Finite Group Databasebeta

← C2×C5×D3×Dic3 #10624183

Subgroups — C2×C5×D3×Dic3 #10624183

Order 720.

232 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C22×C3×C5×D3C211yesmaximal, characteristic
2C2×C5×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3×C5×Dic3C211yesmaximal, characteristic, complemented
4C5×D3×Dic3C214yesmaximal, direct product factor
5C22×C5×Dic3—31nomaximal
6C2×C4×C5×D3—31nomaximal
7C22×C32×C5C2211yescharacteristic
8C2×C3×C5×D3C414yescomplemented
9C2×C3×C5×D3C2212yes—
10C5×C3⋊Dic3C2212yescomplemented
11C3×C5×Dic3C2212yescomplemented
12C2×D3×Dic3C511yesmaximal, characteristic, direct product factor, Hall
13C23×C3×C5—31no—
14C22×C5×D3D311yescharacteristic
15C2×C3×C4×C5—31no—
16C2×C5×Dic3—31no—
17C2×C5×Dic3—31no—
18C2×C5×Dic3—34no—
19C2×C5×Dic3—61no—
20C2×C5×Dic3D311yescharacteristic, direct product factor
21C4×C5×D3—34no—
22C2×C32×C5C2×C412yescomplemented
23C2×C32×C5C2311yescharacteristic
24C3×C5×D3C2×C414yescomplemented
25C22×C4×C5—91noHall
26C22×C3×D3C2×C511yescharacteristic
27C2×C3⋊Dic3C2×C511yescharacteristic, complemented
28C2×C3×Dic3C2×C511yescharacteristic, complemented
29D3×Dic3C2×C514yesdirect product factor
30C22×C3×C5—21no—
31C22×C3×C5—34no—
32C22×C3×C5—32no—
33C22×C3×C5C2×D311yescharacteristic
34C22×C3×C5C2×D311yescharacteristic
35C2×C5×D3Dic314yesdirect product factor
36C2×C5×D3C2×D312yes—
37C3×C4×C5—32no—
38C5×Dic3—32no—
39C5×Dic3—32no—
40C5×Dic3—62no—
41C5×Dic3C2×D312yesdirect product factor
42C22×Dic3—31no—
43C2×C4×D3—31no—
44C32×C5C22×C411yescharacteristic, complemented, Hall
45C23×C5—31no—
46C2×C4×C5—31no—
47C2×C4×C5—94no—
48C2×C4×C5—91no—
49C22×C32C22×C511yescharacteristic
50C2×C3×D3C4×C514yescomplemented
51C2×C3×D3C22×C512yes—
52C3⋊Dic3C22×C512yescomplemented
53C3×Dic3C22×C512yescomplemented
54C2×C3×C5—22no—
55C2×C3×C5—21no—
56C2×C3×C5—34no—
57C2×C3×C5C4×D312yescomplemented
58C2×C3×C5C2×Dic312yescomplemented
59C2×C3×C5C22×D311yescharacteristic
60C2×C3×C5C22×D311yescharacteristic
61C5×D3C2×Dic314yesdirect product factor
62C23×C3—31no—
63C22×D3C5×D311yescharacteristic
64C2×C3×C4—31no—
65C2×Dic3—31no—
66C2×Dic3—31no—
67C2×Dic3—34no—
68C2×Dic3—61no—
69C2×Dic3C5×D311yescharacteristic, direct product factor
70C4×D3—34no—
71C22×C5—34no—
72C22×C5—32no—
73C22×C5D3211yescharacteristic
74C4×C5—32no—
75C4×C5—92no—
76C2×C32C2×C4×C512yescomplemented
77C2×C32C23×C511yescharacteristic
78C3×D3C2×C4×C514yescomplemented
79C22×C4—91noSylow
80C3×C5—21no—
81C3×C5C2×C4×D311yescharacteristic, complemented
82C3×C5C22×Dic311yescharacteristic, complemented
83C22×C3—21no—
84C22×C3—34no—
85C22×C3—32no—
86C22×C3C2×C5×D311yescharacteristic
87C22×C3C2×C5×D311yescharacteristic
88C2×D3C5×Dic314yesdirect product factor
89C2×D3C2×C5×D312yes—
90C3×C4—32no—
91Dic3—32no—
92Dic3—32no—
93Dic3—62no—
94Dic3C2×C5×D312yesdirect product factor
95C2×C5—34no—
96C2×C5D3×Dic312yesdirect product factor
97C2×C5C2×D3211yescharacteristic
98C32C22×C4×C511yescharacteristic, complemented, Sylow
99C23—31no—
100C2×C4—31no—
101C2×C4—94no—
102C2×C4—91no—
103C2×C3—22no—
104C2×C3—21no—
105C2×C3—34no—
106C2×C3C4×C5×D312yescomplemented
107C2×C3C2×C5×Dic312yescomplemented
108C2×C3C22×C5×D311yescharacteristic
109C2×C3C22×C5×D311yescharacteristic
110D3C2×C5×Dic314yesdirect product factor
111C5C2×D3×Dic311yescharacteristic, direct product factor, Sylow
112C22—34no—
113C22—32no—
114C22C5×D3211yescharacteristic
115C4—32no—
116C4—92no—
117C3—21no—
118C3C2×C4×C5×D311yescharacteristic, complemented
119C3C22×C5×Dic311yescharacteristic, complemented
120C2—34no—
121C2C5×D3×Dic312yesdirect product factor
122C2C2×C5×D3211yescharacteristic
subgroupG/H
center, hypercenterC22×C5D32
commutator, hypocenterC32C22×C4×C5
FrattiniC2C2×C5×D32
Fitting, socleC22×C32×C5C22
prime radicalC22×C3×C5×D3C2
cosocleC2×C32C23×C5
LCSG ▷ C32
UCSC1 ◁ C22×C5

Complements

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

NH ≅ G/Ntype
2. C2×C5×C3⋊Dic3C2semidirect
3. C2×C3×C5×Dic3C2semidirect
4. C5×D3×Dic3C2direct (N × H)
8. C2×C3×C5×D3C4semidirect
10. C5×C3⋊Dic3C22semidirect
11. C3×C5×Dic3C22semidirect
12. C2×D3×Dic3C5direct (N × H)
20. C2×C5×Dic3D3direct (N × H)
22. C2×C32×C5C2×C4semidirect
24. C3×C5×D3C2×C4semidirect
27. C2×C3⋊Dic3C2×C5semidirect
28. C2×C3×Dic3C2×C5semidirect
29. D3×Dic3C2×C5direct (N × H)
35. C2×C5×D3Dic3direct (N × H)
41. C5×Dic3C2×D3direct (N × H)
44. C32×C5C22×C4semidirect
50. C2×C3×D3C4×C5semidirect
52. C3⋊Dic3C22×C5semidirect
53. C3×Dic3C22×C5semidirect
57. C2×C3×C5C4×D3semidirect
58. C2×C3×C5C2×Dic3semidirect
61. C5×D3C2×Dic3direct (N × H)
69. C2×Dic3C5×D3direct (N × H)
76. C2×C32C2×C4×C5semidirect
78. C3×D3C2×C4×C5semidirect
81. C3×C5C2×C4×D3semidirect
82. C3×C5C22×Dic3semidirect
88. C2×D3C5×Dic3direct (N × H)
94. Dic3C2×C5×D3direct (N × H)
96. C2×C5D3×Dic3direct (N × H)
98. C32C22×C4×C5semidirect
106. C2×C3C4×C5×D3semidirect
107. C2×C3C2×C5×Dic3semidirect
110. D3C2×C5×Dic3direct (N × H)
111. C5C2×D3×Dic3direct (N × H)
118. C3C2×C4×C5×D3semidirect
119. C3C22×C5×Dic3semidirect
121. C2C5×D3×Dic3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×D32
C2×C5×C3⋊Dic32
C2×C3×C5×Dic32
C5×D3×Dic32
C22×C5×Dic33
C2×C4×C5×D33
C2×D3×Dic35