Finite Group Databasebeta

← D123⋊(C3⋊(C5⋊D5)) #49147160

Subgroups — D123⋊(C3⋊(C5⋊D5)) #49147160

Order 36900.

352 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×C52×D123C211yesmaximal, characteristic, complemented
2C3×C3⋊(C5⋊D205)C211yesmaximal, characteristic, complemented
3C41×C3⋊(C3⋊(C5⋊D5))C211yesmaximal, characteristic, complemented
4D123⋊(C5⋊D5)—31nomaximal
5D41×C3⋊(C5⋊D5)—31nomaximal
6C32×C52×C41C2211yescharacteristic, complemented, Hall
7D123⋊D15—56nomaximal
8C3⋊(C5⋊D205)D311yescharacteristic, complemented
9C41×C3⋊(C5⋊D5)—31no—
10C41×C3⋊(C5⋊D5)—31no—
11C41×C3⋊(C5⋊D5)—61no—
12C52×D123D311yescharacteristic, complemented
13C3×C5⋊D205—31no—
14C3×C52×D41—31no—
15C5⋊D5×D41—91noHall
16C41×C3⋊D15—56no—
17C3×D615—56no—
18C3×C5×D123D516yescomplemented
19C3×C52×C41—21no—
20C3×C52×C41C2×D311yescharacteristic, complemented
21C3×C52×C41C2×D311yescharacteristic, complemented
22D15⋊D41—156no—
23D15×D41—156no—
24C5⋊D205—31no—
25C41×C5⋊D5—91no—
26C52×D41—31no—
27C32×C5×C41C2×D516yescomplemented
28(C3⋊D3)⋊D41—251noHall
29D615—56no—
30D15×C41—156no—
31D15×C41—156no—
32D15×C41—306no—
33C5×D123D1516yescomplemented
34C3×D205—156no—
35C3×C5×D41—36no—
36C52×C41D3211yescharacteristic, complemented, Hall
37D3×C3⋊(C5⋊D5)—411nomaximal, Hall
38D5×D41—456no—
39C3⋊D3×C41—251no—
40C3×D123—251no—
41C3×D123C5⋊D511yescharacteristic, complemented
42C3×C5×C41—26no—
43C3×C5×C41D3×D516yescomplemented
44C3×C5×C41C2×D1516yescomplemented
45D3×D41—751no—
46D3×D41—751no—
47C3⋊(C3⋊(C5⋊D5))D4111yescharacteristic, complemented
48C3×C3⋊(C5⋊D5)—411no—
49C3×C52×D3—411no—
50D205—156no—
51C5×D41—36no—
52D5×C41—456no—
53C32×C41C2×C5⋊D511yescharacteristic, complemented, Hall
54C2×C3⋊(C5⋊D5)—1231no—
55D3×C5⋊D5—1231no—
56D123—251no—
57D123C3⋊(C5⋊D5)11yescharacteristic, complemented
58C3×D41—31no—
59C3×D41—751no—
60D3×C41—751no—
61D3×C41—751no—
62D3×C41—1501no—
63C32×C52C2×D4111yescharacteristic, complemented, Hall
64C5×C41D3×D1516yescomplemented
65D3×D15—2056no—
66C2×D41—2251noHall
67C2×C3×C52—1231no—
68C3⋊(C5⋊D5)—31no—
69C3⋊(C5⋊D5)—31no—
70C3⋊(C5⋊D5)—61no—
71C3⋊(C5⋊D5)—411no—
72C52×D3—411no—
73C3×C5⋊D5—1231no—
74C3×C41—21no—
75C3×C41D3×C5⋊D511yescharacteristic, complemented
76C3×C41C2×C3⋊(C5⋊D5)11yescharacteristic, complemented
77C2×C5⋊D5—3691noHall
78C3⋊D15—56no—
79C3×D15—2056no—
80C3×C5×D3—416no—
81C2×C41—2251no—
82D41—31no—
83D41—751no—
84C3×C52—21no—
85C3×C52D3×D4111yescharacteristic, complemented
86C3×C52D3×D4111yescharacteristic, complemented
87C2×D15—6156no—
88D3×D5—6156no—
89C2×C52—1231no—
90C5⋊D5—91no—
91C5⋊D5—1231no—
92C32×C5D5×D4116yescomplemented
93C41D3×C3⋊(C5⋊D5)11yescharacteristic, complemented, Sylow
94D32—10251noHall
95C2×C3×C5—1236no—
96D15—156no—
97D15—156no—
98D15—306no—
99D15—2056no—
100C3×D5—6156no—
101C5×D3—416no—
102C52(C3⋊D3)⋊D4111yescharacteristic, complemented, Sylow
103C2×D5—18456no—
104C3⋊D3—251no—
105C3×D3—10251no—
106C3×D3—411no—
107C3×C5—26no—
108C3×C5D15×D4116yescomplemented
109C3×C5D15⋊D4116yescomplemented
110C2×D3—30751no—
111C2×D3—30751no—
112C2×C5—1236no—
113D5—456no—
114D5—6156no—
115C32C5⋊D5×D4111yescharacteristic, complemented, Sylow
116C2×C3—30751no—
117C2×C3—1231no—
118D3—10251no—
119D3—411no—
120D3—751no—
121D3—751no—
122D3—1501no—
123C5D123⋊D1516yescomplemented
124C22—92251noSylow
125C3—21no—
126C3D41×C3⋊(C5⋊D5)11yescharacteristic, complemented
127C3D123⋊(C5⋊D5)11yescharacteristic, complemented
128C2—30751no—
129C2—1231no—
130C2—2251no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C52×C41C22
LCSG ▷ C32×C52×C41
UCSC1

Complements

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

NH ≅ G/Ntype
1. C3×C52×D123C2semidirect
2. C3×C3⋊(C5⋊D205)C2semidirect
3. C41×C3⋊(C3⋊(C5⋊D5))C2semidirect
6. C32×C52×C41C22semidirect
8. C3⋊(C5⋊D205)D3semidirect
12. C52×D123D3semidirect
18. C3×C5×D123D5semidirect
20. C3×C52×C41C2×D3semidirect
21. C3×C52×C41C2×D3semidirect
27. C32×C5×C41C2×D5semidirect
33. C5×D123D15semidirect
36. C52×C41D32semidirect
41. C3×D123C5⋊D5semidirect
43. C3×C5×C41D3×D5semidirect
44. C3×C5×C41C2×D15semidirect
47. C3⋊(C3⋊(C5⋊D5))D41semidirect
53. C32×C41C2×C5⋊D5semidirect
57. D123C3⋊(C5⋊D5)semidirect
63. C32×C52C2×D41semidirect
64. C5×C41D3×D15semidirect
75. C3×C41D3×C5⋊D5semidirect
76. C3×C41C2×C3⋊(C5⋊D5)semidirect
85. C3×C52D3×D41semidirect
86. C3×C52D3×D41semidirect
92. C32×C5D5×D41semidirect
93. C41D3×C3⋊(C5⋊D5)semidirect
102. C52(C3⋊D3)⋊D41semidirect
108. C3×C5D15×D41semidirect
109. C3×C5D15⋊D41semidirect
115. C32C5⋊D5×D41semidirect
123. C5D123⋊D15semidirect
126. C3D41×C3⋊(C5⋊D5)semidirect
127. C3D123⋊(C5⋊D5)semidirect

Maximal subgroups

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

subgroupindex
C3×C52×D1232
C3×C3⋊(C5⋊D205)2
C41×C3⋊(C3⋊(C5⋊D5))2
D123⋊(C5⋊D5)3
D41×C3⋊(C5⋊D5)3
D123⋊D155
D3×C3⋊(C5⋊D5)41