Subgroups — #14725281

Order 2816.

644 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077634	C₂	1	2	yes	maximal, complemented
2	C₄×#66957	C₂	1	1	yes	maximal, characteristic
3	#13067531	C₂	1	2	yes	maximal, complemented
4	#13067496	C₂	1	4	yes	maximal, complemented
5	#13067436	C₂	1	2	yes	maximal, complemented
6	#13067379	C₂	1	1	yes	maximal, characteristic, complemented
7	#13067051	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	#10622741	—	2	4	no	—
15	#10622726	C₄	1	2	yes	—
16	#10622719	C₄	1	1	yes	characteristic
17	#10622539	C₂²	1	2	yes	—
18	#10622320	—	2	2	no	—
19	#10622318	—	2	2	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	#10622301	C₂²	1	4	yes	—
27	C₈×Dic₂₂	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	#10622096	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	D₄	1	2	yes	—
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×Dic₄₄	C₂×C₄	1	1	yes	characteristic
43	#66957	—	2	4	no	—
44	C₂×D₈₈	C₂×C₄	1	1	yes	characteristic
45	C₂×#4314	C₂×C₄	1	2	yes	—
46	C₂×#4313	—	2	4	no	—
47	C₄×D₄₄	C₂×C₄	1	2	yes	—
48	C₄×D₄₄	C₂³	1	2	yes	—
49	#66925	—	2	2	no	—
50	C₄²×D₁₁	—	2	2	no	—
51	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
52	C₄×Dic₂₂	C₂³	1	2	yes	—
53	C₁₁×C₄⋊C₈	—	2	1	no	—
54	C₁₁×C₂²⋊C₈	—	2	1	no	—
55	C₁₁×#97	D₄	1	2	yes	—
56	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₁	C₂³	1	2	yes	—
59	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₁	C₂³	1	2	yes	—
62	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
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	Dic₁₁⋊C₈	—	2	4	no	—
72	C₈×Dic₁₁	—	2	4	no	—
73	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
74	#66867	—	2	2	no	—
75	#18043	—	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	C₂×C₁₁⋊C₈	—	2	4	no	—
111	Dic₄₄	M₁₆	1	4	yes	complemented
112	D₈₈	—	2	2	no	—
113	#4314	—	2	4	no	—
114	#4313	—	4	8	no	—
115	#2971	—	11	2	no	—
116	C₄×#135	—	11	1	no	—
117	#1594	—	11	2	no	—
118	#1588	—	11	4	no	—
119	#1579	—	11	2	no	—
120	#1568	—	11	1	no	—
121	#1488	—	11	2	no	—
122	#1451	D₁₁	1	1	yes	characteristic, complemented
123	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
124	C₂²×D₁₁	—	2	2	no	—
125	C₂×C₄×C₁₁	—	2	1	no	—
126	C₂×C₄×C₁₁	—	2	1	no	—
127	C₂×C₄×C₁₁	—	2	2	no	—
128	C₂×C₄×C₁₁	—	4	1	no	—
129	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
135	C₂²⋊D₁₁	—	4	4	no	—
136	C₂×Dic₁₁	—	2	2	no	—
137	C₂×Dic₁₁	—	2	2	no	—
138	C₂×Dic₁₁	—	2	2	no	—
139	D₄₄	—	2	2	no	—
140	D₄₄	—	2	2	no	—
141	C₄×D₁₁	—	4	4	no	—
142	C₄×D₁₁	—	4	4	no	—
143	C₄×D₁₁	—	8	2	no	—
144	Dic₂₂	—	2	2	no	—
145	Dic₂₂	C₂×M₁₆	1	4	yes	—
146	C₈×C₁₁	—	2	1	no	—
147	C₈×C₁₁	—	2	1	no	—
148	C₈×C₁₁	—	2	1	no	—
149	C₈×C₁₁	—	2	4	no	—
150	C₈×C₁₁	—	4	2	no	—
151	C₈×C₁₁	C₂×M₁₆	1	2	yes	—
152	C₁₁⋊C₈	—	4	4	no	—
153	C₂×#135	—	11	1	no	—
154	C₄×#41	—	11	2	no	—
155	C₈×Q₈	—	11	4	no	—
156	C₄×Q₁₆	—	11	1	no	—
157	C₄×Q₁₆	—	11	1	no	—
158	C₄×SD₁₆	—	11	2	no	—
159	C₄×SD₁₆	—	11	2	no	—
160	C₄×D₈	—	11	1	no	—
161	C₄×D₈	—	11	1	no	—
162	#436	—	11	4	no	—
163	#436	—	22	4	no	—
164	#432	—	11	2	no	—
165	#432	—	22	2	no	—
166	#431	C₂×D₁₁	1	1	yes	characteristic
167	#427	—	11	1	no	—
168	#416	—	11	2	no	—
169	C₄×M₁₆	—	22	2	no	—
170	C₄×M₁₆	C₂×D₁₁	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
172	#335	—	11	2	no	—
173	#334	—	11	2	no	—
174	Q₈⋊C₈	—	11	4	no	—
175	D₄⋊C₈	—	11	4	no	—
176	#322	C₂×D₁₁	1	2	yes	—
177	#322	C₂×D₁₁	1	2	yes	—
178	C₂²×C₁₁	—	2	1	no	—
179	C₂²×C₁₁	—	2	1	no	—
180	C₂²×C₁₁	—	2	1	no	—
181	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
182	C₂×D₁₁	—	4	2	no	—
183	C₂×D₁₁	—	4	2	no	—
184	C₂×D₁₁	—	4	2	no	—
185	C₄×C₁₁	—	2	1	no	—
186	C₄×C₁₁	—	2	1	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	—	2	1	no	—
189	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
190	C₄×C₁₁	#568	1	1	yes	characteristic
191	C₄×C₁₁	#575	1	2	yes	—
192	Dic₁₁	—	2	4	no	—
193	Dic₁₁	—	4	2	no	—
194	Dic₁₁	—	4	2	no	—
195	C₂×#41	—	11	2	no	—
196	#135	—	22	4	no	—
197	C₂×Q₁₆	—	11	1	no	—
198	C₂×SD₁₆	—	11	2	no	—
199	C₂×D₈	—	11	1	no	—
200	C₂×M₁₆	—	22	4	no	—
201	C₂×M₁₆	D₄₄	1	2	yes	—
202	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
203	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
204	C₄×Q₈	—	11	2	no	—
205	C₄×Q₈	—	11	2	no	—
206	C₄×D₄	—	11	2	no	—
207	C₄×D₄	—	11	2	no	—
208	C₄×D₄	—	22	2	no	—
209	C₄×D₄	—	22	2	no	—
210	#117	—	11	2	no	—
211	#117	—	22	2	no	—
212	C₂×C₄²	—	22	2	no	—
213	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
214	#107	—	11	1	no	—
215	#107	—	11	1	no	—
216	#106	—	11	2	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	11	4	no	—
219	C₄⋊C₈	—	22	4	no	—
220	#103	—	11	2	no	—
221	#103	—	11	2	no	—
222	#102	—	11	2	no	—
223	#102	—	11	2	no	—
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	—	22	2	no	—
226	C₂²⋊C₈	—	22	4	no	—
227	#97	—	22	2	no	—
228	#97	D₄₄	1	2	yes	—
229	C₄×C₈	—	22	4	no	—
230	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
231	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₁₁	1	2	yes	—
233	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₁₁	1	2	yes	—
236	C₂×C₁₁	—	4	1	no	—
237	C₂×C₁₁	D₄×M₁₆	1	1	yes	characteristic
238	C₂×C₁₁	#2943	1	1	yes	characteristic
239	C₂×C₁₁	#2957	1	1	yes	characteristic
240	D₁₁	—	8	2	no	—
241	#41	—	22	4	no	—
242	#41	—	44	2	no	—
243	C₂×Q₈	—	11	2	no	—
244	C₂×D₄	—	11	2	no	—
245	C₂×D₄	—	22	2	no	—
246	C₂²×C₄	—	22	2	no	—
247	C₂²×C₄	—	22	2	no	—
248	C₂²×C₄	—	22	2	no	—
249	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
251	C₂²×C₄	#4339	1	1	yes	characteristic
252	Q₁₆	—	11	4	no	—
253	SD₁₆	—	22	4	no	—
254	D₈	—	22	2	no	—
255	M₁₆	—	4	2	no	—
256	M₁₆	—	44	8	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	22	4	no	—
261	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
263	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
264	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄₄	1	4	yes	—
266	C₄⋊C₄	—	11	2	no	—
267	C₄⋊C₄	—	11	2	no	—
268	C₄⋊C₄	—	22	2	no	—
269	C₄⋊C₄	—	22	2	no	—
270	C₂²⋊C₄	—	22	2	no	—
271	C₂²⋊C₄	—	22	2	no	—
272	C₂²⋊C₄	—	22	2	no	—
273	C₄²	—	22	2	no	—
274	C₄²	—	22	2	no	—
275	C₄²	—	22	2	no	—
276	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
277	C₄²	C₂×D₄₄	1	1	yes	characteristic
278	C₄²	#4339	1	1	yes	characteristic
279	C₄²	C₂³×D₁₁	1	1	yes	characteristic
280	C₁₁	#18043	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	22	2	no	—
282	C₂³	C₄×D₄₄	1	1	yes	characteristic
283	Q₈	—	11	4	no	—
284	Q₈	—	22	2	no	—
285	D₄	—	22	2	no	—
286	D₄	—	22	2	no	—
287	D₄	—	44	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	22	2	no	—
293	C₂×C₄	—	22	2	no	—
294	C₂×C₄	—	22	2	no	—
295	C₂×C₄	—	44	4	no	—
296	C₂×C₄	—	44	4	no	—
297	C₂×C₄	—	88	2	no	—
298	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
299	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
300	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
303	C₂×C₄	C₂×#4339	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	4	no	—
308	C₈	—	4	2	no	—
309	C₈	—	44	4	no	—
310	C₈	M₁₆×D₁₁	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	44	2	no	—
315	C₂²	—	44	2	no	—
316	C₂²	—	44	2	no	—
317	C₂²	C₂×C₄×D₄₄	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	22	4	no	—
323	C₄	—	44	2	no	—
324	C₄	—	44	2	no	—
325	C₄	C₂×M₁₆×D₁₁	1	1	yes	characteristic
326	C₄	#10623276	1	1	yes	characteristic
327	C₄	#10623279	1	2	yes	—
328	C₂	—	4	1	no	—
329	C₂	—	88	2	no	—
330	C₂	M₁₆×D₄₄	1	1	yes	characteristic
331	C₂	#13077418	1	1	yes	characteristic
332	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	C₁₁×#1451	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂×#66957	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#1451	D₁₁
hypocenter	C₁₁	#18043
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
1. #13077634	C₂	semidirect
3. #13067531	C₂	semidirect
4. #13067496	C₂	semidirect
5. #13067436	C₂	semidirect
6. #13067379	C₂	semidirect
7. #13067051	C₂	semidirect
8. C₁₁×#1451	C₂	semidirect
111. Dic₄₄	M₁₆	semidirect
122. #1451	D₁₁	semidirect
280. C₁₁	#18043	semidirect

Maximal subgroups

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

subgroup	index
#13077634	2
C₄×#66957	2
#13067531	2
#13067496	2
#13067436	2
#13067379	2
#13067051	2
C₁₁×#1451	2
#18043	11