Subgroups — Q₁₆×C₄⋊Dic₁₁ #14772205

Order 2816.

926 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×Dic₁₁	C₂	1	2	yes	maximal
2	Q₈×C₄⋊Dic₁₁	C₂	1	2	yes	maximal
3	#13069350	C₂	1	2	yes	maximal
4	#13068872	C₂	1	4	yes	maximal
5	#13068776	C₂	1	1	yes	maximal, characteristic
6	#13068621	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₁₁×Q₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₁	C₂²	1	2	yes	—
11	C₂×Q₈×Dic₁₁	C₂²	1	4	yes	—
12	C₄×C₁₁×Q₁₆	C₄	1	8	yes	complemented
13	C₂×C₁₁×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₁×#103	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
16	C₂×#66899	C₂²	1	4	yes	—
17	Q₁₆×Dic₁₁	—	2	8	no	—
18	C₂×#66882	C₂²	1	2	yes	—
19	C₂×#66882	C₂²	1	2	yes	—
20	C₂×C₈×Dic₁₁	C₂²	1	2	yes	—
21	#10622598	—	2	4	no	—
22	#10622598	C₂²	1	4	yes	—
23	#10622548	—	2	2	no	—
24	#10622548	C₂²	1	2	yes	—
25	C₄×C₄⋊Dic₁₁	—	2	2	no	—
26	C₄×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
28	#10622188	C₂²	1	2	yes	—
29	#10622167	C₂²	1	4	yes	—
30	C₂²×Q₈×C₁₁	C₂³	1	2	yes	—
31	C₂×C₁₁×Q₁₆	C₂×C₄	1	4	yes	—
32	C₂×C₁₁×Q₁₆	D₄	1	4	yes	—
33	C₂×C₁₁×Q₁₆	Q₈	1	4	yes	—
34	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₁₁	—	2	4	no	—
37	C₄×Q₈×C₁₁	C₂×C₄	1	8	yes	—
38	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
40	C₂×C₄²×C₁₁	—	2	2	no	—
41	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
42	Q₈×Dic₁₁	—	2	8	no	—
43	Q₈×Dic₁₁	—	4	4	no	—
44	C₂×C₄⋊Dic₁₁	—	2	2	no	—
45	C₂×C₄⋊Dic₁₁	—	2	4	no	—
46	C₂×C₄⋊Dic₁₁	—	2	2	no	—
47	C₂×C₄⋊Dic₁₁	—	2	4	no	—
48	C₂×C₄⋊Dic₁₁	D₄	1	1	yes	characteristic
49	C₂×C₄⋊Dic₁₁	D₄	1	1	yes	characteristic
50	C₂×C₄⋊Dic₁₁	C₂³	1	4	yes	—
51	C₂×C₄×Dic₁₁	—	2	4	no	—
52	C₂×C₄×Dic₁₁	C₂³	1	2	yes	—
53	C₂²×C₁₁⋊C₈	C₂³	1	2	yes	—
54	C₁₁×#107	C₂×C₄	1	4	yes	—
55	C₁₁×#103	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₁₁	C₂×C₄	1	4	yes	—
57	#66899	—	2	8	no	—
58	C₁₁⋊#95	—	2	4	no	—
59	C₁₁⋊#95	D₄	1	2	yes	—
60	#66882	—	2	4	no	—
61	#66882	—	4	2	no	—
62	C₈×Dic₁₁	—	2	4	no	—
63	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
64	C₄⋊C₄×Q₁₆	—	11	1	no	maximal, Sylow
65	C₂×Q₈×C₁₁	—	2	2	no	—
66	C₂×Q₈×C₁₁	—	2	2	no	—
67	C₂×Q₈×C₁₁	—	2	2	no	—
68	C₂×Q₈×C₁₁	C₂²×C₄	1	4	yes	—
69	C₂×Q₈×C₁₁	C₂×D₄	1	4	yes	—
70	C₂×Q₈×C₁₁	C₂×Q₈	1	4	yes	—
71	C₂²×C₄×C₁₁	—	2	2	no	—
72	C₂²×C₄×C₁₁	—	2	2	no	—
73	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₁₁	—	2	4	no	—
77	C₂²×Dic₁₁	C₂×D₄	1	2	yes	—
78	C₂²×Dic₁₁	C₂×D₄	1	2	yes	—
79	C₁₁×Q₁₆	C₄⋊C₄	1	16	yes	complemented
80	C₂×C₈×C₁₁	—	2	1	no	—
81	C₂×C₈×C₁₁	—	2	1	no	—
82	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
86	C₁₁×C₄⋊C₄	—	2	8	no	—
87	C₁₁×C₄⋊C₄	—	2	2	no	—
88	C₁₁×C₄⋊C₄	C₂²×C₄	1	4	yes	—
89	C₄²×C₁₁	—	2	8	no	—
90	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
91	C₄²×C₁₁	#41	1	2	yes	—
92	C₄⋊Dic₁₁	—	2	2	no	—
93	C₄⋊Dic₁₁	—	2	4	no	—
94	C₄⋊Dic₁₁	—	2	4	no	—
95	C₄⋊Dic₁₁	—	4	8	no	—
96	C₄⋊Dic₁₁	—	4	4	no	—
97	C₄⋊Dic₁₁	—	8	2	no	—
98	C₄⋊Dic₁₁	—	8	4	no	—
99	C₄⋊Dic₁₁	Q₁₆	1	4	yes	direct product factor
100	C₄×Dic₁₁	—	2	2	no	—
101	C₄×Dic₁₁	—	2	2	no	—
102	C₄×Dic₁₁	—	4	8	no	—
103	C₂×C₁₁⋊C₈	—	2	2	no	—
104	C₂×C₁₁⋊C₈	—	2	2	no	—
105	C₂×C₁₁⋊C₈	—	2	2	no	—
106	C₂×C₄×Q₁₆	—	11	2	no	—
107	C₂×C₄×Q₁₆	D₁₁	1	1	yes	characteristic
108	Q₈×C₄⋊C₄	—	11	2	no	—
109	#1942	—	11	2	no	—
110	#1863	—	11	4	no	—
111	#1845	—	11	1	no	—
112	#1803	—	11	2	no	—
113	C₈×C₄⋊C₄	—	11	1	no	—
114	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₁₁	—	2	4	no	—
116	Q₈×C₁₁	—	2	4	no	—
117	Q₈×C₁₁	—	2	4	no	—
118	Q₈×C₁₁	C₂×C₄⋊C₄	1	8	yes	—
119	C₂×C₄×C₁₁	—	2	4	no	—
120	C₂×C₄×C₁₁	—	2	4	no	—
121	C₂×C₄×C₁₁	—	2	4	no	—
122	C₂×C₄×C₁₁	—	2	1	no	—
123	C₂×C₄×C₁₁	—	2	1	no	—
124	C₂×C₄×C₁₁	—	2	4	no	—
125	C₂×C₄×C₁₁	—	4	2	no	—
126	C₂×C₄×C₁₁	—	4	2	no	—
127	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
130	C₂×C₄×C₁₁	#135	1	2	yes	—
131	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂×Dic₁₁	—	2	2	no	—
138	C₂×Dic₁₁	—	2	2	no	—
139	C₂×Dic₁₁	—	2	2	no	—
140	C₂×Dic₁₁	—	2	2	no	—
141	C₂×Dic₁₁	—	2	2	no	—
142	C₂×Dic₁₁	—	4	4	no	—
143	C₂×Dic₁₁	—	4	4	no	—
144	C₂×Dic₁₁	—	4	4	no	—
145	C₂×Dic₁₁	C₂×Q₁₆	1	4	yes	—
146	C₈×C₁₁	—	2	2	no	—
147	C₈×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
148	C₁₁⋊C₈	—	4	2	no	—
149	C₂²×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
150	C₂×C₄×Q₈	—	11	4	no	—
151	C₂×C₄×Q₈	C₂×D₁₁	1	2	yes	—
152	C₄×Q₁₆	—	22	8	no	—
153	C₄×Q₁₆	Dic₁₁	1	8	yes	complemented
154	C₂×#107	—	11	2	no	—
155	C₂×#107	—	11	2	no	—
156	C₂×#107	C₂×D₁₁	1	1	yes	characteristic
157	C₂×C₄⋊C₈	—	11	1	no	—
158	C₂×#103	—	11	4	no	—
159	C₂×#103	C₂×D₁₁	1	2	yes	—
160	C₂×C₄×C₈	—	11	2	no	—
161	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
162	#389	—	11	4	no	—
163	#389	—	22	4	no	—
164	#384	—	11	2	no	—
165	#384	—	22	2	no	—
166	C₄×C₄⋊C₄	—	11	1	no	—
167	C₄×C₄⋊C₄	—	22	2	no	—
168	#340	—	11	4	no	—
169	#336	—	11	2	no	—
170	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂²×C₁₁	D₄²	1	1	yes	characteristic
172	C₂²×C₁₁	D₄×Q₈	1	1	yes	characteristic
173	C₂²×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
174	C₂²×C₁₁	C₂×#135	1	1	yes	characteristic
175	C₂²×C₁₁	#576	1	1	yes	characteristic
176	C₂²×C₁₁	#578	1	1	yes	characteristic
177	C₄×C₁₁	—	2	8	no	—
178	C₄×C₁₁	—	2	2	no	—
179	C₄×C₁₁	—	4	4	no	—
180	C₄×C₁₁	C₄×Q₁₆	1	4	yes	complemented
181	C₄×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
182	C₄×C₁₁	#520	1	1	yes	characteristic
183	C₄×C₁₁	#547	1	1	yes	characteristic
184	C₄×C₁₁	#554	1	1	yes	characteristic
185	Dic₁₁	—	2	4	no	—
186	Dic₁₁	—	4	2	no	—
187	Dic₁₁	—	8	4	no	—
188	C₂²×Q₈	C₂²×D₁₁	1	2	yes	—
189	C₂×Q₁₆	Dic₂₂	1	4	yes	—
190	C₂×Q₁₆	D₄₄	1	4	yes	—
191	C₂×Q₁₆	C₂×Dic₁₁	1	4	yes	—
192	C₂²×C₈	—	11	2	no	—
193	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
195	C₄×Q₈	—	2	4	no	—
196	C₄×Q₈	—	22	8	no	—
197	C₄×Q₈	—	44	4	no	—
198	C₄×Q₈	C₂×Dic₁₁	1	8	yes	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	11	1	no	—
201	C₂×C₄⋊C₄	—	11	1	no	—
202	C₂×C₄⋊C₄	—	11	4	no	—
203	C₂×C₄⋊C₄	—	22	4	no	—
204	C₂×C₄⋊C₄	—	22	2	no	—
205	C₂×C₄⋊C₄	—	22	4	no	—
206	C₂×C₄⋊C₄	—	22	2	no	—
207	C₂×C₄⋊C₄	C₂²×D₁₁	1	2	yes	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	—	11	2	no	—
210	C₂×C₄²	—	22	4	no	—
211	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
212	#107	—	22	4	no	—
213	#107	—	44	2	no	—
214	#107	C₂×Dic₁₁	1	4	yes	—
215	C₄⋊C₈	—	22	2	no	—
216	#103	—	22	8	no	—
217	#103	C₂×Dic₁₁	1	8	yes	—
218	C₄×C₈	—	22	4	no	—
219	C₄×C₈	C₂×Dic₁₁	1	4	yes	—
220	#95	—	11	2	no	—
221	#95	—	22	4	no	—
222	C₂×C₁₁	D₄×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₁₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
224	C₂×C₁₁	#2951	1	1	yes	characteristic
225	C₂×C₁₁	D₄×Q₁₆	1	1	yes	characteristic
226	C₂×C₁₁	#3287	1	1	yes	characteristic
227	C₂×C₁₁	Q₈×Q₁₆	1	1	yes	characteristic
228	C₂×C₁₁	#3383	1	1	yes	characteristic
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	C₂×Dic₂₂	1	4	yes	—
233	C₂×Q₈	C₂×D₄₄	1	4	yes	—
234	C₂×Q₈	C₂²×Dic₁₁	1	4	yes	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	11	2	no	—
238	C₂²×C₄	—	11	2	no	—
239	C₂²×C₄	—	22	4	no	—
240	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
241	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
242	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
243	Q₁₆	C₄⋊Dic₁₁	1	16	yes	direct product factor
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	22	2	no	—
247	C₂×C₈	—	22	2	no	—
248	C₂×C₈	—	22	2	no	—
249	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
250	C₂×C₈	C₂×D₄₄	1	2	yes	—
251	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
252	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
253	C₄⋊C₄	—	2	8	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	11	4	no	—
256	C₄⋊C₄	—	22	2	no	—
257	C₄⋊C₄	—	22	4	no	—
258	C₄⋊C₄	—	22	4	no	—
259	C₄⋊C₄	—	44	8	no	—
260	C₄⋊C₄	—	44	4	no	—
261	C₄⋊C₄	—	88	2	no	—
262	C₄⋊C₄	—	88	4	no	—
263	C₄⋊C₄	C₂²×Dic₁₁	1	4	yes	—
264	C₄²	—	2	8	no	—
265	C₄²	—	22	2	no	—
266	C₄²	—	22	2	no	—
267	C₄²	—	44	8	no	—
268	C₄²	#4341	1	2	yes	—
269	C₄²	C₂²×Dic₁₁	1	2	yes	—
270	C₁₁	C₄⋊C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
271	C₂³	C₂×D₄×D₁₁	1	1	yes	characteristic
272	Q₈	—	2	4	no	—
273	Q₈	—	2	4	no	—
274	Q₈	—	2	4	no	—
275	Q₈	C₂×C₄⋊Dic₁₁	1	8	yes	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	11	4	no	—
285	C₂×C₄	—	22	2	no	—
286	C₂×C₄	—	22	2	no	—
287	C₂×C₄	—	22	2	no	—
288	C₂×C₄	—	22	2	no	—
289	C₂×C₄	—	22	2	no	—
290	C₂×C₄	—	44	4	no	—
291	C₂×C₄	—	44	4	no	—
292	C₂×C₄	—	44	4	no	—
293	C₂×C₄	Q₁₆×D₁₁	1	2	yes	—
294	C₂×C₄	#66972	1	2	yes	—
295	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
296	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
297	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
298	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
299	C₂×C₄	C₂×#4341	1	1	yes	characteristic
300	C₂×C₄	#67042	1	1	yes	characteristic
301	C₂×C₄	#67043	1	1	yes	characteristic
302	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	44	2	no	—
305	C₈	C₂×C₄⋊Dic₁₁	1	4	yes	—
306	C₂²	D₄×Dic₂₂	1	1	yes	characteristic
307	C₂²	D₄×D₄₄	1	1	yes	characteristic
308	C₂²	C₂×Q₁₆×D₁₁	1	1	yes	characteristic
309	C₂²	C₂×#66972	1	1	yes	characteristic
310	C₂²	#10623300	1	1	yes	characteristic
311	C₂²	#10623302	1	1	yes	characteristic
312	C₂²	C₂×D₄×Dic₁₁	1	1	yes	characteristic
313	C₄	—	2	8	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	4	no	—
316	C₄	—	22	4	no	—
317	C₄	—	44	2	no	—
318	C₄	—	88	4	no	—
319	C₄	Q₁₆×Dic₁₁	1	4	yes	complemented
320	C₄	#10623074	1	1	yes	characteristic
321	C₄	#10623086	1	1	yes	characteristic
322	C₄	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
323	C₄	#10623358	1	1	yes	characteristic
324	C₂	D₄×C₄⋊Dic₁₁	1	1	yes	characteristic
325	C₂	C₂×Q₁₆×Dic₁₁	1	1	yes	characteristic
326	C₂	#13077492	1	1	yes	characteristic
327	C₂	Q₁₆×D₄₄	1	1	yes	characteristic
328	C₂	#13080375	1	1	yes	characteristic
329	C₂	Q₁₆×Dic₂₂	1	1	yes	characteristic
330	C₂	#13081505	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₂×C₄×C₁₁×Q₁₆	C₂
socle, prime radical	C₂³×C₁₁	C₂²×D₄
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₁₁
hypocenter	C₁₁	C₄⋊C₄×Q₁₆
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
12. C₄×C₁₁×Q₁₆	C₄	semidirect
79. C₁₁×Q₁₆	C₄⋊C₄	semidirect
99. C₄⋊Dic₁₁	Q₁₆	direct (N × H)
153. C₄×Q₁₆	Dic₁₁	semidirect
180. C₄×C₁₁	C₄×Q₁₆	semidirect
243. Q₁₆	C₄⋊Dic₁₁	direct (N × H)
270. C₁₁	C₄⋊C₄×Q₁₆	semidirect
319. C₄	Q₁₆×Dic₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×Dic₁₁	2
Q₈×C₄⋊Dic₁₁	2
#13069350	2
#13068872	2
#13068776	2
#13068621	2
C₈×C₄⋊Dic₁₁	2
C₂×C₄×C₁₁×Q₁₆	2
C₄⋊C₄×Q₁₆	11

Subgroups — Q16×C4⋊Dic11 #14772205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×C₄⋊Dic₁₁ #14772205