Subgroups — #14643249

Order 2816.

674 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622537	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#13066451	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₁×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
6	C₂²×#4318	—	2	1	no	—
7	C₂³×C₁₁⋊C₈	—	2	1	no	—
8	C₁₁×#431	—	2	2	no	—
9	C₁₁×#431	—	2	2	no	—
10	C₂×C₁₁×C₄⋊C₈	C₄	1	2	yes	—
11	C₂×C₁₁×C₂²⋊C₈	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
13	#10622537	—	2	2	no	—
14	#10622537	C₄	1	4	yes	—
15	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
16	C₂×C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
17	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
18	C₂×C₄⋊(C₁₁⋊C₈)	C₄	1	4	yes	—
19	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
20	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
21	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₁	Q₈	1	1	yes	characteristic
24	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₁	D₄	1	2	yes	—
30	C₂×C₄²×C₁₁	D₄	1	2	yes	—
31	C₂×C₄²×C₁₁	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
33	C₂×#4318	—	2	4	no	—
34	C₂×#4318	—	4	2	no	—
35	C₂×#4318	—	4	2	no	—
36	C₂²×C₁₁⋊C₈	—	2	4	no	—
37	C₂²×C₁₁⋊C₈	—	2	2	no	—
38	C₂²×C₁₁⋊C₈	—	2	2	no	—
39	C₂²×C₁₁⋊C₈	—	4	2	no	—
40	C₂²×C₁₁⋊C₈	—	4	2	no	—
41	C₁₁×C₄⋊C₈	—	2	4	no	—
42	C₁₁×C₂²⋊C₈	—	2	4	no	—
43	C₄×C₈×C₁₁	—	2	2	no	—
44	C₄×C₈×C₁₁	—	2	2	no	—
45	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
46	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
47	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
48	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
49	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	4	yes	—
50	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	4	yes	—
51	#7877	—	11	1	no	maximal, Sylow
52	C₂⁴×C₁₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₁	—	2	4	no	—
54	C₂²×C₄×C₁₁	—	4	2	no	—
55	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
56	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₁	D₈	1	2	yes	—
65	C₂²×C₄×C₁₁	SD₁₆	1	2	yes	—
66	C₂²×C₄×C₁₁	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₁₁	Q₁₆	1	2	yes	—
68	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₁₁	—	2	4	no	—
72	C₂×C₈×C₁₁	—	2	8	no	—
73	C₄²×C₁₁	C₄²	1	2	yes	—
74	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
78	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
79	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
83	#4318	—	4	8	no	—
84	C₂×C₁₁⋊C₈	—	2	4	no	—
85	C₂×C₁₁⋊C₈	—	2	4	no	—
86	C₂×C₁₁⋊C₈	—	4	8	no	—
87	C₂×C₁₁⋊C₈	—	4	2	no	—
88	C₂×C₁₁⋊C₈	—	4	2	no	—
89	C₂×C₁₁⋊C₈	—	4	2	no	—
90	C₂×C₁₁⋊C₈	—	4	2	no	—
91	C₂×#431	D₁₁	1	1	yes	characteristic
92	C₂×#423	—	11	1	no	—
93	C₂²×C₄⋊C₈	—	11	1	no	—
94	#1277	—	11	4	no	—
95	C₂³×C₁₁	—	2	2	no	—
96	C₂³×C₁₁	—	2	2	no	—
97	C₂³×C₁₁	#95	1	2	yes	—
98	C₂³×C₁₁	#106	1	2	yes	—
99	C₂³×C₁₁	#107	1	2	yes	—
100	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₁	—	2	2	no	—
102	C₂×C₄×C₁₁	—	2	2	no	—
103	C₂×C₄×C₁₁	—	2	2	no	—
104	C₂×C₄×C₁₁	—	2	4	no	—
105	C₂×C₄×C₁₁	—	2	4	no	—
106	C₂×C₄×C₁₁	—	4	2	no	—
107	C₂×C₄×C₁₁	—	4	2	no	—
108	C₂×C₄×C₁₁	#95	1	2	yes	—
109	C₂×C₄×C₁₁	#95	1	2	yes	—
110	C₂×C₄×C₁₁	#95	1	2	yes	—
111	C₂×C₄×C₁₁	#102	1	4	yes	—
112	C₂×C₄×C₁₁	#102	1	4	yes	—
113	C₂×C₄×C₁₁	#103	1	4	yes	—
114	C₂×C₄×C₁₁	#103	1	4	yes	—
115	C₂×C₄×C₁₁	#106	1	2	yes	—
116	C₂×C₄×C₁₁	#107	1	2	yes	—
117	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁	#131	1	4	yes	—
124	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
128	C₈×C₁₁	—	4	4	no	—
129	C₁₁⋊C₈	—	4	4	no	—
130	C₁₁⋊C₈	—	4	4	no	—
131	C₂²×M₁₆	—	22	1	no	—
132	C₂³×C₈	—	22	1	no	—
133	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
134	#431	—	2	2	no	—
135	#431	—	2	2	no	—
136	#423	—	11	4	no	—
137	#423	—	22	2	no	—
138	C₂×C₄⋊C₈	—	11	2	no	—
139	C₂×C₄⋊C₈	—	11	2	no	—
140	C₂×C₄⋊C₈	—	11	4	no	—
141	C₂×C₄⋊C₈	—	22	1	no	—
142	C₂×C₄⋊C₈	—	22	1	no	—
143	C₂×C₄⋊C₈	—	22	2	no	—
144	C₂×C₄⋊C₈	Dic₁₁	1	2	yes	—
145	C₂×C₂²⋊C₈	Dic₁₁	1	2	yes	—
146	C₂×C₄×C₈	C₂×D₁₁	1	2	yes	—
147	C₂²×C₁₁	—	2	4	no	—
148	C₂²×C₁₁	—	2	4	no	—
149	C₂²×C₁₁	—	2	2	no	—
150	C₂²×C₁₁	—	2	2	no	—
151	C₂²×C₁₁	#340	1	4	yes	—
152	C₂²×C₁₁	C₂×#95	1	1	yes	characteristic
153	C₂²×C₁₁	#424	1	2	yes	—
154	C₂²×C₁₁	C₂×#106	1	1	yes	characteristic
155	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
156	C₂²×C₁₁	#430	1	2	yes	—
157	C₄×C₁₁	—	2	2	no	—
158	C₄×C₁₁	—	2	2	no	—
159	C₄×C₁₁	—	2	1	no	—
160	C₄×C₁₁	—	2	1	no	—
161	C₄×C₁₁	#340	1	4	yes	—
162	C₄×C₁₁	#405	1	2	yes	—
163	C₄×C₁₁	#408	1	2	yes	—
164	C₄×C₁₁	C₂×#102	1	1	yes	characteristic
165	C₄×C₁₁	C₂×#102	1	1	yes	characteristic
166	C₄×C₁₁	C₂×#103	1	1	yes	characteristic
167	C₄×C₁₁	C₂×#103	1	1	yes	characteristic
168	C₂³×C₄	Dic₂₂	1	1	yes	characteristic
169	C₂³×C₄	D₄₄	1	1	yes	characteristic
170	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
171	C₂×M₁₆	—	22	4	no	—
172	C₂×M₁₆	—	44	2	no	—
173	C₂×M₁₆	—	44	2	no	—
174	C₂²×C₈	—	22	4	no	—
175	C₂²×C₈	—	22	2	no	—
176	C₂²×C₈	—	22	2	no	—
177	C₂²×C₈	—	44	2	no	—
178	C₂²×C₈	—	44	2	no	—
179	C₂²×C₈	C₂×Dic₁₁	1	4	yes	—
180	C₂×C₄²	Dic₂₂	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₁₁	1	2	yes	—
182	C₂×C₄²	C₄×D₁₁	1	2	yes	—
183	C₂×C₄²	D₄₄	1	1	yes	characteristic
184	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₁₁	1	2	yes	—
186	C₂×C₄²	C₂²⋊D₁₁	1	2	yes	—
187	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
188	C₄⋊C₈	—	2	4	no	—
189	C₄⋊C₈	—	11	4	no	—
190	C₄⋊C₈	—	11	4	no	—
191	C₄⋊C₈	—	22	4	no	—
192	C₄⋊C₈	—	22	4	no	—
193	C₄⋊C₈	—	22	2	no	—
194	C₄⋊C₈	—	22	2	no	—
195	C₂²⋊C₈	—	2	4	no	—
196	C₄×C₈	—	2	2	no	—
197	C₄×C₈	—	2	2	no	—
198	C₂×C₁₁	—	2	4	no	—
199	C₂×C₁₁	#1570	1	2	yes	—
200	C₂×C₁₁	#1571	1	2	yes	—
201	C₂×C₁₁	#1729	1	1	yes	characteristic
202	C₂×C₁₁	C₂×#340	1	1	yes	characteristic
203	C₂×C₁₁	#1738	1	1	yes	characteristic
204	C₂⁴	C₄⋊Dic₁₁	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	#4314	1	2	yes	—
208	C₂²×C₄	#4314	1	2	yes	—
209	C₂²×C₄	D₈₈	1	2	yes	—
210	C₂²×C₄	Dic₄₄	1	2	yes	—
211	C₂²×C₄	C₄×Dic₁₁	1	2	yes	—
212	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
213	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
214	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
215	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
216	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
217	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	2	yes	—
218	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	2	yes	—
219	C₂²×C₄	C₂²⋊Dic₁₁	1	2	yes	—
220	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
222	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
223	M₁₆	—	44	8	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	22	4	no	—
227	C₂×C₈	—	22	4	no	—
228	C₂×C₈	—	44	8	no	—
229	C₂×C₈	—	44	2	no	—
230	C₂×C₈	—	44	2	no	—
231	C₂×C₈	—	44	2	no	—
232	C₂×C₈	—	44	2	no	—
233	C₄²	C₄×Dic₁₁	1	2	yes	—
234	C₄²	Dic₁₁⋊C₄	1	2	yes	—
235	C₄²	Dic₁₁⋊C₄	1	2	yes	—
236	C₄²	(C₂×D₁₁)⋊C₄	1	2	yes	—
237	C₄²	(C₂×D₁₁)⋊C₄	1	2	yes	—
238	C₄²	C₂²⋊Dic₁₁	1	2	yes	—
239	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
240	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
241	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
242	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
243	C₁₁	#7877	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	#66881	1	2	yes	—
247	C₂³	#66882	1	2	yes	—
248	C₂³	C₁₁⋊#95	1	2	yes	—
249	C₂³	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	#66880	1	4	yes	—
258	C₂×C₄	#66880	1	4	yes	—
259	C₂×C₄	#66881	1	2	yes	—
260	C₂×C₄	#66882	1	2	yes	—
261	C₂×C₄	#66885	1	4	yes	—
262	C₂×C₄	#66885	1	4	yes	—
263	C₂×C₄	C₁₁⋊#95	1	2	yes	—
264	C₂×C₄	C₁₁⋊#95	1	2	yes	—
265	C₂×C₄	C₁₁⋊#95	1	2	yes	—
266	C₂×C₄	C₂×#4314	1	1	yes	characteristic
267	C₂×C₄	C₂×#4314	1	1	yes	characteristic
268	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
269	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
271	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
274	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
275	C₂×C₄	#67001	1	4	yes	—
276	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
277	C₈	—	4	4	no	—
278	C₈	—	44	4	no	—
279	C₈	—	44	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	#10622189	1	4	yes	—
285	C₂²	#10622612	1	2	yes	—
286	C₂²	C₂×#66881	1	1	yes	characteristic
287	C₂²	C₂×#66882	1	1	yes	characteristic
288	C₂²	#10622738	1	2	yes	—
289	C₂²	C₂×C₁₁⋊#95	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	#10622189	1	4	yes	—
295	C₄	#10622610	1	2	yes	—
296	C₄	C₂×#66880	1	1	yes	characteristic
297	C₄	C₂×#66880	1	1	yes	characteristic
298	C₄	C₂×#66885	1	1	yes	characteristic
299	C₄	C₂×#66885	1	1	yes	characteristic
300	C₄	#10622847	1	2	yes	—
301	C₂	—	2	4	no	—
302	C₂	#13067387	1	2	yes	—
303	C₂	#13067391	1	2	yes	—
304	C₂	#13068313	1	1	yes	characteristic
305	C₂	C₂×#10622189	1	1	yes	characteristic
306	C₂	#13068343	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄₄
commutator	C₂×C₄×C₁₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₂×C₁₁×#431	C₂
socle	C₂³×C₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₁	C₄⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂×#431	D₁₁
hypocenter	C₁₁	#7877
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #13066451	C₂	semidirect
243. C₁₁	#7877	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622537	2
C₂²×C₄⋊(C₁₁⋊C₈)	2
#13066451	2
C₂×C₁₁×#431	2
#7877	11