Subgroups — #14725596

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	#13079195	C₂	1	1	yes	maximal, characteristic
2	#13077359	C₂	1	2	yes	maximal, complemented
3	#13067529	C₂	1	2	yes	maximal, complemented
4	#13067479	C₂	1	4	yes	maximal, complemented
5	#13067439	C₂	1	2	yes	maximal, complemented
6	#13067014	C₂	1	2	yes	maximal, complemented
7	#13066905	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#66954	—	2	2	no	—
10	#10623008	C₂²	1	2	yes	—
11	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	#10622728	C₄	1	2	yes	—
14	#10622728	C₄	1	2	yes	—
15	#10622723	—	2	2	no	—
16	#10622538	C₂²	1	1	yes	characteristic
17	C₄×#4318	C₂²	1	1	yes	characteristic
18	#10622451	C₂²	1	4	yes	—
19	#10622449	—	2	4	no	—
20	#10622447	C₂²	1	4	yes	—
21	#10622308	C₄	1	1	yes	characteristic
22	#10622307	C₄	1	2	yes	—
23	C₄×D₈₈	C₂²	1	2	yes	—
24	C₄×D₈₈	C₂²	1	2	yes	—
25	#10622295	C₄	1	1	yes	characteristic
26	C₁₁×#334	C₂²	1	1	yes	characteristic
27	C₁₁×#334	C₂²	1	1	yes	characteristic
28	C₁₁×#334	C₂²	1	2	yes	—
29	#10622121	C₂²	1	4	yes	—
30	#10622121	C₂²	1	4	yes	—
31	#10622093	C₂²	1	2	yes	—
32	#10622092	C₂²	1	2	yes	—
33	C₂×#4339	—	2	2	no	—
34	C₂²×C₈×C₁₁	—	2	2	no	—
35	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
38	#66983	—	2	2	no	—
39	#66983	C₂×C₄	1	4	yes	—
40	C₂×#4318	D₄	1	2	yes	—
41	C₂×D₈₈	C₂×C₄	1	2	yes	—
42	C₂×D₈₈	C₂×C₄	1	2	yes	—
43	#66954	—	4	8	no	—
44	C₂×#4313	—	2	4	no	—
45	C₂×C₈×D₁₁	—	2	4	no	—
46	#66928	—	2	2	no	—
47	#66927	C₂×C₄	1	2	yes	—
48	C₄×D₄₄	—	2	2	no	—
49	C₄×D₄₄	C₂³	1	4	yes	—
50	#66922	C₂×C₄	1	2	yes	—
51	C₁₁×C₄⋊C₈	—	2	2	no	—
52	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
53	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
54	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
58	C₂²⋊(C₁₁⋊C₈)	—	2	1	no	—
59	#66885	C₂×C₄	1	4	yes	—
60	#66885	C₂×C₄	1	4	yes	—
61	(C₂×D₁₁)⋊C₈	—	2	4	no	—
62	(C₂×D₁₁)⋊C₈	—	2	4	no	—
63	#66882	C₂×C₄	1	2	yes	—
64	#66882	C₂×C₄	1	2	yes	—
65	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
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	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
73	C₂×C₂²⋊D₁₁	—	2	4	no	—
74	#4339	—	4	4	no	—
75	#4339	—	4	4	no	—
76	C₂×D₄₄	—	2	2	no	—
77	C₂×D₄₄	—	2	2	no	—
78	C₂×D₄₄	C₂²×C₄	1	4	yes	—
79	C₂×C₄×D₁₁	—	2	4	no	—
80	C₂×Dic₂₂	—	2	2	no	—
81	C₂×C₈×C₁₁	—	2	2	no	—
82	C₂×C₈×C₁₁	—	2	2	no	—
83	C₂×C₈×C₁₁	—	2	2	no	—
84	C₂×C₈×C₁₁	—	4	1	no	—
85	C₂×C₈×C₁₁	—	4	4	no	—
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	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₁	#41	1	1	yes	characteristic
93	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
94	(C₂×D₁₁)⋊C₄	—	2	4	no	—
95	(C₂×D₁₁)⋊C₄	—	2	4	no	—
96	C₄⋊Dic₁₁	—	2	2	no	—
97	C₄⋊Dic₁₁	C₂²×C₄	1	4	yes	—
98	#4318	—	2	4	no	—
99	C₂×C₁₁⋊C₈	—	2	2	no	—
100	C₂×C₁₁⋊C₈	C₂×D₄	1	4	yes	—
101	D₈₈	—	2	4	no	—
102	D₈₈	—	4	2	no	—
103	#4313	—	4	8	no	—
104	C₈×D₁₁	—	4	8	no	—
105	#3169	—	11	1	no	—
106	#2935	—	11	2	no	—
107	#1594	—	11	2	no	—
108	#1586	—	11	4	no	—
109	#1580	—	11	2	no	—
110	C₂×#334	D₁₁	1	1	yes	characteristic, complemented
111	#1480	—	11	2	no	—
112	#1451	—	11	1	no	—
113	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
114	C₂²×D₁₁	—	2	4	no	—
115	C₂×C₄×C₁₁	—	2	1	no	—
116	C₂×C₄×C₁₁	—	2	1	no	—
117	C₂×C₄×C₁₁	—	4	1	no	—
118	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
125	C₂²⋊D₁₁	—	4	8	no	—
126	C₂×Dic₁₁	—	2	4	no	—
127	D₄₄	—	2	4	no	—
128	D₄₄	—	2	4	no	—
129	D₄₄	—	4	4	no	—
130	D₄₄	—	8	2	no	—
131	C₄×D₁₁	—	4	4	no	—
132	C₄×D₁₁	—	4	4	no	—
133	Dic₂₂	—	4	4	no	—
134	C₈×C₁₁	—	2	1	no	—
135	C₈×C₁₁	—	2	1	no	—
136	C₈×C₁₁	—	2	1	no	—
137	C₈×C₁₁	—	4	2	no	—
138	C₈×C₁₁	—	4	2	no	—
139	C₈×C₁₁	#131	1	2	yes	complemented
140	C₁₁⋊C₈	—	2	4	no	—
141	C₁₁⋊C₈	—	4	2	no	—
142	C₂×#131	—	22	2	no	—
143	#532	—	11	2	no	—
144	#500	—	11	1	no	—
145	#493	—	11	1	no	—
146	#486	—	11	2	no	—
147	#466	—	11	2	no	—
148	#466	—	11	2	no	—
149	C₄×D₈	—	11	2	no	—
150	C₄×D₈	—	11	2	no	—
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	C₂×D₁₁	1	2	yes	—
156	#408	—	22	2	no	—
157	C₄×M₁₆	—	11	1	no	—
158	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
159	#334	C₂×D₁₁	1	1	yes	characteristic
160	#334	C₂×D₁₁	1	2	yes	—
161	#334	C₂×D₁₁	1	1	yes	characteristic
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	#442	1	2	yes	—
169	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×D₁₁	—	4	4	no	—
171	C₂×D₁₁	—	4	4	no	—
172	C₂×D₁₁	—	4	4	no	—
173	C₄×C₁₁	—	2	1	no	—
174	C₄×C₁₁	—	2	1	no	—
175	C₄×C₁₁	—	2	1	no	—
176	C₄×C₁₁	#442	1	2	yes	—
177	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
178	C₄×C₁₁	#568	1	1	yes	characteristic
179	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
180	Dic₁₁	—	4	4	no	—
181	C₂×#41	—	22	2	no	—
182	C₂×D₈	—	11	2	no	—
183	C₂×D₈	—	11	2	no	—
184	#131	—	44	8	no	—
185	C₂×M₁₆	—	11	2	no	—
186	C₂×M₁₆	—	22	4	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	22	4	no	—
189	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
190	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
191	C₄⋊Q₈	—	11	2	no	—
192	#127	—	11	2	no	—
193	#124	—	22	2	no	—
194	#121	—	11	4	no	—
195	#121	—	22	2	no	—
196	C₄×D₄	—	11	4	no	—
197	C₄×D₄	—	22	2	no	—
198	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
199	#107	—	11	2	no	—
200	#107	—	11	2	no	—
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	22	1	no	—
203	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
204	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
205	#102	—	11	4	no	—
206	#102	—	11	4	no	—
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	C₄×D₁₁	1	1	yes	characteristic
214	C₄×C₈	C₄×D₁₁	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	#2935	1	1	yes	characteristic
219	C₂×C₁₁	C₂×#442	1	1	yes	characteristic
220	C₂×C₁₁	#2957	1	1	yes	characteristic
221	D₁₁	—	8	4	no	—
222	#41	—	44	4	no	—
223	#41	—	44	4	no	—
224	C₂×Q₈	—	22	2	no	—
225	C₂×D₄	—	11	4	no	—
226	C₂×D₄	—	22	4	no	—
227	C₂×D₄	—	22	2	no	—
228	C₂×D₄	—	22	2	no	—
229	C₂²×C₄	—	22	4	no	—
230	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
232	C₂²×C₄	#4343	1	1	yes	characteristic
233	D₈	—	22	4	no	—
234	D₈	—	44	2	no	—
235	M₁₆	—	22	4	no	—
236	M₁₆	—	44	8	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	4	no	—
242	C₂×C₈	—	11	4	no	—
243	C₂×C₈	—	22	2	no	—
244	C₂×C₈	—	44	8	no	—
245	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
249	C₄⋊C₄	—	11	4	no	—
250	C₄⋊C₄	—	22	2	no	—
251	C₂²⋊C₄	—	22	4	no	—
252	C₂²⋊C₄	—	22	4	no	—
253	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
254	C₄²	D₄×D₁₁	1	1	yes	characteristic
255	C₄²	#4343	1	1	yes	characteristic
256	C₄²	C₂³×D₁₁	1	1	yes	characteristic
257	C₁₁	#18073	1	1	yes	characteristic, complemented, Sylow
258	C₂³	—	22	4	no	—
259	C₂³	#66946	1	1	yes	characteristic
260	Q₈	—	44	4	no	—
261	D₄	—	22	4	no	—
262	D₄	—	22	4	no	—
263	D₄	—	44	8	no	—
264	D₄	—	44	4	no	—
265	D₄	—	88	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	22	4	no	—
270	C₂×C₄	—	44	4	no	—
271	C₂×C₄	—	44	4	no	—
272	C₂×C₄	#66946	1	1	yes	characteristic
273	C₂×C₄	#66946	1	1	yes	characteristic
274	C₂×C₄	#66946	1	1	yes	characteristic
275	C₂×C₄	#66967	1	4	yes	—
276	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
278	C₂×C₄	C₂×#4343	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	2	no	—
284	C₈	—	22	4	no	—
285	C₈	—	44	2	no	—
286	C₈	#66954	1	2	yes	complemented
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	44	4	no	—
290	C₂²	—	44	4	no	—
291	C₂²	—	44	4	no	—
292	C₂²	#10622482	1	2	yes	—
293	C₂²	C₂×#66946	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	44	4	no	—
298	C₄	#10622482	1	2	yes	—
299	C₄	C₂×#66954	1	1	yes	characteristic
300	C₄	#10623276	1	1	yes	characteristic
301	C₄	C₂×#66967	1	2	yes	—
302	C₂	—	2	2	no	—
303	C₂	—	88	4	no	—
304	C₂	#13077359	1	1	yes	characteristic
305	C₂	C₂×#10622482	1	1	yes	characteristic
306	C₂	#13077532	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#334	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	#13079195	C₂
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#334	D₁₁
hypocenter	C₁₁	#18073
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
2. #13077359	C₂	semidirect
3. #13067529	C₂	semidirect
4. #13067479	C₂	semidirect
5. #13067439	C₂	semidirect
6. #13067014	C₂	semidirect
7. #13066905	C₂	semidirect
8. C₂×C₁₁×#334	C₂	semidirect
110. C₂×#334	D₁₁	semidirect
139. C₈×C₁₁	#131	semidirect
257. C₁₁	#18073	semidirect
286. C₈	#66954	semidirect

Maximal subgroups

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

subgroup	index
#13079195	2
#13077359	2
#13067529	2
#13067479	2
#13067439	2
#13067014	2
#13066905	2
C₂×C₁₁×#334	2
#18073	11