Subgroups — #14723263

Order 2816.

714 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077727	C₂	1	2	yes	maximal, complemented
2	#13067440	C₂	1	2	yes	maximal, complemented
3	#13067435	C₂	1	2	yes	maximal, complemented
4	SD₁₆×C₁₁⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂×#10622095	C₂	1	1	yes	maximal, characteristic, complemented
6	#13067003	C₂	1	2	yes	maximal, complemented
7	#13066899	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₁×#135	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₁×#135	C₄	1	1	yes	characteristic
10	C₄×C₁₁×#41	C₂²	1	2	yes	—
11	C₄×C₁₁×Q₁₆	C₄	1	1	yes	characteristic
12	C₄×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₁₁×SD₁₆	C₄	1	2	yes	—
14	C₄×C₁₁×SD₁₆	C₂²	1	2	yes	—
15	C₄×C₁₁×D₈	C₄	1	1	yes	characteristic
16	C₄×C₁₁×D₈	C₂²	1	1	yes	characteristic
17	C₁₁×#427	C₄	1	1	yes	characteristic
18	C₁₁×#416	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
20	Q₈×C₁₁⋊C₈	C₂²	1	4	yes	—
21	D₄×C₁₁⋊C₈	—	2	4	no	—
22	D₄×C₁₁⋊C₈	C₂²	1	4	yes	—
23	#10622538	—	2	2	no	—
24	#10622538	C₂²	1	1	yes	characteristic
25	#10622538	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
27	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
28	Q₈⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
29	D₄⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
30	#10622095	C₂²	1	2	yes	—
31	#10622095	C₂²	1	2	yes	—
32	#10622093	C₂²	1	2	yes	—
33	C₈×C₁₁⋊C₈	C₂²	1	2	yes	—
34	C₂×C₁₁×#41	C₂×C₄	1	2	yes	—
35	C₁₁×#135	C₈	1	8	yes	complemented
36	C₂×C₁₁×Q₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₁×SD₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₁₁×D₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
41	C₄×Q₈×C₁₁	C₂×C₄	1	2	yes	—
42	C₄×Q₈×C₁₁	C₂³	1	2	yes	—
43	C₄×D₄×C₁₁	—	2	2	no	—
44	C₄×D₄×C₁₁	—	2	2	no	—
45	C₄×D₄×C₁₁	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₁₁	C₂³	1	2	yes	—
47	C₁₁×#117	—	2	2	no	—
48	C₁₁×#117	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₁	—	2	2	no	—
50	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
51	C₂²×C₁₁⋊C₈	—	2	2	no	—
52	C₂²×C₁₁⋊C₈	—	2	4	no	—
53	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
54	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
55	C₁₁×#106	C₂×C₄	1	2	yes	—
56	C₁₁×#103	C₂×C₄	1	2	yes	—
57	C₁₁×#103	C₂×C₄	1	2	yes	—
58	C₁₁×#102	C₂×C₄	1	2	yes	—
59	C₁₁×#102	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
64	C₂²⋊(C₁₁⋊C₈)	—	2	1	no	—
65	C₂²⋊(C₁₁⋊C₈)	—	2	4	no	—
66	C₂²⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
67	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
68	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
69	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
70	C₄⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
71	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
72	C₄×C₁₁⋊C₈	—	2	4	no	—
73	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
74	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
75	#17858	—	11	1	no	maximal, Sylow
76	C₁₁×#41	—	2	4	no	—
77	C₁₁×#41	C₂×C₈	1	8	yes	complemented
78	C₂×Q₈×C₁₁	C₂²×C₄	1	2	yes	—
79	C₂×D₄×C₁₁	—	2	2	no	—
80	C₂×D₄×C₁₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₁	—	2	2	no	—
82	C₂²×C₄×C₁₁	—	2	2	no	—
83	C₂²×C₄×C₁₁	—	2	2	no	—
84	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
87	C₁₁×Q₁₆	C₂×C₈	1	4	yes	complemented
88	C₁₁×SD₁₆	C₂×C₈	1	8	yes	complemented
89	C₁₁×D₈	C₂×C₈	1	4	yes	complemented
90	C₂×C₈×C₁₁	—	2	2	no	—
91	C₂×C₈×C₁₁	C₂×C₈	1	4	yes	complemented
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	C₂²×C₄	1	1	yes	characteristic
96	C₁₁×C₄⋊C₄	—	2	2	no	—
97	C₁₁×C₄⋊C₄	—	2	2	no	—
98	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
99	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
100	C₁₁×C₂²⋊C₄	—	2	2	no	—
101	C₁₁×C₂²⋊C₄	—	2	2	no	—
102	C₁₁×C₂²⋊C₄	—	2	2	no	—
103	C₄²×C₁₁	—	2	2	no	—
104	C₄²×C₁₁	—	2	2	no	—
105	C₄²×C₁₁	—	2	2	no	—
106	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₁	#41	1	1	yes	characteristic
109	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
110	C₂×C₁₁⋊C₈	—	2	4	no	—
111	C₂×C₁₁⋊C₈	—	2	2	no	—
112	C₂×C₁₁⋊C₈	—	8	2	no	—
113	C₂×C₁₁⋊C₈	—	8	4	no	—
114	C₂×C₁₁⋊C₈	C₂×D₄	1	4	yes	—
115	#2988	—	11	2	no	—
116	C₄×#135	D₁₁	1	1	yes	characteristic
117	#1580	—	11	2	no	—
118	#1579	—	11	2	no	—
119	C₈×SD₁₆	—	11	4	no	—
120	C₂×#334	—	11	1	no	—
121	#1477	—	11	2	no	—
122	#1450	—	11	1	no	—
123	C₂³×C₁₁	—	2	2	no	—
124	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
125	Q₈×C₁₁	—	2	2	no	—
126	Q₈×C₁₁	C₂²×C₈	1	4	yes	—
127	D₄×C₁₁	—	2	8	no	—
128	D₄×C₁₁	—	2	2	no	—
129	D₄×C₁₁	C₂²×C₈	1	4	yes	—
130	C₂×C₄×C₁₁	—	2	2	no	—
131	C₂×C₄×C₁₁	—	2	2	no	—
132	C₂×C₄×C₁₁	—	2	8	no	—
133	C₂×C₄×C₁₁	—	2	2	no	—
134	C₂×C₄×C₁₁	—	2	2	no	—
135	C₂×C₄×C₁₁	—	4	1	no	—
136	C₂×C₄×C₁₁	—	4	2	no	—
137	C₂×C₄×C₁₁	—	4	2	no	—
138	C₂×C₄×C₁₁	—	8	2	no	—
139	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
143	C₂×C₄×C₁₁	#131	1	2	yes	—
144	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
147	C₈×C₁₁	—	2	1	no	—
148	C₈×C₁₁	—	2	1	no	—
149	C₈×C₁₁	C₂²×C₈	1	2	yes	—
150	C₈×C₁₁	C₂²×C₈	1	2	yes	—
151	C₁₁⋊C₈	—	2	4	no	—
152	C₁₁⋊C₈	—	4	2	no	—
153	C₁₁⋊C₈	—	4	4	no	—
154	C₂×#135	Dic₁₁	1	1	yes	characteristic
155	C₄×#41	C₂×D₁₁	1	2	yes	—
156	C₈×Q₈	—	11	4	no	—
157	C₄×Q₁₆	Dic₁₁	1	1	yes	characteristic
158	C₄×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
159	C₄×SD₁₆	Dic₁₁	1	2	yes	—
160	C₄×SD₁₆	C₂×D₁₁	1	2	yes	—
161	C₄×D₈	Dic₁₁	1	1	yes	characteristic
162	C₄×D₈	C₂×D₁₁	1	1	yes	characteristic
163	C₈×D₄	—	11	4	no	—
164	C₈×D₄	—	22	4	no	—
165	#431	—	11	1	no	—
166	#431	—	11	1	no	—
167	#431	—	22	2	no	—
168	#427	Dic₁₁	1	1	yes	characteristic
169	C₂×C₄⋊C₈	—	11	2	no	—
170	C₂×C₄⋊C₈	—	22	2	no	—
171	#416	Dic₁₁	1	2	yes	—
172	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
173	#334	—	11	2	no	—
174	#334	—	11	2	no	—
175	Q₈⋊C₈	—	11	4	no	—
176	D₄⋊C₈	—	11	4	no	—
177	#322	—	11	2	no	—
178	C₈²	—	11	2	no	—
179	C₂²×C₁₁	—	2	1	no	—
180	C₂²×C₁₁	—	2	1	no	—
181	C₂²×C₁₁	—	2	4	no	—
182	C₂²×C₁₁	—	4	2	no	—
183	C₂²×C₁₁	—	4	2	no	—
184	C₂²×C₁₁	C₈×D₄	1	2	yes	—
185	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄×C₁₁	—	2	4	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	—	2	1	no	—
189	C₄×C₁₁	—	2	1	no	—
190	C₄×C₁₁	—	4	2	no	—
191	C₄×C₁₁	—	4	2	no	—
192	C₄×C₁₁	C₈×D₄	1	2	yes	—
193	C₄×C₁₁	C₂³×C₈	1	1	yes	characteristic
194	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
195	C₄×C₁₁	#577	1	2	yes	—
196	C₂×#41	C₂×Dic₁₁	1	2	yes	—
197	#135	C₁₁⋊C₈	1	8	yes	complemented
198	C₂×Q₁₆	C₂×Dic₁₁	1	1	yes	characteristic
199	C₂×SD₁₆	C₂×Dic₁₁	1	2	yes	—
200	C₂×D₈	C₂×Dic₁₁	1	1	yes	characteristic
201	C₂²×C₈	—	22	4	no	—
202	C₂²×C₈	—	22	2	no	—
203	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
204	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
205	C₄×Q₈	C₂×Dic₁₁	1	2	yes	—
206	C₄×Q₈	C₂²×D₁₁	1	2	yes	—
207	C₄×D₄	—	2	2	no	—
208	C₄×D₄	—	2	2	no	—
209	C₄×D₄	C₂×Dic₁₁	1	2	yes	—
210	C₄×D₄	C₂²×D₁₁	1	2	yes	—
211	#117	—	2	2	no	—
212	#117	C₂×Dic₁₁	1	2	yes	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
215	#107	C₂×Dic₁₁	1	1	yes	characteristic
216	#107	C₂×Dic₁₁	1	1	yes	characteristic
217	#106	C₂×Dic₁₁	1	2	yes	—
218	C₄⋊C₈	—	11	2	no	—
219	C₄⋊C₈	—	11	4	no	—
220	C₄⋊C₈	—	22	4	no	—
221	C₄⋊C₈	—	22	2	no	—
222	C₄⋊C₈	—	22	1	no	—
223	#103	C₂×Dic₁₁	1	2	yes	—
224	#103	C₂×Dic₁₁	1	2	yes	—
225	#102	C₂×Dic₁₁	1	2	yes	—
226	#102	C₂×Dic₁₁	1	2	yes	—
227	C₂²⋊C₈	—	11	2	no	—
228	C₂²⋊C₈	—	22	4	no	—
229	C₂²⋊C₈	—	22	1	no	—
230	C₄×C₈	—	11	2	no	—
231	C₄×C₈	—	11	2	no	—
232	C₄×C₈	—	22	4	no	—
233	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
234	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
237	C₂×C₁₁	—	2	2	no	—
238	C₂×C₁₁	—	4	4	no	—
239	C₂×C₁₁	C₂×C₈×D₄	1	1	yes	characteristic
240	C₂×C₁₁	#2952	1	1	yes	characteristic
241	C₂×C₁₁	#2957	1	1	yes	characteristic
242	#41	—	2	4	no	—
243	#41	C₂×C₁₁⋊C₈	1	8	yes	complemented
244	C₂×Q₈	C₂²×Dic₁₁	1	2	yes	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	C₂²×Dic₁₁	1	2	yes	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
251	C₂²×C₄	#4341	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
253	Q₁₆	C₂×C₁₁⋊C₈	1	4	yes	complemented
254	SD₁₆	C₂×C₁₁⋊C₈	1	8	yes	complemented
255	D₈	C₂×C₁₁⋊C₈	1	4	yes	complemented
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	11	4	no	—
258	C₂×C₈	—	22	4	no	—
259	C₂×C₈	—	22	2	no	—
260	C₂×C₈	—	88	4	no	—
261	C₂×C₈	—	88	2	no	—
262	C₂×C₈	C₂×C₁₁⋊C₈	1	4	yes	complemented
263	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
264	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
265	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
266	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	C₂²×Dic₁₁	1	2	yes	—
270	C₄⋊C₄	C₂²×Dic₁₁	1	2	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	D₄×D₁₁	1	1	yes	characteristic
278	C₄²	#4341	1	1	yes	characteristic
279	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
280	C₄²	C₂³×D₁₁	1	1	yes	characteristic
281	C₁₁	#17858	1	1	yes	characteristic, complemented, Sylow
282	C₂³	—	2	2	no	—
283	C₂³	D₄×Dic₁₁	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	C₂²×C₁₁⋊C₈	1	4	yes	—
286	D₄	—	2	8	no	—
287	D₄	—	2	2	no	—
288	D₄	C₂²×C₁₁⋊C₈	1	4	yes	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
299	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
300	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
301	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
302	C₂×C₄	#67001	1	2	yes	—
303	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
304	C₂×C₄	C₂×#4341	1	1	yes	characteristic
305	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	22	4	no	—
309	C₈	—	44	2	no	—
310	C₈	—	44	4	no	—
311	C₈	C₂²×C₁₁⋊C₈	1	2	yes	—
312	C₈	C₂²×C₁₁⋊C₈	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	D₄×C₁₁⋊C₈	1	2	yes	—
319	C₂²	C₂×D₄×Dic₁₁	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	4	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	D₄×C₁₁⋊C₈	1	2	yes	—
327	C₄	#10623301	1	2	yes	—
328	C₄	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
329	C₄	C₂×#67001	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	4	4	no	—
332	C₂	C₂×D₄×C₁₁⋊C₈	1	1	yes	characteristic
333	C₂	#13077500	1	1	yes	characteristic
334	C₂	#13077533	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₄×C₁₁	C₂³×C₈
Frattini	C₄²	C₂³×D₁₁
Fitting	C₄×C₁₁×#135	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂×C₁₁×#135	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₄×#135	D₁₁
hypocenter	C₁₁	#17858
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #13077727	C₂	semidirect
2. #13067440	C₂	semidirect
3. #13067435	C₂	semidirect
4. SD₁₆×C₁₁⋊C₈	C₂	semidirect
5. C₂×#10622095	C₂	semidirect
6. #13067003	C₂	semidirect
7. #13066899	C₂	semidirect
35. C₁₁×#135	C₈	semidirect
77. C₁₁×#41	C₂×C₈	semidirect
87. C₁₁×Q₁₆	C₂×C₈	semidirect
88. C₁₁×SD₁₆	C₂×C₈	semidirect
89. C₁₁×D₈	C₂×C₈	semidirect
91. C₂×C₈×C₁₁	C₂×C₈	semidirect
197. #135	C₁₁⋊C₈	semidirect
243. #41	C₂×C₁₁⋊C₈	semidirect
253. Q₁₆	C₂×C₁₁⋊C₈	semidirect
254. SD₁₆	C₂×C₁₁⋊C₈	semidirect
255. D₈	C₂×C₁₁⋊C₈	semidirect
262. C₂×C₈	C₂×C₁₁⋊C₈	semidirect
281. C₁₁	#17858	semidirect

Maximal subgroups

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

subgroup	index
#13077727	2
#13067440	2
#13067435	2
SD₁₆×C₁₁⋊C₈	2
C₂×#10622095	2
#13067003	2
#13066899	2
C₄×C₁₁×#135	2
#17858	11