Subgroups — #14738494

Order 2816.

568 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13080007	C₂	1	2	yes	maximal
2	#13077550	C₂	1	1	yes	maximal, characteristic
3	#13068131	C₂	1	2	yes	maximal, complemented
4	#13068107	C₂	1	2	yes	maximal, complemented
5	#13068043	C₂	1	4	yes	maximal, complemented
6	#13067372	C₂	1	1	yes	maximal, characteristic
7	#13067354	C₂	1	2	yes	maximal
8	C₁₁×#3170	C₂	1	1	yes	maximal, characteristic
9	C₁₁×#533	C₂²	1	2	yes	—
10	#10623001	C₂²	1	1	yes	characteristic
11	C₁₁×#500	C₂²	1	1	yes	characteristic
12	C₁₁×#494	C₂²	1	1	yes	characteristic
13	C₁₁×#487	C₂²	1	2	yes	—
14	C₁₁×#467	—	2	1	no	—
15	C₁₁×#467	—	2	1	no	—
16	C₄×C₁₁×Q₁₆	—	2	1	no	—
17	C₄×C₁₁×Q₁₆	—	2	1	no	—
18	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
19	#10622715	—	2	1	no	—
20	#10622714	—	2	1	no	—
21	#10622707	C₂²	1	4	yes	—
22	#10622703	C₂²	1	4	yes	—
23	#10622665	—	2	4	no	—
24	#10622644	—	2	2	no	—
25	#10622633	—	2	2	no	—
26	#10622537	C₂²	1	1	yes	characteristic
27	#10622537	C₂²	1	2	yes	—
28	#10622292	—	2	1	no	—
29	#10622292	C₂²	1	2	yes	—
30	C₈×Dic₂₂	—	2	1	no	—
31	C₈×Dic₂₂	C₂²	1	2	yes	—
32	#10622185	C₂²	1	4	yes	—
33	#10622185	C₂²	1	4	yes	—
34	#10622095	C₂²	1	2	yes	—
35	#10622095	C₂²	1	2	yes	—
36	C₂×C₁₁×Q₁₆	—	2	1	no	—
37	C₂×C₁₁×Q₁₆	—	2	1	no	—
38	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
40	C₁₁×C₄⋊Q₈	C₂³	1	2	yes	—
41	C₁₁×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₁₁×#125	—	2	2	no	—
43	C₁₁×C₂²⋊Q₈	—	2	2	no	—
44	C₁₁×C₂²⋊Q₈	—	2	2	no	—
45	C₄×Q₈×C₁₁	—	2	2	no	—
46	C₄×Q₈×C₁₁	—	2	2	no	—
47	C₁₁×#117	—	2	2	no	—
48	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
49	C₂×#4326	—	2	4	no	—
50	#66979	—	2	1	no	—
51	#66977	—	2	1	no	—
52	#66977	D₄	1	2	yes	—
53	C₂×#4318	—	2	1	no	—
54	C₂×#4318	—	2	2	no	—
55	#66923	—	2	1	no	—
56	#66922	D₄	1	1	yes	characteristic
57	#66922	D₄	1	1	yes	characteristic
58	C₄×Dic₂₂	—	2	1	no	—
59	C₄×Dic₂₂	C₂³	1	2	yes	—
60	C₁₁×#107	—	2	1	no	—
61	C₁₁×#107	—	2	1	no	—
62	C₁₁×#103	—	2	2	no	—
63	C₁₁×#103	—	2	2	no	—
64	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
65	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
66	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
68	#66899	—	2	4	no	—
69	#66879	—	2	2	no	—
70	Dic₁₁⋊C₈	—	2	2	no	—
71	Dic₁₁⋊C₈	—	2	2	no	—
72	C₈×Dic₁₁	—	2	2	no	—
73	#66874	—	2	4	no	—
74	#66872	—	2	4	no	—
75	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
76	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
77	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
78	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
79	#19350	—	11	1	no	maximal, Sylow
80	C₂×Q₈×C₁₁	—	2	2	no	—
81	C₂×Q₈×C₁₁	—	2	2	no	—
82	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
85	C₂×Dic₂₂	—	2	1	no	—
86	C₂×Dic₂₂	C₂×D₄	1	2	yes	—
87	C₁₁×Q₁₆	—	4	2	no	—
88	C₁₁×Q₁₆	—	4	2	no	—
89	C₂×C₈×C₁₁	—	2	2	no	—
90	C₂×C₈×C₁₁	—	4	1	no	—
91	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
95	C₁₁×C₄⋊C₄	—	2	2	no	—
96	C₁₁×C₄⋊C₄	—	2	2	no	—
97	C₁₁×C₄⋊C₄	—	2	2	no	—
98	C₁₁×C₄⋊C₄	—	2	2	no	—
99	C₁₁×C₄⋊C₄	—	4	2	no	—
100	C₁₁×C₄⋊C₄	—	4	2	no	—
101	C₁₁×C₂²⋊C₄	—	4	2	no	—
102	C₄²×C₁₁	—	4	2	no	—
103	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₁	#41	1	1	yes	characteristic
106	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₁₁	—	2	2	no	—
108	#4326	—	8	4	no	—
109	C₄⋊Dic₁₁	—	2	1	no	—
110	C₄⋊Dic₁₁	—	2	1	no	—
111	C₄⋊Dic₁₁	—	2	1	no	—
112	C₄⋊Dic₁₁	C₂×D₄	1	2	yes	—
113	Dic₁₁⋊C₄	—	2	2	no	—
114	Dic₁₁⋊C₄	—	2	2	no	—
115	C₄×Dic₁₁	—	2	2	no	—
116	#4318	—	4	2	no	—
117	#4318	—	8	2	no	—
118	C₂×C₁₁⋊C₈	—	2	4	no	—
119	C₂×C₁₁⋊C₈	—	2	2	no	—
120	#3255	—	11	2	no	—
121	#3170	D₁₁	1	1	yes	characteristic
122	#2960	—	11	1	no	—
123	#1694	—	11	2	no	—
124	#1690	—	11	2	no	—
125	#1680	—	11	4	no	—
126	#1566	—	11	1	no	—
127	#1561	—	11	2	no	—
128	C₂³×C₁₁	#121	1	1	yes	characteristic
129	Q₈×C₁₁	—	4	2	no	—
130	Q₈×C₁₁	—	4	2	no	—
131	Q₈×C₁₁	—	4	2	no	—
132	Q₈×C₁₁	—	4	2	no	—
133	C₂×C₄×C₁₁	—	2	1	no	—
134	C₂×C₄×C₁₁	—	2	1	no	—
135	C₂×C₄×C₁₁	—	2	2	no	—
136	C₂×C₄×C₁₁	—	2	2	no	—
137	C₂×C₄×C₁₁	—	4	2	no	—
138	C₂×C₄×C₁₁	—	4	2	no	—
139	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₁₁	—	2	2	no	—
146	C₂×Dic₁₁	—	2	2	no	—
147	Dic₂₂	—	2	4	no	—
148	Dic₂₂	—	4	2	no	—
149	C₈×C₁₁	—	2	1	no	—
150	C₈×C₁₁	—	2	1	no	—
151	C₈×C₁₁	—	2	1	no	—
152	C₈×C₁₁	#137	1	2	yes	—
153	C₁₁⋊C₈	—	4	2	no	—
154	C₁₁⋊C₈	—	4	4	no	—
155	#533	—	11	1	no	—
156	#533	C₂×D₁₁	1	2	yes	—
157	#500	C₂×D₁₁	1	1	yes	characteristic
158	#494	C₂×D₁₁	1	1	yes	characteristic
159	#487	C₂×D₁₁	1	2	yes	—
160	#475	—	11	4	no	—
161	#475	—	22	2	no	—
162	#470	—	22	4	no	—
163	#467	—	2	1	no	—
164	#467	—	2	1	no	—
165	#462	—	11	4	no	—
166	#462	—	22	2	no	—
167	#446	—	11	2	no	—
168	#446	—	22	1	no	—
169	C₈×Q₈	—	11	2	no	—
170	C₈×Q₈	—	22	1	no	—
171	C₄×Q₁₆	—	2	1	no	—
172	C₄×Q₁₆	—	2	1	no	—
173	#424	—	22	1	no	—
174	#423	—	11	1	no	—
175	#423	—	11	2	no	—
176	#405	—	22	1	no	—
177	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
178	#334	—	11	2	no	—
179	#334	—	11	2	no	—
180	#332	—	11	4	no	—
181	#332	—	11	4	no	—
182	C₂²×C₁₁	—	2	1	no	—
183	C₂²×C₁₁	—	2	1	no	—
184	C₂²×C₁₁	—	2	1	no	—
185	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
186	C₄×C₁₁	—	2	1	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	—	2	1	no	—
189	C₄×C₁₁	—	8	2	no	—
190	C₄×C₁₁	—	8	2	no	—
191	C₄×C₁₁	#473	1	2	yes	—
192	C₄×C₁₁	C₂×#137	1	1	yes	characteristic
193	C₄×C₁₁	#575	1	1	yes	characteristic
194	C₄×C₁₁	#577	1	2	yes	—
195	Dic₁₁	—	4	2	no	—
196	Dic₁₁	—	4	2	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	—	22	4	no	—
200	C₂×M₁₆	—	22	1	no	—
201	C₂×M₁₆	—	22	2	no	—
202	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
203	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
204	C₄⋊Q₈	—	11	1	no	—
205	C₄⋊Q₈	—	11	1	no	—
206	C₄⋊Q₈	C₂²×D₁₁	1	2	yes	—
207	C₄⋊Q₈	C₂²×D₁₁	1	2	yes	—
208	#125	—	2	2	no	—
209	#125	—	22	1	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₂²⋊Q₈	—	2	2	no	—
212	C₂²⋊Q₈	—	11	2	no	—
213	C₂²⋊Q₈	—	22	1	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	—	11	2	no	—
217	C₄×Q₈	—	22	1	no	—
218	#117	—	2	2	no	—
219	#117	—	22	1	no	—
220	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
221	#107	—	2	1	no	—
222	#107	—	2	1	no	—
223	#106	—	22	4	no	—
224	C₄⋊C₈	—	11	2	no	—
225	C₄⋊C₈	—	11	4	no	—
226	C₄⋊C₈	—	22	2	no	—
227	C₄⋊C₈	—	22	2	no	—
228	C₄⋊C₈	—	22	2	no	—
229	C₄⋊C₈	—	22	1	no	—
230	#103	—	2	2	no	—
231	#103	—	2	2	no	—
232	#103	—	22	4	no	—
233	#103	—	22	4	no	—
234	#97	—	22	2	no	—
235	C₄×C₈	—	22	2	no	—
236	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
237	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
240	C₂×C₁₁	—	4	1	no	—
241	C₂×C₁₁	#3037	1	1	yes	characteristic
242	C₂×C₁₁	#3060	1	1	yes	characteristic
243	C₂×C₁₁	C₂×#473	1	1	yes	characteristic
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	11	2	no	—
247	C₂×Q₈	—	22	1	no	—
248	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
249	C₂²×C₄	#4341	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
251	Q₁₆	—	4	2	no	—
252	Q₁₆	—	4	2	no	—
253	Q₁₆	—	88	4	no	—
254	M₁₆	—	44	2	no	—
255	M₁₆	—	88	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	22	4	no	—
259	C₂×C₈	—	22	2	no	—
260	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	11	2	no	—
271	C₄⋊C₄	—	22	1	no	—
272	C₄⋊C₄	—	22	1	no	—
273	C₄⋊C₄	—	22	2	no	—
274	C₄⋊C₄	—	22	1	no	—
275	C₄⋊C₄	—	22	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	22	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	22	2	no	—
280	C₄²	D₄×D₁₁	1	1	yes	characteristic
281	C₄²	#4341	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
283	C₄²	C₂³×D₁₁	1	1	yes	characteristic
284	C₁₁	#19350	1	1	yes	characteristic, complemented, Sylow
285	C₂³	#66991	1	1	yes	characteristic
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	22	4	no	—
291	Q₈	—	44	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	22	2	no	—
299	C₂×C₄	—	22	2	no	—
300	C₂×C₄	#66991	1	1	yes	characteristic
301	C₂×C₄	#66991	1	1	yes	characteristic
302	C₂×C₄	#66991	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
304	C₂×C₄	C₂×#4341	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	44	4	no	—
310	C₈	—	44	2	no	—
311	C₈	#67004	1	2	yes	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	C₂×#66991	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	44	2	no	—
322	C₄	—	44	2	no	—
323	C₄	#10622805	1	2	yes	—
324	C₄	#10623301	1	2	yes	—
325	C₄	#10623353	1	1	yes	characteristic
326	C₄	C₂×#67004	1	1	yes	characteristic
327	C₂	—	4	1	no	—
328	C₂	#13078084	1	1	yes	characteristic
329	C₂	#13078274	1	1	yes	characteristic
330	C₂	C₂×#10622805	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#66991
commutator, cosocle	C₄²×C₁₁	C₂⁴
Frattini	C₄²	C₂³×D₁₁
Fitting	C₁₁×#3170	C₂
socle	C₂²×C₁₁	C₂×#121
prime radical	C₂³×C₁₁	#121
hypercenter	#3170	D₁₁
hypocenter	C₁₁	#19350
LCS	G ▷ C₄²×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #13068131	C₂	semidirect
4. #13068107	C₂	semidirect
5. #13068043	C₂	semidirect
284. C₁₁	#19350	semidirect

Maximal subgroups

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

subgroup	index
#13080007	2
#13077550	2
#13068131	2
#13068107	2
#13068043	2
#13067372	2
#13067354	2
C₁₁×#3170	2
#19350	11