Subgroups — #14669694

Order 2816.

566 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊(C₁₁⋊C₁₆)	C₂	1	2	yes	maximal
3	#13066694	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₁	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₁	—	2	4	no	—
9	C₂×C₄×C₈×C₁₁	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂²×C₁₁⋊C₁₆	C₂²	1	4	yes	—
14	C₂²⋊(C₁₁⋊C₁₆)	—	2	4	no	—
15	C₄⋊(C₁₁⋊C₁₆)	—	2	4	no	—
16	C₂³×C₄×C₁₁	C₈	1	1	yes	characteristic
17	C₂³×C₄×C₁₁	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₁	—	2	2	no	—
20	C₂²×C₈×C₁₁	—	2	2	no	—
21	C₂²×C₈×C₁₁	—	2	2	no	—
22	C₂²×C₈×C₁₁	—	2	2	no	—
23	C₂²×C₈×C₁₁	—	2	2	no	—
24	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₁	D₄	1	2	yes	—
28	C₂²×C₈×C₁₁	Q₈	1	2	yes	—
29	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₁₁	—	2	2	no	—
31	C₂×C₄²×C₁₁	—	2	2	no	—
32	C₂×C₄²×C₁₁	C₈	1	1	yes	characteristic
33	C₂×C₄²×C₁₁	C₈	1	1	yes	characteristic
34	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₁	—	2	2	no	—
37	C₄×C₈×C₁₁	—	2	2	no	—
38	C₄×C₈×C₁₁	—	2	2	no	—
39	C₄×C₈×C₁₁	D₄	1	4	yes	—
40	C₂×C₁₁⋊C₁₆	—	2	4	no	—
41	C₂×C₁₁⋊C₁₆	—	2	4	no	—
42	C₂×C₁₁⋊C₁₆	—	2	4	no	—
43	#12783	—	11	1	no	maximal, Sylow
44	C₂⁴×C₁₁	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₁	—	2	2	no	—
46	C₂²×C₄×C₁₁	—	2	2	no	—
47	C₂²×C₄×C₁₁	—	2	2	no	—
48	C₂²×C₄×C₁₁	—	2	2	no	—
49	C₂²×C₄×C₁₁	—	2	2	no	—
50	C₂²×C₄×C₁₁	—	2	1	no	—
51	C₂²×C₄×C₁₁	—	2	1	no	—
52	C₂²×C₄×C₁₁	—	2	1	no	—
53	C₂²×C₄×C₁₁	—	2	1	no	—
54	C₂²×C₄×C₁₁	C₁₆	1	4	yes	complemented
55	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	M₁₆	1	2	yes	—
62	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₁	—	2	8	no	—
64	C₂×C₈×C₁₁	—	2	4	no	—
65	C₂×C₈×C₁₁	—	2	4	no	—
66	C₂×C₈×C₁₁	—	2	1	no	—
67	C₂×C₈×C₁₁	—	2	1	no	—
68	C₂×C₈×C₁₁	—	2	1	no	—
69	C₂×C₈×C₁₁	—	2	1	no	—
70	C₂×C₈×C₁₁	—	2	1	no	—
71	C₂×C₈×C₁₁	—	2	1	no	—
72	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
75	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₁₁	#41	1	2	yes	—
79	C₄²×C₁₁	—	2	2	no	—
80	C₄²×C₁₁	—	2	2	no	—
81	C₄²×C₁₁	—	2	1	no	—
82	C₄²×C₁₁	—	2	1	no	—
83	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₁₁	M₁₆	1	2	yes	—
85	C₁₁⋊C₁₆	—	4	4	no	—
86	C₂²×C₄×C₈	D₁₁	1	1	yes	characteristic
87	C₂×C₄⋊C₁₆	—	11	2	no	—
88	C₂×C₂²⋊C₁₆	—	11	2	no	—
89	#1374	—	11	2	no	—
90	C₂³×C₁₁	—	2	1	no	—
91	C₂³×C₁₁	—	2	1	no	—
92	C₂³×C₁₁	—	2	1	no	—
93	C₂³×C₁₁	—	2	1	no	—
94	C₂³×C₁₁	C₄⋊C₈	1	2	yes	—
95	C₂³×C₁₁	C₂×C₁₆	1	2	yes	—
96	C₂³×C₁₁	M₃₂	1	2	yes	—
97	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₁	—	2	8	no	—
99	C₂×C₄×C₁₁	—	2	1	no	—
100	C₂×C₄×C₁₁	—	2	1	no	—
101	C₂×C₄×C₁₁	—	2	1	no	—
102	C₂×C₄×C₁₁	—	2	1	no	—
103	C₂×C₄×C₁₁	—	2	1	no	—
104	C₂×C₄×C₁₁	—	2	1	no	—
105	C₂×C₄×C₁₁	—	2	1	no	—
106	C₂×C₄×C₁₁	—	2	1	no	—
107	C₂×C₄×C₁₁	—	2	8	no	—
108	C₂×C₄×C₁₁	—	2	2	no	—
109	C₂×C₄×C₁₁	—	2	2	no	—
110	C₂×C₄×C₁₁	—	2	2	no	—
111	C₂×C₄×C₁₁	—	2	2	no	—
112	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
113	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
114	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
115	C₂×C₄×C₁₁	C₂×C₁₆	1	2	yes	—
116	C₂×C₄×C₁₁	C₂×C₁₆	1	2	yes	—
117	C₂×C₄×C₁₁	M₃₂	1	2	yes	—
118	C₂×C₄×C₁₁	M₃₂	1	2	yes	—
119	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
125	C₈×C₁₁	—	2	2	no	—
126	C₈×C₁₁	—	2	2	no	—
127	C₈×C₁₁	—	2	2	no	—
128	C₈×C₁₁	#121	1	2	yes	—
129	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
130	C₂³×C₈	Dic₁₁	1	1	yes	characteristic
131	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
132	C₂²×C₄²	Dic₁₁	1	1	yes	characteristic
133	C₂²×C₁₆	—	11	4	no	—
134	C₂×C₄×C₈	—	2	4	no	—
135	C₂×C₄×C₈	Dic₁₁	1	1	yes	characteristic
136	C₂×C₄×C₈	Dic₁₁	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
139	C₄⋊C₁₆	—	22	4	no	—
140	C₂²⋊C₁₆	—	22	4	no	—
141	C₂²×C₁₁	—	2	2	no	—
142	C₂²×C₁₁	—	2	2	no	—
143	C₂²×C₁₁	—	2	2	no	—
144	C₂²×C₁₁	—	2	2	no	—
145	C₂²×C₁₁	—	2	2	no	—
146	C₂²×C₁₁	—	2	2	no	—
147	C₂²×C₁₁	C₄⋊C₁₆	1	4	yes	complemented
148	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
150	C₂²×C₁₁	#431	1	1	yes	characteristic
151	C₂²×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
152	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
153	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
154	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
155	C₄×C₁₁	—	2	2	no	—
156	C₄×C₁₁	—	2	2	no	—
157	C₄×C₁₁	—	2	2	no	—
158	C₄×C₁₁	—	2	2	no	—
159	C₄×C₁₁	C₂²⋊C₁₆	1	4	yes	complemented
160	C₄×C₁₁	#380	1	1	yes	characteristic
161	C₄×C₁₁	#407	1	1	yes	characteristic
162	C₄×C₁₁	#423	1	1	yes	characteristic
163	C₄×C₁₁	#432	1	1	yes	characteristic
164	C₂³×C₄	C₁₁⋊C₈	1	1	yes	characteristic
165	C₂³×C₄	C₁₁⋊C₈	1	1	yes	characteristic
166	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	Dic₂₂	1	2	yes	—
173	C₂²×C₈	D₄₄	1	2	yes	—
174	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
175	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
176	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₁₁⋊C₈	1	1	yes	characteristic
181	C₂×C₄²	C₁₁⋊C₈	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
183	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
184	C₂×C₁₆	—	22	4	no	—
185	C₂×C₁₆	—	22	4	no	—
186	C₂×C₁₆	—	22	4	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	C₂²⋊D₁₁	1	4	yes	—
191	C₂×C₁₁	—	2	4	no	—
192	C₂×C₁₁	#1797	1	1	yes	characteristic
193	C₂×C₁₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
194	C₂×C₁₁	#2112	1	1	yes	characteristic
195	C₂×C₁₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
196	C₂×C₁₁	#2150	1	1	yes	characteristic
197	C₂×C₁₁	#2162	1	1	yes	characteristic
198	C₂×C₁₁	#2163	1	1	yes	characteristic
199	C₂⁴	C₂×C₁₁⋊C₈	1	1	yes	characteristic
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	C₁₁⋊C₁₆	1	4	yes	complemented
210	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
215	C₂²×C₄	#4318	1	2	yes	—
216	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
218	C₂×C₈	—	2	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	#4318	1	2	yes	—
228	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
229	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
230	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
232	C₂×C₈	#4339	1	2	yes	—
233	C₂×C₈	C₂×C₂²⋊D₁₁	1	2	yes	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	#4318	1	2	yes	—
239	C₄²	C₂²⋊Dic₁₁	1	2	yes	—
240	C₁₆	—	44	4	no	—
241	C₁₁	#12783	1	1	yes	characteristic, complemented, Sylow
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
247	C₂³	C₂×C₁₁⋊C₁₆	1	2	yes	—
248	C₂³	#66876	1	2	yes	—
249	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
265	C₂×C₄	C₂×C₁₁⋊C₁₆	1	2	yes	—
266	C₂×C₄	C₂×C₁₁⋊C₁₆	1	2	yes	—
267	C₂×C₄	#66876	1	2	yes	—
268	C₂×C₄	#66876	1	2	yes	—
269	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	2	yes	—
270	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	2	yes	—
271	C₂×C₄	C₂×#4318	1	1	yes	characteristic
272	C₂×C₄	C₂×#4318	1	1	yes	characteristic
273	C₂×C₄	C₂×#4318	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
275	C₂×C₄	#66979	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	2	no	—
279	C₈	—	2	2	no	—
280	C₈	#66977	1	2	yes	—
281	C₈	#66983	1	2	yes	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₄⋊(C₁₁⋊C₁₆)	1	4	yes	complemented
289	C₂²	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
290	C₂²	#10622538	1	1	yes	characteristic
291	C₂²	C₂²×C₁₁⋊C₁₆	1	1	yes	characteristic
292	C₂²	C₂×#66876	1	1	yes	characteristic
293	C₂²	C₂×#66876	1	1	yes	characteristic
294	C₂²	C₂×#66876	1	1	yes	characteristic
295	C₂²	C₂×C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	C₂²⋊(C₁₁⋊C₁₆)	1	4	yes	complemented
301	C₄	#10622537	1	1	yes	characteristic
302	C₄	#10622539	1	1	yes	characteristic
303	C₄	#10622820	1	1	yes	characteristic
304	C₄	#10622825	1	1	yes	characteristic
305	C₂	—	2	4	no	—
306	C₂	#13068596	1	1	yes	characteristic
307	C₂	C₂×C₂²⋊(C₁₁⋊C₁₆)	1	1	yes	characteristic
308	C₂	#13070227	1	1	yes	characteristic
309	C₂	C₂×C₄⋊(C₁₁⋊C₁₆)	1	1	yes	characteristic
310	C₂	#13070372	1	1	yes	characteristic
311	C₂	#13070415	1	1	yes	characteristic
312	C₂	#13070419	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂²×C₁₆
Fitting	C₂²×C₄×C₈×C₁₁	C₂
socle	C₂³×C₁₁	C₂²×C₈
prime radical	C₂⁴×C₁₁	C₂×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂²×C₄×C₈	D₁₁
hypocenter	C₁₁	#12783
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	semidirect
3. #13066694	C₂	semidirect
54. C₂²×C₄×C₁₁	C₁₆	semidirect
147. C₂²×C₁₁	C₄⋊C₁₆	semidirect
159. C₄×C₁₁	C₂²⋊C₁₆	semidirect
209. C₂²×C₄	C₁₁⋊C₁₆	semidirect
241. C₁₁	#12783	semidirect
288. C₂²	C₄⋊(C₁₁⋊C₁₆)	semidirect
300. C₄	C₂²⋊(C₁₁⋊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⋊(C₁₁⋊C₁₆)	2
C₂×C₂²⋊(C₁₁⋊C₁₆)	2
#13066694	2
C₂²×C₄×C₈×C₁₁	2
#12783	11