Subgroups — #13077271

Order 1408.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₁×#117	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
3	#10622739	C₂	1	1	yes	maximal, characteristic
4	#10622732	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×(C₂×D₁₁)⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#10622715	C₂	1	1	yes	maximal, characteristic, complemented
7	#10622610	C₂	1	1	yes	maximal, characteristic, complemented
8	#10622322	C₂	1	2	yes	maximal, complemented
9	#10622318	C₂	1	2	yes	maximal, complemented
10	#10622303	C₂	1	2	yes	maximal, complemented
11	#10622298	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×D₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₁₁×#117	C₂²	1	1	yes	characteristic
16	#66979	C₂²	1	1	yes	characteristic
17	C₂×#4318	C₂²	1	1	yes	characteristic
18	C₂²×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
19	#66945	C₄	1	1	yes	characteristic
20	#66945	C₄	1	1	yes	characteristic
21	C₄⋊C₄×D₁₁	C₄	1	1	yes	characteristic
22	C₄⋊C₄×D₁₁	C₄	1	1	yes	characteristic
23	C₂²⋊C₄×D₁₁	C₄	1	2	yes	—
24	#66930	C₄	1	2	yes	—
25	#66925	C₂²	1	2	yes	—
26	C₄²×D₁₁	C₂²	1	2	yes	—
27	C₁₁×#97	C₂²	1	2	yes	complemented
28	C₄×C₈×C₁₁	C₂²	1	2	yes	complemented
29	(C₂×D₁₁)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₁₁)⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₁₁)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₁₁)⋊C₈	C₂²	1	1	yes	characteristic
33	(C₂×D₁₁)⋊C₈	C₂²	1	2	yes	—
34	(C₂×D₁₁)⋊C₈	C₂²	1	2	yes	—
35	Dic₁₁⋊C₈	C₂²	1	2	yes	complemented
36	Dic₁₁⋊C₈	C₂²	1	2	yes	complemented
37	Dic₁₁⋊C₈	C₂²	1	2	yes	complemented
38	Dic₁₁⋊C₈	C₂²	1	2	yes	complemented
39	#66867	C₂²	1	2	yes	complemented
40	C₄×C₁₁⋊C₈	C₂²	1	2	yes	complemented
41	C₂³×D₁₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₁	—	2	2	no	—
45	C₂×C₄×D₁₁	—	2	2	no	—
46	C₂×C₄×D₁₁	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₁	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₁	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₁₁	C₂³	1	1	yes	characteristic
52	C₁₁×M₁₆	—	2	2	no	—
53	C₂×C₈×C₁₁	—	2	2	no	—
54	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₁	C₂³	1	2	yes	—
56	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
57	C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
58	C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
59	C₁₁×C₂²⋊C₄	C₂×C₄	1	2	yes	—
60	C₄²×C₁₁	C₂³	1	2	yes	—
61	C₂²⋊Dic₁₁	C₂×C₄	1	2	yes	—
62	(C₂×D₁₁)⋊C₄	C₂×C₄	1	2	yes	—
63	(C₂×D₁₁)⋊C₄	C₂×C₄	1	2	yes	—
64	C₄⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
65	C₄⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
66	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
67	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₁	C₂×C₄	1	2	yes	—
69	C₄×Dic₁₁	C₂×C₄	1	2	yes	—
70	C₄×Dic₁₁	C₂³	1	2	yes	—
71	#4318	—	2	2	no	—
72	C₂×C₁₁⋊C₈	—	2	2	no	—
73	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
75	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
76	#2924	—	11	1	no	maximal, Sylow
77	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×D₁₁	—	2	2	no	—
79	C₂²×D₁₁	—	2	2	no	—
80	C₂²×D₁₁	—	4	1	no	—
81	C₂²×D₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×D₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₁	#41	1	2	yes	—
87	C₂×C₄×C₁₁	#41	1	2	yes	—
88	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
89	C₂×Dic₁₁	—	2	2	no	—
90	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
92	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
93	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₄×D₁₁	—	2	4	no	—
95	C₄×D₁₁	—	2	2	no	—
96	C₄×D₁₁	—	2	2	no	—
97	C₄×D₁₁	—	2	1	no	—
98	C₄×D₁₁	—	2	1	no	—
99	C₄×D₁₁	—	2	1	no	—
100	C₄×D₁₁	—	2	1	no	—
101	C₄×D₁₁	—	4	2	no	—
102	C₈×C₁₁	—	2	1	no	—
103	C₈×C₁₁	—	2	1	no	—
104	C₈×C₁₁	—	2	2	no	—
105	C₁₁⋊C₈	—	2	1	no	—
106	C₁₁⋊C₈	—	2	1	no	—
107	C₁₁⋊C₈	—	2	2	no	—
108	C₂×#117	—	11	1	no	—
109	#433	—	11	2	no	—
110	#433	—	11	2	no	—
111	#432	—	11	2	no	—
112	#431	—	11	2	no	—
113	#424	—	11	1	no	—
114	#424	—	11	1	no	—
115	#407	—	11	1	no	—
116	C₂×C₂²⋊C₈	—	11	1	no	—
117	#405	—	11	1	no	—
118	#405	D₁₁	1	1	yes	characteristic, complemented
119	C₂²×C₁₁	—	2	1	no	—
120	C₂²×C₁₁	—	2	1	no	—
121	C₂²×C₁₁	#117	1	2	yes	—
122	C₂²×C₁₁	C₂³×C₄	1	1	yes	characteristic
123	C₂×D₁₁	—	2	1	no	—
124	C₂×D₁₁	—	2	1	no	—
125	C₂×D₁₁	—	2	1	no	—
126	C₂×D₁₁	—	2	1	no	—
127	C₂×D₁₁	—	2	1	no	—
128	C₂×D₁₁	—	4	4	no	—
129	C₂×D₁₁	#131	1	2	yes	—
130	C₄×C₁₁	—	2	2	no	—
131	C₄×C₁₁	—	2	2	no	—
132	C₄×C₁₁	#117	1	2	yes	—
133	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
134	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
135	Dic₁₁	—	2	1	no	—
136	Dic₁₁	—	2	1	no	—
137	Dic₁₁	—	2	2	no	—
138	Dic₁₁	—	2	2	no	—
139	C₂³×C₄	—	11	1	no	—
140	C₂×M₁₆	—	11	1	no	—
141	C₂×M₁₆	C₂×D₁₁	1	1	yes	characteristic
142	C₂²×C₈	—	11	1	no	—
143	C₂²×C₈	C₂×D₁₁	1	1	yes	characteristic
144	#117	—	11	1	no	—
145	#117	—	11	1	no	—
146	#117	—	11	1	no	—
147	#117	—	11	2	no	—
148	#117	—	11	2	no	—
149	#117	C₂×D₁₁	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	11	1	no	—
151	C₂×C₄⋊C₄	—	11	1	no	—
152	C₂×C₂²⋊C₄	—	11	2	no	—
153	C₂×C₄²	—	11	2	no	—
154	C₄⋊C₈	—	11	2	no	—
155	C₄⋊C₈	—	11	2	no	—
156	C₄⋊C₈	—	11	2	no	—
157	C₄⋊C₈	—	11	2	no	—
158	C₂²⋊C₈	—	11	1	no	—
159	C₂²⋊C₈	—	11	1	no	—
160	C₂²⋊C₈	—	11	1	no	—
161	C₂²⋊C₈	—	11	1	no	—
162	C₂²⋊C₈	—	11	2	no	—
163	C₂²⋊C₈	—	11	2	no	—
164	#97	—	11	2	no	—
165	#97	C₂×D₁₁	1	2	yes	complemented
166	C₄×C₈	—	11	2	no	—
167	C₄×C₈	C₂×D₁₁	1	2	yes	complemented
168	C₂×C₁₁	—	2	2	no	—
169	C₂×C₁₁	C₂×#117	1	1	yes	characteristic
170	C₂×C₁₁	C₂×#131	1	1	yes	characteristic
171	C₂×C₁₁	#568	1	1	yes	characteristic
172	D₁₁	—	2	2	no	—
173	D₁₁	—	4	1	no	—
174	C₂⁴	—	11	1	no	—
175	C₂²×C₄	—	11	1	no	—
176	C₂²×C₄	—	11	2	no	—
177	C₂²×C₄	—	11	1	no	—
178	C₂²×C₄	—	11	1	no	—
179	C₂²×C₄	—	11	1	no	—
180	C₂²×C₄	—	11	1	no	—
181	C₂²×C₄	—	11	2	no	—
182	C₂²×C₄	—	22	2	no	—
183	C₂²×C₄	—	22	2	no	—
184	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
185	M₁₆	—	2	2	no	—
186	M₁₆	—	22	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	11	1	no	—
189	C₂×C₈	—	11	2	no	—
190	C₂×C₈	—	11	1	no	—
191	C₂×C₈	—	22	2	no	—
192	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₁₁	1	2	yes	—
194	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
195	C₄⋊C₄	—	11	2	no	—
196	C₄⋊C₄	—	11	2	no	—
197	C₄⋊C₄	—	11	1	no	—
198	C₄⋊C₄	—	11	1	no	—
199	C₄⋊C₄	C₄×D₁₁	1	1	yes	characteristic
200	C₄⋊C₄	C₄×D₁₁	1	1	yes	characteristic
201	C₂²⋊C₄	—	11	2	no	—
202	C₂²⋊C₄	—	11	2	no	—
203	C₂²⋊C₄	—	11	2	no	—
204	C₂²⋊C₄	C₄×D₁₁	1	2	yes	—
205	C₄²	—	11	2	no	—
206	C₄²	—	11	2	no	—
207	C₄²	—	11	2	no	—
208	C₄²	C₂²×D₁₁	1	2	yes	—
209	C₁₁	#2924	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	11	1	no	—
211	C₂³	—	11	1	no	—
212	C₂³	—	22	2	no	—
213	C₂³	—	22	2	no	—
214	C₂³	—	44	1	no	—
215	C₂³	C₂×C₄×D₁₁	1	1	yes	characteristic
216	C₂×C₄	—	11	1	no	—
217	C₂×C₄	—	11	2	no	—
218	C₂×C₄	—	11	2	no	—
219	C₂×C₄	—	11	1	no	—
220	C₂×C₄	—	22	4	no	—
221	C₂×C₄	—	22	2	no	—
222	C₂×C₄	—	22	2	no	—
223	C₂×C₄	—	22	2	no	—
224	C₂×C₄	—	22	1	no	—
225	C₂×C₄	—	22	1	no	—
226	C₂×C₄	—	22	1	no	—
227	C₂×C₄	—	22	1	no	—
228	C₂×C₄	—	44	2	no	—
229	C₂×C₄	C₂×C₄×D₁₁	1	2	yes	—
230	C₂×C₄	C₂×C₄×D₁₁	1	2	yes	—
231	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
232	C₂×C₄	#4339	1	2	yes	—
233	C₂×C₄	#4339	1	2	yes	—
234	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	22	1	no	—
239	C₈	—	22	1	no	—
240	C₈	—	22	2	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	11	2	no	—
244	C₂²	—	22	1	no	—
245	C₂²	—	22	1	no	—
246	C₂²	—	22	1	no	—
247	C₂²	—	22	1	no	—
248	C₂²	—	22	1	no	—
249	C₂²	—	44	4	no	—
250	C₂²	#66925	1	2	yes	—
251	C₂²	C₂²×C₄×D₁₁	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	22	1	no	—
255	C₄	—	22	1	no	—
256	C₄	—	22	2	no	—
257	C₄	—	22	2	no	—
258	C₄	#66925	1	2	yes	—
259	C₄	C₂×#4339	1	1	yes	characteristic
260	C₄	C₂×#4339	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	22	2	no	—
263	C₂	—	44	1	no	—
264	C₂	C₂×#66925	1	1	yes	characteristic
265	C₂	D₁₁×#131	1	1	yes	characteristic
266	C₂	#10623281	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₄
Fitting	C₁₁×#405	C₂
prime radical	C₂³×D₁₁	C₂×C₄
cosocle	C₂×C₄×C₁₁	C₂⁴
hypercenter	#405	D₁₁
hypocenter	C₁₁	#2924
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. C₁₁×#405	C₂	semidirect
4. #10622732	C₂	semidirect
6. #10622715	C₂	semidirect
7. #10622610	C₂	semidirect
8. #10622322	C₂	semidirect
9. #10622318	C₂	semidirect
10. #10622303	C₂	semidirect
11. #10622298	C₂	semidirect
27. C₁₁×#97	C₂²	semidirect
28. C₄×C₈×C₁₁	C₂²	semidirect
35. Dic₁₁⋊C₈	C₂²	semidirect
36. Dic₁₁⋊C₈	C₂²	semidirect
37. Dic₁₁⋊C₈	C₂²	semidirect
38. Dic₁₁⋊C₈	C₂²	semidirect
39. #66867	C₂²	semidirect
40. C₄×C₁₁⋊C₈	C₂²	semidirect
118. #405	D₁₁	semidirect
165. #97	C₂×D₁₁	semidirect
167. C₄×C₈	C₂×D₁₁	semidirect
209. C₁₁	#2924	semidirect

Maximal subgroups

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

subgroup	index
D₁₁×#117	2
C₁₁×#405	2
#10622739	2
#10622732	2
C₂×(C₂×D₁₁)⋊C₈	2
#10622715	2
#10622610	2
#10622322	2
#10622318	2
#10622303	2
#10622298	2
#2924	11