Subgroups — #14907492

Order 2816.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077593	C₂	1	1	yes	maximal, characteristic
2	#13070525	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070449	C₂	1	1	yes	maximal, characteristic, complemented
4	#13070440	C₂	1	2	yes	maximal, complemented
5	C₁₆×D₄₄	C₂	1	2	yes	maximal, complemented
6	#13070435	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070421	C₂	1	2	yes	maximal, complemented
8	#13070413	C₂	1	1	yes	maximal, characteristic, complemented
9	#13070373	C₂	1	2	yes	maximal, complemented
10	D₁₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623111	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₁₁	C₂²	1	1	yes	characteristic
14	#10622637	C₄	1	1	yes	characteristic
15	C₂×#66862	C₂²	1	2	yes	—
16	C₂×C₁₆×D₁₁	C₂²	1	2	yes	—
17	C₂×C₁₆×D₁₁	C₂²	1	2	yes	—
18	#10622453	C₄	1	2	yes	—
19	#10622451	C₄	1	1	yes	characteristic
20	#10622449	C₄	1	2	yes	—
21	#10622448	C₄	1	1	yes	characteristic
22	#10622302	C₂²	1	2	yes	—
23	C₈×D₄₄	C₂²	1	1	yes	characteristic
24	C₈×D₄₄	C₂²	1	2	yes	—
25	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
26	C₁₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₁₁×C₄⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
29	C₄×C₁₁×C₁₆	C₂²	1	2	yes	—
30	(C₂×D₁₁)⋊C₁₆	C₂²	1	1	yes	characteristic
31	(C₂×D₁₁)⋊C₁₆	C₂²	1	1	yes	characteristic
32	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
33	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
34	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
35	Dic₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
36	Dic₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
39	#10622099	C₂²	1	2	yes	—
40	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
42	#67000	C₈	1	1	yes	characteristic
43	#66999	C₈	1	1	yes	characteristic
44	C₂×#4313	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₁₁	C₂³	1	1	yes	characteristic
47	C₂×C₈×D₁₁	C₂³	1	2	yes	—
48	#66951	C₈	1	2	yes	—
49	#66948	C₈	1	1	yes	characteristic
50	#66947	C₈	1	2	yes	—
51	#66943	C₈	1	1	yes	characteristic
52	C₄×D₄₄	C₂×C₄	1	2	yes	—
53	C₄×D₄₄	C₂×C₄	1	1	yes	characteristic
54	#66925	C₂×C₄	1	2	yes	—
55	C₄²×D₁₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
59	C₁₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₁	C₂³	1	2	yes	—
63	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
64	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
65	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
66	#66879	C₂×C₄	1	2	yes	—
67	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
69	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
71	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
72	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
73	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
74	#66867	C₂×C₄	1	2	yes	—
75	C₄×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
76	#66862	—	4	2	no	—
77	C₁₆×D₁₁	—	4	2	no	—
78	C₁₆×D₁₁	—	4	2	no	—
79	#33508	—	11	1	no	maximal, Sylow
80	C₂×Q₈×C₁₁	C₂×C₈	1	1	yes	characteristic
81	C₂×D₄₄	C₂×C₈	1	2	yes	—
82	C₂×D₄₄	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
90	C₁₁×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
91	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
92	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
94	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
95	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
96	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
97	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
98	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
100	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
101	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
102	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
103	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
109	#4313	—	4	2	no	—
110	C₈×D₁₁	—	2	1	no	—
111	C₈×D₁₁	—	2	2	no	—
112	C₈×D₁₁	—	2	1	no	—
113	C₈×D₁₁	—	2	2	no	—
114	C₈×D₁₁	—	4	1	no	—
115	C₁₁×C₁₆	—	2	2	no	—
116	C₁₁×C₁₆	—	2	1	no	—
117	C₁₁×C₁₆	—	2	1	no	—
118	C₁₁⋊C₁₆	—	2	1	no	—
119	C₁₁⋊C₁₆	—	2	2	no	—
120	C₁₁⋊C₁₆	—	2	1	no	—
121	#2966	—	11	1	no	—
122	#2183	—	11	1	no	—
123	Q₈×C₁₆	D₁₁	1	1	yes	characteristic, complemented
124	#2169	—	11	1	no	—
125	#2168	—	11	2	no	—
126	D₄×C₁₆	—	11	1	no	—
127	D₄×C₁₆	—	11	2	no	—
128	#2164	—	11	2	no	—
129	#2162	—	11	1	no	—
130	#2151	—	11	2	no	—
131	C₂×C₄⋊C₁₆	—	11	1	no	—
132	C₂²×D₁₁	C₂²×C₈	1	2	yes	—
133	C₂²×D₁₁	C₂²×C₈	1	1	yes	characteristic
134	Q₈×C₁₁	—	2	2	no	—
135	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
140	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
141	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
142	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
143	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
144	D₄₄	—	2	4	no	—
145	D₄₄	—	2	2	no	—
146	C₄×D₁₁	—	2	2	no	—
147	C₄×D₁₁	—	2	1	no	—
148	C₄×D₁₁	—	2	2	no	—
149	C₄×D₁₁	—	2	1	no	—
150	C₄×D₁₁	—	2	2	no	—
151	C₄×D₁₁	—	2	2	no	—
152	C₈×C₁₁	—	2	1	no	—
153	C₈×C₁₁	—	2	2	no	—
154	C₈×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
155	C₈×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
156	C₁₁⋊C₈	—	2	1	no	—
157	C₁₁⋊C₈	—	2	2	no	—
158	C₁₁⋊C₈	—	2	1	no	—
159	#529	—	11	1	no	—
160	C₂×M₃₂	—	11	2	no	—
161	C₂²×C₁₆	—	11	2	no	—
162	C₂²×C₁₆	—	11	2	no	—
163	#446	—	11	1	no	—
164	C₈×Q₈	C₂×D₁₁	1	1	yes	characteristic
165	#436	—	11	1	no	—
166	#435	—	11	2	no	—
167	C₈×D₄	—	11	2	no	—
168	C₈×D₄	—	11	1	no	—
169	#433	—	11	2	no	—
170	#432	—	11	1	no	—
171	#405	—	11	2	no	—
172	C₂×C₄×C₈	—	11	1	no	—
173	C₄⋊C₁₆	—	11	2	no	—
174	C₄⋊C₁₆	—	11	1	no	—
175	C₄⋊C₁₆	—	11	1	no	—
176	C₄⋊C₁₆	—	11	1	no	—
177	C₄⋊C₁₆	—	11	2	no	—
178	C₄⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
179	C₄⋊C₁₆	C₂×D₁₁	1	2	yes	—
180	C₂²⋊C₁₆	—	11	1	no	—
181	C₂²⋊C₁₆	—	11	1	no	—
182	C₂²⋊C₁₆	—	11	2	no	—
183	C₂²⋊C₁₆	—	11	2	no	—
184	#346	—	11	2	no	—
185	C₄×C₁₆	—	11	1	no	—
186	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
187	C₄×C₁₆	C₂×D₁₁	1	2	yes	—
188	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
189	C₂×D₁₁	—	2	1	no	—
190	C₂×D₁₁	—	2	1	no	—
191	C₂×D₁₁	—	2	2	no	—
192	C₂×D₁₁	—	2	2	no	—
193	C₂×D₁₁	—	2	2	no	—
194	C₂×D₁₁	#504	1	2	yes	—
195	C₄×C₁₁	—	2	2	no	—
196	C₄×C₁₁	—	2	1	no	—
197	C₄×C₁₁	—	2	2	no	—
198	C₄×C₁₁	#504	1	2	yes	—
199	C₄×C₁₁	#520	1	1	yes	characteristic
200	C₄×C₁₁	#568	1	1	yes	characteristic
201	Dic₁₁	—	2	1	no	—
202	Dic₁₁	—	2	1	no	—
203	Dic₁₁	—	2	1	no	—
204	Dic₁₁	—	2	2	no	—
205	C₂×M₁₆	—	11	2	no	—
206	C₂²×C₈	—	11	1	no	—
207	C₂²×C₈	—	11	2	no	—
208	C₂²×C₈	—	11	1	no	—
209	#126	—	11	2	no	—
210	#125	—	11	1	no	—
211	#124	—	11	1	no	—
212	#123	—	11	2	no	—
213	C₂²⋊Q₈	—	11	1	no	—
214	#121	—	11	1	no	—
215	C₄×Q₈	C₄×D₁₁	1	1	yes	characteristic
216	C₄×D₄	—	11	2	no	—
217	C₄×D₄	—	11	1	no	—
218	#117	—	11	2	no	—
219	C₂×C₄²	—	11	1	no	—
220	M₃₂	—	44	2	no	—
221	C₂×C₁₆	—	11	1	no	—
222	C₂×C₁₆	—	11	1	no	—
223	C₂×C₁₆	—	11	2	no	—
224	C₂×C₁₆	—	44	2	no	—
225	C₂×C₁₆	—	44	2	no	—
226	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
227	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
228	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
229	C₄⋊C₈	—	11	1	no	—
230	C₄⋊C₈	—	11	2	no	—
231	C₄⋊C₈	C₄×D₁₁	1	1	yes	characteristic
232	C₄⋊C₈	C₄×D₁₁	1	2	yes	—
233	C₂²⋊C₈	—	11	2	no	—
234	C₂²⋊C₈	—	11	2	no	—
235	C₂²⋊C₈	—	11	2	no	—
236	#97	—	11	2	no	—
237	#97	—	11	2	no	—
238	C₄×C₈	—	11	1	no	—
239	C₄×C₈	—	11	1	no	—
240	C₄×C₈	—	11	1	no	—
241	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₁₁	1	2	yes	—
243	C₂×C₁₁	#2988	1	1	yes	characteristic
244	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
245	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
246	D₁₁	—	2	2	no	—
247	D₁₁	—	4	2	no	—
248	C₂×Q₈	C₈×D₁₁	1	1	yes	characteristic
249	C₂×D₄	—	11	2	no	—
250	C₂×D₄	—	11	1	no	—
251	C₂²×C₄	—	11	1	no	—
252	C₂²×C₄	—	11	1	no	—
253	C₂²×C₄	—	11	1	no	—
254	C₂²×C₄	—	11	2	no	—
255	M₁₆	—	44	2	no	—
256	C₂×C₈	—	11	1	no	—
257	C₂×C₈	—	11	1	no	—
258	C₂×C₈	—	11	2	no	—
259	C₂×C₈	—	22	1	no	—
260	C₂×C₈	—	22	2	no	—
261	C₂×C₈	—	22	1	no	—
262	C₂×C₈	—	22	2	no	—
263	C₂×C₈	—	44	1	no	—
264	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
266	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
267	C₄⋊C₄	—	11	2	no	—
268	C₄⋊C₄	—	11	2	no	—
269	C₄⋊C₄	—	11	1	no	—
270	C₄⋊C₄	—	11	2	no	—
271	C₄⋊C₄	C₈×D₁₁	1	1	yes	characteristic
272	C₄⋊C₄	C₈×D₁₁	1	2	yes	—
273	C₂²⋊C₄	—	11	2	no	—
274	C₂²⋊C₄	—	11	2	no	—
275	C₂²⋊C₄	—	11	2	no	—
276	C₂²⋊C₄	—	11	2	no	—
277	C₂²⋊C₄	—	11	2	no	—
278	C₄²	—	11	1	no	—
279	C₄²	—	11	1	no	—
280	C₄²	—	11	1	no	—
281	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
282	C₄²	C₂×C₄×D₁₁	1	2	yes	—
283	C₁₆	—	2	1	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	1	no	—
286	C₁₆	—	22	1	no	—
287	C₁₆	—	22	1	no	—
288	C₁₆	—	22	2	no	—
289	C₁₁	#33508	1	1	yes	characteristic, complemented, Sylow
290	C₂³	—	11	1	no	—
291	C₂³	—	11	2	no	—
292	Q₈	—	2	2	no	—
293	D₄	—	22	4	no	—
294	D₄	—	22	2	no	—
295	C₂×C₄	—	11	1	no	—
296	C₂×C₄	—	11	1	no	—
297	C₂×C₄	—	11	1	no	—
298	C₂×C₄	—	11	2	no	—
299	C₂×C₄	—	22	2	no	—
300	C₂×C₄	—	22	1	no	—
301	C₂×C₄	—	22	2	no	—
302	C₂×C₄	—	22	2	no	—
303	C₂×C₄	—	22	1	no	—
304	C₂×C₄	—	22	2	no	—
305	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
306	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
307	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
308	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	—	22	1	no	—
313	C₈	—	22	1	no	—
314	C₈	—	22	2	no	—
315	C₈	#67040	1	1	yes	characteristic
316	C₈	#67042	1	1	yes	characteristic
317	C₂²	—	11	2	no	—
318	C₂²	—	22	1	no	—
319	C₂²	—	22	1	no	—
320	C₂²	—	22	2	no	—
321	C₂²	—	22	2	no	—
322	C₂²	—	22	2	no	—
323	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	22	1	no	—
328	C₄	—	22	1	no	—
329	C₄	—	22	1	no	—
330	C₄	—	22	2	no	—
331	C₄	#10622514	1	2	yes	—
332	C₄	#10623105	1	1	yes	characteristic
333	C₄	#10623281	1	1	yes	characteristic
334	C₂	—	22	2	no	—
335	C₂	—	44	2	no	—
336	C₂	#13077726	1	1	yes	characteristic
337	C₂	D₁₁×#504	1	1	yes	characteristic
338	C₂	C₂×#10622514	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	Q₈×C₁₁×C₁₆	C₂
prime radical	#66948	C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	Q₈×C₁₆	D₁₁
hypocenter	C₁₁	#33508
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13070525	C₂	semidirect
3. #13070449	C₂	semidirect
4. #13070440	C₂	semidirect
5. C₁₆×D₄₄	C₂	semidirect
6. #13070435	C₂	semidirect
7. #13070421	C₂	semidirect
8. #13070413	C₂	semidirect
9. #13070373	C₂	semidirect
10. D₁₁×C₄⋊C₁₆	C₂	semidirect
11. Q₈×C₁₁×C₁₆	C₂	semidirect
123. Q₈×C₁₆	D₁₁	semidirect
289. C₁₁	#33508	semidirect

Maximal subgroups

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

subgroup	index
#13077593	2
#13070525	2
#13070449	2
#13070440	2
C₁₆×D₄₄	2
#13070435	2
#13070421	2
#13070413	2
#13070373	2
D₁₁×C₄⋊C₁₆	2
Q₈×C₁₁×C₁₆	2
#33508	11