Subgroups — #14670256

Order 2816.

384 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622514	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622149	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10622153	C₂	1	1	yes	maximal, characteristic
4	#13068455	C₂	1	1	yes	maximal, characteristic
5	#13066705	C₂	1	1	yes	maximal, characteristic, complemented
6	#13066693	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₁×#2151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#66954	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
11	C₁₁×#405	C₂²	1	1	yes	characteristic
12	C₂×#66889	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
14	C₂×#66876	C₂²	1	1	yes	characteristic
15	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	#10622615	C₄	1	1	yes	characteristic
17	#10622514	—	2	2	no	—
18	#10622514	—	2	2	no	—
19	C₂×#66862	—	2	1	no	—
20	C₂×C₁₆×D₁₁	—	2	1	no	—
21	C₁₁×C₄⋊C₁₆	—	2	1	no	—
22	C₁₁×C₄⋊C₁₆	—	2	1	no	—
23	#10622196	C₄	1	1	yes	characteristic
24	#10622170	C₄	1	1	yes	characteristic
25	#10622153	—	2	2	no	—
26	#10622149	—	2	2	no	—
27	C₂×#4339	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₁×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
30	C₁₁×#117	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂×#4318	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
34	#66954	—	2	1	no	—
35	#66954	—	2	1	no	—
36	#66954	D₄	1	2	yes	—
37	#66954	Q₈	1	2	yes	—
38	C₂×#4313	—	2	1	no	—
39	C₂×C₈×D₁₁	—	2	1	no	—
40	C₁₁×M₃₂	—	2	1	no	—
41	C₁₁×M₃₂	—	2	1	no	—
42	C₂×C₁₁×C₁₆	—	2	1	no	—
43	C₂×C₁₁×C₁₆	—	2	1	no	—
44	C₂×C₁₁×C₁₆	—	2	1	no	—
45	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
46	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
47	C₁₁×#97	—	2	1	no	—
48	C₄×C₈×C₁₁	—	2	1	no	—
49	#66889	—	2	2	no	—
50	C₈×Dic₁₁	—	2	1	no	—
51	C₈×Dic₁₁	—	2	1	no	—
52	#66876	—	2	1	no	—
53	#66876	D₄	1	2	yes	—
54	C₂×C₁₁⋊C₁₆	—	2	1	no	—
55	C₂×C₁₁⋊C₁₆	—	2	1	no	—
56	C₂×C₁₁⋊C₁₆	—	2	1	no	—
57	C₂×C₁₁⋊C₁₆	—	2	1	no	—
58	#66874	C₈	1	1	yes	characteristic
59	#66874	C₈	1	1	yes	characteristic
60	#66873	C₈	1	1	yes	characteristic
61	#66873	C₈	1	1	yes	characteristic
62	#66862	—	2	4	no	—
63	C₁₆×D₁₁	—	2	4	no	—
64	#12917	—	11	1	no	maximal, Sylow
65	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₁	—	2	1	no	—
67	C₂²×Dic₁₁	—	2	1	no	—
68	#4339	—	2	1	no	—
69	#4339	—	2	1	no	—
70	#4339	C₄⋊C₄	1	2	yes	—
71	#4339	M₁₆	1	2	yes	—
72	C₂×D₄₄	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₁₁	—	2	1	no	—
74	C₂×Dic₂₂	C₂×C₈	1	1	yes	characteristic
75	C₁₁×M₁₆	—	2	1	no	—
76	C₁₁×M₁₆	—	2	1	no	—
77	C₂×C₈×C₁₁	—	2	1	no	—
78	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
84	C₁₁×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
85	C₁₁×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
86	C₁₁×C₂²⋊C₄	—	2	1	no	—
87	C₄²×C₁₁	—	2	1	no	—
88	C₄×Dic₁₁	—	2	1	no	—
89	C₄×Dic₁₁	—	2	1	no	—
90	#4318	—	2	1	no	—
91	#4318	C₄⋊C₄	1	2	yes	—
92	C₂×C₁₁⋊C₈	—	2	1	no	—
93	C₂×C₁₁⋊C₈	—	2	1	no	—
94	C₂×C₁₁⋊C₈	—	2	1	no	—
95	C₂×C₁₁⋊C₈	C₂×C₈	1	1	yes	characteristic
96	C₂×C₁₁⋊C₈	C₂×C₈	1	1	yes	characteristic
97	#4313	—	2	2	no	—
98	#4313	—	2	2	no	—
99	C₈×D₁₁	—	2	2	no	—
100	C₈×D₁₁	—	2	2	no	—
101	C₁₁×C₁₆	—	2	1	no	—
102	C₁₁×C₁₆	—	2	1	no	—
103	C₁₁×C₁₆	—	4	1	no	—
104	C₁₁⋊C₁₆	—	2	2	no	—
105	C₁₁⋊C₁₆	—	4	1	no	—
106	C₂×#504	—	11	1	no	—
107	C₂×#364	—	11	1	no	—
108	#2151	D₁₁	1	1	yes	characteristic, complemented
109	C₂×#350	—	11	1	no	—
110	#1764	—	11	1	no	—
111	#1379	—	11	1	no	—
112	#1374	—	11	1	no	—
113	C₂³×C₁₁	#131	1	1	yes	characteristic
114	C₂²×D₁₁	—	2	1	no	—
115	C₂×C₄×C₁₁	—	2	1	no	—
116	C₂×C₄×C₁₁	—	2	1	no	—
117	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₁	#131	1	1	yes	characteristic
123	C₂²⋊D₁₁	—	2	2	no	—
124	C₂²⋊D₁₁	—	2	2	no	—
125	C₂×Dic₁₁	—	2	1	no	—
126	C₂×Dic₁₁	—	2	2	no	—
127	C₂×Dic₁₁	—	4	1	no	—
128	C₂×Dic₁₁	—	4	1	no	—
129	D₄₄	—	2	1	no	—
130	D₄₄	C₄⋊C₈	1	2	yes	—
131	C₄×D₁₁	—	2	2	no	—
132	C₄×D₁₁	—	2	2	no	—
133	Dic₂₂	—	2	1	no	—
134	Dic₂₂	C₄⋊C₈	1	2	yes	—
135	C₈×C₁₁	—	4	1	no	—
136	C₈×C₁₁	#120	1	1	yes	characteristic
137	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
138	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
139	C₈×C₁₁	#123	1	1	yes	characteristic
140	C₁₁⋊C₈	—	2	2	no	—
141	C₁₁⋊C₈	—	2	1	no	—
142	C₁₁⋊C₈	—	2	1	no	—
143	C₂×#131	—	11	1	no	—
144	#504	—	22	2	no	—
145	#504	—	22	2	no	—
146	C₂×M₃₂	—	11	1	no	—
147	C₂×M₃₂	—	22	1	no	—
148	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
149	C₂²×C₁₆	—	11	1	no	—
150	C₂²×C₁₆	—	22	1	no	—
151	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
152	C₂×#104	—	11	1	no	—
153	#416	—	11	1	no	—
154	#405	C₂×D₁₁	1	1	yes	characteristic
155	C₂×C₄×C₈	—	11	1	no	—
156	#364	—	22	2	no	—
157	C₄⋊C₁₆	—	2	1	no	—
158	C₄⋊C₁₆	—	2	1	no	—
159	#350	—	22	2	no	—
160	#341	—	11	1	no	—
161	#339	—	11	1	no	—
162	C₂²×C₁₁	—	2	1	no	—
163	C₂²×C₁₁	—	2	1	no	—
164	C₂²×C₁₁	#424	1	1	yes	characteristic
165	C₂²×C₁₁	C₈×D₄	1	1	yes	characteristic
166	C₂²×C₁₁	#435	1	1	yes	characteristic
167	C₂×D₁₁	—	2	2	no	—
168	C₂×D₁₁	—	4	1	no	—
169	C₂×D₁₁	—	4	1	no	—
170	C₄×C₁₁	—	4	1	no	—
171	C₄×C₁₁	—	4	1	no	—
172	C₄×C₁₁	#385	1	1	yes	characteristic
173	C₄×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
174	C₄×C₁₁	C₈×D₄	1	1	yes	characteristic
175	C₄×C₁₁	#435	1	1	yes	characteristic
176	Dic₁₁	—	2	2	no	—
177	Dic₁₁	—	4	2	no	—
178	C₂×#41	—	11	1	no	—
179	#131	—	11	2	no	—
180	#131	—	11	2	no	—
181	#131	—	22	1	no	—
182	#131	—	22	1	no	—
183	C₂×M₁₆	—	11	1	no	—
184	C₂×M₁₆	—	22	1	no	—
185	C₂×M₁₆	C₄×D₁₁	1	1	yes	characteristic
186	C₂²×C₈	—	11	1	no	—
187	C₂²×C₈	—	22	1	no	—
188	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
189	#117	C₄×D₁₁	1	1	yes	characteristic
190	C₂×C₄²	—	11	1	no	—
191	M₃₂	—	2	1	no	—
192	M₃₂	—	2	1	no	—
193	M₃₂	—	11	2	no	—
194	M₃₂	—	22	4	no	—
195	M₃₂	—	22	1	no	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	22	4	no	—
200	C₂×C₁₆	—	22	1	no	—
201	C₂×C₁₆	—	22	1	no	—
202	C₂×C₁₆	—	22	1	no	—
203	C₂×C₁₆	—	22	1	no	—
204	C₂×C₁₆	C₂²⋊D₁₁	1	1	yes	characteristic
205	C₂×C₁₆	C₂²⋊D₁₁	1	1	yes	characteristic
206	#104	—	22	2	no	—
207	#103	—	11	1	no	—
208	#103	—	11	1	no	—
209	#102	—	11	1	no	—
210	#102	—	11	1	no	—
211	#97	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	22	1	no	—
214	C₄×C₈	—	22	1	no	—
215	C₂×C₁₁	—	2	1	no	—
216	C₂×C₁₁	—	2	1	no	—
217	C₂×C₁₁	#1850	1	1	yes	characteristic
218	C₂×C₁₁	#2170	1	1	yes	characteristic
219	C₂×C₁₁	#2171	1	1	yes	characteristic
220	D₁₁	—	4	2	no	—
221	#41	—	11	2	no	—
222	#41	—	11	2	no	—
223	#41	—	22	1	no	—
224	#41	—	22	1	no	—
225	C₂×Q₈	—	11	1	no	—
226	C₂×D₄	—	11	1	no	—
227	C₂×D₄	—	22	1	no	—
228	C₂²×C₄	—	22	1	no	—
229	C₂²×C₄	—	22	1	no	—
230	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	11	2	no	—
234	M₁₆	—	22	1	no	—
235	M₁₆	—	22	2	no	—
236	M₁₆	—	22	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	11	1	no	—
239	C₂×C₈	—	11	1	no	—
240	C₂×C₈	—	22	1	no	—
241	C₂×C₈	—	22	1	no	—
242	C₂×C₈	—	22	1	no	—
243	C₂×C₈	—	22	2	no	—
244	C₂×C₈	—	22	2	no	—
245	C₂×C₈	#4339	1	1	yes	characteristic
246	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
247	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
248	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
249	C₂×C₈	#4343	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
251	C₄⋊C₄	C₈×D₁₁	1	1	yes	characteristic
252	C₄⋊C₄	C₈×D₁₁	1	1	yes	characteristic
253	C₂²⋊C₄	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	22	1	no	—
256	C₄²	—	22	1	no	—
257	C₁₆	—	2	1	no	—
258	C₁₆	—	2	1	no	—
259	C₁₆	—	4	1	no	—
260	C₁₆	—	22	2	no	—
261	C₁₆	—	44	1	no	—
262	C₁₁	#12917	1	1	yes	characteristic, complemented, Sylow
263	C₂³	—	22	1	no	—
264	C₂³	#66954	1	1	yes	characteristic
265	Q₈	—	11	2	no	—
266	Q₈	—	22	1	no	—
267	D₄	—	11	2	no	—
268	D₄	—	22	2	no	—
269	D₄	—	22	2	no	—
270	D₄	—	22	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	22	1	no	—
274	C₂×C₄	—	22	2	no	—
275	C₂×C₄	—	22	2	no	—
276	C₂×C₄	—	22	2	no	—
277	C₂×C₄	—	44	1	no	—
278	C₂×C₄	—	44	1	no	—
279	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
280	C₂×C₄	#66946	1	1	yes	characteristic
281	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
282	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
283	C₂×C₄	#66960	1	1	yes	characteristic
284	C₂×C₄	C₄×C₂²⋊D₁₁	1	1	yes	characteristic
285	C₈	—	4	1	no	—
286	C₈	—	22	2	no	—
287	C₈	—	22	1	no	—
288	C₈	—	22	1	no	—
289	C₈	#66947	1	1	yes	characteristic
290	C₈	#66949	1	1	yes	characteristic
291	C₈	#66990	1	1	yes	characteristic
292	C₈	#66999	1	1	yes	characteristic
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	22	2	no	—
296	C₂²	—	44	1	no	—
297	C₂²	—	44	1	no	—
298	C₂²	#10622449	1	1	yes	characteristic
299	C₂²	#10622452	1	1	yes	characteristic
300	C₂²	C₈×C₂²⋊D₁₁	1	1	yes	characteristic
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	22	2	no	—
304	C₄	—	44	2	no	—
305	C₄	D₁₁×C₄⋊C₈	1	1	yes	characteristic
306	C₄	#10622447	1	1	yes	characteristic
307	C₄	#10622602	1	1	yes	characteristic
308	C₄	#10622740	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	44	2	no	—
312	C₂	#13068792	1	1	yes	characteristic
313	C₂	#13070451	1	1	yes	characteristic
314	C₂	#13070456	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₁
commutator	C₂×C₄×C₁₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#2151	C₂
socle	C₂²×C₁₁	C₈×D₄
prime radical	C₂×#4339	C₂×C₄
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#2151	D₁₁
hypocenter	C₁₁	#12917
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ #2151

Complements

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

N	H ≅ G/N	type
2. C₂×#10622149	C₂	semidirect
5. #13066705	C₂	semidirect
6. #13066693	C₂	semidirect
7. C₁₁×#2151	C₂	semidirect
108. #2151	D₁₁	semidirect
262. C₁₁	#12917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622514	2
C₂×#10622149	2
C₂×#10622153	2
#13068455	2
#13066705	2
#13066693	2
C₁₁×#2151	2
#12917	11