Subgroups — #14907441

Order 2816.

570 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#4339	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₂₂	C₂	1	2	yes	maximal, complemented
3	C₁₆×D₄₄	C₂	1	2	yes	maximal, complemented
4	(C₂²⋊D₁₁)⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	#13070414	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₁×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
8	C₁₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#4339	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₈×C₂²⋊D₁₁	C₄	1	2	yes	—
12	C₈×C₂²⋊D₁₁	C₂²	1	2	yes	—
13	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
14	#10622538	C₄	1	1	yes	characteristic
15	C₂×C₁₆×D₁₁	C₂²	1	4	yes	—
16	C₈×D₄₄	C₄	1	1	yes	characteristic
17	C₈×D₄₄	C₂²	1	1	yes	characteristic
18	#10622298	C₄	1	2	yes	—
19	C₄×C₈×D₁₁	C₂²	1	2	yes	—
20	C₈×Dic₂₂	C₄	1	1	yes	characteristic
21	C₈×Dic₂₂	C₂²	1	1	yes	characteristic
22	C₁₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₁₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₁₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
25	C₄×C₁₁×C₁₆	C₂²	1	2	yes	—
26	(C₂×D₁₁)⋊C₁₆	C₂²	1	4	yes	—
27	Dic₁₁⋊C₁₆	C₂²	1	4	yes	—
28	C₁₆×Dic₁₁	C₂²	1	4	yes	—
29	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
30	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
31	C₂×#4339	C₈	1	1	yes	characteristic
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	C₄×C₂²⋊D₁₁	C₈	1	2	yes	—
36	C₄×C₂²⋊D₁₁	C₂×C₄	1	2	yes	—
37	#66979	C₈	1	1	yes	characteristic
38	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₁	C₂³	1	2	yes	—
40	C₄×D₄₄	C₈	1	1	yes	characteristic
41	C₄×D₄₄	C₂×C₄	1	1	yes	characteristic
42	#66925	C₈	1	2	yes	—
43	C₄²×D₁₁	C₂×C₄	1	2	yes	—
44	C₄×Dic₂₂	C₈	1	1	yes	characteristic
45	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
47	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
48	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
52	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
53	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
54	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
55	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
56	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
57	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
58	C₈×Dic₁₁	C₂³	1	2	yes	—
59	C₂×C₁₁⋊C₁₆	—	4	2	no	—
60	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
61	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
62	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
63	C₄×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
64	C₁₆×D₁₁	—	4	4	no	—
65	#33502	—	11	1	no	maximal, Sylow
66	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₁	C₂×C₈	1	2	yes	—
70	#4339	C₁₆	1	8	yes	complemented
71	C₂×D₄₄	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₁₁	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₁₁	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
75	C₂×Dic₂₂	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₁₁	—	2	1	no	—
77	C₂×C₈×C₁₁	—	2	1	no	—
78	C₂×C₈×C₁₁	—	4	1	no	—
79	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
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	C₂²⋊Dic₁₁	C₂×C₈	1	2	yes	—
88	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
89	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
90	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
91	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
92	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
93	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
94	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
95	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
96	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
97	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
99	C₈×D₁₁	—	2	2	no	—
100	C₈×D₁₁	—	2	2	no	—
101	C₈×D₁₁	—	4	2	no	—
102	C₁₁×C₁₆	—	2	2	no	—
103	C₁₁×C₁₆	—	2	2	no	—
104	C₁₁⋊C₁₆	—	2	4	no	—
105	C₈×#41	—	11	1	no	—
106	Q₈×C₁₆	—	11	2	no	—
107	D₄×C₁₆	—	11	4	no	—
108	D₄×C₁₆	—	11	2	no	—
109	#2162	—	11	2	no	—
110	#2162	D₁₁	1	1	yes	characteristic, complemented
111	C₂×C₄⋊C₁₆	—	11	1	no	—
112	C₂×C₄⋊C₁₆	—	11	2	no	—
113	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
114	C₂²×D₁₁	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₁₁	—	2	1	no	—
116	C₂×C₄×C₁₁	—	2	1	no	—
117	C₂×C₄×C₁₁	—	4	1	no	—
118	C₂×C₄×C₁₁	C₂×C₁₆	1	4	yes	complemented
119	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
125	C₂²⋊D₁₁	C₂×C₁₆	1	8	yes	complemented
126	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
127	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
128	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
129	D₄₄	C₂×C₁₆	1	4	yes	complemented
130	C₄×D₁₁	—	2	2	no	—
131	C₄×D₁₁	—	2	2	no	—
132	C₄×D₁₁	—	4	2	no	—
133	C₄×D₁₁	C₂×C₁₆	1	8	yes	complemented
134	Dic₂₂	C₂×C₁₆	1	4	yes	complemented
135	C₈×C₁₁	—	2	1	no	—
136	C₈×C₁₁	—	2	1	no	—
137	C₈×C₁₁	—	2	1	no	—
138	C₈×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
139	C₈×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
140	C₁₁⋊C₈	—	2	2	no	—
141	C₁₁⋊C₈	—	2	2	no	—
142	C₄×#41	—	11	1	no	—
143	C₂²×C₁₆	—	11	2	no	—
144	C₂²×C₁₆	—	11	4	no	—
145	C₈×Q₈	—	11	1	no	—
146	C₈×Q₈	—	11	1	no	—
147	C₈×D₄	—	11	2	no	—
148	C₈×D₄	—	11	2	no	—
149	C₈×D₄	—	11	1	no	—
150	C₈×D₄	—	11	1	no	—
151	#431	—	11	1	no	—
152	#431	—	11	2	no	—
153	C₂×C₄×C₈	—	11	2	no	—
154	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
155	C₄⋊C₁₆	—	11	4	no	—
156	C₄⋊C₁₆	—	11	2	no	—
157	C₄⋊C₁₆	—	11	2	no	—
158	C₄⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
159	C₄⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
160	C₂²⋊C₁₆	—	11	4	no	—
161	C₂²⋊C₁₆	C₂×D₁₁	1	2	yes	—
162	C₄×C₁₆	—	11	4	no	—
163	C₄×C₁₆	C₂×D₁₁	1	2	yes	—
164	C₂²×C₁₁	—	2	1	no	—
165	C₂²×C₁₁	—	2	1	no	—
166	C₂²×C₁₁	C₂²×C₁₆	1	2	yes	—
167	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
168	C₂×D₁₁	—	2	2	no	—
169	C₂×D₁₁	—	2	2	no	—
170	C₂×D₁₁	C₂²×C₁₆	1	4	yes	—
171	C₄×C₁₁	—	2	1	no	—
172	C₄×C₁₁	—	2	1	no	—
173	C₄×C₁₁	—	2	1	no	—
174	C₄×C₁₁	C₂²×C₁₆	1	2	yes	—
175	C₄×C₁₁	C₂²×C₁₆	1	2	yes	—
176	C₄×C₁₁	#520	1	1	yes	characteristic
177	C₄×C₁₁	#568	1	1	yes	characteristic
178	Dic₁₁	—	2	2	no	—
179	Dic₁₁	—	2	2	no	—
180	Dic₁₁	C₂²×C₁₆	1	4	yes	—
181	C₂×#41	—	11	1	no	—
182	C₂²×C₈	—	11	2	no	—
183	C₂²×C₈	—	11	2	no	—
184	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
186	C₄×Q₈	—	11	1	no	—
187	C₄×Q₈	—	11	1	no	—
188	C₄×D₄	—	11	2	no	—
189	C₄×D₄	—	11	2	no	—
190	C₄×D₄	—	11	1	no	—
191	C₄×D₄	—	11	1	no	—
192	#117	—	11	1	no	—
193	#117	—	11	2	no	—
194	C₂×C₄²	—	11	2	no	—
195	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
196	C₂×C₁₆	—	11	4	no	—
197	C₂×C₁₆	—	44	4	no	—
198	C₂×C₁₆	—	44	2	no	—
199	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
200	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
201	C₄⋊C₈	—	11	2	no	—
202	C₄⋊C₈	—	11	2	no	—
203	C₄⋊C₈	—	11	1	no	—
204	C₄⋊C₈	—	11	1	no	—
205	C₂²⋊C₈	—	11	2	no	—
206	C₂²⋊C₈	—	11	2	no	—
207	C₂²⋊C₈	—	11	2	no	—
208	C₄×C₈	—	11	2	no	—
209	C₄×C₈	—	11	2	no	—
210	C₄×C₈	—	11	2	no	—
211	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
212	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
215	C₂×C₁₁	—	2	2	no	—
216	C₂×C₁₁	#2988	1	1	yes	characteristic
217	C₂×C₁₁	C₂³×C₁₆	1	1	yes	characteristic
218	C₂×C₁₁	#3407	1	1	yes	characteristic
219	D₁₁	—	2	4	no	—
220	#41	—	11	8	no	—
221	C₂×Q₈	—	11	1	no	—
222	C₂×D₄	—	11	2	no	—
223	C₂×D₄	—	11	1	no	—
224	C₂²×C₄	—	11	2	no	—
225	C₂²×C₄	—	11	2	no	—
226	C₂²×C₄	—	11	2	no	—
227	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
228	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	11	2	no	—
234	C₂×C₈	—	11	2	no	—
235	C₂×C₈	—	22	2	no	—
236	C₂×C₈	—	22	2	no	—
237	C₂×C₈	—	44	2	no	—
238	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
240	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
241	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
242	C₄⋊C₄	—	11	2	no	—
243	C₄⋊C₄	—	11	2	no	—
244	C₄⋊C₄	—	11	1	no	—
245	C₄⋊C₄	—	11	1	no	—
246	C₂²⋊C₄	—	11	2	no	—
247	C₂²⋊C₄	—	11	2	no	—
248	C₂²⋊C₄	—	11	2	no	—
249	C₄²	—	11	2	no	—
250	C₄²	—	11	2	no	—
251	C₄²	—	11	2	no	—
252	C₄²	C₈×D₁₁	1	1	yes	characteristic
253	C₄²	C₈×D₁₁	1	1	yes	characteristic
254	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
255	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
256	C₁₆	—	2	2	no	—
257	C₁₆	—	2	2	no	—
258	C₁₆	—	22	4	no	—
259	C₁₁	#33502	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	11	2	no	—
261	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
262	Q₈	—	11	4	no	—
263	D₄	—	11	8	no	—
264	D₄	—	11	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	11	2	no	—
269	C₂×C₄	—	11	8	no	—
270	C₂×C₄	—	11	2	no	—
271	C₂×C₄	—	11	2	no	—
272	C₂×C₄	—	22	2	no	—
273	C₂×C₄	—	22	2	no	—
274	C₂×C₄	—	44	2	no	—
275	C₂×C₄	C₁₆×D₁₁	1	4	yes	complemented
276	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
277	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
278	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
279	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
280	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	22	2	no	—
286	C₈	—	22	2	no	—
287	C₈	#67042	1	1	yes	characteristic
288	C₈	#67043	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	11	4	no	—
292	C₂²	—	22	2	no	—
293	C₂²	—	22	2	no	—
294	C₂²	C₂×C₁₆×D₁₁	1	2	yes	—
295	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	11	4	no	—
300	C₄	—	22	2	no	—
301	C₄	—	22	2	no	—
302	C₄	C₂×C₁₆×D₁₁	1	2	yes	—
303	C₄	C₂×C₁₆×D₁₁	1	2	yes	—
304	C₄	#10623054	1	1	yes	characteristic
305	C₄	#10623276	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	22	4	no	—
308	C₂	#13077724	1	1	yes	characteristic
309	C₂	C₂²×C₁₆×D₁₁	1	1	yes	characteristic
310	C₂	#13081718	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator	C₂×C₁₁	C₂³×C₁₆
Fitting	C₁₁×#2162	C₂
socle	C₂²×C₁₁	C₂³×C₈
prime radical	C₂×#4339	C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2162	D₁₁
hypocenter	C₁₁	#33502
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. C₁₆×Dic₂₂	C₂	semidirect
3. C₁₆×D₄₄	C₂	semidirect
4. (C₂²⋊D₁₁)⋊C₁₆	C₂	semidirect
5. #13070414	C₂	semidirect
6. C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	semidirect
7. D₁₁×C₄⋊C₁₆	C₂	semidirect
8. C₁₁×#2162	C₂	semidirect
70. #4339	C₁₆	semidirect
110. #2162	D₁₁	semidirect
118. C₂×C₄×C₁₁	C₂×C₁₆	semidirect
125. C₂²⋊D₁₁	C₂×C₁₆	semidirect
129. D₄₄	C₂×C₁₆	semidirect
133. C₄×D₁₁	C₂×C₁₆	semidirect
134. Dic₂₂	C₂×C₁₆	semidirect
259. C₁₁	#33502	semidirect
275. C₂×C₄	C₁₆×D₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₈×#4339	2
C₁₆×Dic₂₂	2
C₁₆×D₄₄	2
(C₂²⋊D₁₁)⋊C₁₆	2
#13070414	2
C₂×C₄⋊(C₁₁⋊C₁₆)	2
D₁₁×C₄⋊C₁₆	2
C₁₁×#2162	2
#33502	11