Subgroups — #14725616

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	#13077621	C₂	1	2	yes	maximal, complemented
2	#13067530	C₂	1	1	yes	maximal, characteristic, complemented
3	#13067525	C₂	1	1	yes	maximal, characteristic, complemented
4	#13067495	C₂	1	2	yes	maximal, complemented
5	#13067474	C₂	1	2	yes	maximal, complemented
6	#13067440	C₂	1	1	yes	maximal, characteristic, complemented
7	#13067435	C₂	1	1	yes	maximal, characteristic, complemented
8	#13067364	C₂	1	1	yes	maximal, characteristic, complemented
9	#13067024	C₂	1	2	yes	maximal, complemented
10	#13066905	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₁×#3168	C₂	1	1	yes	maximal, characteristic
12	C₁₁×#529	C₂²	1	2	yes	—
13	C₁₁×#499	C₄	1	1	yes	characteristic
14	C₁₁×#495	C₄	1	1	yes	characteristic
15	C₁₁×#488	C₄	1	2	yes	—
16	C₁₁×#467	C₄	1	1	yes	characteristic
17	C₁₁×#466	C₄	1	1	yes	characteristic
18	C₁₁×#465	C₄	1	2	yes	—
19	C₄×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
20	C₄×C₁₁×SD₁₆	C₂²	1	2	yes	—
21	C₄×C₁₁×D₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
23	#10622637	C₂²	1	2	yes	—
24	Q₈×C₁₁⋊C₈	C₂²	1	2	yes	—
25	#10622625	—	2	2	no	—
26	#10622625	C₂²	1	2	yes	—
27	#10622624	—	2	2	no	—
28	D₄×C₁₁⋊C₈	C₂²	1	2	yes	—
29	#10622613	—	2	2	no	—
30	#10622610	—	2	2	no	—
31	#10622538	C₂²	1	2	yes	—
32	#10622538	C₂²	1	1	yes	characteristic
33	C₄×#4318	C₂²	1	1	yes	characteristic
34	Q₈⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
35	Q₈⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
36	D₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
37	D₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
38	#10622096	C₂²	1	2	yes	—
39	#10622095	C₂²	1	1	yes	characteristic
40	#10622095	C₂²	1	1	yes	characteristic
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	C₂×C₁₁×Q₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₁×SD₁₆	C₂×C₄	1	2	yes	—
47	C₂×C₁₁×D₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
50	C₁₁×#126	—	2	2	no	—
51	C₁₁×#125	C₂×C₄	1	2	yes	—
52	C₁₁×#124	C₂×C₄	1	2	yes	—
53	C₁₁×#123	—	2	2	no	—
54	C₁₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
55	C₁₁×#121	C₂×C₄	1	2	yes	—
56	C₄×Q₈×C₁₁	C₂³	1	2	yes	—
57	C₄×D₄×C₁₁	—	2	2	no	—
58	C₄×D₄×C₁₁	C₂³	1	2	yes	—
59	C₁₁×#117	—	2	2	no	—
60	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
61	C₂×#4318	—	2	2	no	—
62	C₂×#4318	D₄	1	2	yes	—
63	C₂²×C₁₁⋊C₈	—	2	2	no	—
64	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
65	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
66	C₁₁×#106	C₂×C₄	1	2	yes	—
67	C₁₁×#103	C₂×C₄	1	2	yes	—
68	C₁₁×#103	C₂×C₄	1	2	yes	—
69	C₁₁×#102	C₂×C₄	1	2	yes	—
70	C₁₁×#102	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
75	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
76	C₂²⋊(C₁₁⋊C₈)	—	2	1	no	—
77	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
78	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
79	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
80	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
81	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
82	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
83	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
84	#66867	—	2	2	no	—
85	#66867	D₄	1	2	yes	—
86	C₄×C₁₁⋊C₈	—	2	2	no	—
87	C₄×C₁₁⋊C₈	—	2	2	no	—
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	C₂×Q₈×C₁₁	C₂²×C₄	1	2	yes	—
94	C₂×D₄×C₁₁	—	2	2	no	—
95	C₂×D₄×C₁₁	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₁₁	—	2	2	no	—
97	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
100	C₁₁×Q₁₆	—	2	2	no	—
101	C₁₁×SD₁₆	—	4	2	no	—
102	C₁₁×D₈	—	2	2	no	—
103	C₂×C₈×C₁₁	—	2	2	no	—
104	C₂×C₈×C₁₁	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
109	C₁₁×C₄⋊C₄	—	2	2	no	—
110	C₁₁×C₄⋊C₄	—	2	2	no	—
111	C₁₁×C₄⋊C₄	—	2	2	no	—
112	C₁₁×C₄⋊C₄	—	2	2	no	—
113	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
114	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
115	C₁₁×C₂²⋊C₄	—	2	2	no	—
116	C₁₁×C₂²⋊C₄	—	2	2	no	—
117	C₁₁×C₂²⋊C₄	—	2	2	no	—
118	C₁₁×C₂²⋊C₄	—	2	2	no	—
119	C₁₁×C₂²⋊C₄	—	2	2	no	—
120	C₄²×C₁₁	—	2	2	no	—
121	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₁₁	#41	1	1	yes	characteristic
124	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
125	#4318	—	4	2	no	—
126	#4318	—	4	4	no	—
127	C₂×C₁₁⋊C₈	—	2	2	no	—
128	C₂×C₁₁⋊C₈	—	2	2	no	—
129	C₂×C₁₁⋊C₈	—	2	1	no	—
130	C₂×C₁₁⋊C₈	—	2	1	no	—
131	C₂×C₁₁⋊C₈	—	4	4	no	—
132	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
133	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
134	#3168	D₁₁	1	1	yes	characteristic
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	—	11	1	no	—
143	#1482	—	11	2	no	—
144	#1451	—	11	1	no	—
145	C₂³×C₁₁	—	2	2	no	—
146	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
147	Q₈×C₁₁	—	2	2	no	—
148	Q₈×C₁₁	—	2	2	no	—
149	D₄×C₁₁	—	2	2	no	—
150	D₄×C₁₁	—	2	2	no	—
151	D₄×C₁₁	—	8	2	no	—
152	C₂×C₄×C₁₁	—	2	1	no	—
153	C₂×C₄×C₁₁	—	2	1	no	—
154	C₂×C₄×C₁₁	—	2	2	no	—
155	C₂×C₄×C₁₁	—	2	2	no	—
156	C₂×C₄×C₁₁	—	2	2	no	—
157	C₂×C₄×C₁₁	—	2	2	no	—
158	C₂×C₄×C₁₁	—	4	1	no	—
159	C₂×C₄×C₁₁	—	4	2	no	—
160	C₂×C₄×C₁₁	—	4	2	no	—
161	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
167	C₈×C₁₁	—	2	1	no	—
168	C₈×C₁₁	—	2	1	no	—
169	C₈×C₁₁	—	2	1	no	—
170	C₈×C₁₁	#131	1	2	yes	—
171	C₁₁⋊C₈	—	2	2	no	—
172	C₁₁⋊C₈	—	2	2	no	—
173	C₁₁⋊C₈	—	4	1	no	—
174	C₁₁⋊C₈	—	4	1	no	—
175	C₁₁⋊C₈	—	4	2	no	—
176	C₁₁⋊C₈	—	4	2	no	—
177	#529	C₂×D₁₁	1	2	yes	—
178	#499	Dic₁₁	1	1	yes	characteristic
179	#495	Dic₁₁	1	1	yes	characteristic
180	#488	Dic₁₁	1	2	yes	—
181	#467	Dic₁₁	1	1	yes	characteristic
182	#466	Dic₁₁	1	1	yes	characteristic
183	#465	Dic₁₁	1	2	yes	—
184	#446	—	11	2	no	—
185	C₈×Q₈	—	11	2	no	—
186	C₄×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
187	C₄×SD₁₆	C₂×D₁₁	1	2	yes	—
188	C₄×D₈	C₂×D₁₁	1	1	yes	characteristic
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	—	11	2	no	—
196	#405	—	22	2	no	—
197	C₄×M₁₆	—	11	1	no	—
198	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
199	#335	—	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	—	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	—	4	2	no	—
214	C₂²×C₁₁	—	4	2	no	—
215	C₂²×C₁₁	—	4	2	no	—
216	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
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	—	4	2	no	—
222	C₄×C₁₁	—	4	2	no	—
223	C₄×C₁₁	—	4	2	no	—
224	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
225	C₄×C₁₁	#568	1	1	yes	characteristic
226	C₄×C₁₁	#575	1	2	yes	—
227	C₂×Q₁₆	C₂×Dic₁₁	1	1	yes	characteristic
228	C₂×SD₁₆	C₂×Dic₁₁	1	2	yes	—
229	C₂×D₈	C₂×Dic₁₁	1	1	yes	characteristic
230	C₂×M₁₆	—	11	2	no	—
231	C₂×M₁₆	—	22	2	no	—
232	C₂²×C₈	—	22	2	no	—
233	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
234	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
235	#126	—	2	2	no	—
236	#125	C₂×Dic₁₁	1	2	yes	—
237	#124	C₂×Dic₁₁	1	2	yes	—
238	#123	—	2	2	no	—
239	C₂²⋊Q₈	C₂×Dic₁₁	1	2	yes	—
240	#121	C₂×Dic₁₁	1	2	yes	—
241	C₄×Q₈	C₂²×D₁₁	1	2	yes	—
242	C₄×D₄	—	2	2	no	—
243	C₄×D₄	C₂²×D₁₁	1	2	yes	—
244	#117	—	2	2	no	—
245	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
246	#107	C₂×Dic₁₁	1	1	yes	characteristic
247	#107	C₂×Dic₁₁	1	1	yes	characteristic
248	#106	C₂×Dic₁₁	1	2	yes	—
249	C₄⋊C₈	—	11	2	no	—
250	C₄⋊C₈	—	11	2	no	—
251	C₄⋊C₈	—	22	2	no	—
252	C₄⋊C₈	—	22	2	no	—
253	C₄⋊C₈	—	22	1	no	—
254	#103	C₂×Dic₁₁	1	2	yes	—
255	#103	C₂×Dic₁₁	1	2	yes	—
256	#102	C₂×Dic₁₁	1	2	yes	—
257	#102	C₂×Dic₁₁	1	2	yes	—
258	C₂²⋊C₈	—	22	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	—	11	1	no	—
265	C₄×C₈	—	11	1	no	—
266	C₄×C₈	—	11	1	no	—
267	C₄×C₈	—	11	1	no	—
268	C₄×C₈	—	22	2	no	—
269	C₄×C₈	—	22	2	no	—
270	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
271	C₄×C₈	C₂×Dic₁₁	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	—	8	2	no	—
276	C₂×C₁₁	#2935	1	1	yes	characteristic
277	C₂×C₁₁	#2943	1	1	yes	characteristic
278	C₂×C₁₁	#2957	1	1	yes	characteristic
279	C₂×Q₈	C₂²×Dic₁₁	1	2	yes	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	C₂²×Dic₁₁	1	2	yes	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
284	C₂²×C₄	#4341	1	1	yes	characteristic
285	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
286	Q₁₆	—	2	2	no	—
287	SD₁₆	—	4	2	no	—
288	D₈	—	2	2	no	—
289	M₁₆	—	44	2	no	—
290	M₁₆	—	44	4	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	4	1	no	—
293	C₂×C₈	—	11	2	no	—
294	C₂×C₈	—	11	2	no	—
295	C₂×C₈	—	22	2	no	—
296	C₂×C₈	—	22	2	no	—
297	C₂×C₈	—	22	1	no	—
298	C₂×C₈	—	22	1	no	—
299	C₂×C₈	—	44	4	no	—
300	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
301	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
302	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
303	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	C₂²×Dic₁₁	1	2	yes	—
309	C₄⋊C₄	C₂²×Dic₁₁	1	2	yes	—
310	C₂²⋊C₄	—	2	2	no	—
311	C₂²⋊C₄	—	2	2	no	—
312	C₂²⋊C₄	—	2	2	no	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	D₄×D₁₁	1	1	yes	characteristic
317	C₄²	#4341	1	1	yes	characteristic
318	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
319	C₄²	C₂³×D₁₁	1	1	yes	characteristic
320	C₁₁	#18075	1	1	yes	characteristic, complemented, Sylow
321	C₂³	—	2	2	no	—
322	C₂³	D₄×Dic₁₁	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	8	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
338	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
339	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
341	C₂×C₄	C₂×#4341	1	1	yes	characteristic
342	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	22	2	no	—
347	C₈	—	22	2	no	—
348	C₈	—	44	1	no	—
349	C₈	—	44	1	no	—
350	C₈	—	44	2	no	—
351	C₈	—	44	2	no	—
352	C₈	#67001	1	2	yes	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	C₂×D₄×Dic₁₁	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	—	4	2	no	—
365	C₄	—	4	2	no	—
366	C₄	—	4	2	no	—
367	C₄	#10623299	1	2	yes	—
368	C₄	C₂×#67001	1	1	yes	characteristic
369	C₄	#10623350	1	1	yes	characteristic
370	C₂	—	4	1	no	—
371	C₂	—	8	2	no	—
372	C₂	#13077360	1	1	yes	characteristic
373	C₂	#13077417	1	1	yes	characteristic
374	C₂	#13077533	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×#3168	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₁₁×#466	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#3168	D₁₁
hypocenter	C₁₁	#18075
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
1. #13077621	C₂	semidirect
2. #13067530	C₂	semidirect
3. #13067525	C₂	semidirect
4. #13067495	C₂	semidirect
5. #13067474	C₂	semidirect
6. #13067440	C₂	semidirect
7. #13067435	C₂	semidirect
8. #13067364	C₂	semidirect
9. #13067024	C₂	semidirect
10. #13066905	C₂	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
#13077621	2
#13067530	2
#13067525	2
#13067495	2
#13067474	2
#13067440	2
#13067435	2
#13067364	2
#13067024	2
#13066905	2
C₁₁×#3168	2
#18075	11