Subgroups — C₂²×C₁₁×#560 #14639880

Order 2816.

9606 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#560	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₁₁×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₁×#126	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₁×#123	C₂	1	3	yes	maximal, complemented
5	C₂²×C₁₁×#121	C₂	1	6	yes	maximal, complemented
6	C₂²×C₁₁×#120	C₂	1	3	yes	maximal, complemented
7	C₂⁶×C₁₁	C₂²	1	1	yes	characteristic
8	C₂³×D₄×C₁₁	C₂²	1	6	yes	—
9	C₂³×D₄×C₁₁	C₂²	1	3	yes	—
10	C₂³×D₄×C₁₁	C₂²	1	3	yes	complemented
11	C₂⁴×C₄×C₁₁	C₂²	1	3	yes	complemented
12	C₁₁×#560	C₂²	1	256	yes	direct product factor
13	C₂×C₁₁×#127	C₂²	1	24	yes	complemented
14	C₂×C₁₁×#126	C₂²	1	48	yes	complemented
15	C₂×C₁₁×#123	C₂²	1	72	yes	complemented
16	C₂×C₁₁×#121	C₂²	1	144	yes	complemented
17	C₂×C₁₁×#120	C₂²	1	72	yes	complemented
18	C₂²×C₁₁×C₄⋊C₄	C₂²	1	6	yes	complemented
19	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	6	yes	—
20	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	3	yes	—
21	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	3	yes	complemented
22	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic, complemented
23	C₂⁵×C₁₁	—	4	12	no	—
24	C₂⁵×C₁₁	C₂³	1	2	yes	—
25	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
26	C₂⁵×C₁₁	C₂³	1	3	yes	—
27	C₂⁵×C₁₁	C₂³	1	12	yes	—
28	C₂²×D₄×C₁₁	—	4	24	no	—
29	C₂²×D₄×C₁₁	—	4	12	no	—
30	C₂²×D₄×C₁₁	—	4	12	no	—
31	C₂²×D₄×C₁₁	C₂³	1	72	yes	—
32	C₂²×D₄×C₁₁	C₂³	1	36	yes	—
33	C₂²×D₄×C₁₁	C₂³	1	36	yes	complemented
34	C₂³×C₄×C₁₁	—	4	12	no	—
35	C₂³×C₄×C₁₁	C₂³	1	3	yes	—
36	C₂³×C₄×C₁₁	C₂³	1	6	yes	—
37	C₂³×C₄×C₁₁	C₂³	1	36	yes	complemented
38	C₁₁×#127	C₂³	1	64	yes	complemented
39	C₁₁×#126	C₂³	1	128	yes	complemented
40	C₁₁×#123	C₂³	1	192	yes	complemented
41	C₁₁×#121	C₂³	1	384	yes	complemented
42	C₁₁×#120	C₂³	1	192	yes	complemented
43	C₂×C₁₁×C₄⋊C₄	C₂³	1	72	yes	complemented
44	C₂×C₁₁×C₂²⋊C₄	C₂³	1	72	yes	—
45	C₂×C₁₁×C₂²⋊C₄	C₂³	1	36	yes	—
46	C₂×C₁₁×C₂²⋊C₄	C₂³	1	36	yes	complemented
47	C₂×C₄²×C₁₁	C₂³	1	12	yes	complemented
48	C₂²×#560	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₁	—	2	12	no	—
50	C₂⁴×C₁₁	—	2	24	no	—
51	C₂⁴×C₁₁	—	2	24	no	—
52	C₂⁴×C₁₁	—	2	12	no	—
53	C₂⁴×C₁₁	—	4	72	no	—
54	C₂⁴×C₁₁	—	4	64	no	—
55	C₂⁴×C₁₁	C₂⁴	1	12	yes	—
56	C₂⁴×C₁₁	C₂⁴	1	6	yes	—
57	C₂⁴×C₁₁	C₂⁴	1	18	yes	—
58	C₂⁴×C₁₁	C₂⁴	1	16	yes	—
59	C₂⁴×C₁₁	C₂⁴	1	1	yes	characteristic
60	C₂×D₄×C₁₁	—	4	144	no	—
61	C₂×D₄×C₁₁	—	4	72	no	—
62	C₂×D₄×C₁₁	—	4	72	no	—
63	C₂×D₄×C₁₁	C₂⁴	1	96	yes	—
64	C₂×D₄×C₁₁	C₂⁴	1	48	yes	—
65	C₂×D₄×C₁₁	C₂⁴	1	48	yes	complemented
66	C₂²×C₄×C₁₁	—	2	24	no	—
67	C₂²×C₄×C₁₁	—	2	12	no	—
68	C₂²×C₄×C₁₁	—	4	72	no	—
69	C₂²×C₄×C₁₁	C₂⁴	1	18	yes	—
70	C₂²×C₄×C₁₁	C₂⁴	1	36	yes	—
71	C₂²×C₄×C₁₁	C₂⁴	1	48	yes	complemented
72	C₁₁×C₄⋊C₄	C₂⁴	1	96	yes	complemented
73	C₁₁×C₂²⋊C₄	C₂⁴	1	96	yes	—
74	C₁₁×C₂²⋊C₄	C₂⁴	1	48	yes	—
75	C₁₁×C₂²⋊C₄	C₂⁴	1	48	yes	complemented
76	C₄²×C₁₁	C₂⁴	1	16	yes	complemented
77	C₂×#560	C₂×C₁₁	1	48	yes	direct product factor
78	C₂²×#127	C₂×C₁₁	1	1	yes	characteristic, complemented
79	C₂²×#126	C₂×C₁₁	1	2	yes	complemented
80	C₂²×#123	C₂×C₁₁	1	3	yes	complemented
81	C₂²×#121	C₂×C₁₁	1	6	yes	complemented
82	C₂²×#120	C₂×C₁₁	1	3	yes	complemented
83	C₂³×C₁₁	—	2	36	no	—
84	C₂³×C₁₁	—	2	72	no	—
85	C₂³×C₁₁	—	2	72	no	—
86	C₂³×C₁₁	—	2	36	no	—
87	C₂³×C₁₁	—	4	48	no	—
88	C₂³×C₁₁	—	4	192	no	—
89	C₂³×C₁₁	—	4	48	no	—
90	C₂³×C₁₁	—	4	32	no	—
91	C₂³×C₁₁	—	4	16	no	—
92	C₂³×C₁₁	H(5,𝔽₂)	1	12	yes	—
93	C₂³×C₁₁	C₂⁵	1	8	yes	—
94	C₂³×C₁₁	C₂⁵	1	4	yes	—
95	C₂³×C₁₁	C₂⁵	1	12	yes	—
96	C₂³×C₁₁	C₂⁵	1	3	yes	—
97	D₄×C₁₁	—	4	96	no	—
98	D₄×C₁₁	—	4	48	no	—
99	D₄×C₁₁	—	4	48	no	—
100	C₂×C₄×C₁₁	—	2	72	no	—
101	C₂×C₄×C₁₁	—	2	36	no	—
102	C₂×C₄×C₁₁	—	4	48	no	—
103	C₂×C₄×C₁₁	C₂⁵	1	12	yes	—
104	C₂×C₄×C₁₁	C₂⁵	1	24	yes	—
105	C₂⁶	C₂²×C₁₁	1	1	yes	characteristic
106	C₂³×D₄	C₂²×C₁₁	1	6	yes	—
107	C₂³×D₄	C₂²×C₁₁	1	3	yes	—
108	C₂³×D₄	C₂²×C₁₁	1	3	yes	complemented
109	C₂⁴×C₄	C₂²×C₁₁	1	3	yes	complemented
110	#560	C₂²×C₁₁	1	256	yes	direct product factor
111	C₂×#127	C₂²×C₁₁	1	24	yes	complemented
112	C₂×#126	C₂²×C₁₁	1	48	yes	complemented
113	C₂×#123	C₂²×C₁₁	1	72	yes	complemented
114	C₂×#121	C₂²×C₁₁	1	144	yes	complemented
115	C₂×#120	C₂²×C₁₁	1	72	yes	complemented
116	C₂²×C₄⋊C₄	C₂²×C₁₁	1	6	yes	complemented
117	C₂²×C₂²⋊C₄	C₂²×C₁₁	1	6	yes	—
118	C₂²×C₂²⋊C₄	C₂²×C₁₁	1	3	yes	—
119	C₂²×C₂²⋊C₄	C₂²×C₁₁	1	3	yes	complemented
120	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic, complemented
121	C₂²×C₁₁	—	2	12	no	—
122	C₂²×C₁₁	—	2	24	no	—
123	C₂²×C₁₁	—	2	24	no	—
124	C₂²×C₁₁	—	2	12	no	—
125	C₂²×C₁₁	—	4	64	no	—
126	C₂²×C₁₁	—	4	72	no	—
127	C₂²×C₁₁	—	4	48	no	—
128	C₂²×C₁₁	—	4	24	no	—
129	C₂²×C₁₁	#560	1	16	yes	direct product factor
130	C₂²×C₁₁	C₂×H(5,𝔽₂)	1	18	yes	—
131	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
132	C₄×C₁₁	—	2	24	no	—
133	C₄×C₁₁	—	2	12	no	—
134	C₂⁵	—	4	12	no	—
135	C₂⁵	C₂³×C₁₁	1	2	yes	—
136	C₂⁵	C₂³×C₁₁	1	1	yes	characteristic
137	C₂⁵	C₂³×C₁₁	1	3	yes	—
138	C₂⁵	C₂³×C₁₁	1	12	yes	—
139	C₂²×D₄	—	4	24	no	—
140	C₂²×D₄	—	4	12	no	—
141	C₂²×D₄	—	4	12	no	—
142	C₂²×D₄	C₂³×C₁₁	1	72	yes	—
143	C₂²×D₄	C₂³×C₁₁	1	36	yes	—
144	C₂²×D₄	C₂³×C₁₁	1	36	yes	complemented
145	C₂³×C₄	—	4	12	no	—
146	C₂³×C₄	C₂³×C₁₁	1	6	yes	—
147	C₂³×C₄	C₂³×C₁₁	1	3	yes	—
148	C₂³×C₄	C₂³×C₁₁	1	36	yes	complemented
149	#127	C₂³×C₁₁	1	64	yes	complemented
150	#126	C₂³×C₁₁	1	128	yes	complemented
151	#123	C₂³×C₁₁	1	192	yes	complemented
152	#121	C₂³×C₁₁	1	384	yes	complemented
153	#120	C₂³×C₁₁	1	192	yes	complemented
154	C₂×C₄⋊C₄	C₂³×C₁₁	1	72	yes	complemented
155	C₂×C₂²⋊C₄	C₂³×C₁₁	1	72	yes	—
156	C₂×C₂²⋊C₄	C₂³×C₁₁	1	36	yes	—
157	C₂×C₂²⋊C₄	C₂³×C₁₁	1	36	yes	complemented
158	C₂×C₄²	C₂³×C₁₁	1	12	yes	complemented
159	C₂×C₁₁	—	4	12	no	—
160	C₂×C₁₁	—	4	8	no	—
161	C₂×C₁₁	—	4	4	no	—
162	C₂×C₁₁	C₂×#560	1	12	yes	direct product factor
163	C₂×C₁₁	C₂²×H(5,𝔽₂)	1	3	yes	—
164	C₂⁴	—	2	12	no	—
165	C₂⁴	—	2	24	no	—
166	C₂⁴	—	2	24	no	—
167	C₂⁴	—	2	12	no	—
168	C₂⁴	—	4	72	no	—
169	C₂⁴	—	4	64	no	—
170	C₂⁴	C₂⁴×C₁₁	1	12	yes	—
171	C₂⁴	C₂⁴×C₁₁	1	6	yes	—
172	C₂⁴	C₂⁴×C₁₁	1	18	yes	—
173	C₂⁴	C₂⁴×C₁₁	1	16	yes	—
174	C₂⁴	C₂⁴×C₁₁	1	1	yes	characteristic
175	C₂×D₄	—	4	144	no	—
176	C₂×D₄	—	4	72	no	—
177	C₂×D₄	—	4	72	no	—
178	C₂×D₄	C₂⁴×C₁₁	1	96	yes	—
179	C₂×D₄	C₂⁴×C₁₁	1	48	yes	—
180	C₂×D₄	C₂⁴×C₁₁	1	48	yes	complemented
181	C₂²×C₄	—	2	24	no	—
182	C₂²×C₄	—	2	12	no	—
183	C₂²×C₄	—	4	72	no	—
184	C₂²×C₄	C₂⁴×C₁₁	1	18	yes	—
185	C₂²×C₄	C₂⁴×C₁₁	1	36	yes	—
186	C₂²×C₄	C₂⁴×C₁₁	1	48	yes	complemented
187	C₄⋊C₄	C₂⁴×C₁₁	1	96	yes	complemented
188	C₂²⋊C₄	C₂⁴×C₁₁	1	96	yes	—
189	C₂²⋊C₄	C₂⁴×C₁₁	1	48	yes	—
190	C₂²⋊C₄	C₂⁴×C₁₁	1	48	yes	complemented
191	C₄²	C₂⁴×C₁₁	1	16	yes	complemented
192	C₁₁	C₂²×#560	1	1	yes	characteristic, direct product factor, Sylow
193	C₂³	—	2	36	no	—
194	C₂³	—	2	72	no	—
195	C₂³	—	2	72	no	—
196	C₂³	—	2	36	no	—
197	C₂³	—	4	48	no	—
198	C₂³	—	4	192	no	—
199	C₂³	—	4	48	no	—
200	C₂³	—	4	32	no	—
201	C₂³	—	4	16	no	—
202	C₂³	C₁₁×H(5,𝔽₂)	1	12	yes	—
203	C₂³	C₂⁵×C₁₁	1	8	yes	—
204	C₂³	C₂⁵×C₁₁	1	4	yes	—
205	C₂³	C₂⁵×C₁₁	1	12	yes	—
206	C₂³	C₂⁵×C₁₁	1	3	yes	—
207	D₄	—	4	96	no	—
208	D₄	—	4	48	no	—
209	D₄	—	4	48	no	—
210	C₂×C₄	—	2	72	no	—
211	C₂×C₄	—	2	36	no	—
212	C₂×C₄	—	4	48	no	—
213	C₂×C₄	C₂⁵×C₁₁	1	12	yes	—
214	C₂×C₄	C₂⁵×C₁₁	1	24	yes	—
215	C₂²	—	2	12	no	—
216	C₂²	—	2	24	no	—
217	C₂²	—	2	24	no	—
218	C₂²	—	2	12	no	—
219	C₂²	—	4	64	no	—
220	C₂²	—	4	48	no	—
221	C₂²	—	4	24	no	—
222	C₂²	—	4	72	no	—
223	C₂²	C₁₁×#560	1	16	yes	direct product factor
224	C₂²	C₂×C₁₁×H(5,𝔽₂)	1	18	yes	—
225	C₂²	C₂⁶×C₁₁	1	1	yes	characteristic
226	C₄	—	2	24	no	—
227	C₄	—	2	12	no	—
228	C₂	—	4	8	no	—
229	C₂	—	4	4	no	—
230	C₂	—	4	12	no	—
231	C₂	C₂×C₁₁×#560	1	12	yes	direct product factor
232	C₂	C₂²×C₁₁×H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₁×#560	C₂	direct (N × H)
2. C₂²×C₁₁×#127	C₂	semidirect
3. C₂²×C₁₁×#126	C₂	semidirect
4. C₂²×C₁₁×#123	C₂	semidirect
5. C₂²×C₁₁×#121	C₂	semidirect
6. C₂²×C₁₁×#120	C₂	semidirect
10. C₂³×D₄×C₁₁	C₂²	semidirect
11. C₂⁴×C₄×C₁₁	C₂²	semidirect
12. C₁₁×#560	C₂²	direct (N × H)
13. C₂×C₁₁×#127	C₂²	semidirect
14. C₂×C₁₁×#126	C₂²	semidirect
15. C₂×C₁₁×#123	C₂²	semidirect
16. C₂×C₁₁×#121	C₂²	semidirect
17. C₂×C₁₁×#120	C₂²	semidirect
18. C₂²×C₁₁×C₄⋊C₄	C₂²	semidirect
21. C₂²×C₁₁×C₂²⋊C₄	C₂²	semidirect
22. C₂²×C₄²×C₁₁	C₂²	semidirect
33. C₂²×D₄×C₁₁	C₂³	semidirect
37. C₂³×C₄×C₁₁	C₂³	semidirect
38. C₁₁×#127	C₂³	semidirect
39. C₁₁×#126	C₂³	semidirect
40. C₁₁×#123	C₂³	semidirect
41. C₁₁×#121	C₂³	semidirect
42. C₁₁×#120	C₂³	semidirect
43. C₂×C₁₁×C₄⋊C₄	C₂³	semidirect
46. C₂×C₁₁×C₂²⋊C₄	C₂³	semidirect
47. C₂×C₄²×C₁₁	C₂³	semidirect
48. C₂²×#560	C₁₁	direct (N × H)
65. C₂×D₄×C₁₁	C₂⁴	semidirect
71. C₂²×C₄×C₁₁	C₂⁴	semidirect
72. C₁₁×C₄⋊C₄	C₂⁴	semidirect
75. C₁₁×C₂²⋊C₄	C₂⁴	semidirect
76. C₄²×C₁₁	C₂⁴	semidirect
77. C₂×#560	C₂×C₁₁	direct (N × H)
78. C₂²×#127	C₂×C₁₁	semidirect
79. C₂²×#126	C₂×C₁₁	semidirect
80. C₂²×#123	C₂×C₁₁	semidirect
81. C₂²×#121	C₂×C₁₁	semidirect
82. C₂²×#120	C₂×C₁₁	semidirect
108. C₂³×D₄	C₂²×C₁₁	semidirect
109. C₂⁴×C₄	C₂²×C₁₁	semidirect
110. #560	C₂²×C₁₁	direct (N × H)
111. C₂×#127	C₂²×C₁₁	semidirect
112. C₂×#126	C₂²×C₁₁	semidirect
113. C₂×#123	C₂²×C₁₁	semidirect
114. C₂×#121	C₂²×C₁₁	semidirect
115. C₂×#120	C₂²×C₁₁	semidirect
116. C₂²×C₄⋊C₄	C₂²×C₁₁	semidirect
119. C₂²×C₂²⋊C₄	C₂²×C₁₁	semidirect
120. C₂²×C₄²	C₂²×C₁₁	semidirect
129. C₂²×C₁₁	#560	direct (N × H)
144. C₂²×D₄	C₂³×C₁₁	semidirect
148. C₂³×C₄	C₂³×C₁₁	semidirect
149. #127	C₂³×C₁₁	semidirect
150. #126	C₂³×C₁₁	semidirect
151. #123	C₂³×C₁₁	semidirect
152. #121	C₂³×C₁₁	semidirect
153. #120	C₂³×C₁₁	semidirect
154. C₂×C₄⋊C₄	C₂³×C₁₁	semidirect
157. C₂×C₂²⋊C₄	C₂³×C₁₁	semidirect
158. C₂×C₄²	C₂³×C₁₁	semidirect
162. C₂×C₁₁	C₂×#560	direct (N × H)
180. C₂×D₄	C₂⁴×C₁₁	semidirect
186. C₂²×C₄	C₂⁴×C₁₁	semidirect
187. C₄⋊C₄	C₂⁴×C₁₁	semidirect
190. C₂²⋊C₄	C₂⁴×C₁₁	semidirect
191. C₄²	C₂⁴×C₁₁	semidirect
192. C₁₁	C₂²×#560	direct (N × H)
223. C₂²	C₁₁×#560	direct (N × H)
231. C₂	C₂×C₁₁×#560	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×#560	2
C₂²×C₁₁×#127	2
C₂²×C₁₁×#126	2
C₂²×C₁₁×#123	2
C₂²×C₁₁×#121	2
C₂²×C₁₁×#120	2
C₂²×#560	11

Subgroups — C22×C11×#560 #14639880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₁×#560 #14639880