Subgroups — #14907408

Order 2816.

564 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077329	C₂	1	1	yes	maximal, characteristic
2	#13070449	C₂	1	2	yes	maximal, complemented
3	#13070440	C₂	1	4	yes	maximal, complemented
4	C₁₆×D₄₄	C₂	1	2	yes	maximal, complemented
5	#13070419	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070228	C₂	1	4	yes	maximal, complemented
7	C₁₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#66954	C₂²	1	2	yes	—
9	#10623008	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	#10622723	C₄	1	2	yes	—
12	#10622537	C₄	1	1	yes	characteristic
13	C₂×#66862	C₂²	1	4	yes	—
14	C₂×C₁₆×D₁₁	C₂²	1	4	yes	—
15	#10622301	C₄	1	2	yes	—
16	#10622301	C₄	1	2	yes	—
17	C₈×D₄₄	C₂²	1	2	yes	—
18	C₈×D₄₄	C₂²	1	2	yes	—
19	C₁₁×C₄⋊C₁₆	C₂²	1	2	yes	—
20	C₁₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
21	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
22	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
23	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
24	(C₂×D₁₁)⋊C₁₆	C₂²	1	4	yes	—
25	(C₂×D₁₁)⋊C₁₆	C₂²	1	4	yes	—
26	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
27	#10622099	C₂²	1	2	yes	—
28	C₂×#4339	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
32	#66983	C₈	1	2	yes	—
33	#66983	C₈	1	2	yes	—
34	C₂×#4318	C₂×C₄	1	2	yes	—
35	#66954	—	4	4	no	—
36	C₂×#4313	C₂×C₄	1	4	yes	—
37	C₂×C₈×D₁₁	C₂³	1	4	yes	—
38	#66928	C₈	1	2	yes	—
39	#66927	C₈	1	1	yes	characteristic
40	C₄×D₄₄	C₂×C₄	1	2	yes	—
41	C₄×D₄₄	C₂×C₄	1	2	yes	—
42	#66922	C₈	1	1	yes	characteristic
43	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
44	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
45	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
49	(C₂×D₁₁)⋊C₈	C₂×C₄	1	4	yes	—
50	(C₂×D₁₁)⋊C₈	C₂×C₄	1	4	yes	—
51	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
52	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
53	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
54	#66862	—	4	4	no	—
55	C₁₆×D₁₁	—	4	4	no	—
56	#33495	—	11	1	no	maximal, Sylow
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×C₂²⋊D₁₁	C₂×C₈	1	4	yes	—
61	#4339	—	4	2	no	—
62	#4339	—	4	2	no	—
63	C₂×D₄₄	C₂×C₈	1	2	yes	—
64	C₂×D₄₄	C₂×C₈	1	2	yes	—
65	C₂×D₄₄	C₂×C₈	1	2	yes	—
66	C₂×C₄×D₁₁	C₂²×C₄	1	4	yes	—
67	C₂×Dic₂₂	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₁₁	—	2	2	no	—
69	C₂×C₈×C₁₁	—	4	1	no	—
70	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
74	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
78	(C₂×D₁₁)⋊C₄	C₂×C₈	1	4	yes	—
79	(C₂×D₁₁)⋊C₄	C₂×C₈	1	4	yes	—
80	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
81	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
82	#4318	—	4	2	no	—
83	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
84	#4313	—	4	4	no	—
85	C₈×D₁₁	—	2	4	no	—
86	C₈×D₁₁	—	2	4	no	—
87	C₁₁×C₁₆	—	2	2	no	—
88	C₁₁×C₁₆	—	2	2	no	—
89	C₁₁⋊C₁₆	—	2	4	no	—
90	#2931	—	11	1	no	—
91	#2169	—	11	2	no	—
92	#2168	—	11	4	no	—
93	D₄×C₁₆	—	11	2	no	—
94	#2163	—	11	1	no	—
95	#2162	D₁₁	1	1	yes	characteristic, complemented
96	#2113	—	11	4	no	—
97	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
98	C₂²×D₁₁	C₂²×C₈	1	4	yes	—
99	C₂×C₄×C₁₁	—	2	1	no	—
100	C₂×C₄×C₁₁	—	2	1	no	—
101	C₂×C₄×C₁₁	—	2	2	no	—
102	C₂×C₄×C₁₁	—	4	1	no	—
103	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
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	C₂³×C₄	1	1	yes	characteristic
109	C₂²⋊D₁₁	—	4	4	no	—
110	C₂×Dic₁₁	C₂²×C₈	1	4	yes	—
111	D₄₄	—	2	4	no	—
112	D₄₄	—	4	2	no	—
113	D₄₄	—	4	2	no	—
114	C₄×D₁₁	—	2	4	no	—
115	C₄×D₁₁	—	2	4	no	—
116	Dic₂₂	—	4	2	no	—
117	C₈×C₁₁	—	2	1	no	—
118	C₈×C₁₁	—	2	1	no	—
119	C₈×C₁₁	—	2	1	no	—
120	C₈×C₁₁	H(5,𝔽₂)	1	2	yes	—
121	C₁₁⋊C₈	—	2	4	no	—
122	C₂×#131	—	11	2	no	—
123	#532	—	11	1	no	—
124	C₂×M₃₂	—	11	4	no	—
125	C₂²×C₁₆	—	11	4	no	—
126	#436	—	11	2	no	—
127	#436	—	11	2	no	—
128	C₈×D₄	—	11	2	no	—
129	C₈×D₄	—	11	2	no	—
130	#423	—	11	1	no	—
131	#408	—	11	2	no	—
132	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
133	C₄⋊C₁₆	—	11	2	no	—
134	C₄⋊C₁₆	C₂×D₁₁	1	2	yes	—
135	C₂²⋊C₁₆	—	11	2	no	—
136	C₂²⋊C₁₆	—	11	4	no	—
137	C₂²⋊C₁₆	—	11	4	no	—
138	C₂²⋊C₁₆	C₂×D₁₁	1	2	yes	—
139	#346	—	11	2	no	—
140	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
141	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
142	C₂²×C₁₁	—	2	1	no	—
143	C₂²×C₁₁	—	2	1	no	—
144	C₂²×C₁₁	—	2	1	no	—
145	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
146	C₂×D₁₁	—	2	4	no	—
147	C₂×D₁₁	—	2	4	no	—
148	C₂×D₁₁	—	2	4	no	—
149	C₄×C₁₁	—	2	1	no	—
150	C₄×C₁₁	—	2	1	no	—
151	C₄×C₁₁	—	2	1	no	—
152	C₄×C₁₁	—	2	1	no	—
153	C₄×C₁₁	#504	1	2	yes	—
154	C₄×C₁₁	#518	1	1	yes	characteristic
155	C₄×C₁₁	#568	1	1	yes	characteristic
156	Dic₁₁	—	2	4	no	—
157	C₂×#41	—	11	2	no	—
158	#131	—	44	4	no	—
159	C₂×M₁₆	—	11	2	no	—
160	C₂×M₁₆	—	11	4	no	—
161	C₂²×C₈	—	11	4	no	—
162	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
164	C₄⋊Q₈	—	11	1	no	—
165	#127	—	11	1	no	—
166	#124	—	11	2	no	—
167	#121	—	11	2	no	—
168	#121	—	11	2	no	—
169	C₄×D₄	—	11	2	no	—
170	C₄×D₄	—	11	2	no	—
171	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
172	M₃₂	—	44	4	no	—
173	C₂×C₁₆	—	11	4	no	—
174	C₂×C₁₆	—	44	4	no	—
175	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
176	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
177	C₄⋊C₈	—	11	2	no	—
178	C₄⋊C₈	—	11	2	no	—
179	C₂²⋊C₈	—	11	4	no	—
180	C₂²⋊C₈	—	11	4	no	—
181	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
182	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
183	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
184	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
185	C₂×C₁₁	—	4	1	no	—
186	C₂×C₁₁	#2972	1	1	yes	characteristic
187	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
188	C₂×C₁₁	#3407	1	1	yes	characteristic
189	D₁₁	—	4	4	no	—
190	#41	—	44	2	no	—
191	#41	—	44	2	no	—
192	C₂×Q₈	—	11	2	no	—
193	C₂×D₄	—	11	4	no	—
194	C₂×D₄	—	11	2	no	—
195	C₂×D₄	—	11	2	no	—
196	C₂×D₄	—	11	2	no	—
197	C₂²×C₄	—	11	4	no	—
198	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
199	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
201	M₁₆	—	44	4	no	—
202	M₁₆	—	44	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	11	4	no	—
206	C₂×C₈	—	22	4	no	—
207	C₂×C₈	—	22	4	no	—
208	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
209	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
211	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
212	C₄⋊C₄	—	11	2	no	—
213	C₄⋊C₄	—	11	2	no	—
214	C₂²⋊C₄	—	11	4	no	—
215	C₂²⋊C₄	—	11	4	no	—
216	C₄²	C₈×D₁₁	1	1	yes	characteristic
217	C₄²	C₈×D₁₁	1	1	yes	characteristic
218	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
219	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
220	C₁₆	—	2	2	no	—
221	C₁₆	—	2	2	no	—
222	C₁₆	—	22	4	no	—
223	C₁₁	#33495	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	11	4	no	—
225	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
226	Q₈	—	44	2	no	—
227	D₄	—	22	4	no	—
228	D₄	—	44	2	no	—
229	D₄	—	44	2	no	—
230	D₄	—	44	4	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	11	4	no	—
236	C₂×C₄	—	22	4	no	—
237	C₂×C₄	—	22	4	no	—
238	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
239	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
240	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
241	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
242	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	22	4	no	—
248	C₈	#67042	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	22	4	no	—
253	C₂²	—	22	4	no	—
254	C₂²	—	22	4	no	—
255	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	22	4	no	—
261	C₄	#10622514	1	2	yes	—
262	C₄	#10623052	1	1	yes	characteristic
263	C₄	#10623276	1	1	yes	characteristic
264	C₂	—	4	1	no	—
265	C₂	—	44	4	no	—
266	C₂	#13077639	1	1	yes	characteristic
267	C₂	C₂×#10622514	1	1	yes	characteristic
268	C₂	#13081718	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2162	C₂
prime radical	#10623008	C₄
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2162	D₁₁
hypocenter	C₁₁	#33495
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. #13070449	C₂	semidirect
3. #13070440	C₂	semidirect
4. C₁₆×D₄₄	C₂	semidirect
5. #13070419	C₂	semidirect
6. #13070228	C₂	semidirect
7. C₁₁×#2162	C₂	semidirect
95. #2162	D₁₁	semidirect
223. C₁₁	#33495	semidirect

Maximal subgroups

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

subgroup	index
#13077329	2
#13070449	2
#13070440	2
C₁₆×D₄₄	2
#13070419	2
#13070228	2
C₁₁×#2162	2
#33495	11