Finite Group Databasebeta

← (C3⋊(D31⋊C3))⋊(C5⋊D5) #42124816

Subgroups — (C3⋊(D31⋊C3))⋊(C5⋊D5) #42124816

Order 27900.

352 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C31⋊C3×C3⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
2C52×C3⋊(D31⋊C3)C211yesmaximal, characteristic, complemented
3C3×C5⋊(C5⋊(D31⋊C3))C211yesmaximal, characteristic, complemented
4D93⋊(C5⋊D5)C311yesmaximal, characteristic, complemented
5C5⋊D5×D31⋊C3—31nomaximal
6C3×C52×C31⋊C3C2211yescharacteristic, complemented, Hall
7D15⋊(D31⋊C3)—56nomaximal
8C31×C3⋊(C5⋊D5)C2×C311yescharacteristic, complemented
9C52×D93C2×C311yescharacteristic, complemented
10C3×C5⋊D155C2×C311yescharacteristic, complemented
11C5⋊D5×C31⋊C3—31no—
12C5⋊(C5⋊(D31⋊C3))—21no—
13C5⋊(C5⋊(D31⋊C3))D311yescharacteristic, complemented
14C52×D31⋊C3—31no—
15C5⋊D5×D31—31noHall
16D15×C31⋊C3—56no—
17C5×C3⋊(D31⋊C3)D516yescomplemented
18C3×C5⋊(D31⋊C3)—56no—
19C52×C31⋊C3—21no—
20C52×C31⋊C3C2×D311yescharacteristic, complemented
21C3×C52×C31C22×C311yescharacteristic, complemented
22D15⋊D31—56no—
23D5×D31⋊C3—156no—
24C5⋊D155C3×D311yescharacteristic, complemented
25C31×C5⋊D5—31no—
26C52×D31—31no—
27C3×C5×C31⋊C3C2×D516yescomplemented
28D3×D31⋊C3—251noHall
29D15×C31—56no—
30C5×D93C3×D516yescomplemented
31C3×D155—56no—
32C5⋊(D31⋊C3)—56no—
33C5⋊(D31⋊C3)—106no—
34D5×C31⋊C3—156no—
35C5×D31⋊C3—36no—
36C3×D3×C5⋊D5—311nomaximal, Hall
37C52×C31C2×C3×D311yescharacteristic, complemented, Hall
38D5×D31—156no—
39C3⋊(D31⋊C3)C5⋊D511yescharacteristic, complemented
40D3×C31⋊C3—251no—
41C3×D31⋊C3—251no—
42C3×C5×C31C2×C3×D516yescomplemented
43C5×C31⋊C3—26no—
44C5×C31⋊C3D3×D516yescomplemented
45C3×C3⋊(C5⋊D5)—311no—
46C3×C52×D3—311no—
47C32×C5⋊D5—311no—
48D3×D31—251no—
49C2×D31⋊C3—751no—
50D155—56no—
51C5×D31—36no—
52D5×C31—156no—
53C2×C3×C5⋊D5—931no—
54D3×C5⋊D5—311no—
55C3×C31⋊C3C2×C5⋊D511yescharacteristic, complemented, Hall
56C32×C52—311noHall
57D93C3×C5⋊D511yescharacteristic, complemented
58C3×D31—251no—
59D3×C31—251no—
60C2×C31⋊C3—751no—
61D31⋊C3—31no—
62D31⋊C3—251no—
63D31⋊C3—501no—
64C3×D3×D5—1556no—
65C5×C31C3×D3×D516yescomplemented
66C2×C3×C52—931no—
67C3⋊(C5⋊D5)D31⋊C311yescharacteristic, complemented
68C52×D3—311no—
69C3×C5⋊D5—311no—
70C3×C5⋊D5—311no—
71C3×C5⋊D5—621no—
72C3×C5⋊D5—931no—
73C2×D31—751noHall
74C2×C5⋊D5—931noHall
75C3×C31C2×C3×C5⋊D511yescharacteristic, complemented
76C31⋊C3—21no—
77C31⋊C3D3×C5⋊D511yescharacteristic, complemented
78C3×D15—1556no—
79C3×C5×D3—316no—
80C32×D5—1556no—
81C3×C52—311no—
82C3×C52—621no—
83C3×C52C2×D31⋊C311yescharacteristic, complemented
84C2×C31—751no—
85D31—31no—
86D31—251no—
87C2×C3×D5—4656no—
88D3×D5—1556no—
89C2×C52—931no—
90C5⋊D5—31no—
91C5⋊D5—311no—
92C32×C5—316no—
93C2×C3×D3—7751noHall
94C31C3×D3×C5⋊D511yescharacteristic, complemented, Sylow
95C2×C3×C5—936no—
96D15—56no—
97C3×D5—3106no—
98C3×D5—1556no—
99C3×D5—1556no—
100C3×D5—4656no—
101C5×D3—316no—
102C52D3×D31⋊C311yescharacteristic, complemented, Sylow
103C2×D5—4656no—
104C2×C32—7751no—
105C3×D3—7751no—
106C3×D3—311no—
107C3×C5—316no—
108C3×C5—626no—
109C3×C5D5×D31⋊C316yescomplemented
110C22×C3—23251no—
111C2×D3—7751no—
112C2×C5—936no—
113D5—156no—
114D5—1556no—
115C32—311noSylow
116C2×C3—7751no—
117C2×C3—7751no—
118C2×C3—15501no—
119C2×C3—23251no—
120C2×C3—931no—
121D3—251no—
122D3—311no—
123C5D15⋊(D31⋊C3)16yescomplemented
124C22—23251noSylow
125C3—311no—
126C3—621no—
127C3C5⋊D5×D31⋊C311yescharacteristic, complemented
128C2—7751no—
129C2—751no—
130C2—931no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C52×C31C22×C3
LCSG ▷ C3×C52×C31
UCSC1

Complements

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

NH ≅ G/Ntype
1. C31⋊C3×C3⋊(C5⋊D5)C2semidirect
2. C52×C3⋊(D31⋊C3)C2semidirect
3. C3×C5⋊(C5⋊(D31⋊C3))C2semidirect
4. D93⋊(C5⋊D5)C3semidirect
6. C3×C52×C31⋊C3C22semidirect
8. C31×C3⋊(C5⋊D5)C2×C3semidirect
9. C52×D93C2×C3semidirect
10. C3×C5⋊D155C2×C3semidirect
13. C5⋊(C5⋊(D31⋊C3))D3semidirect
17. C5×C3⋊(D31⋊C3)D5semidirect
20. C52×C31⋊C3C2×D3semidirect
21. C3×C52×C31C22×C3semidirect
24. C5⋊D155C3×D3semidirect
27. C3×C5×C31⋊C3C2×D5semidirect
30. C5×D93C3×D5semidirect
37. C52×C31C2×C3×D3semidirect
39. C3⋊(D31⋊C3)C5⋊D5semidirect
42. C3×C5×C31C2×C3×D5semidirect
44. C5×C31⋊C3D3×D5semidirect
55. C3×C31⋊C3C2×C5⋊D5semidirect
57. D93C3×C5⋊D5semidirect
65. C5×C31C3×D3×D5semidirect
67. C3⋊(C5⋊D5)D31⋊C3semidirect
75. C3×C31C2×C3×C5⋊D5semidirect
77. C31⋊C3D3×C5⋊D5semidirect
83. C3×C52C2×D31⋊C3semidirect
94. C31C3×D3×C5⋊D5semidirect
102. C52D3×D31⋊C3semidirect
109. C3×C5D5×D31⋊C3semidirect
123. C5D15⋊(D31⋊C3)semidirect
127. C3C5⋊D5×D31⋊C3semidirect

Maximal subgroups

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

subgroupindex
C31⋊C3×C3⋊(C5⋊D5)2
C52×C3⋊(D31⋊C3)2
C3×C5⋊(C5⋊(D31⋊C3))2
D93⋊(C5⋊D5)3
C5⋊D5×D31⋊C33
D15⋊(D31⋊C3)5
C3×D3×C5⋊D531