Subgroups — #14725552

Order 2816.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079202	C₂	1	1	yes	maximal, characteristic
2	#13077563	C₂	1	2	yes	maximal
3	#13067514	C₂	1	2	yes	maximal, complemented
4	#13067468	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₄₄	C₂	1	2	yes	maximal, complemented
6	C₂×#10622096	C₂	1	1	yes	maximal, characteristic
7	#13067020	C₂	1	2	yes	maximal
8	C₈×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
9	#10623001	C₂²	1	2	yes	—
10	C₁₁×#431	C₂²	1	1	yes	characteristic
11	C₄×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	#10622735	—	2	2	no	—
14	#10622732	—	2	2	no	—
15	#10622729	C₄	1	2	yes	—
16	#10622729	C₄	1	2	yes	—
17	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
18	#10622314	—	2	4	no	—
19	#10622312	C₄	1	1	yes	characteristic
20	C₄×Dic₄₄	C₂²	1	2	yes	—
21	C₄×Dic₄₄	C₂²	1	2	yes	—
22	#10622295	C₄	1	1	yes	characteristic
23	#10622294	C₄	1	2	yes	—
24	#10622292	C₂²	1	4	yes	—
25	C₈×Dic₂₂	C₂²	1	4	yes	—
26	C₁₁×#322	C₂²	1	2	yes	—
27	C₈²×C₁₁	C₂²	1	2	yes	—
28	#10622096	C₂²	1	1	yes	characteristic
29	#10622096	C₂²	1	2	yes	—
30	#10622096	C₂²	1	1	yes	characteristic
31	#10622094	C₂²	1	4	yes	—
32	#10622094	C₂²	1	4	yes	—
33	C₂×C₁₁×M₁₆	D₄	1	1	yes	characteristic
34	C₂×C₁₁×M₁₆	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
38	#66979	—	2	2	no	—
39	#66977	—	2	2	no	—
40	#66977	C₂×C₄	1	4	yes	—
41	C₂²×C₁₁⋊C₈	—	2	2	no	—
42	C₂×Dic₄₄	C₂×C₄	1	2	yes	—
43	C₂×Dic₄₄	C₂×C₄	1	2	yes	—
44	#66923	—	2	2	no	—
45	#66922	C₂×C₄	1	2	yes	—
46	#66922	C₂×C₄	1	2	yes	—
47	C₄×Dic₂₂	—	2	2	no	—
48	C₄×Dic₂₂	C₂³	1	4	yes	—
49	C₁₁×C₄⋊C₈	—	2	1	no	—
50	C₁₁×C₂²⋊C₈	—	2	1	no	—
51	C₁₁×#97	D₄	1	2	yes	—
52	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₁	C₂³	1	2	yes	—
55	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂³	1	2	yes	—
58	#66882	C₂×C₄	1	2	yes	—
59	#66882	C₂×C₄	1	2	yes	—
60	#66880	C₂×C₄	1	4	yes	—
61	#66880	C₂×C₄	1	4	yes	—
62	#66879	—	2	4	no	—
63	Dic₁₁⋊C₈	—	2	4	no	—
64	Dic₁₁⋊C₈	—	2	4	no	—
65	C₈×Dic₁₁	—	2	4	no	—
66	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
67	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
68	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
69	#18068	—	11	1	no	maximal, Sylow
70	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
73	C₂×Dic₂₂	—	2	2	no	—
74	C₂×Dic₂₂	C₂²×C₄	1	4	yes	—
75	C₁₁×M₁₆	—	2	2	no	—
76	C₁₁×M₁₆	Q₁₆	1	4	yes	complemented
77	C₂×C₈×C₁₁	—	2	2	no	—
78	C₂×C₈×C₁₁	—	2	2	no	—
79	C₂×C₈×C₁₁	—	4	1	no	—
80	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
86	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₁	#41	1	1	yes	characteristic
89	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
90	C₂²⋊Dic₁₁	—	2	4	no	—
91	C₄⋊Dic₁₁	—	2	2	no	—
92	C₄⋊Dic₁₁	—	2	2	no	—
93	C₄⋊Dic₁₁	—	2	2	no	—
94	C₄⋊Dic₁₁	C₂²×C₄	1	4	yes	—
95	Dic₁₁⋊C₄	—	2	4	no	—
96	Dic₁₁⋊C₄	—	2	4	no	—
97	C₄×Dic₁₁	—	2	4	no	—
98	C₂×C₁₁⋊C₈	—	2	2	no	—
99	C₂×C₁₁⋊C₈	—	2	2	no	—
100	C₂×C₁₁⋊C₈	—	4	4	no	—
101	Dic₄₄	—	2	4	no	—
102	Dic₄₄	—	4	2	no	—
103	#3170	—	11	1	no	—
104	#2962	—	11	2	no	—
105	#1591	—	11	2	no	—
106	#1584	—	11	4	no	—
107	C₈×Q₁₆	—	11	2	no	—
108	C₂×#335	—	11	1	no	—
109	#1481	—	11	2	no	—
110	C₈×M₁₆	D₁₁	1	1	yes	characteristic
111	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₁	—	2	1	no	—
113	C₂×C₄×C₁₁	—	2	1	no	—
114	C₂×C₄×C₁₁	—	4	1	no	—
115	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
119	C₂×C₄×C₁₁	#135	1	2	yes	—
120	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₁₁	—	2	4	no	—
124	C₂×Dic₁₁	—	2	4	no	—
125	Dic₂₂	—	2	4	no	—
126	Dic₂₂	—	2	4	no	—
127	Dic₂₂	—	8	2	no	—
128	C₈×C₁₁	—	2	1	no	—
129	C₈×C₁₁	—	2	1	no	—
130	C₈×C₁₁	—	2	1	no	—
131	C₈×C₁₁	—	2	2	no	—
132	C₈×C₁₁	—	4	2	no	—
133	C₈×C₁₁	#131	1	2	yes	—
134	C₈×C₁₁	C₂×Q₁₆	1	4	yes	—
135	C₁₁⋊C₈	—	4	2	no	—
136	C₁₁⋊C₈	—	4	2	no	—
137	#533	—	11	2	no	—
138	#500	—	11	1	no	—
139	#494	—	11	1	no	—
140	#487	—	11	2	no	—
141	#467	—	11	2	no	—
142	#467	—	11	2	no	—
143	#446	—	11	4	no	—
144	#446	—	22	4	no	—
145	C₈×Q₈	—	11	4	no	—
146	C₄×Q₁₆	—	11	2	no	—
147	C₄×Q₁₆	—	11	2	no	—
148	#431	C₂×D₁₁	1	1	yes	characteristic
149	#424	—	22	2	no	—
150	C₂×C₄⋊C₈	—	11	2	no	—
151	#405	—	22	2	no	—
152	C₄×M₁₆	C₂×D₁₁	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
154	#335	—	11	1	no	—
155	#335	—	11	2	no	—
156	#335	—	11	1	no	—
157	Q₈⋊C₈	—	11	4	no	—
158	Q₈⋊C₈	—	11	4	no	—
159	#322	C₂×D₁₁	1	2	yes	—
160	C₈²	C₂×D₁₁	1	2	yes	—
161	C₂²×C₁₁	—	2	1	no	—
162	C₂²×C₁₁	—	2	1	no	—
163	C₂²×C₁₁	C₄×Q₁₆	1	2	yes	—
164	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
165	C₄×C₁₁	—	2	1	no	—
166	C₄×C₁₁	—	2	1	no	—
167	C₄×C₁₁	—	2	1	no	—
168	C₄×C₁₁	C₄×Q₁₆	1	2	yes	—
169	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
170	C₄×C₁₁	#568	1	1	yes	characteristic
171	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
172	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
173	Dic₁₁	—	4	4	no	—
174	Dic₁₁	—	4	4	no	—
175	C₂×Q₁₆	—	11	2	no	—
176	C₂×Q₁₆	—	11	2	no	—
177	C₂×M₁₆	D₄₄	1	1	yes	characteristic
178	C₂×M₁₆	D₄₄	1	1	yes	characteristic
179	C₂²×C₈	—	22	2	no	—
180	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
181	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
182	C₄⋊Q₈	—	11	2	no	—
183	C₄⋊Q₈	—	11	2	no	—
184	#125	—	22	2	no	—
185	C₂²⋊Q₈	—	11	4	no	—
186	C₂²⋊Q₈	—	22	2	no	—
187	C₄×Q₈	—	11	4	no	—
188	C₄×Q₈	—	22	2	no	—
189	#117	—	22	2	no	—
190	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
191	#107	—	11	2	no	—
192	#107	—	11	2	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	11	2	no	—
195	C₄⋊C₈	—	11	2	no	—
196	C₄⋊C₈	—	22	4	no	—
197	C₄⋊C₈	—	22	4	no	—
198	C₄⋊C₈	—	22	2	no	—
199	#103	—	11	4	no	—
200	#103	—	11	4	no	—
201	C₂²⋊C₈	—	2	1	no	—
202	#97	—	22	4	no	—
203	#97	D₄₄	1	2	yes	—
204	C₄×C₈	—	22	4	no	—
205	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
206	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₁₁	1	2	yes	—
208	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₁₁	1	2	yes	—
211	C₂×C₁₁	—	2	2	no	—
212	C₂×C₁₁	#2935	1	1	yes	characteristic
213	C₂×C₁₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
214	C₂×C₁₁	#2957	1	1	yes	characteristic
215	C₂×Q₈	—	11	4	no	—
216	C₂×Q₈	—	22	2	no	—
217	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
219	C₂²×C₄	#4339	1	1	yes	characteristic
220	Q₁₆	—	22	4	no	—
221	Q₁₆	—	44	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	Dic₄₄	1	4	yes	complemented
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	22	2	no	—
228	C₂×C₈	—	22	2	no	—
229	C₂×C₈	—	44	4	no	—
230	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
234	C₂×C₈	C₂×D₄₄	1	2	yes	—
235	C₂×C₈	C₂×D₄₄	1	2	yes	—
236	C₄⋊C₄	—	11	4	no	—
237	C₄⋊C₄	—	22	2	no	—
238	C₄⋊C₄	—	22	2	no	—
239	C₄⋊C₄	—	22	4	no	—
240	C₄⋊C₄	—	22	2	no	—
241	C₄⋊C₄	—	22	4	no	—
242	C₂²⋊C₄	—	22	4	no	—
243	C₄²	—	22	4	no	—
244	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
245	C₄²	C₂×D₄₄	1	1	yes	characteristic
246	C₄²	#4339	1	1	yes	characteristic
247	C₄²	C₂³×D₁₁	1	1	yes	characteristic
248	C₁₁	#18068	1	1	yes	characteristic, complemented, Sylow
249	C₂³	C₄×D₄₄	1	1	yes	characteristic
250	Q₈	—	22	4	no	—
251	Q₈	—	22	4	no	—
252	Q₈	—	88	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	22	4	no	—
257	C₂×C₄	—	22	4	no	—
258	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
259	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
260	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
261	C₂×C₄	#66957	1	2	yes	—
262	C₂×C₄	C₂×Dic₄₄	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
265	C₂×C₄	C₂×#4339	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	4	2	no	—
271	C₈	—	44	2	no	—
272	C₈	—	44	2	no	—
273	C₈	C₂×Dic₄₄	1	4	yes	—
274	C₈	#66960	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₄×Dic₄₄	1	2	yes	—
278	C₂²	C₂×C₄×D₄₄	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	44	4	no	—
283	C₄	—	44	4	no	—
284	C₄	C₄×Dic₄₄	1	2	yes	—
285	C₄	C₂×#66957	1	1	yes	characteristic
286	C₄	C₂²×Dic₄₄	1	1	yes	characteristic
287	C₄	C₂×#66960	1	1	yes	characteristic
288	C₄	#10623276	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#13077362	1	1	yes	characteristic
291	C₂	C₂×C₄×Dic₄₄	1	1	yes	characteristic
292	C₂	#13077534	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₈×C₁₁×M₁₆	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂³×C₁₁	C₄×D₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₈×M₁₆	D₁₁
hypocenter	C₁₁	#18068
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13067514	C₂	semidirect
4. #13067468	C₂	semidirect
5. C₈×Dic₄₄	C₂	semidirect
76. C₁₁×M₁₆	Q₁₆	semidirect
223. M₁₆	Dic₄₄	semidirect
248. C₁₁	#18068	semidirect

Maximal subgroups

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

subgroup	index
#13079202	2
#13077563	2
#13067514	2
#13067468	2
C₈×Dic₄₄	2
C₂×#10622096	2
#13067020	2
C₈×C₁₁×M₁₆	2
#18068	11