Subgroups — #14723094

Order 2816.

594 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077743	C₂	1	2	yes	maximal
2	#13077582	C₂	1	1	yes	maximal, characteristic
3	#13067402	C₂	1	2	yes	maximal, complemented
4	#13067393	C₂	1	2	yes	maximal, complemented
5	#13067385	C₂	1	2	yes	maximal, complemented
6	#13067382	C₂	1	2	yes	maximal, complemented
7	#13067367	C₂	1	2	yes	maximal, complemented
8	C₂×#10622095	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#4339	C₂²	1	1	yes	characteristic
11	C₁₁×#432	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	#10622724	C₄	1	2	yes	—
14	#10622724	C₄	1	2	yes	—
15	#10622538	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
17	#10622450	C₂²	1	2	yes	—
18	#10622448	C₂²	1	2	yes	—
19	#10622446	C₂²	1	2	yes	—
20	D₁₁×C₄⋊C₈	C₂²	1	2	yes	—
21	#10622343	—	2	2	no	—
22	(C₂²⋊D₁₁)⋊C₈	—	2	2	no	—
23	#10622319	—	2	2	no	—
24	#10622314	—	2	2	no	—
25	#10622301	C₄	1	1	yes	characteristic
26	#10622301	C₄	1	1	yes	characteristic
27	C₄×#4313	C₂²	1	2	yes	—
28	#10622298	C₂²	1	2	yes	—
29	#10622292	C₄	1	1	yes	characteristic
30	#10622292	C₄	1	1	yes	characteristic
31	C₁₁×#335	C₂²	1	2	yes	—
32	C₁₁×#334	C₂²	1	1	yes	characteristic
33	C₁₁×#334	C₂²	1	1	yes	characteristic
34	#10622118	C₂²	1	2	yes	—
35	#10622118	C₂²	1	2	yes	—
36	#10622117	C₂²	1	2	yes	—
37	#10622117	C₂²	1	2	yes	—
38	#10622095	C₂²	1	1	yes	characteristic
39	#10622095	C₂²	1	1	yes	characteristic
40	#10622095	C₂²	1	2	yes	—
41	C₂×#4339	Q₈	1	1	yes	characteristic
42	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
45	C₄×C₂²⋊D₁₁	C₂×C₄	1	2	yes	—
46	C₄×C₂²⋊D₁₁	D₄	1	2	yes	—
47	#66979	Q₈	1	1	yes	characteristic
48	C₂²×C₁₁⋊C₈	—	2	2	no	—
49	C₂×#4313	—	2	2	no	—
50	C₂×#4313	C₂×C₄	1	2	yes	—
51	C₂×#4313	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₁₁	—	2	2	no	—
53	C₄×D₄₄	C₂×C₄	1	1	yes	characteristic
54	C₄×D₄₄	D₄	1	1	yes	characteristic
55	#66925	Q₈	1	2	yes	—
56	C₄²×D₁₁	C₂³	1	2	yes	—
57	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₂₂	D₄	1	1	yes	characteristic
59	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
60	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
61	C₁₁×C₂²⋊C₈	—	2	2	no	—
62	C₁₁×#97	—	2	2	no	—
63	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
67	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
68	(C₂×D₁₁)⋊C₈	—	2	2	no	—
69	(C₂×D₁₁)⋊C₈	—	2	2	no	—
70	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
71	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
72	#66879	—	2	2	no	—
73	#66879	C₂×C₄	1	2	yes	—
74	#66879	C₂×C₄	1	2	yes	—
75	Dic₁₁⋊C₈	—	2	2	no	—
76	Dic₁₁⋊C₈	—	2	2	no	—
77	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
78	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
79	C₈×Dic₁₁	—	2	2	no	—
80	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
81	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
82	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
83	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
84	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
85	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
86	#17838	—	11	1	no	maximal, Sylow
87	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₂²⋊D₁₁	C₄⋊C₄	1	2	yes	—
91	#4339	—	4	2	no	—
92	C₂×D₄₄	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₁₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×D₁₁	C₂×Q₈	1	2	yes	—
96	C₂×Dic₂₂	C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₁	—	2	2	no	—
98	C₂×C₈×C₁₁	—	2	2	no	—
99	C₂×C₈×C₁₁	—	2	2	no	—
100	C₂×C₈×C₁₁	—	4	1	no	—
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	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
108	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
109	C₂²⋊Dic₁₁	C₄⋊C₄	1	2	yes	—
110	(C₂×D₁₁)⋊C₄	C₄⋊C₄	1	2	yes	—
111	(C₂×D₁₁)⋊C₄	C₄⋊C₄	1	2	yes	—
112	C₄⋊Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
113	C₄⋊Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
114	Dic₁₁⋊C₄	C₄⋊C₄	1	2	yes	—
115	Dic₁₁⋊C₄	C₄⋊C₄	1	2	yes	—
116	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
117	C₄×Dic₁₁	C₂×D₄	1	2	yes	—
118	C₄×Dic₁₁	C₂×Q₈	1	2	yes	—
119	C₂×C₁₁⋊C₈	—	2	2	no	—
120	C₂×C₁₁⋊C₈	—	2	2	no	—
121	C₂×C₁₁⋊C₈	—	8	2	no	—
122	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
123	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
124	#4313	—	2	4	no	—
125	#4313	—	4	2	no	—
126	#4313	—	8	2	no	—
127	C₈×D₁₁	—	8	2	no	—
128	#2990	—	11	2	no	—
129	#2965	—	11	1	no	—
130	#1573	—	11	2	no	—
131	#1572	—	11	2	no	—
132	#1570	—	11	2	no	—
133	#1569	—	11	2	no	—
134	#1568	D₁₁	1	1	yes	characteristic, complemented
135	#1565	—	11	2	no	—
136	C₂×#334	—	11	1	no	—
137	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×D₁₁	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₁₁	—	2	2	no	—
140	C₂×C₄×C₁₁	—	2	1	no	—
141	C₂×C₄×C₁₁	—	2	1	no	—
142	C₂×C₄×C₁₁	—	2	2	no	—
143	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
149	C₂²⋊D₁₁	—	2	4	no	—
150	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
152	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
153	D₄₄	—	4	1	no	—
154	C₄×D₁₁	—	2	2	no	—
155	C₄×D₁₁	—	2	2	no	—
156	C₄×D₁₁	—	2	2	no	—
157	C₄×D₁₁	—	2	2	no	—
158	C₄×D₁₁	—	4	2	no	—
159	Dic₂₂	—	4	1	no	—
160	C₈×C₁₁	—	2	1	no	—
161	C₈×C₁₁	—	2	1	no	—
162	C₈×C₁₁	—	2	1	no	—
163	C₈×C₁₁	—	4	2	no	—
164	C₈×C₁₁	—	4	2	no	—
165	C₈×C₁₁	#131	1	2	yes	—
166	C₁₁⋊C₈	—	2	2	no	—
167	C₁₁⋊C₈	—	2	2	no	—
168	C₁₁⋊C₈	—	4	2	no	—
169	C₁₁⋊C₈	—	4	2	no	—
170	C₄×#41	—	11	1	no	—
171	#446	—	11	1	no	—
172	#446	—	11	1	no	—
173	#446	—	22	2	no	—
174	#436	—	11	1	no	—
175	#436	—	11	1	no	—
176	#435	—	11	2	no	—
177	#435	—	11	2	no	—
178	#435	—	22	2	no	—
179	#435	—	22	2	no	—
180	C₈×D₄	—	22	2	no	—
181	#432	—	11	2	no	—
182	#432	C₂×D₁₁	1	2	yes	—
183	#431	—	11	1	no	—
184	#431	—	11	2	no	—
185	#431	—	11	2	no	—
186	#423	—	11	2	no	—
187	C₂×C₄⋊C₈	—	11	2	no	—
188	C₂×C₄⋊C₈	—	11	2	no	—
189	C₄×M₁₆	—	11	2	no	—
190	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
191	#335	—	11	2	no	—
192	#335	—	11	2	no	—
193	#335	C₂×D₁₁	1	2	yes	—
194	#334	—	11	2	no	—
195	#334	—	11	2	no	—
196	#334	—	11	1	no	—
197	#334	—	11	1	no	—
198	#334	—	11	2	no	—
199	#334	C₂×D₁₁	1	1	yes	characteristic
200	#334	C₂×D₁₁	1	1	yes	characteristic
201	C₂²×C₁₁	—	2	1	no	—
202	C₂²×C₁₁	—	2	1	no	—
203	C₂²×C₁₁	#430	1	2	yes	—
204	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
205	C₂×D₁₁	—	2	2	no	—
206	C₂×D₁₁	—	2	2	no	—
207	C₂×D₁₁	—	2	2	no	—
208	C₄×C₁₁	—	2	1	no	—
209	C₄×C₁₁	—	2	1	no	—
210	C₄×C₁₁	—	2	1	no	—
211	C₄×C₁₁	—	2	1	no	—
212	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
213	C₄×C₁₁	#568	1	1	yes	characteristic
214	C₄×C₁₁	#577	1	2	yes	—
215	Dic₁₁	—	2	2	no	—
216	Dic₁₁	—	2	2	no	—
217	Dic₁₁	—	2	2	no	—
218	C₂×#41	—	11	1	no	—
219	C₂×M₁₆	—	11	2	no	—
220	C₂×M₁₆	—	11	2	no	—
221	C₂×M₁₆	—	22	2	no	—
222	C₂²×C₈	—	22	2	no	—
223	C₂²×C₈	—	22	2	no	—
224	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
225	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
226	C₄×Q₈	—	11	1	no	—
227	C₄×Q₈	—	11	1	no	—
228	C₄×D₄	—	11	2	no	—
229	C₄×D₄	—	11	2	no	—
230	C₄×D₄	—	11	1	no	—
231	C₄×D₄	—	11	1	no	—
232	#117	—	11	1	no	—
233	#117	—	11	2	no	—
234	C₂×C₄²	—	11	2	no	—
235	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
236	C₄⋊C₈	—	11	2	no	—
237	C₄⋊C₈	—	11	2	no	—
238	C₄⋊C₈	—	11	1	no	—
239	C₄⋊C₈	—	11	1	no	—
240	C₄⋊C₈	—	11	2	no	—
241	C₄⋊C₈	—	11	2	no	—
242	C₄⋊C₈	—	22	2	no	—
243	C₄⋊C₈	—	22	2	no	—
244	C₄⋊C₈	—	22	2	no	—
245	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
246	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
247	C₂²⋊C₈	—	2	2	no	—
248	C₂²⋊C₈	—	11	2	no	—
249	C₂²⋊C₈	—	11	2	no	—
250	C₂²⋊C₈	—	11	2	no	—
251	C₂²⋊C₈	—	22	2	no	—
252	C₂²⋊C₈	—	22	2	no	—
253	#97	—	2	2	no	—
254	#97	—	11	2	no	—
255	#97	—	11	2	no	—
256	#97	—	22	2	no	—
257	C₄×C₈	—	11	2	no	—
258	C₄×C₈	—	22	2	no	—
259	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
260	C₄×C₈	C₄×D₁₁	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	#2906	1	1	yes	characteristic
265	C₂×C₁₁	#2912	1	1	yes	characteristic
266	C₂×C₁₁	C₂×#430	1	1	yes	characteristic
267	D₁₁	—	4	2	no	—
268	#41	—	44	2	no	—
269	C₂×Q₈	—	11	1	no	—
270	C₂×D₄	—	11	2	no	—
271	C₂×D₄	—	11	1	no	—
272	C₂²×C₄	—	11	2	no	—
273	C₂²×C₄	—	11	2	no	—
274	C₂²×C₄	—	11	2	no	—
275	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
276	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
277	C₂²×C₄	Q₈×D₁₁	1	1	yes	characteristic
278	M₁₆	—	22	4	no	—
279	M₁₆	—	44	2	no	—
280	M₁₆	—	88	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	11	2	no	—
286	C₂×C₈	—	11	2	no	—
287	C₂×C₈	—	22	2	no	—
288	C₂×C₈	—	22	2	no	—
289	C₂×C₈	—	88	2	no	—
290	C₂×C₈	—	88	2	no	—
291	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
293	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
294	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
295	C₄⋊C₄	—	11	2	no	—
296	C₄⋊C₄	—	11	2	no	—
297	C₄⋊C₄	—	11	1	no	—
298	C₄⋊C₄	—	11	1	no	—
299	C₂²⋊C₄	—	11	2	no	—
300	C₂²⋊C₄	—	11	2	no	—
301	C₂²⋊C₄	—	11	2	no	—
302	C₄²	—	11	2	no	—
303	C₄²	—	11	2	no	—
304	C₄²	—	11	2	no	—
305	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
306	C₄²	D₄×D₁₁	1	1	yes	characteristic
307	C₄²	Q₈×D₁₁	1	1	yes	characteristic
308	C₄²	C₂³×D₁₁	1	1	yes	characteristic
309	C₁₁	#17838	1	1	yes	characteristic, complemented, Sylow
310	C₂³	—	11	2	no	—
311	C₂³	C₄⋊C₄×D₁₁	1	1	yes	characteristic
312	Q₈	—	44	1	no	—
313	D₄	—	22	4	no	—
314	D₄	—	44	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	11	2	no	—
320	C₂×C₄	—	11	2	no	—
321	C₂×C₄	—	11	2	no	—
322	C₂×C₄	—	22	2	no	—
323	C₂×C₄	—	22	2	no	—
324	C₂×C₄	—	22	2	no	—
325	C₂×C₄	—	22	2	no	—
326	C₂×C₄	—	44	2	no	—
327	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
328	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
329	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
331	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
332	C₂×C₄	C₂×Q₈×D₁₁	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₈	—	22	2	no	—
339	C₈	—	22	2	no	—
340	C₈	—	44	2	no	—
341	C₈	—	44	2	no	—
342	C₈	#66960	1	2	yes	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	22	2	no	—
346	C₂²	—	22	2	no	—
347	C₂²	—	22	2	no	—
348	C₂²	#10622506	1	2	yes	—
349	C₂²	C₂×C₄⋊C₄×D₁₁	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	22	2	no	—
355	C₄	—	22	2	no	—
356	C₄	—	22	2	no	—
357	C₄	C₂×#66960	1	1	yes	characteristic
358	C₄	#10623276	1	1	yes	characteristic
359	C₄	#10623301	1	2	yes	—
360	C₂	—	2	2	no	—
361	C₂	—	44	2	no	—
362	C₂	#13077185	1	1	yes	characteristic
363	C₂	#13077208	1	1	yes	characteristic
364	C₂	C₂×#10622506	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×#1568	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂×#4339	Q₈
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1568	D₁₁
hypocenter	C₁₁	#17838
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #13067402	C₂	semidirect
4. #13067393	C₂	semidirect
5. #13067385	C₂	semidirect
6. #13067382	C₂	semidirect
7. #13067367	C₂	semidirect
8. C₂×#10622095	C₂	semidirect
9. C₁₁×#1568	C₂	semidirect
134. #1568	D₁₁	semidirect
309. C₁₁	#17838	semidirect

Maximal subgroups

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

subgroup	index
#13077743	2
#13077582	2
#13067402	2
#13067393	2
#13067385	2
#13067382	2
#13067367	2
C₂×#10622095	2
C₁₁×#1568	2
#17838	11