Subgroups — #14724475

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	#13077767	C₂	1	2	yes	maximal, complemented
3	#13067531	C₂	1	1	yes	maximal, characteristic, complemented
4	#13067526	C₂	1	1	yes	maximal, characteristic, complemented
5	#13067508	C₂	1	2	yes	maximal, complemented
6	#13067441	C₂	1	1	yes	maximal, characteristic, complemented
7	#13067436	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#4314	C₂	1	2	yes	maximal, complemented
9	#13067372	C₂	1	1	yes	maximal, characteristic, complemented
10	#13067082	C₂	1	2	yes	maximal, complemented
11	C₁₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623010	C₂²	1	2	yes	—
13	C₁₁×#431	C₂²	1	1	yes	characteristic
14	C₁₁×#431	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
16	#10622729	C₄	1	1	yes	characteristic
17	#10622728	C₄	1	1	yes	characteristic
18	#10622727	C₄	1	2	yes	—
19	#10622537	C₂²	1	2	yes	—
20	#10622453	—	2	2	no	—
21	#10622452	—	2	2	no	—
22	#10622342	—	2	2	no	—
23	#10622341	—	2	2	no	—
24	C₄×Dic₄₄	C₂²	1	1	yes	characteristic
25	#10622310	C₄	1	2	yes	—
26	#10622309	C₄	1	1	yes	characteristic
27	C₄×D₈₈	C₂²	1	1	yes	characteristic
28	C₄×#4314	C₂²	1	2	yes	—
29	#10622301	C₂²	1	2	yes	—
30	C₈×D₄₄	C₂²	1	2	yes	—
31	#10622296	C₄	1	1	yes	characteristic
32	#10622292	C₂²	1	2	yes	—
33	C₈×Dic₂₂	C₂²	1	2	yes	—
34	C₁₁×#322	C₂²	1	1	yes	characteristic
35	C₁₁×#322	C₂²	1	1	yes	characteristic
36	C₈²×C₁₁	C₂²	1	1	yes	characteristic
37	C₈²×C₁₁	C₂²	1	1	yes	characteristic
38	#10622097	C₂²	1	2	yes	—
39	#10622097	C₂²	1	2	yes	—
40	#10622095	C₂²	1	1	yes	characteristic
41	#10622095	C₂²	1	1	yes	characteristic
42	#10622095	C₂²	1	2	yes	—
43	#10622094	C₂²	1	2	yes	—
44	#10622094	C₂²	1	2	yes	—
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	—	2	2	no	—
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	—	2	1	no	—
65	C₁₁×C₄⋊C₈	D₄	1	2	yes	—
66	C₁₁×C₂²⋊C₈	—	2	1	no	—
67	C₁₁×C₂²⋊C₈	D₄	1	2	yes	—
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	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
76	#66885	C₂×C₄	1	2	yes	—
77	#66885	C₂×C₄	1	2	yes	—
78	(C₂×D₁₁)⋊C₈	—	2	2	no	—
79	(C₂×D₁₁)⋊C₈	—	2	2	no	—
80	#66882	C₂×C₄	1	1	yes	characteristic
81	#66882	C₂×C₄	1	1	yes	characteristic
82	#66881	C₂×C₄	1	2	yes	—
83	#66880	C₂×C₄	1	2	yes	—
84	#66880	C₂×C₄	1	2	yes	—
85	#66879	—	2	2	no	—
86	Dic₁₁⋊C₈	—	2	2	no	—
87	Dic₁₁⋊C₈	—	2	2	no	—
88	C₈×Dic₁₁	—	2	2	no	—
89	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
90	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
91	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
92	#17974	—	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	1	no	—
103	C₂×C₈×C₁₁	—	2	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	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
110	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₁	#41	1	1	yes	characteristic
113	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
114	C₂²⋊Dic₁₁	—	2	2	no	—
115	(C₂×D₁₁)⋊C₄	—	2	2	no	—
116	(C₂×D₁₁)⋊C₄	—	2	2	no	—
117	(C₂×D₁₁)⋊C₄	—	2	2	no	—
118	(C₂×D₁₁)⋊C₄	—	2	2	no	—
119	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
120	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
121	Dic₁₁⋊C₄	—	2	2	no	—
122	Dic₁₁⋊C₄	—	2	2	no	—
123	Dic₁₁⋊C₄	—	2	2	no	—
124	Dic₁₁⋊C₄	—	2	2	no	—
125	C₄×Dic₁₁	—	2	2	no	—
126	#4318	—	8	2	no	—
127	C₂×C₁₁⋊C₈	—	2	2	no	—
128	C₂×C₁₁⋊C₈	—	2	2	no	—
129	Dic₄₄	—	2	2	no	—
130	D₈₈	—	2	2	no	—
131	#4314	—	2	4	no	—
132	#4313	—	8	2	no	—
133	C₈×D₁₁	—	8	2	no	—
134	#3168	—	11	1	no	—
135	#2993	—	11	2	no	—
136	#1594	—	11	1	no	—
137	#1593	—	11	1	no	—
138	#1590	—	11	2	no	—
139	#1580	—	11	1	no	—
140	#1579	—	11	1	no	—
141	C₈×SD₁₆	—	11	2	no	—
142	#1566	—	11	1	no	—
143	#1495	—	11	2	no	—
144	#1450	D₁₁	1	1	yes	characteristic, complemented
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	2	no	—
170	C₈×C₁₁	—	2	2	no	—
171	C₈×C₁₁	—	4	1	no	—
172	C₈×C₁₁	—	4	1	no	—
173	C₈×C₁₁	#131	1	2	yes	—
174	C₈×C₁₁	#131	1	2	yes	—
175	C₁₁⋊C₈	—	4	2	no	—
176	C₁₁⋊C₈	—	4	2	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	C₂×D₁₁	1	1	yes	characteristic
195	#431	C₂×D₁₁	1	1	yes	characteristic
196	#424	—	22	2	no	—
197	#423	—	11	2	no	—
198	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
199	#334	—	11	2	no	—
200	#334	—	11	1	no	—
201	#334	—	11	1	no	—
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	C₂×D₁₁	1	1	yes	characteristic
207	#322	C₂×D₁₁	1	1	yes	characteristic
208	C₈²	C₂×D₁₁	1	1	yes	characteristic
209	C₈²	C₂×D₁₁	1	1	yes	characteristic
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	C₂×#131	1	1	yes	characteristic
223	C₄×C₁₁	#577	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	—	22	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	—	2	1	no	—
250	C₄⋊C₈	—	11	2	no	—
251	C₄⋊C₈	—	11	2	no	—
252	C₄⋊C₈	—	22	2	no	—
253	C₄⋊C₈	—	22	2	no	—
254	C₄⋊C₈	—	22	2	no	—
255	C₄⋊C₈	D₄₄	1	2	yes	—
256	#103	—	11	2	no	—
257	#103	—	11	2	no	—
258	#102	—	11	2	no	—
259	#102	—	11	2	no	—
260	C₂²⋊C₈	—	2	1	no	—
261	C₂²⋊C₈	—	22	2	no	—
262	C₂²⋊C₈	—	22	2	no	—
263	C₂²⋊C₈	D₄₄	1	2	yes	—
264	#97	—	22	2	no	—
265	C₄×C₈	—	22	2	no	—
266	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
267	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
271	C₄×C₈	C₂²×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	#2931	1	1	yes	characteristic
276	C₂×C₁₁	#2952	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	C₂×D₄₄	1	1	yes	characteristic
285	C₂²×C₄	#4339	1	1	yes	characteristic
286	Q₁₆	—	22	2	no	—
287	SD₁₆	—	22	4	no	—
288	D₈	—	22	2	no	—
289	M₁₆	—	88	2	no	—
290	M₁₆	—	88	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	22	2	no	—
296	C₂×C₈	—	22	2	no	—
297	C₂×C₈	—	88	2	no	—
298	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
299	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
300	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
302	C₂×C₈	C₂×D₄₄	1	2	yes	—
303	C₂×C₈	C₂×D₄₄	1	2	yes	—
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	C₂×D₄₄	1	1	yes	characteristic
318	C₄²	#4339	1	1	yes	characteristic
319	C₄²	C₂³×D₁₁	1	1	yes	characteristic
320	C₁₁	#17974	1	1	yes	characteristic, complemented, Sylow
321	C₂³	—	22	2	no	—
322	C₂³	C₄×D₄₄	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	C₄×D₄₄	1	1	yes	characteristic
338	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
339	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
340	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
341	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
342	C₂×C₄	C₂×#4339	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	2	no	—
346	C₈	—	2	2	no	—
347	C₈	—	4	1	no	—
348	C₈	—	4	1	no	—
349	C₈	—	44	2	no	—
350	C₈	—	44	2	no	—
351	C₈	#66954	1	2	yes	—
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×C₄×D₄₄	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	C₂×#66954	1	1	yes	characteristic
369	C₄	#10623282	1	2	yes	—
370	C₂	—	4	1	no	—
371	C₂	—	88	2	no	—
372	C₂	#13077329	1	1	yes	characteristic
373	C₂	#13077501	1	1	yes	characteristic
374	C₂	#13077534	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₁₁×#1450	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	#10622728	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1450	D₁₁
hypocenter	C₁₁	#17974
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
2. #13077767	C₂	semidirect
3. #13067531	C₂	semidirect
4. #13067526	C₂	semidirect
5. #13067508	C₂	semidirect
6. #13067441	C₂	semidirect
7. #13067436	C₂	semidirect
8. C₈×#4314	C₂	semidirect
9. #13067372	C₂	semidirect
10. #13067082	C₂	semidirect
11. C₁₁×#1450	C₂	semidirect
144. #1450	D₁₁	semidirect
320. C₁₁	#17974	semidirect

Maximal subgroups

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

subgroup	index
#13079186	2
#13077767	2
#13067531	2
#13067526	2
#13067508	2
#13067441	2
#13067436	2
C₈×#4314	2
#13067372	2
#13067082	2
C₁₁×#1450	2
#17974	11