Subgroups — #14722982

Order 2816.

572 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077733	C₂	1	2	yes	maximal
2	#13067405	C₂	1	2	yes	maximal, complemented
3	#13067401	C₂	1	2	yes	maximal, complemented
4	#13067396	C₂	1	2	yes	maximal, complemented
5	#13067387	C₂	1	2	yes	maximal, complemented
6	#13067383	C₂	1	2	yes	maximal, complemented
7	#13067379	C₂	1	1	yes	maximal, characteristic, complemented
8	#13067372	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁×#2922	C₂	1	1	yes	maximal, characteristic
10	C₄×C₁₁×#41	C₂²	1	1	yes	characteristic
11	C₁₁×#433	C₄	1	4	yes	—
12	C₁₁×#431	C₂²	1	2	yes	—
13	C₁₁×#431	C₂²	1	2	yes	—
14	C₁₁×#424	C₄	1	2	yes	—
15	C₁₁×#408	C₄	1	2	yes	—
16	C₄×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
18	#10622637	—	2	2	no	—
19	#10622625	—	2	2	no	—
20	#10622624	—	2	2	no	—
21	#10622624	—	2	2	no	—
22	#10622539	C₂²	1	2	yes	—
23	#10622539	C₂²	1	2	yes	—
24	#10622538	C₂²	1	2	yes	—
25	#10622537	C₂²	1	2	yes	—
26	#10622537	C₂²	1	2	yes	—
27	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
28	#10622096	C₂²	1	2	yes	—
29	#10622096	C₂²	1	2	yes	—
30	#10622096	C₂²	1	2	yes	—
31	#10622095	C₂²	1	2	yes	—
32	#10622095	C₂²	1	1	yes	characteristic
33	#10622095	C₂²	1	1	yes	characteristic
34	#10622095	C₂²	1	1	yes	characteristic
35	#10622095	C₂²	1	1	yes	characteristic
36	#10622095	C₂²	1	2	yes	—
37	#10622095	C₂²	1	2	yes	—
38	C₂×C₁₁×#41	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
41	C₄×Q₈×C₁₁	D₄	1	1	yes	characteristic
42	C₄×Q₈×C₁₁	Q₈	1	1	yes	characteristic
43	C₄×D₄×C₁₁	D₄	1	1	yes	characteristic
44	C₄×D₄×C₁₁	D₄	1	2	yes	—
45	C₄×D₄×C₁₁	Q₈	1	2	yes	—
46	C₄×D₄×C₁₁	Q₈	1	1	yes	characteristic
47	C₁₁×#117	C₂×C₄	1	1	yes	characteristic
48	C₁₁×#117	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
51	C₂×#4318	—	2	2	no	—
52	C₂×#4318	—	2	2	no	—
53	C₂²×C₁₁⋊C₈	—	2	2	no	—
54	C₁₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
55	C₁₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
56	C₁₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
57	C₁₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
58	C₁₁×#97	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₁	C₂³	1	2	yes	—
63	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
64	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
65	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
66	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
67	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
68	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
69	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
70	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
71	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
72	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
73	#66867	—	2	2	no	—
74	#66867	—	2	2	no	—
75	C₄×C₁₁⋊C₈	—	2	2	no	—
76	#17824	—	11	1	no	maximal, Sylow
77	C₁₁×#41	—	4	2	no	—
78	C₂×Q₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
79	C₂×D₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
80	C₂×D₄×C₁₁	C₄⋊C₄	1	2	yes	—
81	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
87	C₁₁×M₁₆	—	4	2	no	—
88	C₂×C₈×C₁₁	—	4	2	no	—
89	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
92	C₁₁×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
93	C₁₁×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
94	C₁₁×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
95	C₁₁×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
96	C₁₁×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
97	C₁₁×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
98	C₁₁×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
99	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
101	C₄²×C₁₁	C₂×D₄	1	2	yes	—
102	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₁	C₂×Q₈	1	2	yes	—
104	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
105	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
106	#4318	—	8	2	no	—
107	#4318	—	8	2	no	—
108	C₂×C₁₁⋊C₈	—	2	2	no	—
109	C₂×C₁₁⋊C₈	—	2	2	no	—
110	C₂×C₁₁⋊C₈	—	2	2	no	—
111	C₂×C₁₁⋊C₈	—	2	2	no	—
112	C₂×C₁₁⋊C₈	—	8	2	no	—
113	#2989	—	11	2	no	—
114	#2922	D₁₁	1	1	yes	characteristic
115	#1574	—	11	2	no	—
116	#1573	—	11	2	no	—
117	#1572	—	11	2	no	—
118	#1570	—	11	2	no	—
119	#1569	—	11	2	no	—
120	#1568	—	11	1	no	—
121	#1566	—	11	1	no	—
122	C₂³×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
123	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
124	Q₈×C₁₁	—	4	1	no	—
125	D₄×C₁₁	—	4	1	no	—
126	D₄×C₁₁	—	4	2	no	—
127	C₂×C₄×C₁₁	—	2	2	no	—
128	C₂×C₄×C₁₁	—	2	4	no	—
129	C₂×C₄×C₁₁	—	2	2	no	—
130	C₂×C₄×C₁₁	—	2	2	no	—
131	C₂×C₄×C₁₁	—	2	2	no	—
132	C₂×C₄×C₁₁	—	4	2	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	2	yes	—
136	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
143	C₈×C₁₁	—	2	2	no	—
144	C₈×C₁₁	—	2	2	no	—
145	C₈×C₁₁	—	2	4	no	—
146	C₁₁⋊C₈	—	4	2	no	—
147	C₁₁⋊C₈	—	4	2	no	—
148	C₁₁⋊C₈	—	4	2	no	—
149	C₁₁⋊C₈	—	4	2	no	—
150	C₄×#41	C₂×D₁₁	1	1	yes	characteristic
151	#446	—	22	2	no	—
152	#436	—	22	2	no	—
153	#435	—	22	2	no	—
154	#435	—	22	2	no	—
155	#433	Dic₁₁	1	4	yes	—
156	#432	—	11	2	no	—
157	#432	—	11	2	no	—
158	#431	—	11	2	no	—
159	#431	C₂×D₁₁	1	2	yes	—
160	#431	C₂×D₁₁	1	2	yes	—
161	#424	Dic₁₁	1	2	yes	—
162	#423	—	11	2	no	—
163	#423	—	11	2	no	—
164	C₂×C₄⋊C₈	—	11	2	no	—
165	#408	Dic₁₁	1	2	yes	—
166	C₄×M₁₆	C₂×D₁₁	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
168	#335	—	11	2	no	—
169	#335	—	11	2	no	—
170	#335	—	11	2	no	—
171	#334	—	11	2	no	—
172	#334	—	11	1	no	—
173	#334	—	11	1	no	—
174	#334	—	11	1	no	—
175	#334	—	11	1	no	—
176	#334	—	11	2	no	—
177	#334	—	11	2	no	—
178	C₂²×C₁₁	—	2	1	no	—
179	C₂²×C₁₁	—	2	2	no	—
180	C₂²×C₁₁	—	2	2	no	—
181	C₂²×C₁₁	—	2	1	no	—
182	C₂²×C₁₁	—	2	2	no	—
183	C₂²×C₁₁	—	2	1	no	—
184	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₄×C₁₁	—	2	1	no	—
186	C₄×C₁₁	—	2	2	no	—
187	C₄×C₁₁	—	2	2	no	—
188	C₄×C₁₁	—	2	1	no	—
189	C₄×C₁₁	—	2	1	no	—
190	C₄×C₁₁	—	2	1	no	—
191	C₄×C₁₁	—	2	2	no	—
192	C₄×C₁₁	#568	1	2	yes	—
193	C₄×C₁₁	#577	1	2	yes	—
194	C₂×#41	C₂×Dic₁₁	1	1	yes	characteristic
195	C₂×M₁₆	—	22	2	no	—
196	C₂×M₁₆	—	22	2	no	—
197	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
198	C₂²×C₈	—	22	2	no	—
199	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
200	C₄×Q₈	Dic₂₂	1	1	yes	characteristic
201	C₄×Q₈	D₄₄	1	1	yes	characteristic
202	C₄×D₄	Dic₂₂	1	1	yes	characteristic
203	C₄×D₄	Dic₂₂	1	2	yes	—
204	C₄×D₄	D₄₄	1	1	yes	characteristic
205	C₄×D₄	D₄₄	1	2	yes	—
206	#117	C₂×Dic₁₁	1	1	yes	characteristic
207	#117	C₂×Dic₁₁	1	2	yes	—
208	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
210	C₄⋊C₈	—	11	2	no	—
211	C₄⋊C₈	—	11	2	no	—
212	C₄⋊C₈	—	11	2	no	—
213	C₄⋊C₈	—	11	2	no	—
214	C₄⋊C₈	—	22	2	no	—
215	C₄⋊C₈	—	22	2	no	—
216	C₄⋊C₈	—	22	2	no	—
217	C₄⋊C₈	C₂×Dic₁₁	1	4	yes	—
218	C₄⋊C₈	C₂×Dic₁₁	1	4	yes	—
219	C₂²⋊C₈	—	22	2	no	—
220	C₂²⋊C₈	—	22	2	no	—
221	C₂²⋊C₈	—	22	2	no	—
222	C₂²⋊C₈	C₂×Dic₁₁	1	4	yes	—
223	C₂²⋊C₈	C₂×Dic₁₁	1	4	yes	—
224	#97	—	22	2	no	—
225	#97	—	22	2	no	—
226	#97	C₂×Dic₁₁	1	2	yes	—
227	C₄×C₈	—	22	2	no	—
228	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
229	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₁₁	1	2	yes	—
232	C₂×C₁₁	—	4	1	no	—
233	C₂×C₁₁	—	4	2	no	—
234	C₂×C₁₁	#2905	1	1	yes	characteristic
235	C₂×C₁₁	#2912	1	1	yes	characteristic
236	C₂×C₁₁	#2916	1	1	yes	characteristic
237	#41	—	4	2	no	—
238	C₂×Q₈	C₄⋊Dic₁₁	1	1	yes	characteristic
239	C₂×D₄	C₄⋊Dic₁₁	1	1	yes	characteristic
240	C₂×D₄	C₄⋊Dic₁₁	1	2	yes	—
241	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
242	C₂²×C₄	C₂×Dic₂₂	1	2	yes	—
243	C₂²×C₄	C₂×D₄₄	1	2	yes	—
244	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
246	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
247	M₁₆	—	4	2	no	—
248	M₁₆	—	88	2	no	—
249	M₁₆	—	88	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	22	2	no	—
252	C₂×C₈	—	22	2	no	—
253	C₂×C₈	—	22	2	no	—
254	C₂×C₈	—	22	2	no	—
255	C₂×C₈	—	88	2	no	—
256	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
257	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
258	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
259	C₄⋊C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
260	C₄⋊C₄	C₄⋊Dic₁₁	1	2	yes	—
261	C₄⋊C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
262	C₄⋊C₄	C₄⋊Dic₁₁	1	2	yes	—
263	C₂²⋊C₄	C₄⋊Dic₁₁	1	2	yes	—
264	C₂²⋊C₄	C₄⋊Dic₁₁	1	2	yes	—
265	C₂²⋊C₄	C₄⋊Dic₁₁	1	2	yes	—
266	C₄²	C₂×Dic₂₂	1	1	yes	characteristic
267	C₄²	C₂×Dic₂₂	1	2	yes	—
268	C₄²	C₂×D₄₄	1	2	yes	—
269	C₄²	C₂×D₄₄	1	1	yes	characteristic
270	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
271	C₄²	C₂²×Dic₁₁	1	2	yes	—
272	C₄²	C₂³×D₁₁	1	1	yes	characteristic
273	C₁₁	#17824	1	1	yes	characteristic, complemented, Sylow
274	C₂³	C₂×C₄⋊Dic₁₁	1	2	yes	—
275	C₂³	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
276	Q₈	—	4	1	no	—
277	D₄	—	4	1	no	—
278	D₄	—	4	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₂×C₄⋊Dic₁₁	1	2	yes	—
287	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
289	C₂×C₄	C₂×C₄⋊Dic₁₁	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊Dic₁₁	1	2	yes	—
291	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
292	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
294	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	4	no	—
298	C₈	—	44	2	no	—
299	C₈	—	44	2	no	—
300	C₈	—	44	2	no	—
301	C₈	—	44	2	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	1	no	—
308	C₂²	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	#10623282	1	2	yes	—
317	C₄	#10623350	1	2	yes	—
318	C₂	—	4	1	no	—
319	C₂	—	4	2	no	—
320	C₂	#13077180	1	1	yes	characteristic
321	C₂	#13077211	1	1	yes	characteristic
322	C₂	#13077227	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₁₁×#2922	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂×C₁₁×#41	C₂×C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#2922	D₁₁
hypocenter	C₁₁	#17824
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2922

Complements

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

N	H ≅ G/N	type
2. #13067405	C₂	semidirect
3. #13067401	C₂	semidirect
4. #13067396	C₂	semidirect
5. #13067387	C₂	semidirect
6. #13067383	C₂	semidirect
7. #13067379	C₂	semidirect
8. #13067372	C₂	semidirect
273. C₁₁	#17824	semidirect

Maximal subgroups

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

subgroup	index
#13077733	2
#13067405	2
#13067401	2
#13067396	2
#13067387	2
#13067383	2
#13067379	2
#13067372	2
C₁₁×#2922	2
#17824	11