Subgroups — #14725784

Order 2816.

606 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077776	C₂	1	2	yes	maximal, complemented
2	#13067507	C₂	1	2	yes	maximal, complemented
3	#13067488	C₂	1	2	yes	maximal, complemented
4	#13067481	C₂	1	2	yes	maximal, complemented
5	SD₁₆×C₁₁⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×#10622095	C₂	1	1	yes	maximal, characteristic, complemented
7	#13067029	C₂	1	2	yes	maximal, complemented
8	C₈×#4318	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁×#3168	C₂	1	1	yes	maximal, characteristic
10	C₁₁×#529	C₂²	1	2	yes	—
11	C₁₁×#499	C₄	1	1	yes	characteristic
12	C₁₁×#495	C₄	1	1	yes	characteristic
13	C₁₁×#488	C₄	1	2	yes	—
14	C₁₁×#467	C₄	1	1	yes	characteristic
15	C₁₁×#466	C₄	1	1	yes	characteristic
16	C₁₁×#465	C₄	1	2	yes	—
17	C₄×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
18	C₄×C₁₁×SD₁₆	C₂²	1	2	yes	—
19	C₄×C₁₁×D₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
21	#10622637	C₂²	1	2	yes	—
22	Q₈×C₁₁⋊C₈	C₂²	1	2	yes	—
23	#10622625	C₂²	1	2	yes	—
24	#10622624	—	2	2	no	—
25	D₄×C₁₁⋊C₈	—	2	2	no	—
26	D₄×C₁₁⋊C₈	C₂²	1	2	yes	—
27	#10622613	—	2	2	no	—
28	#10622612	—	2	2	no	—
29	#10622538	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
31	C₄×#4318	C₂²	1	1	yes	characteristic
32	Q₈⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
33	Q₈⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
34	D₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
35	D₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
36	#10622095	C₂²	1	1	yes	characteristic
37	#10622095	C₂²	1	2	yes	—
38	#10622095	C₂²	1	1	yes	characteristic
39	#10622092	C₂²	1	2	yes	—
40	C₈×C₁₁⋊C₈	C₂²	1	2	yes	—
41	C₂×C₁₁×Q₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₁×SD₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₁₁×D₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
46	C₁₁×#126	—	2	2	no	—
47	C₁₁×#125	C₂×C₄	1	2	yes	—
48	C₁₁×#124	C₂×C₄	1	2	yes	—
49	C₁₁×#123	—	2	2	no	—
50	C₁₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
51	C₁₁×#121	C₂×C₄	1	2	yes	—
52	C₄×Q₈×C₁₁	C₂³	1	2	yes	—
53	C₄×D₄×C₁₁	—	2	2	no	—
54	C₄×D₄×C₁₁	C₂³	1	2	yes	—
55	C₁₁×#117	—	2	2	no	—
56	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
57	C₂×#4318	—	2	1	no	—
58	C₂×#4318	—	2	2	no	—
59	C₂²×C₁₁⋊C₈	—	2	2	no	—
60	C₂²×C₁₁⋊C₈	—	2	2	no	—
61	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
62	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
63	C₁₁×#106	C₂×C₄	1	2	yes	—
64	C₁₁×#103	C₂×C₄	1	2	yes	—
65	C₁₁×#103	C₂×C₄	1	2	yes	—
66	C₁₁×#102	C₂×C₄	1	2	yes	—
67	C₁₁×#102	C₂×C₄	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	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
73	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
74	C₂²⋊(C₁₁⋊C₈)	D₄	1	1	yes	characteristic
75	C₂²⋊(C₁₁⋊C₈)	D₄	1	1	yes	characteristic
76	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
77	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
78	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
79	C₄⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
80	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
81	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
82	#66867	—	2	1	no	—
83	#66867	—	2	2	no	—
84	C₄×C₁₁⋊C₈	—	2	2	no	—
85	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
86	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
87	#18088	—	11	1	no	maximal, Sylow
88	C₂×Q₈×C₁₁	C₂²×C₄	1	2	yes	—
89	C₂×D₄×C₁₁	—	2	2	no	—
90	C₂×D₄×C₁₁	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₁₁	—	2	2	no	—
92	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
95	C₁₁×Q₁₆	—	4	1	no	—
96	C₁₁×SD₁₆	—	2	4	no	—
97	C₁₁×D₈	—	4	1	no	—
98	C₂×C₈×C₁₁	—	2	2	no	—
99	C₂×C₈×C₁₁	—	4	1	no	—
100	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
104	C₁₁×C₄⋊C₄	—	2	2	no	—
105	C₁₁×C₄⋊C₄	—	2	2	no	—
106	C₁₁×C₄⋊C₄	—	2	2	no	—
107	C₁₁×C₄⋊C₄	—	2	2	no	—
108	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
109	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
110	C₁₁×C₂²⋊C₄	—	2	2	no	—
111	C₁₁×C₂²⋊C₄	—	2	2	no	—
112	C₁₁×C₂²⋊C₄	—	2	2	no	—
113	C₁₁×C₂²⋊C₄	—	2	2	no	—
114	C₁₁×C₂²⋊C₄	—	2	2	no	—
115	C₄²×C₁₁	—	2	2	no	—
116	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₁	#41	1	1	yes	characteristic
119	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
120	#4318	—	2	4	no	—
121	#4318	—	8	2	no	—
122	C₂×C₁₁⋊C₈	—	2	2	no	—
123	C₂×C₁₁⋊C₈	—	2	2	no	—
124	C₂×C₁₁⋊C₈	—	2	2	no	—
125	C₂×C₁₁⋊C₈	—	8	2	no	—
126	C₂×C₁₁⋊C₈	—	8	2	no	—
127	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
128	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
129	#3168	D₁₁	1	1	yes	characteristic
130	#2994	—	11	2	no	—
131	#1590	—	11	2	no	—
132	#1587	—	11	2	no	—
133	#1586	—	11	2	no	—
134	C₈×SD₁₆	—	11	2	no	—
135	C₂×#334	—	11	1	no	—
136	#1483	—	11	2	no	—
137	C₈×M₁₆	—	11	1	no	—
138	C₂³×C₁₁	—	2	2	no	—
139	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
140	Q₈×C₁₁	—	2	2	no	—
141	Q₈×C₁₁	—	2	2	no	—
142	D₄×C₁₁	—	2	2	no	—
143	D₄×C₁₁	—	2	2	no	—
144	D₄×C₁₁	—	4	4	no	—
145	C₂×C₄×C₁₁	—	2	2	no	—
146	C₂×C₄×C₁₁	—	2	1	no	—
147	C₂×C₄×C₁₁	—	2	1	no	—
148	C₂×C₄×C₁₁	—	2	2	no	—
149	C₂×C₄×C₁₁	—	2	2	no	—
150	C₂×C₄×C₁₁	—	2	2	no	—
151	C₂×C₄×C₁₁	—	2	2	no	—
152	C₂×C₄×C₁₁	—	4	2	no	—
153	C₂×C₄×C₁₁	—	4	2	no	—
154	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
160	C₈×C₁₁	—	2	1	no	—
161	C₈×C₁₁	—	2	1	no	—
162	C₈×C₁₁	—	2	1	no	—
163	C₈×C₁₁	#131	1	2	yes	—
164	C₁₁⋊C₈	—	2	4	no	—
165	C₁₁⋊C₈	—	2	2	no	—
166	C₁₁⋊C₈	—	2	2	no	—
167	C₁₁⋊C₈	—	4	2	no	—
168	C₁₁⋊C₈	—	4	2	no	—
169	#529	C₂×D₁₁	1	2	yes	—
170	#499	Dic₁₁	1	1	yes	characteristic
171	#495	Dic₁₁	1	1	yes	characteristic
172	#488	Dic₁₁	1	2	yes	—
173	#467	Dic₁₁	1	1	yes	characteristic
174	#466	Dic₁₁	1	1	yes	characteristic
175	#465	Dic₁₁	1	2	yes	—
176	#446	—	11	2	no	—
177	C₈×Q₈	—	11	2	no	—
178	C₄×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
179	C₄×SD₁₆	C₂×D₁₁	1	2	yes	—
180	C₄×D₈	C₂×D₁₁	1	1	yes	characteristic
181	#436	—	11	2	no	—
182	#435	—	22	2	no	—
183	C₈×D₄	—	11	2	no	—
184	C₈×D₄	—	22	2	no	—
185	#433	—	22	2	no	—
186	#431	—	11	1	no	—
187	#424	—	22	2	no	—
188	C₂×C₄⋊C₈	—	11	2	no	—
189	C₄×M₁₆	—	11	1	no	—
190	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
191	#334	—	11	1	no	—
192	#334	—	11	2	no	—
193	#334	—	11	1	no	—
194	Q₈⋊C₈	—	11	2	no	—
195	Q₈⋊C₈	—	11	2	no	—
196	D₄⋊C₈	—	11	2	no	—
197	D₄⋊C₈	—	11	2	no	—
198	#322	—	11	2	no	—
199	C₈²	—	11	2	no	—
200	C₂²×C₁₁	—	2	1	no	—
201	C₂²×C₁₁	—	2	1	no	—
202	C₂²×C₁₁	—	4	2	no	—
203	C₂²×C₁₁	—	4	2	no	—
204	C₂²×C₁₁	—	4	2	no	—
205	C₂²×C₁₁	#443	1	2	yes	—
206	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
207	C₄×C₁₁	—	2	1	no	—
208	C₄×C₁₁	—	2	1	no	—
209	C₄×C₁₁	—	2	1	no	—
210	C₄×C₁₁	—	2	1	no	—
211	C₄×C₁₁	—	4	2	no	—
212	C₄×C₁₁	—	4	2	no	—
213	C₄×C₁₁	—	4	2	no	—
214	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
215	C₄×C₁₁	#568	1	1	yes	characteristic
216	C₄×C₁₁	#577	1	2	yes	—
217	C₂×Q₁₆	C₂×Dic₁₁	1	1	yes	characteristic
218	C₂×SD₁₆	C₂×Dic₁₁	1	2	yes	—
219	C₂×D₈	C₂×Dic₁₁	1	1	yes	characteristic
220	C₂×M₁₆	—	22	2	no	—
221	C₂×M₁₆	—	22	1	no	—
222	C₂²×C₈	—	22	2	no	—
223	C₂²×C₈	—	22	2	no	—
224	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
225	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
226	#126	—	2	2	no	—
227	#125	C₂×Dic₁₁	1	2	yes	—
228	#124	C₂×Dic₁₁	1	2	yes	—
229	#123	—	2	2	no	—
230	C₂²⋊Q₈	C₂×Dic₁₁	1	2	yes	—
231	#121	C₂×Dic₁₁	1	2	yes	—
232	C₄×Q₈	C₂²×D₁₁	1	2	yes	—
233	C₄×D₄	—	2	2	no	—
234	C₄×D₄	C₂²×D₁₁	1	2	yes	—
235	#117	—	2	2	no	—
236	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
237	#107	C₂×Dic₁₁	1	1	yes	characteristic
238	#107	C₂×Dic₁₁	1	1	yes	characteristic
239	#106	C₂×Dic₁₁	1	2	yes	—
240	C₄⋊C₈	—	11	2	no	—
241	C₄⋊C₈	—	11	2	no	—
242	C₄⋊C₈	—	11	2	no	—
243	C₄⋊C₈	—	22	2	no	—
244	C₄⋊C₈	—	22	2	no	—
245	C₄⋊C₈	—	22	2	no	—
246	#103	C₂×Dic₁₁	1	2	yes	—
247	#103	C₂×Dic₁₁	1	2	yes	—
248	#102	C₂×Dic₁₁	1	2	yes	—
249	#102	C₂×Dic₁₁	1	2	yes	—
250	C₂²⋊C₈	—	11	1	no	—
251	C₂²⋊C₈	—	11	1	no	—
252	C₂²⋊C₈	—	22	2	no	—
253	C₂²⋊C₈	—	22	2	no	—
254	#97	—	22	2	no	—
255	#97	—	22	1	no	—
256	C₄×C₈	—	11	2	no	—
257	C₄×C₈	—	11	2	no	—
258	C₄×C₈	—	22	2	no	—
259	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
260	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
263	C₂×C₁₁	—	2	2	no	—
264	C₂×C₁₁	—	8	2	no	—
265	C₂×C₁₁	#2935	1	1	yes	characteristic
266	C₂×C₁₁	#2952	1	1	yes	characteristic
267	C₂×C₁₁	C₂×#443	1	1	yes	characteristic
268	C₂×Q₈	C₂²×Dic₁₁	1	2	yes	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	C₂²×Dic₁₁	1	2	yes	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
273	C₂²×C₄	#4341	1	1	yes	characteristic
274	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
275	Q₁₆	—	4	1	no	—
276	SD₁₆	—	2	4	no	—
277	D₈	—	4	1	no	—
278	M₁₆	—	22	4	no	—
279	M₁₆	—	88	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	11	2	no	—
283	C₂×C₈	—	11	2	no	—
284	C₂×C₈	—	22	2	no	—
285	C₂×C₈	—	22	2	no	—
286	C₂×C₈	—	22	2	no	—
287	C₂×C₈	—	88	2	no	—
288	C₂×C₈	—	88	2	no	—
289	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
290	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
291	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
292	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	C₂²×Dic₁₁	1	2	yes	—
298	C₄⋊C₄	C₂²×Dic₁₁	1	2	yes	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₂²⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	D₄×D₁₁	1	1	yes	characteristic
306	C₄²	#4341	1	1	yes	characteristic
307	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
308	C₄²	C₂³×D₁₁	1	1	yes	characteristic
309	C₁₁	#18088	1	1	yes	characteristic, complemented, Sylow
310	C₂³	—	2	2	no	—
311	C₂³	D₄×Dic₁₁	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	2	2	no	—
316	D₄	—	4	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
327	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
328	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
330	C₂×C₄	C₂×#4341	1	1	yes	characteristic
331	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	22	4	no	—
336	C₈	—	22	2	no	—
337	C₈	—	22	2	no	—
338	C₈	—	44	2	no	—
339	C₈	—	44	2	no	—
340	C₈	#67001	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	#10622809	1	2	yes	—
347	C₂²	C₂×D₄×Dic₁₁	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	#10623301	1	2	yes	—
356	C₄	C₂×#67001	1	1	yes	characteristic
357	C₄	#10623350	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	8	2	no	—
360	C₂	#13077360	1	1	yes	characteristic
361	C₂	#13077500	1	1	yes	characteristic
362	C₂	C₂×#10622809	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	#18088
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. #13077776	C₂	semidirect
2. #13067507	C₂	semidirect
3. #13067488	C₂	semidirect
4. #13067481	C₂	semidirect
5. SD₁₆×C₁₁⋊C₈	C₂	semidirect
6. C₂×#10622095	C₂	semidirect
7. #13067029	C₂	semidirect
8. C₈×#4318	C₂	semidirect
309. C₁₁	#18088	semidirect

Maximal subgroups

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

subgroup	index
#13077776	2
#13067507	2
#13067488	2
#13067481	2
SD₁₆×C₁₁⋊C₈	2
C₂×#10622095	2
#13067029	2
C₈×#4318	2
C₁₁×#3168	2
#18088	11