Subgroups — #14725525

Order 2816.

652 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	#13077737	C₂	1	2	yes	maximal, complemented
2	C₄×#66957	C₂	1	1	yes	maximal, characteristic
3	#13067526	C₂	1	2	yes	maximal, complemented
4	#13067454	C₂	1	4	yes	maximal, complemented
5	#13067441	C₂	1	2	yes	maximal, complemented
6	#13067372	C₂	1	1	yes	maximal, characteristic, complemented
7	#13067056	C₂	1	2	yes	maximal, complemented
8	C₁₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#66957	C₄	1	1	yes	characteristic
10	C₄×#4339	C₂²	1	2	yes	—
11	C₁₁×#431	C₂²	1	1	yes	characteristic
12	C₄×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	#10622726	C₄	1	2	yes	—
15	#10622719	C₄	1	1	yes	characteristic
16	#10622537	C₂²	1	2	yes	—
17	#10622448	—	2	2	no	—
18	D₁₁×C₄⋊C₈	—	2	2	no	—
19	#10622341	—	2	4	no	—
20	C₄×Dic₄₄	C₄	1	1	yes	characteristic
21	C₄×Dic₄₄	C₂²	1	1	yes	characteristic
22	C₄×D₈₈	C₄	1	1	yes	characteristic
23	C₄×D₈₈	C₂²	1	1	yes	characteristic
24	C₄×#4314	C₄	1	2	yes	—
25	C₄×#4314	C₂²	1	2	yes	—
26	C₈×D₄₄	C₂²	1	4	yes	—
27	#10622292	C₂²	1	4	yes	—
28	C₁₁×#322	C₂²	1	2	yes	—
29	C₁₁×#322	C₂²	1	2	yes	—
30	#10622097	C₂²	1	4	yes	—
31	#10622095	C₂²	1	2	yes	—
32	#10622095	C₂²	1	2	yes	—
33	#10622094	C₂²	1	4	yes	—
34	C₂×#4339	C₂×C₄	1	2	yes	—
35	C₂×C₁₁×M₁₆	—	2	1	no	—
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
39	C₄×C₂²⋊D₁₁	—	2	2	no	—
40	C₄×C₂²⋊D₁₁	—	2	2	no	—
41	#66979	C₂×C₄	1	2	yes	—
42	C₂×#4318	—	2	2	no	—
43	C₂×Dic₄₄	C₂×C₄	1	1	yes	characteristic
44	#66957	—	2	4	no	—
45	C₂×D₈₈	C₂×C₄	1	1	yes	characteristic
46	C₂×#4314	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₁₁	—	2	4	no	—
48	C₄×D₄₄	C₂×C₄	1	2	yes	—
49	C₄×D₄₄	C₂³	1	2	yes	—
50	#66925	—	2	2	no	—
51	C₄²×D₁₁	—	2	2	no	—
52	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
53	C₄×Dic₂₂	C₂³	1	2	yes	—
54	C₁₁×C₄⋊C₈	D₄	1	2	yes	—
55	C₁₁×C₂²⋊C₈	D₄	1	2	yes	—
56	C₁₁×#97	—	2	1	no	—
57	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₁	C₂³	1	2	yes	—
60	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₁	C₂³	1	2	yes	—
63	#66885	C₂×C₄	1	2	yes	—
64	#66885	C₂×C₄	1	2	yes	—
65	(C₂×D₁₁)⋊C₈	—	2	4	no	—
66	#66882	C₂×C₄	1	1	yes	characteristic
67	#66882	C₂×C₄	1	1	yes	characteristic
68	#66881	C₂×C₄	1	2	yes	—
69	#66880	C₂×C₄	1	2	yes	—
70	#66880	C₂×C₄	1	2	yes	—
71	#66879	—	2	4	no	—
72	Dic₁₁⋊C₈	—	2	4	no	—
73	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
74	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
75	#18065	—	11	1	no	maximal, Sylow
76	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₁₁	—	2	2	no	—
80	#4339	—	2	4	no	—
81	#4339	—	4	2	no	—
82	C₂×D₄₄	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₁₁	—	2	2	no	—
84	C₂×C₄×D₁₁	—	2	2	no	—
85	C₂×C₄×D₁₁	—	2	2	no	—
86	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
87	C₁₁×M₁₆	—	4	2	no	—
88	C₂×C₈×C₁₁	—	2	2	no	—
89	C₂×C₈×C₁₁	—	2	2	no	—
90	C₂×C₈×C₁₁	—	4	1	no	—
91	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₁	C₂×D₄	1	4	yes	—
96	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₁	#41	1	1	yes	characteristic
99	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
100	C₂²⋊Dic₁₁	—	2	2	no	—
101	(C₂×D₁₁)⋊C₄	—	2	2	no	—
102	(C₂×D₁₁)⋊C₄	—	2	2	no	—
103	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
104	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
105	Dic₁₁⋊C₄	—	2	2	no	—
106	Dic₁₁⋊C₄	—	2	2	no	—
107	C₄×Dic₁₁	—	2	2	no	—
108	C₄×Dic₁₁	—	2	2	no	—
109	C₄×Dic₁₁	—	2	2	no	—
110	#4318	—	8	2	no	—
111	C₂×C₁₁⋊C₈	—	2	4	no	—
112	Dic₄₄	—	2	2	no	—
113	D₈₈	M₁₆	1	4	yes	complemented
114	#4314	—	2	4	no	—
115	C₈×D₁₁	—	8	4	no	—
116	#2990	—	11	2	no	—
117	C₄×#135	—	11	1	no	—
118	#1593	—	11	2	no	—
119	#1582	—	11	4	no	—
120	#1580	—	11	2	no	—
121	#1566	—	11	1	no	—
122	#1489	—	11	2	no	—
123	#1451	D₁₁	1	1	yes	characteristic, complemented
124	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
125	C₂²×D₁₁	—	2	2	no	—
126	C₂×C₄×C₁₁	—	2	1	no	—
127	C₂×C₄×C₁₁	—	2	1	no	—
128	C₂×C₄×C₁₁	—	2	2	no	—
129	C₂×C₄×C₁₁	—	2	2	no	—
130	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
136	C₂²⋊D₁₁	—	4	4	no	—
137	C₂×Dic₁₁	—	2	2	no	—
138	C₂×Dic₁₁	—	2	2	no	—
139	C₂×Dic₁₁	—	2	2	no	—
140	D₄₄	—	2	2	no	—
141	D₄₄	C₂×M₁₆	1	4	yes	—
142	C₄×D₁₁	—	4	4	no	—
143	C₄×D₁₁	—	4	2	no	—
144	C₄×D₁₁	—	4	2	no	—
145	C₄×D₁₁	—	4	4	no	—
146	Dic₂₂	—	2	2	no	—
147	Dic₂₂	—	2	2	no	—
148	C₈×C₁₁	—	2	1	no	—
149	C₈×C₁₁	—	2	1	no	—
150	C₈×C₁₁	—	2	1	no	—
151	C₈×C₁₁	—	2	4	no	—
152	C₈×C₁₁	—	4	2	no	—
153	C₈×C₁₁	C₂×M₁₆	1	2	yes	—
154	C₁₁⋊C₈	—	4	4	no	—
155	C₂×#135	—	11	1	no	—
156	C₄×#41	—	11	2	no	—
157	#446	—	11	4	no	—
158	C₄×Q₁₆	—	11	1	no	—
159	C₄×Q₁₆	—	11	1	no	—
160	C₄×SD₁₆	—	11	2	no	—
161	C₄×SD₁₆	—	11	2	no	—
162	C₄×D₈	—	11	1	no	—
163	C₄×D₈	—	11	1	no	—
164	#435	—	22	4	no	—
165	C₈×D₄	—	11	4	no	—
166	#432	—	22	2	no	—
167	#431	C₂×D₁₁	1	1	yes	characteristic
168	#427	—	11	1	no	—
169	#423	—	11	2	no	—
170	C₂×C₄⋊C₈	—	22	2	no	—
171	#416	—	11	2	no	—
172	C₄×M₁₆	C₂×D₁₁	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
174	#334	—	11	2	no	—
175	#334	—	11	2	no	—
176	Q₈⋊C₈	—	11	4	no	—
177	D₄⋊C₈	—	11	4	no	—
178	#322	C₂×D₁₁	1	2	yes	—
179	#322	C₂×D₁₁	1	2	yes	—
180	C₂²×C₁₁	—	2	1	no	—
181	C₂²×C₁₁	—	2	1	no	—
182	C₂²×C₁₁	—	2	1	no	—
183	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
184	C₂×D₁₁	—	2	4	no	—
185	C₂×D₁₁	—	4	2	no	—
186	C₂×D₁₁	—	4	2	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	—	2	1	no	—
189	C₄×C₁₁	—	2	1	no	—
190	C₄×C₁₁	#444	1	2	yes	—
191	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
192	C₄×C₁₁	#568	1	1	yes	characteristic
193	C₄×C₁₁	#577	1	2	yes	—
194	Dic₁₁	—	4	2	no	—
195	Dic₁₁	—	4	2	no	—
196	Dic₁₁	—	4	2	no	—
197	C₂×#41	—	11	2	no	—
198	#135	—	22	4	no	—
199	C₂×Q₁₆	—	11	1	no	—
200	C₂×SD₁₆	—	11	2	no	—
201	C₂×D₈	—	11	1	no	—
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	22	2	no	—
204	C₂²×C₈	—	22	4	no	—
205	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
206	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
207	C₄×Q₈	—	11	2	no	—
208	C₄×Q₈	—	11	2	no	—
209	C₄×D₄	—	11	2	no	—
210	C₄×D₄	—	11	2	no	—
211	C₄×D₄	—	22	2	no	—
212	C₄×D₄	—	22	2	no	—
213	#117	—	11	2	no	—
214	#117	—	22	2	no	—
215	C₂×C₄²	—	22	2	no	—
216	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
217	#107	—	11	1	no	—
218	#107	—	11	1	no	—
219	#106	—	11	2	no	—
220	C₄⋊C₈	—	11	4	no	—
221	C₄⋊C₈	—	22	4	no	—
222	C₄⋊C₈	—	22	2	no	—
223	C₄⋊C₈	D₄₄	1	2	yes	—
224	#103	—	11	2	no	—
225	#103	—	11	2	no	—
226	#102	—	11	2	no	—
227	#102	—	11	2	no	—
228	C₂²⋊C₈	—	22	4	no	—
229	C₂²⋊C₈	D₄₄	1	2	yes	—
230	#97	—	2	1	no	—
231	#97	—	22	4	no	—
232	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
233	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₁₁	1	2	yes	—
235	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₁₁	1	2	yes	—
238	C₂×C₁₁	—	4	1	no	—
239	C₂×C₁₁	D₄×M₁₆	1	1	yes	characteristic
240	C₂×C₁₁	#2952	1	1	yes	characteristic
241	C₂×C₁₁	C₂×#444	1	1	yes	characteristic
242	D₁₁	—	4	4	no	—
243	#41	—	22	4	no	—
244	#41	—	44	2	no	—
245	C₂×Q₈	—	11	2	no	—
246	C₂×D₄	—	11	2	no	—
247	C₂×D₄	—	22	2	no	—
248	C₂²×C₄	—	22	2	no	—
249	C₂²×C₄	—	22	2	no	—
250	C₂²×C₄	—	22	2	no	—
251	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
253	C₂²×C₄	#4339	1	1	yes	characteristic
254	Q₁₆	—	22	2	no	—
255	SD₁₆	—	22	4	no	—
256	D₈	—	11	4	no	—
257	M₁₆	—	4	2	no	—
258	M₁₆	—	88	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	22	4	no	—
263	C₂×C₈	—	88	4	no	—
264	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄₄	1	4	yes	—
269	C₄⋊C₄	—	11	2	no	—
270	C₄⋊C₄	—	11	2	no	—
271	C₄⋊C₄	—	22	2	no	—
272	C₄⋊C₄	—	22	2	no	—
273	C₂²⋊C₄	—	22	2	no	—
274	C₂²⋊C₄	—	22	2	no	—
275	C₂²⋊C₄	—	22	2	no	—
276	C₄²	—	22	2	no	—
277	C₄²	—	22	2	no	—
278	C₄²	—	22	2	no	—
279	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
280	C₄²	C₂×D₄₄	1	1	yes	characteristic
281	C₄²	#4339	1	1	yes	characteristic
282	C₄²	C₂³×D₁₁	1	1	yes	characteristic
283	C₁₁	#18065	1	1	yes	characteristic, complemented, Sylow
284	C₂³	—	22	2	no	—
285	C₂³	C₄×D₄₄	1	1	yes	characteristic
286	Q₈	—	22	2	no	—
287	Q₈	—	22	2	no	—
288	D₄	—	11	4	no	—
289	D₄	—	22	2	no	—
290	D₄	—	44	4	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	22	2	no	—
296	C₂×C₄	—	22	2	no	—
297	C₂×C₄	—	22	2	no	—
298	C₂×C₄	—	44	4	no	—
299	C₂×C₄	—	44	2	no	—
300	C₂×C₄	—	44	2	no	—
301	C₂×C₄	—	44	4	no	—
302	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
303	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
304	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
307	C₂×C₄	C₂×#4339	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	4	no	—
312	C₈	—	4	2	no	—
313	C₈	—	44	4	no	—
314	C₈	M₁₆×D₁₁	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	22	4	no	—
319	C₂²	—	44	2	no	—
320	C₂²	—	44	2	no	—
321	C₂²	C₂×C₄×D₄₄	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	44	2	no	—
326	C₄	—	44	2	no	—
327	C₄	—	44	2	no	—
328	C₄	#10622330	1	2	yes	—
329	C₄	C₂×M₁₆×D₁₁	1	1	yes	characteristic
330	C₄	#10623276	1	1	yes	characteristic
331	C₄	#10623282	1	2	yes	—
332	C₂	—	4	1	no	—
333	C₂	—	44	4	no	—
334	C₂	M₁₆×D₄₄	1	1	yes	characteristic
335	C₂	#13077501	1	1	yes	characteristic
336	C₂	C₂×#10622330	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₁₁×#1451	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂×#66957	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1451	D₁₁
hypocenter	C₁₁	#18065
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
1. #13077737	C₂	semidirect
3. #13067526	C₂	semidirect
4. #13067454	C₂	semidirect
5. #13067441	C₂	semidirect
6. #13067372	C₂	semidirect
7. #13067056	C₂	semidirect
8. C₁₁×#1451	C₂	semidirect
113. D₈₈	M₁₆	semidirect
123. #1451	D₁₁	semidirect
283. C₁₁	#18065	semidirect

Maximal subgroups

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

subgroup	index
#13077737	2
C₄×#66957	2
#13067526	2
#13067454	2
#13067441	2
#13067372	2
#13067056	2
C₁₁×#1451	2
#18065	11