Subgroups — #14725597

Order 2816.

714 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077360	C₂	1	2	yes	maximal, complemented
2	#13067530	C₂	1	2	yes	maximal, complemented
3	#13067481	C₂	1	4	yes	maximal, complemented
4	#13067440	C₂	1	2	yes	maximal, complemented
5	C₂×#10622095	C₂	1	1	yes	maximal, characteristic, complemented
6	#13067015	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	C₂×#67001	—	2	2	no	—
11	C₁₁×#500	C₄	1	1	yes	characteristic
12	C₁₁×#493	C₄	1	1	yes	characteristic
13	C₁₁×#486	C₄	1	2	yes	—
14	C₁₁×#466	C₄	1	2	yes	—
15	C₁₁×#466	C₄	1	2	yes	—
16	C₄×C₁₁×D₈	C₂²	1	2	yes	—
17	C₄×C₁₁×D₈	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
19	#10622847	—	2	2	no	—
20	#10622625	C₂²	1	4	yes	—
21	#10622624	—	2	4	no	—
22	D₄×C₁₁⋊C₈	C₂²	1	4	yes	—
23	#10622538	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
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	#10622095	C₂²	1	1	yes	characteristic
29	#10622095	C₂²	1	1	yes	characteristic
30	#10622095	C₂²	1	2	yes	—
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	#67001	—	4	8	no	—
47	C₂×#4318	—	2	4	no	—
48	C₂×#4318	D₄	1	2	yes	—
49	C₂²×C₁₁⋊C₈	—	2	2	no	—
50	C₂²×C₁₁⋊C₈	—	2	4	no	—
51	C₁₁×#107	C₂×C₄	1	2	yes	—
52	C₁₁×#107	C₂×C₄	1	2	yes	—
53	C₁₁×#102	C₂×C₄	1	4	yes	—
54	C₁₁×#102	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
59	C₂²⋊(C₁₁⋊C₈)	—	2	1	no	—
60	C₂²⋊(C₁₁⋊C₈)	—	2	4	no	—
61	C₂²⋊(C₁₁⋊C₈)	—	2	4	no	—
62	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
63	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
64	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
65	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
66	#66867	D₄	1	2	yes	—
67	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
68	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
69	#18073	—	11	1	no	maximal, Sylow
70	C₁₁×#41	—	4	4	no	—
71	C₁₁×#41	—	4	4	no	—
72	C₂×Q₈×C₁₁	—	2	2	no	—
73	C₂×D₄×C₁₁	—	2	4	no	—
74	C₂×D₄×C₁₁	—	2	2	no	—
75	C₂×D₄×C₁₁	—	2	2	no	—
76	C₂×D₄×C₁₁	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₁₁	—	2	4	no	—
78	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
81	C₁₁×D₈	—	2	4	no	—
82	C₁₁×D₈	—	4	2	no	—
83	C₂×C₈×C₁₁	—	2	2	no	—
84	C₂×C₈×C₁₁	—	4	1	no	—
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	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
89	C₁₁×C₄⋊C₄	—	2	2	no	—
90	C₁₁×C₄⋊C₄	C₂²×C₄	1	4	yes	—
91	C₁₁×C₂²⋊C₄	—	2	4	no	—
92	C₁₁×C₂²⋊C₄	—	2	4	no	—
93	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₁	#41	1	1	yes	characteristic
96	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
97	#4318	—	2	4	no	—
98	#4318	—	4	8	no	—
99	C₂×C₁₁⋊C₈	—	2	2	no	—
100	C₂×C₁₁⋊C₈	—	2	2	no	—
101	C₂×C₁₁⋊C₈	—	2	2	no	—
102	C₂×C₁₁⋊C₈	—	4	8	no	—
103	C₂×C₁₁⋊C₈	—	4	4	no	—
104	C₂×C₁₁⋊C₈	C₂×D₄	1	4	yes	—
105	#3169	D₁₁	1	1	yes	characteristic
106	#2935	—	11	2	no	—
107	#1594	—	11	2	no	—
108	#1586	—	11	4	no	—
109	#1580	—	11	2	no	—
110	C₂×#334	—	11	1	no	—
111	#1480	—	11	2	no	—
112	#1451	—	11	1	no	—
113	C₂³×C₁₁	—	2	4	no	—
114	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
115	Q₈×C₁₁	—	4	4	no	—
116	D₄×C₁₁	—	2	4	no	—
117	D₄×C₁₁	—	2	4	no	—
118	D₄×C₁₁	—	4	8	no	—
119	D₄×C₁₁	—	4	4	no	—
120	D₄×C₁₁	—	8	2	no	—
121	C₂×C₄×C₁₁	—	2	1	no	—
122	C₂×C₄×C₁₁	—	2	1	no	—
123	C₂×C₄×C₁₁	—	2	4	no	—
124	C₂×C₄×C₁₁	—	4	1	no	—
125	C₂×C₄×C₁₁	—	4	4	no	—
126	C₂×C₄×C₁₁	—	4	4	no	—
127	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	4	yes	—
131	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
134	C₈×C₁₁	—	2	1	no	—
135	C₈×C₁₁	—	2	1	no	—
136	C₈×C₁₁	—	2	1	no	—
137	C₈×C₁₁	#131	1	2	yes	complemented
138	C₁₁⋊C₈	—	2	4	no	—
139	C₁₁⋊C₈	—	4	2	no	—
140	C₁₁⋊C₈	—	4	2	no	—
141	C₁₁⋊C₈	—	4	2	no	—
142	C₂×#131	—	22	2	no	—
143	#532	C₂×D₁₁	1	2	yes	—
144	#500	Dic₁₁	1	1	yes	characteristic
145	#493	Dic₁₁	1	1	yes	characteristic
146	#486	Dic₁₁	1	2	yes	—
147	#466	Dic₁₁	1	2	yes	—
148	#466	Dic₁₁	1	2	yes	—
149	C₄×D₈	C₂×D₁₁	1	2	yes	—
150	C₄×D₈	C₂×D₁₁	1	2	yes	—
151	#436	—	11	4	no	—
152	#435	—	22	4	no	—
153	C₈×D₄	—	11	4	no	—
154	#431	—	11	1	no	—
155	C₂×C₄⋊C₈	—	11	2	no	—
156	#408	—	22	2	no	—
157	C₄×M₁₆	—	11	1	no	—
158	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
159	#334	—	11	1	no	—
160	#334	—	11	2	no	—
161	#334	—	11	1	no	—
162	D₄⋊C₈	—	11	4	no	—
163	D₄⋊C₈	—	11	4	no	—
164	#322	—	11	2	no	—
165	#322	—	11	2	no	—
166	C₂²×C₁₁	—	2	1	no	—
167	C₂²×C₁₁	—	2	1	no	—
168	C₂²×C₁₁	—	4	4	no	—
169	C₂²×C₁₁	—	4	4	no	—
170	C₂²×C₁₁	—	4	4	no	—
171	C₂²×C₁₁	#442	1	2	yes	—
172	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
173	C₄×C₁₁	—	2	1	no	—
174	C₄×C₁₁	—	2	1	no	—
175	C₄×C₁₁	—	2	1	no	—
176	C₄×C₁₁	—	4	4	no	—
177	C₄×C₁₁	#442	1	2	yes	—
178	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
179	C₄×C₁₁	#568	1	1	yes	characteristic
180	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
181	C₂×#41	—	2	2	no	—
182	C₂×D₈	C₂×Dic₁₁	1	2	yes	—
183	C₂×D₈	C₂×Dic₁₁	1	2	yes	—
184	#131	—	44	8	no	—
185	C₂×M₁₆	—	11	2	no	—
186	C₂×M₁₆	—	22	4	no	—
187	C₂²×C₈	—	22	4	no	—
188	C₂²×C₈	—	22	2	no	—
189	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
190	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
191	C₄⋊Q₈	C₂×Dic₁₁	1	2	yes	—
192	#127	C₂×Dic₁₁	1	2	yes	—
193	#124	—	2	2	no	—
194	#121	—	2	2	no	—
195	#121	C₂×Dic₁₁	1	4	yes	—
196	C₄×D₄	—	2	2	no	—
197	C₄×D₄	C₂²×D₁₁	1	4	yes	—
198	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
199	#107	C₂×Dic₁₁	1	2	yes	—
200	#107	C₂×Dic₁₁	1	2	yes	—
201	C₄⋊C₈	—	11	2	no	—
202	C₄⋊C₈	—	11	2	no	—
203	C₄⋊C₈	—	22	2	no	—
204	C₄⋊C₈	—	22	1	no	—
205	#102	C₂×Dic₁₁	1	4	yes	—
206	#102	C₂×Dic₁₁	1	4	yes	—
207	C₂²⋊C₈	—	22	4	no	—
208	C₂²⋊C₈	—	22	4	no	—
209	C₂²⋊C₈	—	22	1	no	—
210	#97	—	11	2	no	—
211	C₄×C₈	—	11	2	no	—
212	C₄×C₈	—	11	2	no	—
213	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
214	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
217	C₂×C₁₁	—	2	2	no	—
218	C₂×C₁₁	—	8	4	no	—
219	C₂×C₁₁	#2935	1	1	yes	characteristic
220	C₂×C₁₁	C₂×#442	1	1	yes	characteristic
221	C₂×C₁₁	#2957	1	1	yes	characteristic
222	#41	—	4	4	no	—
223	#41	—	4	4	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₂²×Dic₁₁	1	4	yes	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
231	C₂²×C₄	#4341	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
233	D₈	—	2	4	no	—
234	D₈	—	4	2	no	—
235	M₁₆	—	22	4	no	—
236	M₁₆	—	44	8	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	11	4	no	—
240	C₂×C₈	—	22	2	no	—
241	C₂×C₈	—	22	2	no	—
242	C₂×C₈	—	22	2	no	—
243	C₂×C₈	—	44	8	no	—
244	C₂×C₈	—	44	4	no	—
245	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
248	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	C₂²×Dic₁₁	1	4	yes	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₄²	D₄×D₁₁	1	1	yes	characteristic
254	C₄²	#4341	1	1	yes	characteristic
255	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
256	C₄²	C₂³×D₁₁	1	1	yes	characteristic
257	C₁₁	#18073	1	1	yes	characteristic, complemented, Sylow
258	C₂³	—	2	4	no	—
259	C₂³	D₄×Dic₁₁	1	1	yes	characteristic
260	Q₈	—	4	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	4	no	—
263	D₄	—	4	8	no	—
264	D₄	—	4	4	no	—
265	D₄	—	8	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	#66964	1	4	yes	—
273	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
274	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
275	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
276	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
277	C₂×C₄	C₂×#4341	1	1	yes	characteristic
278	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	22	4	no	—
283	C₈	—	44	2	no	—
284	C₈	—	44	2	no	—
285	C₈	—	44	2	no	—
286	C₈	#67001	1	2	yes	complemented
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	#10622765	1	2	yes	—
293	C₂²	C₂×D₄×Dic₁₁	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	4	4	no	—
298	C₄	#10622765	1	2	yes	—
299	C₄	C₂×#66964	1	2	yes	—
300	C₄	C₂×#67001	1	1	yes	characteristic
301	C₄	#10623350	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	8	4	no	—
304	C₂	#13077360	1	1	yes	characteristic
305	C₂	C₂×#10622765	1	1	yes	characteristic
306	C₂	#13077533	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	#18073
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. #13077360	C₂	semidirect
2. #13067530	C₂	semidirect
3. #13067481	C₂	semidirect
4. #13067440	C₂	semidirect
5. C₂×#10622095	C₂	semidirect
6. #13067015	C₂	semidirect
7. #13066905	C₂	semidirect
137. C₈×C₁₁	#131	semidirect
257. C₁₁	#18073	semidirect
286. C₈	#67001	semidirect

Maximal subgroups

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

subgroup	index
#13077360	2
#13067530	2
#13067481	2
#13067440	2
C₂×#10622095	2
#13067015	2
#13066905	2
C₁₁×#3169	2
#18073	11