Subgroups — #14754139

Order 2816.

616 conjugacy classes of subgroups.

The subgroup lattice has 318 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	#13068495	C₂	1	2	yes	maximal, complemented
7	#13068416	C₂	1	2	yes	maximal
8	#13068371	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₁×#131	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×C₄×C₈×C₁₁	C₄	1	2	yes	complemented
20	C₂×C₄×C₈×C₁₁	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	2	no	—
33	C₁₁×#131	—	2	2	no	—
34	C₁₁×#131	—	2	2	no	—
35	C₁₁×#131	—	2	2	no	—
36	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₁₁×M₁₆	C₂³	1	1	yes	characteristic
39	C₂×C₁₁×M₁₆	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₁₁	—	2	1	no	—
45	C₄×D₄×C₁₁	—	2	2	no	—
46	C₄×D₄×C₁₁	—	2	1	no	—
47	C₁₁×#117	C₂×C₄	1	2	yes	—
48	C₁₁×#117	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
51	#66979	C₂³	1	4	yes	—
52	C₂×#4318	C₂³	1	4	yes	—
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	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
63	#66900	—	2	2	no	—
64	#66898	—	2	4	no	—
65	#66897	—	2	2	no	—
66	#66883	—	2	2	no	—
67	#66883	—	2	2	no	—
68	#66882	—	2	2	no	—
69	#66881	—	2	2	no	—
70	#66879	—	2	4	no	—
71	C₈×Dic₁₁	—	2	4	no	—
72	#20787	—	11	1	no	maximal, Sylow
73	C₁₁×#41	—	2	2	no	—
74	C₁₁×#41	—	2	2	no	—
75	C₂×Q₈×C₁₁	#41	1	1	yes	characteristic
76	C₂×D₄×C₁₁	#41	1	2	yes	—
77	C₂×D₄×C₁₁	#41	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	—	2	2	no	—
79	C₂²×C₄×C₁₁	—	2	1	no	—
80	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
82	C₁₁×M₁₆	—	2	4	no	—
83	C₁₁×M₁₆	—	2	4	no	—
84	C₁₁×M₁₆	—	2	2	no	—
85	C₁₁×M₁₆	—	2	2	no	—
86	C₂×C₈×C₁₁	—	2	2	no	—
87	C₂×C₈×C₁₁	—	2	2	no	—
88	C₂×C₈×C₁₁	—	2	2	no	—
89	C₂×C₈×C₁₁	—	2	2	no	—
90	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₁	#41	1	2	yes	—
97	C₂×C₈×C₁₁	#41	1	2	yes	—
98	C₂×C₈×C₁₁	#41	1	2	yes	—
99	C₂×C₈×C₁₁	#41	1	2	yes	—
100	C₁₁×C₄⋊C₄	—	2	2	no	—
101	C₁₁×C₄⋊C₄	—	2	1	no	—
102	C₁₁×C₂²⋊C₄	—	2	2	no	—
103	C₁₁×C₂²⋊C₄	—	2	1	no	—
104	C₄²×C₁₁	—	2	2	no	—
105	C₄²×C₁₁	—	2	1	no	—
106	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
108	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
109	C₂²⋊Dic₁₁	—	2	4	no	—
110	C₄⋊Dic₁₁	—	2	4	no	—
111	C₄×Dic₁₁	—	2	4	no	—
112	#4318	—	2	4	no	—
113	#4318	—	2	4	no	—
114	C₂×C₁₁⋊C₈	—	2	4	no	—
115	C₄×#131	D₁₁	1	1	yes	characteristic
116	#1841	—	11	2	no	—
117	#1833	—	11	2	no	—
118	#1827	—	11	2	no	—
119	#1813	—	11	1	no	—
120	#1812	—	11	1	no	—
121	#1772	—	11	2	no	—
122	#1757	—	11	2	no	—
123	#1747	—	11	2	no	—
124	C₂³×C₁₁	#117	1	2	yes	—
125	C₂³×C₁₁	C₂×#41	1	1	yes	characteristic
126	Q₈×C₁₁	—	2	2	no	—
127	D₄×C₁₁	—	2	4	no	—
128	D₄×C₁₁	—	2	2	no	—
129	C₂×C₄×C₁₁	—	2	1	no	—
130	C₂×C₄×C₁₁	—	2	2	no	—
131	C₂×C₄×C₁₁	—	2	2	no	—
132	C₂×C₄×C₁₁	—	2	2	no	—
133	C₂×C₄×C₁₁	—	2	1	no	—
134	C₂×C₄×C₁₁	—	4	2	no	—
135	C₂×C₄×C₁₁	—	4	4	no	—
136	C₂×C₄×C₁₁	#117	1	2	yes	—
137	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
139	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
140	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₁₁	—	4	4	no	—
142	C₈×C₁₁	—	2	2	no	—
143	C₈×C₁₁	—	2	2	no	—
144	C₈×C₁₁	#117	1	1	yes	characteristic
145	C₈×C₁₁	#117	1	1	yes	characteristic
146	C₈×C₁₁	#117	1	1	yes	characteristic
147	C₈×C₁₁	#117	1	2	yes	—
148	C₈×C₁₁	#117	1	2	yes	—
149	C₈×C₁₁	#117	1	1	yes	characteristic
150	C₁₁⋊C₈	—	4	4	no	—
151	C₂×#131	C₂×D₁₁	1	1	yes	characteristic
152	C₂×#131	C₂×D₁₁	1	1	yes	characteristic
153	C₄×#41	C₂×D₁₁	1	1	yes	characteristic
154	C₂×#108	—	11	2	no	—
155	#427	—	11	2	no	—
156	#413	—	11	2	no	—
157	#410	—	11	2	no	—
158	#405	—	11	4	no	—
159	#405	Dic₁₁	1	2	yes	complemented
160	#405	Dic₁₁	1	2	yes	complemented
161	#405	C₂×D₁₁	1	1	yes	characteristic
162	#405	C₂×D₁₁	1	1	yes	characteristic
163	C₄×M₁₆	Dic₁₁	1	2	yes	complemented
164	C₄×M₁₆	C₂×D₁₁	1	1	yes	characteristic
165	C₂×C₄×C₈	Dic₁₁	1	2	yes	complemented
166	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
167	#344	—	11	4	no	—
168	#339	—	11	4	no	—
169	#338	—	11	2	no	—
170	H(3,ℤ/4)	—	11	2	no	—
171	H(3,ℤ/4)	—	11	4	no	—
172	C₂²×C₁₁	—	2	2	no	—
173	C₂²×C₁₁	—	4	1	no	—
174	C₂²×C₁₁	—	4	2	no	—
175	C₂²×C₁₁	C₂×#117	1	1	yes	characteristic
176	C₂²×C₁₁	#529	1	2	yes	—
177	C₄×C₁₁	—	2	2	no	—
178	C₄×C₁₁	—	2	2	no	—
179	C₄×C₁₁	—	4	2	no	—
180	C₄×C₁₁	—	4	1	no	—
181	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
182	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
183	C₄×C₁₁	#529	1	2	yes	—
184	Dic₁₁	—	8	4	no	—
185	C₂×#41	C₂²×D₁₁	1	1	yes	characteristic
186	#131	—	2	2	no	—
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	#131	—	2	2	no	—
190	C₂×M₁₆	—	11	4	no	—
191	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
192	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
193	C₂×M₁₆	C₂²×D₁₁	1	1	yes	characteristic
194	C₂×M₁₆	C₂²×D₁₁	1	1	yes	characteristic
195	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
196	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
197	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
199	C₄×Q₈	—	2	1	no	—
200	C₄×D₄	—	2	2	no	—
201	C₄×D₄	—	2	1	no	—
202	#117	—	11	4	no	—
203	#117	C₂×Dic₁₁	1	2	yes	—
204	#117	C₂²×D₁₁	1	1	yes	characteristic
205	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
206	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
207	#108	—	22	2	no	—
208	#108	—	22	2	no	—
209	#107	—	22	2	no	—
210	#106	—	22	2	no	—
211	#101	—	22	2	no	—
212	#100	—	22	2	no	—
213	#99	—	22	4	no	—
214	#97	—	22	4	no	—
215	#97	C₂×Dic₁₁	1	2	yes	—
216	#97	C₂×Dic₁₁	1	2	yes	—
217	#97	C₂×Dic₁₁	1	2	yes	—
218	C₄×C₈	—	22	4	no	—
219	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
220	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
221	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
222	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
223	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
224	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
225	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
226	C₂×C₁₁	—	2	2	no	—
227	C₂×C₁₁	—	2	1	no	—
228	C₂×C₁₁	—	4	2	no	—
229	C₂×C₁₁	#2283	1	1	yes	characteristic
230	#41	—	2	2	no	—
231	#41	—	2	2	no	—
232	C₂×Q₈	#4339	1	1	yes	characteristic
233	C₂×D₄	#4339	1	2	yes	—
234	C₂×D₄	#4339	1	1	yes	characteristic
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
238	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
239	M₁₆	—	2	2	no	—
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	22	4	no	—
244	M₁₆	—	22	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	22	4	no	—
250	C₂×C₈	#4339	1	2	yes	—
251	C₂×C₈	#4339	1	2	yes	—
252	C₂×C₈	#4339	1	2	yes	—
253	C₂×C₈	#4339	1	2	yes	—
254	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
255	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
256	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
257	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
258	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
259	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	22	4	no	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	22	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	22	4	no	—
269	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
270	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
271	C₄²	C₂²×Dic₁₁	1	2	yes	—
272	C₁₁	#20787	1	1	yes	characteristic, complemented, Sylow
273	C₂³	#66979	1	2	yes	—
274	C₂³	C₂×#4339	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	D₄	—	2	4	no	—
277	D₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	44	4	no	—
286	C₂×C₄	#66979	1	2	yes	—
287	C₂×C₄	C₂×#4339	1	1	yes	characteristic
288	C₂×C₄	C₂×#4339	1	2	yes	—
289	C₂×C₄	C₂×#4339	1	2	yes	—
290	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	44	4	no	—
294	C₈	#66979	1	1	yes	characteristic
295	C₈	#66979	1	1	yes	characteristic
296	C₈	#66979	1	1	yes	characteristic
297	C₈	#66979	1	2	yes	—
298	C₈	#66979	1	2	yes	—
299	C₈	#66979	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#10623010	1	2	yes	—
304	C₂²	C₂×#66979	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	88	4	no	—
310	C₄	#10623010	1	2	yes	—
311	C₄	C₂×#66979	1	1	yes	characteristic
312	C₄	C₂×#66979	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	4	2	no	—
316	C₂	#13071014	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#66979
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₄×C₁₁×#131	C₂
socle	C₂×C₁₁	#2283
prime radical	C₂×C₁₁×#41	C₂³
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₄×#131	D₁₁
hypocenter	C₁₁	#20787
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₈ ◁ #405 ◁ C₄×#131

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. #13068495	C₂	semidirect
8. #13068371	C₂	semidirect
13. C₁₁×#405	C₄	semidirect
14. C₁₁×#405	C₄	semidirect
17. C₄×C₁₁×M₁₆	C₄	semidirect
19. C₂×C₄×C₈×C₁₁	C₄	semidirect
159. #405	Dic₁₁	semidirect
160. #405	Dic₁₁	semidirect
163. C₄×M₁₆	Dic₁₁	semidirect
165. C₂×C₄×C₈	Dic₁₁	semidirect
272. C₁₁	#20787	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
#13068495	2
#13068416	2
#13068371	2
C₄×C₁₁×#131	2
#20787	11