Subgroups — #14908465

Order 2816.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079156	C₂	1	1	yes	maximal, characteristic, complemented
2	#13077467	C₂	1	1	yes	maximal, characteristic
3	#13070776	C₂	1	2	yes	maximal, complemented
4	#13070772	C₂	1	2	yes	maximal, complemented
5	#13070762	C₂	1	2	yes	maximal, complemented
6	#13070752	C₂	1	2	yes	maximal, complemented
7	#13070405	C₂	1	1	yes	maximal, characteristic, complemented
8	#13070374	C₂	1	1	yes	maximal, characteristic, complemented
9	#13070331	C₂	1	1	yes	maximal, characteristic
10	C₂×#10622147	C₂	1	1	yes	maximal, characteristic
11	C₁₁×#2158	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈₈	C₂²	1	1	yes	characteristic
13	#10623052	—	2	1	no	—
14	#10623050	—	2	1	no	—
15	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
16	C₁₁×#427	C₂²	1	1	yes	characteristic
17	C₁₁×#427	C₂²	1	1	yes	characteristic
18	C₂×#66882	C₂²	1	1	yes	characteristic
19	#10622714	C₂²	1	1	yes	characteristic
20	C₂×#66876	C₂²	1	1	yes	characteristic
21	#10622614	—	2	1	no	—
22	#10622608	—	2	1	no	—
23	#10622491	C₂²	1	2	yes	complemented
24	#10622487	C₂²	1	2	yes	complemented
25	#10622485	C₂²	1	2	yes	—
26	#10622485	C₂²	1	2	yes	—
27	#10622482	—	2	2	no	—
28	#10622469	—	2	2	no	—
29	C₁₁×#367	C₂²	1	1	yes	characteristic, complemented
30	C₁₁×#367	C₂²	1	1	yes	characteristic, complemented
31	C₁₁×#366	C₂²	1	1	yes	characteristic, complemented
32	C₁₁×#366	C₂²	1	1	yes	characteristic, complemented
33	#10622147	C₂²	1	2	yes	—
34	#10622147	C₂²	1	2	yes	—
35	Dic₁₁⋊C₁₆	C₂²	1	2	yes	complemented
36	Dic₁₁⋊C₁₆	C₂²	1	2	yes	complemented
37	#10622129	C₂²	1	2	yes	—
38	#10622129	C₂²	1	2	yes	—
39	#10622128	C₂²	1	2	yes	complemented
40	#10622127	C₂²	1	2	yes	complemented
41	C₂²×D₄₄	—	2	1	no	—
42	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
43	C₁₁×#117	—	2	1	no	—
44	C₁₁×#117	—	2	1	no	—
45	#66979	D₄	1	1	yes	characteristic
46	C₂×C₄⋊Dic₁₁	—	2	1	no	—
47	C₂×#4318	D₄	1	1	yes	characteristic
48	C₂×D₈₈	—	2	1	no	—
49	C₂×D₈₈	—	2	2	no	—
50	C₂×D₈₈	Q₈	1	4	yes	complemented
51	C₂×D₈₈	C₂³	1	2	yes	—
52	#66946	—	2	2	no	—
53	#66946	—	2	2	no	—
54	#66943	—	2	2	no	—
55	#66941	—	2	2	no	—
56	C₂²⋊C₄×D₁₁	—	4	2	no	—
57	#66931	—	4	2	no	—
58	C₄×D₄₄	—	4	1	no	—
59	C₄×D₄₄	—	4	1	no	—
60	#66923	—	4	1	no	—
61	#66922	—	4	1	no	—
62	C₂×C₁₁×C₁₆	—	2	2	no	—
63	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
65	C₁₁×#107	C₂³	1	1	yes	characteristic
66	C₁₁×#107	C₂³	1	1	yes	characteristic
67	C₁₁×#107	C₂³	1	1	yes	characteristic
68	C₁₁×#107	C₂³	1	1	yes	characteristic
69	C₁₁×#106	—	2	1	no	—
70	C₁₁×#106	—	2	1	no	—
71	#66882	—	2	1	no	—
72	#66882	C₂³	1	2	yes	—
73	#66879	D₄	1	2	yes	—
74	C₈×Dic₁₁	C₂³	1	2	yes	—
75	#66876	—	2	2	no	—
76	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
77	#66873	—	2	2	no	—
78	#66873	—	2	2	no	—
79	#66872	—	2	2	no	—
80	#66871	—	2	2	no	—
81	#33631	—	11	1	no	maximal, Sylow
82	C₂³×D₁₁	—	4	1	no	—
83	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×Dic₁₁	—	4	1	no	—
85	C₂×D₄₄	—	2	4	no	—
86	C₂×D₄₄	—	2	2	no	—
87	C₂×D₄₄	—	4	1	no	—
88	C₂×D₄₄	—	4	2	no	—
89	C₂×C₄×D₁₁	—	4	2	no	—
90	C₂×C₄×D₁₁	—	4	2	no	—
91	C₂×Dic₂₂	—	4	2	no	—
92	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₁₁	#41	1	2	yes	—
95	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
96	C₁₁×C₄⋊C₄	—	2	1	no	—
97	C₁₁×C₄⋊C₄	—	2	1	no	—
98	C₁₁×C₄⋊C₄	—	2	1	no	—
99	C₁₁×C₄⋊C₄	—	2	1	no	—
100	C₁₁×C₂²⋊C₄	—	4	1	no	—
101	C₁₁×C₂²⋊C₄	—	4	1	no	—
102	C₄²×C₁₁	—	4	1	no	—
103	C₄²×C₁₁	—	4	1	no	—
104	C₂²⋊Dic₁₁	D₈	1	2	yes	—
105	(C₂×D₁₁)⋊C₄	—	4	2	no	—
106	(C₂×D₁₁)⋊C₄	—	4	2	no	—
107	C₄⋊Dic₁₁	—	2	2	no	—
108	C₄⋊Dic₁₁	—	4	2	no	—
109	C₄⋊Dic₁₁	—	4	1	no	—
110	C₄⋊Dic₁₁	D₈	1	1	yes	characteristic
111	C₄⋊Dic₁₁	D₈	1	1	yes	characteristic
112	Dic₁₁⋊C₄	—	4	2	no	—
113	Dic₁₁⋊C₄	—	4	2	no	—
114	C₄×Dic₁₁	C₂×D₄	1	2	yes	—
115	#4318	—	2	2	no	—
116	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
117	D₈₈	—	2	4	no	—
118	D₈₈	—	2	2	no	—
119	D₈₈	C₂×Q₈	1	4	yes	—
120	C₁₁×C₁₆	—	2	1	no	—
121	C₁₁×C₁₆	—	2	1	no	—
122	C₁₁⋊C₁₆	—	2	2	no	—
123	#3164	—	11	1	no	—
124	#2948	—	11	1	no	—
125	#2228	—	11	2	no	—
126	#2228	—	11	2	no	—
127	#2226	—	11	2	no	—
128	#2224	—	11	2	no	—
129	#2159	—	11	1	no	—
130	#2158	D₁₁	1	1	yes	characteristic, complemented
131	#2151	—	11	1	no	—
132	#2140	—	11	1	no	—
133	C₂×#357	—	11	1	no	—
134	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₁₁	—	4	2	no	—
136	C₂²×D₁₁	—	4	4	no	—
137	C₂²×D₁₁	—	16	1	no	—
138	C₂²×D₁₁	—	16	1	no	—
139	C₂×C₄×C₁₁	—	4	1	no	—
140	C₂×C₄×C₁₁	—	4	1	no	—
141	C₂×C₄×C₁₁	—	4	1	no	—
142	C₂×C₄×C₁₁	—	4	1	no	—
143	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₁	#137	1	2	yes	—
146	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×Dic₁₁	—	4	2	no	—
148	C₂×Dic₁₁	—	16	1	no	—
149	C₂×Dic₁₁	C₂×D₈	1	2	yes	—
150	C₂×Dic₁₁	C₂×D₈	1	2	yes	—
151	D₄₄	—	2	4	no	—
152	D₄₄	—	4	2	no	—
153	D₄₄	—	4	4	no	—
154	C₄×D₁₁	—	16	2	no	—
155	C₄×D₁₁	—	16	2	no	—
156	Dic₂₂	—	16	2	no	—
157	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
158	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
159	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
160	C₁₁⋊C₈	—	2	2	no	—
161	C₂²×D₈	—	11	1	no	—
162	#544	—	22	1	no	—
163	#518	—	22	1	no	—
164	C₂×M₃₂	—	11	1	no	—
165	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
166	#501	—	22	2	no	—
167	#500	—	11	2	no	—
168	#499	—	11	2	no	—
169	#442	—	22	2	no	—
170	C₄×D₈	—	11	2	no	—
171	C₄×D₈	—	11	2	no	—
172	#428	—	22	1	no	—
173	#427	C₂×D₁₁	1	1	yes	characteristic
174	#427	C₂×D₁₁	1	1	yes	characteristic
175	C₂×#107	—	11	1	no	—
176	#418	—	22	1	no	—
177	#405	—	11	1	no	—
178	#367	—	11	2	no	—
179	#367	C₂×D₁₁	1	1	yes	characteristic, complemented
180	#367	C₂×D₁₁	1	1	yes	characteristic, complemented
181	#366	—	11	2	no	—
182	#366	C₂×D₁₁	1	1	yes	characteristic, complemented
183	#366	C₂×D₁₁	1	1	yes	characteristic, complemented
184	C₄⋊C₁₆	—	11	2	no	—
185	C₄⋊C₁₆	—	11	2	no	—
186	#357	—	11	2	no	—
187	#357	—	11	2	no	—
188	#357	—	11	2	no	—
189	#357	—	11	2	no	—
190	C₂²×C₁₁	—	2	1	no	—
191	C₂²×C₁₁	—	2	1	no	—
192	C₂²×C₁₁	#474	1	2	yes	—
193	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
194	C₂×D₁₁	—	4	4	no	—
195	C₂×D₁₁	—	8	2	no	—
196	C₂×D₁₁	—	8	2	no	—
197	C₂×D₁₁	—	16	4	no	—
198	C₄×C₁₁	—	8	1	no	—
199	C₄×C₁₁	—	8	1	no	—
200	C₄×C₁₁	—	8	1	no	—
201	C₄×C₁₁	—	8	1	no	—
202	C₄×C₁₁	#474	1	2	yes	—
203	C₄×C₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
204	C₄×C₁₁	C₂×#137	1	1	yes	characteristic
205	Dic₁₁	—	2	2	no	—
206	Dic₁₁	—	2	2	no	—
207	Dic₁₁	—	8	2	no	—
208	C₂²×D₄	—	22	1	no	—
209	C₂×D₈	—	11	4	no	—
210	C₂×D₈	—	11	2	no	—
211	C₂×D₈	—	22	2	no	—
212	C₂×D₈	—	22	1	no	—
213	C₂×M₁₆	—	11	1	no	—
214	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
215	C₄⋊Q₈	—	22	2	no	—
216	C₄⋊Q₈	—	44	1	no	—
217	#125	—	22	2	no	—
218	#125	—	44	1	no	—
219	C₂²⋊Q₈	—	44	2	no	—
220	C₄×D₄	—	22	2	no	—
221	C₄×D₄	—	22	2	no	—
222	C₄×D₄	—	44	1	no	—
223	C₄×D₄	—	44	1	no	—
224	#117	—	2	1	no	—
225	#117	—	2	1	no	—
226	#117	—	11	1	no	—
227	C₂×C₄⋊C₄	—	22	1	no	—
228	C₂×C₂²⋊C₄	—	44	2	no	—
229	M₃₂	—	22	2	no	—
230	C₂×C₁₆	—	2	2	no	—
231	C₂×C₁₆	—	11	2	no	—
232	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
233	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
234	#107	—	11	2	no	—
235	#107	—	22	2	no	—
236	#107	—	22	1	no	—
237	#107	C₂²×D₁₁	1	1	yes	characteristic
238	#107	C₂²×D₁₁	1	1	yes	characteristic
239	#107	C₂²×D₁₁	1	1	yes	characteristic
240	#107	C₂²×D₁₁	1	1	yes	characteristic
241	#106	—	2	1	no	—
242	#106	—	2	1	no	—
243	#106	—	22	2	no	—
244	#102	—	22	2	no	—
245	#102	—	22	2	no	—
246	#97	—	11	2	no	—
247	C₄×C₈	—	11	2	no	—
248	C₂×C₁₁	—	2	2	no	—
249	C₂×C₁₁	C₂×#474	1	1	yes	characteristic
250	C₂×C₁₁	#3415	1	1	yes	characteristic
251	C₂×C₁₁	#3416	1	1	yes	characteristic
252	D₁₁	—	8	4	no	—
253	C₂⁴	—	44	1	no	—
254	C₂×Q₈	—	44	2	no	—
255	C₂×D₄	—	22	4	no	—
256	C₂×D₄	—	22	2	no	—
257	C₂×D₄	—	44	1	no	—
258	C₂×D₄	—	44	2	no	—
259	C₂²×C₄	—	44	2	no	—
260	C₂²×C₄	—	44	2	no	—
261	C₂²×C₄	—	44	1	no	—
262	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
263	D₈	—	11	4	no	—
264	D₈	—	22	4	no	—
265	D₈	—	22	2	no	—
266	M₁₆	—	22	2	no	—
267	C₂×C₈	—	11	2	no	—
268	C₂×C₈	#4339	1	2	yes	—
269	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
270	C₂×C₈	Q₈×D₁₁	1	2	yes	—
271	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	11	1	no	—
277	C₄⋊C₄	—	11	1	no	—
278	C₄⋊C₄	—	22	2	no	—
279	C₄⋊C₄	—	44	2	no	—
280	C₄⋊C₄	—	44	2	no	—
281	C₄⋊C₄	—	44	2	no	—
282	C₄⋊C₄	—	44	1	no	—
283	C₂²⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	11	2	no	—
286	C₂²⋊C₄	—	44	2	no	—
287	C₂²⋊C₄	—	44	2	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	4	1	no	—
290	C₄²	—	11	2	no	—
291	C₁₆	—	2	1	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	22	2	no	—
294	C₁₁	#33631	1	1	yes	characteristic, complemented, Sylow
295	C₂³	—	44	2	no	—
296	C₂³	—	44	4	no	—
297	C₂³	—	176	1	no	—
298	C₂³	—	176	1	no	—
299	C₂³	D₈×D₁₁	1	1	yes	characteristic
300	Q₈	—	176	2	no	—
301	D₄	—	22	4	no	—
302	D₄	—	44	4	no	—
303	D₄	—	44	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	11	2	no	—
309	C₂×C₄	—	11	2	no	—
310	C₂×C₄	—	44	2	no	—
311	C₂×C₄	—	176	2	no	—
312	C₂×C₄	—	176	2	no	—
313	C₂×C₄	—	176	1	no	—
314	C₂×C₄	#66949	1	2	yes	—
315	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
316	C₂×C₄	#66971	1	2	yes	—
317	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
318	C₈	—	22	2	no	—
319	C₈	#66949	1	2	yes	—
320	C₈	C₂×#4339	1	1	yes	characteristic
321	C₈	C₂×Q₈×D₁₁	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	44	4	no	—
325	C₂²	—	88	2	no	—
326	C₂²	—	88	2	no	—
327	C₂²	—	176	4	no	—
328	C₂²	#10622503	1	2	yes	—
329	C₂²	C₂×D₈×D₁₁	1	1	yes	characteristic
330	C₄	—	8	1	no	—
331	C₄	—	8	1	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	22	2	no	—
335	C₄	—	22	2	no	—
336	C₄	—	88	2	no	—
337	C₄	#10622503	1	2	yes	—
338	C₄	C₂×#66949	1	1	yes	characteristic
339	C₄	C₂×#66971	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	88	4	no	—
342	C₂	C₂×#10622503	1	1	yes	characteristic
343	C₂	#13081754	1	1	yes	characteristic
344	C₂	#13081759	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₁
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#2158	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	C₂²×D₈₈	C₂²
hypercenter	#2158	D₁₁
hypocenter	C₁₁	#33631
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
1. #13079156	C₂	semidirect
3. #13070776	C₂	semidirect
4. #13070772	C₂	semidirect
5. #13070762	C₂	semidirect
6. #13070752	C₂	semidirect
7. #13070405	C₂	semidirect
8. #13070374	C₂	semidirect
11. C₁₁×#2158	C₂	semidirect
23. #10622491	C₂²	semidirect
24. #10622487	C₂²	semidirect
29. C₁₁×#367	C₂²	semidirect
30. C₁₁×#367	C₂²	semidirect
31. C₁₁×#366	C₂²	semidirect
32. C₁₁×#366	C₂²	semidirect
35. Dic₁₁⋊C₁₆	C₂²	semidirect
36. Dic₁₁⋊C₁₆	C₂²	semidirect
39. #10622128	C₂²	semidirect
40. #10622127	C₂²	semidirect
50. C₂×D₈₈	Q₈	semidirect
130. #2158	D₁₁	semidirect
179. #367	C₂×D₁₁	semidirect
180. #367	C₂×D₁₁	semidirect
182. #366	C₂×D₁₁	semidirect
183. #366	C₂×D₁₁	semidirect
294. C₁₁	#33631	semidirect

Maximal subgroups

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

subgroup	index
#13079156	2
#13077467	2
#13070776	2
#13070772	2
#13070762	2
#13070752	2
#13070405	2
#13070374	2
#13070331	2
C₂×#10622147	2
C₁₁×#2158	2
#33631	11