Subgroups — #14719516

Order 2816.

798 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077582	C₂	1	1	yes	maximal, characteristic
2	#13077582	C₂	1	1	yes	maximal, characteristic
3	C₄×#66954	C₂	1	1	yes	maximal, characteristic
4	#13066920	C₂	1	2	yes	maximal, complemented
5	#13066912	C₂	1	2	yes	maximal, complemented
6	#13066908	C₂	1	1	yes	maximal, characteristic, complemented
7	#13066904	C₂	1	2	yes	maximal, complemented
8	C₈×#4318	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×#4318	C₂	1	1	yes	maximal, characteristic, complemented
10	#13066895	C₂	1	2	yes	maximal, complemented
11	C₂×C₁₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#66954	C₄	1	2	yes	—
13	C₄×#4339	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
17	#10622724	C₄	1	4	yes	—
18	#10622724	C₄	1	4	yes	—
19	#10622714	C₄	1	2	yes	—
20	#10622538	C₂²	1	1	yes	characteristic
21	#10622538	C₂²	1	1	yes	characteristic
22	C₄×#4318	C₂²	1	1	yes	characteristic
23	#10622302	C₄	1	4	yes	—
24	#10622301	C₄	1	2	yes	—
25	#10622301	C₄	1	2	yes	—
26	C₄×#4313	C₂²	1	2	yes	—
27	C₄×#4313	C₂²	1	2	yes	—
28	C₄×#4313	C₂²	1	2	yes	—
29	#10622298	C₂²	1	2	yes	—
30	#10622298	C₂²	1	2	yes	—
31	C₄×C₈×D₁₁	C₂²	1	2	yes	—
32	#10622292	C₄	1	2	yes	—
33	#10622292	C₄	1	2	yes	—
34	C₁₁×#322	C₂²	1	1	yes	characteristic
35	C₁₁×#322	C₂²	1	1	yes	characteristic
36	C₁₁×#322	C₂²	1	1	yes	characteristic
37	C₁₁×#322	C₂²	1	1	yes	characteristic
38	#10622092	C₂²	1	2	yes	—
39	#10622092	C₂²	1	2	yes	—
40	#10622092	C₂²	1	2	yes	—
41	#10622092	C₂²	1	2	yes	—
42	C₈×C₁₁⋊C₈	C₂²	1	2	yes	—
43	C₈×C₁₁⋊C₈	C₂²	1	2	yes	—
44	C₂×#4339	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
49	C₄×C₂²⋊D₁₁	C₂×C₄	1	2	yes	—
50	C₄×C₂²⋊D₁₁	C₂×C₄	1	2	yes	—
51	#66979	C₂×C₄	1	1	yes	characteristic
52	C₂×#4318	C₂×C₄	1	2	yes	—
53	#66954	—	2	8	no	—
54	C₂×#4313	C₂×C₄	1	4	yes	—
55	C₂×#4313	C₂×C₄	1	4	yes	—
56	C₂×#4313	C₂×C₄	1	4	yes	—
57	C₂×C₈×D₁₁	C₂×C₄	1	4	yes	—
58	C₄×D₄₄	C₂×C₄	1	1	yes	characteristic
59	C₄×D₄₄	C₂×C₄	1	1	yes	characteristic
60	#66925	C₂×C₄	1	2	yes	—
61	C₄²×D₁₁	C₂³	1	2	yes	—
62	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
73	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
74	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
75	(C₂×D₁₁)⋊C₈	C₂×C₄	1	4	yes	—
76	(C₂×D₁₁)⋊C₈	C₂×C₄	1	4	yes	—
77	#66879	C₂×C₄	1	4	yes	—
78	#66879	C₂×C₄	1	4	yes	—
79	#66879	C₂×C₄	1	4	yes	—
80	Dic₁₁⋊C₈	C₂×C₄	1	4	yes	—
81	Dic₁₁⋊C₈	C₂×C₄	1	4	yes	—
82	C₈×Dic₁₁	C₂×C₄	1	4	yes	—
83	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
84	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
85	#66867	C₂×C₄	1	2	yes	—
86	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
87	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
88	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
89	#17337	—	11	1	no	maximal, Sylow
90	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₂²⋊D₁₁	C₄²	1	2	yes	—
94	#4339	—	2	4	no	—
95	C₂×D₄₄	C₄²	1	1	yes	characteristic
96	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
99	C₂×Dic₂₂	C₄²	1	1	yes	characteristic
100	C₂×C₈×C₁₁	—	2	4	no	—
101	C₂×C₈×C₁₁	—	2	4	no	—
102	C₂×C₈×C₁₁	—	4	2	no	—
103	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
109	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
113	C₂²⋊Dic₁₁	C₄²	1	2	yes	—
114	(C₂×D₁₁)⋊C₄	C₄²	1	2	yes	—
115	(C₂×D₁₁)⋊C₄	C₄²	1	2	yes	—
116	C₄⋊Dic₁₁	C₄²	1	1	yes	characteristic
117	C₄⋊Dic₁₁	C₄²	1	1	yes	characteristic
118	Dic₁₁⋊C₄	C₄²	1	2	yes	—
119	Dic₁₁⋊C₄	C₄²	1	2	yes	—
120	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
121	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
122	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
123	#4318	M₁₆	1	8	yes	complemented
124	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
125	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
126	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
127	#4313	—	2	8	no	—
128	#4313	—	2	8	no	—
129	#4313	—	4	4	no	—
130	C₈×D₁₁	—	2	8	no	—
131	#2965	—	11	1	no	—
132	#2965	—	11	1	no	—
133	C₄×#131	—	11	1	no	—
134	#1455	—	11	2	no	—
135	#1453	—	11	2	no	—
136	#1452	—	11	1	no	—
137	#1451	—	11	2	no	—
138	C₈×M₁₆	—	11	1	no	—
139	C₈×M₁₆	—	11	1	no	—
140	C₈×M₁₆	—	11	2	no	—
141	C₂×#322	D₁₁	1	1	yes	characteristic, complemented
142	C₂³×C₁₁	C₂×C₄²	1	1	yes	characteristic
143	C₂²×D₁₁	C₂×C₄²	1	2	yes	—
144	C₂×C₄×C₁₁	—	2	2	no	—
145	C₂×C₄×C₁₁	—	2	2	no	—
146	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
149	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
150	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
153	C₂²⋊D₁₁	—	2	4	no	—
154	C₂×Dic₁₁	C₂×C₄²	1	2	yes	—
155	C₂×Dic₁₁	C₂×C₄²	1	2	yes	—
156	C₂×Dic₁₁	C₂×C₄²	1	2	yes	—
157	D₄₄	—	2	2	no	—
158	C₄×D₁₁	—	2	4	no	—
159	C₄×D₁₁	—	2	4	no	—
160	C₄×D₁₁	—	2	4	no	—
161	Dic₂₂	—	2	2	no	—
162	C₈×C₁₁	—	2	2	no	—
163	C₈×C₁₁	—	2	2	no	—
164	C₈×C₁₁	—	2	2	no	—
165	C₈×C₁₁	—	2	2	no	—
166	C₈×C₁₁	—	2	2	no	—
167	C₈×C₁₁	#131	1	4	yes	complemented
168	C₁₁⋊C₈	—	2	4	no	—
169	C₁₁⋊C₈	—	2	4	no	—
170	C₁₁⋊C₈	C₂×M₁₆	1	8	yes	—
171	C₂×#131	—	11	2	no	—
172	C₄×#41	—	11	1	no	—
173	#446	—	11	2	no	—
174	#446	—	11	2	no	—
175	#436	—	11	2	no	—
176	#436	—	11	2	no	—
177	#435	—	11	4	no	—
178	#435	—	11	4	no	—
179	#431	—	11	1	no	—
180	#431	—	11	1	no	—
181	#431	—	11	2	no	—
182	#431	—	11	2	no	—
183	#405	—	11	2	no	—
184	#405	—	11	4	no	—
185	C₄×M₁₆	—	11	1	no	—
186	C₄×M₁₆	—	11	2	no	—
187	C₄×M₁₆	—	11	2	no	—
188	C₄×M₁₆	—	11	2	no	—
189	C₂×C₄×C₈	—	11	2	no	—
190	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
191	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
192	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
193	#322	—	11	2	no	—
194	#322	—	11	2	no	—
195	#322	—	11	2	no	—
196	#322	—	11	2	no	—
197	#322	C₂×D₁₁	1	1	yes	characteristic
198	#322	C₂×D₁₁	1	1	yes	characteristic
199	#322	C₂×D₁₁	1	1	yes	characteristic
200	#322	C₂×D₁₁	1	1	yes	characteristic
201	C₈²	—	11	2	no	—
202	C₈²	—	11	2	no	—
203	C₂²×C₁₁	—	2	1	no	—
204	C₂²×C₁₁	—	2	1	no	—
205	C₂²×C₁₁	C₄×M₁₆	1	2	yes	—
206	C₂²×C₁₁	C₂²×C₄²	1	1	yes	characteristic
207	C₂×D₁₁	—	2	2	no	—
208	C₂×D₁₁	—	2	2	no	—
209	C₂×D₁₁	—	2	2	no	—
210	C₄×C₁₁	—	2	1	no	—
211	C₄×C₁₁	—	2	1	no	—
212	C₄×C₁₁	C₄×M₁₆	1	2	yes	—
213	C₄×C₁₁	C₂²×M₁₆	1	2	yes	—
214	C₄×C₁₁	C₂×#131	1	2	yes	—
215	C₄×C₁₁	#568	1	2	yes	—
216	Dic₁₁	—	2	2	no	—
217	Dic₁₁	—	2	2	no	—
218	Dic₁₁	—	2	2	no	—
219	C₂×#41	—	11	1	no	—
220	#131	—	22	8	no	—
221	C₂×M₁₆	—	11	2	no	—
222	C₂×M₁₆	—	11	4	no	—
223	C₂×M₁₆	—	11	4	no	—
224	C₂×M₁₆	—	11	4	no	—
225	C₂²×C₈	—	11	4	no	—
226	C₂²×C₈	C₄×D₁₁	1	2	yes	—
227	C₂²×C₈	C₄×D₁₁	1	2	yes	—
228	C₂²×C₈	C₄×D₁₁	1	2	yes	—
229	C₄×Q₈	—	11	1	no	—
230	C₄×Q₈	—	11	1	no	—
231	C₄×D₄	—	11	2	no	—
232	C₄×D₄	—	11	2	no	—
233	C₄×D₄	—	11	1	no	—
234	C₄×D₄	—	11	1	no	—
235	#117	—	11	1	no	—
236	#117	—	11	2	no	—
237	C₂×C₄²	—	11	2	no	—
238	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
239	C₄⋊C₈	—	11	4	no	—
240	C₄⋊C₈	—	11	4	no	—
241	C₄⋊C₈	—	11	2	no	—
242	C₄⋊C₈	—	11	2	no	—
243	C₂²⋊C₈	—	11	2	no	—
244	C₂²⋊C₈	—	11	2	no	—
245	C₂²⋊C₈	—	11	4	no	—
246	C₂²⋊C₈	—	11	4	no	—
247	#97	—	11	4	no	—
248	#97	—	11	4	no	—
249	#97	—	11	4	no	—
250	#97	—	11	2	no	—
251	C₄×C₈	—	11	4	no	—
252	C₄×C₈	—	11	2	no	—
253	C₄×C₈	—	11	2	no	—
254	C₄×C₈	—	11	2	no	—
255	C₄×C₈	C₄×D₁₁	1	2	yes	—
256	C₄×C₈	C₄×D₁₁	1	2	yes	—
257	C₄×C₈	C₄×D₁₁	1	2	yes	—
258	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
264	C₂×C₁₁	—	2	2	no	—
265	C₂×C₁₁	C₂×C₄×M₁₆	1	1	yes	characteristic
266	C₂×C₁₁	C₄×#131	1	1	yes	characteristic
267	C₂×C₁₁	#2875	1	1	yes	characteristic
268	D₁₁	—	4	2	no	—
269	#41	—	22	4	no	—
270	C₂×Q₈	—	11	1	no	—
271	C₂×D₄	—	11	2	no	—
272	C₂×D₄	—	11	1	no	—
273	C₂²×C₄	—	11	2	no	—
274	C₂²×C₄	—	11	2	no	—
275	C₂²×C₄	—	11	2	no	—
276	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
279	M₁₆	—	11	8	no	—
280	M₁₆	—	22	8	no	—
281	M₁₆	—	22	8	no	—
282	M₁₆	—	44	4	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	11	4	no	—
287	C₂×C₈	—	11	4	no	—
288	C₂×C₈	—	11	4	no	—
289	C₂×C₈	—	22	8	no	—
290	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
291	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
292	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
293	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
294	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
295	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
296	C₄⋊C₄	—	11	2	no	—
297	C₄⋊C₄	—	11	2	no	—
298	C₄⋊C₄	—	11	1	no	—
299	C₄⋊C₄	—	11	1	no	—
300	C₂²⋊C₄	—	11	2	no	—
301	C₂²⋊C₄	—	11	2	no	—
302	C₂²⋊C₄	—	11	2	no	—
303	C₄²	—	11	2	no	—
304	C₄²	—	11	2	no	—
305	C₄²	—	11	2	no	—
306	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
307	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
308	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
309	C₄²	C₂³×D₁₁	1	1	yes	characteristic
310	C₁₁	#17337	1	1	yes	characteristic, complemented, Sylow
311	C₂³	—	11	2	no	—
312	C₂³	C₄²×D₁₁	1	1	yes	characteristic
313	Q₈	—	22	2	no	—
314	D₄	—	22	4	no	—
315	D₄	—	22	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	11	2	no	—
319	C₂×C₄	—	11	2	no	—
320	C₂×C₄	—	11	2	no	—
321	C₂×C₄	—	22	4	no	—
322	C₂×C₄	—	22	4	no	—
323	C₂×C₄	—	22	4	no	—
324	C₂×C₄	C₄²×D₁₁	1	1	yes	characteristic
325	C₂×C₄	C₄²×D₁₁	1	1	yes	characteristic
326	C₂×C₄	C₄²×D₁₁	1	1	yes	characteristic
327	C₂×C₄	M₁₆×D₁₁	1	4	yes	—
328	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	2	no	—
333	C₈	—	2	2	no	—
334	C₈	—	2	2	no	—
335	C₈	—	2	2	no	—
336	C₈	—	11	8	no	—
337	C₈	—	22	4	no	—
338	C₈	—	22	4	no	—
339	C₈	#66954	1	4	yes	complemented
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	22	2	no	—
343	C₂²	—	22	2	no	—
344	C₂²	—	22	2	no	—
345	C₂²	#10622320	1	2	yes	—
346	C₂²	C₂×C₄²×D₁₁	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	22	2	no	—
350	C₄	—	22	2	no	—
351	C₄	—	22	2	no	—
352	C₄	#10622320	1	2	yes	—
353	C₄	C₂×#66954	1	2	yes	—
354	C₄	C₂×M₁₆×D₁₁	1	2	yes	—
355	C₄	#10623276	1	2	yes	—
356	C₂	—	2	2	no	—
357	C₂	—	44	2	no	—
358	C₂	C₂×#10622320	1	1	yes	characteristic
359	C₂	C₄×#66954	1	1	yes	characteristic
360	C₂	#13077041	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂²×C₄²
Fitting	C₂×C₁₁×#322	C₂
prime radical	C₂×#4339	C₂×C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#322	D₁₁
hypocenter	C₁₁	#17337
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. #13066920	C₂	semidirect
5. #13066912	C₂	semidirect
6. #13066908	C₂	semidirect
7. #13066904	C₂	semidirect
8. C₈×#4318	C₂	semidirect
9. C₈×#4318	C₂	semidirect
10. #13066895	C₂	semidirect
11. C₂×C₁₁×#322	C₂	semidirect
123. #4318	M₁₆	semidirect
141. C₂×#322	D₁₁	semidirect
167. C₈×C₁₁	#131	semidirect
310. C₁₁	#17337	semidirect
339. C₈	#66954	semidirect

Maximal subgroups

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

subgroup	index
#13077582	2
C₄×#66954	2
#13066920	2
#13066912	2
#13066908	2
#13066904	2
C₈×#4318	2
#13066895	2
C₂×C₁₁×#322	2
#17337	11