Finite Group Databasebeta

← C22×C3×D5×Dic3 #13089240

Subgroups — C22×C3×D5×Dic3 #13089240

Order 1440.

1078 conjugacy classes of subgroups; only the 456 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 456 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×C32×D5C211yesmaximal, characteristic
2C22×C3×Dic15C211yesmaximal, characteristic, complemented
3C22×C3×C5×Dic3C211yesmaximal, characteristic, complemented
4C2×C3×D5×Dic3C2112yesmaximal, direct product factor
5C22×C3×C4×D5—31nomaximal
6C22×D5×Dic3C311yesmaximal, characteristic, direct product factor
7C23×C32×C5C2211yescharacteristic
8C22×C32×D5C418yescomplemented
9C22×C32×D5C2216yes—
10C2×C3×Dic15C2216yescomplemented
11C2×C3×C5×Dic3C2216yescomplemented
12C3×D5×Dic3C22116yesdirect product factor
13C23×C3×Dic3—51nomaximal, Hall
14C23×C3×D5—21no—
15C23×C3×D5D311yescharacteristic
16C23×C3×D5C2×C311yescharacteristic
17C22×C3×C4×C5—31no—
18C22×Dic15C2×C311yescharacteristic, complemented
19C22×C5×Dic3C2×C311yescharacteristic, complemented
20C22×C3×Dic5—31no—
21C2×C3×C4×D5—312no—
22C2×D5×Dic3C2×C3112yesdirect product factor
23C22×C32×C5C2×C414yescomplemented
24C22×C32×C5C2313yes—
25C2×C32×D5C2×C4124yescomplemented
26C2×C32×D5C2314yes—
27C3×Dic15C2314yescomplemented
28C3×C5×Dic3C2314yescomplemented
29C22×C4×D5—31noHall
30C24×C32—51no—
31C22×C3×Dic3—51no—
32C22×C3×Dic3—512no—
33C22×C3×Dic3D511yescharacteristic, direct product factor
34C23×C3×C5—21no—
35C23×C3×C5C2×D311yescharacteristic
36C23×C3×C5C22×C311yescharacteristic
37C22×C3×D5—28no—
38C22×C3×D5—26no—
39C22×C3×D5Dic318yesdirect product factor
40C22×C3×D5C3×C418yescomplemented
41C22×C3×D5C2×D316yes—
42C22×C3×D5C22×C316yes—
43C2×C3×C4×C5—36no—
44C2×Dic15C22×C316yescomplemented
45C2×C5×Dic3C22×C316yescomplemented
46C2×C3×Dic5—36no—
47C3×C4×D5—316no—
48D5×Dic3C22×C3116yesdirect product factor
49C23×C3×C4—151no—
50C23×Dic3—51no—
51C2×C32×C5C22×C416yescomplemented
52C2×C32×C5C2411yescharacteristic
53C32×D5C22×C418yescomplemented
54C23×D5C3×D311yescharacteristic
55C22×C4×C5—31no—
56C22×Dic5—31no—
57C2×C4×D5—312no—
58C23×C32—58no—
59C23×C32—56no—
60C23×C32C2×D511yescharacteristic
61C2×C3×Dic3—56no—
62C2×C3×Dic3—516no—
63C2×C3×Dic3C2×D516yesdirect product factor
64C22×C3×C5—24no—
65C22×C3×C5—23no—
66C22×C3×C5C2×Dic314yescomplemented
67C22×C3×C5C2×C3×C414yescomplemented
68C22×C3×C5C22×D313yes—
69C22×C3×C5C23×C313yes—
70C2×C3×D5—224no—
71C2×C3×D5—24no—
72C2×C3×D5C2×Dic3124yesdirect product factor
73C2×C3×D5C2×C3×C4124yescomplemented
74C2×C3×D5C22×D314yes—
75C2×C3×D5C23×C314yes—
76C3×C4×C5—34no—
77Dic15C23×C314yescomplemented
78C3×Dic5—34no—
79C5×Dic3C23×C314yescomplemented
80C24×C3—51no—
81C24×C3—51no—
82C24×C3—101no—
83C22×C3×C4—31no—
84C22×C3×C4—1512no—
85C22×C3×C4—151no—
86C22×Dic3—51no—
87C22×Dic3—512no—
88C22×Dic3C3×D511yescharacteristic, direct product factor
89C32×C5C23×C411yescharacteristic, complemented, Hall
90C23×C5C2×C3×D311yescharacteristic
91C22×D5C3×Dic318yesdirect product factor
92C22×D5C2×C3×D316yes—
93C2×C4×C5—36no—
94C2×Dic5—36no—
95C4×D5—316no—
96C22×C32—524no—
97C22×C32—54no—
98C22×C32C4×D514yescomplemented
99C22×C32C22×D513yes—
100C3×Dic3—54no—
101C3×Dic3C22×D514yesdirect product factor
102C23×C4—151noSylow
103C2×C3×C5—21no—
104C2×C3×C5—26no—
105C2×C3×C5C22×Dic316yescomplemented
106C2×C3×C5C22×C3×C416yescomplemented
107C2×C3×C5C23×D311yescharacteristic
108C2×C3×C5C24×C311yescharacteristic
109C3×D5—28no—
110C3×D5C22×Dic318yesdirect product factor
111C3×D5C22×C3×C418yescomplemented
112C23×C3—21no—
113C23×C3—58no—
114C23×C3—58no—
115C23×C3—56no—
116C23×C3—56no—
117C23×C3—108no—
118C23×C3—106no—
119C23×C3D3×D511yescharacteristic
120C23×C3C2×C3×D511yescharacteristic
121C2×C3×C4—36no—
122C2×C3×C4—1516no—
123C2×C3×C4—156no—
124C2×Dic3—56no—
125C2×Dic3—516no—
126C2×Dic3C2×C3×D516yesdirect product factor
127C22×C5C2×C3×Dic314yescomplemented
128C22×C5C22×C3×D313yes—
129C2×D5C2×C3×Dic3124yesdirect product factor
130C2×D5C22×C3×D314yes—
131C4×C5—34no—
132Dic5—34no—
133C2×C32—58no—
134C2×C32C2×C4×D516yescomplemented
135C2×C32C23×D511yescharacteristic
136C24—51no—
137C22×C4—31no—
138C22×C4—1512no—
139C22×C4—151no—
140C3×C5—21no—
141C3×C5C23×Dic311yescharacteristic, complemented
142C3×C5C23×C3×C411yescharacteristic, complemented
143C22×C3—24no—
144C22×C3—23no—
145C22×C3—524no—
146C22×C3—524no—
147C22×C3—54no—
148C22×C3—54no—
149C22×C3—104no—
150C22×C3—1024no—
151C22×C3D5×Dic314yesdirect product factor
152C22×C3C3×C4×D514yescomplemented
153C22×C3C2×D3×D513yes—
154C22×C3C22×C3×D513yes—
155C3×C4—34no—
156C3×C4—154no—
157Dic3—54no—
158Dic3C22×C3×D514yesdirect product factor
159C2×C5C22×C3×Dic316yescomplemented
160C2×C5C23×C3×D311yescharacteristic
161D5C22×C3×Dic318yesdirect product factor
162C32C22×C4×D511yescharacteristic, complemented, Sylow
163C23—58no—
164C23—56no—
165C23C3×D3×D511yescharacteristic
166C2×C4—36no—
167C2×C4—1516no—
168C2×C4—156no—
169C2×C3—21no—
170C2×C3—26no—
171C2×C3—58no—
172C2×C3—58no—
173C2×C3—108no—
174C2×C3C2×D5×Dic316yesdirect product factor
175C2×C3C2×C3×C4×D516yescomplemented
176C2×C3C22×D3×D511yescharacteristic
177C2×C3C23×C3×D511yescharacteristic
178C5C23×C3×Dic311yescharacteristic, complemented, Sylow
179C22—54no—
180C22—524no—
181C22C3×D5×Dic314yesdirect product factor
182C22C2×C3×D3×D513yes—
183C4—34no—
184C4—154no—
185C3—21no—
186C3C22×D5×Dic311yescharacteristic, direct product factor
187C3C22×C3×C4×D511yescharacteristic, complemented
188C2—58no—
189C2C2×C3×D5×Dic316yesdirect product factor
190C2C22×C3×D3×D511yescharacteristic
subgroupG/H
center, hypercenterC23×C3D3×D5
commutator, hypocenterC3×C5C23×C3×C4
FrattiniC2C22×C3×D3×D5
Fitting, socleC23×C32×C5C22
prime radicalC23×C32×D5C2
cosocleC2×C3×C5C24×C3
LCSG ▷ C3×C5
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C22×C3×Dic15C2semidirect
3. C22×C3×C5×Dic3C2semidirect
4. C2×C3×D5×Dic3C2direct (N × H)
6. C22×D5×Dic3C3direct (N × H)
8. C22×C32×D5C4semidirect
10. C2×C3×Dic15C22semidirect
11. C2×C3×C5×Dic3C22semidirect
12. C3×D5×Dic3C22direct (N × H)
18. C22×Dic15C2×C3semidirect
19. C22×C5×Dic3C2×C3semidirect
22. C2×D5×Dic3C2×C3direct (N × H)
23. C22×C32×C5C2×C4semidirect
25. C2×C32×D5C2×C4semidirect
27. C3×Dic15C23semidirect
28. C3×C5×Dic3C23semidirect
33. C22×C3×Dic3D5direct (N × H)
39. C22×C3×D5Dic3direct (N × H)
40. C22×C3×D5C3×C4semidirect
44. C2×Dic15C22×C3semidirect
45. C2×C5×Dic3C22×C3semidirect
48. D5×Dic3C22×C3direct (N × H)
51. C2×C32×C5C22×C4semidirect
53. C32×D5C22×C4semidirect
63. C2×C3×Dic3C2×D5direct (N × H)
66. C22×C3×C5C2×Dic3semidirect
67. C22×C3×C5C2×C3×C4semidirect
72. C2×C3×D5C2×Dic3direct (N × H)
73. C2×C3×D5C2×C3×C4semidirect
77. Dic15C23×C3semidirect
79. C5×Dic3C23×C3semidirect
88. C22×Dic3C3×D5direct (N × H)
89. C32×C5C23×C4semidirect
91. C22×D5C3×Dic3direct (N × H)
98. C22×C32C4×D5semidirect
101. C3×Dic3C22×D5direct (N × H)
105. C2×C3×C5C22×Dic3semidirect
106. C2×C3×C5C22×C3×C4semidirect
110. C3×D5C22×Dic3direct (N × H)
111. C3×D5C22×C3×C4semidirect
126. C2×Dic3C2×C3×D5direct (N × H)
127. C22×C5C2×C3×Dic3semidirect
129. C2×D5C2×C3×Dic3direct (N × H)
134. C2×C32C2×C4×D5semidirect
141. C3×C5C23×Dic3semidirect
142. C3×C5C23×C3×C4semidirect
151. C22×C3D5×Dic3direct (N × H)
152. C22×C3C3×C4×D5semidirect
158. Dic3C22×C3×D5direct (N × H)
159. C2×C5C22×C3×Dic3semidirect
161. D5C22×C3×Dic3direct (N × H)
162. C32C22×C4×D5semidirect
174. C2×C3C2×D5×Dic3direct (N × H)
175. C2×C3C2×C3×C4×D5semidirect
178. C5C23×C3×Dic3semidirect
181. C22C3×D5×Dic3direct (N × H)
186. C3C22×D5×Dic3direct (N × H)
187. C3C22×C3×C4×D5semidirect
189. C2C2×C3×D5×Dic3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×D52
C22×C3×Dic152
C22×C3×C5×Dic32
C2×C3×D5×Dic32
C22×C3×C4×D53
C22×D5×Dic33
C23×C3×Dic35