Subgroups — #14668807

Order 2816.

514 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622302	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₁₁)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×(C₂×D₁₁)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×#10622140	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#13066694	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₁×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#4313	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₁₁	C₂²	1	1	yes	characteristic
10	C₂×#66925	C₄	1	1	yes	characteristic
11	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₂×#66879	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
16	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×#66867	C₄	1	1	yes	characteristic
19	#10622302	—	2	2	no	—
20	#10622302	—	2	2	no	—
21	(C₂×D₁₁)⋊C₁₆	C₄	1	4	yes	complemented
22	(C₂×D₁₁)⋊C₁₆	C₄	1	4	yes	complemented
23	#10622140	—	2	2	no	—
24	C₁₆×Dic₁₁	—	2	2	no	—
25	C₂²×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂×(C₂×D₁₁)⋊C₄	C₈	1	2	yes	—
30	C₂×Dic₁₁⋊C₄	C₈	1	2	yes	—
31	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×#4313	—	2	2	no	—
35	C₂×#4313	—	2	2	no	—
36	C₂×#4313	C₈	1	4	yes	—
37	C₂×C₈×D₁₁	—	2	1	no	—
38	C₂×C₈×D₁₁	—	2	1	no	—
39	C₂×C₈×D₁₁	—	2	1	no	—
40	C₂×C₈×D₁₁	—	2	1	no	—
41	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
43	#66925	—	2	2	no	—
44	#66925	—	2	2	no	—
45	C₂×C₁₁×C₁₆	—	2	1	no	—
46	C₂×C₁₁×C₁₆	—	2	1	no	—
47	C₂×C₁₁×C₁₆	—	2	1	no	—
48	C₂×C₁₁×C₁₆	—	2	1	no	—
49	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
50	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₁	—	2	1	no	—
52	C₄×C₈×C₁₁	—	2	1	no	—
53	#66879	—	2	2	no	—
54	C₈×Dic₁₁	D₄	1	2	yes	—
55	C₈×Dic₁₁	D₄	1	2	yes	—
56	C₂×C₁₁⋊C₁₆	—	2	1	no	—
57	C₂×C₁₁⋊C₁₆	—	2	1	no	—
58	C₂×C₁₁⋊C₁₆	—	2	1	no	—
59	C₂×C₁₁⋊C₁₆	—	2	1	no	—
60	C₂×C₁₁⋊C₁₆	C₂×C₄	1	2	yes	—
61	C₂×C₁₁⋊C₁₆	C₂×C₄	1	2	yes	—
62	#66867	—	2	2	no	—
63	#12600	—	11	1	no	maximal, Sylow
64	C₂³×D₁₁	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
68	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₁₁	—	2	1	no	—
70	C₂×C₄×D₁₁	—	2	1	no	—
71	C₂×C₄×D₁₁	—	2	1	no	—
72	C₂×C₄×D₁₁	—	2	1	no	—
73	C₂×C₄×D₁₁	C₄²	1	2	yes	—
74	C₂×C₄×D₁₁	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₁₁	—	2	1	no	—
76	C₂×C₈×C₁₁	—	2	1	no	—
77	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
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	1	yes	characteristic
81	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
84	C₄²×C₁₁	—	2	1	no	—
85	C₄²×C₁₁	—	2	1	no	—
86	(C₂×D₁₁)⋊C₄	—	2	4	no	—
87	Dic₁₁⋊C₄	—	2	4	no	—
88	C₄×Dic₁₁	C₂²⋊C₄	1	2	yes	—
89	C₄×Dic₁₁	M₁₆	1	2	yes	—
90	C₂×C₁₁⋊C₈	—	2	1	no	—
91	C₂×C₁₁⋊C₈	—	2	1	no	—
92	C₂×C₁₁⋊C₈	C₄²	1	2	yes	—
93	C₂×C₁₁⋊C₈	C₂²⋊C₄	1	2	yes	—
94	C₂×C₁₁⋊C₈	C₂×C₈	1	2	yes	—
95	C₂×C₁₁⋊C₈	M₁₆	1	2	yes	—
96	#4313	—	2	8	no	—
97	C₈×D₁₁	—	2	2	no	—
98	C₈×D₁₁	—	2	2	no	—
99	C₈×D₁₁	—	2	2	no	—
100	C₈×D₁₁	—	2	2	no	—
101	C₁₁×C₁₆	—	2	2	no	—
102	C₁₁×C₁₆	—	2	2	no	—
103	C₁₁⋊C₁₆	—	2	2	no	—
104	C₁₁⋊C₁₆	—	2	2	no	—
105	C₂×#405	—	11	1	no	—
106	C₂×C₂²⋊C₁₆	—	11	1	no	—
107	C₂×C₂²⋊C₁₆	—	11	1	no	—
108	C₂×#346	—	11	1	no	—
109	C₂×C₄×C₁₆	—	11	1	no	—
110	#1374	—	11	1	no	—
111	#1374	D₁₁	1	1	yes	characteristic, complemented
112	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
113	C₂²×D₁₁	—	2	1	no	—
114	C₂²×D₁₁	—	2	1	no	—
115	C₂²×D₁₁	—	2	2	no	—
116	C₂²×D₁₁	—	2	2	no	—
117	C₂²×D₁₁	C₄×C₈	1	2	yes	—
118	C₂×C₄×C₁₁	—	2	2	no	—
119	C₂×C₄×C₁₁	—	2	2	no	—
120	C₂×C₄×C₁₁	—	2	2	no	—
121	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
127	C₂×Dic₁₁	—	2	1	no	—
128	C₂×Dic₁₁	—	2	1	no	—
129	C₂×Dic₁₁	—	2	2	no	—
130	C₂×Dic₁₁	—	2	2	no	—
131	C₂×Dic₁₁	C₄×C₈	1	2	yes	—
132	C₂×Dic₁₁	C₂²⋊C₈	1	4	yes	—
133	C₄×D₁₁	—	2	2	no	—
134	C₄×D₁₁	—	2	2	no	—
135	C₄×D₁₁	—	2	2	no	—
136	C₄×D₁₁	—	2	2	no	—
137	C₈×C₁₁	—	2	2	no	—
138	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
139	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
140	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
141	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
142	C₁₁⋊C₈	—	2	2	no	—
143	C₁₁⋊C₈	C₂²⋊C₈	1	4	yes	—
144	C₂²×M₁₆	—	11	1	no	—
145	C₂³×C₈	—	11	1	no	—
146	C₂×#117	—	11	1	no	—
147	C₂²×C₁₆	—	11	1	no	—
148	C₂²×C₁₆	—	11	1	no	—
149	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
150	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
151	#405	—	22	2	no	—
152	#405	—	22	2	no	—
153	C₂×#97	—	11	1	no	—
154	C₂×#97	—	11	1	no	—
155	C₂×C₄×C₈	—	11	1	no	—
156	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
157	C₂²⋊C₁₆	—	11	4	no	—
158	C₂²⋊C₁₆	—	11	4	no	—
159	#346	—	22	2	no	—
160	C₄×C₁₆	—	22	2	no	—
161	C₂²×C₁₁	C₂×C₄×C₈	1	1	yes	characteristic
162	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₂²×C₁₁	#431	1	1	yes	characteristic
164	C₂²×C₁₁	#504	1	2	yes	—
165	C₂²×C₁₁	#504	1	2	yes	—
166	C₂×D₁₁	—	2	2	no	—
167	C₂×D₁₁	—	2	2	no	—
168	C₂×D₁₁	—	2	2	no	—
169	C₂×D₁₁	—	4	2	no	—
170	C₂×D₁₁	—	4	2	no	—
171	C₄×C₁₁	—	4	2	no	—
172	C₄×C₁₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
173	C₄×C₁₁	C₄×M₁₆	1	1	yes	characteristic
174	C₄×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
175	C₄×C₁₁	#431	1	1	yes	characteristic
176	Dic₁₁	—	2	4	no	—
177	Dic₁₁	—	2	2	no	—
178	C₂³×C₄	—	11	1	no	—
179	C₂×M₁₆	—	11	4	no	—
180	C₂×M₁₆	—	22	2	no	—
181	C₂×M₁₆	—	22	2	no	—
182	C₂²×C₈	—	11	1	no	—
183	C₂²×C₈	—	11	1	no	—
184	C₂²×C₈	—	11	2	no	—
185	C₂²×C₈	—	11	2	no	—
186	C₂²×C₈	—	22	1	no	—
187	C₂²×C₈	—	22	1	no	—
188	C₂²×C₈	—	22	1	no	—
189	C₂²×C₈	—	22	1	no	—
190	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
192	#117	—	22	2	no	—
193	#117	—	22	2	no	—
194	C₂×C₄⋊C₄	—	11	2	no	—
195	C₂×C₂²⋊C₄	—	11	2	no	—
196	C₂×C₄²	—	11	1	no	—
197	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	2	1	no	—
202	C₂×C₁₆	—	11	2	no	—
203	C₂×C₁₆	—	11	2	no	—
204	C₂×C₁₆	—	22	1	no	—
205	C₂×C₁₆	—	22	1	no	—
206	C₂×C₁₆	—	22	1	no	—
207	C₂×C₁₆	—	22	1	no	—
208	C₂×C₁₆	C₄×D₁₁	1	2	yes	—
209	C₂×C₁₆	C₄×D₁₁	1	2	yes	—
210	#97	—	22	2	no	—
211	#97	—	22	2	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	11	2	no	—
215	C₄×C₈	—	11	2	no	—
216	C₂×C₁₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
217	C₂×C₁₁	#2108	1	1	yes	characteristic
218	C₂×C₁₁	#2108	1	1	yes	characteristic
219	C₂×C₁₁	#2113	1	2	yes	—
220	C₂×C₁₁	#2164	1	2	yes	—
221	D₁₁	—	4	2	no	—
222	C₂⁴	—	11	1	no	—
223	C₂²×C₄	—	11	2	no	—
224	C₂²×C₄	—	11	1	no	—
225	C₂²×C₄	—	11	2	no	—
226	C₂²×C₄	—	11	2	no	—
227	C₂²×C₄	—	22	1	no	—
228	C₂²×C₄	—	22	1	no	—
229	C₂²×C₄	—	22	1	no	—
230	C₂²×C₄	—	22	1	no	—
231	C₂²×C₄	C₈×D₁₁	1	2	yes	—
232	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
233	M₁₆	—	22	8	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	11	2	no	—
237	C₂×C₈	—	11	2	no	—
238	C₂×C₈	—	11	2	no	—
239	C₂×C₈	—	11	2	no	—
240	C₂×C₈	—	22	1	no	—
241	C₂×C₈	—	22	1	no	—
242	C₂×C₈	—	22	2	no	—
243	C₂×C₈	—	22	2	no	—
244	C₂×C₈	—	22	2	no	—
245	C₂×C₈	—	22	2	no	—
246	C₂×C₈	C₈×D₁₁	1	2	yes	—
247	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
249	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
250	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
251	C₂×C₈	#4341	1	1	yes	characteristic
252	C₂×C₈	#4343	1	1	yes	characteristic
253	C₄⋊C₄	—	22	4	no	—
254	C₂²⋊C₄	—	22	4	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	11	2	no	—
258	C₄²	—	11	2	no	—
259	C₁₆	—	2	2	no	—
260	C₁₆	—	2	2	no	—
261	C₁₆	—	22	2	no	—
262	C₁₆	—	22	2	no	—
263	C₁₁	#12600	1	1	yes	characteristic, complemented, Sylow
264	C₂³	—	11	2	no	—
265	C₂³	—	22	2	no	—
266	C₂³	—	22	2	no	—
267	C₂³	—	22	1	no	—
268	C₂³	—	22	1	no	—
269	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	11	2	no	—
274	C₂×C₄	—	11	4	no	—
275	C₂×C₄	—	22	2	no	—
276	C₂×C₄	—	22	2	no	—
277	C₂×C₄	—	22	1	no	—
278	C₂×C₄	—	22	1	no	—
279	C₂×C₄	—	22	2	no	—
280	C₂×C₄	—	22	2	no	—
281	C₂×C₄	—	22	2	no	—
282	C₂×C₄	—	22	2	no	—
283	C₂×C₄	C₄²×D₁₁	1	1	yes	characteristic
284	C₂×C₄	C₂²⋊C₄×D₁₁	1	1	yes	characteristic
285	C₂×C₄	#66945	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
287	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	11	4	no	—
291	C₈	—	22	2	no	—
292	C₈	(C₂²⋊D₁₁)⋊C₄	1	1	yes	characteristic
293	C₈	(C₂²⋊D₁₁)⋊C₄	1	1	yes	characteristic
294	C₈	#66946	1	1	yes	characteristic
295	C₈	#66946	1	1	yes	characteristic
296	C₂²	—	22	2	no	—
297	C₂²	—	22	2	no	—
298	C₂²	—	22	2	no	—
299	C₂²	—	44	2	no	—
300	C₂²	—	44	2	no	—
301	C₂²	C₄×C₈×D₁₁	1	1	yes	characteristic
302	C₂²	D₁₁×C₂²⋊C₈	1	1	yes	characteristic
303	C₂²	#10622446	1	1	yes	characteristic
304	C₂²	#10622514	1	2	yes	—
305	C₂²	#10622520	1	2	yes	—
306	C₄	—	4	2	no	—
307	C₄	—	22	2	no	—
308	C₄	—	22	4	no	—
309	C₄	#10622279	1	1	yes	characteristic
310	C₄	#10622320	1	1	yes	characteristic
311	C₄	D₁₁×C₂²⋊C₈	1	1	yes	characteristic
312	C₄	#10622446	1	1	yes	characteristic
313	C₂	—	44	2	no	—
314	C₂	#13068373	1	1	yes	characteristic
315	C₂	#13070211	1	1	yes	characteristic
316	C₂	#13070214	1	1	yes	characteristic
317	C₂	#13070228	1	2	yes	—
318	C₂	#13070422	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂×C₄×C₈
Fitting	C₁₁×#1374	C₂
socle	C₂³×C₁₁	C₂²×C₈
prime radical	C₂³×D₁₁	C₂×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1374	D₁₁
hypocenter	C₁₁	#12600
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
4. C₂×#10622140	C₂	semidirect
5. C₂×C₁₆×Dic₁₁	C₂	semidirect
6. #13066694	C₂	semidirect
7. C₁₁×#1374	C₂	semidirect
21. (C₂×D₁₁)⋊C₁₆	C₄	semidirect
22. (C₂×D₁₁)⋊C₁₆	C₄	semidirect
111. #1374	D₁₁	semidirect
263. C₁₁	#12600	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622302	2
C₂×(C₂×D₁₁)⋊C₁₆	2
C₂×#10622140	2
C₂×C₁₆×Dic₁₁	2
#13066694	2
C₁₁×#1374	2
#12600	11