Finite Group Databasebeta

← C2×C32×C4⋊(C5⋊D5) #72230639

Subgroups — C2×C32×C4⋊(C5⋊D5) #72230639

Order 3600.

1296 conjugacy classes of subgroups; only the 450 normal subgroups are shown.

normal subgroups only

This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 450 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×C32×C5⋊D5C212yesmaximal, complemented
2C2×C32×C4×C52C211yesmaximal, characteristic, complemented
3C32×C4⋊(C5⋊D5)C214yesmaximal, direct product factor
4C2×C3×C4⋊(C5⋊D5)C314yesmaximal, direct product factor
5C22×C32×C52C2211yescharacteristic
6C2×C32×C5⋊D5—24no—
7C2×C32×C5⋊D5C2214yescomplemented
8C32×C4×C52C2212yescomplemented
9C2×C32×D20—56nomaximal
10C22×C3×C5⋊D5C2×C318yescomplemented
11C2×C3×C4×C52C2×C314yescomplemented
12C3×C4⋊(C5⋊D5)C2×C3116yesdirect product factor
13C2×C32×C52D412yescomplemented
14C2×C32×C52C2311yescharacteristic
15C32×C5⋊D5—24no—
16C2×C4⋊(C5⋊D5)C3211yescharacteristic, direct product factor, Hall
17C22×C32×D5—512no—
18C2×C32×C4×C5D516yescomplemented
19C32×D20—524no—
20C22×C3×C52C22×C314yes—
21C2×C3×C5⋊D5—216no—
22C2×C3×C5⋊D5C22×C3116yescomplemented
23C3×C4×C52C22×C318yescomplemented
24C2×C3×D20—524no—
25C32×C52C2×D411yescharacteristic, complemented, Hall
26C22×C5⋊D5C2×C3212yescomplemented
27C2×C4×C52C2×C3211yescharacteristic, complemented
28C4⋊(C5⋊D5)C2×C3214yesdirect product factor
29C22×C32×C5C2×D516yes—
30C2×C32×D5—524no—
31C2×C32×D5—1024no—
32C32×C4×C5C2×D5112yescomplemented
33C2×C3×C52C3×D418yescomplemented
34C2×C3×C52C23×C314yes—
35C3×C5⋊D5—216no—
36C2×C32×D4—251noHall
37C22×C3×D5—548no—
38C2×C3×C4×C5C3×D5124yescomplemented
39C3×D20—596no—
40C22×C52C22×C3211yescharacteristic
41C2×C5⋊D5—24no—
42C2×C5⋊D5C22×C3214yescomplemented
43C4×C52C22×C3212yescomplemented
44C2×C32×C5D20112yescomplemented
45C2×C32×C5C22×D516yes—
46C32×D5—1024no—
47C2×D20—56no—
48C3×C52C2×C3×D414yescomplemented
49C23×C32—252no—
50C32×D4—254no—
51C2×C32×C4C5⋊D511yescharacteristic, complemented
52C22×C3×C5C2×C3×D5124yes—
53C2×C3×D5—596no—
54C2×C3×D5—1096no—
55C3×C4×C5C2×C3×D5148yescomplemented
56C2×C52C32×D412yescomplemented
57C2×C52C23×C3211yescharacteristic
58C5⋊D5—24no—
59C2×C3×D4—254no—
60C32×C5C2×D2016yescomplemented
61C22×D5—512no—
62C2×C4×C5C32×D516yescomplemented
63D20—524no—
64C22×C32—254no—
65C22×C32—504no—
66C22×C32C2×C5⋊D511yescharacteristic
67C32×C4C2×C5⋊D512yescomplemented
68C2×C3×C5C3×D20148yescomplemented
69C2×C3×C5C22×C3×D5124yes—
70C3×D5—1096no—
71C52C2×C32×D411yescharacteristic, complemented, Sylow
72C23×C3—258no—
73C3×D4—2516no—
74C2×C3×C4C3×C5⋊D514yescomplemented
75C22×C5C2×C32×D516yes—
76C2×D5—524no—
77C2×D5—1024no—
78C4×C5C2×C32×D5112yescomplemented
79C2×C32—504no—
80C2×C32C4⋊(C5⋊D5)12yesdirect product factor
81C2×C32C22×C5⋊D511yescharacteristic
82C2×D4—251noSylow
83C3×C5C2×C3×D20124yescomplemented
84C22×C3—2516no—
85C22×C3—5016no—
86C22×C3C2×C3×C5⋊D514yes—
87C3×C4C2×C3×C5⋊D518yescomplemented
88C2×C5C32×D20112yescomplemented
89C2×C5C22×C32×D516yes—
90D5—1024no—
91C32C2×C4⋊(C5⋊D5)11yescharacteristic, direct product factor, Sylow
92C23—252no—
93D4—254no—
94C2×C4C32×C5⋊D511yescharacteristic, complemented
95C2×C3—5016no—
96C2×C3C3×C4⋊(C5⋊D5)18yesdirect product factor
97C2×C3C22×C3×C5⋊D514yes—
98C5C2×C32×D2016yescomplemented
99C22—254no—
100C22—504no—
101C22C2×C32×C5⋊D511yescharacteristic
102C4C2×C32×C5⋊D512yescomplemented
103C3C2×C3×C4⋊(C5⋊D5)14yesdirect product factor
104C2—504no—
105C2C32×C4⋊(C5⋊D5)12yesdirect product factor
106C2C22×C32×C5⋊D511yescharacteristic
subgroupG/H
centerC22×C32C2×C5⋊D5
commutator, cosocleC2×C52C23×C32
FrattiniC2C22×C32×C5⋊D5
FittingC2×C32×C4×C52C2
socleC22×C32×C52C22
hypercenterC2×C32×C4C5⋊D5
hypocenterC52C2×C32×D4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C22×C32 ◁ C2×C32×C4

Complements

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

NH ≅ G/Ntype
1. C22×C32×C5⋊D5C2semidirect
2. C2×C32×C4×C52C2semidirect
3. C32×C4⋊(C5⋊D5)C2direct (N × H)
4. C2×C3×C4⋊(C5⋊D5)C3direct (N × H)
7. C2×C32×C5⋊D5C22semidirect
8. C32×C4×C52C22semidirect
10. C22×C3×C5⋊D5C2×C3semidirect
11. C2×C3×C4×C52C2×C3semidirect
12. C3×C4⋊(C5⋊D5)C2×C3direct (N × H)
13. C2×C32×C52D4semidirect
16. C2×C4⋊(C5⋊D5)C32direct (N × H)
18. C2×C32×C4×C5D5semidirect
22. C2×C3×C5⋊D5C22×C3semidirect
23. C3×C4×C52C22×C3semidirect
25. C32×C52C2×D4semidirect
26. C22×C5⋊D5C2×C32semidirect
27. C2×C4×C52C2×C32semidirect
28. C4⋊(C5⋊D5)C2×C32direct (N × H)
32. C32×C4×C5C2×D5semidirect
33. C2×C3×C52C3×D4semidirect
38. C2×C3×C4×C5C3×D5semidirect
42. C2×C5⋊D5C22×C32semidirect
43. C4×C52C22×C32semidirect
44. C2×C32×C5D20semidirect
48. C3×C52C2×C3×D4semidirect
51. C2×C32×C4C5⋊D5semidirect
55. C3×C4×C5C2×C3×D5semidirect
56. C2×C52C32×D4semidirect
60. C32×C5C2×D20semidirect
62. C2×C4×C5C32×D5semidirect
67. C32×C4C2×C5⋊D5semidirect
68. C2×C3×C5C3×D20semidirect
71. C52C2×C32×D4semidirect
74. C2×C3×C4C3×C5⋊D5semidirect
78. C4×C5C2×C32×D5semidirect
80. C2×C32C4⋊(C5⋊D5)direct (N × H)
83. C3×C5C2×C3×D20semidirect
87. C3×C4C2×C3×C5⋊D5semidirect
88. C2×C5C32×D20semidirect
91. C32C2×C4⋊(C5⋊D5)direct (N × H)
94. C2×C4C32×C5⋊D5semidirect
96. C2×C3C3×C4⋊(C5⋊D5)direct (N × H)
98. C5C2×C32×D20semidirect
102. C4C2×C32×C5⋊D5semidirect
103. C3C2×C3×C4⋊(C5⋊D5)direct (N × H)
105. C2C32×C4⋊(C5⋊D5)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C5⋊D52
C2×C32×C4×C522
C32×C4⋊(C5⋊D5)2
C2×C3×C4⋊(C5⋊D5)3
C2×C32×D205