Finite Group Databasebeta

← C3×D31×C3⋊(C5⋊D5) #42124914

Subgroups — C3×D31×C3⋊(C5⋊D5) #42124914

Order 27900.

352 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C32×C52×D31C211yesmaximal, characteristic, complemented
2C3×C31×C3⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
3C3×C3⋊(C5⋊D155)C211yesmaximal, characteristic, complemented
4D31×C3⋊(C5⋊D5)C311yesmaximal, characteristic, direct product factor
5C3×C5⋊D5×D31—31nomaximal
6C32×C52×C31C2211yescharacteristic, complemented, Hall
7C3×D15×D31—56nomaximal
8C3⋊(C5⋊D155)C2×C311yescharacteristic, complemented
9C31×C3⋊(C5⋊D5)C2×C311yescharacteristic, complemented
10C3×C5⋊D155—31no—
11C3×C31×C5⋊D5—31no—
12C3×C52×D31—21no—
13C3×C52×D31D311yescharacteristic, complemented
14C3×C52×D31C2×C311yescharacteristic, complemented
15C5⋊D5×D31—31noHall
16C3×D465—56no—
17C3×D15×C31—56no—
18C32×C5×D31D516yescomplemented
19C3×C52×C31—21no—
20C3×C52×C31C2×D311yescharacteristic, complemented
21C3×C52×C31C22×C311yescharacteristic, complemented
22D15×D31—56no—
23C3×D5×D31—156no—
24C5⋊D155—31no—
25C31×C5⋊D5—31no—
26C52×D31C3×D311yescharacteristic, complemented
27C32×C5×C31C2×D516yescomplemented
28C3×D3×D31—251noHall
29D465—56no—
30D15×C31—56no—
31C3×D155—156no—
32C3×D5×C31—156no—
33C3×C5×D31—26no—
34C3×C5×D31C3×D516yescomplemented
35C3×C5×D31D1516yescomplemented
36C2×C3×C3⋊(C5⋊D5)—311nomaximal, Hall
37C52×C31C2×C3×D311yescharacteristic, complemented, Hall
38D5×D31—156no—
39C3×D93—251no—
40C3×D3×C31—251no—
41C32×D31C5⋊D511yescharacteristic, complemented
42C3×C5×C31—26no—
43C3×C5×C31C2×C3×D516yescomplemented
44C3×C5×C31C2×D1516yescomplemented
45C2×C32×C52—311no—
46C3×C3⋊(C5⋊D5)—311no—
47C3×C3⋊(C5⋊D5)D3111yescharacteristic, direct product factor
48C2×C3×D31—751no—
49D3×D31—251no—
50D155—156no—
51C5×D31C3×D1516yescomplemented
52D5×C31—156no—
53C2×C3⋊(C5⋊D5)—311no—
54C2×C3×C5⋊D5—931no—
55C32×C31C2×C5⋊D511yescharacteristic, complemented, Hall
56C32×C52C2×D3111yescharacteristic, complemented, Hall
57C2×C3×C31—751no—
58D93—251no—
59C3×D31—21no—
60C3×D31—751no—
61C3×D31C3×C5⋊D511yescharacteristic, complemented
62C3×D31C3⋊(C5⋊D5)11yescharacteristic, direct product factor
63D3×C31—251no—
64C2×C3×D15—1556no—
65C5×C31C2×C3×D1516yescomplemented
66C2×C3×C52—311no—
67C2×C3×C52—311no—
68C2×C3×C52—621no—
69C3⋊(C5⋊D5)—311no—
70C3⋊(C5⋊D5)C3×D3111yescharacteristic, direct product factor
71C3×C5⋊D5—31no—
72C3×C5⋊D5—931no—
73C2×D31—751noHall
74C2×C5⋊D5—931noHall
75C3×C31—21no—
76C3×C31C2×C3×C5⋊D511yescharacteristic, complemented
77C3×C31C2×C3⋊(C5⋊D5)11yescharacteristic, complemented
78C2×C32×C5—316no—
79C3×D15—56no—
80C3×D15—1556no—
81C3×C52—21no—
82C3×C52D3×D3111yescharacteristic, complemented
83C3×C52C2×C3×D3111yescharacteristic, complemented
84C2×C31—751no—
85D31—751no—
86D31C3×C3⋊(C5⋊D5)11yescharacteristic, direct product factor
87C2×D15—1556no—
88C2×C3×D5—4656no—
89C2×C52—311no—
90C5⋊D5—31no—
91C5⋊D5—931no—
92C32×C5D5×D3116yescomplemented
93C2×C3×D3—7751noHall
94C31C2×C3×C3⋊(C5⋊D5)11yescharacteristic, complemented, Sylow
95C2×C3×C5—316no—
96C2×C3×C5—316no—
97C2×C3×C5—626no—
98D15—56no—
99D15—1556no—
100C3×D5—156no—
101C3×D5—4656no—
102C52C3×D3×D3111yescharacteristic, complemented, Sylow
103C2×D5—4656no—
104C2×C32—311no—
105C3×D3—7751no—
106C3×D3—251no—
107C3×C5—26no—
108C3×C5C3×D5×D3116yescomplemented
109C3×C5D15×D3116yescomplemented
110C22×C3—23251no—
111C2×D3—7751no—
112C2×C5—316no—
113D5—156no—
114D5—4656no—
115C32C5⋊D5×D3111yescharacteristic, complemented, Sylow
116C2×C3—23251no—
117C2×C3—311no—
118C2×C3—311no—
119C2×C3—621no—
120C2×C3—751no—
121D3—7751no—
122D3—251no—
123C5C3×D15×D3116yescomplemented
124C22—23251noSylow
125C3—21no—
126C3C3×C5⋊D5×D3111yescharacteristic, complemented
127C3D31×C3⋊(C5⋊D5)11yescharacteristic, direct product factor
128C2—23251no—
129C2—311no—
130C2—751no—
subgroupG/H
center, hypercenterC3D31×C3⋊(C5⋊D5)
commutator, cosocle, hypocenterC3×C52×C31C22×C3
FrattiniC1G
Fitting, socleC32×C52×C31C22
LCSG ▷ C3×C52×C31
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C32×C52×D31C2semidirect
2. C3×C31×C3⋊(C5⋊D5)C2semidirect
3. C3×C3⋊(C5⋊D155)C2semidirect
4. D31×C3⋊(C5⋊D5)C3direct (N × H)
6. C32×C52×C31C22semidirect
8. C3⋊(C5⋊D155)C2×C3semidirect
9. C31×C3⋊(C5⋊D5)C2×C3semidirect
13. C3×C52×D31D3semidirect
14. C3×C52×D31C2×C3semidirect
18. C32×C5×D31D5semidirect
20. C3×C52×C31C2×D3semidirect
21. C3×C52×C31C22×C3semidirect
26. C52×D31C3×D3semidirect
27. C32×C5×C31C2×D5semidirect
34. C3×C5×D31C3×D5semidirect
35. C3×C5×D31D15semidirect
37. C52×C31C2×C3×D3semidirect
41. C32×D31C5⋊D5semidirect
43. C3×C5×C31C2×C3×D5semidirect
44. C3×C5×C31C2×D15semidirect
47. C3×C3⋊(C5⋊D5)D31direct (N × H)
51. C5×D31C3×D15semidirect
55. C32×C31C2×C5⋊D5semidirect
56. C32×C52C2×D31semidirect
61. C3×D31C3×C5⋊D5semidirect
62. C3×D31C3⋊(C5⋊D5)direct (N × H)
65. C5×C31C2×C3×D15semidirect
70. C3⋊(C5⋊D5)C3×D31direct (N × H)
76. C3×C31C2×C3×C5⋊D5semidirect
77. C3×C31C2×C3⋊(C5⋊D5)semidirect
82. C3×C52D3×D31semidirect
83. C3×C52C2×C3×D31semidirect
86. D31C3×C3⋊(C5⋊D5)direct (N × H)
92. C32×C5D5×D31semidirect
94. C31C2×C3×C3⋊(C5⋊D5)semidirect
102. C52C3×D3×D31semidirect
108. C3×C5C3×D5×D31semidirect
109. C3×C5D15×D31semidirect
115. C32C5⋊D5×D31semidirect
123. C5C3×D15×D31semidirect
126. C3C3×C5⋊D5×D31semidirect
127. C3D31×C3⋊(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
C32×C52×D312
C3×C31×C3⋊(C5⋊D5)2
C3×C3⋊(C5⋊D155)2
D31×C3⋊(C5⋊D5)3
C3×C5⋊D5×D313
C3×D15×D315
C2×C3×C3⋊(C5⋊D5)31