Subgroups — #14724450

Order 2816.

560 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079199	C₂	1	1	yes	maximal, characteristic
2	#13077707	C₂	1	2	yes	maximal
3	#13067525	C₂	1	1	yes	maximal, characteristic, complemented
4	#13067524	C₂	1	1	yes	maximal, characteristic, complemented
5	#13067510	C₂	1	2	yes	maximal, complemented
6	#13067436	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#4326	C₂	1	2	yes	maximal, complemented
8	#13067376	C₂	1	1	yes	maximal, characteristic
9	#13067087	C₂	1	2	yes	maximal
10	#13066901	C₂	1	1	yes	maximal, characteristic
11	C₁₁×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623109	C₂²	1	2	yes	—
13	C₈×Q₈×C₁₁	C₂²	1	2	yes	—
14	C₄×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
15	#10622801	C₄	1	1	yes	characteristic
16	#10622794	C₄	1	1	yes	characteristic
17	C₄×#4326	C₂²	1	2	yes	—
18	#10622637	C₂²	1	2	yes	—
19	#10622500	C₄	1	1	yes	characteristic
20	#10622487	C₄	1	1	yes	characteristic
21	#10622453	—	2	2	no	—
22	#10622448	C₂²	1	2	yes	—
23	#10622444	C₂²	1	2	yes	—
24	#10622441	—	2	2	no	—
25	#10622426	C₄	1	2	yes	—
26	#10622405	C₄	1	2	yes	—
27	C₄×Dic₄₄	C₂²	1	1	yes	characteristic
28	#10622303	—	2	2	no	—
29	#10622298	C₂²	1	1	yes	characteristic
30	#10622298	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
32	#10622292	—	2	2	no	—
33	C₈×Dic₂₂	C₂²	1	2	yes	—
34	C₁₁×#334	C₂²	1	1	yes	characteristic
35	C₁₁×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₁₁×#322	C₂²	1	1	yes	characteristic
37	Q₈⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
38	#10622122	C₂²	1	2	yes	—
39	#10622117	C₂²	1	2	yes	—
40	#10622095	C₂²	1	1	yes	characteristic
41	#10622094	C₂²	1	2	yes	—
42	#10622093	C₂²	1	1	yes	characteristic
43	C₈×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
44	C₈×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
45	C₂×C₁₁×Q₁₆	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₁₁	C₂³	1	2	yes	—
47	#66999	C₂×C₄	1	2	yes	—
48	#66997	C₂×C₄	1	2	yes	—
49	C₂×#4326	C₂×C₄	1	2	yes	—
50	C₂×Dic₄₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
53	#66950	C₂×C₄	1	2	yes	—
54	#66949	—	2	2	no	—
55	#66942	—	2	2	no	—
56	#66941	C₂×C₄	1	2	yes	—
57	#66925	—	2	2	no	—
58	C₄²×D₁₁	C₂³	1	1	yes	characteristic
59	C₄×Dic₂₂	—	2	2	no	—
60	C₄×Dic₂₂	C₂³	1	2	yes	—
61	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
62	C₁₁×C₄⋊C₈	—	2	2	no	—
63	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
64	C₁₁×#103	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₁₁	—	2	2	no	—
66	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
69	#66899	C₂×C₄	1	2	yes	—
70	(C₂×D₁₁)⋊C₈	—	2	2	no	—
71	(C₂×D₁₁)⋊C₈	—	2	1	no	—
72	(C₂×D₁₁)⋊C₈	D₄	1	2	yes	—
73	#66882	C₂×C₄	1	1	yes	characteristic
74	#66880	C₂×C₄	1	2	yes	—
75	#66879	—	2	2	no	—
76	#66879	—	2	2	no	—
77	Dic₁₁⋊C₈	—	2	2	no	—
78	Dic₁₁⋊C₈	—	2	1	no	—
79	Dic₁₁⋊C₈	—	2	2	no	—
80	Dic₁₁⋊C₈	D₄	1	2	yes	—
81	C₈×Dic₁₁	—	2	2	no	—
82	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
84	#66874	C₂×C₄	1	2	yes	—
85	#66871	C₂×C₄	1	2	yes	—
86	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
87	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
88	#66867	—	2	2	no	—
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	#17972	—	11	1	no	maximal, Sylow
93	C₂×Q₈×C₁₁	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₁₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×D₁₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₁₁	#41	1	1	yes	characteristic
97	C₂×Dic₂₂	—	2	2	no	—
98	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
99	C₁₁×Q₁₆	—	2	2	no	—
100	C₂×C₈×C₁₁	—	2	2	no	—
101	C₂×C₈×C₁₁	—	2	1	no	—
102	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
105	C₁₁×C₄⋊C₄	—	2	2	no	—
106	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
107	C₄²×C₁₁	—	2	2	no	—
108	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
109	#4326	—	2	4	no	—
110	(C₂×D₁₁)⋊C₄	—	2	2	no	—
111	(C₂×D₁₁)⋊C₄	—	2	2	no	—
112	C₄⋊Dic₁₁	—	2	2	no	—
113	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
114	Dic₁₁⋊C₄	—	2	2	no	—
115	Dic₁₁⋊C₄	—	2	2	no	—
116	Dic₁₁⋊C₄	—	2	2	no	—
117	Dic₁₁⋊C₄	—	2	2	no	—
118	Dic₁₁⋊C₄	—	2	2	no	—
119	C₄×Dic₁₁	—	2	2	no	—
120	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
121	C₄×Dic₁₁	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₁₁	#41	1	1	yes	characteristic
123	C₂×C₁₁⋊C₈	—	2	2	no	—
124	C₂×C₁₁⋊C₈	—	2	1	no	—
125	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
128	Dic₄₄	—	2	2	no	—
129	C₈×D₁₁	—	2	2	no	—
130	C₈×D₁₁	—	2	2	no	—
131	#3170	—	11	1	no	—
132	#2985	—	11	2	no	—
133	#1593	—	11	1	no	—
134	#1593	—	11	1	no	—
135	#1591	—	11	2	no	—
136	#1579	—	11	1	no	—
137	#1579	D₁₁	1	1	yes	characteristic, complemented
138	C₈×Q₁₆	—	11	2	no	—
139	#1568	—	11	1	no	—
140	#1497	—	11	2	no	—
141	#1450	—	11	1	no	—
142	C₂²×D₁₁	C₄×D₄	1	1	yes	characteristic
143	Q₈×C₁₁	—	2	2	no	—
144	Q₈×C₁₁	—	2	2	no	—
145	C₂×C₄×C₁₁	—	2	2	no	—
146	C₂×C₄×C₁₁	—	2	2	no	—
147	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
150	C₂×Dic₁₁	—	2	2	no	—
151	C₂×Dic₁₁	—	2	2	no	—
152	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
154	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
155	C₄×D₁₁	—	2	1	no	—
156	C₄×D₁₁	—	2	1	no	—
157	C₄×D₁₁	—	2	2	no	—
158	C₄×D₁₁	—	4	1	no	—
159	Dic₂₂	—	2	2	no	—
160	Dic₂₂	—	2	2	no	—
161	Dic₂₂	—	8	2	no	—
162	C₈×C₁₁	—	2	1	no	—
163	C₈×C₁₁	—	2	2	no	—
164	C₈×C₁₁	—	4	1	no	—
165	C₈×C₁₁	—	4	2	no	—
166	C₈×C₁₁	#131	1	2	yes	—
167	C₁₁⋊C₈	—	2	1	no	—
168	C₁₁⋊C₈	—	2	2	no	—
169	C₁₁⋊C₈	—	4	1	no	—
170	C₁₁⋊C₈	—	4	2	no	—
171	C₁₁⋊C₈	#131	1	2	yes	—
172	#533	—	11	2	no	—
173	#500	—	11	1	no	—
174	#494	—	11	1	no	—
175	#487	—	11	2	no	—
176	#467	—	11	1	no	—
177	#467	—	11	1	no	—
178	#467	—	11	2	no	—
179	#446	—	11	2	no	—
180	#446	—	11	2	no	—
181	#446	—	22	2	no	—
182	#446	—	22	2	no	—
183	C₈×Q₈	—	11	2	no	—
184	C₈×Q₈	C₂×D₁₁	1	2	yes	—
185	C₄×Q₁₆	—	11	2	no	—
186	C₄×Q₁₆	—	11	1	no	—
187	C₄×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
188	#433	—	22	2	no	—
189	#433	—	22	2	no	—
190	#432	—	11	2	no	—
191	#431	—	11	1	no	—
192	#431	—	11	1	no	—
193	C₂×C₄×C₈	—	11	1	no	—
194	#335	—	11	2	no	—
195	#334	—	11	1	no	—
196	#334	C₂×D₁₁	1	1	yes	characteristic
197	Q₈⋊C₈	—	11	2	no	—
198	Q₈⋊C₈	—	11	2	no	—
199	Q₈⋊C₈	—	11	2	no	—
200	Q₈⋊C₈	C₂×D₁₁	1	2	yes	—
201	#322	—	11	1	no	—
202	#322	C₂×D₁₁	1	1	yes	characteristic
203	C₈²	—	11	1	no	—
204	C₈²	—	11	1	no	—
205	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
206	C₂×D₁₁	—	2	1	no	—
207	C₂×D₁₁	—	2	1	no	—
208	C₂×D₁₁	—	2	1	no	—
209	C₄×C₁₁	—	2	1	no	—
210	C₄×C₁₁	—	4	2	no	—
211	C₄×C₁₁	—	4	2	no	—
212	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
213	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
214	C₄×C₁₁	#578	1	2	yes	—
215	Dic₁₁	—	2	1	no	—
216	Dic₁₁	—	2	1	no	—
217	Dic₁₁	—	2	1	no	—
218	Dic₁₁	—	4	2	no	—
219	Dic₁₁	—	4	2	no	—
220	C₂×Q₁₆	—	11	2	no	—
221	C₂×Q₁₆	—	11	1	no	—
222	C₂×Q₁₆	C₄×D₁₁	1	1	yes	characteristic
223	C₂²×C₈	—	11	1	no	—
224	C₂²×C₈	—	11	1	no	—
225	C₄⋊Q₈	—	11	2	no	—
226	C₄⋊Q₈	—	11	2	no	—
227	#125	—	22	2	no	—
228	C₂²⋊Q₈	—	11	2	no	—
229	C₂²⋊Q₈	—	11	2	no	—
230	C₂²⋊Q₈	—	22	2	no	—
231	C₄×Q₈	—	11	2	no	—
232	C₄×Q₈	—	22	2	no	—
233	C₄×Q₈	C₂²×D₁₁	1	2	yes	—
234	#117	—	22	2	no	—
235	C₂×C₄²	—	11	1	no	—
236	#107	—	11	1	no	—
237	#107	—	11	2	no	—
238	#107	C₄×D₁₁	1	1	yes	characteristic
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	11	2	no	—
241	C₄⋊C₈	—	11	2	no	—
242	C₄⋊C₈	—	22	2	no	—
243	C₄⋊C₈	—	22	1	no	—
244	C₄⋊C₈	—	22	2	no	—
245	C₄⋊C₈	—	22	2	no	—
246	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
247	#103	—	11	2	no	—
248	#103	—	11	2	no	—
249	#103	—	11	2	no	—
250	#103	C₄×D₁₁	1	2	yes	—
251	C₂²⋊C₈	—	11	2	no	—
252	C₂²⋊C₈	—	22	1	no	—
253	C₂²⋊C₈	—	22	2	no	—
254	#97	—	22	2	no	—
255	#97	—	22	2	no	—
256	#97	—	22	2	no	—
257	C₄×C₈	—	2	2	no	—
258	C₄×C₈	—	11	1	no	—
259	C₄×C₈	—	11	1	no	—
260	C₄×C₈	—	11	1	no	—
261	C₄×C₈	—	11	1	no	—
262	C₄×C₈	—	11	1	no	—
263	C₄×C₈	—	22	2	no	—
264	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
267	C₂×C₁₁	#2931	1	1	yes	characteristic
268	C₂×C₁₁	#2950	1	1	yes	characteristic
269	C₂×C₁₁	#2957	1	1	yes	characteristic
270	D₁₁	—	4	1	no	—
271	C₂×Q₈	—	11	2	no	—
272	C₂×Q₈	—	22	2	no	—
273	C₂×Q₈	C₂×C₄×D₁₁	1	2	yes	—
274	C₂²×C₄	—	11	1	no	—
275	C₂²×C₄	—	11	1	no	—
276	C₂²×C₄	—	11	1	no	—
277	Q₁₆	—	2	2	no	—
278	Q₁₆	—	22	4	no	—
279	Q₁₆	—	22	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	11	1	no	—
283	C₂×C₈	—	11	1	no	—
284	C₂×C₈	—	11	2	no	—
285	C₂×C₈	—	22	2	no	—
286	C₂×C₈	—	22	2	no	—
287	C₂×C₈	—	22	2	no	—
288	C₂×C₈	—	22	1	no	—
289	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
291	C₂×C₈	D₄×D₁₁	1	2	yes	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	11	2	no	—
294	C₄⋊C₄	—	22	2	no	—
295	C₄⋊C₄	—	22	2	no	—
296	C₄⋊C₄	—	22	2	no	—
297	C₄⋊C₄	—	22	2	no	—
298	C₄⋊C₄	—	22	2	no	—
299	C₄⋊C₄	—	22	2	no	—
300	C₄⋊C₄	C₂×C₄×D₁₁	1	2	yes	—
301	C₂²⋊C₄	—	22	2	no	—
302	C₂²⋊C₄	—	22	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	11	1	no	—
305	C₄²	—	11	1	no	—
306	C₄²	—	11	1	no	—
307	C₄²	—	22	2	no	—
308	C₄²	C₂³×D₁₁	1	1	yes	characteristic
309	C₁₁	#17972	1	1	yes	characteristic, complemented, Sylow
310	C₂³	—	11	1	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	2	2	no	—
313	Q₈	—	22	2	no	—
314	Q₈	—	22	2	no	—
315	Q₈	—	88	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	11	1	no	—
319	C₂×C₄	—	11	1	no	—
320	C₂×C₄	—	11	1	no	—
321	C₂×C₄	—	22	1	no	—
322	C₂×C₄	—	22	1	no	—
323	C₂×C₄	—	22	2	no	—
324	C₂×C₄	—	22	2	no	—
325	C₂×C₄	—	22	2	no	—
326	C₂×C₄	—	44	1	no	—
327	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
329	C₂×C₄	#41×D₁₁	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	2	no	—
332	C₈	—	4	1	no	—
333	C₈	—	4	2	no	—
334	C₈	—	11	2	no	—
335	C₈	—	22	1	no	—
336	C₈	—	22	2	no	—
337	C₈	—	44	1	no	—
338	C₈	—	44	2	no	—
339	C₈	#66954	1	2	yes	—
340	C₂²	—	22	1	no	—
341	C₂²	—	22	1	no	—
342	C₂²	—	22	1	no	—
343	C₂²	C₄×D₄×D₁₁	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	22	1	no	—
348	C₄	—	22	1	no	—
349	C₄	—	22	1	no	—
350	C₄	—	44	2	no	—
351	C₄	—	44	2	no	—
352	C₄	C₂×#66954	1	1	yes	characteristic
353	C₄	D₁₁×#131	1	1	yes	characteristic
354	C₄	#10623310	1	2	yes	—
355	C₂	—	44	1	no	—
356	C₂	#13077323	1	1	yes	characteristic
357	C₂	#13077481	1	1	yes	characteristic
358	C₂	#13077530	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×#1579	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂²×D₁₁	C₄×D₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1579	D₁₁
hypocenter	C₁₁	#17972
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #13067525	C₂	semidirect
4. #13067524	C₂	semidirect
5. #13067510	C₂	semidirect
6. #13067436	C₂	semidirect
7. C₈×#4326	C₂	semidirect
11. C₁₁×#1579	C₂	semidirect
137. #1579	D₁₁	semidirect
309. C₁₁	#17972	semidirect

Maximal subgroups

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

subgroup	index
#13079199	2
#13077707	2
#13067525	2
#13067524	2
#13067510	2
#13067436	2
C₈×#4326	2
#13067376	2
#13067087	2
#13066901	2
C₁₁×#1579	2
#17972	11