Finite Group Databasebeta

← C22×C3×#1201 #13089246

Subgroups — C22×C3×#1201 #13089246

Order 1440.

1078 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 296 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C5×D3C211yesmaximal, characteristic, complemented
2C23×C32×D5C211yesmaximal, characteristic, complemented
3C22×C3×Dic15C211yesmaximal, characteristic, complemented
4C2×C3×#1201C2112yesmaximal, direct product factor
5C22×C3×C22⋊D5—31nomaximal
6C22×#1201C311yesmaximal, characteristic, direct product factor
7C23×C32×C5C2211yescharacteristic
8C22×C3×C5×D3—24no—
9C22×C3×C5×D3C2216yescomplemented
10C22×C32×D5—24no—
11C22×C32×D5C2216yescomplemented
12C2×C3×Dic15C2216yescomplemented
13C3×#1201C22116yesdirect product factor
14C22×C3×#67—51nomaximal, Hall
15C24×C3×C5—31no—
16C23×C5×D3C2×C311yescharacteristic, complemented
17C23×C3×D5—21no—
18C23×C3×D5D311yescharacteristic, complemented
19C23×C3×D5C2×C311yescharacteristic, complemented
20C22×Dic15C2×C311yescharacteristic, complemented
21C2×C3×C22⋊D5—312no—
22C22×C3×Dic5—31no—
23C2×#1201C2×C3112yesdirect product factor
24C22×C32×C5D414yescomplemented
25C22×C32×C5C2313yes—
26C2×C3×C5×D3—212no—
27C2×C3×C5×D3C2314yescomplemented
28C2×C32×D5—212no—
29C2×C32×D5C2314yescomplemented
30C3×Dic15C2314yescomplemented
31C22×C22⋊D5—31noHall
32C24×C32—51no—
33C23×C3×D3D511yescharacteristic, complemented
34C2×C3×#67—512no—
35C22×C3×Dic3—51no—
36C23×C3×C5—21no—
37C23×C3×C5—36no—
38C23×C3×C5—64no—
39C23×C3×C5C2×D311yescharacteristic
40C23×C3×C5C22×C311yescharacteristic
41C22×C5×D3—24no—
42C22×C5×D3C22×C316yescomplemented
43C22×C3×D5—28no—
44C22×C3×D5—26no—
45C22×C3×D5—24no—
46C22×C3×D5—24no—
47C22×C3×D5C2×D316yescomplemented
48C22×C3×D5C22×C316yescomplemented
49C2×Dic15C22×C316yescomplemented
50C3×C22⋊D5—316no—
51C2×C3×Dic5—36no—
52#1201C22×C3116yesdirect product factor
53C22×C3×D4—151no—
54C22×#67—51no—
55C2×C32×C5C2×D416yescomplemented
56C2×C32×C5C2411yescharacteristic
57C3×C5×D3—24no—
58C32×D5—24no—
59C24×C5—31no—
60C23×D5C3×D311yescharacteristic, complemented
61C2×C22⋊D5—312no—
62C22×Dic5—31no—
63C23×C32—56no—
64C23×C32—104no—
65C23×C32C2×D511yescharacteristic
66C22×C3×D3—24no—
67C22×C3×D3C2×D516yescomplemented
68C3×#67—516no—
69C2×C3×Dic3—56no—
70C22×C3×C5—24no—
71C22×C3×C5—23no—
72C22×C3×C5—34no—
73C22×C3×C5—612no—
74C22×C3×C5#6714yescomplemented
75C22×C3×C5C3×D414yescomplemented
76C22×C3×C5C22×D313yes—
77C22×C3×C5C23×C313yes—
78C2×C5×D3—212no—
79C2×C5×D3C23×C314yescomplemented
80C2×C3×D5—224no—
81C2×C3×D5—212no—
82C2×C3×D5—212no—
83C2×C3×D5—24no—
84C2×C3×D5C22×D314yescomplemented
85C2×C3×D5C23×C314yescomplemented
86Dic15C23×C314yescomplemented
87C3×Dic5—34no—
88C24×C3—31no—
89C24×C3—51no—
90C24×C3—51no—
91C24×C3—101no—
92C23×D3C3×D511yescharacteristic, complemented
93C2×C3×D4—1512no—
94C22×C3×C4—151no—
95C2×#67—512no—
96C22×Dic3—51no—
97C32×C5C22×D411yescharacteristic, complemented, Hall
98C23×C5—36no—
99C23×C5—64no—
100C23×C5C2×C3×D311yescharacteristic
101C22×D5—24no—
102C22×D5C2×C3×D316yescomplemented
103C22⋊D5—316no—
104C2×Dic5—36no—
105C22×C32—54no—
106C22×C32—1012no—
107C22×C32C22⋊D514yescomplemented
108C22×C32C22×D513yes—
109C2×C3×D3—212no—
110C2×C3×D3C22×D514yescomplemented
111C3×Dic3—54no—
112C22×D4—151noSylow
113C2×C3×C5—21no—
114C2×C3×C5—26no—
115C2×C3×C5—64no—
116C2×C3×C5C2×#6716yescomplemented
117C2×C3×C5C2×C3×D416yescomplemented
118C2×C3×C5C23×D311yescharacteristic
119C2×C3×C5C24×C311yescharacteristic
120C3×D5—28no—
121C3×D5—24no—
122C3×D5—24no—
123C5×D3—24no—
124C23×C3—21no—
125C23×C3—36no—
126C23×C3—56no—
127C23×C3—56no—
128C23×C3—64no—
129C23×C3—106no—
130C23×C3—108no—
131C23×C3—104no—
132C23×C3—104no—
133C23×C3D3×D511yescharacteristic
134C23×C3C2×C3×D511yescharacteristic
135C22×D3—24no—
136C22×D3C2×C3×D516yescomplemented
137C3×D4—1516no—
138C2×C3×C4—156no—
139#67—516no—
140C2×Dic3—56no—
141C22×C5—34no—
142C22×C5—612no—
143C22×C5C3×#6714yescomplemented
144C22×C5C22×C3×D313yes—
145C2×D5—212no—
146C2×D5C22×C3×D314yescomplemented
147Dic5—34no—
148C2×C32—104no—
149C2×C32C2×C22⋊D516yescomplemented
150C2×C32C23×D511yescharacteristic
151C3×D3—24no—
152C24—31no—
153C24—51no—
154C2×D4—1512no—
155C22×C4—151no—
156C3×C5—21no—
157C3×C5C22×#6711yescharacteristic, complemented
158C3×C5C22×C3×D411yescharacteristic, complemented
159C22×C3—24no—
160C22×C3—23no—
161C22×C3—34no—
162C22×C3—54no—
163C22×C3—54no—
164C22×C3—612no—
165C22×C3—104no—
166C22×C3—1024no—
167C22×C3—1012no—
168C22×C3—1012no—
169C22×C3#120114yesdirect product factor
170C22×C3C3×C22⋊D514yescomplemented
171C22×C3C2×D3×D513yes—
172C22×C3C22×C3×D513yes—
173C2×D3—212no—
174C2×D3C22×C3×D514yescomplemented
175C3×C4—154no—
176Dic3—54no—
177C2×C5—64no—
178C2×C5C2×C3×#6716yescomplemented
179C2×C5C23×C3×D311yescharacteristic
180D5—24no—
181C32C22×C22⋊D511yescharacteristic, complemented, Sylow
182C23—36no—
183C23—56no—
184C23—64no—
185C23—104no—
186C23C3×D3×D511yescharacteristic
187D4—1516no—
188C2×C4—156no—
189C2×C3—21no—
190C2×C3—26no—
191C2×C3—64no—
192C2×C3—108no—
193C2×C3—104no—
194C2×C3—104no—
195C2×C3C2×#120116yesdirect product factor
196C2×C3C2×C3×C22⋊D516yescomplemented
197C2×C3C22×D3×D511yescharacteristic
198C2×C3C23×C3×D511yescharacteristic
199D3—24no—
200C5C22×C3×#6711yescharacteristic, complemented, Sylow
201C22—34no—
202C22—54no—
203C22—612no—
204C22—1012no—
205C22C3×#120114yesdirect product factor
206C22C2×C3×D3×D513yes—
207C4—154no—
208C3—21no—
209C3C22×#120111yescharacteristic, direct product factor
210C3C22×C3×C22⋊D511yescharacteristic, complemented
211C2—64no—
212C2—104no—
213C2C2×C3×#120116yesdirect product factor
214C2C22×C3×D3×D511yescharacteristic
subgroupG/H
center, hypercenterC23×C3D3×D5
commutator, cosocleC2×C3×C5C24×C3
FrattiniC2C22×C3×D3×D5
Fitting, socleC23×C32×C5C22
hypocenterC3×C5C22×C3×D4
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×C5×D3C2semidirect
2. C23×C32×D5C2semidirect
3. C22×C3×Dic15C2semidirect
4. C2×C3×#1201C2direct (N × H)
6. C22×#1201C3direct (N × H)
9. C22×C3×C5×D3C22semidirect
11. C22×C32×D5C22semidirect
12. C2×C3×Dic15C22semidirect
13. C3×#1201C22direct (N × H)
16. C23×C5×D3C2×C3semidirect
18. C23×C3×D5D3semidirect
19. C23×C3×D5C2×C3semidirect
20. C22×Dic15C2×C3semidirect
23. C2×#1201C2×C3direct (N × H)
24. C22×C32×C5D4semidirect
27. C2×C3×C5×D3C23semidirect
29. C2×C32×D5C23semidirect
30. C3×Dic15C23semidirect
33. C23×C3×D3D5semidirect
42. C22×C5×D3C22×C3semidirect
47. C22×C3×D5C2×D3semidirect
48. C22×C3×D5C22×C3semidirect
49. C2×Dic15C22×C3semidirect
52. #1201C22×C3direct (N × H)
55. C2×C32×C5C2×D4semidirect
60. C23×D5C3×D3semidirect
67. C22×C3×D3C2×D5semidirect
74. C22×C3×C5#67semidirect
75. C22×C3×C5C3×D4semidirect
79. C2×C5×D3C23×C3semidirect
84. C2×C3×D5C22×D3semidirect
85. C2×C3×D5C23×C3semidirect
86. Dic15C23×C3semidirect
92. C23×D3C3×D5semidirect
97. C32×C5C22×D4semidirect
102. C22×D5C2×C3×D3semidirect
107. C22×C32C22⋊D5semidirect
110. C2×C3×D3C22×D5semidirect
116. C2×C3×C5C2×#67semidirect
117. C2×C3×C5C2×C3×D4semidirect
136. C22×D3C2×C3×D5semidirect
143. C22×C5C3×#67semidirect
146. C2×D5C22×C3×D3semidirect
149. C2×C32C2×C22⋊D5semidirect
157. C3×C5C22×#67semidirect
158. C3×C5C22×C3×D4semidirect
169. C22×C3#1201direct (N × H)
170. C22×C3C3×C22⋊D5semidirect
174. C2×D3C22×C3×D5semidirect
178. C2×C5C2×C3×#67semidirect
181. C32C22×C22⋊D5semidirect
195. C2×C3C2×#1201direct (N × H)
196. C2×C3C2×C3×C22⋊D5semidirect
200. C5C22×C3×#67semidirect
205. C22C3×#1201direct (N × H)
209. C3C22×#1201direct (N × H)
210. C3C22×C3×C22⋊D5semidirect
213. C2C2×C3×#1201direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C5×D32
C23×C32×D52
C22×C3×Dic152
C2×C3×#12012
C22×C3×C22⋊D53
C22×#12013
C22×C3×#675