Subgroups — #14662971

Order 2816.

386 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070415	C₂	1	2	yes	maximal, complemented
2	#13066556	C₂	1	2	yes	maximal, complemented
3	#13066540	C₂	1	2	yes	maximal
4	C₂×C₁₁×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₁	C₄	1	1	yes	characteristic
6	C₁₁×#431	—	2	2	no	—
7	C₁₁×#431	—	2	2	no	—
8	C₂×C₁₁×C₄⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×C₂²⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₁₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	4	yes	—
15	C₄⋊(C₁₁⋊C₁₆)	—	2	2	no	—
16	C₄×C₁₁⋊C₁₆	—	2	2	no	—
17	C₂³×C₄×C₁₁	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₁	C₈	1	1	yes	characteristic
19	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₁₁	—	2	1	no	—
25	C₂×C₄²×C₁₁	—	2	1	no	—
26	C₂×C₄²×C₁₁	—	2	1	no	—
27	C₂×C₄²×C₁₁	—	2	1	no	—
28	C₂×C₄²×C₁₁	C₈	1	1	yes	characteristic
29	C₂×C₄²×C₁₁	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₁₁×C₄⋊C₈	—	2	2	no	—
33	C₁₁×C₄⋊C₈	—	2	2	no	—
34	C₁₁×C₂²⋊C₈	—	2	2	no	—
35	C₁₁×C₂²⋊C₈	—	2	2	no	—
36	C₄×C₈×C₁₁	—	2	1	no	—
37	C₄×C₈×C₁₁	—	2	1	no	—
38	C₄×C₈×C₁₁	D₄	1	2	yes	—
39	C₄×C₈×C₁₁	D₄	1	2	yes	—
40	C₂×C₁₁⋊C₁₆	—	2	4	no	—
41	#11459	—	11	1	no	maximal, Sylow
42	C₂⁴×C₁₁	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	—	2	1	no	—
44	C₂²×C₄×C₁₁	—	2	1	no	—
45	C₂²×C₄×C₁₁	—	2	2	no	—
46	C₂²×C₄×C₁₁	—	2	1	no	—
47	C₂²×C₄×C₁₁	—	2	1	no	—
48	C₂²×C₄×C₁₁	—	2	1	no	—
49	C₂²×C₄×C₁₁	—	2	1	no	—
50	C₂²×C₄×C₁₁	—	4	2	no	—
51	C₂²×C₄×C₁₁	—	4	1	no	—
52	C₂²×C₄×C₁₁	—	4	1	no	—
53	C₂²×C₄×C₁₁	C₁₆	1	4	yes	complemented
54	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₁	—	2	1	no	—
61	C₂×C₈×C₁₁	—	2	1	no	—
62	C₂×C₈×C₁₁	—	4	1	no	—
63	C₂×C₈×C₁₁	—	4	1	no	—
64	C₂×C₈×C₁₁	—	4	1	no	—
65	C₂×C₈×C₁₁	—	4	1	no	—
66	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
68	C₄²×C₁₁	—	2	1	no	—
69	C₄²×C₁₁	—	2	1	no	—
70	C₄²×C₁₁	—	2	1	no	—
71	C₄²×C₁₁	—	2	1	no	—
72	C₄²×C₁₁	—	2	1	no	—
73	C₄²×C₁₁	—	2	1	no	—
74	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₁₁	M₁₆	1	1	yes	characteristic
77	C₄²×C₁₁	M₁₆	1	1	yes	characteristic
78	C₁₁⋊C₁₆	—	4	4	no	—
79	C₂×#431	D₁₁	1	1	yes	characteristic
80	#2162	—	11	2	no	—
81	#1322	—	11	2	no	—
82	#1314	—	11	2	no	—
83	C₂³×C₁₁	—	2	1	no	—
84	C₂³×C₁₁	—	2	1	no	—
85	C₂³×C₁₁	—	4	1	no	—
86	C₂³×C₁₁	—	4	1	no	—
87	C₂³×C₁₁	C₂×C₁₆	1	2	yes	—
88	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₁	—	2	1	no	—
90	C₂×C₄×C₁₁	—	2	1	no	—
91	C₂×C₄×C₁₁	—	2	4	no	—
92	C₂×C₄×C₁₁	—	2	4	no	—
93	C₂×C₄×C₁₁	—	2	2	no	—
94	C₂×C₄×C₁₁	—	2	1	no	—
95	C₂×C₄×C₁₁	—	2	1	no	—
96	C₂×C₄×C₁₁	—	2	1	no	—
97	C₂×C₄×C₁₁	—	2	1	no	—
98	C₂×C₄×C₁₁	—	4	2	no	—
99	C₂×C₄×C₁₁	—	4	2	no	—
100	C₂×C₄×C₁₁	—	4	2	no	—
101	C₂×C₄×C₁₁	—	4	1	no	—
102	C₂×C₄×C₁₁	—	4	1	no	—
103	C₂×C₄×C₁₁	—	4	1	no	—
104	C₂×C₄×C₁₁	—	4	1	no	—
105	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₁	C₂×C₁₆	1	2	yes	—
110	C₂×C₄×C₁₁	C₂×C₁₆	1	2	yes	—
111	C₂×C₄×C₁₁	M₃₂	1	2	yes	—
112	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
114	C₈×C₁₁	—	2	2	no	—
115	C₈×C₁₁	—	2	2	no	—
116	C₈×C₁₁	—	4	1	no	—
117	C₈×C₁₁	—	4	1	no	—
118	C₂²×C₄²	Dic₁₁	1	1	yes	characteristic
119	#431	—	2	2	no	—
120	#431	—	2	2	no	—
121	C₂×C₄⋊C₈	Dic₁₁	1	1	yes	characteristic
122	C₂×C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
123	C₂×C₂²⋊C₈	Dic₁₁	1	1	yes	characteristic
124	C₂×C₂²⋊C₈	C₂×D₁₁	1	1	yes	characteristic
125	C₂×C₄×C₈	Dic₁₁	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
127	C₄⋊C₁₆	—	22	2	no	—
128	C₂²⋊C₁₆	—	11	4	no	—
129	C₄×C₁₆	—	22	2	no	—
130	C₂²×C₁₁	—	2	1	no	—
131	C₂²×C₁₁	—	2	1	no	—
132	C₂²×C₁₁	—	2	2	no	—
133	C₂²×C₁₁	—	4	2	no	—
134	C₂²×C₁₁	—	4	2	no	—
135	C₂²×C₁₁	—	4	1	no	—
136	C₂²×C₁₁	—	4	1	no	—
137	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
139	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
140	C₄×C₁₁	—	2	2	no	—
141	C₄×C₁₁	—	2	1	no	—
142	C₄×C₁₁	—	2	1	no	—
143	C₄×C₁₁	—	4	1	no	—
144	C₄×C₁₁	—	4	1	no	—
145	C₄×C₁₁	C₂²⋊C₁₆	1	2	yes	—
146	C₄×C₁₁	C₂²⋊C₁₆	1	2	yes	—
147	C₄×C₁₁	#349	1	2	yes	—
148	C₄×C₁₁	#410	1	1	yes	characteristic
149	C₄×C₁₁	#413	1	1	yes	characteristic
150	C₂³×C₄	C₁₁⋊C₈	1	1	yes	characteristic
151	C₂³×C₄	C₁₁⋊C₈	1	1	yes	characteristic
152	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
153	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
154	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
155	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	—	2	1	no	—
160	C₂×C₄²	—	2	1	no	—
161	C₂×C₄²	C₁₁⋊C₈	1	1	yes	characteristic
162	C₂×C₄²	C₁₁⋊C₈	1	1	yes	characteristic
163	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
165	C₂×C₁₆	—	22	4	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₂²⋊C₈	—	2	2	no	—
169	C₂²⋊C₈	—	2	2	no	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
173	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
174	C₂×C₁₁	—	2	1	no	—
175	C₂×C₁₁	—	2	1	no	—
176	C₂×C₁₁	—	4	2	no	—
177	C₂×C₁₁	#1458	1	1	yes	characteristic
178	C₂×C₁₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
179	C₂×C₁₁	C₂×#349	1	1	yes	characteristic
180	C₂⁴	C₂×C₁₁⋊C₈	1	1	yes	characteristic
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	1	no	—
190	C₂²×C₄	—	4	1	no	—
191	C₂²×C₄	C₁₁⋊C₁₆	1	4	yes	complemented
192	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
197	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
205	C₂×C₈	C₂×C₂²⋊D₁₁	1	2	yes	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	#4318	1	1	yes	characteristic
213	C₄²	#4318	1	1	yes	characteristic
214	C₄²	C₂²⋊Dic₁₁	1	1	yes	characteristic
215	C₄²	C₂²⋊Dic₁₁	1	1	yes	characteristic
216	C₁₆	—	44	4	no	—
217	C₁₁	#11459	1	1	yes	characteristic, complemented, Sylow
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	4	1	no	—
221	C₂³	—	4	1	no	—
222	C₂³	C₂×C₁₁⋊C₁₆	1	2	yes	—
223	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	C₂×C₁₁⋊C₁₆	1	2	yes	—
241	C₂×C₄	C₂×C₁₁⋊C₁₆	1	2	yes	—
242	C₂×C₄	#66876	1	2	yes	—
243	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
244	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
245	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
246	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
247	C₂×C₄	C₂×#4318	1	1	yes	characteristic
248	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	4	1	no	—
252	C₈	—	4	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	C₂²×C₁₁⋊C₁₆	1	1	yes	characteristic
261	C₂²	C₂×#66876	1	1	yes	characteristic
262	C₂²	C₂×C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	C₂²⋊(C₁₁⋊C₁₆)	1	2	yes	—
269	C₄	C₂²⋊(C₁₁⋊C₁₆)	1	2	yes	—
270	C₄	#10622191	1	2	yes	—
271	C₄	#10622850	1	1	yes	characteristic
272	C₄	#10622853	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	2	1	no	—
275	C₂	—	4	2	no	—
276	C₂	#13066930	1	1	yes	characteristic
277	C₂	C₂×C₂²⋊(C₁₁⋊C₁₆)	1	1	yes	characteristic
278	C₂	C₂×#10622191	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²⋊Dic₁₁
commutator	C₂²×C₁₁	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#431	C₂
socle	C₂²×C₁₁	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₁	C₂×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#431	D₁₁
hypocenter	C₁₁	#11459
LCS	G ▷ C₂²×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
1. #13070415	C₂	semidirect
2. #13066556	C₂	semidirect
53. C₂²×C₄×C₁₁	C₁₆	semidirect
191. C₂²×C₄	C₁₁⋊C₁₆	semidirect
217. C₁₁	#11459	semidirect

Maximal subgroups

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

subgroup	index
#13070415	2
#13066556	2
#13066540	2
C₂×C₁₁×#431	2
#11459	11