Subgroups — #14738451

Order 2816.

568 conjugacy classes of subgroups.

The subgroup lattice has 338 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	#13068167	C₂	1	4	yes	maximal, complemented
4	#13068007	C₂	1	2	yes	maximal, complemented
5	#13067983	C₂	1	2	yes	maximal, complemented
6	#13067375	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	#10622096	C₂²	1	2	yes	—
35	#10622096	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	#19345	—	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	#1700	—	11	4	no	—
124	#1674	—	11	2	no	—
125	#1670	—	11	2	no	—
126	#1567	—	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	1	no	—
136	C₂×C₄×C₁₁	—	2	1	no	—
137	C₂×C₄×C₁₁	—	2	2	no	—
138	C₂×C₄×C₁₁	—	4	2	no	—
139	C₂×C₄×C₁₁	—	4	2	no	—
140	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₁₁	—	2	2	no	—
147	C₂×Dic₁₁	—	2	2	no	—
148	Dic₂₂	—	2	2	no	—
149	Dic₂₂	—	2	2	no	—
150	Dic₂₂	—	4	2	no	—
151	C₈×C₁₁	—	2	1	no	—
152	C₈×C₁₁	—	2	1	no	—
153	C₈×C₁₁	—	2	1	no	—
154	C₈×C₁₁	#137	1	2	yes	—
155	C₁₁⋊C₈	—	4	2	no	—
156	C₁₁⋊C₈	—	4	4	no	—
157	#533	—	11	1	no	—
158	#533	C₂×D₁₁	1	2	yes	—
159	#500	C₂×D₁₁	1	1	yes	characteristic
160	#494	C₂×D₁₁	1	1	yes	characteristic
161	#487	C₂×D₁₁	1	2	yes	—
162	#475	—	11	4	no	—
163	#475	—	22	2	no	—
164	#470	—	22	4	no	—
165	#467	—	2	1	no	—
166	#467	—	2	1	no	—
167	#462	—	11	4	no	—
168	#462	—	22	2	no	—
169	#446	—	11	2	no	—
170	#446	—	22	1	no	—
171	C₈×Q₈	—	11	2	no	—
172	C₈×Q₈	—	22	1	no	—
173	C₄×Q₁₆	—	2	1	no	—
174	C₄×Q₁₆	—	2	1	no	—
175	#424	—	22	1	no	—
176	#423	—	11	1	no	—
177	#423	—	11	2	no	—
178	#405	—	22	1	no	—
179	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
180	#335	—	11	2	no	—
181	#335	—	11	2	no	—
182	#332	—	11	4	no	—
183	#332	—	11	4	no	—
184	C₂²×C₁₁	—	2	1	no	—
185	C₂²×C₁₁	—	2	1	no	—
186	C₂²×C₁₁	—	2	1	no	—
187	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
188	C₄×C₁₁	—	2	1	no	—
189	C₄×C₁₁	—	2	1	no	—
190	C₄×C₁₁	—	2	1	no	—
191	C₄×C₁₁	—	8	2	no	—
192	C₄×C₁₁	—	8	2	no	—
193	C₄×C₁₁	#473	1	2	yes	—
194	C₄×C₁₁	C₂×#137	1	1	yes	characteristic
195	C₄×C₁₁	#575	1	1	yes	characteristic
196	C₄×C₁₁	#576	1	1	yes	characteristic
197	C₄×C₁₁	#578	1	1	yes	characteristic
198	Dic₁₁	—	4	2	no	—
199	Dic₁₁	—	4	2	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	22	4	no	—
203	C₂×M₁₆	—	22	1	no	—
204	C₂×M₁₆	—	22	2	no	—
205	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
206	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
207	C₄⋊Q₈	—	11	1	no	—
208	C₄⋊Q₈	—	11	1	no	—
209	C₄⋊Q₈	C₂²×D₁₁	1	2	yes	—
210	C₄⋊Q₈	C₂²×D₁₁	1	2	yes	—
211	#125	—	2	2	no	—
212	#125	—	22	1	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	11	2	no	—
216	C₂²⋊Q₈	—	22	1	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	11	2	no	—
220	C₄×Q₈	—	22	1	no	—
221	#117	—	2	2	no	—
222	#117	—	22	1	no	—
223	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
224	#107	—	2	1	no	—
225	#107	—	2	1	no	—
226	#106	—	22	4	no	—
227	C₄⋊C₈	—	11	2	no	—
228	C₄⋊C₈	—	11	4	no	—
229	C₄⋊C₈	—	22	2	no	—
230	C₄⋊C₈	—	22	2	no	—
231	C₄⋊C₈	—	22	2	no	—
232	C₄⋊C₈	—	22	1	no	—
233	#103	—	2	2	no	—
234	#103	—	2	2	no	—
235	#103	—	22	4	no	—
236	#103	—	22	4	no	—
237	#97	—	22	2	no	—
238	C₄×C₈	—	22	2	no	—
239	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
240	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
243	C₂×C₁₁	—	4	1	no	—
244	C₂×C₁₁	#3037	1	1	yes	characteristic
245	C₂×C₁₁	#3061	1	1	yes	characteristic
246	C₂×C₁₁	C₂×#473	1	1	yes	characteristic
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	11	2	no	—
250	C₂×Q₈	—	22	1	no	—
251	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
252	C₂²×C₄	#4341	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
254	Q₁₆	—	4	2	no	—
255	Q₁₆	—	4	2	no	—
256	Q₁₆	—	88	4	no	—
257	M₁₆	—	44	2	no	—
258	M₁₆	—	88	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	22	4	no	—
262	C₂×C₈	—	22	2	no	—
263	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
264	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
266	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	11	2	no	—
274	C₄⋊C₄	—	22	1	no	—
275	C₄⋊C₄	—	22	1	no	—
276	C₄⋊C₄	—	22	2	no	—
277	C₄⋊C₄	—	22	1	no	—
278	C₄⋊C₄	—	22	2	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	22	2	no	—
281	C₄²	—	4	2	no	—
282	C₄²	—	22	2	no	—
283	C₄²	D₄×D₁₁	1	1	yes	characteristic
284	C₄²	#4341	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
286	C₄²	C₂³×D₁₁	1	1	yes	characteristic
287	C₁₁	#19345	1	1	yes	characteristic, complemented, Sylow
288	C₂³	#66991	1	1	yes	characteristic
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	22	2	no	—
294	Q₈	—	22	2	no	—
295	Q₈	—	44	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	22	2	no	—
304	C₂×C₄	—	22	2	no	—
305	C₂×C₄	#66991	1	1	yes	characteristic
306	C₂×C₄	#66991	1	1	yes	characteristic
307	C₂×C₄	#66991	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
309	C₂×C₄	C₂×#4341	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	44	4	no	—
315	C₈	—	44	2	no	—
316	C₈	#67004	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	C₂×#66991	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	44	2	no	—
327	C₄	—	44	2	no	—
328	C₄	#10622805	1	2	yes	—
329	C₄	#10623300	1	1	yes	characteristic
330	C₄	#10623302	1	1	yes	characteristic
331	C₄	#10623353	1	1	yes	characteristic
332	C₄	C₂×#67004	1	1	yes	characteristic
333	C₂	—	4	1	no	—
334	C₂	#13078084	1	1	yes	characteristic
335	C₂	#13078282	1	1	yes	characteristic
336	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	#19345
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. #13068167	C₂	semidirect
4. #13068007	C₂	semidirect
5. #13067983	C₂	semidirect
287. C₁₁	#19345	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
#13068167	2
#13068007	2
#13067983	2
#13067375	2
#13067354	2
C₁₁×#3170	2
#19345	11