Subgroups — #14724327

Order 2816.

598 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	#13079199	C₂	1	1	yes	maximal, characteristic
2	#13077545	C₂	1	2	yes	maximal
3	#13067525	C₂	1	1	yes	maximal, characteristic, complemented
4	#13067524	C₂	1	1	yes	maximal, characteristic, complemented
5	#13067522	C₂	1	2	yes	maximal, complemented
6	#13067436	C₂	1	1	yes	maximal, characteristic, complemented
7	#13067433	C₂	1	2	yes	maximal, complemented
8	#13067370	C₂	1	1	yes	maximal, characteristic
9	#13067069	C₂	1	2	yes	maximal
10	D₁₁×#322	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	#10622452	—	2	2	no	—
22	#10622450	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	#10622302	—	2	2	no	—
29	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
32	C₈×Dic₂₂	—	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	#10622118	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	#10622092	C₂²	1	1	yes	characteristic
44	#10622092	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×#4313	—	2	2	no	—
52	C₂×C₈×D₁₁	—	2	1	no	—
53	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₁₁	D₄	1	2	yes	—
56	#66950	C₂×C₄	1	2	yes	—
57	#66949	—	2	2	no	—
58	#66942	—	2	2	no	—
59	#66941	C₂×C₄	1	2	yes	—
60	#66925	—	2	2	no	—
61	C₄²×D₁₁	C₂³	1	1	yes	characteristic
62	C₄×Dic₂₂	—	2	2	no	—
63	C₄×Dic₂₂	C₂³	1	2	yes	—
64	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
65	C₁₁×C₄⋊C₈	—	2	2	no	—
66	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
67	C₁₁×#103	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₁	—	2	2	no	—
69	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
72	#66899	C₂×C₄	1	2	yes	—
73	#66882	C₂×C₄	1	1	yes	characteristic
74	#66880	C₂×C₄	1	2	yes	—
75	#66879	—	2	2	no	—
76	Dic₁₁⋊C₈	—	2	2	no	—
77	Dic₁₁⋊C₈	—	2	2	no	—
78	Dic₁₁⋊C₈	—	2	2	no	—
79	C₈×Dic₁₁	—	2	2	no	—
80	C₈×Dic₁₁	—	2	1	no	—
81	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₁₁	D₄	1	2	yes	—
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	#17961	—	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	#4313	—	4	4	no	—
130	C₈×D₁₁	—	2	4	no	—
131	C₈×D₁₁	—	2	2	no	—
132	C₈×D₁₁	—	2	2	no	—
133	C₈×D₁₁	—	4	2	no	—
134	#3170	—	11	1	no	—
135	#2960	—	11	2	no	—
136	#1593	—	11	1	no	—
137	#1593	—	11	1	no	—
138	#1593	—	11	2	no	—
139	#1579	—	11	2	no	—
140	#1579	—	11	1	no	—
141	#1579	D₁₁	1	1	yes	characteristic, complemented
142	#1566	—	11	1	no	—
143	#1493	—	11	2	no	—
144	C₂×#322	—	11	1	no	—
145	C₂²×D₁₁	C₄×D₄	1	1	yes	characteristic
146	Q₈×C₁₁	—	2	2	no	—
147	Q₈×C₁₁	—	2	2	no	—
148	C₂×C₄×C₁₁	—	2	2	no	—
149	C₂×C₄×C₁₁	—	2	2	no	—
150	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
153	C₂×Dic₁₁	—	2	2	no	—
154	C₂×Dic₁₁	—	2	2	no	—
155	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
156	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
157	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
158	C₄×D₁₁	—	2	1	no	—
159	C₄×D₁₁	—	2	1	no	—
160	C₄×D₁₁	—	2	2	no	—
161	C₄×D₁₁	#137	1	4	yes	—
162	Dic₂₂	—	2	2	no	—
163	Dic₂₂	—	2	2	no	—
164	Dic₂₂	—	4	4	no	—
165	C₈×C₁₁	—	2	1	no	—
166	C₈×C₁₁	—	2	2	no	—
167	C₈×C₁₁	—	4	1	no	—
168	C₈×C₁₁	—	4	2	no	—
169	C₈×C₁₁	#131	1	2	yes	—
170	C₁₁⋊C₈	—	2	1	no	—
171	C₁₁⋊C₈	—	2	2	no	—
172	C₁₁⋊C₈	—	4	1	no	—
173	C₁₁⋊C₈	—	4	2	no	—
174	C₁₁⋊C₈	#131	1	2	yes	—
175	#533	—	11	2	no	—
176	#500	—	11	1	no	—
177	#494	—	11	1	no	—
178	#487	—	11	2	no	—
179	#467	—	11	1	no	—
180	#467	—	11	1	no	—
181	#467	—	11	2	no	—
182	#446	—	11	2	no	—
183	#446	—	11	2	no	—
184	#446	—	22	2	no	—
185	C₈×Q₈	—	11	2	no	—
186	C₈×Q₈	—	22	2	no	—
187	C₈×Q₈	C₂×D₁₁	1	2	yes	—
188	C₄×Q₁₆	—	11	2	no	—
189	C₄×Q₁₆	—	11	1	no	—
190	C₄×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
191	#424	—	22	2	no	—
192	#423	—	11	2	no	—
193	#405	—	22	2	no	—
194	C₂×C₄×C₈	—	11	1	no	—
195	C₂×C₄×C₈	—	11	1	no	—
196	C₂×C₄×C₈	—	11	1	no	—
197	#334	—	11	2	no	—
198	#334	—	11	1	no	—
199	#334	C₂×D₁₁	1	1	yes	characteristic
200	Q₈⋊C₈	—	11	2	no	—
201	Q₈⋊C₈	—	11	2	no	—
202	Q₈⋊C₈	—	11	2	no	—
203	Q₈⋊C₈	C₂×D₁₁	1	2	yes	—
204	#322	—	11	1	no	—
205	#322	—	11	1	no	—
206	#322	—	11	1	no	—
207	#322	C₂×D₁₁	1	1	yes	characteristic
208	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
209	C₂×D₁₁	—	2	1	no	—
210	C₂×D₁₁	—	2	1	no	—
211	C₂×D₁₁	#441	1	2	yes	—
212	C₄×C₁₁	—	2	1	no	—
213	C₄×C₁₁	—	4	2	no	—
214	C₄×C₁₁	—	4	2	no	—
215	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
216	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
217	C₄×C₁₁	C₂×#137	1	2	yes	—
218	Dic₁₁	—	2	1	no	—
219	Dic₁₁	—	4	2	no	—
220	Dic₁₁	—	4	2	no	—
221	Dic₁₁	#441	1	2	yes	—
222	Dic₁₁	#444	1	2	yes	—
223	C₂×Q₁₆	—	11	2	no	—
224	C₂×Q₁₆	—	11	1	no	—
225	C₂×Q₁₆	C₄×D₁₁	1	1	yes	characteristic
226	C₂×M₁₆	—	22	2	no	—
227	C₂²×C₈	—	11	1	no	—
228	C₂²×C₈	—	11	1	no	—
229	C₂²×C₈	—	11	2	no	—
230	C₂²×C₈	—	22	1	no	—
231	C₄⋊Q₈	—	11	2	no	—
232	C₄⋊Q₈	—	11	2	no	—
233	#125	—	22	2	no	—
234	C₂²⋊Q₈	—	11	2	no	—
235	C₂²⋊Q₈	—	11	2	no	—
236	C₂²⋊Q₈	—	22	2	no	—
237	C₄×Q₈	—	11	2	no	—
238	C₄×Q₈	—	22	2	no	—
239	C₄×Q₈	C₂²×D₁₁	1	2	yes	—
240	#117	—	22	2	no	—
241	C₂×C₄²	—	11	1	no	—
242	#107	—	11	1	no	—
243	#107	—	11	2	no	—
244	#107	C₄×D₁₁	1	1	yes	characteristic
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	11	2	no	—
247	C₄⋊C₈	—	22	2	no	—
248	C₄⋊C₈	—	22	2	no	—
249	C₄⋊C₈	—	22	2	no	—
250	C₄⋊C₈	—	22	2	no	—
251	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
252	#103	—	11	2	no	—
253	#103	—	11	2	no	—
254	#103	—	11	2	no	—
255	#103	C₄×D₁₁	1	2	yes	—
256	#97	—	22	2	no	—
257	#97	—	22	2	no	—
258	C₄×C₈	—	2	2	no	—
259	C₄×C₈	—	11	1	no	—
260	C₄×C₈	—	11	1	no	—
261	C₄×C₈	—	11	2	no	—
262	C₄×C₈	—	11	1	no	—
263	C₄×C₈	—	11	1	no	—
264	C₄×C₈	—	11	1	no	—
265	C₄×C₈	—	22	1	no	—
266	C₄×C₈	—	22	2	no	—
267	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
270	C₂×C₁₁	#2931	1	1	yes	characteristic
271	C₂×C₁₁	C₂×#441	1	1	yes	characteristic
272	C₂×C₁₁	C₂×#444	1	1	yes	characteristic
273	D₁₁	—	2	2	no	—
274	C₂×Q₈	—	11	2	no	—
275	C₂×Q₈	—	22	2	no	—
276	C₂×Q₈	C₂×C₄×D₁₁	1	2	yes	—
277	C₂²×C₄	—	11	1	no	—
278	C₂²×C₄	—	11	1	no	—
279	C₂²×C₄	—	11	1	no	—
280	Q₁₆	—	2	2	no	—
281	Q₁₆	—	22	4	no	—
282	Q₁₆	—	22	2	no	—
283	M₁₆	—	44	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	11	1	no	—
287	C₂×C₈	—	11	1	no	—
288	C₂×C₈	—	11	2	no	—
289	C₂×C₈	—	22	2	no	—
290	C₂×C₈	—	22	4	no	—
291	C₂×C₈	—	22	2	no	—
292	C₂×C₈	—	22	2	no	—
293	C₂×C₈	—	22	1	no	—
294	C₂×C₈	—	44	2	no	—
295	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
297	C₂×C₈	D₄×D₁₁	1	2	yes	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	11	2	no	—
300	C₄⋊C₄	—	22	2	no	—
301	C₄⋊C₄	—	22	2	no	—
302	C₄⋊C₄	—	22	2	no	—
303	C₄⋊C₄	—	22	2	no	—
304	C₄⋊C₄	—	22	2	no	—
305	C₄⋊C₄	—	22	2	no	—
306	C₄⋊C₄	C₂×C₄×D₁₁	1	2	yes	—
307	C₂²⋊C₄	—	22	2	no	—
308	C₂²⋊C₄	—	22	2	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	11	1	no	—
311	C₄²	—	11	1	no	—
312	C₄²	—	11	1	no	—
313	C₄²	—	22	2	no	—
314	C₄²	C₂³×D₁₁	1	1	yes	characteristic
315	C₁₁	#17961	1	1	yes	characteristic, complemented, Sylow
316	C₂³	—	11	1	no	—
317	Q₈	—	2	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	22	2	no	—
320	Q₈	—	22	2	no	—
321	Q₈	—	44	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	11	1	no	—
325	C₂×C₄	—	11	1	no	—
326	C₂×C₄	—	11	4	no	—
327	C₂×C₄	—	11	1	no	—
328	C₂×C₄	—	22	2	no	—
329	C₂×C₄	—	22	1	no	—
330	C₂×C₄	—	22	1	no	—
331	C₂×C₄	—	22	2	no	—
332	C₂×C₄	—	22	2	no	—
333	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
335	C₂×C₄	#41×D₁₁	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	11	2	no	—
341	C₈	—	22	1	no	—
342	C₈	—	22	2	no	—
343	C₈	—	44	1	no	—
344	C₈	—	44	2	no	—
345	C₈	#66954	1	2	yes	—
346	C₂²	—	11	2	no	—
347	C₂²	—	22	1	no	—
348	C₂²	—	22	1	no	—
349	C₂²	C₄×D₄×D₁₁	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	11	2	no	—
354	C₄	—	11	2	no	—
355	C₄	—	22	1	no	—
356	C₄	—	44	2	no	—
357	C₄	—	44	2	no	—
358	C₄	C₂×#66954	1	1	yes	characteristic
359	C₄	D₁₁×#131	1	1	yes	characteristic
360	C₄	D₁₁×#137	1	2	yes	—
361	C₂	—	22	2	no	—
362	C₂	#13077323	1	1	yes	characteristic
363	C₂	D₁₁×#441	1	1	yes	characteristic
364	C₂	D₁₁×#444	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	#17961
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. #13067522	C₂	semidirect
6. #13067436	C₂	semidirect
7. #13067433	C₂	semidirect
11. C₁₁×#1579	C₂	semidirect
141. #1579	D₁₁	semidirect
315. C₁₁	#17961	semidirect

Maximal subgroups

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

subgroup	index
#13079199	2
#13077545	2
#13067525	2
#13067524	2
#13067522	2
#13067436	2
#13067433	2
#13067370	2
#13067069	2
D₁₁×#322	2
C₁₁×#1579	2
#17961	11