Subgroups — #14907424

Order 2816.

452 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	#13077545	C₂	1	1	yes	maximal, characteristic
2	#13070525	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070524	C₂	1	2	yes	maximal, complemented
4	#13070523	C₂	1	2	yes	maximal, complemented
5	#13070521	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₂₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070422	C₂	1	2	yes	maximal
8	#13070421	C₂	1	2	yes	maximal
9	#13070418	C₂	1	1	yes	maximal, characteristic
10	#13070413	C₂	1	1	yes	maximal, characteristic
11	Q₈×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623109	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₁₁	C₂²	1	1	yes	characteristic
14	#10622637	C₄	1	1	yes	characteristic
15	#10622452	C₄	1	2	yes	—
16	#10622450	C₄	1	1	yes	characteristic
17	#10622444	C₄	1	1	yes	characteristic
18	#10622441	C₄	1	2	yes	—
19	#10622302	C₂²	1	2	yes	—
20	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
21	C₈×Dic₂₂	C₂²	1	1	yes	characteristic
22	C₈×Dic₂₂	C₂²	1	2	yes	—
23	C₁₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₁₁×C₄⋊C₁₆	C₂²	1	2	yes	—
25	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₁₁×C₁₆	C₂²	1	2	yes	—
27	(C₂×D₁₁)⋊C₁₆	C₂²	1	1	yes	characteristic
28	(C₂×D₁₁)⋊C₁₆	C₂²	1	1	yes	characteristic
29	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
30	#10622140	C₂²	1	2	yes	—
31	#10622140	C₂²	1	2	yes	—
32	Dic₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
33	Dic₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
34	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
35	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
36	C₁₆×Dic₁₁	C₂²	1	2	yes	—
37	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
39	#10622099	C₂²	1	2	yes	—
40	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
42	#66999	C₈	1	1	yes	characteristic
43	#66997	C₈	1	1	yes	characteristic
44	C₂×#4313	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₁₁	C₂³	1	1	yes	characteristic
47	#66950	C₈	1	1	yes	characteristic
48	#66949	C₈	1	2	yes	—
49	#66942	C₈	1	2	yes	—
50	#66941	C₈	1	1	yes	characteristic
51	#66925	C₂×C₄	1	2	yes	—
52	C₄²×D₁₁	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
54	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
58	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
59	C₁₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₁	C₂³	1	2	yes	—
62	#66879	C₂×C₄	1	2	yes	—
63	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
64	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
65	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
66	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
68	C₈×Dic₁₁	C₂³	1	2	yes	—
69	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
72	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
73	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
74	#66867	C₂×C₄	1	2	yes	—
75	C₄×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
76	#33499	—	11	1	no	maximal, Sylow
77	C₂×Q₈×C₁₁	C₂×C₈	1	1	yes	characteristic
78	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₁₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₂₂	C₂×C₈	1	1	yes	characteristic
82	C₂×Dic₂₂	C₂×C₈	1	2	yes	—
83	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
86	C₁₁×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
87	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
88	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
90	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
91	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
92	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
93	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
94	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
95	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
96	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
97	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
98	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
99	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
100	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
102	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
103	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
106	#4313	—	4	2	no	—
107	C₈×D₁₁	—	2	1	no	—
108	C₈×D₁₁	—	2	1	no	—
109	C₈×D₁₁	—	4	1	no	—
110	C₁₁×C₁₆	—	2	1	no	—
111	C₁₁×C₁₆	—	2	1	no	—
112	C₁₁×C₁₆	—	2	2	no	—
113	C₁₁⋊C₁₆	—	2	1	no	—
114	C₁₁⋊C₁₆	—	2	2	no	—
115	C₁₁⋊C₁₆	—	2	1	no	—
116	#2960	—	11	1	no	—
117	#2183	—	11	1	no	—
118	#2183	—	11	1	no	—
119	#2183	—	11	2	no	—
120	#2183	—	11	2	no	—
121	Q₈×C₁₆	—	11	1	no	—
122	Q₈×C₁₆	D₁₁	1	1	yes	characteristic, complemented
123	#2164	—	11	2	no	—
124	#2164	—	11	2	no	—
125	#2163	—	11	1	no	—
126	#2162	—	11	1	no	—
127	C₂²×D₁₁	C₂²×C₈	1	1	yes	characteristic
128	Q₈×C₁₁	—	2	2	no	—
129	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
134	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
135	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
136	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
137	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
138	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
139	C₄×D₁₁	—	2	1	no	—
140	C₄×D₁₁	—	2	1	no	—
141	C₄×D₁₁	—	2	2	no	—
142	C₄×D₁₁	—	4	1	no	—
143	Dic₂₂	—	2	2	no	—
144	Dic₂₂	—	4	2	no	—
145	C₈×C₁₁	—	2	1	no	—
146	C₈×C₁₁	—	2	2	no	—
147	C₈×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
148	C₈×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
149	C₁₁⋊C₈	—	2	1	no	—
150	C₁₁⋊C₈	—	2	1	no	—
151	C₁₁⋊C₈	—	2	2	no	—
152	#533	—	11	1	no	—
153	#446	—	11	1	no	—
154	#446	—	11	2	no	—
155	#446	—	11	1	no	—
156	C₈×Q₈	—	11	1	no	—
157	C₈×Q₈	—	11	2	no	—
158	C₈×Q₈	C₂×D₁₁	1	1	yes	characteristic
159	#424	—	11	2	no	—
160	#423	—	11	1	no	—
161	#405	—	11	2	no	—
162	C₂×C₄×C₈	—	11	1	no	—
163	C₄⋊C₁₆	—	11	1	no	—
164	C₄⋊C₁₆	—	11	1	no	—
165	C₄⋊C₁₆	—	11	2	no	—
166	C₄⋊C₁₆	—	11	2	no	—
167	C₄⋊C₁₆	—	11	1	no	—
168	C₄⋊C₁₆	—	11	2	no	—
169	C₄⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
170	C₄⋊C₁₆	C₂×D₁₁	1	2	yes	—
171	C₂²⋊C₁₆	—	11	1	no	—
172	C₂²⋊C₁₆	—	11	1	no	—
173	C₂²⋊C₁₆	—	11	2	no	—
174	#346	—	11	2	no	—
175	#346	—	11	2	no	—
176	#346	—	11	2	no	—
177	C₄×C₁₆	—	11	2	no	—
178	C₄×C₁₆	—	11	1	no	—
179	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
180	C₄×C₁₆	C₂×D₁₁	1	2	yes	—
181	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
182	C₂×D₁₁	—	2	1	no	—
183	C₂×D₁₁	—	2	1	no	—
184	C₂×D₁₁	—	2	1	no	—
185	C₄×C₁₁	—	2	2	no	—
186	C₄×C₁₁	—	2	2	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	#504	1	2	yes	—
189	C₄×C₁₁	#519	1	1	yes	characteristic
190	C₄×C₁₁	#568	1	1	yes	characteristic
191	Dic₁₁	—	2	2	no	—
192	Dic₁₁	—	2	1	no	—
193	Dic₁₁	—	2	1	no	—
194	Dic₁₁	—	2	2	no	—
195	Dic₁₁	—	2	1	no	—
196	C₂×M₁₆	—	11	2	no	—
197	C₂²×C₈	—	11	1	no	—
198	C₂²×C₈	—	11	1	no	—
199	C₄⋊Q₈	—	11	1	no	—
200	C₄⋊Q₈	—	11	1	no	—
201	#125	—	11	2	no	—
202	C₂²⋊Q₈	—	11	1	no	—
203	C₂²⋊Q₈	—	11	2	no	—
204	C₂²⋊Q₈	—	11	1	no	—
205	C₄×Q₈	—	11	2	no	—
206	C₄×Q₈	—	11	1	no	—
207	C₄×Q₈	C₄×D₁₁	1	1	yes	characteristic
208	#117	—	11	2	no	—
209	C₂×C₄²	—	11	1	no	—
210	C₂×C₁₆	—	11	1	no	—
211	C₂×C₁₆	—	11	2	no	—
212	C₂×C₁₆	—	11	1	no	—
213	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
214	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
215	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
216	C₄⋊C₈	—	11	2	no	—
217	C₄⋊C₈	—	11	2	no	—
218	C₄⋊C₈	—	11	2	no	—
219	C₄⋊C₈	—	11	1	no	—
220	C₄⋊C₈	—	11	2	no	—
221	C₄⋊C₈	C₄×D₁₁	1	1	yes	characteristic
222	C₄⋊C₈	C₄×D₁₁	1	2	yes	—
223	#97	—	11	2	no	—
224	#97	—	11	2	no	—
225	C₄×C₈	—	11	1	no	—
226	C₄×C₈	—	11	2	no	—
227	C₄×C₈	—	11	1	no	—
228	C₄×C₈	—	11	1	no	—
229	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₁₁	1	2	yes	—
231	C₂×C₁₁	#2982	1	1	yes	characteristic
232	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
233	C₂×C₁₁	#3407	1	1	yes	characteristic
234	D₁₁	—	4	1	no	—
235	C₂×Q₈	—	11	1	no	—
236	C₂×Q₈	—	11	2	no	—
237	C₂×Q₈	C₈×D₁₁	1	1	yes	characteristic
238	C₂²×C₄	—	11	1	no	—
239	C₂²×C₄	—	11	1	no	—
240	C₂²×C₄	—	11	1	no	—
241	M₁₆	—	44	2	no	—
242	C₂×C₈	—	11	1	no	—
243	C₂×C₈	—	11	1	no	—
244	C₂×C₈	—	11	2	no	—
245	C₂×C₈	—	22	1	no	—
246	C₂×C₈	—	22	1	no	—
247	C₂×C₈	—	44	1	no	—
248	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
250	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
251	C₄⋊C₄	—	11	2	no	—
252	C₄⋊C₄	—	11	2	no	—
253	C₄⋊C₄	—	11	2	no	—
254	C₄⋊C₄	—	11	2	no	—
255	C₄⋊C₄	—	11	2	no	—
256	C₄⋊C₄	—	11	2	no	—
257	C₄⋊C₄	—	11	1	no	—
258	C₄⋊C₄	C₈×D₁₁	1	2	yes	—
259	C₄⋊C₄	C₈×D₁₁	1	1	yes	characteristic
260	C₂²⋊C₄	—	11	2	no	—
261	C₂²⋊C₄	—	11	2	no	—
262	C₄²	—	11	1	no	—
263	C₄²	—	11	1	no	—
264	C₄²	—	11	1	no	—
265	C₄²	—	11	2	no	—
266	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
267	C₄²	C₂×C₄×D₁₁	1	2	yes	—
268	C₁₆	—	2	1	no	—
269	C₁₆	—	2	1	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	—	22	1	no	—
272	C₁₆	—	22	2	no	—
273	C₁₆	—	22	1	no	—
274	C₁₁	#33499	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	11	1	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	22	2	no	—
278	Q₈	—	44	2	no	—
279	C₂×C₄	—	11	1	no	—
280	C₂×C₄	—	11	1	no	—
281	C₂×C₄	—	11	1	no	—
282	C₂×C₄	—	11	2	no	—
283	C₂×C₄	—	11	2	no	—
284	C₂×C₄	—	22	1	no	—
285	C₂×C₄	—	22	2	no	—
286	C₂×C₄	—	22	1	no	—
287	C₂×C₄	—	44	1	no	—
288	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
289	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
290	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
291	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	2	no	—
295	C₈	—	22	1	no	—
296	C₈	—	22	2	no	—
297	C₈	—	22	1	no	—
298	C₈	#67040	1	1	yes	characteristic
299	C₈	#67043	1	1	yes	characteristic
300	C₂²	—	22	1	no	—
301	C₂²	—	22	1	no	—
302	C₂²	—	22	1	no	—
303	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	22	1	no	—
308	C₄	—	22	2	no	—
309	C₄	—	22	1	no	—
310	C₄	—	22	2	no	—
311	C₄	—	22	1	no	—
312	C₄	#10622514	1	2	yes	—
313	C₄	#10623103	1	1	yes	characteristic
314	C₄	#10623281	1	1	yes	characteristic
315	C₂	—	44	1	no	—
316	C₂	#13077695	1	1	yes	characteristic
317	C₂	C₂×#10622514	1	1	yes	characteristic
318	C₂	#13081717	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	Q₈×C₁₁×C₁₆	C₂
prime radical	C₂²×D₁₁	C₂²×C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	Q₈×C₁₆	D₁₁
hypocenter	C₁₁	#33499
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13070525	C₂	semidirect
3. #13070524	C₂	semidirect
4. #13070523	C₂	semidirect
5. #13070521	C₂	semidirect
6. C₁₆×Dic₂₂	C₂	semidirect
11. Q₈×C₁₁×C₁₆	C₂	semidirect
122. Q₈×C₁₆	D₁₁	semidirect
274. C₁₁	#33499	semidirect

Maximal subgroups

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

subgroup	index
#13077545	2
#13070525	2
#13070524	2
#13070523	2
#13070521	2
C₁₆×Dic₂₂	2
#13070422	2
#13070421	2
#13070418	2
#13070413	2
Q₈×C₁₁×C₁₆	2
#33499	11