Subgroups — C₁₁×#3011 #13065849

Order 1408.

700 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₁×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₁×#536	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁×#452	C₂	1	4	yes	maximal, complemented
6	C₁₁×#451	C₂	1	4	yes	maximal, complemented
7	C₁₁×#419	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₁×2₋¹⁺⁴	—	2	4	no	—
11	C₁₁×2₋¹⁺⁴	—	2	4	no	—
12	C₂×C₁₁×#41	—	2	2	no	—
13	C₂×C₁₁×#41	—	2	1	no	—
14	C₂×C₁₁×#41	C₂²	1	2	yes	—
15	C₂×C₁₁×#41	C₂²	1	1	yes	characteristic
16	C₂×C₁₁×#41	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₁₁	—	2	1	no	—
18	C₂²×Q₈×C₁₁	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₁₁	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₁	C₂²	1	1	yes	characteristic
21	C₁₁×#137	—	2	4	no	—
22	C₂×C₁₁×Q₁₆	—	2	4	no	—
23	C₂×C₁₁×Q₁₆	C₂²	1	4	yes	—
24	C₂×C₁₁×Q₁₆	C₂²	1	2	yes	—
25	C₂×C₁₁×SD₁₆	C₂²	1	2	yes	—
26	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
27	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
28	C₁₁×C₄⋊Q₈	—	2	1	no	—
29	C₁₁×#124	—	2	1	no	—
30	C₁₁×#123	—	2	2	no	—
31	C₁₁×C₂²⋊Q₈	C₂²	1	4	yes	—
32	C₁₁×#117	C₂²	1	1	yes	characteristic
33	C₁₁×#103	C₂²	1	2	yes	—
34	C₁₁×#103	C₂²	1	4	yes	—
35	C₁₁×#102	C₂²	1	2	yes	—
36	C₁₁×C₂²⋊C₈	C₂²	1	4	yes	—
37	C₁₁×#41	—	2	16	no	—
38	C₁₁×#41	—	2	8	no	—
39	C₁₁×#41	—	2	4	no	—
40	C₁₁×#41	—	2	4	no	—
41	C₁₁×#41	—	2	2	no	—
42	C₁₁×#41	—	2	2	no	—
43	C₁₁×#41	—	2	2	no	—
44	C₁₁×#41	D₄	1	4	yes	—
45	C₂×Q₈×C₁₁	—	2	8	no	—
46	C₂×Q₈×C₁₁	—	2	4	no	—
47	C₂×Q₈×C₁₁	—	2	4	no	—
48	C₂×Q₈×C₁₁	—	2	2	no	—
49	C₂×Q₈×C₁₁	—	2	1	no	—
50	C₂×Q₈×C₁₁	—	2	2	no	—
51	C₂×Q₈×C₁₁	—	2	1	no	—
52	C₂×Q₈×C₁₁	—	2	2	no	—
53	C₂×Q₈×C₁₁	—	2	2	no	—
54	C₂×Q₈×C₁₁	D₄	1	4	yes	—
55	C₂×Q₈×C₁₁	D₄	1	1	yes	characteristic
56	C₂×Q₈×C₁₁	C₂³	1	2	yes	—
57	C₂×Q₈×C₁₁	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₁₁	C₂³	1	2	yes	—
59	C₂×D₄×C₁₁	—	2	2	no	—
60	C₂×D₄×C₁₁	—	2	1	no	—
61	C₂×D₄×C₁₁	D₄	1	2	yes	—
62	C₂×D₄×C₁₁	D₄	1	1	yes	characteristic
63	C₂×D₄×C₁₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₁	—	2	2	no	—
65	C₂²×C₄×C₁₁	—	2	1	no	—
66	C₂²×C₄×C₁₁	—	2	2	no	—
67	C₂²×C₄×C₁₁	—	2	1	no	—
68	C₂²×C₄×C₁₁	—	2	1	no	—
69	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
70	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
71	C₁₁×Q₁₆	—	2	8	no	—
72	C₁₁×Q₁₆	—	2	4	no	—
73	C₁₁×SD₁₆	—	2	4	no	—
74	C₁₁×M₁₆	—	2	2	no	—
75	C₂×C₈×C₁₁	—	2	2	no	—
76	C₂×C₈×C₁₁	C₂³	1	2	yes	—
77	C₂×C₈×C₁₁	C₂³	1	2	yes	—
78	C₁₁×C₄⋊C₄	—	2	4	no	—
79	C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
80	C₁₁×C₂²⋊C₄	—	2	1	no	—
81	C₁₁×C₂²⋊C₄	—	2	4	no	—
82	C₄²×C₁₁	—	2	1	no	—
83	#3011	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₁₁	—	2	1	no	—
85	C₂³×C₁₁	C₂×D₄	1	2	yes	—
86	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₁₁	—	2	8	no	—
88	Q₈×C₁₁	—	2	4	no	—
89	Q₈×C₁₁	—	2	2	no	—
90	Q₈×C₁₁	—	2	1	no	—
91	Q₈×C₁₁	—	2	2	no	—
92	Q₈×C₁₁	—	2	2	no	—
93	Q₈×C₁₁	—	2	2	no	—
94	Q₈×C₁₁	—	4	2	no	—
95	Q₈×C₁₁	C₂×D₄	1	2	yes	—
96	Q₈×C₁₁	C₂×D₄	1	4	yes	—
97	D₄×C₁₁	—	2	8	no	—
98	D₄×C₁₁	—	2	4	no	—
99	D₄×C₁₁	—	2	4	no	—
100	D₄×C₁₁	—	2	2	no	—
101	D₄×C₁₁	—	2	1	no	—
102	D₄×C₁₁	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₁	—	2	8	no	—
104	C₂×C₄×C₁₁	—	2	8	no	—
105	C₂×C₄×C₁₁	—	2	4	no	—
106	C₂×C₄×C₁₁	—	2	2	no	—
107	C₂×C₄×C₁₁	—	2	1	no	—
108	C₂×C₄×C₁₁	—	2	4	no	—
109	C₂×C₄×C₁₁	—	2	2	no	—
110	C₂×C₄×C₁₁	—	2	2	no	—
111	C₂×C₄×C₁₁	—	2	2	no	—
112	C₂×C₄×C₁₁	—	2	2	no	—
113	C₂×C₄×C₁₁	—	4	2	no	—
114	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
119	C₈×C₁₁	—	2	2	no	—
120	C₈×C₁₁	—	2	2	no	—
121	C₂×2₋¹⁺⁴	C₂×C₁₁	1	1	yes	characteristic
122	C₂×#137	C₂×C₁₁	1	1	yes	characteristic, complemented
123	C₂²×Q₁₆	C₂×C₁₁	1	1	yes	characteristic, complemented
124	#536	C₂×C₁₁	1	1	yes	characteristic, complemented
125	#452	C₂×C₁₁	1	4	yes	complemented
126	#451	C₂×C₁₁	1	4	yes	complemented
127	#419	C₂×C₁₁	1	1	yes	characteristic, complemented
128	#416	C₂×C₁₁	1	1	yes	characteristic, complemented
129	#408	C₂×C₁₁	1	1	yes	characteristic, complemented
130	C₂²×C₁₁	—	2	2	no	—
131	C₂²×C₁₁	—	2	2	no	—
132	C₂²×C₁₁	—	2	2	no	—
133	C₂²×C₁₁	—	2	4	no	—
134	C₂²×C₁₁	—	4	2	no	—
135	C₂²×C₁₁	#120	1	2	yes	—
136	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₁₁	—	2	4	no	—
138	C₄×C₁₁	—	2	2	no	—
139	C₄×C₁₁	—	2	2	no	—
140	C₄×C₁₁	—	4	2	no	—
141	C₄×C₁₁	—	4	2	no	—
142	C₄×C₁₁	#120	1	2	yes	—
143	C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
144	C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
145	2₋¹⁺⁴	—	2	4	no	—
146	2₋¹⁺⁴	—	2	4	no	—
147	C₂×#41	—	2	2	no	—
148	C₂×#41	—	2	1	no	—
149	C₂×#41	C₂²×C₁₁	1	2	yes	—
150	C₂×#41	C₂²×C₁₁	1	1	yes	characteristic
151	C₂×#41	C₂²×C₁₁	1	1	yes	characteristic
152	C₂²×Q₈	—	2	1	no	—
153	C₂²×Q₈	C₂²×C₁₁	1	1	yes	characteristic
154	C₂²×Q₈	C₂²×C₁₁	1	2	yes	—
155	C₂²×Q₈	C₂²×C₁₁	1	1	yes	characteristic
156	#137	—	2	4	no	—
157	C₂×Q₁₆	—	2	4	no	—
158	C₂×Q₁₆	C₂²×C₁₁	1	4	yes	—
159	C₂×Q₁₆	C₂²×C₁₁	1	2	yes	—
160	C₂×SD₁₆	C₂²×C₁₁	1	2	yes	—
161	C₂×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
162	C₂²×C₈	C₂²×C₁₁	1	1	yes	characteristic
163	C₄⋊Q₈	—	2	1	no	—
164	#124	—	2	1	no	—
165	#123	—	2	2	no	—
166	C₂²⋊Q₈	C₂²×C₁₁	1	4	yes	—
167	#117	C₂²×C₁₁	1	1	yes	characteristic
168	#103	C₂²×C₁₁	1	2	yes	—
169	#103	C₂²×C₁₁	1	4	yes	—
170	#102	C₂²×C₁₁	1	2	yes	—
171	C₂²⋊C₈	C₂²×C₁₁	1	4	yes	—
172	C₂×C₁₁	—	2	2	no	—
173	C₂×C₁₁	—	4	2	no	—
174	C₂×C₁₁	—	4	2	no	—
175	C₂×C₁₁	C₂×#120	1	1	yes	characteristic
176	C₂×C₁₁	#578	1	2	yes	—
177	#41	—	2	16	no	—
178	#41	—	2	8	no	—
179	#41	—	2	4	no	—
180	#41	—	2	4	no	—
181	#41	—	2	2	no	—
182	#41	—	2	2	no	—
183	#41	—	2	2	no	—
184	#41	D₄×C₁₁	1	4	yes	—
185	C₂×Q₈	—	2	8	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	1	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	D₄×C₁₁	1	4	yes	—
195	C₂×Q₈	D₄×C₁₁	1	1	yes	characteristic
196	C₂×Q₈	C₂³×C₁₁	1	2	yes	—
197	C₂×Q₈	C₂³×C₁₁	1	1	yes	characteristic
198	C₂×Q₈	C₂³×C₁₁	1	2	yes	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	D₄×C₁₁	1	2	yes	—
202	C₂×D₄	D₄×C₁₁	1	1	yes	characteristic
203	C₂×D₄	C₂³×C₁₁	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	C₂³×C₁₁	1	2	yes	—
210	C₂²×C₄	C₂³×C₁₁	1	1	yes	characteristic
211	Q₁₆	—	2	8	no	—
212	Q₁₆	—	2	4	no	—
213	SD₁₆	—	2	4	no	—
214	M₁₆	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	C₂³×C₁₁	1	2	yes	—
217	C₂×C₈	C₂³×C₁₁	1	2	yes	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	C₂³×C₁₁	1	2	yes	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₄²	—	2	1	no	—
223	C₁₁	#3011	1	1	yes	characteristic, direct product factor, Sylow
224	C₂³	—	2	1	no	—
225	C₂³	C₂×D₄×C₁₁	1	2	yes	—
226	C₂³	C₂×D₄×C₁₁	1	1	yes	characteristic
227	Q₈	—	2	8	no	—
228	Q₈	—	2	4	no	—
229	Q₈	—	2	2	no	—
230	Q₈	—	2	1	no	—
231	Q₈	—	2	2	no	—
232	Q₈	—	2	2	no	—
233	Q₈	—	2	2	no	—
234	Q₈	—	4	2	no	—
235	Q₈	C₂×D₄×C₁₁	1	2	yes	—
236	Q₈	C₂×D₄×C₁₁	1	4	yes	—
237	D₄	—	2	8	no	—
238	D₄	—	2	4	no	—
239	D₄	—	2	4	no	—
240	D₄	—	2	2	no	—
241	D₄	—	2	1	no	—
242	D₄	C₂×D₄×C₁₁	1	2	yes	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	C₂×D₄×C₁₁	1	2	yes	—
255	C₂×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₁₁	1	2	yes	—
257	C₂×C₄	C₂×D₄×C₁₁	1	2	yes	—
258	C₂×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₁₁×#120	1	2	yes	—
267	C₂²	C₂²×D₄×C₁₁	1	1	yes	characteristic
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	C₁₁×#120	1	2	yes	—
274	C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
275	C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	—	4	2	no	—
278	C₂	—	4	2	no	—
279	C₂	C₂×C₁₁×#120	1	1	yes	characteristic
280	C₂	C₁₁×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₁
prime radical	C₂×C₁₁×2₋¹⁺⁴	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₂×C₁₁×#41 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₁×#137	C₂	semidirect
3. C₂²×C₁₁×Q₁₆	C₂	semidirect
4. C₁₁×#536	C₂	semidirect
5. C₁₁×#452	C₂	semidirect
6. C₁₁×#451	C₂	semidirect
7. C₁₁×#419	C₂	semidirect
8. C₁₁×#416	C₂	semidirect
9. C₁₁×#408	C₂	semidirect
83. #3011	C₁₁	direct (N × H)
122. C₂×#137	C₂×C₁₁	semidirect
123. C₂²×Q₁₆	C₂×C₁₁	semidirect
124. #536	C₂×C₁₁	semidirect
125. #452	C₂×C₁₁	semidirect
126. #451	C₂×C₁₁	semidirect
127. #419	C₂×C₁₁	semidirect
128. #416	C₂×C₁₁	semidirect
129. #408	C₂×C₁₁	semidirect
223. C₁₁	#3011	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₁×2₋¹⁺⁴	2
C₂×C₁₁×#137	2
C₂²×C₁₁×Q₁₆	2
C₁₁×#536	2
C₁₁×#452	2
C₁₁×#451	2
C₁₁×#419	2
C₁₁×#416	2
C₁₁×#408	2
#3011	11

Subgroups — C11×#3011 #13065849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#3011 #13065849