Finite Group Databasebeta

← C2×C5×D5×C31⋊C9 #42124524

Subgroups — C2×C5×D5×C31⋊C9 #42124524

Order 27900.

144 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 70 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×C52×C31⋊C9C211yesmaximal, characteristic, complemented
2C5×D5×C31⋊C9C212yesmaximal, direct product factor
3C2×C3×C5×D5×C31C311yesmaximal, characteristic
4C52×C31⋊C9C2211yescharacteristic, complemented, Hall
5C22×C5×C31⋊C9—51nomaximal
6C2×D5×C31⋊C9C511yesmaximal, characteristic, direct product factor
7C3×C5×D5×C31C2×C312yes—
8C2×C3×C52×C31C2×C311yescharacteristic
9C2×C5×D5×C31C911yescharacteristic, complemented, Hall
10C2×C5×C31⋊C9—22no—
11C2×C5×C31⋊C9—52no—
12C2×C5×C31⋊C9D511yescharacteristic, direct product factor
13C2×C5×C31⋊C9C2×C511yescharacteristic, complemented
14D5×C31⋊C9C2×C512yesdirect product factor
15C3×C52×C31C22×C311yescharacteristic
16C22×C3×C5×C31—51no—
17C2×C3×D5×C31C3×C511yescharacteristic
18C2×C52×C31C2×C911yescharacteristic, complemented
19C5×D5×C31C2×C912yescomplemented
20C5×C31⋊C9—22no—
21C5×C31⋊C9C2×D511yescharacteristic, direct product factor
22C5×C31⋊C9C22×C511yescharacteristic, complemented
23C22×C31⋊C9—51noHall
24C2×C3×C5×C31—22no—
25C2×C3×C5×C31—52no—
26C2×C3×C5×C31C3×D511yescharacteristic
27C2×C3×C5×C31C2×C3×C511yescharacteristic
28C3×D5×C31C2×C3×C512yes—
29C2×C5×C9×D5—311nomaximal, Hall
30C52×C31C22×C911yescharacteristic, complemented, Hall
31C22×C5×C31—51no—
32C2×D5×C31C5×C911yescharacteristic, complemented
33C2×C31⋊C9—52no—
34C2×C31⋊C9C5×D511yescharacteristic, direct product factor
35C3×C5×C31—22no—
36C3×C5×C31C2×C3×D511yescharacteristic
37C3×C5×C31C22×C3×C511yescharacteristic
38C2×C52×C9—311no—
39C5×C9×D5—312no—
40C22×C3×C31—51no—
41C2×C5×C31—22no—
42C2×C5×C31—52no—
43C2×C5×C31C9×D511yescharacteristic, complemented
44C2×C5×C31C2×C5×C911yescharacteristic, complemented
45D5×C31C2×C5×C912yescomplemented
46C2×C3×C5×D5C31⋊C311yescharacteristic
47C31⋊C9C2×C5×D511yescharacteristic, direct product factor, Hall
48C52×C9—311noHall
49C2×C3×C31—52no—
50C2×C3×C31C3×C5×D511yescharacteristic
51C22×C5×C9—1551no—
52C2×C9×D5—311no—
53C5×C31—22no—
54C5×C31C2×C9×D511yescharacteristic, complemented
55C5×C31C22×C5×C911yescharacteristic, complemented
56C2×C3×C52C2×C31⋊C311yescharacteristic
57C3×C5×D5C2×C31⋊C312yes—
58C22×C31—51noHall
59C2×C5×D5C31⋊C911yescharacteristic, direct product factor, Hall
60C3×C31C2×C3×C5×D511yescharacteristic
61C2×C5×C9—311no—
62C2×C5×C9—311no—
63C2×C5×C9—622no—
64C2×C5×C9—1552no—
65C9×D5—312no—
66C3×C52C22×C31⋊C311yescharacteristic
67C2×C31—52no—
68C2×C31C5×C9×D511yescharacteristic, complemented
69C22×C3×C5—51no—
70C2×C3×D5C5×C31⋊C311yescharacteristic
71C2×C52C2×C31⋊C911yescharacteristic, complemented
72C5×D5C2×C31⋊C912yesdirect product factor
73C5×C9—311no—
74C5×C9—311no—
75C5×C9—622no—
76C22×C9—1551noHall
77C31C2×C5×C9×D511yescharacteristic, complemented, Sylow
78C2×C3×C5—22no—
79C2×C3×C5—52no—
80C2×C3×C5D5×C31⋊C311yescharacteristic
81C2×C3×C5C2×C5×C31⋊C311yescharacteristic
82C3×D5C2×C5×C31⋊C312yes—
83C52C22×C31⋊C911yescharacteristic, complemented, Sylow
84C22×C5—51no—
85C2×D5C5×C31⋊C911yescharacteristic, direct product factor
86C2×C9—311no—
87C2×C9—1552no—
88C3×C5—22no—
89C3×C5C2×D5×C31⋊C311yescharacteristic
90C3×C5C22×C5×C31⋊C311yescharacteristic
91C22×C3—51no—
92C2×C5—22no—
93C2×C5—52no—
94C2×C5D5×C31⋊C911yescharacteristic, direct product factor
95C2×C5C2×C5×C31⋊C911yescharacteristic, complemented
96D5C2×C5×C31⋊C912yesdirect product factor
97C9—311noSylow
98C2×C3—52no—
99C2×C3C5×D5×C31⋊C311yescharacteristic
100C5—22no—
101C5C2×D5×C31⋊C911yescharacteristic, direct product factor
102C5C22×C5×C31⋊C911yescharacteristic, complemented
103C22—51noSylow
104C3C2×C5×D5×C31⋊C311yescharacteristic
105C2—52no—
106C2C5×D5×C31⋊C911yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5D5×C31⋊C3
commutator, hypocenterC5×C31C22×C5×C9
FrattiniC3C2×C5×D5×C31⋊C3
Fitting, socleC2×C3×C52×C31C2×C3
prime radicalC2×C3×C5×D5×C31C3
cosocleC3×C5×C31C22×C3×C5
LCSG ▷ C5×C31
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
1. C2×C52×C31⋊C9C2semidirect
2. C5×D5×C31⋊C9C2direct (N × H)
4. C52×C31⋊C9C22semidirect
6. C2×D5×C31⋊C9C5direct (N × H)
9. C2×C5×D5×C31C9semidirect
12. C2×C5×C31⋊C9D5direct (N × H)
13. C2×C5×C31⋊C9C2×C5semidirect
14. D5×C31⋊C9C2×C5direct (N × H)
18. C2×C52×C31C2×C9semidirect
19. C5×D5×C31C2×C9semidirect
21. C5×C31⋊C9C2×D5direct (N × H)
22. C5×C31⋊C9C22×C5semidirect
30. C52×C31C22×C9semidirect
32. C2×D5×C31C5×C9semidirect
34. C2×C31⋊C9C5×D5direct (N × H)
43. C2×C5×C31C9×D5semidirect
44. C2×C5×C31C2×C5×C9semidirect
45. D5×C31C2×C5×C9semidirect
47. C31⋊C9C2×C5×D5direct (N × H)
54. C5×C31C2×C9×D5semidirect
55. C5×C31C22×C5×C9semidirect
59. C2×C5×D5C31⋊C9direct (N × H)
68. C2×C31C5×C9×D5semidirect
71. C2×C52C2×C31⋊C9semidirect
72. C5×D5C2×C31⋊C9direct (N × H)
77. C31C2×C5×C9×D5semidirect
83. C52C22×C31⋊C9semidirect
85. C2×D5C5×C31⋊C9direct (N × H)
94. C2×C5D5×C31⋊C9direct (N × H)
95. C2×C5C2×C5×C31⋊C9semidirect
96. D5C2×C5×C31⋊C9direct (N × H)
101. C5C2×D5×C31⋊C9direct (N × H)
102. C5C22×C5×C31⋊C9semidirect
106. C2C5×D5×C31⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C52×C31⋊C92
C5×D5×C31⋊C92
C2×C3×C5×D5×C313
C22×C5×C31⋊C95
C2×D5×C31⋊C95
C2×C5×C9×D531