Subgroups — #14722819

Order 2816.

666 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077734	C₂	1	2	yes	maximal
2	C₄×#66954	C₂	1	1	yes	maximal, characteristic
3	#13067382	C₂	1	4	yes	maximal, complemented
4	#13067378	C₂	1	2	yes	maximal, complemented
5	#13067373	C₂	1	2	yes	maximal, complemented
6	#13067366	C₂	1	2	yes	maximal, complemented
7	#13067364	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#66954	C₄	1	1	yes	characteristic
10	C₂×#66954	C₄	1	1	yes	characteristic
11	C₄×#4339	C₂²	1	1	yes	characteristic
12	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	#10622724	—	2	4	no	—
15	#10622714	C₄	1	1	yes	characteristic
16	#10622714	C₄	1	1	yes	characteristic
17	#10622538	C₂²	1	2	yes	—
18	C₄×#4318	C₂²	1	1	yes	characteristic
19	#10622451	—	2	2	no	—
20	#10622450	C₂²	1	4	yes	—
21	#10622448	C₂²	1	4	yes	—
22	#10622444	—	2	2	no	—
23	#10622302	C₄	1	2	yes	—
24	#10622302	C₄	1	2	yes	—
25	C₄×#4313	C₂²	1	2	yes	—
26	C₄×C₈×D₁₁	C₂²	1	2	yes	—
27	C₁₁×#335	C₂²	1	2	yes	—
28	C₁₁×#335	C₂²	1	2	yes	—
29	#10622118	C₂²	1	4	yes	—
30	#10622117	C₂²	1	4	yes	—
31	#10622096	C₂²	1	1	yes	characteristic
32	#10622096	C₂²	1	1	yes	characteristic
33	#10622095	C₂²	1	2	yes	—
34	C₂×#4339	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₁	—	2	2	no	—
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
39	C₄×C₂²⋊D₁₁	D₄	1	2	yes	—
40	C₄×C₂²⋊D₁₁	Q₈	1	2	yes	—
41	#66979	C₂×C₄	1	1	yes	characteristic
42	C₂×#4318	C₂×C₄	1	1	yes	characteristic
43	C₂×#4318	C₂×C₄	1	1	yes	characteristic
44	#66954	—	2	4	no	—
45	#66954	—	2	4	no	—
46	C₂×#4313	—	2	4	no	—
47	C₂×#4313	C₂×C₄	1	2	yes	—
48	C₂×#4313	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
51	C₄×D₄₄	D₄	1	1	yes	characteristic
52	C₄×D₄₄	Q₈	1	1	yes	characteristic
53	#66925	C₂×C₄	1	2	yes	—
54	C₄²×D₁₁	C₂³	1	2	yes	—
55	C₄×Dic₂₂	D₄	1	1	yes	characteristic
56	C₄×Dic₂₂	Q₈	1	1	yes	characteristic
57	C₁₁×C₄⋊C₈	—	2	2	no	—
58	C₁₁×C₄⋊C₈	C₂³	1	4	yes	—
59	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
63	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
64	(C₂×D₁₁)⋊C₈	—	2	4	no	—
65	#66879	—	2	4	no	—
66	#66879	C₂×C₄	1	2	yes	—
67	#66879	C₂×C₄	1	2	yes	—
68	Dic₁₁⋊C₈	—	2	4	no	—
69	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
70	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
71	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
72	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
73	#66867	C₂×C₄	1	2	yes	—
74	C₄×C₁₁⋊C₈	—	2	2	no	—
75	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
76	#17799	—	11	1	no	maximal, Sylow
77	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₂²⋊D₁₁	C₄⋊C₄	1	2	yes	—
81	#4339	—	2	2	no	—
82	#4339	—	2	2	no	—
83	C₂×D₄₄	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
85	C₂×C₄×D₁₁	C₂×D₄	1	2	yes	—
86	C₂×C₄×D₁₁	C₂×Q₈	1	2	yes	—
87	C₂×Dic₂₂	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	—	2	2	no	—
89	C₂×C₈×C₁₁	—	2	4	no	—
90	C₂×C₈×C₁₁	—	8	2	no	—
91	C₂×C₈×C₁₁	M₁₆	1	4	yes	complemented
92	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
99	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
100	C₂²⋊Dic₁₁	C₄⋊C₄	1	2	yes	—
101	(C₂×D₁₁)⋊C₄	C₄⋊C₄	1	2	yes	—
102	(C₂×D₁₁)⋊C₄	C₄⋊C₄	1	2	yes	—
103	C₄⋊Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
104	C₄⋊Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
105	Dic₁₁⋊C₄	C₄⋊C₄	1	2	yes	—
106	Dic₁₁⋊C₄	C₄⋊C₄	1	2	yes	—
107	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
108	C₄×Dic₁₁	C₂×D₄	1	2	yes	—
109	C₄×Dic₁₁	C₂×Q₈	1	2	yes	—
110	#4318	—	2	2	no	—
111	#4318	M₁₆	1	4	yes	complemented
112	C₂×C₁₁⋊C₈	—	2	2	no	—
113	C₂×C₁₁⋊C₈	—	2	2	no	—
114	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
115	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
116	#4313	—	2	4	no	—
117	#4313	—	2	4	no	—
118	#4313	—	8	4	no	—
119	C₈×D₁₁	—	2	4	no	—
120	C₈×D₁₁	—	2	4	no	—
121	#2989	—	11	2	no	—
122	C₄×#131	—	11	1	no	—
123	#1569	—	11	4	no	—
124	#1568	—	11	2	no	—
125	#1567	—	11	2	no	—
126	#1565	—	11	2	no	—
127	#1564	—	11	1	no	—
128	C₂×#335	D₁₁	1	1	yes	characteristic, complemented
129	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×D₁₁	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₁	—	2	1	no	—
132	C₂×C₄×C₁₁	—	2	1	no	—
133	C₂×C₄×C₁₁	—	4	1	no	—
134	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
142	C₂²⋊D₁₁	—	2	4	no	—
143	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
145	C₂×Dic₁₁	C₂×C₄⋊C₄	1	2	yes	—
146	D₄₄	—	2	2	no	—
147	C₄×D₁₁	—	2	2	no	—
148	C₄×D₁₁	—	2	2	no	—
149	C₄×D₁₁	—	2	2	no	—
150	C₄×D₁₁	—	2	2	no	—
151	C₄×D₁₁	—	4	2	no	—
152	Dic₂₂	—	2	2	no	—
153	C₈×C₁₁	—	2	1	no	—
154	C₈×C₁₁	—	2	1	no	—
155	C₈×C₁₁	—	4	4	no	—
156	C₈×C₁₁	C₂×M₁₆	1	2	yes	—
157	C₈×C₁₁	C₂×M₁₆	1	2	yes	—
158	C₁₁⋊C₈	—	2	2	no	—
159	C₁₁⋊C₈	—	4	2	no	—
160	C₁₁⋊C₈	—	4	2	no	—
161	C₁₁⋊C₈	C₂×M₁₆	1	4	yes	—
162	C₂×#131	—	11	1	no	—
163	C₂×#131	—	11	1	no	—
164	C₄×#41	—	11	1	no	—
165	#446	—	22	2	no	—
166	#436	—	22	2	no	—
167	#435	—	22	4	no	—
168	#432	—	11	4	no	—
169	#431	—	11	2	no	—
170	#423	—	11	4	no	—
171	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
172	#405	—	11	1	no	—
173	#405	—	11	1	no	—
174	#405	—	11	2	no	—
175	#405	—	11	2	no	—
176	C₄×M₁₆	—	11	1	no	—
177	C₄×M₁₆	—	11	2	no	—
178	C₂×C₄×C₈	—	11	2	no	—
179	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
180	#335	—	11	4	no	—
181	#335	—	11	1	no	—
182	#335	—	11	1	no	—
183	#335	C₂×D₁₁	1	2	yes	—
184	#335	C₂×D₁₁	1	2	yes	—
185	#334	—	11	4	no	—
186	#334	—	11	2	no	—
187	C₂²×C₁₁	—	2	1	no	—
188	C₂²×C₁₁	—	2	1	no	—
189	C₂²×C₁₁	#423	1	2	yes	—
190	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
191	C₂×D₁₁	—	2	2	no	—
192	C₂×D₁₁	—	2	2	no	—
193	C₂×D₁₁	—	2	2	no	—
194	C₄×C₁₁	—	2	1	no	—
195	C₄×C₁₁	—	2	1	no	—
196	C₄×C₁₁	—	2	1	no	—
197	C₄×C₁₁	#423	1	2	yes	—
198	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
199	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
200	C₄×C₁₁	#576	1	1	yes	characteristic
201	C₄×C₁₁	#578	1	1	yes	characteristic
202	Dic₁₁	—	2	2	no	—
203	Dic₁₁	—	2	2	no	—
204	Dic₁₁	—	2	2	no	—
205	C₂×#41	—	11	1	no	—
206	#131	—	22	4	no	—
207	#131	—	22	4	no	—
208	C₂×M₁₆	—	11	1	no	—
209	C₂×M₁₆	—	11	1	no	—
210	C₂×M₁₆	—	11	2	no	—
211	C₂×M₁₆	—	11	2	no	—
212	C₂×M₁₆	—	22	4	no	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	—	11	2	no	—
215	C₂²×C₈	—	11	2	no	—
216	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
217	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
218	C₄×Q₈	—	11	1	no	—
219	C₄×Q₈	—	11	1	no	—
220	C₄×D₄	—	11	2	no	—
221	C₄×D₄	—	11	2	no	—
222	C₄×D₄	—	11	1	no	—
223	C₄×D₄	—	11	1	no	—
224	#117	—	11	1	no	—
225	#117	—	11	2	no	—
226	C₂×C₄²	—	11	2	no	—
227	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	11	2	no	—
230	C₄⋊C₈	—	11	2	no	—
231	C₄⋊C₈	—	22	4	no	—
232	C₄⋊C₈	C₂²×D₁₁	1	4	yes	—
233	C₂²⋊C₈	—	22	4	no	—
234	C₂²⋊C₈	—	22	2	no	—
235	#97	—	11	2	no	—
236	#97	—	11	2	no	—
237	#97	—	11	2	no	—
238	#97	—	22	4	no	—
239	C₄×C₈	—	11	2	no	—
240	C₄×C₈	—	11	2	no	—
241	C₄×C₈	—	11	2	no	—
242	C₄×C₈	—	22	2	no	—
243	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
244	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
247	C₂×C₁₁	—	2	2	no	—
248	C₂×C₁₁	C₂×#423	1	1	yes	characteristic
249	C₂×C₁₁	#2913	1	1	yes	characteristic
250	C₂×C₁₁	#2916	1	1	yes	characteristic
251	D₁₁	—	4	2	no	—
252	#41	—	22	2	no	—
253	#41	—	22	2	no	—
254	C₂×Q₈	—	11	1	no	—
255	C₂×D₄	—	11	2	no	—
256	C₂×D₄	—	11	1	no	—
257	C₂²×C₄	—	11	2	no	—
258	C₂²×C₄	—	11	2	no	—
259	C₂²×C₄	—	11	2	no	—
260	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
261	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
262	C₂²×C₄	Q₈×D₁₁	1	1	yes	characteristic
263	M₁₆	—	11	4	no	—
264	M₁₆	—	22	4	no	—
265	M₁₆	—	22	2	no	—
266	M₁₆	—	22	4	no	—
267	M₁₆	—	88	4	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	8	2	no	—
271	C₂×C₈	—	11	2	no	—
272	C₂×C₈	—	11	2	no	—
273	C₂×C₈	—	22	4	no	—
274	C₂×C₈	—	22	4	no	—
275	C₂×C₈	—	22	2	no	—
276	C₂×C₈	—	22	2	no	—
277	C₂×C₈	#4313	1	4	yes	complemented
278	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
279	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
280	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
281	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
282	C₄⋊C₄	—	11	2	no	—
283	C₄⋊C₄	—	11	2	no	—
284	C₄⋊C₄	—	11	1	no	—
285	C₄⋊C₄	—	11	1	no	—
286	C₂²⋊C₄	—	11	2	no	—
287	C₂²⋊C₄	—	11	2	no	—
288	C₂²⋊C₄	—	11	2	no	—
289	C₄²	—	11	2	no	—
290	C₄²	—	11	2	no	—
291	C₄²	—	11	2	no	—
292	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
293	C₄²	D₄×D₁₁	1	1	yes	characteristic
294	C₄²	Q₈×D₁₁	1	1	yes	characteristic
295	C₄²	C₂³×D₁₁	1	1	yes	characteristic
296	C₁₁	#17799	1	1	yes	characteristic, complemented, Sylow
297	C₂³	—	11	2	no	—
298	C₂³	C₄⋊C₄×D₁₁	1	1	yes	characteristic
299	Q₈	—	22	2	no	—
300	D₄	—	22	4	no	—
301	D₄	—	22	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	11	2	no	—
306	C₂×C₄	—	11	2	no	—
307	C₂×C₄	—	11	2	no	—
308	C₂×C₄	—	22	2	no	—
309	C₂×C₄	—	22	2	no	—
310	C₂×C₄	—	22	2	no	—
311	C₂×C₄	—	22	2	no	—
312	C₂×C₄	—	44	2	no	—
313	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
314	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
315	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
316	C₂×C₄	C₂×#4313	1	2	yes	—
317	C₂×C₄	M₁₆×D₁₁	1	2	yes	—
318	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
320	C₂×C₄	C₂×Q₈×D₁₁	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	4	4	no	—
324	C₈	—	11	4	no	—
325	C₈	—	22	2	no	—
326	C₈	—	44	2	no	—
327	C₈	—	44	2	no	—
328	C₈	C₂×#4313	1	2	yes	—
329	C₈	C₂×#4313	1	2	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	22	2	no	—
333	C₂²	—	22	2	no	—
334	C₂²	—	22	2	no	—
335	C₂²	#10622450	1	2	yes	—
336	C₂²	C₂×C₄⋊C₄×D₁₁	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	22	2	no	—
341	C₄	—	22	2	no	—
342	C₄	—	22	2	no	—
343	C₄	#10622450	1	2	yes	—
344	C₄	C₂²×#4313	1	1	yes	characteristic
345	C₄	C₂×M₁₆×D₁₁	1	1	yes	characteristic
346	C₄	#10623288	1	1	yes	characteristic
347	C₄	#10623297	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	44	2	no	—
350	C₂	C₂×#10622450	1	1	yes	characteristic
351	C₂	#13077212	1	1	yes	characteristic
352	C₂	#13077224	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#335	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂×#4339	C₂×C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#335	D₁₁
hypocenter	C₁₁	#17799
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #13067382	C₂	semidirect
4. #13067378	C₂	semidirect
5. #13067373	C₂	semidirect
6. #13067366	C₂	semidirect
7. #13067364	C₂	semidirect
8. C₂×C₁₁×#335	C₂	semidirect
91. C₂×C₈×C₁₁	M₁₆	semidirect
111. #4318	M₁₆	semidirect
128. C₂×#335	D₁₁	semidirect
277. C₂×C₈	#4313	semidirect
296. C₁₁	#17799	semidirect

Maximal subgroups

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

subgroup	index
#13077734	2
C₄×#66954	2
#13067382	2
#13067378	2
#13067373	2
#13067366	2
#13067364	2
C₂×C₁₁×#335	2
#17799	11