Subgroups — #14907337

Order 2816.

516 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	#13077591	C₂	1	1	yes	maximal, characteristic
2	#13070524	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₆×Dic₂₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#13070449	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070446	C₂	1	2	yes	maximal, complemented
6	#13070439	C₂	1	2	yes	maximal, complemented
7	C₁₆×D₄₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#13070424	C₂	1	2	yes	maximal, complemented
9	#13070415	C₂	1	1	yes	maximal, characteristic, complemented
10	#13070214	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623010	C₄	1	1	yes	characteristic
13	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
16	#10622724	C₄	1	2	yes	—
17	C₈×C₂²⋊D₁₁	C₂²	1	2	yes	—
18	#10622539	C₄	1	1	yes	characteristic
19	C₂×#66862	C₂²	1	2	yes	—
20	C₂×C₁₆×D₁₁	C₂²	1	2	yes	—
21	#10622303	C₄	1	2	yes	—
22	#10622302	C₂²	1	2	yes	—
23	#10622301	C₄	1	1	yes	characteristic
24	C₈×D₄₄	C₂²	1	1	yes	characteristic
25	#10622292	C₄	1	1	yes	characteristic
26	C₈×Dic₂₂	C₂²	1	1	yes	characteristic
27	C₁₁×#346	C₂²	1	1	yes	characteristic
28	C₁₁×#346	C₂²	1	1	yes	characteristic
29	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
30	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
32	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
33	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
34	#10622140	C₂²	1	2	yes	—
35	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
36	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
37	C₁₆×Dic₁₁	C₂²	1	2	yes	—
38	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
39	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
40	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	—
41	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
44	#66983	C₈	1	1	yes	characteristic
45	#66982	C₈	1	2	yes	—
46	C₄×C₂²⋊D₁₁	C₂×C₄	1	2	yes	—
47	#66977	C₈	1	1	yes	characteristic
48	C₂×#4313	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₁₁	C₂³	1	2	yes	—
50	#66929	C₈	1	2	yes	—
51	#66928	C₈	1	1	yes	characteristic
52	C₄×D₄₄	C₂×C₄	1	1	yes	characteristic
53	#66925	C₂×C₄	1	2	yes	—
54	#66923	C₈	1	1	yes	characteristic
55	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
56	C₁₁×M₃₂	—	2	2	no	—
57	C₁₁×M₃₂	—	2	2	no	—
58	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
66	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
67	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
68	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
69	#66879	C₂×C₄	1	2	yes	—
70	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
71	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
72	C₈×Dic₁₁	C₂³	1	2	yes	—
73	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
74	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
75	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
76	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
77	#66867	C₂×C₄	1	2	yes	—
78	#66862	—	2	4	no	—
79	C₁₆×D₁₁	—	2	4	no	—
80	#33483	—	11	1	no	maximal, Sylow
81	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₁₁	C₂×C₈	1	2	yes	—
85	C₂×D₄₄	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
87	C₂×Dic₂₂	C₂×C₈	1	1	yes	characteristic
88	C₂×C₈×C₁₁	—	2	2	no	—
89	C₂×C₈×C₁₁	—	2	1	no	—
90	C₂×C₈×C₁₁	—	2	1	no	—
91	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
95	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²⋊Dic₁₁	C₂×C₈	1	2	yes	—
100	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
101	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
102	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
103	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
104	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
105	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
106	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
107	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
108	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
109	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
110	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
111	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
112	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
113	#4313	—	2	4	no	—
114	C₈×D₁₁	—	2	2	no	—
115	C₈×D₁₁	—	2	2	no	—
116	C₁₁×C₁₆	—	2	1	no	—
117	C₁₁×C₁₆	—	2	1	no	—
118	C₁₁×C₁₆	#41	1	2	yes	—
119	C₁₁×C₁₆	#41	1	2	yes	—
120	C₁₁⋊C₁₆	—	2	2	no	—
121	C₁₁⋊C₁₆	—	2	2	no	—
122	#2966	—	11	1	no	—
123	#2183	—	11	1	no	—
124	Q₈×C₁₆	—	11	1	no	—
125	#2169	—	11	2	no	—
126	#2169	—	11	1	no	—
127	#2168	—	11	2	no	—
128	D₄×C₁₆	—	11	1	no	—
129	#2164	—	11	2	no	—
130	#2162	—	11	1	no	—
131	#2108	—	11	2	no	—
132	C₄×M₃₂	D₁₁	1	1	yes	characteristic, complemented
133	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
134	C₂²×D₁₁	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₁₁	—	2	1	no	—
136	C₂×C₄×C₁₁	—	2	1	no	—
137	C₂×C₄×C₁₁	—	2	2	no	—
138	C₂×C₄×C₁₁	—	4	1	no	—
139	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
144	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
145	C₂²⋊D₁₁	—	4	2	no	—
146	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
147	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
148	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
149	D₄₄	—	2	2	no	—
150	C₄×D₁₁	—	2	2	no	—
151	C₄×D₁₁	—	2	2	no	—
152	Dic₂₂	—	2	2	no	—
153	C₈×C₁₁	—	2	1	no	—
154	C₈×C₁₁	—	2	1	no	—
155	C₈×C₁₁	#131	1	2	yes	—
156	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
157	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
158	C₁₁⋊C₈	—	2	2	no	—
159	C₁₁⋊C₈	—	2	2	no	—
160	#529	—	11	1	no	—
161	C₂×M₃₂	—	11	2	no	—
162	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
163	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
164	C₂²×C₁₆	—	11	2	no	—
165	#446	—	11	1	no	—
166	C₈×Q₈	—	11	1	no	—
167	#436	—	11	1	no	—
168	#435	—	11	2	no	—
169	C₈×D₄	—	11	2	no	—
170	C₈×D₄	—	11	1	no	—
171	#433	—	11	2	no	—
172	#432	—	11	1	no	—
173	#405	—	11	2	no	—
174	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
175	C₄⋊C₁₆	—	11	2	no	—
176	C₄⋊C₁₆	—	11	2	no	—
177	C₄⋊C₁₆	—	11	1	no	—
178	C₄⋊C₁₆	—	11	1	no	—
179	C₂²⋊C₁₆	—	11	2	no	—
180	C₂²⋊C₁₆	—	11	2	no	—
181	C₂²⋊C₁₆	—	11	2	no	—
182	#346	—	11	2	no	—
183	#346	C₂×D₁₁	1	1	yes	characteristic
184	#346	C₂×D₁₁	1	1	yes	characteristic
185	C₄×C₁₆	—	11	2	no	—
186	C₄×C₁₆	—	11	2	no	—
187	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
188	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
189	C₂²×C₁₁	—	2	1	no	—
190	C₂²×C₁₁	—	2	1	no	—
191	C₂²×C₁₁	—	2	1	no	—
192	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
193	C₂×D₁₁	—	2	2	no	—
194	C₂×D₁₁	—	2	2	no	—
195	C₂×D₁₁	—	2	2	no	—
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	#504	1	2	yes	—
201	C₄×C₁₁	C₄×#41	1	1	yes	characteristic
202	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
203	Dic₁₁	—	2	2	no	—
204	Dic₁₁	—	2	2	no	—
205	Dic₁₁	—	2	2	no	—
206	C₂×M₁₆	—	11	2	no	—
207	C₂²×C₈	—	11	2	no	—
208	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
210	#126	—	11	2	no	—
211	#125	—	11	1	no	—
212	#124	—	11	1	no	—
213	#123	—	11	2	no	—
214	C₂²⋊Q₈	—	11	1	no	—
215	#121	—	11	1	no	—
216	C₄×Q₈	—	11	1	no	—
217	C₄×D₄	—	11	2	no	—
218	C₄×D₄	—	11	1	no	—
219	#117	—	11	2	no	—
220	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
221	M₃₂	—	2	2	no	—
222	M₃₂	—	2	2	no	—
223	M₃₂	—	22	4	no	—
224	C₂×C₁₆	—	11	2	no	—
225	C₂×C₁₆	—	11	2	no	—
226	C₂×C₁₆	—	22	4	no	—
227	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
228	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
229	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
230	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
231	C₄⋊C₈	—	11	2	no	—
232	C₄⋊C₈	—	11	2	no	—
233	C₄⋊C₈	—	11	1	no	—
234	C₄⋊C₈	—	11	1	no	—
235	C₂²⋊C₈	—	11	2	no	—
236	C₂²⋊C₈	—	11	2	no	—
237	C₂²⋊C₈	—	11	2	no	—
238	#97	—	11	2	no	—
239	#97	—	11	2	no	—
240	C₄×C₈	—	11	2	no	—
241	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
242	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
245	C₂×C₁₁	—	4	1	no	—
246	C₂×C₁₁	C₈×#41	1	1	yes	characteristic
247	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
248	C₂×C₁₁	#3407	1	1	yes	characteristic
249	D₁₁	—	4	2	no	—
250	C₂×Q₈	—	11	1	no	—
251	C₂×D₄	—	11	2	no	—
252	C₂×D₄	—	11	1	no	—
253	C₂²×C₄	—	11	2	no	—
254	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
255	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
257	M₁₆	—	22	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	11	2	no	—
262	C₂×C₈	—	11	2	no	—
263	C₂×C₈	—	22	2	no	—
264	C₂×C₈	—	22	2	no	—
265	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
269	C₄⋊C₄	—	11	2	no	—
270	C₄⋊C₄	—	11	2	no	—
271	C₄⋊C₄	—	11	1	no	—
272	C₄⋊C₄	—	11	2	no	—
273	C₄⋊C₄	—	11	1	no	—
274	C₄⋊C₄	—	11	2	no	—
275	C₂²⋊C₄	—	11	2	no	—
276	C₂²⋊C₄	—	11	2	no	—
277	C₂²⋊C₄	—	11	2	no	—
278	C₂²⋊C₄	—	11	2	no	—
279	C₂²⋊C₄	—	11	2	no	—
280	C₄²	—	11	2	no	—
281	C₄²	C₈×D₁₁	1	1	yes	characteristic
282	C₄²	C₈×D₁₁	1	1	yes	characteristic
283	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
284	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	22	2	no	—
288	C₁₆	—	22	2	no	—
289	C₁₆	#4339	1	2	yes	—
290	C₁₆	#4339	1	2	yes	—
291	C₁₁	#33483	1	1	yes	characteristic, complemented, Sylow
292	C₂³	—	11	2	no	—
293	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
294	Q₈	—	22	2	no	—
295	D₄	—	22	2	no	—
296	D₄	—	44	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	11	2	no	—
302	C₂×C₄	—	11	2	no	—
303	C₂×C₄	—	11	2	no	—
304	C₂×C₄	—	22	2	no	—
305	C₂×C₄	—	22	2	no	—
306	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
309	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
310	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	22	2	no	—
315	C₈	—	22	2	no	—
316	C₈	#66954	1	2	yes	—
317	C₈	C₂×#4339	1	1	yes	characteristic
318	C₈	C₂×#4339	1	1	yes	characteristic
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	22	2	no	—
323	C₂²	—	22	2	no	—
324	C₂²	—	22	2	no	—
325	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	22	2	no	—
331	C₄	—	22	2	no	—
332	C₄	—	22	2	no	—
333	C₄	#10622520	1	2	yes	—
334	C₄	C₄×#4339	1	1	yes	characteristic
335	C₄	C₂×#66954	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	44	2	no	—
338	C₂	C₈×#4339	1	1	yes	characteristic
339	C₂	C₂×#10622520	1	1	yes	characteristic
340	C₂	#13081718	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₄×C₁₁×M₃₂	C₂
prime radical	#66983	C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₄×M₃₂	D₁₁
hypocenter	C₁₁	#33483
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #13070524	C₂	semidirect
3. C₁₆×Dic₂₂	C₂	semidirect
4. #13070449	C₂	semidirect
5. #13070446	C₂	semidirect
6. #13070439	C₂	semidirect
7. C₁₆×D₄₄	C₂	semidirect
8. #13070424	C₂	semidirect
9. #13070415	C₂	semidirect
10. #13070214	C₂	semidirect
11. C₄×C₁₁×M₃₂	C₂	semidirect
132. C₄×M₃₂	D₁₁	semidirect
291. C₁₁	#33483	semidirect

Maximal subgroups

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

subgroup	index
#13077591	2
#13070524	2
C₁₆×Dic₂₂	2
#13070449	2
#13070446	2
#13070439	2
C₁₆×D₄₄	2
#13070424	2
#13070415	2
#13070214	2
C₄×C₁₁×M₃₂	2
#33483	11