Subgroups — #14907183

Order 2816.

652 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13081749	C₂	1	1	yes	maximal, characteristic
2	D₁₁×#443	C₂	1	2	yes	maximal, complemented
3	#13070503	C₂	1	1	yes	maximal, characteristic, complemented
4	#13070500	C₂	1	2	yes	maximal, complemented
5	#13070497	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070496	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070494	C₂	1	2	yes	maximal, complemented
8	#13070409	C₂	1	1	yes	maximal, characteristic, complemented
9	#13070342	C₂	1	2	yes	maximal, complemented
10	#13070209	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₁×#2179	C₂	1	1	yes	maximal, characteristic, complemented
12	D₁₁×#135	C₂²	1	2	yes	—
13	D₁₁×#131	—	2	2	no	—
14	C₁₁×#508	C₂²	1	1	yes	characteristic
15	C₁₁×#443	C₂²	1	2	yes	complemented
16	#10622809	C₂²	1	2	yes	complemented
17	#10622807	C₂²	1	2	yes	—
18	#10622757	—	2	2	no	—
19	#10622531	C₄	1	2	yes	complemented
20	#10622528	C₄	1	2	yes	complemented
21	#10622527	C₄	1	2	yes	complemented
22	#10622524	C₄	1	2	yes	complemented
23	#10622517	C₂²	1	1	yes	characteristic
24	C₂×#66862	C₂²	1	1	yes	characteristic
25	C₂×#66862	C₂²	1	1	yes	characteristic
26	#10622508	C₂²	1	2	yes	complemented
27	D₁₁×#108	C₂²	1	2	yes	—
28	#10622437	—	2	2	no	—
29	D₁₁×#104	—	2	2	no	—
30	#10622313	C₂²	1	2	yes	complemented
31	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
32	C₁₁×#368	C₂²	1	1	yes	characteristic, complemented
33	C₁₁×#359	C₂²	1	2	yes	complemented
34	C₁₁×#346	C₂²	1	1	yes	characteristic, complemented
35	#10622205	C₂²	1	2	yes	complemented
36	#10622148	C₂²	1	2	yes	complemented
37	#10622144	C₂²	1	1	yes	characteristic, complemented
38	C₁₆×Dic₁₁	C₂²	1	1	yes	characteristic, complemented
39	C₁₆×Dic₁₁	C₂²	1	1	yes	characteristic, complemented
40	#10622135	C₂²	1	2	yes	complemented
41	#10622131	C₂²	1	2	yes	complemented
42	#10622099	C₂²	1	1	yes	characteristic, complemented
43	#41×D₁₁	—	2	2	no	—
44	C₁₁×#135	C₂³	1	2	yes	—
45	C₁₁×#131	—	2	2	no	—
46	#67003	—	2	2	no	—
47	#67001	—	2	2	no	—
48	#66972	C₂×C₄	1	2	yes	—
49	Q₁₆×D₁₁	C₂×C₄	1	2	yes	—
50	#66969	—	2	2	no	—
51	SD₁₆×D₁₁	—	2	2	no	—
52	#66965	C₂×C₄	1	2	yes	—
53	D₈×D₁₁	C₂×C₄	1	2	yes	—
54	#66960	—	2	2	no	—
55	#66960	—	4	2	no	—
56	M₁₆×D₁₁	—	2	2	no	—
57	M₁₆×D₁₁	—	4	2	no	—
58	#66957	C₂³	1	2	yes	—
59	#66954	—	2	2	no	—
60	#66954	—	4	2	no	—
61	C₂×C₈×D₁₁	—	4	2	no	—
62	C₂×C₈×D₁₁	D₄	1	1	yes	characteristic
63	C₂×C₈×D₁₁	C₂³	1	1	yes	characteristic
64	C₄²×D₁₁	D₄	1	1	yes	characteristic
65	C₁₁×Q₃₂	C₂×C₄	1	1	yes	characteristic, complemented
66	C₁₁×SD₃₂	C₂×C₄	1	2	yes	complemented
67	C₁₁×D₁₆	C₂×C₄	1	1	yes	characteristic, complemented
68	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
70	C₁₁×#108	C₂³	1	2	yes	—
71	C₁₁×#104	—	2	2	no	—
72	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
73	#66901	—	2	2	no	—
74	#66893	C₂×C₄	1	2	yes	complemented
75	#66892	C₂×C₄	1	2	yes	complemented
76	#66891	C₂×C₄	1	2	yes	complemented
77	#66890	C₂×C₄	1	2	yes	complemented
78	#66889	—	2	2	no	—
79	#66886	—	2	2	no	—
80	#66883	C₂³	1	2	yes	—
81	C₈×Dic₁₁	D₄	1	1	yes	characteristic
82	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
83	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
84	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
85	#66869	—	2	2	no	—
86	C₄×C₁₁⋊C₈	D₄	1	1	yes	characteristic
87	Dic₈₈	C₂×C₄	1	1	yes	characteristic, complemented
88	#66864	C₂×C₄	1	2	yes	complemented
89	D₁₇₆	C₂×C₄	1	1	yes	characteristic, complemented
90	#66862	—	2	2	no	—
91	#66862	C₂×C₄	1	2	yes	—
92	#66862	C₂×C₄	1	2	yes	—
93	#33463	—	11	1	no	maximal, Sylow
94	C₁₁×#41	—	2	2	no	—
95	#4343	—	2	2	no	—
96	Q₈×D₁₁	—	2	2	no	—
97	#4341	—	2	2	no	—
98	#4341	—	4	2	no	—
99	D₄×D₁₁	—	2	2	no	—
100	D₄×D₁₁	—	4	2	no	—
101	#4339	—	2	2	no	—
102	#4339	—	4	2	no	—
103	C₂×C₄×D₁₁	—	2	1	no	—
104	C₂×C₄×D₁₁	—	4	2	no	—
105	C₂×C₄×D₁₁	C₂×D₄	1	1	yes	characteristic
106	C₁₁×Q₁₆	C₂²×C₄	1	2	yes	—
107	C₁₁×SD₁₆	—	2	2	no	—
108	C₁₁×D₈	C₂²×C₄	1	2	yes	—
109	C₁₁×M₁₆	—	2	2	no	—
110	C₁₁×M₁₆	—	4	2	no	—
111	C₂×C₈×C₁₁	—	4	2	no	—
112	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
114	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
115	#4326	—	2	2	no	—
116	#4325	—	2	2	no	—
117	#4324	—	2	2	no	—
118	#4323	—	2	2	no	—
119	C₄×Dic₁₁	—	2	1	no	—
120	C₄×Dic₁₁	C₂×D₄	1	1	yes	characteristic
121	#4318	—	2	2	no	—
122	#4318	—	4	2	no	—
123	C₂×C₁₁⋊C₈	—	4	2	no	—
124	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
126	Dic₄₄	C₂²×C₄	1	2	yes	—
127	D₈₈	C₂²×C₄	1	2	yes	—
128	#4314	—	2	2	no	—
129	#4313	—	4	2	no	—
130	#4313	—	4	2	no	—
131	#4313	—	4	2	no	—
132	C₈×D₁₁	—	4	2	no	—
133	C₈×D₁₁	—	4	2	no	—
134	C₈×D₁₁	—	4	2	no	—
135	C₈×D₁₁	D₈	1	2	yes	—
136	C₈×D₁₁	D₈	1	2	yes	—
137	C₈×D₁₁	C₂²×C₄	1	2	yes	—
138	C₈×D₁₁	#41	1	2	yes	—
139	C₁₁×C₁₆	—	2	1	no	—
140	C₁₁×C₁₆	C₂²×C₄	1	1	yes	characteristic
141	C₁₁×C₁₆	C₂²×C₄	1	1	yes	characteristic
142	C₁₁⋊C₁₆	—	2	1	no	—
143	C₁₁⋊C₁₆	C₂²×C₄	1	2	yes	—
144	#3414	—	11	1	no	—
145	C₂×#443	—	11	2	no	—
146	#2179	—	11	2	no	—
147	#2179	—	11	1	no	—
148	#2179	D₁₁	1	1	yes	characteristic, complemented
149	#2178	—	11	1	no	—
150	#2178	—	11	1	no	—
151	#2178	—	11	2	no	—
152	#2161	—	11	1	no	—
153	#2143	—	11	2	no	—
154	C₄×M₃₂	—	11	1	no	—
155	C₂²×D₁₁	—	4	2	no	—
156	C₂²×D₁₁	#135	1	1	yes	characteristic
157	Q₈×C₁₁	—	2	2	no	—
158	D₄×C₁₁	—	2	2	no	—
159	D₄×C₁₁	—	4	2	no	—
160	C₂×C₄×C₁₁	—	2	1	no	—
161	C₂×C₄×C₁₁	—	4	2	no	—
162	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
163	C₂²⋊D₁₁	—	4	2	no	—
164	C₂²⋊D₁₁	—	4	2	no	—
165	C₂²⋊D₁₁	—	4	2	no	—
166	C₂×Dic₁₁	—	2	1	no	—
167	C₂×Dic₁₁	—	4	2	no	—
168	C₂×Dic₁₁	#135	1	1	yes	characteristic
169	D₄₄	—	2	2	no	—
170	D₄₄	—	4	2	no	—
171	C₄×D₁₁	—	4	2	no	—
172	C₄×D₁₁	—	4	2	no	—
173	C₄×D₁₁	—	4	2	no	—
174	C₄×D₁₁	—	4	2	no	—
175	C₄×D₁₁	C₄×D₄	1	2	yes	—
176	C₄×D₁₁	C₂×D₈	1	2	yes	—
177	Dic₂₂	—	2	2	no	—
178	Dic₂₂	—	4	2	no	—
179	C₈×C₁₁	—	4	2	no	—
180	C₈×C₁₁	C₂×D₈	1	2	yes	—
181	C₈×C₁₁	C₂³×C₄	1	1	yes	characteristic
182	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
183	C₁₁⋊C₈	—	4	2	no	—
184	C₁₁⋊C₈	C₄×D₄	1	2	yes	—
185	C₁₁⋊C₈	C₂×D₈	1	1	yes	characteristic
186	C₁₁⋊C₈	C₂×D₈	1	1	yes	characteristic
187	C₂×#135	—	11	2	no	—
188	C₂×#131	—	22	2	no	—
189	#510	—	11	2	no	—
190	#510	—	11	2	no	—
191	#509	—	11	2	no	—
192	#509	—	11	2	no	—
193	#508	—	11	2	no	—
194	#508	—	11	1	no	—
195	#508	C₂×D₁₁	1	1	yes	characteristic
196	C₂×M₃₂	—	11	1	no	—
197	C₂×M₃₂	—	11	1	no	—
198	#443	—	11	2	no	—
199	#443	—	11	2	no	—
200	#443	—	11	2	no	—
201	#443	—	22	2	no	—
202	#443	—	22	2	no	—
203	#443	C₂×D₁₁	1	2	yes	complemented
204	C₂×#108	—	11	2	no	—
205	C₂×#104	—	22	2	no	—
206	C₂×C₄×C₈	—	11	1	no	—
207	#368	—	11	1	no	—
208	#368	—	11	2	no	—
209	#368	C₂×D₁₁	1	1	yes	characteristic, complemented
210	#359	—	11	2	no	—
211	#359	—	11	2	no	—
212	#359	—	11	2	no	—
213	#359	C₂×D₁₁	1	2	yes	complemented
214	#346	—	11	1	no	—
215	#346	C₂×D₁₁	1	1	yes	characteristic, complemented
216	C₄×C₁₆	—	11	1	no	—
217	C₄×C₁₆	—	11	1	no	—
218	C₂²×C₁₁	—	4	2	no	—
219	C₂²×C₁₁	C₂×#135	1	1	yes	characteristic
220	C₂×D₁₁	—	4	1	no	—
221	C₂×D₁₁	—	4	2	no	—
222	C₂×D₁₁	—	8	4	no	—
223	C₂×D₁₁	C₄×D₈	1	2	yes	—
224	C₄×C₁₁	—	4	1	no	—
225	C₄×C₁₁	—	4	2	no	—
226	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
227	C₄×C₁₁	C₂²×D₈	1	1	yes	characteristic
228	Dic₁₁	—	2	2	no	—
229	Dic₁₁	—	4	2	no	—
230	Dic₁₁	C₄×D₈	1	2	yes	—
231	C₂×#41	—	22	2	no	—
232	#135	—	11	2	no	—
233	#135	—	11	2	no	—
234	#135	—	11	2	no	—
235	#135	—	22	2	no	—
236	#135	—	22	2	no	—
237	#135	C₂²×D₁₁	1	2	yes	—
238	C₂×Q₁₆	—	11	2	no	—
239	C₂×SD₁₆	—	22	2	no	—
240	C₂×D₈	—	11	2	no	—
241	#131	—	2	2	no	—
242	#131	—	22	2	no	—
243	#131	—	22	2	no	—
244	#131	—	22	2	no	—
245	#131	—	44	2	no	—
246	#131	—	44	2	no	—
247	C₂×M₁₆	—	22	2	no	—
248	C₂×M₁₆	—	44	2	no	—
249	C₂²×C₈	—	11	1	no	—
250	C₂²×C₈	—	11	1	no	—
251	C₂²×C₈	—	44	2	no	—
252	C₂×C₄²	—	11	1	no	—
253	Q₃₂	—	11	2	no	—
254	Q₃₂	—	11	1	no	—
255	Q₃₂	C₄×D₁₁	1	1	yes	characteristic, complemented
256	SD₃₂	—	11	2	no	—
257	SD₃₂	—	11	2	no	—
258	SD₃₂	—	11	2	no	—
259	SD₃₂	C₄×D₁₁	1	2	yes	complemented
260	D₁₆	—	11	2	no	—
261	D₁₆	—	11	1	no	—
262	D₁₆	C₄×D₁₁	1	1	yes	characteristic, complemented
263	M₃₂	—	11	2	no	—
264	M₃₂	—	11	2	no	—
265	M₃₂	—	22	2	no	—
266	C₂×C₁₆	—	11	1	no	—
267	C₂×C₁₆	—	11	1	no	—
268	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
269	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
270	#108	—	11	2	no	—
271	#108	—	22	2	no	—
272	#108	C₂²×D₁₁	1	2	yes	—
273	#104	—	2	2	no	—
274	#104	—	22	2	no	—
275	#104	—	22	2	no	—
276	#104	—	22	2	no	—
277	C₄×C₈	—	11	1	no	—
278	C₄×C₈	—	11	1	no	—
279	C₄×C₈	—	11	1	no	—
280	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
281	C₂×C₁₁	—	2	1	no	—
282	C₂×C₁₁	—	8	2	no	—
283	C₂×C₁₁	C₂×C₄×D₈	1	1	yes	characteristic
284	D₁₁	—	2	2	no	—
285	D₁₁	—	8	2	no	—
286	#41	—	2	2	no	—
287	#41	—	22	2	no	—
288	#41	—	22	2	no	—
289	#41	—	22	2	no	—
290	#41	—	44	2	no	—
291	#41	—	44	2	no	—
292	C₂×Q₈	—	22	2	no	—
293	C₂×D₄	—	22	2	no	—
294	C₂×D₄	—	44	2	no	—
295	C₂²×C₄	—	11	1	no	—
296	C₂²×C₄	—	22	1	no	—
297	C₂²×C₄	—	44	2	no	—
298	Q₁₆	—	11	2	no	—
299	Q₁₆	—	22	2	no	—
300	Q₁₆	C₂×C₄×D₁₁	1	2	yes	—
301	SD₁₆	—	2	2	no	—
302	SD₁₆	—	22	2	no	—
303	SD₁₆	—	22	2	no	—
304	SD₁₆	—	22	2	no	—
305	D₈	—	11	2	no	—
306	D₈	—	22	2	no	—
307	D₈	C₂×C₄×D₁₁	1	2	yes	—
308	M₁₆	—	2	2	no	—
309	M₁₆	—	4	2	no	—
310	M₁₆	—	22	2	no	—
311	M₁₆	—	44	2	no	—
312	M₁₆	—	44	2	no	—
313	M₁₆	—	44	2	no	—
314	M₁₆	—	44	2	no	—
315	C₂×C₈	—	4	2	no	—
316	C₂×C₈	—	11	1	no	—
317	C₂×C₈	—	11	1	no	—
318	C₂×C₈	—	11	2	no	—
319	C₂×C₈	—	11	2	no	—
320	C₂×C₈	—	11	2	no	—
321	C₂×C₈	—	11	2	no	—
322	C₂×C₈	—	44	2	no	—
323	C₂×C₈	—	44	2	no	—
324	C₂×C₈	—	44	2	no	—
325	C₂×C₈	—	44	2	no	—
326	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
327	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
328	C₄²	—	11	1	no	—
329	C₄²	—	22	1	no	—
330	C₄²	D₄×D₁₁	1	1	yes	characteristic
331	C₁₆	—	2	1	no	—
332	C₁₆	—	11	2	no	—
333	C₁₆	—	22	1	no	—
334	C₁₆	C₂×C₄×D₁₁	1	1	yes	characteristic
335	C₁₆	C₂×C₄×D₁₁	1	1	yes	characteristic
336	C₁₁	#33463	1	1	yes	characteristic, complemented, Sylow
337	C₂³	—	11	1	no	—
338	C₂³	—	44	2	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	22	2	no	—
341	Q₈	—	44	2	no	—
342	D₄	—	2	2	no	—
343	D₄	—	4	2	no	—
344	D₄	—	22	2	no	—
345	D₄	—	44	2	no	—
346	D₄	—	44	2	no	—
347	D₄	—	44	2	no	—
348	D₄	—	44	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	11	1	no	—
352	C₂×C₄	—	11	2	no	—
353	C₂×C₄	—	11	2	no	—
354	C₂×C₄	—	22	1	no	—
355	C₂×C₄	—	44	2	no	—
356	C₂×C₄	—	44	2	no	—
357	C₂×C₄	—	44	2	no	—
358	C₂×C₄	—	44	2	no	—
359	C₂×C₄	—	44	2	no	—
360	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
361	C₈	—	4	2	no	—
362	C₈	—	11	1	no	—
363	C₈	—	11	1	no	—
364	C₈	—	11	2	no	—
365	C₈	—	44	2	no	—
366	C₈	D₈×D₁₁	1	2	yes	—
367	C₈	C₂²×C₄×D₁₁	1	1	yes	characteristic
368	C₈	#41×D₁₁	1	1	yes	characteristic
369	C₂²	—	4	2	no	—
370	C₂²	—	11	2	no	—
371	C₂²	—	44	1	no	—
372	C₂²	—	44	2	no	—
373	C₂²	—	88	4	no	—
374	C₂²	D₁₁×#135	1	1	yes	characteristic
375	C₄	—	4	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	11	2	no	—
378	C₄	—	22	2	no	—
379	C₄	—	44	2	no	—
380	C₄	C₄×D₄×D₁₁	1	1	yes	characteristic
381	C₄	C₂×D₈×D₁₁	1	1	yes	characteristic
382	C₂	—	2	1	no	—
383	C₂	—	8	2	no	—
384	C₂	—	22	2	no	—
385	C₂	—	88	2	no	—
386	C₂	C₄×D₈×D₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₁₁
commutator	C₈×C₁₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#2179	C₂
socle	C₂×C₁₁	C₂×C₄×D₈
prime radical	#13081749	C₂
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2179	D₁₁
hypocenter	C₁₁	#33463
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2179

Complements

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

N	H ≅ G/N	type
2. D₁₁×#443	C₂	semidirect
3. #13070503	C₂	semidirect
4. #13070500	C₂	semidirect
5. #13070497	C₂	semidirect
6. #13070496	C₂	semidirect
7. #13070494	C₂	semidirect
8. #13070409	C₂	semidirect
9. #13070342	C₂	semidirect
10. #13070209	C₂	semidirect
11. C₁₁×#2179	C₂	semidirect
15. C₁₁×#443	C₂²	semidirect
16. #10622809	C₂²	semidirect
19. #10622531	C₄	semidirect
20. #10622528	C₄	semidirect
21. #10622527	C₄	semidirect
22. #10622524	C₄	semidirect
26. #10622508	C₂²	semidirect
30. #10622313	C₂²	semidirect
32. C₁₁×#368	C₂²	semidirect
33. C₁₁×#359	C₂²	semidirect
34. C₁₁×#346	C₂²	semidirect
35. #10622205	C₂²	semidirect
36. #10622148	C₂²	semidirect
37. #10622144	C₂²	semidirect
38. C₁₆×Dic₁₁	C₂²	semidirect
39. C₁₆×Dic₁₁	C₂²	semidirect
40. #10622135	C₂²	semidirect
41. #10622131	C₂²	semidirect
42. #10622099	C₂²	semidirect
65. C₁₁×Q₃₂	C₂×C₄	semidirect
66. C₁₁×SD₃₂	C₂×C₄	semidirect
67. C₁₁×D₁₆	C₂×C₄	semidirect
74. #66893	C₂×C₄	semidirect
75. #66892	C₂×C₄	semidirect
76. #66891	C₂×C₄	semidirect
77. #66890	C₂×C₄	semidirect
87. Dic₈₈	C₂×C₄	semidirect
88. #66864	C₂×C₄	semidirect
89. D₁₇₆	C₂×C₄	semidirect
148. #2179	D₁₁	semidirect
203. #443	C₂×D₁₁	semidirect
209. #368	C₂×D₁₁	semidirect
213. #359	C₂×D₁₁	semidirect
215. #346	C₂×D₁₁	semidirect
255. Q₃₂	C₄×D₁₁	semidirect
259. SD₃₂	C₄×D₁₁	semidirect
262. D₁₆	C₄×D₁₁	semidirect
336. C₁₁	#33463	semidirect

Maximal subgroups

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

subgroup	index
#13081749	2
D₁₁×#443	2
#13070503	2
#13070500	2
#13070497	2
#13070496	2
#13070494	2
#13070409	2
#13070342	2
#13070209	2
C₁₁×#2179	2
#33463	11