Subgroups — #14723232

Order 2816.

782 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077639	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₈₈	C₂	1	1	yes	maximal, characteristic
3	#13067441	C₂	1	4	yes	maximal, complemented
4	C₈×D₈₈	C₂	1	4	yes	maximal, complemented
5	#13067364	C₂	1	1	yes	maximal, characteristic, complemented
6	#13067008	C₂	1	2	yes	maximal, complemented
7	C₁₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈₈	C₄	1	1	yes	characteristic
9	C₂×C₄×D₄₄	C₂²	1	2	yes	—
10	C₁₁×#431	C₂²	1	1	yes	characteristic
11	C₁₁×#431	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×#66885	C₄	1	2	yes	—
14	C₂×#66882	C₄	1	1	yes	characteristic
15	#10622538	C₂²	1	2	yes	—
16	#10622447	—	2	4	no	—
17	D₁₁×C₂²⋊C₈	—	2	4	no	—
18	C₄×D₈₈	C₄	1	2	yes	—
19	C₄×D₈₈	C₄	1	2	yes	—
20	C₄×D₈₈	C₂²	1	2	yes	—
21	C₄×D₈₈	C₂²	1	2	yes	—
22	C₈×D₄₄	C₂²	1	8	yes	complemented
23	C₁₁×#322	C₂²	1	2	yes	complemented
24	C₈²×C₁₁	C₂²	1	2	yes	complemented
25	#10622097	C₂²	1	8	yes	complemented
26	#10622096	C₂²	1	2	yes	complemented
27	#10622095	C₂²	1	2	yes	complemented
28	C₂²×D₄₄	C₂×C₄	1	2	yes	—
29	C₂²×C₄×D₁₁	—	2	2	no	—
30	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
33	C₂×(C₂×D₁₁)⋊C₄	—	2	2	no	—
34	C₂×C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
35	C₂×D₈₈	C₈	1	8	yes	complemented
36	C₂×D₈₈	C₂×C₄	1	2	yes	—
37	C₂×D₈₈	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₁	—	2	8	no	—
39	C₄×D₄₄	—	2	2	no	—
40	C₄×D₄₄	—	2	2	no	—
41	C₄×D₄₄	C₂×C₄	1	4	yes	—
42	C₄×D₄₄	C₂³	1	4	yes	—
43	C₁₁×C₄⋊C₈	—	2	1	no	—
44	C₁₁×C₄⋊C₈	D₄	1	2	yes	—
45	C₁₁×C₂²⋊C₈	—	2	1	no	—
46	C₁₁×C₂²⋊C₈	D₄	1	2	yes	—
47	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₁	C₂³	1	2	yes	—
50	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₁₁	C₂³	1	2	yes	—
53	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
54	#66885	C₂×C₄	1	4	yes	—
55	#66885	C₂×C₄	1	4	yes	—
56	(C₂×D₁₁)⋊C₈	—	2	8	no	—
57	#66882	C₂×C₄	1	2	yes	—
58	#66882	C₂×C₄	1	2	yes	—
59	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
60	C₄×C₁₁⋊C₈	—	2	2	no	—
61	#17854	—	11	1	no	maximal, Sylow
62	C₂³×D₁₁	—	2	2	no	—
63	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
66	C₂²×Dic₁₁	—	2	2	no	—
67	C₂×D₄₄	—	2	2	no	—
68	C₂×D₄₄	—	2	4	no	—
69	C₂×D₄₄	C₂×C₈	1	8	yes	complemented
70	C₂×D₄₄	C₂²×C₄	1	4	yes	—
71	C₂×C₄×D₁₁	—	2	4	no	—
72	C₂×C₄×D₁₁	—	2	4	no	—
73	C₂×C₄×D₁₁	—	8	4	no	—
74	C₂×C₈×C₁₁	—	2	2	no	—
75	C₂×C₈×C₁₁	—	2	2	no	—
76	C₂×C₈×C₁₁	C₂×C₈	1	4	yes	complemented
77	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₁	C₂×D₄	1	4	yes	—
82	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₁	#41	1	1	yes	characteristic
85	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
86	(C₂×D₁₁)⋊C₄	—	2	4	no	—
87	(C₂×D₁₁)⋊C₄	—	2	4	no	—
88	C₄⋊Dic₁₁	—	2	2	no	—
89	C₄⋊Dic₁₁	C₂²×C₄	1	4	yes	—
90	C₂×C₁₁⋊C₈	—	2	4	no	—
91	D₈₈	C₂×C₈	1	8	yes	complemented
92	D₈₈	C₂×C₈	1	8	yes	complemented
93	C₈×D₁₁	—	8	8	no	—
94	#2972	—	11	2	no	—
95	C₂×C₄×D₈	—	11	1	no	—
96	#1580	—	11	4	no	—
97	C₈×D₈	—	11	4	no	—
98	#1564	—	11	1	no	—
99	#1478	—	11	2	no	—
100	#1450	D₁₁	1	1	yes	characteristic, complemented
101	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
102	C₂²×D₁₁	—	2	4	no	—
103	C₂²×D₁₁	—	2	8	no	—
104	C₂²×D₁₁	—	8	2	no	—
105	C₂²×D₁₁	—	8	2	no	—
106	C₂×C₄×C₁₁	—	2	2	no	—
107	C₂×C₄×C₁₁	—	4	1	no	—
108	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₁₁	#131	1	2	yes	—
113	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
116	C₂×Dic₁₁	—	2	4	no	—
117	C₂×Dic₁₁	—	8	2	no	—
118	D₄₄	—	2	4	no	—
119	D₄₄	—	2	8	no	—
120	D₄₄	C₂²×C₈	1	8	yes	—
121	C₄×D₁₁	—	4	8	no	—
122	C₄×D₁₁	—	8	4	no	—
123	C₈×C₁₁	—	2	1	no	—
124	C₈×C₁₁	—	2	1	no	—
125	C₈×C₁₁	—	2	4	no	—
126	C₈×C₁₁	—	4	2	no	—
127	C₈×C₁₁	C₂²×C₈	1	2	yes	—
128	C₈×C₁₁	C₂²×C₈	1	2	yes	—
129	C₁₁⋊C₈	—	4	4	no	—
130	C₂²×D₈	—	11	1	no	—
131	C₂×C₄×D₄	—	11	2	no	—
132	C₄×D₈	—	11	2	no	—
133	C₄×D₈	—	11	2	no	—
134	C₄×D₈	—	11	2	no	—
135	C₄×D₈	—	11	2	no	—
136	C₈×D₄	—	11	8	no	—
137	C₈×D₄	—	22	4	no	—
138	#431	—	11	2	no	—
139	#431	C₂×D₁₁	1	1	yes	characteristic
140	#431	C₂×D₁₁	1	1	yes	characteristic
141	C₂×#107	—	11	1	no	—
142	C₂×#102	—	11	2	no	—
143	C₂×C₂²⋊C₈	—	22	4	no	—
144	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
145	#335	—	11	2	no	—
146	#334	—	11	2	no	—
147	D₄⋊C₈	—	11	8	no	—
148	#322	C₂×D₁₁	1	2	yes	complemented
149	C₈²	C₂×D₁₁	1	2	yes	complemented
150	C₂²×C₁₁	—	2	1	no	—
151	C₂²×C₁₁	—	2	1	no	—
152	C₂²×C₁₁	C₈×D₄	1	2	yes	—
153	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
154	C₂×D₁₁	—	2	8	no	—
155	C₂×D₁₁	—	4	4	no	—
156	C₂×D₁₁	—	4	4	no	—
157	C₂×D₁₁	—	8	8	no	—
158	C₄×C₁₁	—	2	1	no	—
159	C₄×C₁₁	—	2	1	no	—
160	C₄×C₁₁	—	2	1	no	—
161	C₄×C₁₁	C₈×D₄	1	2	yes	—
162	C₄×C₁₁	C₂³×C₈	1	1	yes	characteristic
163	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
164	C₄×C₁₁	#576	1	2	yes	—
165	Dic₁₁	—	4	4	no	—
166	C₂²×D₄	—	11	2	no	—
167	C₂³×C₄	—	22	2	no	—
168	C₂×D₈	—	11	8	no	—
169	C₂×D₈	—	11	2	no	—
170	C₂×D₈	—	11	2	no	—
171	C₂²×C₈	—	22	8	no	—
172	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
173	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
174	C₄×D₄	—	11	4	no	—
175	C₄×D₄	—	11	4	no	—
176	C₄×D₄	—	22	2	no	—
177	C₄×D₄	—	22	2	no	—
178	C₂×C₄⋊C₄	—	11	2	no	—
179	C₂×C₂²⋊C₄	—	22	2	no	—
180	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
181	#107	—	11	2	no	—
182	#107	—	11	2	no	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	11	4	no	—
185	C₄⋊C₈	D₄₄	1	2	yes	—
186	#102	—	11	4	no	—
187	#102	—	11	4	no	—
188	C₂²⋊C₈	—	2	1	no	—
189	C₂²⋊C₈	—	22	2	no	—
190	C₂²⋊C₈	—	22	8	no	—
191	C₂²⋊C₈	D₄₄	1	2	yes	—
192	C₄×C₈	—	22	2	no	—
193	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
194	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
195	C₄×C₈	C₂²×D₁₁	1	2	yes	—
196	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
197	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₁₁	1	2	yes	—
199	C₂×C₁₁	—	2	2	no	—
200	C₂×C₁₁	C₂×C₈×D₄	1	1	yes	characteristic
201	C₂×C₁₁	#2948	1	1	yes	characteristic
202	C₂×C₁₁	#2957	1	1	yes	characteristic
203	D₁₁	—	4	8	no	—
204	C₂⁴	—	22	2	no	—
205	C₂×D₄	—	11	8	no	—
206	C₂×D₄	—	11	4	no	—
207	C₂×D₄	—	22	2	no	—
208	C₂×D₄	—	22	4	no	—
209	C₂²×C₄	—	22	2	no	—
210	C₂²×C₄	—	22	4	no	—
211	C₂²×C₄	—	22	4	no	—
212	C₂²×C₄	—	88	4	no	—
213	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
215	C₂²×C₄	#4339	1	1	yes	characteristic
216	D₈	—	11	8	no	—
217	D₈	—	11	8	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	22	4	no	—
221	C₂×C₈	—	88	8	no	—
222	C₂×C₈	C₈×D₁₁	1	4	yes	complemented
223	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄₄	1	4	yes	—
228	C₄⋊C₄	—	11	4	no	—
229	C₄⋊C₄	—	22	2	no	—
230	C₂²⋊C₄	—	22	4	no	—
231	C₂²⋊C₄	—	22	4	no	—
232	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
233	C₄²	C₂×D₄₄	1	1	yes	characteristic
234	C₄²	#4339	1	1	yes	characteristic
235	C₄²	C₂³×D₁₁	1	1	yes	characteristic
236	C₁₁	#17854	1	1	yes	characteristic, complemented, Sylow
237	C₂³	—	22	4	no	—
238	C₂³	—	22	8	no	—
239	C₂³	—	88	2	no	—
240	C₂³	—	88	2	no	—
241	C₂³	C₄×D₄₄	1	1	yes	characteristic
242	D₄	—	11	8	no	—
243	D₄	—	22	4	no	—
244	D₄	—	22	8	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	22	4	no	—
248	C₂×C₄	—	44	8	no	—
249	C₂×C₄	—	88	4	no	—
250	C₂×C₄	—	88	2	no	—
251	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
252	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
255	C₂×C₄	#66954	1	2	yes	—
256	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
258	C₂×C₄	C₂×#4339	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	4	no	—
262	C₈	—	4	2	no	—
263	C₈	—	44	4	no	—
264	C₈	C₂×C₈×D₁₁	1	2	yes	—
265	C₈	C₂×C₈×D₁₁	1	2	yes	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	22	8	no	—
269	C₂²	—	44	4	no	—
270	C₂²	—	44	4	no	—
271	C₂²	—	88	8	no	—
272	C₂²	C₈×D₄₄	1	2	yes	—
273	C₂²	C₂×C₄×D₄₄	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	44	4	no	—
278	C₄	C₈×D₄₄	1	2	yes	—
279	C₄	C₂²×C₈×D₁₁	1	1	yes	characteristic
280	C₄	C₂×#66954	1	1	yes	characteristic
281	C₄	#10623283	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	44	8	no	—
284	C₂	C₂×C₈×D₄₄	1	1	yes	characteristic
285	C₂	#13077469	1	1	yes	characteristic
286	C₂	#13077534	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₄×C₁₁	C₂³×C₈
Frattini	C₄²	C₂³×D₁₁
Fitting	C₁₁×#1450	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂²×D₈₈	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1450	D₁₁
hypocenter	C₁₁	#17854
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
1. #13077639	C₂	semidirect
3. #13067441	C₂	semidirect
4. C₈×D₈₈	C₂	semidirect
5. #13067364	C₂	semidirect
6. #13067008	C₂	semidirect
7. C₁₁×#1450	C₂	semidirect
22. C₈×D₄₄	C₂²	semidirect
23. C₁₁×#322	C₂²	semidirect
24. C₈²×C₁₁	C₂²	semidirect
25. #10622097	C₂²	semidirect
26. #10622096	C₂²	semidirect
27. #10622095	C₂²	semidirect
35. C₂×D₈₈	C₈	semidirect
69. C₂×D₄₄	C₂×C₈	semidirect
76. C₂×C₈×C₁₁	C₂×C₈	semidirect
91. D₈₈	C₂×C₈	semidirect
92. D₈₈	C₂×C₈	semidirect
100. #1450	D₁₁	semidirect
148. #322	C₂×D₁₁	semidirect
149. C₈²	C₂×D₁₁	semidirect
222. C₂×C₈	C₈×D₁₁	semidirect
236. C₁₁	#17854	semidirect

Maximal subgroups

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

subgroup	index
#13077639	2
C₂×C₄×D₈₈	2
#13067441	2
C₈×D₈₈	2
#13067364	2
#13067008	2
C₁₁×#1450	2
#17854	11