Subgroups — #14725811

Order 2816.

644 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077683	C₂	1	2	yes	maximal, complemented
2	#13067530	C₂	1	2	yes	maximal, complemented
3	#13067446	C₂	1	4	yes	maximal, complemented
4	#13067440	C₂	1	2	yes	maximal, complemented
5	#13067364	C₂	1	1	yes	maximal, characteristic, complemented
6	#13067033	C₂	1	2	yes	maximal, complemented
7	#13066905	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₁×#3169	C₂	1	1	yes	maximal, characteristic
9	C₁₁×#532	C₂²	1	2	yes	—
10	C₁₁×#500	C₄	1	1	yes	characteristic
11	C₁₁×#493	C₄	1	1	yes	characteristic
12	C₁₁×#486	C₄	1	2	yes	—
13	C₁₁×#466	C₄	1	2	yes	—
14	C₁₁×#466	C₄	1	2	yes	—
15	C₄×C₁₁×D₈	C₂²	1	2	yes	—
16	C₄×C₁₁×D₈	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
18	#10622847	—	2	4	no	—
19	#10622625	—	2	2	no	—
20	#10622625	C₂²	1	4	yes	—
21	D₄×C₁₁⋊C₈	—	2	2	no	—
22	D₄×C₁₁⋊C₈	C₂²	1	4	yes	—
23	#10622538	C₂²	1	2	yes	—
24	#10622538	C₂²	1	1	yes	characteristic
25	C₄×#4318	C₂²	1	1	yes	characteristic
26	D₄⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
27	D₄⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
28	#10622096	C₂²	1	2	yes	—
29	#10622095	C₂²	1	1	yes	characteristic
30	#10622095	C₂²	1	1	yes	characteristic
31	#10622093	C₂²	1	2	yes	—
32	#10622092	C₂²	1	2	yes	—
33	C₂×C₁₁×#41	—	2	2	no	—
34	C₂×C₁₁×D₈	C₂×C₄	1	2	yes	—
35	C₂×C₁₁×D₈	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₁₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₁₁×#127	C₂×C₄	1	2	yes	—
40	C₁₁×#124	—	2	2	no	—
41	C₁₁×#121	—	2	2	no	—
42	C₁₁×#121	C₂×C₄	1	4	yes	—
43	C₄×D₄×C₁₁	—	2	2	no	—
44	C₄×D₄×C₁₁	C₂³	1	4	yes	—
45	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
46	C₂×#4318	—	2	1	no	—
47	C₂×#4318	—	2	4	no	—
48	C₂²×C₁₁⋊C₈	—	2	4	no	—
49	C₁₁×#107	C₂×C₄	1	2	yes	—
50	C₁₁×#107	C₂×C₄	1	2	yes	—
51	C₁₁×#102	C₂×C₄	1	4	yes	—
52	C₁₁×#102	C₂×C₄	1	4	yes	—
53	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
57	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
58	C₂²⋊(C₁₁⋊C₈)	—	2	4	no	—
59	C₂²⋊(C₁₁⋊C₈)	—	2	4	no	—
60	C₂²⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
61	C₄⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
62	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
63	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
64	#66867	—	2	1	no	—
65	C₄×C₁₁⋊C₈	—	2	2	no	—
66	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
67	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
68	#18091	—	11	1	no	maximal, Sylow
69	C₁₁×#41	—	8	2	no	—
70	C₁₁×#41	—	8	2	no	—
71	C₂×Q₈×C₁₁	—	2	2	no	—
72	C₂×D₄×C₁₁	—	2	4	no	—
73	C₂×D₄×C₁₁	—	2	2	no	—
74	C₂×D₄×C₁₁	—	2	2	no	—
75	C₂×D₄×C₁₁	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₁₁	—	2	4	no	—
77	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
80	C₁₁×D₈	—	2	4	no	—
81	C₁₁×D₈	—	4	2	no	—
82	C₂×C₈×C₁₁	—	2	2	no	—
83	C₂×C₈×C₁₁	—	4	1	no	—
84	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
88	C₁₁×C₄⋊C₄	—	2	2	no	—
89	C₁₁×C₄⋊C₄	C₂²×C₄	1	4	yes	—
90	C₁₁×C₂²⋊C₄	—	2	4	no	—
91	C₁₁×C₂²⋊C₄	—	2	4	no	—
92	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₁	#41	1	1	yes	characteristic
95	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
96	#4318	—	4	2	no	—
97	#4318	—	8	4	no	—
98	C₂×C₁₁⋊C₈	—	2	2	no	—
99	C₂×C₁₁⋊C₈	—	2	2	no	—
100	C₂×C₁₁⋊C₈	—	2	2	no	—
101	C₂×C₁₁⋊C₈	—	8	4	no	—
102	C₂×C₁₁⋊C₈	C₂×D₄	1	4	yes	—
103	#3169	D₁₁	1	1	yes	characteristic
104	#2979	—	11	2	no	—
105	#1594	—	11	2	no	—
106	#1581	—	11	4	no	—
107	#1580	—	11	2	no	—
108	#1564	—	11	1	no	—
109	#1484	—	11	2	no	—
110	#1451	—	11	1	no	—
111	C₂³×C₁₁	—	2	4	no	—
112	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
113	Q₈×C₁₁	—	8	2	no	—
114	D₄×C₁₁	—	2	4	no	—
115	D₄×C₁₁	—	2	4	no	—
116	D₄×C₁₁	—	4	4	no	—
117	D₄×C₁₁	—	8	2	no	—
118	D₄×C₁₁	—	8	4	no	—
119	C₂×C₄×C₁₁	—	2	1	no	—
120	C₂×C₄×C₁₁	—	2	1	no	—
121	C₂×C₄×C₁₁	—	2	2	no	—
122	C₂×C₄×C₁₁	—	2	2	no	—
123	C₂×C₄×C₁₁	—	2	4	no	—
124	C₂×C₄×C₁₁	—	4	4	no	—
125	C₂×C₄×C₁₁	—	4	4	no	—
126	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
132	C₈×C₁₁	—	2	1	no	—
133	C₈×C₁₁	—	2	1	no	—
134	C₈×C₁₁	—	2	1	no	—
135	C₈×C₁₁	#131	1	2	yes	—
136	C₁₁⋊C₈	—	2	4	no	—
137	C₁₁⋊C₈	—	4	2	no	—
138	C₁₁⋊C₈	—	4	2	no	—
139	C₁₁⋊C₈	—	4	2	no	—
140	#532	C₂×D₁₁	1	2	yes	—
141	#500	Dic₁₁	1	1	yes	characteristic
142	#493	Dic₁₁	1	1	yes	characteristic
143	#486	Dic₁₁	1	2	yes	—
144	#466	Dic₁₁	1	2	yes	—
145	#466	Dic₁₁	1	2	yes	—
146	C₄×D₈	C₂×D₁₁	1	2	yes	—
147	C₄×D₈	C₂×D₁₁	1	2	yes	—
148	#436	—	11	4	no	—
149	#436	—	22	2	no	—
150	C₈×D₄	—	11	4	no	—
151	C₈×D₄	—	22	2	no	—
152	#431	—	11	1	no	—
153	#431	—	11	2	no	—
154	#408	—	22	4	no	—
155	C₄×M₁₆	—	11	1	no	—
156	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
157	#335	—	11	2	no	—
158	#334	—	11	1	no	—
159	#334	—	11	1	no	—
160	D₄⋊C₈	—	11	4	no	—
161	D₄⋊C₈	—	11	4	no	—
162	#322	—	11	2	no	—
163	#322	—	11	2	no	—
164	C₂²×C₁₁	—	2	1	no	—
165	C₂²×C₁₁	—	2	1	no	—
166	C₂²×C₁₁	—	2	1	no	—
167	C₂²×C₁₁	—	4	4	no	—
168	C₂²×C₁₁	—	4	4	no	—
169	C₂²×C₁₁	—	4	4	no	—
170	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄×C₁₁	—	2	1	no	—
172	C₄×C₁₁	—	2	1	no	—
173	C₄×C₁₁	—	2	1	no	—
174	C₄×C₁₁	—	4	4	no	—
175	C₄×C₁₁	#444	1	2	yes	—
176	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
177	C₄×C₁₁	#568	1	1	yes	characteristic
178	C₄×C₁₁	#576	1	2	yes	—
179	C₂×#41	—	2	2	no	—
180	C₂×D₈	C₂×Dic₁₁	1	2	yes	—
181	C₂×D₈	C₂×Dic₁₁	1	2	yes	—
182	C₂×M₁₆	—	22	4	no	—
183	C₂×M₁₆	—	22	1	no	—
184	C₂²×C₈	—	22	4	no	—
185	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
187	C₄⋊Q₈	C₂×Dic₁₁	1	2	yes	—
188	#127	C₂×Dic₁₁	1	2	yes	—
189	#124	—	2	2	no	—
190	#121	—	2	2	no	—
191	#121	C₂×Dic₁₁	1	4	yes	—
192	C₄×D₄	—	2	2	no	—
193	C₄×D₄	C₂²×D₁₁	1	4	yes	—
194	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
195	#107	C₂×Dic₁₁	1	2	yes	—
196	#107	C₂×Dic₁₁	1	2	yes	—
197	C₄⋊C₈	—	11	2	no	—
198	C₄⋊C₈	—	11	2	no	—
199	C₄⋊C₈	—	11	2	no	—
200	#102	C₂×Dic₁₁	1	4	yes	—
201	#102	C₂×Dic₁₁	1	4	yes	—
202	C₂²⋊C₈	—	11	2	no	—
203	C₂²⋊C₈	—	22	4	no	—
204	C₂²⋊C₈	—	22	4	no	—
205	C₂²⋊C₈	—	22	2	no	—
206	#97	—	22	1	no	—
207	C₄×C₈	—	11	2	no	—
208	C₄×C₈	—	11	2	no	—
209	C₄×C₈	—	22	2	no	—
210	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
211	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
214	C₂×C₁₁	—	4	1	no	—
215	C₂×C₁₁	—	8	4	no	—
216	C₂×C₁₁	#2935	1	1	yes	characteristic
217	C₂×C₁₁	#2948	1	1	yes	characteristic
218	C₂×C₁₁	C₂×#444	1	1	yes	characteristic
219	#41	—	8	2	no	—
220	#41	—	8	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂²×Dic₁₁	1	4	yes	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
228	C₂²×C₄	#4341	1	1	yes	characteristic
229	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
230	D₈	—	2	4	no	—
231	D₈	—	4	2	no	—
232	M₁₆	—	44	2	no	—
233	M₁₆	—	88	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	11	4	no	—
237	C₂×C₈	—	22	2	no	—
238	C₂×C₈	—	22	2	no	—
239	C₂×C₈	—	22	2	no	—
240	C₂×C₈	—	88	4	no	—
241	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
242	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
243	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
244	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	C₂²×Dic₁₁	1	4	yes	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₄²	D₄×D₁₁	1	1	yes	characteristic
250	C₄²	#4341	1	1	yes	characteristic
251	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
252	C₄²	C₂³×D₁₁	1	1	yes	characteristic
253	C₁₁	#18091	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	2	4	no	—
255	C₂³	D₄×Dic₁₁	1	1	yes	characteristic
256	Q₈	—	8	2	no	—
257	D₄	—	2	4	no	—
258	D₄	—	2	4	no	—
259	D₄	—	4	4	no	—
260	D₄	—	8	2	no	—
261	D₄	—	8	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
270	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
271	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
272	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
273	C₂×C₄	C₂×#4341	1	1	yes	characteristic
274	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	22	4	no	—
279	C₈	—	44	2	no	—
280	C₈	—	44	2	no	—
281	C₈	—	44	2	no	—
282	C₈	#67001	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	C₂×D₄×Dic₁₁	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	4	no	—
294	C₄	#10622810	1	2	yes	—
295	C₄	#10623288	1	2	yes	—
296	C₄	C₂×#67001	1	1	yes	characteristic
297	C₄	#10623350	1	1	yes	characteristic
298	C₂	—	4	1	no	—
299	C₂	—	8	4	no	—
300	C₂	#13077360	1	1	yes	characteristic
301	C₂	#13077468	1	1	yes	characteristic
302	C₂	C₂×#10622810	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting, prime radical	C₁₁×#3169	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#3169	D₁₁
hypocenter	C₁₁	#18091
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #13077683	C₂	semidirect
2. #13067530	C₂	semidirect
3. #13067446	C₂	semidirect
4. #13067440	C₂	semidirect
5. #13067364	C₂	semidirect
6. #13067033	C₂	semidirect
7. #13066905	C₂	semidirect
253. C₁₁	#18091	semidirect

Maximal subgroups

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

subgroup	index
#13077683	2
#13067530	2
#13067446	2
#13067440	2
#13067364	2
#13067033	2
#13066905	2
C₁₁×#3169	2
#18091	11