Subgroups — #14723523

Order 2816.

682 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077583	C₂	1	2	yes	maximal, complemented
2	#13067513	C₂	1	2	yes	maximal, complemented
3	#13067507	C₂	1	4	yes	maximal, complemented
4	#13067501	C₂	1	2	yes	maximal, complemented
5	#13067364	C₂	1	1	yes	maximal, characteristic, complemented
6	#13066998	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×C₁₁⋊C₈	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	#10622637	C₂²	1	4	yes	—
21	#10622625	C₂²	1	4	yes	—
22	#10622624	—	2	4	no	—
23	#10622538	—	2	2	no	—
24	#10622538	C₂²	1	2	yes	—
25	C₄×#4318	—	2	2	no	—
26	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
28	Q₈⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
29	D₄⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
30	#10622096	C₂²	1	2	yes	—
31	#10622095	C₂²	1	2	yes	—
32	C₈×C₁₁⋊C₈	C₂²	1	2	yes	—
33	C₈×C₁₁⋊C₈	C₂²	1	2	yes	—
34	C₂×C₁₁×#41	C₂×C₄	1	2	yes	—
35	C₁₁×#135	—	2	4	no	—
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	C₂×#4318	—	2	4	no	—
52	C₂²×C₁₁⋊C₈	—	2	1	no	—
53	C₂²×C₁₁⋊C₈	D₄	1	2	yes	—
54	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
55	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
56	C₁₁×#106	C₂×C₄	1	2	yes	—
57	C₁₁×#103	C₂×C₄	1	2	yes	—
58	C₁₁×#103	C₂×C₄	1	2	yes	—
59	C₁₁×#102	C₂×C₄	1	2	yes	—
60	C₁₁×#102	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
65	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
66	C₂²⋊(C₁₁⋊C₈)	—	2	4	no	—
67	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
68	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
69	#66867	—	2	4	no	—
70	C₄×C₁₁⋊C₈	—	2	2	no	—
71	C₄×C₁₁⋊C₈	—	2	1	no	—
72	C₄×C₁₁⋊C₈	D₄	1	1	yes	characteristic
73	C₄×C₁₁⋊C₈	D₄	1	1	yes	characteristic
74	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
75	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
76	#17886	—	11	1	no	maximal, Sylow
77	C₁₁×#41	—	2	4	no	—
78	C₁₁×#41	—	2	4	no	—
79	C₂×Q₈×C₁₁	C₂²×C₄	1	2	yes	—
80	C₂×D₄×C₁₁	—	2	2	no	—
81	C₂×D₄×C₁₁	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₁	—	2	2	no	—
83	C₂²×C₄×C₁₁	—	2	2	no	—
84	C₂²×C₄×C₁₁	—	2	2	no	—
85	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
88	C₁₁×Q₁₆	—	2	2	no	—
89	C₁₁×SD₁₆	—	2	4	no	—
90	C₁₁×D₈	—	2	2	no	—
91	C₂×C₈×C₁₁	—	2	2	no	—
92	C₂×C₈×C₁₁	M₁₆	1	4	yes	complemented
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	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
97	C₁₁×C₄⋊C₄	—	2	2	no	—
98	C₁₁×C₄⋊C₄	—	2	2	no	—
99	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
100	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
101	C₁₁×C₂²⋊C₄	—	2	2	no	—
102	C₁₁×C₂²⋊C₄	—	2	2	no	—
103	C₁₁×C₂²⋊C₄	—	2	2	no	—
104	C₄²×C₁₁	—	2	2	no	—
105	C₄²×C₁₁	—	2	2	no	—
106	C₄²×C₁₁	—	2	2	no	—
107	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₁	#41	1	1	yes	characteristic
110	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
111	#4318	—	4	8	no	—
112	C₂×C₁₁⋊C₈	—	2	4	no	—
113	C₂×C₁₁⋊C₈	—	2	4	no	—
114	C₂×C₁₁⋊C₈	—	2	4	no	—
115	C₂×C₁₁⋊C₈	—	2	2	no	—
116	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
117	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
118	#2965	—	11	2	no	—
119	C₄×#135	D₁₁	1	1	yes	characteristic
120	#1591	—	11	2	no	—
121	#1590	—	11	4	no	—
122	#1589	—	11	2	no	—
123	#1564	—	11	1	no	—
124	#1476	—	11	2	no	—
125	C₂×C₈²	—	11	1	no	—
126	C₂³×C₁₁	—	2	2	no	—
127	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
128	Q₈×C₁₁	—	2	2	no	—
129	Q₈×C₁₁	—	2	2	no	—
130	D₄×C₁₁	—	2	2	no	—
131	D₄×C₁₁	—	2	2	no	—
132	D₄×C₁₁	—	4	4	no	—
133	C₂×C₄×C₁₁	—	2	2	no	—
134	C₂×C₄×C₁₁	—	2	2	no	—
135	C₂×C₄×C₁₁	—	2	2	no	—
136	C₂×C₄×C₁₁	—	2	2	no	—
137	C₂×C₄×C₁₁	—	2	2	no	—
138	C₂×C₄×C₁₁	—	4	4	no	—
139	C₂×C₄×C₁₁	—	4	4	no	—
140	C₂×C₄×C₁₁	—	4	4	no	—
141	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
145	C₂×C₄×C₁₁	#131	1	2	yes	—
146	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
149	C₈×C₁₁	—	2	1	no	—
150	C₈×C₁₁	—	2	1	no	—
151	C₈×C₁₁	C₂×M₁₆	1	2	yes	—
152	C₈×C₁₁	C₂×M₁₆	1	2	yes	—
153	C₁₁⋊C₈	—	2	4	no	—
154	C₁₁⋊C₈	—	2	2	no	—
155	C₁₁⋊C₈	—	4	4	no	—
156	C₁₁⋊C₈	#135	1	4	yes	complemented
157	C₂×#135	Dic₁₁	1	1	yes	characteristic
158	C₄×#41	C₂×D₁₁	1	2	yes	—
159	#446	—	11	4	no	—
160	C₄×Q₁₆	Dic₁₁	1	1	yes	characteristic
161	C₄×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
162	C₄×SD₁₆	Dic₁₁	1	2	yes	—
163	C₄×SD₁₆	C₂×D₁₁	1	2	yes	—
164	C₄×D₈	Dic₁₁	1	1	yes	characteristic
165	C₄×D₈	C₂×D₁₁	1	1	yes	characteristic
166	#436	—	11	4	no	—
167	#435	—	22	4	no	—
168	#431	—	11	2	no	—
169	#431	—	22	2	no	—
170	#427	Dic₁₁	1	1	yes	characteristic
171	#416	Dic₁₁	1	2	yes	—
172	C₄×M₁₆	—	22	2	no	—
173	C₂×C₄×C₈	—	11	1	no	—
174	C₂×C₄×C₈	—	11	1	no	—
175	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
176	#335	—	11	2	no	—
177	#334	—	11	2	no	—
178	Q₈⋊C₈	—	11	4	no	—
179	D₄⋊C₈	—	11	4	no	—
180	C₈²	—	11	2	no	—
181	C₈²	—	11	2	no	—
182	C₂²×C₁₁	—	2	1	no	—
183	C₂²×C₁₁	—	2	1	no	—
184	C₂²×C₁₁	—	4	2	no	—
185	C₂²×C₁₁	—	4	2	no	—
186	C₂²×C₁₁	—	4	2	no	—
187	C₂²×C₁₁	#436	1	2	yes	—
188	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
189	C₄×C₁₁	—	2	1	no	—
190	C₄×C₁₁	—	2	1	no	—
191	C₄×C₁₁	—	4	2	no	—
192	C₄×C₁₁	—	4	2	no	—
193	C₄×C₁₁	—	4	2	no	—
194	C₄×C₁₁	#436	1	2	yes	—
195	C₄×C₁₁	#443	1	2	yes	—
196	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
197	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
198	C₄×C₁₁	C₂×#135	1	2	yes	—
199	C₂×#41	C₂×Dic₁₁	1	2	yes	—
200	#135	—	2	4	no	—
201	C₂×Q₁₆	C₂×Dic₁₁	1	1	yes	characteristic
202	C₂×SD₁₆	C₂×Dic₁₁	1	2	yes	—
203	C₂×D₈	C₂×Dic₁₁	1	1	yes	characteristic
204	C₂×M₁₆	—	22	4	no	—
205	C₂²×C₈	—	11	2	no	—
206	C₂²×C₈	—	22	1	no	—
207	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
208	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
209	C₄×Q₈	C₂×Dic₁₁	1	2	yes	—
210	C₄×Q₈	C₂²×D₁₁	1	2	yes	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	C₂×Dic₁₁	1	2	yes	—
214	C₄×D₄	C₂²×D₁₁	1	2	yes	—
215	#117	—	2	2	no	—
216	#117	C₂×Dic₁₁	1	2	yes	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
219	#107	C₂×Dic₁₁	1	1	yes	characteristic
220	#107	C₂×Dic₁₁	1	1	yes	characteristic
221	#106	C₂×Dic₁₁	1	2	yes	—
222	C₄⋊C₈	—	11	4	no	—
223	C₄⋊C₈	—	22	4	no	—
224	#103	C₂×Dic₁₁	1	2	yes	—
225	#103	C₂×Dic₁₁	1	2	yes	—
226	#102	C₂×Dic₁₁	1	2	yes	—
227	#102	C₂×Dic₁₁	1	2	yes	—
228	C₂²⋊C₈	—	22	4	no	—
229	C₂²⋊C₈	—	22	2	no	—
230	#97	—	22	4	no	—
231	C₄×C₈	—	11	1	no	—
232	C₄×C₈	—	11	1	no	—
233	C₄×C₈	—	11	2	no	—
234	C₄×C₈	—	11	2	no	—
235	C₄×C₈	—	22	2	no	—
236	C₄×C₈	—	22	1	no	—
237	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
238	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
241	C₂×C₁₁	—	2	2	no	—
242	C₂×C₁₁	—	8	2	no	—
243	C₂×C₁₁	C₂×#436	1	1	yes	characteristic
244	C₂×C₁₁	C₄×#135	1	1	yes	characteristic
245	C₂×C₁₁	C₂×#443	1	1	yes	characteristic
246	#41	—	2	4	no	—
247	#41	—	2	4	no	—
248	C₂×Q₈	C₂²×Dic₁₁	1	2	yes	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	C₂²×Dic₁₁	1	2	yes	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
255	C₂²×C₄	#4341	1	1	yes	characteristic
256	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
257	Q₁₆	—	2	2	no	—
258	SD₁₆	—	2	4	no	—
259	D₈	—	2	2	no	—
260	M₁₆	—	44	8	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	11	2	no	—
263	C₂×C₈	—	11	2	no	—
264	C₂×C₈	—	22	4	no	—
265	C₂×C₈	—	22	4	no	—
266	C₂×C₈	—	22	2	no	—
267	C₂×C₈	—	22	4	no	—
268	C₂×C₈	#4318	1	4	yes	complemented
269	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
270	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
271	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
272	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	C₂²×Dic₁₁	1	2	yes	—
276	C₄⋊C₄	C₂²×Dic₁₁	1	2	yes	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	D₄×D₁₁	1	1	yes	characteristic
284	C₄²	#4341	1	1	yes	characteristic
285	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
286	C₄²	C₂³×D₁₁	1	1	yes	characteristic
287	C₁₁	#17886	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	2	2	no	—
289	C₂³	D₄×Dic₁₁	1	1	yes	characteristic
290	Q₈	—	2	2	no	—
291	Q₈	—	2	2	no	—
292	D₄	—	2	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	4	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	C₂×#4318	1	2	yes	—
304	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
305	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
306	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
307	C₂×C₄	#67001	1	2	yes	—
308	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
309	C₂×C₄	C₂×#4341	1	1	yes	characteristic
310	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	11	4	no	—
314	C₈	—	22	4	no	—
315	C₈	—	22	2	no	—
316	C₈	—	44	4	no	—
317	C₈	C₂×#4318	1	2	yes	—
318	C₈	C₂×#4318	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	#10622625	1	2	yes	—
325	C₂²	C₂×D₄×Dic₁₁	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	2	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	2	no	—
331	C₄	#10622625	1	2	yes	—
332	C₄	#10622809	1	2	yes	—
333	C₄	D₁₁×#135	1	2	yes	—
334	C₄	C₂²×#4318	1	1	yes	characteristic
335	C₄	C₂×#67001	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	8	2	no	—
338	C₂	C₂×#10622625	1	1	yes	characteristic
339	C₂	#135×Dic₁₁	1	1	yes	characteristic
340	C₂	C₂×#10622809	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
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	#17886
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #13077583	C₂	semidirect
2. #13067513	C₂	semidirect
3. #13067507	C₂	semidirect
4. #13067501	C₂	semidirect
5. #13067364	C₂	semidirect
6. #13066998	C₂	semidirect
7. C₂×C₈×C₁₁⋊C₈	C₂	semidirect
92. C₂×C₈×C₁₁	M₁₆	semidirect
156. C₁₁⋊C₈	#135	semidirect
268. C₂×C₈	#4318	semidirect
287. C₁₁	#17886	semidirect

Maximal subgroups

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

subgroup	index
#13077583	2
#13067513	2
#13067507	2
#13067501	2
#13067364	2
#13066998	2
C₂×C₈×C₁₁⋊C₈	2
C₄×C₁₁×#135	2
#17886	11