Subgroups — #14754148

Order 2816.

608 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13068765	C₂	1	2	yes	maximal, complemented
2	#13068740	C₂	1	2	yes	maximal, complemented
3	#13068720	C₂	1	2	yes	maximal
4	#13068666	C₂	1	1	yes	maximal, characteristic
5	#13068662	C₂	1	1	yes	maximal, characteristic
6	#13068500	C₂	1	2	yes	maximal, complemented
7	#13068416	C₂	1	2	yes	maximal
8	#13068371	C₂	1	2	yes	maximal, complemented
9	C₁₁×#2875	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₁×#131	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×#131	C₂²	1	1	yes	characteristic
12	C₄×C₁₁×#41	C₂²	1	1	yes	characteristic
13	C₁₁×#405	C₄	1	2	yes	complemented
14	C₁₁×#405	C₄	1	2	yes	complemented
15	C₁₁×#405	C₂²	1	1	yes	characteristic
16	C₁₁×#405	C₂²	1	1	yes	characteristic
17	C₄×C₁₁×M₁₆	C₄	1	2	yes	complemented
18	C₄×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₁₁×#97	C₄	1	2	yes	complemented
20	C₂×C₁₁×#97	C₂²	1	1	yes	characteristic
21	#10622853	C₂²	1	2	yes	—
22	#10622850	C₂²	1	2	yes	—
23	C₂×#66883	C₂²	1	2	yes	—
24	#10622719	C₂²	1	2	yes	—
25	#10622714	C₂²	1	4	yes	—
26	C₁₁⋊#338	C₂²	1	2	yes	—
27	#10622195	C₂²	1	2	yes	—
28	#10622194	C₂²	1	4	yes	—
29	#10622169	C₂²	1	4	yes	—
30	#10622161	C₂²	1	4	yes	—
31	C₂×C₁₁×#41	C₂³	1	1	yes	characteristic
32	C₁₁×#131	—	2	4	no	—
33	C₁₁×#131	—	2	2	no	—
34	C₁₁×#131	—	2	2	no	—
35	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₁₁×M₁₆	C₂³	1	1	yes	characteristic
38	C₂×C₁₁×M₁₆	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₁₁	—	2	1	no	—
44	C₄×D₄×C₁₁	—	2	2	no	—
45	C₄×D₄×C₁₁	—	2	1	no	—
46	C₁₁×#117	C₂×C₄	1	2	yes	—
47	C₁₁×#117	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
50	#66979	C₂³	1	4	yes	—
51	C₂×#4318	C₂³	1	4	yes	—
52	C₁₁×#97	C₂×C₄	1	1	yes	characteristic
53	C₁₁×#97	C₂×C₄	1	2	yes	—
54	C₁₁×#97	C₂×C₄	1	2	yes	—
55	C₁₁×#97	C₂×C₄	1	2	yes	—
56	C₁₁×#97	C₂×C₄	1	1	yes	characteristic
57	C₁₁×#97	C₂×C₄	1	1	yes	characteristic
58	C₁₁×#97	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
62	#66900	—	2	2	no	—
63	#66898	—	2	4	no	—
64	#66897	—	2	2	no	—
65	#66883	—	2	2	no	—
66	#66883	—	2	2	no	—
67	#66882	—	2	2	no	—
68	#66881	—	2	2	no	—
69	#66879	—	2	4	no	—
70	C₈×Dic₁₁	—	2	4	no	—
71	#20788	—	11	1	no	maximal, Sylow
72	C₁₁×#41	—	2	2	no	—
73	C₁₁×#41	—	2	2	no	—
74	C₂×Q₈×C₁₁	#41	1	1	yes	characteristic
75	C₂×D₄×C₁₁	#41	1	2	yes	—
76	C₂×D₄×C₁₁	#41	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	—	2	2	no	—
78	C₂²×C₄×C₁₁	—	2	1	no	—
79	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
81	C₁₁×M₁₆	—	2	4	no	—
82	C₁₁×M₁₆	—	2	4	no	—
83	C₁₁×M₁₆	—	2	2	no	—
84	C₁₁×M₁₆	—	2	2	no	—
85	C₂×C₈×C₁₁	—	2	2	no	—
86	C₂×C₈×C₁₁	—	2	2	no	—
87	C₂×C₈×C₁₁	—	2	4	no	—
88	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₁	#41	1	4	yes	—
95	C₂×C₈×C₁₁	#41	1	2	yes	—
96	C₂×C₈×C₁₁	#41	1	2	yes	—
97	C₁₁×C₄⋊C₄	—	2	2	no	—
98	C₁₁×C₄⋊C₄	—	2	1	no	—
99	C₁₁×C₂²⋊C₄	—	2	2	no	—
100	C₁₁×C₂²⋊C₄	—	2	1	no	—
101	C₄²×C₁₁	—	2	2	no	—
102	C₄²×C₁₁	—	2	1	no	—
103	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
104	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
106	C₂²⋊Dic₁₁	—	2	4	no	—
107	C₄⋊Dic₁₁	—	2	4	no	—
108	C₄×Dic₁₁	—	2	4	no	—
109	#4318	—	2	4	no	—
110	#4318	—	2	4	no	—
111	C₂×C₁₁⋊C₈	—	2	4	no	—
112	#2875	D₁₁	1	1	yes	characteristic
113	#1841	—	11	2	no	—
114	#1833	—	11	2	no	—
115	#1827	—	11	2	no	—
116	#1813	—	11	1	no	—
117	#1812	—	11	1	no	—
118	#1773	—	11	2	no	—
119	#1757	—	11	2	no	—
120	#1747	—	11	2	no	—
121	C₂³×C₁₁	#117	1	2	yes	—
122	C₂³×C₁₁	C₂×#41	1	1	yes	characteristic
123	Q₈×C₁₁	—	2	2	no	—
124	D₄×C₁₁	—	2	4	no	—
125	D₄×C₁₁	—	2	2	no	—
126	C₂×C₄×C₁₁	—	2	2	no	—
127	C₂×C₄×C₁₁	—	2	2	no	—
128	C₂×C₄×C₁₁	—	2	1	no	—
129	C₂×C₄×C₁₁	—	2	2	no	—
130	C₂×C₄×C₁₁	—	2	1	no	—
131	C₂×C₄×C₁₁	—	8	1	no	—
132	C₂×C₄×C₁₁	—	8	2	no	—
133	C₂×C₄×C₁₁	#117	1	2	yes	—
134	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
136	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
137	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₁₁	—	4	4	no	—
139	C₈×C₁₁	—	2	2	no	—
140	C₈×C₁₁	—	2	2	no	—
141	C₈×C₁₁	#117	1	1	yes	characteristic
142	C₈×C₁₁	#117	1	1	yes	characteristic
143	C₈×C₁₁	#117	1	2	yes	—
144	C₈×C₁₁	#117	1	2	yes	—
145	C₈×C₁₁	#117	1	2	yes	—
146	C₁₁⋊C₈	—	4	4	no	—
147	C₂×#131	C₂×D₁₁	1	1	yes	characteristic
148	C₂×#131	C₂×D₁₁	1	1	yes	characteristic
149	C₄×#41	C₂×D₁₁	1	1	yes	characteristic
150	C₂×#108	—	11	2	no	—
151	#427	—	11	2	no	—
152	#413	—	11	2	no	—
153	#410	—	11	2	no	—
154	#405	—	11	4	no	—
155	#405	Dic₁₁	1	2	yes	complemented
156	#405	Dic₁₁	1	2	yes	complemented
157	#405	C₂×D₁₁	1	1	yes	characteristic
158	#405	C₂×D₁₁	1	1	yes	characteristic
159	C₄×M₁₆	Dic₁₁	1	2	yes	complemented
160	C₄×M₁₆	C₂×D₁₁	1	1	yes	characteristic
161	C₂×#97	Dic₁₁	1	2	yes	complemented
162	C₂×#97	C₂×D₁₁	1	1	yes	characteristic
163	#344	—	11	4	no	—
164	#339	—	11	4	no	—
165	#338	—	11	2	no	—
166	H(3,ℤ/4)	—	11	2	no	—
167	H(3,ℤ/4)	—	11	4	no	—
168	C₂²×C₁₁	—	2	2	no	—
169	C₂²×C₁₁	—	4	1	no	—
170	C₂²×C₁₁	—	4	2	no	—
171	C₂²×C₁₁	C₂×#117	1	1	yes	characteristic
172	C₂²×C₁₁	#529	1	2	yes	—
173	C₄×C₁₁	—	2	2	no	—
174	C₄×C₁₁	—	4	2	no	—
175	C₄×C₁₁	—	4	1	no	—
176	C₄×C₁₁	—	4	1	no	—
177	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
178	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
179	C₄×C₁₁	#529	1	2	yes	—
180	Dic₁₁	—	8	4	no	—
181	C₂×#41	C₂²×D₁₁	1	1	yes	characteristic
182	#131	—	2	4	no	—
183	#131	—	2	2	no	—
184	#131	—	2	2	no	—
185	C₂×M₁₆	—	11	4	no	—
186	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
187	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
188	C₂×M₁₆	C₂²×D₁₁	1	1	yes	characteristic
189	C₂×M₁₆	C₂²×D₁₁	1	1	yes	characteristic
190	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
191	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
192	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
194	C₄×Q₈	—	2	1	no	—
195	C₄×D₄	—	2	2	no	—
196	C₄×D₄	—	2	1	no	—
197	#117	—	11	4	no	—
198	#117	C₂×Dic₁₁	1	2	yes	—
199	#117	C₂²×D₁₁	1	1	yes	characteristic
200	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
201	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
202	#108	—	22	2	no	—
203	#108	—	22	2	no	—
204	#107	—	22	2	no	—
205	#106	—	22	2	no	—
206	#101	—	22	2	no	—
207	#100	—	22	2	no	—
208	#99	—	22	4	no	—
209	#97	—	22	4	no	—
210	#97	C₂×Dic₁₁	1	1	yes	characteristic
211	#97	C₂×Dic₁₁	1	2	yes	—
212	#97	C₂×Dic₁₁	1	2	yes	—
213	#97	C₂×Dic₁₁	1	2	yes	—
214	#97	C₂×Dic₁₁	1	1	yes	characteristic
215	#97	C₂×Dic₁₁	1	1	yes	characteristic
216	#97	C₂×Dic₁₁	1	1	yes	characteristic
217	C₄×C₈	—	22	4	no	—
218	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
219	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
220	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
221	C₂×C₁₁	—	2	2	no	—
222	C₂×C₁₁	—	2	1	no	—
223	C₂×C₁₁	—	4	2	no	—
224	C₂×C₁₁	#2283	1	1	yes	characteristic
225	#41	—	2	2	no	—
226	#41	—	2	2	no	—
227	C₂×Q₈	#4339	1	1	yes	characteristic
228	C₂×D₄	#4339	1	2	yes	—
229	C₂×D₄	#4339	1	1	yes	characteristic
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
233	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	22	4	no	—
239	M₁₆	—	22	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	22	4	no	—
244	C₂×C₈	#4339	1	4	yes	—
245	C₂×C₈	#4339	1	2	yes	—
246	C₂×C₈	#4339	1	2	yes	—
247	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
248	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
249	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
250	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
252	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	22	4	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	22	4	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	22	4	no	—
262	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
263	C₄²	C₂²×Dic₁₁	1	2	yes	—
264	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
265	C₁₁	#20788	1	1	yes	characteristic, complemented, Sylow
266	C₂³	#66979	1	2	yes	—
267	C₂³	C₂×#4339	1	1	yes	characteristic
268	Q₈	—	2	2	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	44	4	no	—
279	C₂×C₄	#66979	1	2	yes	—
280	C₂×C₄	C₂×#4339	1	1	yes	characteristic
281	C₂×C₄	C₂×#4339	1	2	yes	—
282	C₂×C₄	C₂×#4339	1	2	yes	—
283	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	—	44	4	no	—
287	C₈	#66979	1	1	yes	characteristic
288	C₈	#66979	1	1	yes	characteristic
289	C₈	#66979	1	2	yes	—
290	C₈	#66979	1	2	yes	—
291	C₈	#66979	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	#10623010	1	2	yes	—
296	C₂²	C₂×#66979	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	1	no	—
301	C₄	—	88	4	no	—
302	C₄	#10623010	1	2	yes	—
303	C₄	C₂×#66979	1	1	yes	characteristic
304	C₄	C₂×#66979	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	4	2	no	—
308	C₂	#13071014	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#66979
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₁₁×#2875	C₂
socle	C₂×C₁₁	#2283
prime radical	C₂×C₁₁×#41	C₂³
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	#2875	D₁₁
hypocenter	C₁₁	#20788
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ #405 ◁ #2875

Complements

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

N	H ≅ G/N	type
1. #13068765	C₂	semidirect
2. #13068740	C₂	semidirect
6. #13068500	C₂	semidirect
8. #13068371	C₂	semidirect
13. C₁₁×#405	C₄	semidirect
14. C₁₁×#405	C₄	semidirect
17. C₄×C₁₁×M₁₆	C₄	semidirect
19. C₂×C₁₁×#97	C₄	semidirect
155. #405	Dic₁₁	semidirect
156. #405	Dic₁₁	semidirect
159. C₄×M₁₆	Dic₁₁	semidirect
161. C₂×#97	Dic₁₁	semidirect
265. C₁₁	#20788	semidirect

Maximal subgroups

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

subgroup	index
#13068765	2
#13068740	2
#13068720	2
#13068666	2
#13068662	2
#13068500	2
#13068416	2
#13068371	2
C₁₁×#2875	2
#20788	11