Subgroups — #14907334

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	#13077588	C₂	1	1	yes	maximal, characteristic
2	#13070524	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070518	C₂	1	1	yes	maximal, characteristic, complemented
4	#13070448	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070445	C₂	1	2	yes	maximal, complemented
6	#13070441	C₂	1	2	yes	maximal, complemented
7	#13070420	C₂	1	2	yes	maximal, complemented
8	#13070414	C₂	1	1	yes	maximal, characteristic, complemented
9	#13070212	C₂	1	2	yes	maximal, complemented
10	C₄×#66862	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623088	C₄	1	1	yes	characteristic
13	C₂²×C₁₁×C₁₆	C₂²	1	2	yes	—
14	C₈×D₄×C₁₁	C₂²	1	1	yes	characteristic
15	C₈×C₂²⋊D₁₁	C₂²	1	2	yes	—
16	#10622714	C₂²	1	2	yes	—
17	C₂×#66876	C₂²	1	2	yes	—
18	#10622625	C₄	1	1	yes	characteristic
19	C₂×#66862	C₂²	1	1	yes	characteristic
20	C₂×#66862	C₂²	1	1	yes	characteristic
21	#10622448	C₄	1	1	yes	characteristic
22	#10622444	C₄	1	1	yes	characteristic
23	#10622341	C₄	1	2	yes	—
24	#10622333	C₄	1	2	yes	—
25	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
26	C₈×Dic₂₂	C₂²	1	1	yes	characteristic
27	C₁₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₁₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
29	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
31	(C₂×D₁₁)⋊C₁₆	C₂²	1	2	yes	—
32	#10622140	C₂²	1	1	yes	characteristic
33	#10622140	C₂²	1	2	yes	—
34	#10622140	C₂²	1	1	yes	characteristic
35	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
36	Dic₁₁⋊C₁₆	C₂²	1	2	yes	—
37	C₁₆×Dic₁₁	C₂²	1	2	yes	—
38	C₁₆×Dic₁₁	C₂²	1	2	yes	—
39	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
40	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
42	C₄×D₄×C₁₁	C₂×C₄	1	1	yes	characteristic
43	#66991	C₈	1	1	yes	characteristic
44	#66989	C₈	1	1	yes	characteristic
45	C₄×C₂²⋊D₁₁	C₂×C₄	1	2	yes	—
46	#66979	C₂×C₄	1	2	yes	—
47	C₂×#4318	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₁₁	C₂³	1	1	yes	characteristic
50	#66950	C₈	1	1	yes	characteristic
51	#66943	C₈	1	1	yes	characteristic
52	#66936	C₈	1	2	yes	—
53	#66932	C₈	1	2	yes	—
54	C₄²×D₁₁	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₂₂	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₁×C₁₆	—	2	4	no	—
57	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
60	C₁₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₁₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
63	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
64	(C₂×D₁₁)⋊C₈	C₂×C₄	1	2	yes	—
65	#66879	C₂×C₄	1	2	yes	—
66	#66879	C₂×C₄	1	2	yes	—
67	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
68	Dic₁₁⋊C₈	C₂×C₄	1	2	yes	—
69	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₁₁	C₂³	1	2	yes	—
71	C₈×Dic₁₁	C₂³	1	1	yes	characteristic
72	#66876	—	2	4	no	—
73	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
75	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
76	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
77	C₄×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
78	#66862	—	2	2	no	—
79	#66862	—	2	2	no	—
80	#33483	—	11	1	no	maximal, Sylow
81	C₂×D₄×C₁₁	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
83	C₂×C₂²⋊D₁₁	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₁	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₂₂	C₂×C₈	1	1	yes	characteristic
88	C₂×C₈×C₁₁	—	2	2	no	—
89	C₂×C₈×C₁₁	—	2	2	no	—
90	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
93	C₁₁×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
94	C₁₁×C₂²⋊C₄	C₂×C₈	1	2	yes	—
95	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²⋊Dic₁₁	C₂×C₈	1	2	yes	—
97	C₂²⋊Dic₁₁	C₂×C₈	1	2	yes	—
98	C₂²⋊Dic₁₁	C₂×C₈	1	2	yes	—
99	(C₂×D₁₁)⋊C₄	C₂×C₈	1	2	yes	—
100	C₄⋊Dic₁₁	C₂×C₈	1	1	yes	characteristic
101	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
102	C₄⋊Dic₁₁	C₂×C₈	1	2	yes	—
103	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
104	Dic₁₁⋊C₄	C₂×C₈	1	2	yes	—
105	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
106	C₄×Dic₁₁	C₂×C₈	1	1	yes	characteristic
107	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
108	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
109	#4318	—	2	4	no	—
110	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
112	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
113	C₈×D₁₁	—	2	1	no	—
114	C₈×D₁₁	—	2	1	no	—
115	C₈×D₁₁	—	2	2	no	—
116	C₁₁×C₁₆	—	2	1	no	—
117	C₁₁×C₁₆	—	2	2	no	—
118	C₁₁×C₁₆	#41	1	2	yes	—
119	C₁₁⋊C₁₆	—	2	1	no	—
120	C₁₁⋊C₁₆	—	2	2	no	—
121	C₁₁⋊C₁₆	#41	1	2	yes	—
122	#2966	—	11	1	no	—
123	#2183	—	11	1	no	—
124	Q₈×C₁₆	—	11	1	no	—
125	#2169	—	11	1	no	—
126	#2169	—	11	2	no	—
127	#2168	—	11	2	no	—
128	D₄×C₁₆	D₁₁	1	1	yes	characteristic, complemented
129	#2164	—	11	2	no	—
130	#2162	—	11	1	no	—
131	#2108	—	11	2	no	—
132	C₄×M₃₂	—	11	1	no	—
133	C₂³×C₁₁	C₂²×C₈	1	2	yes	—
134	C₂²×D₁₁	C₂²×C₈	1	1	yes	characteristic
135	D₄×C₁₁	—	2	2	no	—
136	C₂×C₄×C₁₁	—	2	2	no	—
137	C₂×C₄×C₁₁	—	2	2	no	—
138	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
142	C₂²⋊D₁₁	—	4	2	no	—
143	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
144	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
145	C₂×Dic₁₁	C₂²×C₈	1	1	yes	characteristic
146	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
147	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
148	C₄×D₁₁	—	2	1	no	—
149	C₄×D₁₁	—	2	1	no	—
150	C₄×D₁₁	—	2	2	no	—
151	C₄×D₁₁	—	4	1	no	—
152	Dic₂₂	—	2	2	no	—
153	C₈×C₁₁	—	2	2	no	—
154	C₈×C₁₁	—	2	1	no	—
155	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
156	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
157	C₁₁⋊C₈	—	2	1	no	—
158	C₁₁⋊C₈	—	2	2	no	—
159	C₁₁⋊C₈	#131	1	2	yes	—
160	#529	—	11	1	no	—
161	C₂×M₃₂	—	11	2	no	—
162	C₂×M₃₂	—	11	1	no	—
163	C₂×M₃₂	—	11	1	no	—
164	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
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	C₂×D₁₁	1	1	yes	characteristic
171	#433	—	11	2	no	—
172	#432	—	11	1	no	—
173	#405	—	11	2	no	—
174	C₂×C₄×C₈	—	11	1	no	—
175	C₄⋊C₁₆	—	11	2	no	—
176	C₄⋊C₁₆	—	11	2	no	—
177	C₄⋊C₁₆	—	11	1	no	—
178	C₄⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
179	C₂²⋊C₁₆	—	11	2	no	—
180	C₂²⋊C₁₆	—	11	2	no	—
181	C₂²⋊C₁₆	C₂×D₁₁	1	2	yes	—
182	#346	—	11	1	no	—
183	#346	—	11	1	no	—
184	#346	—	11	2	no	—
185	C₄×C₁₆	—	11	2	no	—
186	C₄×C₁₆	—	11	2	no	—
187	C₄×C₁₆	—	11	1	no	—
188	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
189	C₂²×C₁₁	—	2	2	no	—
190	C₂²×C₁₁	—	2	2	no	—
191	C₂²×C₁₁	—	2	2	no	—
192	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
193	C₂×D₁₁	—	2	1	no	—
194	C₂×D₁₁	—	2	1	no	—
195	C₂×D₁₁	—	2	1	no	—
196	C₄×C₁₁	—	2	1	no	—
197	C₄×C₁₁	—	2	2	no	—
198	C₄×C₁₁	#504	1	2	yes	—
199	C₄×C₁₁	C₄×#41	1	1	yes	characteristic
200	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
201	Dic₁₁	—	2	2	no	—
202	Dic₁₁	—	2	1	no	—
203	Dic₁₁	—	2	1	no	—
204	Dic₁₁	—	2	2	no	—
205	Dic₁₁	—	2	1	no	—
206	C₂×M₁₆	—	11	2	no	—
207	C₂²×C₈	—	11	1	no	—
208	C₂²×C₈	—	11	1	no	—
209	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
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	C₄×D₁₁	1	1	yes	characteristic
219	#117	—	11	2	no	—
220	C₂×C₄²	—	11	1	no	—
221	M₃₂	—	22	2	no	—
222	M₃₂	—	22	4	no	—
223	M₃₂	—	22	2	no	—
224	C₂×C₁₆	—	2	4	no	—
225	C₂×C₁₆	—	11	1	no	—
226	C₂×C₁₆	—	11	1	no	—
227	C₂×C₁₆	—	11	2	no	—
228	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
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	C₄×D₁₁	1	1	yes	characteristic
235	C₂²⋊C₈	—	11	2	no	—
236	C₂²⋊C₈	—	11	2	no	—
237	C₂²⋊C₈	C₄×D₁₁	1	2	yes	—
238	#97	—	11	2	no	—
239	#97	—	11	2	no	—
240	C₄×C₈	—	11	1	no	—
241	C₄×C₈	—	11	1	no	—
242	C₄×C₈	—	11	2	no	—
243	C₄×C₈	—	11	1	no	—
244	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
245	C₂×C₁₁	—	4	2	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	1	no	—
250	C₂×Q₈	—	11	1	no	—
251	C₂×D₄	—	11	2	no	—
252	C₂×D₄	C₈×D₁₁	1	1	yes	characteristic
253	C₂²×C₄	—	11	1	no	—
254	C₂²×C₄	—	11	1	no	—
255	C₂²×C₄	—	11	1	no	—
256	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
257	M₁₆	—	22	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	11	1	no	—
261	C₂×C₈	—	11	1	no	—
262	C₂×C₈	—	11	2	no	—
263	C₂×C₈	—	22	2	no	—
264	C₂×C₈	—	22	1	no	—
265	C₂×C₈	—	22	1	no	—
266	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
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	2	no	—
272	C₄⋊C₄	—	11	1	no	—
273	C₄⋊C₄	—	11	2	no	—
274	C₄⋊C₄	C₈×D₁₁	1	1	yes	characteristic
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	C₈×D₁₁	1	2	yes	—
280	C₄²	—	11	1	no	—
281	C₄²	—	11	1	no	—
282	C₄²	—	11	1	no	—
283	C₄²	—	11	2	no	—
284	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	11	2	no	—
288	C₁₆	—	22	1	no	—
289	C₁₆	—	22	2	no	—
290	C₁₆	#4341	1	2	yes	—
291	C₁₁	#33483	1	1	yes	characteristic, complemented, Sylow
292	C₂³	—	11	1	no	—
293	C₂³	C₂×C₈×D₁₁	1	2	yes	—
294	Q₈	—	22	2	no	—
295	D₄	—	2	2	no	—
296	D₄	—	44	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	11	1	no	—
300	C₂×C₄	—	11	1	no	—
301	C₂×C₄	—	11	1	no	—
302	C₂×C₄	—	11	2	no	—
303	C₂×C₄	—	11	2	no	—
304	C₂×C₄	—	22	1	no	—
305	C₂×C₄	—	22	2	no	—
306	C₂×C₄	—	22	1	no	—
307	C₂×C₄	—	44	1	no	—
308	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
309	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
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	2	no	—
314	C₈	—	11	2	no	—
315	C₈	—	22	2	no	—
316	C₈	—	22	1	no	—
317	C₈	C₂×#4341	1	1	yes	characteristic
318	C₈	#41×D₁₁	1	1	yes	characteristic
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	22	1	no	—
323	C₂²	—	22	1	no	—
324	C₂²	—	22	1	no	—
325	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	22	1	no	—
329	C₄	—	22	1	no	—
330	C₄	—	22	2	no	—
331	C₄	—	22	1	no	—
332	C₄	—	22	2	no	—
333	C₄	#10622514	1	2	yes	—
334	C₄	C₄×#4341	1	1	yes	characteristic
335	C₄	D₁₁×#131	1	1	yes	characteristic
336	C₂	—	4	2	no	—
337	C₂	—	44	1	no	—
338	C₂	C₈×#4341	1	1	yes	characteristic
339	C₂	C₂×#10622514	1	1	yes	characteristic
340	C₂	#13081717	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	D₄×C₁₁×C₁₆	C₂
prime radical	#66991	C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	D₄×C₁₆	D₁₁
hypocenter	C₁₁	#33483
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13070524	C₂	semidirect
3. #13070518	C₂	semidirect
4. #13070448	C₂	semidirect
5. #13070445	C₂	semidirect
6. #13070441	C₂	semidirect
7. #13070420	C₂	semidirect
8. #13070414	C₂	semidirect
9. #13070212	C₂	semidirect
10. C₄×#66862	C₂	semidirect
11. D₄×C₁₁×C₁₆	C₂	semidirect
128. D₄×C₁₆	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
#13077588	2
#13070524	2
#13070518	2
#13070448	2
#13070445	2
#13070441	2
#13070420	2
#13070414	2
#13070212	2
C₄×#66862	2
D₄×C₁₁×C₁₆	2
#33483	11