Subgroups — #14724440

Order 2816.

600 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079175	C₂	1	1	yes	maximal, characteristic
2	#13077713	C₂	1	1	yes	maximal, characteristic, complemented
3	#13077694	C₂	1	1	yes	maximal, characteristic
4	#13067517	C₂	1	2	yes	maximal, complemented
5	#13067505	C₂	1	2	yes	maximal, complemented
6	#13067427	C₂	1	2	yes	maximal, complemented
7	#13067415	C₂	1	2	yes	maximal, complemented
8	#13067088	C₂	1	1	yes	maximal, characteristic, complemented
9	#13067084	C₂	1	1	yes	maximal, characteristic
10	#13066899	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623008	C₂²	1	1	yes	characteristic
13	#10623001	C₂²	1	1	yes	characteristic
14	C₁₁×#432	C₂²	1	1	yes	characteristic
15	C₁₁×#432	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
17	#10622727	C₄	1	2	yes	—
18	#10622727	C₄	1	2	yes	—
19	#10622538	C₂²	1	1	yes	characteristic
20	#10622538	C₂²	1	1	yes	characteristic
21	#10622451	C₂²	1	2	yes	—
22	#10622447	C₂²	1	2	yes	—
23	#10622444	C₂²	1	2	yes	—
24	#10622443	C₂²	1	2	yes	—
25	#10622340	—	2	2	no	—
26	#10622333	—	2	2	no	—
27	#10622319	—	2	2	no	—
28	#10622314	—	2	2	no	—
29	#10622307	C₄	1	1	yes	characteristic
30	#10622306	C₄	1	1	yes	characteristic
31	C₄×#4314	C₂²	1	2	yes	—
32	C₄×#4314	C₂²	1	2	yes	—
33	#10622294	C₄	1	1	yes	characteristic
34	#10622293	C₄	1	1	yes	characteristic
35	C₁₁×#335	C₂²	1	2	yes	—
36	C₁₁×#334	C₂²	1	2	yes	—
37	#10622122	C₂²	1	2	yes	—
38	#10622122	C₂²	1	2	yes	—
39	#10622121	C₂²	1	2	yes	—
40	#10622121	C₂²	1	2	yes	—
41	#10622093	C₂²	1	1	yes	characteristic
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×#4339	—	2	1	no	—
46	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
49	#66983	—	2	1	no	—
50	#66983	C₂×C₄	1	2	yes	—
51	#66979	—	2	1	no	—
52	#66977	—	2	1	no	—
53	#66977	C₂×C₄	1	2	yes	—
54	C₂×#4314	C₂×C₄	1	2	yes	—
55	C₂×#4314	C₂×C₄	1	2	yes	—
56	C₂×#4313	—	2	2	no	—
57	C₂×C₈×D₁₁	—	2	2	no	—
58	#66928	—	2	1	no	—
59	#66927	C₂×C₄	1	1	yes	characteristic
60	C₄×D₄₄	—	2	1	no	—
61	C₄×D₄₄	C₂³	1	2	yes	—
62	#66923	—	2	1	no	—
63	#66922	C₂×C₄	1	1	yes	characteristic
64	#66922	C₂×C₄	1	1	yes	characteristic
65	#66922	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₂₂	—	2	1	no	—
67	C₄×Dic₂₂	C₂³	1	2	yes	—
68	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
69	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
70	C₁₁×C₂²⋊C₈	—	2	1	no	—
71	C₁₁×C₂²⋊C₈	—	2	1	no	—
72	C₁₁×#97	—	2	1	no	—
73	C₁₁×#97	—	2	1	no	—
74	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
78	C₂²⋊(C₁₁⋊C₈)	—	2	1	no	—
79	C₂²⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
80	#66885	C₂×C₄	1	2	yes	—
81	#66885	C₂×C₄	1	2	yes	—
82	(C₂×D₁₁)⋊C₈	—	2	2	no	—
83	(C₂×D₁₁)⋊C₈	—	2	2	no	—
84	#66881	C₂×C₄	1	2	yes	—
85	#66881	C₂×C₄	1	2	yes	—
86	#66880	C₂×C₄	1	2	yes	—
87	#66880	C₂×C₄	1	2	yes	—
88	#66879	—	2	2	no	—
89	Dic₁₁⋊C₈	—	2	2	no	—
90	Dic₁₁⋊C₈	—	2	2	no	—
91	C₈×Dic₁₁	—	2	2	no	—
92	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
93	C₄⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
94	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
95	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
96	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
97	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
98	#17971	—	11	1	no	maximal, Sylow
99	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
102	C₂×C₂²⋊D₁₁	—	2	2	no	—
103	#4339	—	8	2	no	—
104	C₂×D₄₄	—	2	1	no	—
105	C₂×D₄₄	—	2	1	no	—
106	C₂×D₄₄	C₂²×C₄	1	2	yes	—
107	C₂×C₄×D₁₁	—	2	2	no	—
108	C₂×Dic₂₂	—	2	1	no	—
109	C₂×Dic₂₂	—	2	1	no	—
110	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₁₁	—	2	2	no	—
112	C₂×C₈×C₁₁	—	2	2	no	—
113	C₂×C₈×C₁₁	—	2	2	no	—
114	C₂×C₈×C₁₁	—	2	2	no	—
115	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₁₁	#41	1	1	yes	characteristic
122	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
123	C₂²⋊Dic₁₁	—	2	2	no	—
124	(C₂×D₁₁)⋊C₄	—	2	2	no	—
125	(C₂×D₁₁)⋊C₄	—	2	2	no	—
126	C₄⋊Dic₁₁	—	2	1	no	—
127	C₄⋊Dic₁₁	—	2	1	no	—
128	C₄⋊Dic₁₁	—	2	1	no	—
129	C₄⋊Dic₁₁	—	2	1	no	—
130	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
131	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
132	Dic₁₁⋊C₄	—	2	2	no	—
133	Dic₁₁⋊C₄	—	2	2	no	—
134	C₄×Dic₁₁	—	2	2	no	—
135	C₂×C₁₁⋊C₈	—	2	1	no	—
136	C₂×C₁₁⋊C₈	—	2	1	no	—
137	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
138	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
139	#4314	—	2	4	no	—
140	#4314	—	2	4	no	—
141	#4313	—	8	2	no	—
142	C₈×D₁₁	—	8	2	no	—
143	#3167	—	11	1	no	—
144	#2985	—	11	1	no	—
145	#2981	—	11	1	no	—
146	#1592	—	11	2	no	—
147	#1590	—	11	2	no	—
148	SD₁₆⋊C₈	—	11	2	no	—
149	C₈×SD₁₆	—	11	2	no	—
150	#1568	D₁₁	1	1	yes	characteristic, complemented
151	#1497	—	11	1	no	—
152	#1496	—	11	1	no	—
153	#1450	—	11	1	no	—
154	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
155	C₂²×D₁₁	—	2	2	no	—
156	C₂×C₄×C₁₁	—	2	1	no	—
157	C₂×C₄×C₁₁	—	2	1	no	—
158	C₂×C₄×C₁₁	—	2	2	no	—
159	C₂×C₄×C₁₁	—	4	1	no	—
160	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
164	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
166	C₂²⋊D₁₁	—	8	2	no	—
167	C₂×Dic₁₁	—	2	2	no	—
168	C₂×Dic₁₁	—	2	2	no	—
169	C₂×Dic₁₁	—	2	2	no	—
170	D₄₄	—	2	2	no	—
171	D₄₄	—	2	2	no	—
172	D₄₄	—	8	1	no	—
173	D₄₄	—	8	1	no	—
174	C₄×D₁₁	—	4	2	no	—
175	C₄×D₁₁	—	4	2	no	—
176	Dic₂₂	—	2	2	no	—
177	Dic₂₂	—	2	2	no	—
178	Dic₂₂	—	8	1	no	—
179	Dic₂₂	—	8	1	no	—
180	C₈×C₁₁	—	2	1	no	—
181	C₈×C₁₁	—	2	1	no	—
182	C₈×C₁₁	—	4	2	no	—
183	C₈×C₁₁	—	4	2	no	—
184	C₈×C₁₁	#131	1	2	yes	—
185	C₈×C₁₁	#131	1	2	yes	—
186	C₁₁⋊C₈	—	2	2	no	—
187	C₁₁⋊C₈	—	2	2	no	—
188	C₁₁⋊C₈	—	4	1	no	—
189	C₁₁⋊C₈	—	4	1	no	—
190	#533	—	11	1	no	—
191	#532	—	11	1	no	—
192	#498	—	11	1	no	—
193	#492	—	11	1	no	—
194	#487	—	11	1	no	—
195	#486	—	11	1	no	—
196	#465	—	11	2	no	—
197	#465	—	11	2	no	—
198	#446	—	11	2	no	—
199	#446	—	22	2	no	—
200	C₈×Q₈	—	11	2	no	—
201	C₄×SD₁₆	—	11	2	no	—
202	C₄×SD₁₆	—	11	2	no	—
203	#436	—	11	2	no	—
204	#435	—	22	2	no	—
205	C₈×D₄	—	11	2	no	—
206	#433	—	22	2	no	—
207	#432	C₂×D₁₁	1	1	yes	characteristic
208	#432	C₂×D₁₁	1	1	yes	characteristic
209	#431	—	11	1	no	—
210	#431	—	11	1	no	—
211	#408	—	22	2	no	—
212	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
213	#335	C₂×D₁₁	1	2	yes	—
214	#334	C₂×D₁₁	1	2	yes	—
215	Q₈⋊C₈	—	11	2	no	—
216	Q₈⋊C₈	—	11	2	no	—
217	D₄⋊C₈	—	11	2	no	—
218	D₄⋊C₈	—	11	2	no	—
219	#322	—	11	1	no	—
220	#322	—	11	1	no	—
221	C₈²	—	11	1	no	—
222	C₈²	—	11	1	no	—
223	C₂²×C₁₁	—	2	1	no	—
224	C₂²×C₁₁	—	2	1	no	—
225	C₂²×C₁₁	—	2	1	no	—
226	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
227	C₂×D₁₁	—	4	2	no	—
228	C₂×D₁₁	—	4	2	no	—
229	C₂×D₁₁	—	4	2	no	—
230	C₄×C₁₁	—	2	1	no	—
231	C₄×C₁₁	—	2	1	no	—
232	C₄×C₁₁	—	2	1	no	—
233	C₄×C₁₁	—	2	1	no	—
234	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
235	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
236	C₄×C₁₁	#577	1	2	yes	—
237	Dic₁₁	—	4	2	no	—
238	Dic₁₁	—	4	2	no	—
239	Dic₁₁	—	4	2	no	—
240	C₂×#41	—	22	1	no	—
241	C₂×SD₁₆	—	11	2	no	—
242	C₂×SD₁₆	—	11	2	no	—
243	C₂×M₁₆	—	22	2	no	—
244	C₂²×C₈	—	22	2	no	—
245	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
246	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
247	C₄⋊Q₈	—	11	1	no	—
248	C₄⋊Q₈	—	11	1	no	—
249	C₄⋊Q₈	—	11	1	no	—
250	#127	—	11	1	no	—
251	#125	—	22	1	no	—
252	#124	—	22	1	no	—
253	C₂²⋊Q₈	—	11	2	no	—
254	C₂²⋊Q₈	—	22	1	no	—
255	#121	—	11	2	no	—
256	#121	—	22	1	no	—
257	C₄×Q₈	—	11	2	no	—
258	C₄×Q₈	—	22	1	no	—
259	C₄×D₄	—	11	2	no	—
260	C₄×D₄	—	22	1	no	—
261	#117	—	22	1	no	—
262	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
263	#106	—	11	2	no	—
264	#106	—	11	2	no	—
265	C₄⋊C₈	—	11	2	no	—
266	C₄⋊C₈	—	22	2	no	—
267	C₄⋊C₈	—	22	2	no	—
268	C₄⋊C₈	—	22	1	no	—
269	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
270	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
271	#103	—	11	2	no	—
272	#103	—	11	2	no	—
273	#102	—	11	2	no	—
274	#102	—	11	2	no	—
275	C₂²⋊C₈	—	2	1	no	—
276	C₂²⋊C₈	—	2	1	no	—
277	C₂²⋊C₈	—	11	2	no	—
278	C₂²⋊C₈	—	22	2	no	—
279	C₂²⋊C₈	—	22	2	no	—
280	C₂²⋊C₈	—	22	1	no	—
281	#97	—	2	1	no	—
282	#97	—	2	1	no	—
283	#97	—	22	2	no	—
284	C₄×C₈	—	11	1	no	—
285	C₄×C₈	—	11	1	no	—
286	C₄×C₈	—	11	1	no	—
287	C₄×C₈	—	11	1	no	—
288	C₄×C₈	—	22	2	no	—
289	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
290	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
291	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
292	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
293	C₂×C₁₁	—	4	1	no	—
294	C₂×C₁₁	#2931	1	1	yes	characteristic
295	C₂×C₁₁	#2949	1	1	yes	characteristic
296	C₂×C₁₁	#2957	1	1	yes	characteristic
297	D₁₁	—	8	2	no	—
298	#41	—	88	2	no	—
299	C₂×Q₈	—	11	2	no	—
300	C₂×Q₈	—	22	1	no	—
301	C₂×Q₈	—	22	1	no	—
302	C₂×D₄	—	11	2	no	—
303	C₂×D₄	—	22	2	no	—
304	C₂×D₄	—	22	1	no	—
305	C₂×D₄	—	22	1	no	—
306	C₂²×C₄	—	22	2	no	—
307	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
308	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
309	C₂²×C₄	#4343	1	1	yes	characteristic
310	SD₁₆	—	22	4	no	—
311	SD₁₆	—	22	4	no	—
312	M₁₆	—	88	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	11	2	no	—
318	C₂×C₈	—	11	2	no	—
319	C₂×C₈	—	22	1	no	—
320	C₂×C₈	—	22	1	no	—
321	C₂×C₈	—	88	2	no	—
322	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
323	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
325	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
326	C₄⋊C₄	—	11	2	no	—
327	C₄⋊C₄	—	11	2	no	—
328	C₄⋊C₄	—	22	1	no	—
329	C₄⋊C₄	—	22	1	no	—
330	C₄⋊C₄	—	22	1	no	—
331	C₄⋊C₄	—	22	2	no	—
332	C₄⋊C₄	—	22	1	no	—
333	C₄⋊C₄	—	22	2	no	—
334	C₂²⋊C₄	—	22	2	no	—
335	C₂²⋊C₄	—	22	2	no	—
336	C₂²⋊C₄	—	22	2	no	—
337	C₄²	—	22	2	no	—
338	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
339	C₄²	D₄×D₁₁	1	1	yes	characteristic
340	C₄²	#4343	1	1	yes	characteristic
341	C₄²	C₂³×D₁₁	1	1	yes	characteristic
342	C₁₁	#17971	1	1	yes	characteristic, complemented, Sylow
343	C₂³	—	22	2	no	—
344	C₂³	#66946	1	1	yes	characteristic
345	Q₈	—	22	2	no	—
346	Q₈	—	22	2	no	—
347	Q₈	—	88	1	no	—
348	Q₈	—	88	1	no	—
349	D₄	—	22	2	no	—
350	D₄	—	22	2	no	—
351	D₄	—	88	1	no	—
352	D₄	—	88	1	no	—
353	D₄	—	88	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	22	2	no	—
359	C₂×C₄	—	22	2	no	—
360	C₂×C₄	—	22	2	no	—
361	C₂×C₄	—	44	2	no	—
362	C₂×C₄	—	44	2	no	—
363	C₂×C₄	#66946	1	1	yes	characteristic
364	C₂×C₄	#66946	1	1	yes	characteristic
365	C₂×C₄	#66946	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
367	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
368	C₂×C₄	C₂×#4343	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	4	2	no	—
372	C₈	—	4	2	no	—
373	C₈	—	22	2	no	—
374	C₈	—	22	2	no	—
375	C₈	—	44	1	no	—
376	C₈	—	44	1	no	—
377	C₈	#66960	1	2	yes	—
378	C₈	#66960	1	2	yes	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	44	2	no	—
383	C₂²	—	44	2	no	—
384	C₂²	—	44	2	no	—
385	C₂²	C₂×#66946	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	44	2	no	—
391	C₄	—	44	2	no	—
392	C₄	—	44	2	no	—
393	C₄	C₂×#66960	1	1	yes	characteristic
394	C₄	C₂×#66960	1	1	yes	characteristic
395	C₄	#10623301	1	2	yes	—
396	C₂	—	4	1	no	—
397	C₂	—	88	2	no	—
398	C₂	#13077327	1	1	yes	characteristic
399	C₂	#13077474	1	1	yes	characteristic
400	C₂	#13077532	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	C₂×C₄×D₄
prime radical	#10623008	C₂²
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1568	D₁₁
hypocenter	C₁₁	#17971
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
2. #13077713	C₂	semidirect
4. #13067517	C₂	semidirect
5. #13067505	C₂	semidirect
6. #13067427	C₂	semidirect
7. #13067415	C₂	semidirect
8. #13067088	C₂	semidirect
10. #13066899	C₂	semidirect
11. C₁₁×#1568	C₂	semidirect
150. #1568	D₁₁	semidirect
342. C₁₁	#17971	semidirect

Maximal subgroups

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

subgroup	index
#13079175	2
#13077713	2
#13077694	2
#13067517	2
#13067505	2
#13067427	2
#13067415	2
#13067088	2
#13067084	2
#13066899	2
C₁₁×#1568	2
#17971	11