Subgroups — #14724514

Order 2816.

658 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079175	C₂	1	1	yes	maximal, characteristic
2	#13077550	C₂	1	1	yes	maximal, characteristic, complemented
3	#13077329	C₂	1	1	yes	maximal, characteristic
4	#13067496	C₂	1	4	yes	maximal, complemented
5	#13067475	C₂	1	4	yes	maximal, complemented
6	#13067375	C₂	1	1	yes	maximal, characteristic, complemented
7	#13067072	C₂	1	1	yes	maximal, characteristic, complemented
8	#13067068	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#66954	—	2	1	no	—
11	#10623008	C₂²	1	1	yes	characteristic
12	#10623001	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₁	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	#10622727	C₄	1	4	yes	—
17	#10622723	—	2	1	no	—
18	#10622715	—	2	1	no	—
19	#10622714	—	2	1	no	—
20	#10622537	C₂²	1	1	yes	characteristic
21	#10622537	C₂²	1	1	yes	characteristic
22	#10622307	C₄	1	1	yes	characteristic
23	#10622306	C₄	1	1	yes	characteristic
24	C₄×#4314	C₂²	1	4	yes	—
25	#10622301	—	2	1	no	—
26	#10622301	C₂²	1	2	yes	—
27	C₈×D₄₄	—	2	1	no	—
28	C₈×D₄₄	C₂²	1	2	yes	—
29	#10622294	C₄	1	1	yes	characteristic
30	#10622293	C₄	1	1	yes	characteristic
31	#10622292	—	2	1	no	—
32	#10622292	C₂²	1	2	yes	—
33	C₈×Dic₂₂	—	2	1	no	—
34	C₈×Dic₂₂	C₂²	1	2	yes	—
35	C₁₁×#322	C₂²	1	2	yes	—
36	C₁₁×#322	C₂²	1	2	yes	—
37	#10622097	C₂²	1	4	yes	—
38	#10622096	C₂²	1	4	yes	—
39	#10622094	C₂²	1	4	yes	—
40	C₂×#4339	—	2	1	no	—
41	C₂²×C₈×C₁₁	—	2	1	no	—
42	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
46	#66983	—	2	1	no	—
47	#66983	C₂×C₄	1	2	yes	—
48	#66979	—	2	1	no	—
49	#66977	—	2	1	no	—
50	#66977	C₂×C₄	1	2	yes	—
51	C₂×#4318	—	2	1	no	—
52	C₂×#4318	—	2	1	no	—
53	C₂×#4314	C₂×C₄	1	4	yes	—
54	#66954	—	4	4	no	—
55	C₂×#4313	—	2	2	no	—
56	C₂×C₈×D₁₁	—	2	2	no	—
57	#66928	—	2	1	no	—
58	#66927	C₂×C₄	1	1	yes	characteristic
59	C₄×D₄₄	—	2	1	no	—
60	C₄×D₄₄	C₂³	1	2	yes	—
61	#66923	—	2	1	no	—
62	#66922	C₂×C₄	1	1	yes	characteristic
63	#66922	C₂×C₄	1	1	yes	characteristic
64	#66922	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₂₂	—	2	1	no	—
66	C₄×Dic₂₂	C₂³	1	2	yes	—
67	C₄×C₈×C₁₁	—	2	1	no	—
68	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
71	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
72	C₄×C₈×C₁₁	C₂³	1	2	yes	—
73	C₄×C₈×C₁₁	C₂³	1	2	yes	—
74	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
76	#66885	C₂×C₄	1	4	yes	—
77	(C₂×D₁₁)⋊C₈	—	2	2	no	—
78	(C₂×D₁₁)⋊C₈	—	2	2	no	—
79	#66881	C₂×C₄	1	4	yes	—
80	#66880	C₂×C₄	1	4	yes	—
81	#66879	—	2	2	no	—
82	Dic₁₁⋊C₈	—	2	2	no	—
83	Dic₁₁⋊C₈	—	2	2	no	—
84	C₈×Dic₁₁	—	2	2	no	—
85	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
86	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
87	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
88	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
89	#17979	—	11	1	no	maximal, Sylow
90	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
93	C₂×C₂²⋊D₁₁	—	2	2	no	—
94	#4339	—	4	2	no	—
95	#4339	—	4	2	no	—
96	C₂×D₄₄	—	2	1	no	—
97	C₂×D₄₄	—	2	1	no	—
98	C₂×D₄₄	C₂²×C₄	1	2	yes	—
99	C₂×C₄×D₁₁	—	2	2	no	—
100	C₂×Dic₂₂	—	2	1	no	—
101	C₂×Dic₂₂	—	2	1	no	—
102	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₁₁	—	2	4	no	—
104	C₂×C₈×C₁₁	—	2	2	no	—
105	C₂×C₈×C₁₁	—	2	2	no	—
106	C₂×C₈×C₁₁	—	2	2	no	—
107	C₂×C₈×C₁₁	—	4	2	no	—
108	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₁	#41	1	1	yes	characteristic
117	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
118	C₂²⋊Dic₁₁	—	2	2	no	—
119	(C₂×D₁₁)⋊C₄	—	2	2	no	—
120	(C₂×D₁₁)⋊C₄	—	2	2	no	—
121	C₄⋊Dic₁₁	—	2	1	no	—
122	C₄⋊Dic₁₁	—	2	1	no	—
123	C₄⋊Dic₁₁	—	2	1	no	—
124	C₄⋊Dic₁₁	—	2	1	no	—
125	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
126	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
127	Dic₁₁⋊C₄	—	2	2	no	—
128	Dic₁₁⋊C₄	—	2	2	no	—
129	C₄×Dic₁₁	—	2	2	no	—
130	#4318	—	4	2	no	—
131	#4318	—	4	2	no	—
132	C₂×C₁₁⋊C₈	—	2	2	no	—
133	C₂×C₁₁⋊C₈	—	2	2	no	—
134	#4314	—	4	4	no	—
135	#4313	—	4	4	no	—
136	C₈×D₁₁	—	4	4	no	—
137	#3167	—	11	1	no	—
138	#2960	—	11	1	no	—
139	#2931	—	11	1	no	—
140	#1588	—	11	4	no	—
141	#1585	—	11	4	no	—
142	#1567	—	11	1	no	—
143	#1493	—	11	1	no	—
144	#1492	—	11	1	no	—
145	C₂×#322	D₁₁	1	1	yes	characteristic, complemented
146	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
147	C₂²×D₁₁	—	2	2	no	—
148	C₂×C₄×C₁₁	—	2	2	no	—
149	C₂×C₄×C₁₁	—	2	2	no	—
150	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
154	C₂×C₄×C₁₁	#137	1	2	yes	—
155	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
158	C₂²⋊D₁₁	—	4	4	no	—
159	C₂×Dic₁₁	—	2	2	no	—
160	C₂×Dic₁₁	—	2	2	no	—
161	C₂×Dic₁₁	—	2	2	no	—
162	D₄₄	—	2	4	no	—
163	D₄₄	—	4	2	no	—
164	D₄₄	—	4	2	no	—
165	C₄×D₁₁	—	4	2	no	—
166	C₄×D₁₁	—	4	2	no	—
167	Dic₂₂	—	2	4	no	—
168	Dic₂₂	—	4	2	no	—
169	Dic₂₂	—	4	2	no	—
170	C₈×C₁₁	—	2	4	no	—
171	C₈×C₁₁	—	2	2	no	—
172	C₈×C₁₁	—	2	2	no	—
173	C₈×C₁₁	—	2	1	no	—
174	C₈×C₁₁	—	2	1	no	—
175	C₈×C₁₁	—	2	1	no	—
176	C₈×C₁₁	—	2	1	no	—
177	C₁₁⋊C₈	—	4	2	no	—
178	C₁₁⋊C₈	—	4	2	no	—
179	C₂×#131	—	22	1	no	—
180	#533	—	11	1	no	—
181	#532	—	11	1	no	—
182	#498	—	11	1	no	—
183	#492	—	11	1	no	—
184	#487	—	11	1	no	—
185	#486	—	11	1	no	—
186	#465	—	11	4	no	—
187	#446	—	11	2	no	—
188	#446	—	22	1	no	—
189	C₈×Q₈	—	11	2	no	—
190	C₈×Q₈	—	22	1	no	—
191	C₄×SD₁₆	—	11	4	no	—
192	#436	—	11	2	no	—
193	#436	—	22	1	no	—
194	C₈×D₄	—	11	2	no	—
195	C₈×D₄	—	22	1	no	—
196	#424	—	22	1	no	—
197	#423	—	11	1	no	—
198	#423	—	11	1	no	—
199	#408	—	22	1	no	—
200	#405	—	22	1	no	—
201	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
202	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
203	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
204	#335	—	11	4	no	—
205	Q₈⋊C₈	—	11	4	no	—
206	D₄⋊C₈	—	11	4	no	—
207	#322	C₂×D₁₁	1	2	yes	—
208	#322	C₂×D₁₁	1	2	yes	—
209	C₂²×C₁₁	—	2	1	no	—
210	C₂²×C₁₁	—	2	1	no	—
211	C₂²×C₁₁	#440	1	2	yes	—
212	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
213	C₂×D₁₁	—	4	2	no	—
214	C₂×D₁₁	—	4	2	no	—
215	C₂×D₁₁	—	4	2	no	—
216	C₄×C₁₁	—	2	1	no	—
217	C₄×C₁₁	—	2	1	no	—
218	C₄×C₁₁	#440	1	2	yes	—
219	C₄×C₁₁	#443	1	2	yes	—
220	C₄×C₁₁	#568	1	2	yes	—
221	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₄×C₁₁	C₂×#137	1	1	yes	characteristic
223	Dic₁₁	—	4	2	no	—
224	Dic₁₁	—	4	2	no	—
225	Dic₁₁	—	4	2	no	—
226	C₂×#41	—	22	1	no	—
227	C₂×SD₁₆	—	11	4	no	—
228	#131	—	44	4	no	—
229	C₂×M₁₆	—	22	1	no	—
230	C₂×M₁₆	—	22	1	no	—
231	C₂×M₁₆	—	22	2	no	—
232	C₂²×C₈	—	2	1	no	—
233	C₂²×C₈	—	22	2	no	—
234	C₂²×C₈	C₄×D₁₁	1	2	yes	—
235	C₂²×C₈	D₄₄	1	1	yes	characteristic
236	C₂²×C₈	D₄₄	1	1	yes	characteristic
237	C₄⋊Q₈	—	11	1	no	—
238	C₄⋊Q₈	—	11	1	no	—
239	C₄⋊Q₈	—	11	1	no	—
240	#127	—	11	1	no	—
241	#125	—	22	1	no	—
242	#124	—	22	1	no	—
243	C₂²⋊Q₈	—	11	2	no	—
244	C₂²⋊Q₈	—	22	1	no	—
245	#121	—	11	2	no	—
246	#121	—	22	1	no	—
247	C₄×Q₈	—	11	2	no	—
248	C₄×Q₈	—	22	1	no	—
249	C₄×D₄	—	11	2	no	—
250	C₄×D₄	—	22	1	no	—
251	#117	—	22	1	no	—
252	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
253	#106	—	11	4	no	—
254	C₄⋊C₈	—	11	2	no	—
255	C₄⋊C₈	—	11	2	no	—
256	C₄⋊C₈	—	22	2	no	—
257	C₄⋊C₈	—	22	2	no	—
258	C₄⋊C₈	—	22	1	no	—
259	C₄⋊C₈	—	22	1	no	—
260	#103	—	11	4	no	—
261	#102	—	11	4	no	—
262	C₂²⋊C₈	—	22	2	no	—
263	C₂²⋊C₈	—	22	2	no	—
264	#97	—	22	2	no	—
265	C₄×C₈	—	2	1	no	—
266	C₄×C₈	—	22	2	no	—
267	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
268	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
269	C₄×C₈	D₄₄	1	1	yes	characteristic
270	C₄×C₈	D₄₄	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₁₁	1	2	yes	—
272	C₄×C₈	C₂²×D₁₁	1	2	yes	—
273	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
275	C₂×C₁₁	—	2	2	no	—
276	C₂×C₁₁	#2932	1	1	yes	characteristic
277	C₂×C₁₁	C₂×#440	1	1	yes	characteristic
278	C₂×C₁₁	C₂×#443	1	1	yes	characteristic
279	D₁₁	—	8	2	no	—
280	#41	—	44	2	no	—
281	#41	—	44	2	no	—
282	C₂×Q₈	—	11	2	no	—
283	C₂×Q₈	—	22	1	no	—
284	C₂×Q₈	—	22	1	no	—
285	C₂×D₄	—	11	2	no	—
286	C₂×D₄	—	22	2	no	—
287	C₂×D₄	—	22	1	no	—
288	C₂×D₄	—	22	1	no	—
289	C₂²×C₄	—	22	2	no	—
290	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
292	C₂²×C₄	#4339	1	1	yes	characteristic
293	SD₁₆	—	44	4	no	—
294	M₁₆	—	44	4	no	—
295	M₁₆	—	44	2	no	—
296	M₁₆	—	44	2	no	—
297	C₂×C₈	—	2	4	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	4	2	no	—
302	C₂×C₈	—	22	2	no	—
303	C₂×C₈	—	22	2	no	—
304	C₂×C₈	—	44	4	no	—
305	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
306	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
307	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
308	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
309	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
310	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
311	C₄⋊C₄	—	11	2	no	—
312	C₄⋊C₄	—	11	2	no	—
313	C₄⋊C₄	—	22	1	no	—
314	C₄⋊C₄	—	22	1	no	—
315	C₄⋊C₄	—	22	1	no	—
316	C₄⋊C₄	—	22	2	no	—
317	C₄⋊C₄	—	22	1	no	—
318	C₄⋊C₄	—	22	2	no	—
319	C₂²⋊C₄	—	22	2	no	—
320	C₂²⋊C₄	—	22	2	no	—
321	C₂²⋊C₄	—	22	2	no	—
322	C₄²	—	22	2	no	—
323	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
324	C₄²	C₂×D₄₄	1	1	yes	characteristic
325	C₄²	#4339	1	1	yes	characteristic
326	C₄²	C₂³×D₁₁	1	1	yes	characteristic
327	C₁₁	#17979	1	1	yes	characteristic, complemented, Sylow
328	C₂³	—	22	2	no	—
329	C₂³	C₄×D₄₄	1	1	yes	characteristic
330	Q₈	—	22	4	no	—
331	Q₈	—	44	2	no	—
332	Q₈	—	44	2	no	—
333	D₄	—	22	4	no	—
334	D₄	—	44	4	no	—
335	D₄	—	44	2	no	—
336	D₄	—	44	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	22	2	no	—
340	C₂×C₄	—	22	2	no	—
341	C₂×C₄	—	22	2	no	—
342	C₂×C₄	—	44	2	no	—
343	C₂×C₄	—	44	2	no	—
344	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
345	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
346	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
347	C₂×C₄	#66961	1	2	yes	—
348	C₂×C₄	#66962	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
350	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
351	C₂×C₄	C₂×#4339	1	1	yes	characteristic
352	C₈	—	2	4	no	—
353	C₈	—	2	2	no	—
354	C₈	—	2	2	no	—
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	44	2	no	—
360	C₈	—	44	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	44	2	no	—
364	C₂²	—	44	2	no	—
365	C₂²	—	44	2	no	—
366	C₂²	#10622323	1	2	yes	—
367	C₂²	C₂×C₄×D₄₄	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	44	2	no	—
371	C₄	—	44	2	no	—
372	C₄	—	44	2	no	—
373	C₄	#10622313	1	2	yes	—
374	C₄	#10622323	1	2	yes	—
375	C₄	#10623276	1	2	yes	—
376	C₄	C₂×#66961	1	1	yes	characteristic
377	C₄	C₂×#66962	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	88	2	no	—
380	C₂	#13077335	1	1	yes	characteristic
381	C₂	C₂×#10622323	1	1	yes	characteristic
382	C₂	C₂×#10622313	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#322	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	#10623008	C₂²
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#322	D₁₁
hypocenter	C₁₁	#17979
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #13077550	C₂	semidirect
4. #13067496	C₂	semidirect
5. #13067475	C₂	semidirect
6. #13067375	C₂	semidirect
7. #13067072	C₂	semidirect
9. C₂×C₁₁×#322	C₂	semidirect
145. C₂×#322	D₁₁	semidirect
327. C₁₁	#17979	semidirect

Maximal subgroups

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

subgroup	index
#13079175	2
#13077550	2
#13077329	2
#13067496	2
#13067475	2
#13067375	2
#13067072	2
#13067068	2
C₂×C₁₁×#322	2
#17979	11