Subgroups — #14909280

Order 2816.

474 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	#13079153	C₂	1	2	yes	maximal
2	#13070903	C₂	1	2	yes	maximal, complemented
3	#13070901	C₂	1	2	yes	maximal, complemented
4	#13070893	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070891	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070890	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070403	C₂	1	2	yes	maximal
8	#13070397	C₂	1	1	yes	maximal, characteristic
9	D₁₁×#366	C₂	1	1	yes	maximal, characteristic
10	#13070211	C₂	1	1	yes	maximal, characteristic
11	C₁₁×#2253	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623220	—	2	2	no	—
13	C₁₁×#499	C₂²	1	2	yes	—
14	#10622678	—	2	2	no	—
15	C₂×#66862	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₁₁	C₂²	1	1	yes	characteristic
17	#10622493	C₂²	1	2	yes	—
18	D₁₁×#107	C₂²	1	2	yes	—
19	#10622491	C₂²	1	2	yes	—
20	#10622487	C₂²	1	2	yes	—
21	#10622474	—	2	2	no	—
22	#10622469	—	2	2	no	—
23	#10622302	C₂²	1	1	yes	characteristic
24	#10622295	C₂²	1	2	yes	—
25	C₁₁×#367	C₂²	1	2	yes	—
26	C₁₁×#366	C₂²	1	1	yes	characteristic
27	C₁₁×#366	C₂²	1	1	yes	characteristic
28	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
29	#10622142	C₂²	1	1	yes	characteristic
30	#10622142	C₂²	1	1	yes	characteristic
31	#10622142	C₂²	1	2	yes	—
32	#10622140	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₁₁	C₂²	1	1	yes	characteristic
34	#10622128	C₂²	1	2	yes	—
35	#10622128	C₂²	1	2	yes	—
36	#10622127	C₂²	1	2	yes	—
37	#10622127	C₂²	1	2	yes	—
38	#10622099	C₂²	1	1	yes	characteristic
39	C₁₁×#125	—	2	2	no	—
40	C₂×#4313	D₄	1	1	yes	characteristic
41	C₂×C₈×D₁₁	C₂³	1	1	yes	characteristic
42	#66950	—	4	2	no	—
43	#66949	—	4	2	no	—
44	#66945	—	2	2	no	—
45	C₄⋊C₄×D₁₁	—	2	2	no	—
46	#66943	—	2	2	no	—
47	#66942	—	4	2	no	—
48	#66941	—	2	2	no	—
49	#66941	—	4	2	no	—
50	#66925	D₄	1	1	yes	characteristic
51	#66922	—	2	2	no	—
52	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
54	C₁₁×#107	C₂³	1	2	yes	—
55	C₁₁×#107	C₂³	1	2	yes	—
56	C₁₁×#106	—	2	2	no	—
57	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
58	#66882	—	2	2	no	—
59	#66882	C₂³	1	2	yes	—
60	#66882	C₂³	1	2	yes	—
61	#66879	D₄	1	1	yes	characteristic
62	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
65	#66872	—	2	2	no	—
66	#66872	—	2	2	no	—
67	#66871	—	2	2	no	—
68	#66871	—	2	2	no	—
69	#66867	D₄	1	1	yes	characteristic
70	#66862	Q₈	1	4	yes	complemented
71	C₁₆×D₁₁	Q₈	1	4	yes	complemented
72	#33729	—	11	1	no	maximal, Sylow
73	C₂×C₄×D₁₁	—	4	2	no	—
74	C₂×C₄×D₁₁	C₂×D₄	1	1	yes	characteristic
75	C₂×Dic₂₂	—	4	2	no	—
76	C₂×Dic₂₂	—	4	2	no	—
77	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
79	C₁₁×C₄⋊C₄	—	2	2	no	—
80	C₁₁×C₄⋊C₄	—	2	2	no	—
81	C₁₁×C₄⋊C₄	—	4	2	no	—
82	C₁₁×C₄⋊C₄	—	4	2	no	—
83	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
84	(C₂×D₁₁)⋊C₄	—	4	2	no	—
85	(C₂×D₁₁)⋊C₄	D₈	1	1	yes	characteristic
86	(C₂×D₁₁)⋊C₄	D₈	1	1	yes	characteristic
87	C₄⋊Dic₁₁	—	2	2	no	—
88	C₄⋊Dic₁₁	—	2	2	no	—
89	C₄⋊Dic₁₁	—	4	2	no	—
90	C₄⋊Dic₁₁	—	4	2	no	—
91	Dic₁₁⋊C₄	—	4	2	no	—
92	Dic₁₁⋊C₄	—	4	2	no	—
93	Dic₁₁⋊C₄	—	4	2	no	—
94	Dic₁₁⋊C₄	D₈	1	1	yes	characteristic
95	Dic₁₁⋊C₄	D₈	1	1	yes	characteristic
96	C₄×Dic₁₁	—	4	2	no	—
97	C₄×Dic₁₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
100	#4313	Q₁₆	1	4	yes	—
101	C₈×D₁₁	C₂×Q₈	1	2	yes	—
102	C₈×D₁₁	C₂×Q₈	1	2	yes	—
103	C₁₁×C₁₆	C₂×Q₈	1	2	yes	—
104	C₁₁×C₁₆	C₂×Q₈	1	2	yes	—
105	C₁₁⋊C₁₆	C₂×Q₈	1	2	yes	—
106	C₁₁⋊C₁₆	C₂×Q₈	1	2	yes	—
107	#3164	—	11	2	no	—
108	#2255	—	11	2	no	—
109	#2255	—	11	2	no	—
110	#2253	—	11	1	no	—
111	#2253	D₁₁	1	1	yes	characteristic, complemented
112	#2252	—	11	1	no	—
113	#2252	—	11	1	no	—
114	#2159	—	11	2	no	—
115	#2158	—	11	1	no	—
116	C₂×#366	—	11	1	no	—
117	#2108	—	11	1	no	—
118	C₂²×D₁₁	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₁₁	—	4	2	no	—
120	C₂×C₄×C₁₁	—	4	2	no	—
121	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₁₁	—	4	2	no	—
125	C₂×Dic₁₁	—	4	2	no	—
126	C₂×Dic₁₁	C₂×D₈	1	1	yes	characteristic
127	C₂×Dic₁₁	C₂×D₈	1	1	yes	characteristic
128	C₂×Dic₁₁	C₂×D₈	1	1	yes	characteristic
129	C₄×D₁₁	—	16	2	no	—
130	C₄×D₁₁	C₄⋊Q₈	1	2	yes	—
131	C₄×D₁₁	C₂×Q₁₆	1	2	yes	—
132	Dic₂₂	—	16	2	no	—
133	Dic₂₂	—	16	2	no	—
134	C₈×C₁₁	C₂×Q₁₆	1	2	yes	—
135	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
136	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
137	C₁₁⋊C₈	C₄⋊Q₈	1	2	yes	—
138	C₁₁⋊C₈	C₂×Q₁₆	1	2	yes	—
139	#544	—	22	2	no	—
140	C₂×M₃₂	—	11	1	no	—
141	C₂²×C₁₆	—	11	1	no	—
142	#501	—	22	2	no	—
143	#501	—	22	2	no	—
144	#500	—	11	2	no	—
145	#500	—	11	2	no	—
146	#499	—	11	2	no	—
147	#499	C₂×D₁₁	1	2	yes	—
148	#428	—	22	2	no	—
149	#427	—	11	2	no	—
150	C₂×#107	—	11	2	no	—
151	#405	—	11	1	no	—
152	#367	—	11	2	no	—
153	#367	—	11	2	no	—
154	#367	C₂×D₁₁	1	2	yes	—
155	#366	—	11	1	no	—
156	#366	—	11	1	no	—
157	#366	—	11	2	no	—
158	#366	—	11	2	no	—
159	#366	—	11	2	no	—
160	#366	C₂×D₁₁	1	1	yes	characteristic
161	#366	C₂×D₁₁	1	1	yes	characteristic
162	#346	—	11	1	no	—
163	#346	—	11	1	no	—
164	C₄×C₁₆	—	11	1	no	—
165	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
166	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₁₁	—	2	1	no	—
168	C₂×D₁₁	—	2	1	no	—
169	C₂×D₁₁	#500	1	2	yes	—
170	C₄×C₁₁	—	2	1	no	—
171	C₄×C₁₁	—	2	1	no	—
172	C₄×C₁₁	—	8	2	no	—
173	C₄×C₁₁	—	8	2	no	—
174	C₄×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
175	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
176	Dic₁₁	—	2	1	no	—
177	Dic₁₁	—	2	1	no	—
178	Dic₁₁	—	8	2	no	—
179	Dic₁₁	—	8	2	no	—
180	Dic₁₁	#500	1	2	yes	—
181	C₂×M₁₆	—	11	1	no	—
182	C₂²×C₈	—	11	1	no	—
183	C₄⋊Q₈	—	22	2	no	—
184	C₄⋊Q₈	—	22	2	no	—
185	C₄⋊Q₈	—	44	2	no	—
186	#125	—	2	2	no	—
187	#125	—	22	2	no	—
188	#125	—	44	2	no	—
189	C₂²⋊Q₈	—	44	2	no	—
190	C₂²⋊Q₈	—	44	2	no	—
191	#117	—	11	1	no	—
192	#117	—	22	2	no	—
193	C₂×C₄⋊C₄	—	22	2	no	—
194	M₃₂	—	11	4	no	—
195	C₂×C₁₆	—	11	1	no	—
196	C₂×C₁₆	—	11	1	no	—
197	C₂×C₁₆	—	11	4	no	—
198	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
199	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
200	#107	—	11	2	no	—
201	#107	—	11	2	no	—
202	#107	—	22	2	no	—
203	#107	—	22	2	no	—
204	#107	—	22	2	no	—
205	#107	C₂²×D₁₁	1	2	yes	—
206	#107	C₂²×D₁₁	1	2	yes	—
207	#106	—	2	2	no	—
208	#106	—	22	2	no	—
209	#106	—	22	2	no	—
210	#97	—	11	1	no	—
211	#97	—	11	1	no	—
212	C₄×C₈	—	11	1	no	—
213	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
214	C₂×C₁₁	C₂×#500	1	1	yes	characteristic
215	C₂×C₁₁	#3415	1	1	yes	characteristic
216	C₂×C₁₁	#3417	1	1	yes	characteristic
217	D₁₁	—	2	2	no	—
218	C₂×Q₈	—	44	2	no	—
219	C₂×Q₈	—	44	2	no	—
220	C₂²×C₄	—	11	1	no	—
221	C₂²×C₄	—	44	2	no	—
222	M₁₆	—	11	4	no	—
223	C₂×C₈	—	11	1	no	—
224	C₂×C₈	—	11	1	no	—
225	C₂×C₈	—	11	2	no	—
226	C₂×C₈	—	11	2	no	—
227	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
228	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	11	1	no	—
234	C₄⋊C₄	—	11	1	no	—
235	C₄⋊C₄	—	22	2	no	—
236	C₄⋊C₄	—	22	2	no	—
237	C₄⋊C₄	—	44	2	no	—
238	C₄⋊C₄	—	44	2	no	—
239	C₄⋊C₄	—	44	2	no	—
240	C₄⋊C₄	—	44	2	no	—
241	C₄⋊C₄	—	44	2	no	—
242	C₂²⋊C₄	—	11	1	no	—
243	C₂²⋊C₄	—	11	1	no	—
244	C₂²⋊C₄	—	44	2	no	—
245	C₄²	—	11	1	no	—
246	C₄²	—	44	2	no	—
247	C₄²	D₄×D₁₁	1	1	yes	characteristic
248	C₁₆	—	11	2	no	—
249	C₁₆	—	11	2	no	—
250	C₁₆	Q₈×D₁₁	1	2	yes	—
251	C₁₆	Q₈×D₁₁	1	2	yes	—
252	C₁₁	#33729	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	11	1	no	—
254	Q₈	—	176	2	no	—
255	Q₈	—	176	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	11	1	no	—
259	C₂×C₄	—	11	2	no	—
260	C₂×C₄	—	11	2	no	—
261	C₂×C₄	—	11	1	no	—
262	C₂×C₄	—	11	1	no	—
263	C₂×C₄	—	44	2	no	—
264	C₂×C₄	—	44	2	no	—
265	C₂×C₄	—	176	2	no	—
266	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
267	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
269	C₈	—	11	2	no	—
270	C₈	—	11	2	no	—
271	C₈	Q₁₆×D₁₁	1	2	yes	—
272	C₈	C₂×Q₈×D₁₁	1	1	yes	characteristic
273	C₈	C₂×Q₈×D₁₁	1	1	yes	characteristic
274	C₂²	—	11	2	no	—
275	C₂²	—	22	1	no	—
276	C₂²	—	22	1	no	—
277	C₂²	C₂×D₈×D₁₁	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	11	2	no	—
283	C₄	—	22	1	no	—
284	C₄	—	22	1	no	—
285	C₄	—	88	2	no	—
286	C₄	—	88	2	no	—
287	C₄	D₁₁×C₄⋊Q₈	1	1	yes	characteristic
288	C₄	C₂×Q₁₆×D₁₁	1	1	yes	characteristic
289	C₂	—	22	2	no	—
290	C₂	D₁₁×#500	1	1	yes	characteristic
291	C₂	#13081754	1	1	yes	characteristic
292	C₂	#13081766	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₁
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#2253	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	C₂²×D₁₁	C₂×D₈
hypercenter	#2253	D₁₁
hypocenter	C₁₁	#33729
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
2. #13070903	C₂	semidirect
3. #13070901	C₂	semidirect
4. #13070893	C₂	semidirect
5. #13070891	C₂	semidirect
6. #13070890	C₂	semidirect
11. C₁₁×#2253	C₂	semidirect
70. #66862	Q₈	semidirect
71. C₁₆×D₁₁	Q₈	semidirect
111. #2253	D₁₁	semidirect
252. C₁₁	#33729	semidirect

Maximal subgroups

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

subgroup	index
#13079153	2
#13070903	2
#13070901	2
#13070893	2
#13070891	2
#13070890	2
#13070403	2
#13070397	2
D₁₁×#366	2
#13070211	2
C₁₁×#2253	2
#33729	11