Subgroups — #14723068

Order 2816.

612 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077745	C₂	1	2	yes	maximal
2	C₈×#4339	C₂	1	1	yes	maximal, characteristic
3	#13067406	C₂	1	4	yes	maximal, complemented
4	#13067404	C₂	1	2	yes	maximal, complemented
5	#13067372	C₂	1	1	yes	maximal, characteristic, complemented
6	#13067370	C₂	1	2	yes	maximal, complemented
7	#13067361	C₂	1	2	yes	maximal, complemented
8	C₁₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#4339	C₂²	1	1	yes	characteristic
10	C₁₁×#431	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₈×C₂²⋊D₁₁	C₄	1	2	yes	—
13	C₈×C₂²⋊D₁₁	C₄	1	2	yes	—
14	#10622538	C₂²	1	1	yes	characteristic
15	#10622537	C₂²	1	2	yes	—
16	#10622450	C₂²	1	4	yes	—
17	#10622446	C₂²	1	4	yes	—
18	#10622342	—	2	4	no	—
19	#10622301	—	2	2	no	—
20	C₈×D₄₄	C₄	1	1	yes	characteristic
21	C₈×D₄₄	C₄	1	1	yes	characteristic
22	#10622298	C₂²	1	2	yes	—
23	C₄×C₈×D₁₁	C₂²	1	2	yes	—
24	C₈×Dic₂₂	—	2	2	no	—
25	C₈×Dic₂₂	C₄	1	1	yes	characteristic
26	C₈×Dic₂₂	C₄	1	1	yes	characteristic
27	C₁₁×#335	C₂²	1	2	yes	—
28	C₁₁×#334	C₂²	1	1	yes	characteristic
29	C₁₁×#334	C₂²	1	1	yes	characteristic
30	#10622118	C₂²	1	4	yes	—
31	#10622117	C₂²	1	4	yes	—
32	#10622095	C₂²	1	2	yes	—
33	#10622095	C₂²	1	2	yes	—
34	C₂×#4339	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
38	C₄×C₂²⋊D₁₁	C₂×C₄	1	2	yes	—
39	C₄×C₂²⋊D₁₁	D₄	1	2	yes	—
40	#66979	Q₈	1	1	yes	characteristic
41	C₂×#4318	—	2	2	no	—
42	C₂×#4313	—	2	4	no	—
43	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
45	C₄×D₄₄	C₂×C₄	1	1	yes	characteristic
46	C₄×D₄₄	D₄	1	1	yes	characteristic
47	#66925	Q₈	1	2	yes	—
48	C₄²×D₁₁	C₂³	1	2	yes	—
49	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₂₂	D₄	1	1	yes	characteristic
51	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
52	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
53	C₁₁×C₂²⋊C₈	—	2	2	no	—
54	C₄×C₈×C₁₁	—	2	2	no	—
55	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
59	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
60	(C₂×D₁₁)⋊C₈	—	2	4	no	—
61	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
62	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
63	Dic₁₁⋊C₈	—	2	4	no	—
64	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
65	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
66	C₈×Dic₁₁	—	2	4	no	—
67	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
68	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
69	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
70	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
71	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
72	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
73	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
74	#17835	—	11	1	no	maximal, Sylow
75	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₂²⋊D₁₁	C₄⋊C₄	1	2	yes	—
79	#4339	—	2	4	no	—
80	C₂×D₄₄	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
82	C₂×C₄×D₁₁	C₂×D₄	1	2	yes	—
83	C₂×C₄×D₁₁	C₂×Q₈	1	2	yes	—
84	C₂×Dic₂₂	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₁	—	2	2	no	—
86	C₂×C₈×C₁₁	—	2	2	no	—
87	C₂×C₈×C₁₁	—	2	2	no	—
88	C₂×C₈×C₁₁	—	4	1	no	—
89	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
96	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
97	C₂²⋊Dic₁₁	C₄⋊C₄	1	2	yes	—
98	(C₂×D₁₁)⋊C₄	C₄⋊C₄	1	2	yes	—
99	(C₂×D₁₁)⋊C₄	C₄⋊C₄	1	2	yes	—
100	C₄⋊Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
102	Dic₁₁⋊C₄	C₄⋊C₄	1	2	yes	—
103	Dic₁₁⋊C₄	C₄⋊C₄	1	2	yes	—
104	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
105	C₄×Dic₁₁	C₂×D₄	1	2	yes	—
106	C₄×Dic₁₁	C₂×Q₈	1	2	yes	—
107	#4318	—	8	2	no	—
108	C₂×C₁₁⋊C₈	—	2	4	no	—
109	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
110	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
111	#4313	—	8	4	no	—
112	C₈×D₁₁	—	2	4	no	—
113	C₈×D₁₁	—	4	2	no	—
114	#2991	—	11	2	no	—
115	C₈×#41	—	11	1	no	—
116	#1574	—	11	2	no	—
117	#1574	—	11	4	no	—
118	#1566	—	11	1	no	—
119	#1566	—	11	2	no	—
120	#1564	—	11	2	no	—
121	#1564	D₁₁	1	1	yes	characteristic, complemented
122	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×D₁₁	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₁₁	—	2	1	no	—
125	C₂×C₄×C₁₁	—	2	1	no	—
126	C₂×C₄×C₁₁	—	2	2	no	—
127	C₂×C₄×C₁₁	—	2	2	no	—
128	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
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	C₂²⋊D₁₁	—	2	4	no	—
135	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
138	D₄₄	—	2	2	no	—
139	C₄×D₁₁	—	2	2	no	—
140	C₄×D₁₁	—	2	2	no	—
141	C₄×D₁₁	—	2	4	no	—
142	C₄×D₁₁	—	2	4	no	—
143	Dic₂₂	#108	1	4	yes	—
144	C₈×C₁₁	—	2	1	no	—
145	C₈×C₁₁	—	2	1	no	—
146	C₈×C₁₁	—	2	1	no	—
147	C₈×C₁₁	—	4	2	no	—
148	C₈×C₁₁	—	4	2	no	—
149	C₈×C₁₁	#131	1	2	yes	—
150	C₁₁⋊C₈	—	2	2	no	—
151	C₁₁⋊C₈	—	2	2	no	—
152	C₁₁⋊C₈	—	4	4	no	—
153	C₄×#41	—	11	1	no	—
154	C₈×Q₈	—	11	1	no	—
155	C₈×Q₈	—	11	1	no	—
156	C₈×Q₈	—	22	2	no	—
157	#436	—	22	2	no	—
158	#436	—	22	4	no	—
159	C₈×D₄	—	11	2	no	—
160	C₈×D₄	—	11	2	no	—
161	C₈×D₄	—	11	1	no	—
162	C₈×D₄	—	11	1	no	—
163	#431	—	11	1	no	—
164	#431	—	11	2	no	—
165	#431	—	11	4	no	—
166	#431	C₂×D₁₁	1	2	yes	—
167	#423	—	11	2	no	—
168	#423	—	11	4	no	—
169	C₂×C₄×C₈	—	11	2	no	—
170	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
171	#335	—	11	4	no	—
172	#335	C₂×D₁₁	1	2	yes	—
173	#334	—	11	4	no	—
174	#334	—	11	2	no	—
175	#334	—	11	2	no	—
176	#334	C₂×D₁₁	1	1	yes	characteristic
177	#334	C₂×D₁₁	1	1	yes	characteristic
178	C₂²×C₁₁	—	2	1	no	—
179	C₂²×C₁₁	—	2	1	no	—
180	C₂²×C₁₁	—	2	1	no	—
181	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
182	C₂×D₁₁	—	2	2	no	—
183	C₂×D₁₁	—	2	2	no	—
184	C₂×D₁₁	—	2	2	no	—
185	C₄×C₁₁	—	2	1	no	—
186	C₄×C₁₁	—	2	1	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	C₂×#108	1	2	yes	—
189	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
190	C₄×C₁₁	#568	1	1	yes	characteristic
191	C₄×C₁₁	#577	1	2	yes	—
192	Dic₁₁	—	2	2	no	—
193	Dic₁₁	—	2	2	no	—
194	Dic₁₁	C₂×#108	1	4	yes	—
195	C₂×#41	—	11	1	no	—
196	C₂×M₁₆	—	22	4	no	—
197	C₂×M₁₆	—	22	2	no	—
198	C₂²×C₈	—	11	2	no	—
199	C₂²×C₈	—	11	2	no	—
200	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
201	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
202	C₄×Q₈	—	11	1	no	—
203	C₄×Q₈	—	11	1	no	—
204	C₄×D₄	—	11	2	no	—
205	C₄×D₄	—	11	2	no	—
206	C₄×D₄	—	11	1	no	—
207	C₄×D₄	—	11	1	no	—
208	#117	—	11	1	no	—
209	#117	—	11	2	no	—
210	C₂×C₄²	—	11	2	no	—
211	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
212	C₄⋊C₈	—	11	2	no	—
213	C₄⋊C₈	—	11	2	no	—
214	C₄⋊C₈	—	11	1	no	—
215	C₄⋊C₈	—	11	1	no	—
216	C₄⋊C₈	—	11	4	no	—
217	C₄⋊C₈	—	22	4	no	—
218	C₄⋊C₈	—	22	2	no	—
219	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
220	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
221	C₂²⋊C₈	—	2	2	no	—
222	C₂²⋊C₈	—	11	2	no	—
223	C₂²⋊C₈	—	11	2	no	—
224	C₂²⋊C₈	—	11	2	no	—
225	C₂²⋊C₈	—	22	4	no	—
226	C₄×C₈	—	2	2	no	—
227	C₄×C₈	—	11	2	no	—
228	C₄×C₈	—	11	2	no	—
229	C₄×C₈	—	11	2	no	—
230	C₄×C₈	—	22	4	no	—
231	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
232	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
233	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
235	C₂×C₁₁	—	4	1	no	—
236	C₂×C₁₁	#2906	1	1	yes	characteristic
237	C₂×C₁₁	#2912	1	1	yes	characteristic
238	C₂×C₁₁	C₂²×#108	1	1	yes	characteristic
239	D₁₁	—	4	2	no	—
240	#41	—	22	4	no	—
241	C₂×Q₈	—	11	1	no	—
242	C₂×D₄	—	11	2	no	—
243	C₂×D₄	—	11	1	no	—
244	C₂²×C₄	—	11	2	no	—
245	C₂²×C₄	—	11	2	no	—
246	C₂²×C₄	—	11	2	no	—
247	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
249	C₂²×C₄	Q₈×D₁₁	1	1	yes	characteristic
250	M₁₆	—	88	4	no	—
251	M₁₆	—	88	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	11	2	no	—
257	C₂×C₈	—	11	2	no	—
258	C₂×C₈	—	22	4	no	—
259	C₂×C₈	—	22	4	no	—
260	C₂×C₈	—	44	2	no	—
261	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
263	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
265	C₄⋊C₄	—	11	2	no	—
266	C₄⋊C₄	—	11	2	no	—
267	C₄⋊C₄	—	11	1	no	—
268	C₄⋊C₄	—	11	1	no	—
269	C₂²⋊C₄	—	11	2	no	—
270	C₂²⋊C₄	—	11	2	no	—
271	C₂²⋊C₄	—	11	2	no	—
272	C₄²	—	11	2	no	—
273	C₄²	—	11	2	no	—
274	C₄²	—	11	2	no	—
275	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
276	C₄²	D₄×D₁₁	1	1	yes	characteristic
277	C₄²	Q₈×D₁₁	1	1	yes	characteristic
278	C₄²	C₂³×D₁₁	1	1	yes	characteristic
279	C₁₁	#17835	1	1	yes	characteristic, complemented, Sylow
280	C₂³	—	11	2	no	—
281	C₂³	C₄⋊C₄×D₁₁	1	1	yes	characteristic
282	Q₈	—	11	4	no	—
283	D₄	—	22	4	no	—
284	D₄	—	22	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	11	2	no	—
290	C₂×C₄	—	11	2	no	—
291	C₂×C₄	—	11	2	no	—
292	C₂×C₄	—	22	4	no	—
293	C₂×C₄	—	22	2	no	—
294	C₂×C₄	—	22	2	no	—
295	C₂×C₄	—	22	4	no	—
296	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
297	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
298	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₁₁	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	4	2	no	—
306	C₈	—	4	2	no	—
307	C₈	—	22	2	no	—
308	C₈	—	22	2	no	—
309	C₈	—	44	4	no	—
310	C₈	#66954	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	22	2	no	—
315	C₂²	—	22	2	no	—
316	C₂²	—	22	2	no	—
317	C₂²	C₂×C₄⋊C₄×D₁₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	11	4	no	—
322	C₄	—	22	2	no	—
323	C₄	—	22	2	no	—
324	C₄	D₁₁×#108	1	2	yes	—
325	C₄	C₂×#66954	1	1	yes	characteristic
326	C₄	#10623276	1	1	yes	characteristic
327	C₄	#10623301	1	2	yes	—
328	C₂	—	4	1	no	—
329	C₂	—	44	2	no	—
330	C₂	#13077181	1	1	yes	characteristic
331	C₂	#13077208	1	1	yes	characteristic
332	C₂	C₂×D₁₁×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₁₁×#1564	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂×#4339	Q₈
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1564	D₁₁
hypocenter	C₁₁	#17835
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #13067406	C₂	semidirect
4. #13067404	C₂	semidirect
5. #13067372	C₂	semidirect
6. #13067370	C₂	semidirect
7. #13067361	C₂	semidirect
8. C₁₁×#1564	C₂	semidirect
121. #1564	D₁₁	semidirect
279. C₁₁	#17835	semidirect

Maximal subgroups

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

subgroup	index
#13077745	2
C₈×#4339	2
#13067406	2
#13067404	2
#13067372	2
#13067370	2
#13067361	2
C₁₁×#1564	2
#17835	11