Finite Group Databasebeta

← C2×C9×C5⋊(D31⋊C5) #42124516

Subgroups — C2×C9×C5⋊(D31⋊C5) #42124516

Order 27900.

144 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C2×C5×C9×ΓL(1,𝔽32)C211yesmaximal, characteristic, complemented
2C9×C5⋊(D31⋊C5)C212yesmaximal, direct product factor
3C2×C3×C5⋊(D31⋊C5)C311yesmaximal, characteristic
4C5×C9×ΓL(1,𝔽32)C2211yescharacteristic, complemented, Hall
5C2×C9×D155C511yesmaximal, characteristic, complemented
6C2×C9×D31⋊C5—51nomaximal
7C2×C3×C5×ΓL(1,𝔽32)C2×C311yescharacteristic
8C3×C5⋊(D31⋊C5)C2×C312yes—
9C2×C5⋊(D31⋊C5)C911yescharacteristic, direct product factor, Hall
10C9×D155C2×C512yescomplemented
11C2×C9×ΓL(1,𝔽32)—22no—
12C2×C9×ΓL(1,𝔽32)D511yescharacteristic, complemented
13C9×D31⋊C5—52no—
14C2×C5×C9×C31C2×C511yescharacteristic, complemented
15C3×C5×ΓL(1,𝔽32)C22×C311yescharacteristic
16C2×C3×D155C3×C511yescharacteristic
17C2×C3×D31⋊C5—51no—
18C2×C5×ΓL(1,𝔽32)C2×C911yescharacteristic, complemented
19C5⋊(D31⋊C5)C2×C912yesdirect product factor
20C5×C9×C31C22×C511yescharacteristic, complemented
21C9×ΓL(1,𝔽32)—22no—
22C9×ΓL(1,𝔽32)C2×D511yescharacteristic, complemented
23C2×C9×D31—51noHall
24C2×C3×C5×C31C2×C3×C511yescharacteristic
25C3×D155C2×C3×C512yes—
26C2×C3×ΓL(1,𝔽32)—22no—
27C2×C3×ΓL(1,𝔽32)C3×D511yescharacteristic
28C3×D31⋊C5—52no—
29C2×C5×C9×D5—311nomaximal, Hall
30C5×ΓL(1,𝔽32)C22×C911yescharacteristic, complemented, Hall
31C2×D155C5×C911yescharacteristic, complemented
32C2×D31⋊C5—51no—
33C2×C9×C31C5×D511yescharacteristic, complemented
34C9×D31—52no—
35C3×C5×C31C22×C3×C511yescharacteristic
36C3×ΓL(1,𝔽32)—22no—
37C3×ΓL(1,𝔽32)C2×C3×D511yescharacteristic
38C2×C52×C9—311no—
39C5×C9×D5—312no—
40C2×C3×D31—51no—
41C2×C5×C31C2×C5×C911yescharacteristic, complemented
42D155C2×C5×C912yescomplemented
43C2×ΓL(1,𝔽32)—22no—
44C2×ΓL(1,𝔽32)C9×D511yescharacteristic, complemented
45D31⋊C5—52no—
46C2×C3×C5×D5—311no—
47C9×C31C2×C5×D511yescharacteristic, complemented, Hall
48C52×C9—311noHall
49C2×C3×C31C3×C5×D511yescharacteristic
50C3×D31—52no—
51C22×C5×C9—1551no—
52C2×C9×D5—311no—
53C5×C31C22×C5×C911yescharacteristic, complemented
54ΓL(1,𝔽32)—22no—
55ΓL(1,𝔽32)C2×C9×D511yescharacteristic, complemented
56C2×C3×C52—311no—
57C3×C5×D5—312no—
58C2×D31—51noHall
59C2×C5×D5—311noHall
60C3×C31C2×C3×C5×D511yescharacteristic
61C2×C5×C9—311no—
62C2×C5×C9—622no—
63C2×C5×C9—1552no—
64C2×C5×C9D31⋊C511yescharacteristic, complemented
65C9×D5—312no—
66C3×C52—311no—
67C2×C31C5×C9×D511yescharacteristic, complemented
68D31—52no—
69C22×C3×C5—1551no—
70C2×C3×D5—311no—
71C2×C52—311no—
72C5×D5—312no—
73C5×C9—311no—
74C5×C9—622no—
75C5×C9C2×D31⋊C511yescharacteristic, complemented
76C22×C9—1551noHall
77C31C2×C5×C9×D511yescharacteristic, complemented, Sylow
78C2×C3×C5—311no—
79C2×C3×C5—622no—
80C2×C3×C5—1552no—
81C2×C3×C5C3×D31⋊C511yescharacteristic
82C3×D5—312no—
83C52—311noSylow
84C22×C5—1551no—
85C2×D5—311no—
86C2×C9—1552no—
87C2×C9C5⋊(D31⋊C5)11yescharacteristic, direct product factor
88C3×C5—311no—
89C3×C5—622no—
90C3×C5C2×C3×D31⋊C511yescharacteristic
91C22×C3—1551no—
92C2×C5—311no—
93C2×C5—622no—
94C2×C5—1552no—
95C2×C5C9×D31⋊C511yescharacteristic, complemented
96D5—312no—
97C9C2×C5⋊(D31⋊C5)11yescharacteristic, direct product factor, Sylow
98C2×C3—1552no—
99C2×C3C3×C5⋊(D31⋊C5)11yescharacteristic
100C5—311no—
101C5—622no—
102C5C2×C9×D31⋊C511yescharacteristic, complemented
103C22—1551noSylow
104C3C2×C3×C5⋊(D31⋊C5)11yescharacteristic
105C2—1552no—
106C2C9×C5⋊(D31⋊C5)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C9C5⋊(D31⋊C5)
commutator, hypocenterC5×C31C22×C5×C9
FrattiniC3C2×C3×C5⋊(D31⋊C5)
FittingC2×C5×C9×C31C2×C5
socleC2×C3×C5×C31C2×C3×C5
prime radicalC2×C3×C5⋊(D31⋊C5)C3
cosocleC3×C5×C31C22×C3×C5
LCSG ▷ C5×C31
UCSC1 ◁ C2×C9

Complements

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

NH ≅ G/Ntype
1. C2×C5×C9×ΓL(1,𝔽32)C2semidirect
2. C9×C5⋊(D31⋊C5)C2direct (N × H)
4. C5×C9×ΓL(1,𝔽32)C22semidirect
5. C2×C9×D155C5semidirect
9. C2×C5⋊(D31⋊C5)C9direct (N × H)
10. C9×D155C2×C5semidirect
12. C2×C9×ΓL(1,𝔽32)D5semidirect
14. C2×C5×C9×C31C2×C5semidirect
18. C2×C5×ΓL(1,𝔽32)C2×C9semidirect
19. C5⋊(D31⋊C5)C2×C9direct (N × H)
20. C5×C9×C31C22×C5semidirect
22. C9×ΓL(1,𝔽32)C2×D5semidirect
30. C5×ΓL(1,𝔽32)C22×C9semidirect
31. C2×D155C5×C9semidirect
33. C2×C9×C31C5×D5semidirect
41. C2×C5×C31C2×C5×C9semidirect
42. D155C2×C5×C9semidirect
44. C2×ΓL(1,𝔽32)C9×D5semidirect
47. C9×C31C2×C5×D5semidirect
53. C5×C31C22×C5×C9semidirect
55. ΓL(1,𝔽32)C2×C9×D5semidirect
64. C2×C5×C9D31⋊C5semidirect
67. C2×C31C5×C9×D5semidirect
75. C5×C9C2×D31⋊C5semidirect
77. C31C2×C5×C9×D5semidirect
87. C2×C9C5⋊(D31⋊C5)direct (N × H)
95. C2×C5C9×D31⋊C5semidirect
97. C9C2×C5⋊(D31⋊C5)direct (N × H)
102. C5C2×C9×D31⋊C5semidirect
106. C2C9×C5⋊(D31⋊C5)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×C9×ΓL(1,𝔽32)2
C9×C5⋊(D31⋊C5)2
C2×C3×C5⋊(D31⋊C5)3
C2×C9×D1555
C2×C9×D31⋊C55
C2×C5×C9×D531