Subgroups — #14909327

Order 2816.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079187	C₂	1	2	yes	maximal, complemented
2	#13070893	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070878	C₂	1	1	yes	maximal, characteristic, complemented
4	#13070849	C₂	1	2	yes	maximal, complemented
5	#13070714	C₂	1	2	yes	maximal, complemented
6	#13070702	C₂	1	1	yes	maximal, characteristic
7	Q₃₂×Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#13070465	C₂	1	2	yes	maximal, complemented
9	#13070462	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₆×Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₁×#2216	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623155	—	2	2	no	—
13	C₂×C₁₁×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
15	C₁₁×#467	C₂²	1	2	yes	—
16	#10622804	C₂²	1	2	yes	—
17	Q₁₆×Dic₁₁	C₂²	1	2	yes	—
18	C₂×#66892	C₂²	1	2	yes	—
19	#10622728	C₂²	1	2	yes	—
20	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
21	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
22	#10622661	—	2	2	no	—
23	C₂×D₁₇₆	C₂²	1	1	yes	characteristic
24	#10622491	C₂²	1	2	yes	—
25	#10622485	C₂²	1	2	yes	—
26	#10622413	—	2	2	no	—
27	#10622401	—	2	2	no	—
28	C₁₁×#366	C₂²	1	1	yes	characteristic
29	C₁₁×#358	C₂²	1	2	yes	—
30	#10622202	C₂²	1	2	yes	—
31	#10622147	C₂²	1	2	yes	—
32	#10622142	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₁₁	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₁₁	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₁₁	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₁₁	C₂²	1	1	yes	characteristic
37	#10622129	C₂²	1	2	yes	—
38	#10622128	C₂²	1	2	yes	—
39	C₂×C₁₁×Q₁₆	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
41	C₁₁×C₂²⋊Q₈	—	2	2	no	—
42	#67000	—	2	2	no	—
43	Q₈×Dic₁₁	—	2	2	no	—
44	C₂×#4325	—	2	2	no	—
45	#66983	—	2	2	no	—
46	C₂×C₄×Dic₁₁	D₄	1	1	yes	characteristic
47	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
48	C₂×D₈₈	C₂³	1	2	yes	—
49	#66951	—	4	2	no	—
50	#66946	—	2	2	no	—
51	#66945	—	4	2	no	—
52	#66943	—	2	2	no	—
53	#66939	—	4	2	no	—
54	(C₂²⋊D₁₁)⋊C₄	—	4	2	no	—
55	C₁₁×Q₃₂	—	2	2	no	—
56	C₂×C₁₁×C₁₆	—	2	2	no	—
57	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
59	C₁₁×#107	C₂³	1	2	yes	—
60	C₁₁×#103	—	2	2	no	—
61	#66899	—	2	2	no	—
62	#66892	—	2	4	no	—
63	#66885	—	2	2	no	—
64	#66882	C₂³	1	2	yes	—
65	C₈×Dic₁₁	D₄	1	1	yes	characteristic
66	C₈×Dic₁₁	D₄	1	1	yes	characteristic
67	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
68	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
69	C₂×C₁₁⋊C₁₆	—	2	2	no	—
70	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
72	#66873	—	2	2	no	—
73	#66872	—	2	2	no	—
74	D₁₇₆	—	2	2	no	—
75	#33735	—	11	1	no	maximal, Sylow
76	C₂×Q₈×C₁₁	—	2	2	no	—
77	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₁₁	—	4	2	no	—
79	C₂²×Dic₁₁	D₈	1	1	yes	characteristic
80	C₂²×Dic₁₁	D₈	1	1	yes	characteristic
81	C₂×D₄₄	—	2	2	no	—
82	C₂×C₄×D₁₁	—	4	2	no	—
83	C₁₁×Q₁₆	—	2	2	no	—
84	C₁₁×Q₁₆	—	2	2	no	—
85	C₂×C₈×C₁₁	—	2	1	no	—
86	C₂×C₈×C₁₁	—	2	1	no	—
87	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
89	C₁₁×C₄⋊C₄	—	2	2	no	—
90	C₁₁×C₄⋊C₄	—	4	2	no	—
91	C₁₁×C₂²⋊C₄	—	4	2	no	—
92	#4325	—	4	4	no	—
93	(C₂×D₁₁)⋊C₄	—	4	2	no	—
94	(C₂×D₁₁)⋊C₄	—	4	2	no	—
95	(C₂×D₁₁)⋊C₄	—	4	2	no	—
96	(C₂×D₁₁)⋊C₄	—	4	2	no	—
97	C₄⋊Dic₁₁	—	2	2	no	—
98	C₄⋊Dic₁₁	—	4	2	no	—
99	C₄⋊Dic₁₁	—	4	2	no	—
100	Dic₁₁⋊C₄	—	4	2	no	—
101	C₄×Dic₁₁	—	4	2	no	—
102	C₄×Dic₁₁	D₈	1	1	yes	characteristic
103	C₄×Dic₁₁	D₈	1	1	yes	characteristic
104	C₄×Dic₁₁	C₂×D₄	1	1	yes	characteristic
105	C₄×Dic₁₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₁₁⋊C₈	—	2	2	no	—
107	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
109	D₈₈	—	2	2	no	—
110	D₈₈	—	2	2	no	—
111	C₁₁×C₁₆	—	2	1	no	—
112	C₁₁×C₁₆	#41	1	2	yes	—
113	C₁₁⋊C₁₆	—	2	1	no	—
114	C₁₁⋊C₁₆	#41	1	2	yes	—
115	#3168	—	11	2	no	—
116	#2253	—	11	1	no	—
117	#2249	—	11	1	no	—
118	#2242	—	11	2	no	—
119	#2217	—	11	2	no	—
120	#2216	D₁₁	1	1	yes	characteristic, complemented
121	#2215	—	11	1	no	—
122	C₄×Q₃₂	—	11	1	no	—
123	C₄×SD₃₂	—	11	2	no	—
124	C₄×D₁₆	—	11	1	no	—
125	C₂×C₄×C₁₆	—	11	1	no	—
126	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
127	C₂²×D₁₁	—	4	2	no	—
128	Q₈×C₁₁	—	4	2	no	—
129	Q₈×C₁₁	—	4	2	no	—
130	C₂×C₄×C₁₁	—	2	1	no	—
131	C₂×C₄×C₁₁	—	2	1	no	—
132	C₂×C₄×C₁₁	—	4	2	no	—
133	C₂×C₄×C₁₁	—	4	2	no	—
134	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₁₁	—	8	4	no	—
137	C₂×Dic₁₁	—	2	2	no	—
138	C₂×Dic₁₁	—	2	2	no	—
139	C₂×Dic₁₁	—	4	2	no	—
140	C₂×Dic₁₁	C₂×D₈	1	1	yes	characteristic
141	C₂×Dic₁₁	C₂×D₈	1	1	yes	characteristic
142	C₂×Dic₁₁	C₂×D₈	1	1	yes	characteristic
143	C₂×Dic₁₁	C₂×D₈	1	1	yes	characteristic
144	D₄₄	—	4	2	no	—
145	D₄₄	—	4	2	no	—
146	C₄×D₁₁	—	8	4	no	—
147	C₈×C₁₁	—	2	1	no	—
148	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
149	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
150	C₁₁⋊C₈	—	2	1	no	—
151	C₁₁⋊C₈	#135	1	2	yes	—
152	#529	—	22	2	no	—
153	C₂×Q₃₂	C₂×D₁₁	1	1	yes	characteristic
154	C₂×SD₃₂	—	11	2	no	—
155	C₂×D₁₆	—	11	1	no	—
156	C₂²×C₁₆	—	11	1	no	—
157	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
158	#499	—	11	2	no	—
159	#495	—	11	2	no	—
160	#488	—	22	2	no	—
161	#467	C₂×D₁₁	1	2	yes	—
162	#466	—	11	2	no	—
163	#465	—	22	2	no	—
164	C₄×Q₁₆	—	11	2	no	—
165	C₄×SD₁₆	—	22	2	no	—
166	C₄×D₈	—	11	2	no	—
167	C₂×C₄×C₈	—	11	1	no	—
168	#367	—	11	2	no	—
169	#366	—	11	1	no	—
170	#366	C₂×D₁₁	1	1	yes	characteristic
171	#358	—	11	2	no	—
172	#358	C₂×D₁₁	1	2	yes	—
173	#357	—	11	2	no	—
174	#357	—	11	2	no	—
175	C₄×C₁₆	—	11	1	no	—
176	C₄×C₁₆	—	11	1	no	—
177	C₄×C₁₆	—	11	1	no	—
178	C₄×C₁₆	—	11	1	no	—
179	C₂²×C₁₁	—	2	1	no	—
180	C₂²×C₁₁	—	2	1	no	—
181	C₂²×C₁₁	#508	1	2	yes	—
182	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
183	C₂×D₁₁	—	8	2	no	—
184	C₂×D₁₁	—	8	2	no	—
185	C₂×D₁₁	—	8	2	no	—
186	C₄×C₁₁	—	2	1	no	—
187	C₄×C₁₁	—	8	2	no	—
188	C₄×C₁₁	—	8	2	no	—
189	C₄×C₁₁	#532	1	1	yes	characteristic
190	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
191	Dic₁₁	—	2	1	no	—
192	Dic₁₁	—	2	1	no	—
193	Dic₁₁	—	2	1	no	—
194	Dic₁₁	—	8	2	no	—
195	Dic₁₁	#508	1	2	yes	—
196	C₂×Q₁₆	C₂²×D₁₁	1	2	yes	—
197	C₂×SD₁₆	—	22	2	no	—
198	C₂×D₈	—	11	2	no	—
199	C₂²×C₈	—	11	1	no	—
200	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
201	#126	—	44	2	no	—
202	#125	—	22	2	no	—
203	#124	—	22	2	no	—
204	#123	—	44	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	#121	—	22	2	no	—
207	C₄×Q₈	—	22	2	no	—
208	C₄×D₄	—	22	2	no	—
209	C₄×D₄	—	44	2	no	—
210	#117	—	44	2	no	—
211	C₂×C₄²	—	11	1	no	—
212	Q₃₂	—	2	2	no	—
213	SD₃₂	—	22	4	no	—
214	D₁₆	—	22	2	no	—
215	C₂×C₁₆	—	2	2	no	—
216	C₂×C₁₆	—	11	1	no	—
217	C₂×C₁₆	—	11	1	no	—
218	C₂×C₁₆	—	22	2	no	—
219	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
220	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
221	#107	—	11	2	no	—
222	#107	C₂²×D₁₁	1	2	yes	—
223	#106	—	22	2	no	—
224	#103	—	2	2	no	—
225	#103	—	22	2	no	—
226	#102	—	22	2	no	—
227	#102	—	22	2	no	—
228	C₄×C₈	—	11	1	no	—
229	C₄×C₈	—	11	1	no	—
230	C₄×C₈	—	11	1	no	—
231	C₄×C₈	—	11	1	no	—
232	C₂×C₁₁	—	2	2	no	—
233	C₂×C₁₁	#3169	1	1	yes	characteristic
234	C₂×C₁₁	C₂×#508	1	1	yes	characteristic
235	C₂×C₁₁	C₂×#508	1	1	yes	characteristic
236	D₁₁	—	16	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×D₄	—	22	2	no	—
239	C₂×D₄	—	44	2	no	—
240	C₂²×C₄	—	11	1	no	—
241	C₂²×C₄	—	11	1	no	—
242	C₂²×C₄	—	44	2	no	—
243	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	2	2	no	—
246	SD₁₆	—	44	4	no	—
247	D₈	—	22	2	no	—
248	D₈	—	22	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	11	1	no	—
252	C₂×C₈	—	11	1	no	—
253	C₂×C₈	—	22	2	no	—
254	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
255	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	22	2	no	—
259	C₄⋊C₄	—	44	2	no	—
260	C₄⋊C₄	—	44	2	no	—
261	C₄⋊C₄	—	44	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	44	2	no	—
264	C₂²⋊C₄	—	44	2	no	—
265	C₂²⋊C₄	—	44	2	no	—
266	C₂²⋊C₄	—	44	2	no	—
267	C₄²	—	11	1	no	—
268	C₄²	—	11	1	no	—
269	C₄²	—	11	1	no	—
270	C₄²	—	11	1	no	—
271	C₄²	—	44	2	no	—
272	C₁₆	—	2	1	no	—
273	C₁₆	—	11	2	no	—
274	C₁₆	—	22	1	no	—
275	C₁₆	#4341	1	2	yes	—
276	C₁₁	#33735	1	1	yes	characteristic, complemented, Sylow
277	C₂³	—	44	2	no	—
278	C₂³	D₈×D₁₁	1	1	yes	characteristic
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	44	2	no	—
282	D₄	—	44	2	no	—
283	D₄	—	88	4	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	11	1	no	—
289	C₂×C₄	—	11	1	no	—
290	C₂×C₄	—	11	1	no	—
291	C₂×C₄	—	11	1	no	—
292	C₂×C₄	—	22	2	no	—
293	C₂×C₄	—	22	2	no	—
294	C₂×C₄	—	44	2	no	—
295	C₂×C₄	—	88	4	no	—
296	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	11	2	no	—
300	C₈	—	22	1	no	—
301	C₈	C₂×#4341	1	1	yes	characteristic
302	C₈	#41×D₁₁	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	88	2	no	—
306	C₂²	—	88	2	no	—
307	C₂²	—	88	2	no	—
308	C₂²	#10622532	1	2	yes	—
309	C₂²	C₂×D₈×D₁₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	11	2	no	—
314	C₄	—	22	1	no	—
315	C₄	—	22	1	no	—
316	C₄	—	22	1	no	—
317	C₄	—	88	2	no	—
318	C₄	#10623127	1	1	yes	characteristic
319	C₄	D₁₁×#135	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	176	2	no	—
322	C₂	#13079190	1	1	yes	characteristic
323	C₂	D₁₁×#508	1	1	yes	characteristic
324	C₂	C₂×#10622532	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×#2216	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	#13070702	C₂
hypercenter	#2216	D₁₁
hypocenter	C₁₁	#33735
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

Complements

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

N	H ≅ G/N	type
1. #13079187	C₂	semidirect
2. #13070893	C₂	semidirect
3. #13070878	C₂	semidirect
4. #13070849	C₂	semidirect
5. #13070714	C₂	semidirect
7. Q₃₂×Dic₁₁	C₂	semidirect
8. #13070465	C₂	semidirect
9. #13070462	C₂	semidirect
10. C₂×C₁₆×Dic₁₁	C₂	semidirect
11. C₁₁×#2216	C₂	semidirect
120. #2216	D₁₁	semidirect
276. C₁₁	#33735	semidirect

Maximal subgroups

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

subgroup	index
#13079187	2
#13070893	2
#13070878	2
#13070849	2
#13070714	2
#13070702	2
Q₃₂×Dic₁₁	2
#13070465	2
#13070462	2
C₂×C₁₆×Dic₁₁	2
C₁₁×#2216	2
#33735	11