Subgroups — #14906996

Order 2816.

682 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622520	C₂	1	2	yes	maximal, complemented
2	#13077329	C₂	1	1	yes	maximal, characteristic
3	#13070449	C₂	1	2	yes	maximal, complemented
4	#13070443	C₂	1	4	yes	maximal, complemented
5	C₁₆×D₄₄	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070231	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#66954	C₂²	1	2	yes	—
10	#10623008	C₄	1	1	yes	characteristic
11	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	#10622723	C₄	1	2	yes	—
15	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
16	#10622537	C₄	1	1	yes	characteristic
17	#10622520	—	2	8	no	—
18	C₂×#66862	C₂²	1	4	yes	—
19	C₂×C₁₆×D₁₁	C₂²	1	4	yes	—
20	#10622301	C₄	1	2	yes	—
21	#10622301	C₄	1	2	yes	—
22	C₈×D₄₄	C₂²	1	2	yes	—
23	C₈×D₄₄	C₂²	1	2	yes	—
24	C₁₁×#346	C₂²	1	2	yes	—
25	C₄×C₁₁×C₁₆	C₂²	1	2	yes	—
26	(C₂×D₁₁)⋊C₁₆	C₂²	1	4	yes	—
27	(C₂×D₁₁)⋊C₁₆	C₂²	1	4	yes	—
28	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
29	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
30	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
31	C₂×#4339	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
35	#66983	C₈	1	2	yes	—
36	#66983	C₈	1	2	yes	—
37	C₂×#4318	C₂×C₄	1	2	yes	—
38	#66954	—	2	4	no	—
39	#66954	—	2	4	no	—
40	C₂×#4313	C₂×C₄	1	4	yes	—
41	C₂×C₈×D₁₁	C₂³	1	4	yes	—
42	#66928	C₈	1	2	yes	—
43	#66927	C₈	1	1	yes	characteristic
44	C₄×D₄₄	C₂×C₄	1	2	yes	—
45	C₄×D₄₄	C₂×C₄	1	2	yes	—
46	#66922	C₈	1	1	yes	characteristic
47	C₁₁×M₃₂	—	2	2	no	—
48	C₁₁×M₃₂	D₄	1	4	yes	complemented
49	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
50	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
51	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
55	(C₂×D₁₁)⋊C₈	C₂×C₄	1	4	yes	—
56	(C₂×D₁₁)⋊C₈	C₂×C₄	1	4	yes	—
57	C₂×C₁₁⋊C₁₆	—	2	4	no	—
58	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
60	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
61	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
62	#66862	—	2	8	no	—
63	C₁₆×D₁₁	—	2	8	no	—
64	#33438	—	11	1	no	maximal, Sylow
65	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₁	C₂×C₈	1	4	yes	—
69	#4339	—	2	4	no	—
70	#4339	—	2	4	no	—
71	C₂×D₄₄	C₂×C₈	1	2	yes	—
72	C₂×D₄₄	C₂×C₈	1	2	yes	—
73	C₂×D₄₄	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₁₁	C₂²×C₄	1	4	yes	—
75	C₂×Dic₂₂	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₁₁	—	2	2	no	—
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	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₁	#41	1	2	yes	—
82	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
83	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
85	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	(C₂×D₁₁)⋊C₄	C₂×C₈	1	4	yes	—
88	(C₂×D₁₁)⋊C₄	C₂×C₈	1	4	yes	—
89	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
90	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
91	#4318	—	2	4	no	—
92	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
93	#4313	—	2	8	no	—
94	C₈×D₁₁	—	2	4	no	—
95	C₈×D₁₁	—	2	4	no	—
96	C₁₁×C₁₆	—	2	2	no	—
97	C₁₁×C₁₆	C₂×D₄	1	4	yes	—
98	C₁₁⋊C₁₆	—	2	2	no	—
99	C₁₁⋊C₁₆	—	2	2	no	—
100	C₂×#504	—	11	2	no	—
101	#2931	—	11	1	no	—
102	#2169	—	11	2	no	—
103	#2168	—	11	4	no	—
104	D₄×C₁₆	—	11	2	no	—
105	C₂×C₄⋊C₁₆	—	11	1	no	—
106	#2113	—	11	2	no	—
107	C₄×M₃₂	D₁₁	1	1	yes	characteristic, complemented
108	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
109	C₂²×D₁₁	C₂²×C₈	1	4	yes	—
110	C₂×C₄×C₁₁	—	2	2	no	—
111	C₂×C₄×C₁₁	—	4	1	no	—
112	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₁	#131	1	2	yes	—
119	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
120	C₂²⋊D₁₁	—	2	8	no	—
121	C₂×Dic₁₁	C₂²×C₈	1	4	yes	—
122	D₄₄	—	2	4	no	—
123	D₄₄	—	2	4	no	—
124	D₄₄	—	4	2	no	—
125	C₄×D₁₁	—	2	4	no	—
126	C₄×D₁₁	—	2	4	no	—
127	Dic₂₂	—	2	4	no	—
128	C₈×C₁₁	—	2	1	no	—
129	C₈×C₁₁	—	2	1	no	—
130	C₈×C₁₁	C₄×D₄	1	2	yes	—
131	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
133	C₁₁⋊C₈	—	2	4	no	—
134	C₂×#131	—	11	2	no	—
135	#532	—	11	1	no	—
136	#504	—	22	8	no	—
137	C₂×M₃₂	—	11	4	no	—
138	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
139	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
140	C₂²×C₁₆	—	11	2	no	—
141	C₂²×C₁₆	—	11	4	no	—
142	#436	—	11	2	no	—
143	#436	—	11	2	no	—
144	C₈×D₄	—	11	2	no	—
145	C₈×D₄	—	11	2	no	—
146	#423	—	11	1	no	—
147	#408	—	11	2	no	—
148	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
149	C₄⋊C₁₆	—	11	1	no	—
150	C₄⋊C₁₆	—	11	2	no	—
151	C₄⋊C₁₆	—	11	1	no	—
152	C₂²⋊C₁₆	—	11	4	no	—
153	C₂²⋊C₁₆	—	11	4	no	—
154	#346	C₂×D₁₁	1	2	yes	—
155	C₄×C₁₆	C₂×D₁₁	1	2	yes	—
156	C₂²×C₁₁	—	2	1	no	—
157	C₂²×C₁₁	—	2	1	no	—
158	C₂²×C₁₁	C₈×D₄	1	2	yes	—
159	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
160	C₂×D₁₁	—	2	4	no	—
161	C₂×D₁₁	—	2	4	no	—
162	C₂×D₁₁	—	2	4	no	—
163	C₄×C₁₁	—	2	1	no	—
164	C₄×C₁₁	—	2	1	no	—
165	C₄×C₁₁	—	2	1	no	—
166	C₄×C₁₁	C₈×D₄	1	2	yes	—
167	C₄×C₁₁	#504	1	2	yes	complemented
168	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
169	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
170	Dic₁₁	—	2	4	no	—
171	C₂×#41	—	11	2	no	—
172	#131	—	22	4	no	—
173	#131	—	22	4	no	—
174	C₂×M₁₆	—	11	2	no	—
175	C₂×M₁₆	—	11	4	no	—
176	C₂²×C₈	—	11	4	no	—
177	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
179	C₄⋊Q₈	—	11	1	no	—
180	#127	—	11	1	no	—
181	#124	—	11	2	no	—
182	#121	—	11	2	no	—
183	#121	—	11	2	no	—
184	C₄×D₄	—	11	2	no	—
185	C₄×D₄	—	11	2	no	—
186	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
187	M₃₂	—	2	2	no	—
188	M₃₂	—	22	8	no	—
189	M₃₂	D₄₄	1	4	yes	complemented
190	C₂×C₁₆	—	11	2	no	—
191	C₂×C₁₆	—	11	2	no	—
192	C₂×C₁₆	—	22	4	no	—
193	C₂×C₁₆	—	22	8	no	—
194	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
195	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
196	C₄⋊C₈	—	11	2	no	—
197	C₄⋊C₈	—	11	2	no	—
198	C₂²⋊C₈	—	11	4	no	—
199	C₂²⋊C₈	—	11	4	no	—
200	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
201	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
204	C₂×C₁₁	—	2	2	no	—
205	C₂×C₁₁	C₂×C₈×D₄	1	1	yes	characteristic
206	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
207	C₂×C₁₁	#3407	1	1	yes	characteristic
208	D₁₁	—	4	4	no	—
209	#41	—	22	4	no	—
210	#41	—	22	4	no	—
211	C₂×Q₈	—	11	2	no	—
212	C₂×D₄	—	11	4	no	—
213	C₂×D₄	—	11	2	no	—
214	C₂×D₄	—	11	2	no	—
215	C₂×D₄	—	11	2	no	—
216	C₂²×C₄	—	11	4	no	—
217	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
218	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
220	M₁₆	—	22	4	no	—
221	M₁₆	—	22	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	11	4	no	—
224	C₂×C₈	—	22	4	no	—
225	C₂×C₈	—	22	4	no	—
226	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄₄	1	2	yes	—
230	C₂×C₈	#4339	1	2	yes	—
231	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
232	C₄⋊C₄	—	11	2	no	—
233	C₄⋊C₄	—	11	2	no	—
234	C₂²⋊C₄	—	11	4	no	—
235	C₂²⋊C₄	—	11	4	no	—
236	C₄²	C₈×D₁₁	1	1	yes	characteristic
237	C₄²	C₈×D₁₁	1	1	yes	characteristic
238	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
239	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
240	C₁₆	—	2	2	no	—
241	C₁₆	—	22	2	no	—
242	C₁₆	—	22	2	no	—
243	C₁₆	C₂×D₄₄	1	4	yes	—
244	C₁₁	#33438	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	11	4	no	—
246	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
247	Q₈	—	22	4	no	—
248	D₄	—	22	8	no	—
249	D₄	—	22	4	no	—
250	D₄	—	22	4	no	—
251	D₄	—	44	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	11	4	no	—
255	C₂×C₄	—	22	4	no	—
256	C₂×C₄	—	22	4	no	—
257	C₂×C₄	C₄×D₄₄	1	2	yes	—
258	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
260	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
261	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
262	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
263	C₂×C₄	#66954	1	2	yes	—
264	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	22	4	no	—
268	C₈	C₄×D₄₄	1	2	yes	—
269	C₈	C₂²×D₄₄	1	1	yes	characteristic
270	C₈	C₂×#4339	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	22	4	no	—
274	C₂²	—	22	4	no	—
275	C₂²	—	22	4	no	—
276	C₂²	C₈×D₄₄	1	2	yes	—
277	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	22	4	no	—
282	C₄	C₈×D₄₄	1	2	yes	—
283	C₄	#10622520	1	2	yes	complemented
284	C₄	C₂×C₄×D₄₄	1	1	yes	characteristic
285	C₄	C₂×#66954	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	44	4	no	—
288	C₂	C₂×C₈×D₄₄	1	1	yes	characteristic
289	C₂	C₂×#10622520	1	1	yes	characteristic
290	C₂	#13081718	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₄×C₁₁×M₃₂	C₂
prime radical	#10623008	C₄
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₄×M₃₂	D₁₁
hypocenter	C₁₁	#33438
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#10622520	C₂	semidirect
3. #13070449	C₂	semidirect
4. #13070443	C₂	semidirect
5. C₁₆×D₄₄	C₂	semidirect
6. C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	semidirect
7. #13070231	C₂	semidirect
8. C₄×C₁₁×M₃₂	C₂	semidirect
48. C₁₁×M₃₂	D₄	semidirect
107. C₄×M₃₂	D₁₁	semidirect
167. C₄×C₁₁	#504	semidirect
189. M₃₂	D₄₄	semidirect
244. C₁₁	#33438	semidirect
283. C₄	#10622520	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622520	2
#13077329	2
#13070449	2
#13070443	2
C₁₆×D₄₄	2
C₂×C₄⋊(C₁₁⋊C₁₆)	2
#13070231	2
C₄×C₁₁×M₃₂	2
#33438	11