Subgroups — #14643156

Order 2816.

798 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#4318	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₁₁⋊#1273	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	C₄	1	2	yes	—
7	C₂³×C₁₁⋊C₈	C₄	1	2	yes	—
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	C₄×#4318	—	2	2	no	—
14	C₄×#4318	—	2	2	no	—
15	C₂×#66867	C₄	1	2	yes	—
16	C₂×C₄×C₁₁⋊C₈	—	2	2	no	—
17	C₂×C₄×C₁₁⋊C₈	—	2	2	no	—
18	C₂×C₄×C₁₁⋊C₈	C₄	1	2	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	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₁	D₄	1	2	yes	—
29	C₂×C₄²×C₁₁	D₄	1	2	yes	—
30	C₂×C₄²×C₁₁	D₄	1	2	yes	—
31	C₂×C₄²×C₁₁	Q₈	1	2	yes	—
32	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
33	C₂×#4318	—	2	4	no	—
34	C₂×#4318	—	2	4	no	—
35	C₂×#4318	C₈	1	8	yes	complemented
36	C₂²×C₁₁⋊C₈	—	2	4	no	—
37	C₂²×C₁₁⋊C₈	—	2	8	no	—
38	C₂²×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
39	C₂²×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
40	C₁₁×C₄⋊C₈	—	2	4	no	—
41	C₁₁×C₂²⋊C₈	—	2	4	no	—
42	C₄×C₈×C₁₁	—	2	2	no	—
43	C₄×C₈×C₁₁	—	2	2	no	—
44	#66867	—	2	4	no	—
45	C₄×C₁₁⋊C₈	—	2	4	no	—
46	C₄×C₁₁⋊C₈	—	2	2	no	—
47	C₄×C₁₁⋊C₈	—	2	2	no	—
48	C₄×C₁₁⋊C₈	—	2	2	no	—
49	C₄×C₁₁⋊C₈	—	2	2	no	—
50	#7848	—	11	1	no	maximal, Sylow
51	C₂⁴×C₁₁	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₁₁	—	2	4	no	—
53	C₂²×C₄×C₁₁	—	2	4	no	—
54	C₂²×C₄×C₁₁	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₁₁	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₄²	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₁	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₁₁	M₁₆	1	4	yes	—
65	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₁	—	2	4	no	—
69	C₂×C₈×C₁₁	—	2	8	no	—
70	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
74	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
75	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
76	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
80	#4318	—	2	16	no	—
81	C₂×C₁₁⋊C₈	—	2	16	no	—
82	C₂×C₁₁⋊C₈	—	2	4	no	—
83	C₂×C₁₁⋊C₈	—	2	4	no	—
84	C₂×C₁₁⋊C₈	—	2	4	no	—
85	C₂×C₁₁⋊C₈	—	2	4	no	—
86	C₂×C₁₁⋊C₈	C₂×C₈	1	8	yes	complemented
87	C₂×C₁₁⋊C₈	M₁₆	1	8	yes	complemented
88	C₂×#431	D₁₁	1	1	yes	characteristic
89	C₂×C₄×M₁₆	—	11	1	no	—
90	C₂²×C₄×C₈	—	11	1	no	—
91	#1273	—	11	4	no	—
92	C₂³×C₁₁	—	2	2	no	—
93	C₂³×C₁₁	—	2	2	no	—
94	C₂³×C₁₁	#95	1	2	yes	—
95	C₂³×C₁₁	C₄×C₈	1	2	yes	—
96	C₂³×C₁₁	#97	1	2	yes	—
97	C₂³×C₁₁	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₁₁	—	2	2	no	—
99	C₂×C₄×C₁₁	—	2	2	no	—
100	C₂×C₄×C₁₁	—	2	4	no	—
101	C₂×C₄×C₁₁	—	2	4	no	—
102	C₂×C₄×C₁₁	—	2	4	no	—
103	C₂×C₄×C₁₁	—	2	4	no	—
104	C₂×C₄×C₁₁	#95	1	2	yes	—
105	C₂×C₄×C₁₁	#95	1	2	yes	—
106	C₂×C₄×C₁₁	#95	1	2	yes	—
107	C₂×C₄×C₁₁	C₄×C₈	1	2	yes	—
108	C₂×C₄×C₁₁	#97	1	2	yes	—
109	C₂×C₄×C₁₁	C₂²⋊C₈	1	8	yes	—
110	C₂×C₄×C₁₁	C₄⋊C₈	1	8	yes	—
111	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₁₁	#131	1	4	yes	—
121	C₈×C₁₁	—	4	4	no	—
122	C₁₁⋊C₈	—	2	8	no	—
123	C₁₁⋊C₈	—	2	8	no	—
124	C₂²×M₁₆	—	11	2	no	—
125	C₂³×C₈	—	11	2	no	—
126	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
127	#431	—	2	2	no	—
128	#431	—	2	2	no	—
129	C₂×C₄⋊C₈	Dic₁₁	1	2	yes	—
130	C₂×C₂²⋊C₈	Dic₁₁	1	2	yes	—
131	C₄×M₁₆	—	22	2	no	—
132	C₄×M₁₆	—	22	2	no	—
133	C₂×#97	—	11	2	no	—
134	C₂×C₄×C₈	—	11	2	no	—
135	C₂×C₄×C₈	—	11	2	no	—
136	C₂×C₄×C₈	—	11	2	no	—
137	C₂×C₄×C₈	—	22	2	no	—
138	C₂×C₄×C₈	—	22	2	no	—
139	C₂×C₄×C₈	C₂×D₁₁	1	2	yes	—
140	C₂²×C₁₁	—	2	4	no	—
141	C₂²×C₁₁	—	2	4	no	—
142	C₂²×C₁₁	—	2	2	no	—
143	C₂²×C₁₁	—	2	2	no	—
144	C₂²×C₁₁	#336	1	4	yes	—
145	C₂²×C₁₁	C₂×#95	1	1	yes	characteristic
146	C₂²×C₁₁	C₂×C₄×C₈	1	1	yes	characteristic
147	C₂²×C₁₁	C₂×#97	1	1	yes	characteristic
148	C₂²×C₁₁	C₄×M₁₆	1	2	yes	—
149	C₂²×C₁₁	#405	1	2	yes	—
150	C₄×C₁₁	—	2	2	no	—
151	C₄×C₁₁	—	2	2	no	—
152	C₄×C₁₁	#336	1	4	yes	—
153	C₄×C₁₁	C₂×C₂²⋊C₈	1	2	yes	—
154	C₄×C₁₁	#407	1	2	yes	—
155	C₄×C₁₁	#408	1	2	yes	—
156	C₄×C₁₁	C₂×C₄⋊C₈	1	2	yes	—
157	C₄×C₁₁	#423	1	2	yes	—
158	C₄×C₁₁	#424	1	2	yes	—
159	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
160	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
161	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
162	C₂×M₁₆	—	11	8	no	—
163	C₂×M₁₆	—	22	4	no	—
164	C₂×M₁₆	—	22	4	no	—
165	C₂²×C₈	—	11	4	no	—
166	C₂²×C₈	—	11	4	no	—
167	C₂²×C₈	—	22	8	no	—
168	C₂²×C₈	—	22	4	no	—
169	C₂²×C₈	C₂×Dic₁₁	1	4	yes	—
170	C₂×C₄²	Dic₂₂	1	2	yes	—
171	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
172	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
173	C₂×C₄²	D₄₄	1	2	yes	—
174	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
175	C₂×C₄²	C₂²⋊D₁₁	1	2	yes	—
176	C₂×C₄²	C₂²⋊D₁₁	1	2	yes	—
177	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
178	C₄⋊C₈	—	2	4	no	—
179	C₂²⋊C₈	—	2	4	no	—
180	#97	—	22	4	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	22	4	no	—
184	C₄×C₈	—	22	2	no	—
185	C₄×C₈	—	22	2	no	—
186	C₄×C₈	—	22	2	no	—
187	C₄×C₈	—	22	2	no	—
188	C₂×C₁₁	—	2	4	no	—
189	C₂×C₁₁	C₈×M₁₆	1	2	yes	—
190	C₂×C₁₁	#1451	1	2	yes	—
191	C₂×C₁₁	C₂×#336	1	1	yes	characteristic
192	C₂×C₁₁	#1728	1	1	yes	characteristic
193	C₂×C₁₁	#1729	1	1	yes	characteristic
194	C₂⁴	C₄×Dic₁₁	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	C₈×D₁₁	1	4	yes	—
198	C₂²×C₄	#4313	1	4	yes	—
199	C₂²×C₄	C₄×Dic₁₁	1	1	yes	characteristic
200	C₂²×C₄	C₄×Dic₁₁	1	1	yes	characteristic
201	C₂²×C₄	C₄×Dic₁₁	1	1	yes	characteristic
202	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
203	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
204	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
205	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	2	yes	—
206	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	2	yes	—
207	C₂²×C₄	C₂²⋊Dic₁₁	1	2	yes	—
208	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
211	M₁₆	—	22	16	no	—
212	C₂×C₈	—	2	8	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	11	8	no	—
215	C₂×C₈	—	11	8	no	—
216	C₂×C₈	—	22	16	no	—
217	C₂×C₈	—	22	4	no	—
218	C₂×C₈	—	22	4	no	—
219	C₂×C₈	—	22	4	no	—
220	C₂×C₈	—	22	4	no	—
221	C₄²	Dic₁₁⋊C₄	1	2	yes	—
222	C₄²	Dic₁₁⋊C₄	1	2	yes	—
223	C₄²	C₄⋊Dic₁₁	1	2	yes	—
224	C₄²	(C₂×D₁₁)⋊C₄	1	2	yes	—
225	C₄²	(C₂×D₁₁)⋊C₄	1	2	yes	—
226	C₄²	C₂²⋊Dic₁₁	1	2	yes	—
227	C₄²	C₂×Dic₂₂	1	1	yes	characteristic
228	C₄²	C₂×D₄₄	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
230	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
231	C₁₁	#7848	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₈×Dic₁₁	1	2	yes	—
235	C₂³	#66879	1	2	yes	—
236	C₂³	C₁₁⋊#95	1	2	yes	—
237	C₂³	C₂×C₄×Dic₁₁	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	C₈×Dic₁₁	1	2	yes	—
245	C₂×C₄	Dic₁₁⋊C₈	1	8	yes	—
246	C₂×C₄	#66879	1	2	yes	—
247	C₂×C₄	(C₂×D₁₁)⋊C₈	1	8	yes	—
248	C₂×C₄	C₁₁⋊#95	1	2	yes	—
249	C₂×C₄	C₁₁⋊#95	1	2	yes	—
250	C₂×C₄	C₁₁⋊#95	1	2	yes	—
251	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
252	C₂×C₄	C₂×#4313	1	2	yes	—
253	C₂×C₄	M₁₆×D₁₁	1	4	yes	—
254	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
255	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
257	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
259	C₂×C₄	#67001	1	4	yes	—
260	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
261	C₈	—	4	4	no	—
262	C₈	—	22	8	no	—
263	C₈	—	22	8	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	#10622188	1	4	yes	—
269	C₂²	#10622610	1	2	yes	—
270	C₂²	C₂×C₈×Dic₁₁	1	1	yes	characteristic
271	C₂²	C₂×#66879	1	1	yes	characteristic
272	C₂²	M₁₆×Dic₁₁	1	2	yes	—
273	C₂²	C₂×C₁₁⋊#95	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	#10622188	1	4	yes	—
277	C₄	#10622612	1	2	yes	—
278	C₄	C₂×Dic₁₁⋊C₈	1	2	yes	—
279	C₄	C₂×(C₂×D₁₁)⋊C₈	1	2	yes	—
280	C₄	#10622731	1	2	yes	—
281	C₄	#10622739	1	2	yes	—
282	C₄	#10622847	1	2	yes	—
283	C₂	—	2	4	no	—
284	C₂	#13066897	1	2	yes	—
285	C₂	#13066906	1	2	yes	—
286	C₂	C₂×#10622188	1	1	yes	characteristic
287	C₂	#13068309	1	1	yes	characteristic
288	C₂	#13068313	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂×C₄×C₈
Fitting	C₂×C₁₁×#431	C₂
socle	C₂³×C₁₁	C₂×C₄²
prime radical	C₂⁴×C₁₁	C₄²
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂×#431	D₁₁
hypocenter	C₁₁	#7848
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. C₁₁⋊#1273	C₂	semidirect
35. C₂×#4318	C₈	semidirect
86. C₂×C₁₁⋊C₈	C₂×C₈	semidirect
87. C₂×C₁₁⋊C₈	M₁₆	semidirect
231. C₁₁	#7848	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#4318	2
C₂²×C₄×C₁₁⋊C₈	2
C₁₁⋊#1273	2
C₂×C₁₁×#431	2
#7848	11