Subgroups — #14725516

Order 2816.

718 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077651	C₂	1	2	yes	maximal, complemented
2	#13067530	C₂	1	2	yes	maximal, complemented
3	#13067446	C₂	1	4	yes	maximal, complemented
4	#13067440	C₂	1	2	yes	maximal, complemented
5	#13067379	C₂	1	1	yes	maximal, characteristic, complemented
6	#13067059	C₂	1	2	yes	maximal, complemented
7	#13066905	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₁₁×D₈	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₁×D₈	C₄	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₁	C₂²	1	2	yes	—
11	C₄×C₁₁×D₈	C₄	1	2	yes	—
12	C₄×C₁₁×D₈	C₄	1	1	yes	characteristic
13	C₄×C₁₁×D₈	C₄	1	1	yes	characteristic
14	C₄×C₁₁×D₈	C₂²	1	1	yes	characteristic
15	C₄×C₁₁×D₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₁×D₈	C₂²	1	2	yes	—
17	C₂×C₁₁×#107	C₄	1	1	yes	characteristic
18	C₂×C₁₁×#102	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
20	#10622820	—	2	2	no	—
21	C₂×C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
22	#10622625	C₂²	1	4	yes	complemented
23	#10622624	—	2	4	no	—
24	D₄×C₁₁⋊C₈	C₂²	1	4	yes	complemented
25	#10622539	C₂²	1	2	yes	—
26	#10622538	C₂²	1	1	yes	characteristic
27	C₄×#4318	C₂²	1	1	yes	characteristic
28	D₄⋊(C₁₁⋊C₈)	C₂²	1	4	yes	complemented
29	D₄⋊(C₁₁⋊C₈)	C₂²	1	4	yes	complemented
30	#10622096	C₂²	1	2	yes	complemented
31	#10622095	C₂²	1	2	yes	complemented
32	#10622093	C₂²	1	2	yes	complemented
33	#10622092	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₁₁	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₁₁	—	2	2	no	—
36	C₂×C₁₁×D₈	—	2	4	no	—
37	C₂×C₁₁×D₈	C₂×C₄	1	2	yes	—
38	C₂×C₁₁×D₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₁×D₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
42	C₄×D₄×C₁₁	—	2	2	no	—
43	C₄×D₄×C₁₁	—	2	2	no	—
44	C₄×D₄×C₁₁	C₂×C₄	1	2	yes	—
45	C₄×D₄×C₁₁	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₁₁	C₂³	1	2	yes	—
47	C₄×D₄×C₁₁	C₂³	1	2	yes	—
48	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₁₁×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
51	C₂×#4318	—	2	1	no	—
52	C₂×#4318	—	2	4	no	—
53	C₂²×C₁₁⋊C₈	—	2	4	no	—
54	C₁₁×#107	C₂×C₄	1	2	yes	—
55	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
56	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
57	C₁₁×#102	C₂×C₄	1	2	yes	—
58	C₁₁×#102	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	C₂²⋊(C₁₁⋊C₈)	—	2	4	no	—
68	C₂²⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
69	C₄⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
70	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
71	#66867	—	2	2	no	—
72	#66867	—	2	1	no	—
73	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
74	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
75	#18064	—	11	1	no	maximal, Sylow
76	C₂⁴×C₁₁	—	2	2	no	—
77	C₂×D₄×C₁₁	—	2	4	no	—
78	C₂×D₄×C₁₁	—	2	2	no	—
79	C₂×D₄×C₁₁	—	4	2	no	—
80	C₂×D₄×C₁₁	C₂²×C₄	1	2	yes	—
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	—	2	2	no	—
86	C₂²×C₄×C₁₁	—	2	2	no	—
87	C₂²×C₄×C₁₁	—	8	4	no	—
88	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
91	C₁₁×D₈	—	2	2	no	—
92	C₁₁×D₈	—	2	4	no	—
93	C₁₁×D₈	M₁₆	1	4	yes	complemented
94	C₂×C₈×C₁₁	—	2	2	no	—
95	C₂×C₈×C₁₁	—	4	1	no	—
96	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
100	C₁₁×C₄⋊C₄	—	2	2	no	—
101	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
102	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
103	C₁₁×C₂²⋊C₄	—	2	2	no	—
104	C₁₁×C₂²⋊C₄	—	2	2	no	—
105	C₁₁×C₂²⋊C₄	—	2	2	no	—
106	C₁₁×C₂²⋊C₄	—	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	2	no	—
112	#4318	—	8	4	no	—
113	C₂×C₁₁⋊C₈	—	2	4	no	—
114	C₂×C₁₁⋊C₈	—	2	2	no	—
115	C₂×C₁₁⋊C₈	—	8	4	no	—
116	C₂×C₁₁⋊C₈	C₂×D₄	1	4	yes	—
117	#2974	—	11	2	no	—
118	C₂×C₄×D₈	D₁₁	1	1	yes	characteristic
119	#1594	—	11	2	no	—
120	#1581	—	11	4	no	—
121	#1580	—	11	2	no	—
122	#1568	—	11	1	no	—
123	#1490	—	11	2	no	—
124	#1451	—	11	1	no	—
125	C₂³×C₁₁	—	2	2	no	—
126	C₂³×C₁₁	—	2	2	no	—
127	C₂³×C₁₁	—	4	4	no	—
128	C₂³×C₁₁	—	4	4	no	—
129	C₂³×C₁₁	—	8	2	no	—
130	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
131	D₄×C₁₁	—	2	2	no	—
132	D₄×C₁₁	—	2	2	no	—
133	D₄×C₁₁	—	2	2	no	—
134	D₄×C₁₁	—	4	4	no	—
135	D₄×C₁₁	C₂×M₁₆	1	4	yes	—
136	C₂×C₄×C₁₁	—	2	2	no	—
137	C₂×C₄×C₁₁	—	2	1	no	—
138	C₂×C₄×C₁₁	—	2	1	no	—
139	C₂×C₄×C₁₁	—	2	2	no	—
140	C₂×C₄×C₁₁	—	2	2	no	—
141	C₂×C₄×C₁₁	—	2	2	no	—
142	C₂×C₄×C₁₁	—	4	4	no	—
143	C₂×C₄×C₁₁	—	4	4	no	—
144	C₂×C₄×C₁₁	—	8	2	no	—
145	C₂×C₄×C₁₁	—	8	2	no	—
146	C₂×C₄×C₁₁	—	8	2	no	—
147	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
153	C₈×C₁₁	—	2	1	no	—
154	C₈×C₁₁	—	2	1	no	—
155	C₈×C₁₁	—	2	1	no	—
156	C₈×C₁₁	C₂×M₁₆	1	2	yes	—
157	C₁₁⋊C₈	—	2	4	no	—
158	C₁₁⋊C₈	—	4	2	no	—
159	C₁₁⋊C₈	—	4	4	no	—
160	C₂²×D₈	Dic₁₁	1	1	yes	characteristic
161	C₂×C₄×D₄	C₂×D₁₁	1	2	yes	—
162	C₄×D₈	Dic₁₁	1	2	yes	—
163	C₄×D₈	Dic₁₁	1	1	yes	characteristic
164	C₄×D₈	Dic₁₁	1	1	yes	characteristic
165	C₄×D₈	C₂×D₁₁	1	1	yes	characteristic
166	C₄×D₈	C₂×D₁₁	1	2	yes	—
167	C₄×D₈	C₂×D₁₁	1	1	yes	characteristic
168	#436	—	11	4	no	—
169	#435	—	22	4	no	—
170	C₈×D₄	—	11	4	no	—
171	#432	—	11	2	no	—
172	#431	—	11	1	no	—
173	C₂×#107	Dic₁₁	1	1	yes	characteristic
174	C₂×#102	Dic₁₁	1	2	yes	—
175	#407	—	22	2	no	—
176	C₂×C₂²⋊C₈	—	22	2	no	—
177	C₄×M₁₆	—	11	1	no	—
178	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
179	#335	—	11	2	no	—
180	#334	—	11	2	no	—
181	D₄⋊C₈	—	11	4	no	—
182	D₄⋊C₈	—	11	4	no	—
183	#322	—	11	2	no	—
184	#322	—	11	2	no	—
185	C₂²×C₁₁	—	2	1	no	—
186	C₂²×C₁₁	—	2	1	no	—
187	C₂²×C₁₁	—	2	4	no	—
188	C₂²×C₁₁	—	4	2	no	—
189	C₂²×C₁₁	—	4	2	no	—
190	C₂²×C₁₁	—	4	2	no	—
191	C₂²×C₁₁	—	4	2	no	—
192	C₂²×C₁₁	—	4	2	no	—
193	C₂²×C₁₁	—	8	8	no	—
194	C₂²×C₁₁	#444	1	2	yes	—
195	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
196	C₄×C₁₁	—	2	1	no	—
197	C₄×C₁₁	—	2	1	no	—
198	C₄×C₁₁	—	2	1	no	—
199	C₄×C₁₁	—	2	1	no	—
200	C₄×C₁₁	—	4	2	no	—
201	C₄×C₁₁	—	4	2	no	—
202	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
203	C₄×C₁₁	#568	1	1	yes	characteristic
204	C₄×C₁₁	#576	1	2	yes	—
205	C₂²×D₄	C₂×Dic₁₁	1	2	yes	—
206	C₂³×C₄	—	2	2	no	—
207	C₂×D₈	—	2	4	no	—
208	C₂×D₈	C₂×Dic₁₁	1	2	yes	—
209	C₂×D₈	C₂×Dic₁₁	1	1	yes	characteristic
210	C₂×D₈	C₂×Dic₁₁	1	1	yes	characteristic
211	C₂×M₁₆	—	22	4	no	—
212	C₂×M₁₆	—	22	1	no	—
213	C₂²×C₈	—	22	4	no	—
214	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
215	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	C₂×Dic₁₁	1	2	yes	—
219	C₄×D₄	C₂×Dic₁₁	1	2	yes	—
220	C₄×D₄	C₂²×D₁₁	1	2	yes	—
221	C₄×D₄	C₂²×D₁₁	1	2	yes	—
222	C₂×C₄⋊C₄	C₂×Dic₁₁	1	2	yes	—
223	C₂×C₂²⋊C₄	—	2	2	no	—
224	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
225	#107	C₂×Dic₁₁	1	2	yes	—
226	#107	C₂×Dic₁₁	1	1	yes	characteristic
227	#107	C₂×Dic₁₁	1	1	yes	characteristic
228	C₄⋊C₈	—	11	2	no	—
229	C₄⋊C₈	—	11	4	no	—
230	#102	C₂×Dic₁₁	1	2	yes	—
231	#102	C₂×Dic₁₁	1	2	yes	—
232	#102	C₂×Dic₁₁	1	2	yes	—
233	#102	C₂×Dic₁₁	1	2	yes	—
234	C₂²⋊C₈	—	11	2	no	—
235	C₂²⋊C₈	—	22	4	no	—
236	C₂²⋊C₈	—	22	4	no	—
237	C₂²⋊C₈	—	22	2	no	—
238	#97	—	22	2	no	—
239	#97	—	22	1	no	—
240	C₄×C₈	—	11	2	no	—
241	C₄×C₈	—	11	2	no	—
242	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
243	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
246	C₂×C₁₁	—	2	2	no	—
247	C₂×C₁₁	—	4	4	no	—
248	C₂×C₁₁	—	8	2	no	—
249	C₂×C₁₁	D₄×M₁₆	1	1	yes	characteristic
250	C₂×C₁₁	#2948	1	1	yes	characteristic
251	C₂×C₁₁	C₂×#444	1	1	yes	characteristic
252	C₂⁴	—	2	2	no	—
253	C₂×D₄	—	2	4	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	C₂²×Dic₁₁	1	2	yes	—
257	C₂×D₄	C₂²×Dic₁₁	1	2	yes	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	8	4	no	—
264	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
265	C₂²×C₄	#4341	1	1	yes	characteristic
266	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
267	D₈	—	2	4	no	—
268	D₈	—	2	2	no	—
269	D₈	#4318	1	4	yes	complemented
270	M₁₆	—	44	2	no	—
271	M₁₆	—	88	4	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	11	4	no	—
275	C₂×C₈	—	22	4	no	—
276	C₂×C₈	—	22	2	no	—
277	C₂×C₈	—	88	4	no	—
278	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
279	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
280	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
281	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	C₂²×Dic₁₁	1	2	yes	—
284	C₄⋊C₄	C₂²×Dic₁₁	1	2	yes	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₂²⋊C₄	—	2	2	no	—
288	C₂²⋊C₄	—	2	2	no	—
289	C₄²	D₄×D₁₁	1	1	yes	characteristic
290	C₄²	#4341	1	1	yes	characteristic
291	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
292	C₄²	C₂³×D₁₁	1	1	yes	characteristic
293	C₁₁	#18064	1	1	yes	characteristic, complemented, Sylow
294	C₂³	—	2	2	no	—
295	C₂³	—	2	2	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	4	4	no	—
298	C₂³	—	8	2	no	—
299	C₂³	D₄×Dic₁₁	1	1	yes	characteristic
300	D₄	—	2	2	no	—
301	D₄	—	2	2	no	—
302	D₄	—	2	2	no	—
303	D₄	—	4	4	no	—
304	D₄	C₂×#4318	1	4	yes	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
317	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
318	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
320	C₂×C₄	C₂×#4341	1	1	yes	characteristic
321	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	22	4	no	—
326	C₈	—	44	2	no	—
327	C₈	—	44	4	no	—
328	C₈	C₂×#4318	1	2	yes	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	4	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	2	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	8	8	no	—
338	C₂²	#10622810	1	2	yes	—
339	C₂²	C₂×D₄×Dic₁₁	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	2	no	—
346	C₄	#10623288	1	2	yes	—
347	C₄	C₂²×#4318	1	1	yes	characteristic
348	C₄	#10623350	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	4	no	—
351	C₂	—	8	2	no	—
352	C₂	D₄×#4318	1	1	yes	characteristic
353	C₂	#13077468	1	1	yes	characteristic
354	C₂	C₂×#10622810	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₄×C₁₁×D₈	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂²×C₁₁×D₈	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×C₄×D₈	D₁₁
hypocenter	C₁₁	#18064
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #13077651	C₂	semidirect
2. #13067530	C₂	semidirect
3. #13067446	C₂	semidirect
4. #13067440	C₂	semidirect
5. #13067379	C₂	semidirect
6. #13067059	C₂	semidirect
7. #13066905	C₂	semidirect
22. #10622625	C₂²	semidirect
24. D₄×C₁₁⋊C₈	C₂²	semidirect
28. D₄⋊(C₁₁⋊C₈)	C₂²	semidirect
29. D₄⋊(C₁₁⋊C₈)	C₂²	semidirect
30. #10622096	C₂²	semidirect
31. #10622095	C₂²	semidirect
32. #10622093	C₂²	semidirect
33. #10622092	C₂²	semidirect
93. C₁₁×D₈	M₁₆	semidirect
269. D₈	#4318	semidirect
293. C₁₁	#18064	semidirect

Maximal subgroups

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

subgroup	index
#13077651	2
#13067530	2
#13067446	2
#13067440	2
#13067379	2
#13067059	2
#13066905	2
C₂×C₄×C₁₁×D₈	2
#18064	11