Subgroups — C₂²×C₁₁×#508 #14639821

Order 2816.

2118 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×#135	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₁×#508	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₁₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₁₁×SD₃₂	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₁×#41	—	2	2	no	—
8	C₂×C₁₁×#135	—	2	12	no	—
9	C₂×C₁₁×#135	C₂²	1	24	yes	complemented
10	C₂²×C₁₁×Q₁₆	C₂²	1	2	yes	—
11	C₂²×C₁₁×SD₁₆	—	2	2	no	—
12	C₂²×C₁₁×D₈	C₂²	1	2	yes	—
13	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₁₁×#508	C₂²	1	64	yes	direct product factor
15	C₂×C₁₁×Q₃₂	C₂²	1	12	yes	complemented
16	C₂×C₁₁×SD₃₂	C₂²	1	24	yes	complemented
17	C₂×C₁₁×D₁₆	C₂²	1	12	yes	complemented
18	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₁₁×C₁₆	C₂²	1	12	yes	complemented
21	C₂×C₁₁×#41	—	2	24	no	—
22	C₂×C₁₁×#41	—	4	12	no	—
23	C₂²×Q₈×C₁₁	—	2	2	no	—
24	C₂²×D₄×C₁₁	—	2	2	no	—
25	C₂²×D₄×C₁₁	—	4	2	no	—
26	C₂³×C₄×C₁₁	—	4	2	no	—
27	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
28	C₁₁×#135	—	2	48	no	—
29	C₁₁×#135	C₂³	1	32	yes	complemented
30	C₂×C₁₁×Q₁₆	—	2	6	no	—
31	C₂×C₁₁×Q₁₆	C₂³	1	12	yes	—
32	C₂×C₁₁×SD₁₆	—	2	24	no	—
33	C₂×C₁₁×D₈	—	2	6	no	—
34	C₂×C₁₁×D₈	C₂³	1	12	yes	—
35	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₁	D₄	1	6	yes	—
37	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₁	C₂³	1	6	yes	—
39	C₁₁×Q₃₂	C₂³	1	16	yes	complemented
40	C₁₁×SD₃₂	C₂³	1	32	yes	complemented
41	C₁₁×D₁₆	C₂³	1	16	yes	complemented
42	C₂×C₁₁×C₁₆	C₂³	1	6	yes	—
43	C₂×C₁₁×C₁₆	C₂³	1	6	yes	—
44	C₂×C₁₁×C₁₆	C₂³	1	16	yes	complemented
45	C₂²×#508	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂⁴×C₁₁	—	4	2	no	—
47	C₂⁴×C₁₁	D₈	1	1	yes	characteristic
48	C₁₁×#41	—	2	32	no	—
49	C₁₁×#41	—	4	48	no	—
50	C₂×Q₈×C₁₁	—	2	12	no	—
51	C₂×Q₈×C₁₁	—	4	6	no	—
52	C₂×D₄×C₁₁	—	2	12	no	—
53	C₂×D₄×C₁₁	—	4	24	no	—
54	C₂×D₄×C₁₁	—	4	6	no	—
55	C₂²×C₄×C₁₁	—	4	2	no	—
56	C₂²×C₄×C₁₁	—	4	24	no	—
57	C₂²×C₄×C₁₁	D₈	1	6	yes	—
58	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₂×D₄	1	6	yes	—
61	C₁₁×Q₁₆	—	2	12	no	—
62	C₁₁×Q₁₆	C₂⁴	1	8	yes	—
63	C₁₁×SD₁₆	—	2	32	no	—
64	C₁₁×D₈	—	2	12	no	—
65	C₁₁×D₈	C₂⁴	1	8	yes	—
66	C₂×C₈×C₁₁	C₂×D₄	1	3	yes	—
67	C₂×C₈×C₁₁	C₂×D₄	1	6	yes	—
68	C₂×C₈×C₁₁	C₂×D₄	1	12	yes	—
69	C₂×C₈×C₁₁	C₂⁴	1	3	yes	—
70	C₂×C₈×C₁₁	C₂⁴	1	4	yes	—
71	C₁₁×C₁₆	C₂⁴	1	4	yes	—
72	C₁₁×C₁₆	C₂⁴	1	4	yes	—
73	C₂²×#135	C₂×C₁₁	1	2	yes	complemented
74	C₂×#508	C₂×C₁₁	1	24	yes	direct product factor
75	C₂²×Q₃₂	C₂×C₁₁	1	1	yes	characteristic, complemented
76	C₂²×SD₃₂	C₂×C₁₁	1	2	yes	complemented
77	C₂²×D₁₆	C₂×C₁₁	1	1	yes	characteristic, complemented
78	C₂³×C₁₆	C₂×C₁₁	1	1	yes	characteristic, complemented
79	C₂³×C₁₁	—	2	4	no	—
80	C₂³×C₁₁	—	4	12	no	—
81	C₂³×C₁₁	—	8	8	no	—
82	C₂³×C₁₁	C₂×D₈	1	6	yes	—
83	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
84	Q₈×C₁₁	—	2	8	no	—
85	Q₈×C₁₁	—	4	12	no	—
86	D₄×C₁₁	—	2	8	no	—
87	D₄×C₁₁	—	4	32	no	—
88	D₄×C₁₁	—	4	12	no	—
89	C₂×C₄×C₁₁	—	4	12	no	—
90	C₂×C₄×C₁₁	—	4	32	no	—
91	C₂×C₄×C₁₁	C₂×D₈	1	6	yes	—
92	C₂×C₄×C₁₁	C₂×D₈	1	3	yes	—
93	C₂×C₄×C₁₁	C₂×D₈	1	12	yes	—
94	C₂×C₄×C₁₁	C₂²×D₄	1	3	yes	—
95	C₂×C₄×C₁₁	C₂²×D₄	1	4	yes	—
96	C₈×C₁₁	C₂²×D₄	1	3	yes	—
97	C₈×C₁₁	C₂²×D₄	1	4	yes	—
98	C₈×C₁₁	C₂⁵	1	1	yes	characteristic
99	C₂²×#41	—	2	2	no	—
100	C₂×#135	—	2	12	no	—
101	C₂×#135	C₂²×C₁₁	1	24	yes	complemented
102	C₂²×Q₁₆	C₂²×C₁₁	1	2	yes	—
103	C₂²×SD₁₆	—	2	2	no	—
104	C₂²×D₈	C₂²×C₁₁	1	2	yes	—
105	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
106	#508	C₂²×C₁₁	1	64	yes	direct product factor
107	C₂×Q₃₂	C₂²×C₁₁	1	12	yes	complemented
108	C₂×SD₃₂	C₂²×C₁₁	1	24	yes	complemented
109	C₂×D₁₆	C₂²×C₁₁	1	12	yes	complemented
110	C₂²×C₁₆	C₂²×C₁₁	1	1	yes	characteristic
111	C₂²×C₁₆	C₂²×C₁₁	1	1	yes	characteristic
112	C₂²×C₁₆	C₂²×C₁₁	1	12	yes	complemented
113	C₂²×C₁₁	—	2	12	no	—
114	C₂²×C₁₁	—	4	8	no	—
115	C₂²×C₁₁	—	8	24	no	—
116	C₂²×C₁₁	#508	1	4	yes	direct product factor
117	C₂²×C₁₁	C₂²×D₈	1	4	yes	—
118	C₂²×C₁₁	C₂²×D₈	1	3	yes	—
119	C₄×C₁₁	—	4	8	no	—
120	C₄×C₁₁	C₂²×D₈	1	4	yes	—
121	C₄×C₁₁	C₂²×D₈	1	3	yes	—
122	C₄×C₁₁	C₂³×D₄	1	1	yes	characteristic
123	C₂×#41	—	2	24	no	—
124	C₂×#41	—	4	12	no	—
125	C₂²×Q₈	—	2	2	no	—
126	C₂²×D₄	—	2	2	no	—
127	C₂²×D₄	—	4	2	no	—
128	C₂³×C₄	—	4	2	no	—
129	C₂³×C₄	D₄×C₁₁	1	1	yes	characteristic
130	#135	—	2	48	no	—
131	#135	C₂³×C₁₁	1	32	yes	complemented
132	C₂×Q₁₆	—	2	6	no	—
133	C₂×Q₁₆	C₂³×C₁₁	1	12	yes	—
134	C₂×SD₁₆	—	2	24	no	—
135	C₂×D₈	—	2	6	no	—
136	C₂×D₈	C₂³×C₁₁	1	12	yes	—
137	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
138	C₂²×C₈	D₄×C₁₁	1	6	yes	—
139	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₁₁	1	6	yes	—
141	Q₃₂	C₂³×C₁₁	1	16	yes	complemented
142	SD₃₂	C₂³×C₁₁	1	32	yes	complemented
143	D₁₆	C₂³×C₁₁	1	16	yes	complemented
144	C₂×C₁₆	C₂³×C₁₁	1	6	yes	—
145	C₂×C₁₆	C₂³×C₁₁	1	6	yes	—
146	C₂×C₁₆	C₂³×C₁₁	1	16	yes	complemented
147	C₂×C₁₁	—	2	4	no	—
148	C₂×C₁₁	—	8	8	no	—
149	C₂×C₁₁	C₂×#508	1	6	yes	direct product factor
150	C₂×C₁₁	C₂³×D₈	1	1	yes	characteristic
151	C₂⁴	—	4	2	no	—
152	C₂⁴	C₁₁×D₈	1	1	yes	characteristic
153	#41	—	2	32	no	—
154	#41	—	4	48	no	—
155	C₂×Q₈	—	2	12	no	—
156	C₂×Q₈	—	4	6	no	—
157	C₂×D₄	—	2	12	no	—
158	C₂×D₄	—	4	24	no	—
159	C₂×D₄	—	4	6	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	—	4	24	no	—
162	C₂²×C₄	C₁₁×D₈	1	1	yes	characteristic
163	C₂²×C₄	C₁₁×D₈	1	6	yes	—
164	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₁₁	1	6	yes	—
166	Q₁₆	—	2	12	no	—
167	Q₁₆	C₂⁴×C₁₁	1	8	yes	—
168	SD₁₆	—	2	32	no	—
169	D₈	—	2	12	no	—
170	D₈	C₂⁴×C₁₁	1	8	yes	—
171	C₂×C₈	C₂×D₄×C₁₁	1	3	yes	—
172	C₂×C₈	C₂×D₄×C₁₁	1	6	yes	—
173	C₂×C₈	C₂×D₄×C₁₁	1	12	yes	—
174	C₂×C₈	C₂⁴×C₁₁	1	3	yes	—
175	C₂×C₈	C₂⁴×C₁₁	1	4	yes	—
176	C₁₆	C₂⁴×C₁₁	1	4	yes	—
177	C₁₆	C₂⁴×C₁₁	1	4	yes	—
178	C₁₁	C₂²×#508	1	1	yes	characteristic, direct product factor, Sylow
179	C₂³	—	2	4	no	—
180	C₂³	—	4	12	no	—
181	C₂³	—	8	8	no	—
182	C₂³	C₂×C₁₁×D₈	1	6	yes	—
183	C₂³	C₂×C₁₁×D₈	1	1	yes	characteristic
184	Q₈	—	2	8	no	—
185	Q₈	—	4	12	no	—
186	D₄	—	2	8	no	—
187	D₄	—	4	32	no	—
188	D₄	—	4	12	no	—
189	C₂×C₄	—	4	12	no	—
190	C₂×C₄	—	4	32	no	—
191	C₂×C₄	C₂×C₁₁×D₈	1	6	yes	—
192	C₂×C₄	C₂×C₁₁×D₈	1	3	yes	—
193	C₂×C₄	C₂×C₁₁×D₈	1	12	yes	—
194	C₂×C₄	C₂²×D₄×C₁₁	1	3	yes	—
195	C₂×C₄	C₂²×D₄×C₁₁	1	4	yes	—
196	C₈	C₂²×D₄×C₁₁	1	3	yes	—
197	C₈	C₂²×D₄×C₁₁	1	4	yes	—
198	C₈	C₂⁵×C₁₁	1	1	yes	characteristic
199	C₂²	—	2	12	no	—
200	C₂²	—	4	8	no	—
201	C₂²	—	8	24	no	—
202	C₂²	C₁₁×#508	1	4	yes	direct product factor
203	C₂²	C₂²×C₁₁×D₈	1	4	yes	—
204	C₂²	C₂²×C₁₁×D₈	1	3	yes	—
205	C₄	—	4	8	no	—
206	C₄	C₂²×C₁₁×D₈	1	4	yes	—
207	C₄	C₂²×C₁₁×D₈	1	3	yes	—
208	C₄	C₂³×D₄×C₁₁	1	1	yes	characteristic
209	C₂	—	2	4	no	—
210	C₂	—	8	8	no	—
211	C₂	C₂×C₁₁×#508	1	6	yes	direct product factor
212	C₂	C₂³×C₁₁×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₁	D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₁₁
socle	C₂³×C₁₁	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁ ◁ C₂³×C₄×C₁₁ ◁ C₂³×C₈×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₁×#135	C₂	semidirect
2. C₂×C₁₁×#508	C₂	direct (N × H)
3. C₂²×C₁₁×Q₃₂	C₂	semidirect
4. C₂²×C₁₁×SD₃₂	C₂	semidirect
5. C₂²×C₁₁×D₁₆	C₂	semidirect
6. C₂³×C₁₁×C₁₆	C₂	semidirect
9. C₂×C₁₁×#135	C₂²	semidirect
14. C₁₁×#508	C₂²	direct (N × H)
15. C₂×C₁₁×Q₃₂	C₂²	semidirect
16. C₂×C₁₁×SD₃₂	C₂²	semidirect
17. C₂×C₁₁×D₁₆	C₂²	semidirect
20. C₂²×C₁₁×C₁₆	C₂²	semidirect
29. C₁₁×#135	C₂³	semidirect
39. C₁₁×Q₃₂	C₂³	semidirect
40. C₁₁×SD₃₂	C₂³	semidirect
41. C₁₁×D₁₆	C₂³	semidirect
44. C₂×C₁₁×C₁₆	C₂³	semidirect
45. C₂²×#508	C₁₁	direct (N × H)
73. C₂²×#135	C₂×C₁₁	semidirect
74. C₂×#508	C₂×C₁₁	direct (N × H)
75. C₂²×Q₃₂	C₂×C₁₁	semidirect
76. C₂²×SD₃₂	C₂×C₁₁	semidirect
77. C₂²×D₁₆	C₂×C₁₁	semidirect
78. C₂³×C₁₆	C₂×C₁₁	semidirect
101. C₂×#135	C₂²×C₁₁	semidirect
106. #508	C₂²×C₁₁	direct (N × H)
107. C₂×Q₃₂	C₂²×C₁₁	semidirect
108. C₂×SD₃₂	C₂²×C₁₁	semidirect
109. C₂×D₁₆	C₂²×C₁₁	semidirect
112. C₂²×C₁₆	C₂²×C₁₁	semidirect
116. C₂²×C₁₁	#508	direct (N × H)
131. #135	C₂³×C₁₁	semidirect
141. Q₃₂	C₂³×C₁₁	semidirect
142. SD₃₂	C₂³×C₁₁	semidirect
143. D₁₆	C₂³×C₁₁	semidirect
146. C₂×C₁₆	C₂³×C₁₁	semidirect
149. C₂×C₁₁	C₂×#508	direct (N × H)
178. C₁₁	C₂²×#508	direct (N × H)
202. C₂²	C₁₁×#508	direct (N × H)
211. C₂	C₂×C₁₁×#508	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₁×#135	2
C₂×C₁₁×#508	2
C₂²×C₁₁×Q₃₂	2
C₂²×C₁₁×SD₃₂	2
C₂²×C₁₁×D₁₆	2
C₂³×C₁₁×C₁₆	2
C₂²×#508	11

Subgroups — C22×C11×#508 #14639821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₁×#508 #14639821