Subgroups — C₂³×C₁₁×#123 #14639851

Order 2816.

15150 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂⁵×C₄×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₁×#123	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₁₁×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂³×C₁₁×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₁₁×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₂⁶×C₁₁	C₂²	1	1	yes	characteristic
8	C₂⁶×C₁₁	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₁₁	—	2	16	no	—
10	C₂³×D₄×C₁₁	C₂²	1	28	yes	—
11	C₂⁴×C₄×C₁₁	—	2	16	no	—
12	C₂⁴×C₄×C₁₁	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₁₁	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₁₁	C₂²	1	28	yes	complemented
16	C₂×C₁₁×#123	C₂²	1	448	yes	direct product factor
17	C₂²×C₁₁×C₄⋊C₄	C₂²	1	56	yes	complemented
18	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	28	yes	complemented
19	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	56	yes	complemented
20	C₂⁵×C₁₁	—	2	16	no	—
21	C₂⁵×C₁₁	—	2	8	no	—
22	C₂⁵×C₁₁	—	2	16	no	—
23	C₂⁵×C₁₁	D₄	1	16	yes	—
24	C₂⁵×C₁₁	C₂³	1	14	yes	—
25	C₂⁵×C₁₁	C₂³	1	14	yes	—
26	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
27	C₂²×D₄×C₁₁	—	2	224	no	—
28	C₂²×D₄×C₁₁	C₂³	1	112	yes	—
29	C₂³×C₄×C₁₁	—	2	16	no	—
30	C₂³×C₄×C₁₁	—	2	224	no	—
31	C₂³×C₄×C₁₁	—	2	8	no	—
32	C₂³×C₄×C₁₁	—	2	16	no	—
33	C₂³×C₄×C₁₁	C₂³	1	28	yes	—
34	C₂³×C₄×C₁₁	C₂³	1	28	yes	—
35	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
36	C₂³×C₄×C₁₁	C₂³	1	112	yes	complemented
37	C₁₁×#123	C₂³	1	512	yes	direct product factor
38	C₂×C₁₁×C₄⋊C₄	C₂³	1	224	yes	complemented
39	C₂×C₁₁×C₂²⋊C₄	C₂³	1	112	yes	complemented
40	C₂×C₁₁×C₂²⋊C₄	C₂³	1	224	yes	complemented
41	C₂³×#123	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂⁴×C₁₁	—	2	128	no	—
43	C₂⁴×C₁₁	—	2	112	no	—
44	C₂⁴×C₁₁	—	2	56	no	—
45	C₂⁴×C₁₁	—	2	112	no	—
46	C₂⁴×C₁₁	—	4	64	no	—
47	C₂⁴×C₁₁	C₂×D₄	1	112	yes	—
48	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
49	C₂⁴×C₁₁	#41	1	16	yes	—
50	C₂⁴×C₁₁	C₂⁴	1	28	yes	—
51	C₂⁴×C₁₁	C₂⁴	1	28	yes	—
52	C₂⁴×C₁₁	C₂⁴	1	7	yes	—
53	C₂×D₄×C₁₁	—	2	448	no	—
54	C₂×D₄×C₁₁	C₂⁴	1	64	yes	—
55	C₂²×C₄×C₁₁	—	2	112	no	—
56	C₂²×C₄×C₁₁	—	2	448	no	—
57	C₂²×C₄×C₁₁	—	2	56	no	—
58	C₂²×C₄×C₁₁	—	2	112	no	—
59	C₂²×C₄×C₁₁	C₂⁴	1	56	yes	—
60	C₂²×C₄×C₁₁	C₂⁴	1	56	yes	—
61	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
62	C₂²×C₄×C₁₁	C₂⁴	1	64	yes	complemented
63	C₁₁×C₄⋊C₄	C₂⁴	1	128	yes	complemented
64	C₁₁×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
65	C₁₁×C₂²⋊C₄	C₂⁴	1	128	yes	complemented
66	C₂⁴×D₄	C₂×C₁₁	1	1	yes	characteristic
67	C₂⁵×C₄	C₂×C₁₁	1	1	yes	characteristic, complemented
68	C₂²×#123	C₂×C₁₁	1	56	yes	direct product factor
69	C₂³×C₄⋊C₄	C₂×C₁₁	1	2	yes	complemented
70	C₂³×C₂²⋊C₄	C₂×C₁₁	1	2	yes	complemented
71	C₂³×C₂²⋊C₄	C₂×C₁₁	1	1	yes	characteristic, complemented
72	C₂³×C₁₁	—	2	448	no	—
73	C₂³×C₁₁	—	2	112	no	—
74	C₂³×C₁₁	—	2	56	no	—
75	C₂³×C₁₁	—	2	112	no	—
76	C₂³×C₁₁	—	4	224	no	—
77	C₂³×C₁₁	#123	1	64	yes	direct product factor
78	C₂³×C₁₁	C₂²×D₄	1	112	yes	—
79	C₂³×C₁₁	C₂²×D₄	1	28	yes	—
80	C₂³×C₁₁	C₂×#41	1	56	yes	—
81	C₂³×C₁₁	C₂⁵	1	8	yes	—
82	C₂³×C₁₁	C₂⁵	1	8	yes	—
83	C₂³×C₁₁	C₂⁵	1	7	yes	—
84	D₄×C₁₁	—	2	128	no	—
85	C₂×C₄×C₁₁	—	2	112	no	—
86	C₂×C₄×C₁₁	—	2	128	no	—
87	C₂×C₄×C₁₁	—	2	56	no	—
88	C₂×C₄×C₁₁	—	2	112	no	—
89	C₂×C₄×C₁₁	C₂⁵	1	16	yes	—
90	C₂×C₄×C₁₁	C₂⁵	1	16	yes	—
91	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
92	C₂⁶	C₂²×C₁₁	1	1	yes	characteristic
93	C₂⁶	C₂²×C₁₁	1	1	yes	characteristic
94	C₂³×D₄	—	2	16	no	—
95	C₂³×D₄	C₂²×C₁₁	1	28	yes	—
96	C₂⁴×C₄	—	2	16	no	—
97	C₂⁴×C₄	C₂²×C₁₁	1	2	yes	—
98	C₂⁴×C₄	C₂²×C₁₁	1	1	yes	characteristic
99	C₂⁴×C₄	C₂²×C₁₁	1	2	yes	—
100	C₂⁴×C₄	C₂²×C₁₁	1	28	yes	complemented
101	C₂×#123	C₂²×C₁₁	1	448	yes	direct product factor
102	C₂²×C₄⋊C₄	C₂²×C₁₁	1	56	yes	complemented
103	C₂²×C₂²⋊C₄	C₂²×C₁₁	1	28	yes	complemented
104	C₂²×C₂²⋊C₄	C₂²×C₁₁	1	56	yes	complemented
105	C₂²×C₁₁	—	2	16	no	—
106	C₂²×C₁₁	—	2	224	no	—
107	C₂²×C₁₁	—	2	8	no	—
108	C₂²×C₁₁	—	2	16	no	—
109	C₂²×C₁₁	—	4	112	no	—
110	C₂²×C₁₁	C₂×#123	1	112	yes	direct product factor
111	C₂²×C₁₁	C₂³×D₄	1	16	yes	—
112	C₂²×C₁₁	C₂³×D₄	1	14	yes	—
113	C₂²×C₁₁	C₂²×#41	1	28	yes	—
114	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
115	C₄×C₁₁	—	2	16	no	—
116	C₄×C₁₁	—	2	8	no	—
117	C₄×C₁₁	—	2	16	no	—
118	C₂⁵	—	2	16	no	—
119	C₂⁵	—	2	8	no	—
120	C₂⁵	—	2	16	no	—
121	C₂⁵	D₄×C₁₁	1	16	yes	—
122	C₂⁵	C₂³×C₁₁	1	14	yes	—
123	C₂⁵	C₂³×C₁₁	1	14	yes	—
124	C₂⁵	C₂³×C₁₁	1	1	yes	characteristic
125	C₂²×D₄	—	2	224	no	—
126	C₂²×D₄	C₂³×C₁₁	1	112	yes	—
127	C₂³×C₄	—	2	16	no	—
128	C₂³×C₄	—	2	224	no	—
129	C₂³×C₄	—	2	8	no	—
130	C₂³×C₄	—	2	16	no	—
131	C₂³×C₄	C₂³×C₁₁	1	28	yes	—
132	C₂³×C₄	C₂³×C₁₁	1	28	yes	—
133	C₂³×C₄	C₂³×C₁₁	1	14	yes	—
134	C₂³×C₄	C₂³×C₁₁	1	112	yes	complemented
135	#123	C₂³×C₁₁	1	512	yes	direct product factor
136	C₂×C₄⋊C₄	C₂³×C₁₁	1	224	yes	complemented
137	C₂×C₂²⋊C₄	C₂³×C₁₁	1	112	yes	complemented
138	C₂×C₂²⋊C₄	C₂³×C₁₁	1	224	yes	complemented
139	C₂×C₁₁	—	2	16	no	—
140	C₂×C₁₁	—	4	8	no	—
141	C₂×C₁₁	C₂²×#123	1	28	yes	direct product factor
142	C₂×C₁₁	C₂⁴×D₄	1	1	yes	characteristic
143	C₂×C₁₁	C₂³×#41	1	2	yes	—
144	C₂⁴	—	2	128	no	—
145	C₂⁴	—	2	112	no	—
146	C₂⁴	—	2	56	no	—
147	C₂⁴	—	2	112	no	—
148	C₂⁴	—	4	64	no	—
149	C₂⁴	C₂×D₄×C₁₁	1	112	yes	—
150	C₂⁴	C₂×D₄×C₁₁	1	8	yes	—
151	C₂⁴	C₁₁×#41	1	16	yes	—
152	C₂⁴	C₂⁴×C₁₁	1	28	yes	—
153	C₂⁴	C₂⁴×C₁₁	1	28	yes	—
154	C₂⁴	C₂⁴×C₁₁	1	7	yes	—
155	C₂×D₄	—	2	448	no	—
156	C₂×D₄	C₂⁴×C₁₁	1	64	yes	—
157	C₂²×C₄	—	2	112	no	—
158	C₂²×C₄	—	2	448	no	—
159	C₂²×C₄	—	2	56	no	—
160	C₂²×C₄	—	2	112	no	—
161	C₂²×C₄	C₂⁴×C₁₁	1	56	yes	—
162	C₂²×C₄	C₂⁴×C₁₁	1	56	yes	—
163	C₂²×C₄	C₂⁴×C₁₁	1	28	yes	—
164	C₂²×C₄	C₂⁴×C₁₁	1	64	yes	complemented
165	C₄⋊C₄	C₂⁴×C₁₁	1	128	yes	complemented
166	C₂²⋊C₄	C₂⁴×C₁₁	1	64	yes	complemented
167	C₂²⋊C₄	C₂⁴×C₁₁	1	128	yes	complemented
168	C₁₁	C₂³×#123	1	1	yes	characteristic, direct product factor, Sylow
169	C₂³	—	2	448	no	—
170	C₂³	—	2	112	no	—
171	C₂³	—	2	56	no	—
172	C₂³	—	2	112	no	—
173	C₂³	—	4	224	no	—
174	C₂³	C₁₁×#123	1	64	yes	direct product factor
175	C₂³	C₂²×D₄×C₁₁	1	112	yes	—
176	C₂³	C₂²×D₄×C₁₁	1	28	yes	—
177	C₂³	C₂×C₁₁×#41	1	56	yes	—
178	C₂³	C₂⁵×C₁₁	1	8	yes	—
179	C₂³	C₂⁵×C₁₁	1	8	yes	—
180	C₂³	C₂⁵×C₁₁	1	7	yes	—
181	D₄	—	2	128	no	—
182	C₂×C₄	—	2	112	no	—
183	C₂×C₄	—	2	128	no	—
184	C₂×C₄	—	2	56	no	—
185	C₂×C₄	—	2	112	no	—
186	C₂×C₄	C₂⁵×C₁₁	1	16	yes	—
187	C₂×C₄	C₂⁵×C₁₁	1	16	yes	—
188	C₂×C₄	C₂⁵×C₁₁	1	8	yes	—
189	C₂²	—	2	224	no	—
190	C₂²	—	2	16	no	—
191	C₂²	—	2	8	no	—
192	C₂²	—	2	16	no	—
193	C₂²	—	4	112	no	—
194	C₂²	C₂×C₁₁×#123	1	112	yes	direct product factor
195	C₂²	C₂³×D₄×C₁₁	1	16	yes	—
196	C₂²	C₂³×D₄×C₁₁	1	14	yes	—
197	C₂²	C₂²×C₁₁×#41	1	28	yes	—
198	C₂²	C₂⁶×C₁₁	1	1	yes	characteristic
199	C₄	—	2	16	no	—
200	C₄	—	2	8	no	—
201	C₄	—	2	16	no	—
202	C₂	—	2	16	no	—
203	C₂	—	4	8	no	—
204	C₂	C₂²×C₁₁×#123	1	28	yes	direct product factor
205	C₂	C₂⁴×D₄×C₁₁	1	1	yes	characteristic
206	C₂	C₂³×C₁₁×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₁₁	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₁
prime radical	C₂⁴×D₄×C₁₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁵×C₄×C₁₁	C₂	semidirect
3. C₂²×C₁₁×#123	C₂	direct (N × H)
4. C₂³×C₁₁×C₄⋊C₄	C₂	semidirect
5. C₂³×C₁₁×C₂²⋊C₄	C₂	semidirect
6. C₂³×C₁₁×C₂²⋊C₄	C₂	semidirect
15. C₂⁴×C₄×C₁₁	C₂²	semidirect
16. C₂×C₁₁×#123	C₂²	direct (N × H)
17. C₂²×C₁₁×C₄⋊C₄	C₂²	semidirect
18. C₂²×C₁₁×C₂²⋊C₄	C₂²	semidirect
19. C₂²×C₁₁×C₂²⋊C₄	C₂²	semidirect
36. C₂³×C₄×C₁₁	C₂³	semidirect
37. C₁₁×#123	C₂³	direct (N × H)
38. C₂×C₁₁×C₄⋊C₄	C₂³	semidirect
39. C₂×C₁₁×C₂²⋊C₄	C₂³	semidirect
40. C₂×C₁₁×C₂²⋊C₄	C₂³	semidirect
41. C₂³×#123	C₁₁	direct (N × H)
62. C₂²×C₄×C₁₁	C₂⁴	semidirect
63. C₁₁×C₄⋊C₄	C₂⁴	semidirect
64. C₁₁×C₂²⋊C₄	C₂⁴	semidirect
65. C₁₁×C₂²⋊C₄	C₂⁴	semidirect
67. C₂⁵×C₄	C₂×C₁₁	semidirect
68. C₂²×#123	C₂×C₁₁	direct (N × H)
69. C₂³×C₄⋊C₄	C₂×C₁₁	semidirect
70. C₂³×C₂²⋊C₄	C₂×C₁₁	semidirect
71. C₂³×C₂²⋊C₄	C₂×C₁₁	semidirect
77. C₂³×C₁₁	#123	direct (N × H)
100. C₂⁴×C₄	C₂²×C₁₁	semidirect
101. C₂×#123	C₂²×C₁₁	direct (N × H)
102. C₂²×C₄⋊C₄	C₂²×C₁₁	semidirect
103. C₂²×C₂²⋊C₄	C₂²×C₁₁	semidirect
104. C₂²×C₂²⋊C₄	C₂²×C₁₁	semidirect
110. C₂²×C₁₁	C₂×#123	direct (N × H)
134. C₂³×C₄	C₂³×C₁₁	semidirect
135. #123	C₂³×C₁₁	direct (N × H)
136. C₂×C₄⋊C₄	C₂³×C₁₁	semidirect
137. C₂×C₂²⋊C₄	C₂³×C₁₁	semidirect
138. C₂×C₂²⋊C₄	C₂³×C₁₁	semidirect
141. C₂×C₁₁	C₂²×#123	direct (N × H)
164. C₂²×C₄	C₂⁴×C₁₁	semidirect
165. C₄⋊C₄	C₂⁴×C₁₁	semidirect
166. C₂²⋊C₄	C₂⁴×C₁₁	semidirect
167. C₂²⋊C₄	C₂⁴×C₁₁	semidirect
168. C₁₁	C₂³×#123	direct (N × H)
174. C₂³	C₁₁×#123	direct (N × H)
194. C₂²	C₂×C₁₁×#123	direct (N × H)
204. C₂	C₂²×C₁₁×#123	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₁₁	2
C₂⁵×C₄×C₁₁	2
C₂²×C₁₁×#123	2
C₂³×C₁₁×C₄⋊C₄	2
C₂³×C₁₁×C₂²⋊C₄	2
C₂³×#123	11

Subgroups — C23×C11×#123 #14639851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₁₁×#123 #14639851