Subgroups — #14647608

Order 2816.

556 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#13068596	C₂	1	1	yes	maximal, characteristic
3	C₂×#10622095	C₂	1	2	yes	maximal, complemented
4	#13066451	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
7	C₂³×C₁₁⋊C₈	—	2	1	no	—
8	C₂×C₁₁×C₂²⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₁₁×C₂²⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₁₁⋊C₈)	—	2	1	no	—
12	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
13	C₂×C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
14	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
17	C₁₁×#336	C₄	1	2	yes	—
18	#10622162	—	2	1	no	—
19	#10622095	—	2	4	no	—
20	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₁	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₁	—	2	2	no	—
26	C₂×C₄²×C₁₁	—	2	2	no	—
27	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₁	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
31	C₂²×C₁₁⋊C₈	—	2	4	no	—
32	C₂²×C₁₁⋊C₈	—	2	2	no	—
33	C₂²×C₁₁⋊C₈	—	2	1	no	—
34	C₂²×C₁₁⋊C₈	—	2	1	no	—
35	C₂²×C₁₁⋊C₈	—	4	2	no	—
36	C₂²×C₁₁⋊C₈	—	4	2	no	—
37	C₁₁×C₂²⋊C₈	—	2	2	no	—
38	C₁₁×C₂²⋊C₈	C₈	1	4	yes	complemented
39	C₄×C₈×C₁₁	—	2	2	no	—
40	C₄×C₈×C₁₁	—	2	2	no	—
41	C₂²⋊(C₁₁⋊C₈)	—	4	2	no	—
42	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
43	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
44	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
45	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
46	C₄⋊(C₁₁⋊C₈)	D₄	1	4	yes	—
47	#8830	—	11	1	no	maximal, Sylow
48	C₂⁴×C₁₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₁	—	2	2	no	—
50	C₂²×C₄×C₁₁	—	2	1	no	—
51	C₂²×C₄×C₁₁	—	2	1	no	—
52	C₂²×C₄×C₁₁	—	2	2	no	—
53	C₂²×C₄×C₁₁	—	2	1	no	—
54	C₂²×C₄×C₁₁	—	2	1	no	—
55	C₂²×C₄×C₁₁	—	2	2	no	—
56	C₂²×C₄×C₁₁	—	2	2	no	—
57	C₂²×C₄×C₁₁	—	4	1	no	—
58	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₁₁	M₁₆	1	2	yes	—
63	C₂²×C₄×C₁₁	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₁₁	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₁₁	—	2	2	no	—
69	C₂×C₈×C₁₁	—	2	2	no	—
70	C₂×C₈×C₁₁	—	2	2	no	—
71	C₂×C₈×C₁₁	—	2	2	no	—
72	C₂×C₈×C₁₁	—	2	2	no	—
73	C₂×C₈×C₁₁	C₂×C₈	1	4	yes	complemented
74	C₄²×C₁₁	—	2	2	no	—
75	C₄²×C₁₁	—	2	1	no	—
76	C₄²×C₁₁	—	2	2	no	—
77	C₄²×C₁₁	—	2	1	no	—
78	C₄²×C₁₁	C₂×D₄	1	2	yes	—
79	C₄²×C₁₁	#41	1	2	yes	—
80	C₂×C₁₁⋊C₈	—	2	4	no	—
81	C₂×C₁₁⋊C₈	—	4	8	no	—
82	C₂×C₁₁⋊C₈	—	4	1	no	—
83	C₂×C₁₁⋊C₈	—	4	1	no	—
84	C₂×C₁₁⋊C₈	—	4	2	no	—
85	C₂×C₁₁⋊C₈	—	4	1	no	—
86	C₂×C₁₁⋊C₈	—	4	1	no	—
87	C₂×C₁₁⋊C₈	—	4	2	no	—
88	C₂×C₁₁⋊C₈	—	4	1	no	—
89	C₂×C₁₁⋊C₈	—	4	1	no	—
90	C₂²×C₄⋊C₈	—	11	1	no	—
91	#1797	—	11	1	no	—
92	C₄×C₂²⋊C₈	D₁₁	1	1	yes	characteristic
93	C₂×#334	—	11	2	no	—
94	#1277	—	11	2	no	—
95	C₂³×C₁₁	—	2	1	no	—
96	C₂³×C₁₁	—	2	1	no	—
97	C₂³×C₁₁	—	2	1	no	—
98	C₂³×C₁₁	—	2	1	no	—
99	C₂³×C₁₁	C₄⋊C₈	1	2	yes	—
100	C₂³×C₁₁	#106	1	2	yes	—
101	C₂³×C₁₁	#108	1	2	yes	—
102	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁	—	2	1	no	—
104	C₂×C₄×C₁₁	—	2	1	no	—
105	C₂×C₄×C₁₁	—	2	1	no	—
106	C₂×C₄×C₁₁	—	2	1	no	—
107	C₂×C₄×C₁₁	—	2	1	no	—
108	C₂×C₄×C₁₁	—	2	1	no	—
109	C₂×C₄×C₁₁	—	2	4	no	—
110	C₂×C₄×C₁₁	—	2	4	no	—
111	C₂×C₄×C₁₁	—	2	2	no	—
112	C₂×C₄×C₁₁	—	2	2	no	—
113	C₂×C₄×C₁₁	—	2	2	no	—
114	C₂×C₄×C₁₁	—	2	2	no	—
115	C₂×C₄×C₁₁	—	2	1	no	—
116	C₂×C₄×C₁₁	—	2	1	no	—
117	C₂×C₄×C₁₁	—	2	1	no	—
118	C₂×C₄×C₁₁	—	4	4	no	—
119	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₁₁	#106	1	2	yes	—
121	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₁	#120	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₁	#131	1	2	yes	—
128	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₁₁	#137	1	1	yes	characteristic
132	C₈×C₁₁	—	2	4	no	—
133	C₈×C₁₁	—	4	2	no	—
134	C₁₁⋊C₈	—	4	4	no	—
135	C₁₁⋊C₈	—	8	1	no	—
136	C₁₁⋊C₈	—	8	1	no	—
137	C₂³×C₈	—	22	1	no	—
138	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
139	C₂×C₄⋊C₈	—	11	2	no	—
140	C₂×C₄⋊C₈	—	11	1	no	—
141	C₂×C₄⋊C₈	—	11	1	no	—
142	C₂×C₄⋊C₈	—	22	4	no	—
143	C₂×C₄⋊C₈	—	22	1	no	—
144	C₂×C₂²⋊C₈	—	22	1	no	—
145	C₂×C₂²⋊C₈	Dic₁₁	1	1	yes	characteristic
146	C₂×C₂²⋊C₈	Dic₁₁	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂×D₁₁	1	2	yes	—
148	#336	—	22	1	no	—
149	#336	Dic₁₁	1	2	yes	—
150	#334	—	22	4	no	—
151	C₂²×C₁₁	—	2	2	no	—
152	C₂²×C₁₁	—	2	2	no	—
153	C₂²×C₁₁	—	2	2	no	—
154	C₂²×C₁₁	—	2	2	no	—
155	C₂²×C₁₁	—	2	2	no	—
156	C₂²×C₁₁	—	2	2	no	—
157	C₂²×C₁₁	#334	1	4	yes	complemented
158	C₂²×C₁₁	#385	1	1	yes	characteristic
159	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
160	C₂²×C₁₁	#424	1	1	yes	characteristic
161	C₂²×C₁₁	C₂×#106	1	1	yes	characteristic
162	C₂²×C₁₁	#428	1	1	yes	characteristic
163	C₂²×C₁₁	C₂×#108	1	1	yes	characteristic
164	C₂²×C₁₁	#430	1	1	yes	characteristic
165	C₄×C₁₁	—	2	2	no	—
166	C₄×C₁₁	—	2	2	no	—
167	C₄×C₁₁	—	2	2	no	—
168	C₄×C₁₁	—	4	1	no	—
169	C₄×C₁₁	C₈×D₄	1	2	yes	—
170	C₄×C₁₁	#435	1	2	yes	—
171	C₄×C₁₁	#448	1	1	yes	characteristic
172	C₄×C₁₁	#450	1	1	yes	characteristic
173	C₄×C₁₁	#481	1	1	yes	characteristic
174	C₄×C₁₁	#483	1	1	yes	characteristic
175	C₂³×C₄	Dic₂₂	1	1	yes	characteristic
176	C₂³×C₄	D₄₄	1	1	yes	characteristic
177	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
178	C₂²×C₈	—	22	4	no	—
179	C₂²×C₈	—	22	2	no	—
180	C₂²×C₈	—	22	1	no	—
181	C₂²×C₈	—	22	1	no	—
182	C₂²×C₈	—	44	2	no	—
183	C₂²×C₈	—	44	2	no	—
184	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
185	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	Dic₂₂	1	1	yes	characteristic
189	C₂×C₄²	D₄₄	1	1	yes	characteristic
190	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
192	C₄⋊C₈	—	11	4	no	—
193	C₄⋊C₈	—	22	4	no	—
194	C₄⋊C₈	—	22	2	no	—
195	C₄⋊C₈	—	22	2	no	—
196	C₄⋊C₈	—	22	2	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	C₂²⋊C₈	—	44	2	no	—
199	C₂²⋊C₈	C₁₁⋊C₈	1	4	yes	complemented
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	—	2	2	no	—
202	C₂×C₁₁	—	2	4	no	—
203	C₂×C₁₁	C₂×#334	1	1	yes	characteristic
204	C₂×C₁₁	#1565	1	1	yes	characteristic
205	C₂×C₁₁	#1570	1	1	yes	characteristic
206	C₂×C₁₁	#1573	1	1	yes	characteristic
207	C₂×C₁₁	#1850	1	1	yes	characteristic
208	C₂×C₁₁	#1853	1	1	yes	characteristic
209	C₂×C₁₁	#1855	1	1	yes	characteristic
210	C₂⁴	C₄⋊Dic₁₁	1	1	yes	characteristic
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	#4314	1	2	yes	—
221	C₂²×C₄	#4314	1	2	yes	—
222	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
223	C₂²×C₄	#4318	1	2	yes	—
224	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
225	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
226	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
229	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	22	4	no	—
236	C₂×C₈	—	44	8	no	—
237	C₂×C₈	—	44	2	no	—
238	C₂×C₈	—	44	2	no	—
239	C₂×C₈	—	44	1	no	—
240	C₂×C₈	—	44	1	no	—
241	C₂×C₈	—	44	1	no	—
242	C₂×C₈	—	44	1	no	—
243	C₂×C₈	—	44	1	no	—
244	C₂×C₈	—	44	1	no	—
245	C₂×C₈	C₂×C₁₁⋊C₈	1	4	yes	complemented
246	C₄²	—	2	2	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	2	no	—
250	C₄²	D₄×D₁₁	1	2	yes	—
251	C₄²	#4341	1	2	yes	—
252	C₁₁	#8830	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
258	C₂³	#66881	1	2	yes	—
259	C₂³	#66883	1	2	yes	—
260	C₂³	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
278	C₂×C₄	#66881	1	2	yes	—
279	C₂×C₄	#66931	1	2	yes	—
280	C₂×C₄	#66935	1	1	yes	characteristic
281	C₂×C₄	#66939	1	1	yes	characteristic
282	C₂×C₄	C₂×#4314	1	1	yes	characteristic
283	C₂×C₄	C₂×#4314	1	1	yes	characteristic
284	C₂×C₄	#66961	1	1	yes	characteristic
285	C₂×C₄	#66962	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
287	C₂×C₄	C₂×#4318	1	1	yes	characteristic
288	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
289	C₂×C₄	#67001	1	2	yes	—
290	C₈	—	2	4	no	—
291	C₈	—	4	2	no	—
292	C₈	—	44	4	no	—
293	C₈	—	88	1	no	—
294	C₈	—	88	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	#10622095	1	4	yes	complemented
302	C₂²	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
303	C₂²	#10622563	1	1	yes	characteristic
304	C₂²	#10622612	1	1	yes	characteristic
305	C₂²	C₂×#66881	1	1	yes	characteristic
306	C₂²	C₂×#66883	1	1	yes	characteristic
307	C₂²	#10622734	1	1	yes	characteristic
308	C₂²	#10622738	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	1	no	—
313	C₄	#10622334	1	1	yes	characteristic
314	C₄	#10622344	1	1	yes	characteristic
315	C₄	#10622348	1	1	yes	characteristic
316	C₄	#10622351	1	1	yes	characteristic
317	C₄	D₄×C₁₁⋊C₈	1	2	yes	—
318	C₄	#10622624	1	2	yes	—
319	C₂	—	2	4	no	—
320	C₂	C₂×#10622095	1	1	yes	characteristic
321	C₂	#13067368	1	1	yes	characteristic
322	C₂	#13067387	1	1	yes	characteristic
323	C₂	#13067401	1	1	yes	characteristic
324	C₂	#13068796	1	1	yes	characteristic
325	C₂	#13068811	1	1	yes	characteristic
326	C₂	#13068823	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄₄
commutator	C₂×C₄×C₁₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₄×C₁₁×C₂²⋊C₈	C₂
socle	C₂³×C₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₁	C₄⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₄×C₂²⋊C₈	D₁₁
hypocenter	C₁₁	#8830
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#10622095	C₂	semidirect
4. #13066451	C₂	semidirect
38. C₁₁×C₂²⋊C₈	C₈	semidirect
73. C₂×C₈×C₁₁	C₂×C₈	semidirect
157. C₂²×C₁₁	#334	semidirect
199. C₂²⋊C₈	C₁₁⋊C₈	semidirect
245. C₂×C₈	C₂×C₁₁⋊C₈	semidirect
252. C₁₁	#8830	semidirect
301. C₂²	#10622095	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₁₁⋊C₈)	2
#13068596	2
C₂×#10622095	2
#13066451	2
C₄×C₁₁×C₂²⋊C₈	2
#8830	11