Subgroups — #14725614

Order 2816.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079186	C₂	1	1	yes	maximal, characteristic
2	#13077614	C₂	1	2	yes	maximal, complemented
3	#13067529	C₂	1	1	yes	maximal, characteristic, complemented
4	#13067524	C₂	1	1	yes	maximal, characteristic, complemented
5	#13067493	C₂	1	2	yes	maximal, complemented
6	#13067472	C₂	1	2	yes	maximal, complemented
7	#13067439	C₂	1	1	yes	maximal, characteristic, complemented
8	#13067434	C₂	1	1	yes	maximal, characteristic, complemented
9	#13067022	C₂	1	2	yes	maximal, complemented
10	#13066905	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623010	C₂²	1	2	yes	—
13	C₁₁×#431	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	#10622729	C₄	1	1	yes	characteristic
16	#10622728	C₄	1	1	yes	characteristic
17	#10622727	C₄	1	2	yes	—
18	#10622538	C₂²	1	1	yes	characteristic
19	C₄×#4318	C₂²	1	1	yes	characteristic
20	#10622451	C₂²	1	2	yes	—
21	#10622447	C₂²	1	2	yes	—
22	#10622444	C₂²	1	2	yes	—
23	#10622443	C₂²	1	2	yes	—
24	#10622342	—	2	2	no	—
25	#10622341	—	2	2	no	—
26	C₄×Dic₄₄	C₂²	1	1	yes	characteristic
27	#10622310	C₄	1	2	yes	—
28	#10622309	C₄	1	1	yes	characteristic
29	C₄×D₈₈	C₂²	1	1	yes	characteristic
30	C₄×#4314	C₂²	1	2	yes	—
31	#10622303	—	2	2	no	—
32	#10622302	—	2	2	no	—
33	#10622296	C₄	1	1	yes	characteristic
34	C₁₁×#335	C₂²	1	2	yes	—
35	C₁₁×#334	C₂²	1	1	yes	characteristic
36	C₁₁×#334	C₂²	1	1	yes	characteristic
37	#10622122	C₂²	1	2	yes	—
38	#10622122	C₂²	1	2	yes	—
39	#10622121	C₂²	1	2	yes	—
40	#10622121	C₂²	1	2	yes	—
41	#10622093	C₂²	1	1	yes	characteristic
42	#10622093	C₂²	1	1	yes	characteristic
43	#10622092	C₂²	1	1	yes	characteristic
44	#10622092	C₂²	1	1	yes	characteristic
45	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
48	#66983	C₂×C₄	1	2	yes	—
49	#66982	—	2	2	no	—
50	C₄×C₂²⋊D₁₁	—	2	2	no	—
51	#66977	C₂×C₄	1	2	yes	—
52	C₂×#4318	D₄	1	2	yes	—
53	C₂×Dic₄₄	C₂×C₄	1	1	yes	characteristic
54	C₂×D₈₈	C₂×C₄	1	1	yes	characteristic
55	C₂×#4314	C₂×C₄	1	2	yes	—
56	C₂×#4313	—	2	2	no	—
57	C₂×C₈×D₁₁	—	2	2	no	—
58	#66929	—	2	2	no	—
59	#66928	C₂×C₄	1	2	yes	—
60	C₄×D₄₄	C₂³	1	2	yes	—
61	#66925	—	2	2	no	—
62	#66923	C₂×C₄	1	2	yes	—
63	C₄×Dic₂₂	C₂³	1	2	yes	—
64	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
65	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
66	C₁₁×C₂²⋊C₈	—	2	2	no	—
67	C₄×C₈×C₁₁	—	2	2	no	—
68	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
72	C₂²⋊(C₁₁⋊C₈)	—	2	1	no	—
73	#66885	C₂×C₄	1	2	yes	—
74	#66885	C₂×C₄	1	2	yes	—
75	(C₂×D₁₁)⋊C₈	—	2	2	no	—
76	(C₂×D₁₁)⋊C₈	—	2	2	no	—
77	#66882	C₂×C₄	1	1	yes	characteristic
78	#66882	C₂×C₄	1	1	yes	characteristic
79	#66881	C₂×C₄	1	2	yes	—
80	#66880	C₂×C₄	1	2	yes	—
81	#66880	C₂×C₄	1	2	yes	—
82	#66879	—	2	2	no	—
83	Dic₁₁⋊C₈	—	2	2	no	—
84	Dic₁₁⋊C₈	—	2	2	no	—
85	C₈×Dic₁₁	—	2	2	no	—
86	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
87	#66867	D₄	1	2	yes	—
88	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
89	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
90	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
91	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
92	#18075	—	11	1	no	maximal, Sylow
93	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
96	C₂×C₂²⋊D₁₁	—	2	2	no	—
97	C₂×D₄₄	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₁₁	—	2	2	no	—
99	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₁₁	—	2	2	no	—
101	C₂×C₈×C₁₁	—	2	2	no	—
102	C₂×C₈×C₁₁	—	2	2	no	—
103	C₂×C₈×C₁₁	—	4	1	no	—
104	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₁	#41	1	1	yes	characteristic
111	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
112	C₂²⋊Dic₁₁	—	2	2	no	—
113	(C₂×D₁₁)⋊C₄	—	2	2	no	—
114	(C₂×D₁₁)⋊C₄	—	2	2	no	—
115	(C₂×D₁₁)⋊C₄	—	2	2	no	—
116	(C₂×D₁₁)⋊C₄	—	2	2	no	—
117	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
118	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
119	Dic₁₁⋊C₄	—	2	2	no	—
120	Dic₁₁⋊C₄	—	2	2	no	—
121	Dic₁₁⋊C₄	—	2	2	no	—
122	Dic₁₁⋊C₄	—	2	2	no	—
123	C₄×Dic₁₁	—	2	2	no	—
124	#4318	—	4	2	no	—
125	C₂×C₁₁⋊C₈	—	2	1	no	—
126	C₂×C₁₁⋊C₈	—	2	1	no	—
127	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
128	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
129	Dic₄₄	—	2	2	no	—
130	D₈₈	—	2	2	no	—
131	#4314	—	4	2	no	—
132	#4313	—	4	4	no	—
133	C₈×D₁₁	—	4	4	no	—
134	#3168	—	11	1	no	—
135	#2969	—	11	2	no	—
136	#1594	—	11	1	no	—
137	#1593	—	11	1	no	—
138	#1588	—	11	2	no	—
139	#1585	—	11	2	no	—
140	#1580	—	11	1	no	—
141	#1579	—	11	1	no	—
142	#1564	D₁₁	1	1	yes	characteristic, complemented
143	#1482	—	11	2	no	—
144	#1451	—	11	1	no	—
145	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
146	C₂²×D₁₁	—	2	2	no	—
147	C₂×C₄×C₁₁	—	2	1	no	—
148	C₂×C₄×C₁₁	—	2	1	no	—
149	C₂×C₄×C₁₁	—	2	2	no	—
150	C₂×C₄×C₁₁	—	4	1	no	—
151	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
157	C₂²⋊D₁₁	—	8	2	no	—
158	C₂×Dic₁₁	—	2	2	no	—
159	C₂×Dic₁₁	—	2	2	no	—
160	C₂×Dic₁₁	—	2	2	no	—
161	D₄₄	—	2	2	no	—
162	D₄₄	—	2	2	no	—
163	C₄×D₁₁	—	4	2	no	—
164	C₄×D₁₁	—	4	2	no	—
165	Dic₂₂	—	2	2	no	—
166	Dic₂₂	—	2	2	no	—
167	C₈×C₁₁	—	2	1	no	—
168	C₈×C₁₁	—	2	1	no	—
169	C₈×C₁₁	—	2	1	no	—
170	C₈×C₁₁	—	4	2	no	—
171	C₈×C₁₁	—	4	2	no	—
172	C₈×C₁₁	#131	1	2	yes	—
173	C₁₁⋊C₈	—	2	2	no	—
174	C₁₁⋊C₈	—	2	2	no	—
175	C₁₁⋊C₈	—	4	1	no	—
176	C₁₁⋊C₈	—	4	1	no	—
177	#529	—	11	2	no	—
178	#499	—	11	1	no	—
179	#495	—	11	1	no	—
180	#488	—	11	2	no	—
181	#467	—	11	1	no	—
182	#466	—	11	1	no	—
183	#465	—	11	2	no	—
184	#446	—	11	2	no	—
185	C₈×Q₈	—	11	2	no	—
186	C₄×Q₁₆	—	11	1	no	—
187	C₄×SD₁₆	—	11	2	no	—
188	C₄×D₈	—	11	1	no	—
189	#436	—	11	2	no	—
190	#436	—	22	2	no	—
191	#435	—	22	2	no	—
192	C₈×D₄	—	11	2	no	—
193	#433	—	22	2	no	—
194	#431	—	11	1	no	—
195	#431	C₂×D₁₁	1	2	yes	—
196	#405	—	22	2	no	—
197	C₄×M₁₆	—	11	1	no	—
198	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
199	#335	C₂×D₁₁	1	2	yes	—
200	#334	C₂×D₁₁	1	1	yes	characteristic
201	#334	C₂×D₁₁	1	1	yes	characteristic
202	Q₈⋊C₈	—	11	2	no	—
203	Q₈⋊C₈	—	11	2	no	—
204	D₄⋊C₈	—	11	2	no	—
205	D₄⋊C₈	—	11	2	no	—
206	#322	—	11	1	no	—
207	#322	—	11	1	no	—
208	#322	—	11	1	no	—
209	#322	—	11	1	no	—
210	C₂²×C₁₁	—	2	1	no	—
211	C₂²×C₁₁	—	2	1	no	—
212	C₂²×C₁₁	—	2	1	no	—
213	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
214	C₂×D₁₁	—	4	2	no	—
215	C₂×D₁₁	—	4	2	no	—
216	C₂×D₁₁	—	4	2	no	—
217	C₄×C₁₁	—	2	1	no	—
218	C₄×C₁₁	—	2	1	no	—
219	C₄×C₁₁	—	2	1	no	—
220	C₄×C₁₁	—	2	1	no	—
221	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
222	C₄×C₁₁	#568	1	1	yes	characteristic
223	C₄×C₁₁	#575	1	2	yes	—
224	Dic₁₁	—	4	2	no	—
225	Dic₁₁	—	4	2	no	—
226	Dic₁₁	—	4	2	no	—
227	C₂×Q₁₆	—	11	1	no	—
228	C₂×SD₁₆	—	11	2	no	—
229	C₂×D₈	—	11	1	no	—
230	C₂×M₁₆	—	11	2	no	—
231	C₂×M₁₆	—	22	2	no	—
232	C₂²×C₈	—	22	2	no	—
233	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
234	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
235	#126	—	22	2	no	—
236	#125	—	11	2	no	—
237	#124	—	11	2	no	—
238	#123	—	22	2	no	—
239	C₂²⋊Q₈	—	11	2	no	—
240	#121	—	11	2	no	—
241	C₄×Q₈	—	11	2	no	—
242	C₄×D₄	—	11	2	no	—
243	C₄×D₄	—	22	2	no	—
244	#117	—	22	2	no	—
245	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
246	#107	—	11	1	no	—
247	#107	—	11	1	no	—
248	#106	—	11	2	no	—
249	C₄⋊C₈	—	22	2	no	—
250	C₄⋊C₈	—	22	2	no	—
251	C₄⋊C₈	—	22	1	no	—
252	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
253	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
254	#103	—	11	2	no	—
255	#103	—	11	2	no	—
256	#102	—	11	2	no	—
257	#102	—	11	2	no	—
258	C₂²⋊C₈	—	2	2	no	—
259	C₂²⋊C₈	—	22	2	no	—
260	C₂²⋊C₈	—	22	2	no	—
261	C₂²⋊C₈	—	22	1	no	—
262	#97	—	11	2	no	—
263	#97	—	22	2	no	—
264	C₄×C₈	—	2	2	no	—
265	C₄×C₈	—	11	1	no	—
266	C₄×C₈	—	11	1	no	—
267	C₄×C₈	—	11	1	no	—
268	C₄×C₈	—	11	1	no	—
269	C₄×C₈	—	22	2	no	—
270	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
271	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
274	C₂×C₁₁	—	4	1	no	—
275	C₂×C₁₁	#2935	1	1	yes	characteristic
276	C₂×C₁₁	#2943	1	1	yes	characteristic
277	C₂×C₁₁	#2957	1	1	yes	characteristic
278	D₁₁	—	8	2	no	—
279	C₂×Q₈	—	11	2	no	—
280	C₂×D₄	—	11	2	no	—
281	C₂×D₄	—	22	2	no	—
282	C₂²×C₄	—	22	2	no	—
283	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
284	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
285	C₂²×C₄	#4343	1	1	yes	characteristic
286	Q₁₆	—	22	2	no	—
287	SD₁₆	—	44	2	no	—
288	D₈	—	22	2	no	—
289	M₁₆	—	44	4	no	—
290	M₁₆	—	44	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	—	11	2	no	—
296	C₂×C₈	—	11	2	no	—
297	C₂×C₈	—	22	1	no	—
298	C₂×C₈	—	22	1	no	—
299	C₂×C₈	—	44	4	no	—
300	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
304	C₄⋊C₄	—	11	2	no	—
305	C₄⋊C₄	—	11	2	no	—
306	C₄⋊C₄	—	22	2	no	—
307	C₄⋊C₄	—	22	2	no	—
308	C₄⋊C₄	—	22	2	no	—
309	C₄⋊C₄	—	22	2	no	—
310	C₂²⋊C₄	—	22	2	no	—
311	C₂²⋊C₄	—	22	2	no	—
312	C₂²⋊C₄	—	22	2	no	—
313	C₂²⋊C₄	—	22	2	no	—
314	C₂²⋊C₄	—	22	2	no	—
315	C₄²	—	22	2	no	—
316	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
317	C₄²	D₄×D₁₁	1	1	yes	characteristic
318	C₄²	#4343	1	1	yes	characteristic
319	C₄²	C₂³×D₁₁	1	1	yes	characteristic
320	C₁₁	#18075	1	1	yes	characteristic, complemented, Sylow
321	C₂³	—	22	2	no	—
322	C₂³	#66946	1	1	yes	characteristic
323	Q₈	—	22	2	no	—
324	Q₈	—	22	2	no	—
325	D₄	—	22	2	no	—
326	D₄	—	22	2	no	—
327	D₄	—	88	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	22	2	no	—
333	C₂×C₄	—	22	2	no	—
334	C₂×C₄	—	22	2	no	—
335	C₂×C₄	—	44	2	no	—
336	C₂×C₄	—	44	2	no	—
337	C₂×C₄	#66946	1	1	yes	characteristic
338	C₂×C₄	#66946	1	1	yes	characteristic
339	C₂×C₄	#66946	1	1	yes	characteristic
340	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
342	C₂×C₄	C₂×#4343	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	4	2	no	—
347	C₈	—	4	2	no	—
348	C₈	—	22	2	no	—
349	C₈	—	22	2	no	—
350	C₈	—	44	1	no	—
351	C₈	—	44	1	no	—
352	C₈	#66954	1	2	yes	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	44	2	no	—
357	C₂²	—	44	2	no	—
358	C₂²	—	44	2	no	—
359	C₂²	C₂×#66946	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	44	2	no	—
365	C₄	—	44	2	no	—
366	C₄	—	44	2	no	—
367	C₄	C₂×#66954	1	1	yes	characteristic
368	C₄	#10623276	1	1	yes	characteristic
369	C₄	#10623299	1	2	yes	—
370	C₂	—	4	1	no	—
371	C₂	—	88	2	no	—
372	C₂	#13077359	1	1	yes	characteristic
373	C₂	#13077416	1	1	yes	characteristic
374	C₂	#13077532	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	C₂×C₄×D₄
prime radical	#10622728	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1564	D₁₁
hypocenter	C₁₁	#18075
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
2. #13077614	C₂	semidirect
3. #13067529	C₂	semidirect
4. #13067524	C₂	semidirect
5. #13067493	C₂	semidirect
6. #13067472	C₂	semidirect
7. #13067439	C₂	semidirect
8. #13067434	C₂	semidirect
9. #13067022	C₂	semidirect
10. #13066905	C₂	semidirect
11. C₁₁×#1564	C₂	semidirect
142. #1564	D₁₁	semidirect
320. C₁₁	#18075	semidirect

Maximal subgroups

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

subgroup	index
#13079186	2
#13077614	2
#13067529	2
#13067524	2
#13067493	2
#13067472	2
#13067439	2
#13067434	2
#13067022	2
#13066905	2
C₁₁×#1564	2
#18075	11