Subgroups — #14754146

Order 2816.

608 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077041	C₂	1	1	yes	maximal, characteristic
2	#13068762	C₂	1	2	yes	maximal, complemented
3	#13068741	C₂	1	1	yes	maximal, characteristic, complemented
4	#13068739	C₂	1	1	yes	maximal, characteristic, complemented
5	#13068719	C₂	1	2	yes	maximal
6	#13068663	C₂	1	1	yes	maximal, characteristic
7	#13068659	C₂	1	1	yes	maximal, characteristic
8	#13068500	C₂	1	1	yes	maximal, characteristic, complemented
9	#13068417	C₂	1	2	yes	maximal
10	#13068371	C₂	1	2	yes	maximal, complemented
11	C₁₁×#1773	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#66954	C₂²	1	1	yes	characteristic
13	C₂×#66954	C₂²	1	1	yes	characteristic
14	C₄×#4339	C₂²	1	1	yes	characteristic
15	C₂×C₁₁×#108	C₂²	1	1	yes	characteristic
16	C₁₁×#427	C₂²	1	1	yes	characteristic
17	C₁₁×#405	C₂²	1	2	yes	—
18	C₂×C₁₁×#97	C₂²	1	1	yes	characteristic
19	#10622745	C₂²	1	2	yes	—
20	C₂×#66883	C₂²	1	1	yes	characteristic
21	#10622719	C₂²	1	1	yes	characteristic
22	#10622714	C₂²	1	2	yes	—
23	#10622714	C₂²	1	1	yes	characteristic
24	#10622714	C₂²	1	1	yes	characteristic
25	#10622619	C₂²	1	2	yes	—
26	C₄×#4318	C₂²	1	1	yes	characteristic
27	#10622321	C₄	1	2	yes	complemented
28	#10622321	C₄	1	2	yes	complemented
29	#10622320	C₄	1	2	yes	complemented
30	D₁₁×#97	C₄	1	2	yes	complemented
31	C₁₁×#339	C₂²	1	2	yes	—
32	C₁₁⋊#338	C₂²	1	2	yes	—
33	#10622197	C₂²	1	2	yes	—
34	#10622195	C₂²	1	2	yes	—
35	#10622171	C₂²	1	2	yes	—
36	#10622169	C₂²	1	2	yes	—
37	#10622169	C₂²	1	2	yes	—
38	#10622161	C₂²	1	2	yes	—
39	C₂×#4339	C₂³	1	1	yes	characteristic
40	C₂×C₁₁×M₁₆	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
43	C₁₁×#117	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
45	C₄×C₂²⋊D₁₁	—	2	2	no	—
46	#66979	C₂³	1	2	yes	—
47	#66979	C₂³	1	1	yes	characteristic
48	C₂×#4318	C₂³	1	2	yes	—
49	C₂×#4318	C₂³	1	1	yes	characteristic
50	C₂×#4318	C₂³	1	1	yes	characteristic
51	#66954	—	2	1	no	—
52	#66954	—	2	1	no	—
53	#66954	—	2	1	no	—
54	#66954	—	2	1	no	—
55	#66954	—	2	2	no	—
56	#66954	—	2	2	no	—
57	C₂×#4313	C₂×C₄	1	2	yes	—
58	C₂×#4313	C₂×C₄	1	2	yes	—
59	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
61	C₄×D₄₄	—	2	1	no	—
62	#66925	C₂×C₄	1	2	yes	—
63	C₄²×D₁₁	C₂×C₄	1	2	yes	—
64	C₄×Dic₂₂	—	2	1	no	—
65	C₁₁×#108	—	2	1	no	—
66	C₁₁×#108	—	2	1	no	—
67	C₁₁×#107	—	2	1	no	—
68	C₁₁×#106	—	2	1	no	—
69	C₁₁×#97	—	2	2	no	—
70	C₁₁×#97	C₂×C₄	1	1	yes	characteristic
71	C₁₁×#97	C₂×C₄	1	1	yes	characteristic
72	C₁₁×#97	C₂×C₄	1	1	yes	characteristic
73	C₁₁×#97	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₁₁	—	2	2	no	—
75	#66888	—	2	2	no	—
76	#66887	—	2	2	no	—
77	#66883	—	2	1	no	—
78	#66883	—	2	1	no	—
79	#66882	—	2	1	no	—
80	#66881	—	2	1	no	—
81	#66879	—	2	2	no	—
82	#66879	C₂×C₄	1	2	yes	—
83	#66879	C₂×C₄	1	2	yes	—
84	C₈×Dic₁₁	—	2	2	no	—
85	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
86	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
87	#66870	—	2	4	no	—
88	#66867	C₂×C₄	1	2	yes	—
89	C₄×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
90	#20788	—	11	1	no	maximal, Sylow
91	C₂²×C₄×C₁₁	—	2	1	no	—
92	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
93	C₂×C₂²⋊D₁₁	#41	1	2	yes	—
94	#4339	—	2	1	no	—
95	#4339	—	2	1	no	—
96	#4339	—	2	1	no	—
97	#4339	—	2	1	no	—
98	C₂×D₄₄	#41	1	1	yes	characteristic
99	C₂×C₄×D₁₁	—	2	2	no	—
100	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
101	C₂×Dic₂₂	#41	1	1	yes	characteristic
102	C₁₁×M₁₆	—	2	2	no	—
103	C₁₁×M₁₆	—	2	2	no	—
104	C₂×C₈×C₁₁	—	2	2	no	—
105	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
110	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
111	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
112	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
113	C₂×C₈×C₁₁	#41	1	2	yes	—
114	C₂×C₈×C₁₁	#41	1	2	yes	—
115	C₁₁×C₄⋊C₄	—	2	2	no	—
116	C₁₁×C₂²⋊C₄	—	2	2	no	—
117	C₄²×C₁₁	—	2	1	no	—
118	C₄²×C₁₁	—	2	2	no	—
119	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
121	C₂²⋊Dic₁₁	—	2	1	no	—
122	C₂²⋊Dic₁₁	—	2	2	no	—
123	(C₂×D₁₁)⋊C₄	—	2	2	no	—
124	C₄⋊Dic₁₁	—	2	1	no	—
125	C₄⋊Dic₁₁	—	2	2	no	—
126	Dic₁₁⋊C₄	—	2	2	no	—
127	C₄×Dic₁₁	—	2	2	no	—
128	C₄×Dic₁₁	—	2	2	no	—
129	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
130	#4318	—	2	1	no	—
131	#4318	—	2	1	no	—
132	#4318	—	2	1	no	—
133	#4318	—	2	1	no	—
134	#4318	—	2	2	no	—
135	#4318	—	2	2	no	—
136	C₂×C₁₁⋊C₈	—	2	2	no	—
137	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
138	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
139	#4313	—	2	2	no	—
140	#4313	—	2	2	no	—
141	#4313	—	2	4	no	—
142	C₈×D₁₁	—	2	2	no	—
143	C₈×D₁₁	—	2	2	no	—
144	C₈×D₁₁	—	2	4	no	—
145	#2875	—	11	1	no	—
146	#1841	—	11	2	no	—
147	#1833	—	11	1	no	—
148	#1833	—	11	1	no	—
149	#1827	—	11	2	no	—
150	#1813	—	11	1	no	—
151	#1812	—	11	1	no	—
152	#1773	—	11	1	no	—
153	#1773	D₁₁	1	1	yes	characteristic, complemented
154	#1757	—	11	2	no	—
155	#1747	—	11	2	no	—
156	C₂³×C₁₁	C₂×#41	1	1	yes	characteristic
157	C₂²×D₁₁	#117	1	2	yes	—
158	C₂×C₄×C₁₁	—	2	1	no	—
159	C₂×C₄×C₁₁	—	2	1	no	—
160	C₂×C₄×C₁₁	—	4	2	no	—
161	C₂×C₄×C₁₁	—	8	1	no	—
162	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
164	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
168	C₂²⋊D₁₁	—	2	2	no	—
169	C₂²⋊D₁₁	—	2	2	no	—
170	C₂×Dic₁₁	—	2	2	no	—
171	C₂×Dic₁₁	—	4	2	no	—
172	C₂×Dic₁₁	#117	1	2	yes	—
173	D₄₄	—	2	1	no	—
174	D₄₄	—	2	1	no	—
175	C₄×D₁₁	—	2	2	no	—
176	C₄×D₁₁	—	2	2	no	—
177	C₄×D₁₁	—	8	2	no	—
178	Dic₂₂	—	2	1	no	—
179	Dic₂₂	—	2	1	no	—
180	C₈×C₁₁	—	4	2	no	—
181	C₈×C₁₁	#117	1	1	yes	characteristic
182	C₈×C₁₁	#117	1	1	yes	characteristic
183	C₈×C₁₁	#117	1	1	yes	characteristic
184	C₈×C₁₁	#117	1	1	yes	characteristic
185	C₈×C₁₁	#117	1	2	yes	—
186	C₈×C₁₁	#117	1	2	yes	—
187	C₁₁⋊C₈	—	2	2	no	—
188	C₁₁⋊C₈	—	2	2	no	—
189	C₁₁⋊C₈	—	4	2	no	—
190	C₂×#131	—	11	1	no	—
191	C₂×#131	—	11	1	no	—
192	C₄×#41	—	11	1	no	—
193	C₂×#108	—	11	1	no	—
194	C₂×#108	C₂×D₁₁	1	1	yes	characteristic
195	#427	—	11	1	no	—
196	#427	C₂×D₁₁	1	1	yes	characteristic
197	#413	—	11	2	no	—
198	#410	—	11	2	no	—
199	#405	—	11	1	no	—
200	#405	—	11	1	no	—
201	#405	—	11	2	no	—
202	#405	—	11	2	no	—
203	#405	—	11	2	no	—
204	#405	C₂×D₁₁	1	2	yes	—
205	C₄×M₁₆	—	11	1	no	—
206	C₄×M₁₆	—	11	2	no	—
207	C₂×#97	—	11	2	no	—
208	C₂×#97	C₂×D₁₁	1	1	yes	characteristic
209	#344	—	11	2	no	—
210	#344	—	11	2	no	—
211	#339	—	11	2	no	—
212	#339	C₂×D₁₁	1	2	yes	—
213	#338	—	11	2	no	—
214	H(3,ℤ/4)	—	11	2	no	—
215	H(3,ℤ/4)	—	11	2	no	—
216	H(3,ℤ/4)	—	11	2	no	—
217	C₂²×C₁₁	—	4	1	no	—
218	C₂²×C₁₁	C₂×#117	1	1	yes	characteristic
219	C₂²×C₁₁	#529	1	1	yes	characteristic
220	C₂²×C₁₁	#529	1	1	yes	characteristic
221	C₂×D₁₁	—	2	2	no	—
222	C₂×D₁₁	—	4	2	no	—
223	C₄×C₁₁	—	4	1	no	—
224	C₄×C₁₁	—	4	1	no	—
225	C₄×C₁₁	—	8	2	no	—
226	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
227	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
228	C₄×C₁₁	#529	1	1	yes	characteristic
229	C₄×C₁₁	#529	1	1	yes	characteristic
230	Dic₁₁	—	2	2	no	—
231	Dic₁₁	—	4	2	no	—
232	Dic₁₁	—	8	2	no	—
233	C₂×#41	—	11	1	no	—
234	#131	—	22	1	no	—
235	#131	—	22	1	no	—
236	#131	—	22	1	no	—
237	#131	—	22	1	no	—
238	#131	—	22	2	no	—
239	#131	—	22	2	no	—
240	C₂×M₁₆	—	11	1	no	—
241	C₂×M₁₆	—	11	1	no	—
242	C₂×M₁₆	—	11	2	no	—
243	C₂×M₁₆	—	11	2	no	—
244	C₂×M₁₆	—	11	2	no	—
245	C₂×M₁₆	C₂²×D₁₁	1	2	yes	—
246	C₂²×C₈	—	11	2	no	—
247	C₂²×C₈	—	11	2	no	—
248	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
249	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
250	C₄×Q₈	—	22	1	no	—
251	C₄×D₄	—	22	1	no	—
252	C₄×D₄	—	22	2	no	—
253	#117	—	11	1	no	—
254	#117	—	11	2	no	—
255	#117	—	11	2	no	—
256	#117	C₂²×D₁₁	1	2	yes	—
257	C₂×C₄²	—	11	2	no	—
258	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
259	#108	—	2	1	no	—
260	#108	—	2	1	no	—
261	#108	—	22	1	no	—
262	#108	—	22	1	no	—
263	#107	—	2	1	no	—
264	#107	—	22	1	no	—
265	#106	—	2	1	no	—
266	#106	—	22	1	no	—
267	#101	—	22	2	no	—
268	#100	—	22	2	no	—
269	#99	—	22	4	no	—
270	#97	—	2	2	no	—
271	#97	—	11	2	no	—
272	#97	—	11	2	no	—
273	#97	—	11	2	no	—
274	#97	—	22	2	no	—
275	#97	C₄×D₁₁	1	1	yes	characteristic
276	#97	C₄×D₁₁	1	1	yes	characteristic
277	#97	C₄×D₁₁	1	1	yes	characteristic
278	#97	C₄×D₁₁	1	1	yes	characteristic
279	C₄×C₈	—	2	2	no	—
280	C₄×C₈	—	11	2	no	—
281	C₄×C₈	—	11	2	no	—
282	C₄×C₈	—	11	2	no	—
283	C₄×C₈	—	22	2	no	—
284	C₂×C₁₁	—	2	1	no	—
285	C₂×C₁₁	—	2	1	no	—
286	C₂×C₁₁	—	2	1	no	—
287	C₂×C₁₁	#2283	1	1	yes	characteristic
288	D₁₁	—	4	2	no	—
289	#41	—	22	1	no	—
290	#41	—	22	1	no	—
291	#41	—	22	1	no	—
292	#41	—	22	1	no	—
293	C₂×Q₈	—	11	1	no	—
294	C₂×D₄	—	11	2	no	—
295	C₂×D₄	—	11	1	no	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	11	2	no	—
298	C₂²×C₄	—	22	2	no	—
299	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
300	M₁₆	—	2	2	no	—
301	M₁₆	—	2	2	no	—
302	M₁₆	—	22	1	no	—
303	M₁₆	—	22	1	no	—
304	M₁₆	—	22	1	no	—
305	M₁₆	—	22	1	no	—
306	M₁₆	—	22	2	no	—
307	M₁₆	—	22	2	no	—
308	M₁₆	—	22	4	no	—
309	M₁₆	—	22	2	no	—
310	M₁₆	—	22	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	11	2	no	—
313	C₂×C₈	—	11	2	no	—
314	C₂×C₈	—	22	2	no	—
315	C₂×C₈	—	22	2	no	—
316	C₂×C₈	—	22	4	no	—
317	C₂×C₈	—	22	2	no	—
318	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
319	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
320	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
321	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
322	C₂×C₈	#4339	1	1	yes	characteristic
323	C₂×C₈	#4339	1	1	yes	characteristic
324	C₂×C₈	#4339	1	1	yes	characteristic
325	C₂×C₈	#4339	1	1	yes	characteristic
326	C₂×C₈	#4341	1	2	yes	—
327	C₂×C₈	#4343	1	2	yes	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	22	2	no	—
330	C₄⋊C₄	—	22	1	no	—
331	C₄⋊C₄	—	22	2	no	—
332	C₂²⋊C₄	—	2	2	no	—
333	C₂²⋊C₄	—	22	2	no	—
334	C₂²⋊C₄	—	22	1	no	—
335	C₂²⋊C₄	—	22	2	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	11	2	no	—
339	C₄²	—	22	2	no	—
340	C₄²	—	22	2	no	—
341	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
342	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
343	C₁₁	#20788	1	1	yes	characteristic, complemented, Sylow
344	C₂³	—	11	2	no	—
345	C₂³	#41×D₁₁	1	1	yes	characteristic
346	Q₈	—	22	1	no	—
347	Q₈	—	22	1	no	—
348	D₄	—	22	2	no	—
349	D₄	—	22	2	no	—
350	D₄	—	22	1	no	—
351	D₄	—	22	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	8	1	no	—
356	C₂×C₄	—	11	2	no	—
357	C₂×C₄	—	22	2	no	—
358	C₂×C₄	—	22	2	no	—
359	C₂×C₄	—	22	2	no	—
360	C₂×C₄	—	44	2	no	—
361	C₂×C₄	—	88	2	no	—
362	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
363	C₂×C₄	C₂×#4339	1	1	yes	characteristic
364	C₂×C₄	C₂×#4339	1	1	yes	characteristic
365	C₂×C₄	C₂×#4341	1	1	yes	characteristic
366	C₂×C₄	C₂×#4343	1	1	yes	characteristic
367	C₂×C₄	#41×D₁₁	1	1	yes	characteristic
368	C₈	—	4	2	no	—
369	C₈	—	22	2	no	—
370	C₈	—	22	2	no	—
371	C₈	—	44	2	no	—
372	C₈	#66925	1	1	yes	characteristic
373	C₈	#66925	1	1	yes	characteristic
374	C₈	#66925	1	1	yes	characteristic
375	C₈	#66925	1	1	yes	characteristic
376	C₈	#66945	1	2	yes	—
377	C₈	#66945	1	2	yes	—
378	C₂²	—	4	1	no	—
379	C₂²	—	22	2	no	—
380	C₂²	—	44	2	no	—
381	C₂²	D₁₁×#117	1	1	yes	characteristic
382	C₂²	#10623088	1	1	yes	characteristic
383	C₂²	#10623111	1	1	yes	characteristic
384	C₄	—	4	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	8	2	no	—
387	C₄	—	22	2	no	—
388	C₄	—	44	2	no	—
389	C₄	—	88	2	no	—
390	C₄	C₂×#66925	1	1	yes	characteristic
391	C₄	C₂×#66945	1	1	yes	characteristic
392	C₄	#10623088	1	1	yes	characteristic
393	C₄	#10623111	1	1	yes	characteristic
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	2	1	no	—
397	C₂	—	44	2	no	—
398	C₂	#13071012	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#66945
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₁₁×#1773	C₂
socle	C₂×C₁₁	#2283
prime radical	C₂×#4339	C₂³
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	#1773	D₁₁
hypocenter	C₁₁	#20788
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1773

Complements

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

N	H ≅ G/N	type
2. #13068762	C₂	semidirect
3. #13068741	C₂	semidirect
4. #13068739	C₂	semidirect
8. #13068500	C₂	semidirect
10. #13068371	C₂	semidirect
11. C₁₁×#1773	C₂	semidirect
27. #10622321	C₄	semidirect
28. #10622321	C₄	semidirect
29. #10622320	C₄	semidirect
30. D₁₁×#97	C₄	semidirect
153. #1773	D₁₁	semidirect
343. C₁₁	#20788	semidirect

Maximal subgroups

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

subgroup	index
#13077041	2
#13068762	2
#13068741	2
#13068739	2
#13068719	2
#13068663	2
#13068659	2
#13068500	2
#13068417	2
#13068371	2
C₁₁×#1773	2
#20788	11