	22	1	no	—
243	C₂²×C₄	—	22	1	no	—
244	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
245	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	11	2	no	—
250	C₂×C₈	—	11	2	no	—
251	C₂×C₈	—	22	8	no	—
252	C₂×C₈	—	22	1	no	—
253	C₂×C₈	—	22	1	no	—
254	C₂×C₈	—	22	1	no	—
255	C₂×C₈	—	22	1	no	—
256	C₂×C₈	—	22	2	no	—
257	C₂×C₈	—	22	2	no	—
258	C₂×C₈	—	22	2	no	—
259	C₂×C₈	—	22	2	no	—
260	C₂×C₈	(C₂×D₁₁)⋊C₄	1	2	yes	—
261	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
262	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
263	C₂×C₈	#4341	1	1	yes	characteristic
264	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
265	C₂×C₈	#4343	1	1	yes	characteristic
266	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
267	C₄²	—	22	2	no	—
268	C₄²	—	22	2	no	—
269	C₄²	—	22	1	no	—
270	C₄²	—	22	1	no	—
271	C₄²	#4313	1	2	yes	—
272	C₄²	(C₂×D₁₁)⋊C₄	1	2	yes	—
273	C₁₆	—	4	2	no	—
274	C₁₆	—	44	2	no	—
275	C₁₁	#12783	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	—	11	2	no	—
287	C₂×C₄	—	11	2	no	—
288	C₂×C₄	—	11	2	no	—
289	C₂×C₄	—	22	8	no	—
290	C₂×C₄	—	22	1	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	1	no	—
296	C₂×C₄	—	22	2	no	—
297	C₂×C₄	—	22	2	no	—
298	C₂×C₄	—	22	8	no	—
299	C₂×C₄	—	22	2	no	—
300	C₂×C₄	—	22	2	no	—
301	C₂×C₄	C₁₆×D₁₁	1	2	yes	—
302	C₂×C₄	#66862	1	2	yes	—
303	C₂×C₄	(C₂×D₁₁)⋊C₈	1	2	yes	—
304	C₂×C₄	(C₂×D₁₁)⋊C₈	1	2	yes	—
305	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
306	C₂×C₄	#66945	1	1	yes	characteristic
307	C₂×C₄	C₂×#4313	1	1	yes	characteristic
308	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
310	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	22	2	no	—
313	C₈	—	22	2	no	—
314	C₈	#66948	1	1	yes	characteristic
315	C₈	#66950	1	1	yes	characteristic
316	C₈	#66991	1	1	yes	characteristic
317	C₈	#66999	1	1	yes	characteristic
318	C₂²	—	11	4	no	—
319	C₂²	—	22	2	no	—
320	C₂²	—	22	2	no	—
321	C₂²	—	22	2	no	—
322	C₂²	—	22	2	no	—
323	C₂²	—	22	2	no	—
324	C₂²	—	22	2	no	—
325	C₂²	D₁₁×C₄⋊C₈	1	1	yes	characteristic
326	C₂²	#10622446	1	1	yes	characteristic
327	C₂²	C₂×C₁₆×D₁₁	1	1	yes	characteristic
328	C₂²	C₂×#66862	1	1	yes	characteristic
329	C₂²	D₁₁×M₃₂	1	1	yes	characteristic
330	C₂²	D₁₁×M₃₂	1	1	yes	characteristic
331	C₂²	C₂×(C₂×D₁₁)⋊C₈	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	22	2	no	—
334	C₄	—	22	2	no	—
335	C₄	—	22	2	no	—
336	C₄	(C₂×D₁₁)⋊C₁₆	1	4	yes	complemented
337	C₄	#10622448	1	1	yes	characteristic
338	C₄	#10622450	1	1	yes	characteristic
339	C₄	#10622600	1	1	yes	characteristic
340	C₄	#10622739	1	1	yes	characteristic
341	C₂	—	22	4	no	—
342	C₂	#13068593	1	1	yes	characteristic
343	C₂	C₂×(C₂×D₁₁)⋊C₁₆	1	1	yes	characteristic
344	C₂	#13070226	1	1	yes	characteristic
345	C₂	D₁₁×C₄⋊C₁₆	1	1	yes	characteristic
346	C₂	#13070370	1	1	yes	characteristic
347	C₂	#13070414	1	1	yes	characteristic
348	C₂	#13070418	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	#12783
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×C₄⋊(C₁₁⋊C₁₆)	C₂	semidirect
4. #13066693	C₂	semidirect
5. C₂×C₁₁×C₄⋊C₁₆	C₂	semidirect
71. C₂×C₄×D₁₁	C₁₆	semidirect
104. C₂×C₄⋊C₁₆	D₁₁	semidirect
176. C₂×D₁₁	C₄⋊C₁₆	semidirect
178. C₄×C₁₁	C₂²⋊C₁₆	semidirect
275. C₁₁	#12783	semidirect
336. C₄	(C₂×D₁₁)⋊C₁₆	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×C₄⋊(C₁₁⋊C₁₆)	2
C₂×(C₂×D₁₁)⋊C₁₆	2
#13066693	2
C₂×C₁₁×C₄⋊C₁₆	2
#12783	11