Subgroups — #14907432

Order 2816.

460 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077548	C₂	1	1	yes	maximal, characteristic
2	#13070523	C₂	1	2	yes	maximal, complemented
3	#13070522	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₂₂	C₂	1	2	yes	maximal, complemented
5	#13070518	C₂	1	2	yes	maximal, complemented
6	#13070423	C₂	1	1	yes	maximal, characteristic
7	#13070420	C₂	1	2	yes	maximal
8	#13070412	C₂	1	2	yes	maximal
9	C₁₁×#2164	C₂	1	1	yes	maximal, characteristic
10	#10623048	C₄	1	1	yes	characteristic
11	C₁₁×#405	C₂²	1	1	yes	characteristic
12	#10622732	C₄	1	1	yes	characteristic
13	#10622731	C₄	1	1	yes	characteristic
14	#10622714	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
16	#10622612	C₄	1	1	yes	characteristic
17	#10622314	C₄	1	2	yes	—
18	#10622314	C₄	1	2	yes	—
19	C₈×Dic₂₂	C₂²	1	2	yes	—
20	C₈×Dic₂₂	C₂²	1	2	yes	—
21	C₁₁×C₄⋊C₁₆	C₂²	1	2	yes	—
22	C₁₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
23	C₁₁×#346	C₂²	1	1	yes	characteristic
24	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
25	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
26	#10622140	C₂²	1	2	yes	—
27	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
28	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
29	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
30	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
31	C₁₆×Dic₁₁	C₂²	1	2	yes	—
32	C₁₆×Dic₁₁	C₂²	1	2	yes	—
33	C₁₆×Dic₁₁	C₂²	1	2	yes	—
34	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
35	#10622099	C₂²	1	1	yes	characteristic
36	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₂×C₁₁×M₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
39	C₁₁×#117	C₂×C₄	1	1	yes	characteristic
40	#66979	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
42	C₂×#4318	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
44	#66943	C₈	1	2	yes	—
45	#66941	C₈	1	2	yes	—
46	#66931	C₈	1	2	yes	—
47	#66931	C₈	1	2	yes	—
48	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
49	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
50	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
51	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
52	C₁₁×#97	C₂×C₄	1	1	yes	characteristic
53	C₁₁×#97	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
56	#66879	C₂×C₄	1	2	yes	—
57	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
58	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
59	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
60	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
61	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
62	C₈×Dic₁₁	C₂³	1	2	yes	—
63	C₈×Dic₁₁	C₂³	1	2	yes	—
64	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
65	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
66	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
67	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
68	#33500	—	11	1	no	maximal, Sylow
69	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
71	C₂×Dic₂₂	C₂×C₈	1	2	yes	—
72	C₂×Dic₂₂	C₂×C₈	1	2	yes	—
73	C₁₁×M₁₆	—	4	1	no	—
74	C₂×C₈×C₁₁	—	2	2	no	—
75	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
79	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
80	C₁₁×C₂²⋊C₄	C₂×C₈	1	2	yes	—
81	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
84	C₂²⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
85	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
86	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
87	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
88	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
89	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
90	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
91	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
92	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
93	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
94	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
95	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
96	#4318	—	4	1	no	—
97	C₂×C₁₁⋊C₈	—	4	1	no	—
98	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
100	C₁₁×C₁₆	—	2	2	no	—
101	C₁₁×C₁₆	—	2	2	no	—
102	C₁₁⋊C₁₆	—	2	2	no	—
103	C₁₁⋊C₁₆	—	2	2	no	—
104	#2960	—	11	1	no	—
105	#2183	—	11	2	no	—
106	#2183	—	11	2	no	—
107	Q₈×C₁₆	—	11	2	no	—
108	Q₈×C₁₆	—	11	2	no	—
109	#2164	—	11	1	no	—
110	#2164	—	11	2	no	—
111	#2164	D₁₁	1	1	yes	characteristic
112	#2162	—	11	2	no	—
113	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₁	—	2	1	no	—
115	C₂×C₄×C₁₁	—	2	1	no	—
116	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
120	C₂×Dic₁₁	—	2	4	no	—
121	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
122	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
123	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
124	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
125	Dic₂₂	—	2	4	no	—
126	Dic₂₂	—	2	4	no	—
127	C₈×C₁₁	—	2	1	no	—
128	C₈×C₁₁	—	2	1	no	—
129	C₈×C₁₁	—	2	1	no	—
130	C₈×C₁₁	2₋¹⁺⁴	1	2	yes	—
131	C₁₁⋊C₈	—	2	2	no	—
132	C₁₁⋊C₈	—	2	2	no	—
133	#533	—	11	1	no	—
134	#446	—	11	2	no	—
135	#446	—	11	2	no	—
136	C₈×Q₈	—	11	2	no	—
137	C₈×Q₈	—	11	2	no	—
138	#424	—	11	1	no	—
139	#424	—	11	1	no	—
140	#423	—	11	1	no	—
141	#405	—	11	1	no	—
142	#405	C₂×D₁₁	1	1	yes	characteristic
143	C₂×C₄×C₈	—	11	1	no	—
144	C₄⋊C₁₆	—	11	2	no	—
145	C₄⋊C₁₆	—	11	2	no	—
146	C₄⋊C₁₆	—	11	2	no	—
147	C₄⋊C₁₆	—	11	2	no	—
148	C₄⋊C₁₆	—	11	2	no	—
149	C₄⋊C₁₆	C₂×D₁₁	1	2	yes	—
150	C₂²⋊C₁₆	—	11	2	no	—
151	C₂²⋊C₁₆	C₂×D₁₁	1	2	yes	—
152	#346	—	11	2	no	—
153	#346	—	11	1	no	—
154	#346	C₂×D₁₁	1	1	yes	characteristic
155	C₄×C₁₆	—	11	2	no	—
156	C₄×C₁₆	—	11	2	no	—
157	C₄×C₁₆	—	11	2	no	—
158	C₄×C₁₆	—	11	1	no	—
159	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
160	C₂²×C₁₁	—	2	2	no	—
161	C₂²×C₁₁	—	2	1	no	—
162	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
163	C₄×C₁₁	—	2	2	no	—
164	C₄×C₁₁	—	2	2	no	—
165	C₄×C₁₁	—	2	1	no	—
166	C₄×C₁₁	#519	1	1	yes	characteristic
167	C₄×C₁₁	#568	1	1	yes	characteristic
168	Dic₁₁	—	2	2	no	—
169	Dic₁₁	—	2	2	no	—
170	Dic₁₁	—	2	2	no	—
171	Dic₁₁	#504	1	4	yes	—
172	C₂×M₁₆	—	11	1	no	—
173	C₂×M₁₆	C₄×D₁₁	1	1	yes	characteristic
174	C₂²×C₈	—	11	1	no	—
175	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
176	C₄⋊Q₈	—	11	2	no	—
177	#125	—	11	2	no	—
178	C₂²⋊Q₈	—	11	2	no	—
179	C₂²⋊Q₈	—	11	2	no	—
180	C₄×Q₈	—	11	2	no	—
181	C₄×Q₈	—	11	2	no	—
182	#117	—	11	1	no	—
183	#117	C₄×D₁₁	1	1	yes	characteristic
184	C₂×C₄²	—	11	1	no	—
185	C₂×C₁₆	—	11	2	no	—
186	C₂×C₁₆	—	11	2	no	—
187	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
188	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
189	C₄⋊C₈	—	11	2	no	—
190	C₄⋊C₈	—	11	2	no	—
191	C₄⋊C₈	—	11	2	no	—
192	C₄⋊C₈	—	11	2	no	—
193	C₄⋊C₈	—	11	2	no	—
194	C₄⋊C₈	—	11	2	no	—
195	#97	—	11	2	no	—
196	#97	C₄×D₁₁	1	1	yes	characteristic
197	#97	C₄×D₁₁	1	1	yes	characteristic
198	C₄×C₈	—	11	2	no	—
199	C₄×C₈	—	11	2	no	—
200	C₄×C₈	—	11	2	no	—
201	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
203	C₂×C₁₁	—	4	1	no	—
204	C₂×C₁₁	#2982	1	1	yes	characteristic
205	C₂×C₁₁	C₂×#504	1	2	yes	—
206	C₂×Q₈	—	11	2	no	—
207	C₂×Q₈	—	11	2	no	—
208	C₂²×C₄	—	11	2	no	—
209	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
210	M₁₆	—	4	1	no	—
211	M₁₆	—	44	1	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	11	2	no	—
214	C₂×C₈	—	11	2	no	—
215	C₂×C₈	—	44	1	no	—
216	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
217	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
218	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
219	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
220	C₄⋊C₄	—	11	2	no	—
221	C₄⋊C₄	—	11	2	no	—
222	C₄⋊C₄	—	11	2	no	—
223	C₄⋊C₄	—	11	2	no	—
224	C₄⋊C₄	—	11	2	no	—
225	C₄⋊C₄	—	11	2	no	—
226	C₄⋊C₄	—	11	2	no	—
227	C₄⋊C₄	C₈×D₁₁	1	2	yes	—
228	C₂²⋊C₄	—	11	1	no	—
229	C₂²⋊C₄	—	11	1	no	—
230	C₂²⋊C₄	C₈×D₁₁	1	2	yes	—
231	C₄²	—	11	1	no	—
232	C₄²	—	11	1	no	—
233	C₄²	—	11	2	no	—
234	C₄²	—	11	2	no	—
235	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
236	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
237	C₁₆	—	2	2	no	—
238	C₁₆	—	2	2	no	—
239	C₁₆	—	22	2	no	—
240	C₁₆	—	22	2	no	—
241	C₁₁	#33500	1	1	yes	characteristic, complemented, Sylow
242	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
243	Q₈	—	22	4	no	—
244	Q₈	—	22	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	11	2	no	—
248	C₂×C₄	—	11	2	no	—
249	C₂×C₄	—	11	2	no	—
250	C₂×C₄	—	11	2	no	—
251	C₂×C₄	—	22	4	no	—
252	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
253	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
254	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
255	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	22	2	no	—
260	C₈	—	22	2	no	—
261	C₈	#67043	1	2	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	11	4	no	—
269	C₄	—	22	2	no	—
270	C₄	—	22	2	no	—
271	C₄	—	22	2	no	—
272	C₄	#10623047	1	1	yes	characteristic
273	C₄	#10623276	1	1	yes	characteristic
274	C₂	—	4	1	no	—
275	C₂	#13077696	1	1	yes	characteristic
276	C₂	D₁₁×#504	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2164	C₂
prime radical	C₂³×C₁₁	C₂²×C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2164	D₁₁
hypocenter	C₁₁	#33500
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2164

Complements

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

N	H ≅ G/N	type
2. #13070523	C₂	semidirect
3. #13070522	C₂	semidirect
4. C₁₆×Dic₂₂	C₂	semidirect
5. #13070518	C₂	semidirect
241. C₁₁	#33500	semidirect

Maximal subgroups

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

subgroup	index
#13077548	2
#13070523	2
#13070522	2
C₁₆×Dic₂₂	2
#13070518	2
#13070423	2
#13070420	2
#13070412	2
C₁₁×#2164	2
#33500	11