Finite Group Databasebeta

← D3×C17×C3⋊(C5⋊D5) #29506030

Subgroups — D3×C17×C3⋊(C5⋊D5) #29506030

Order 15300.

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
1C17×C3⋊(C3⋊(C5⋊D5))C211yesmaximal, characteristic, complemented
2C3×C17×C3⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
3C3×C52×D3×C17C211yesmaximal, characteristic, complemented
4C2×C17×C3⋊(C5⋊D5)—31nomaximal
5D3×C17×C5⋊D5—31nomaximal
6C32×C52×C17C2211yescharacteristic, complemented, Hall
7D3×C17×D15—56nomaximal
8C17×C3⋊(C5⋊D5)—31no—
9C17×C3⋊(C5⋊D5)—31no—
10C17×C3⋊(C5⋊D5)—61no—
11C17×C3⋊(C5⋊D5)D311yescharacteristic, direct product factor
12C52×D3×C17D311yescharacteristic, complemented
13C3×C17×C5⋊D5—31no—
14C2×C3×C52×C17—31no—
15C2×C17×C5⋊D5—91noHall
16C17×C3⋊D15—56no—
17C3×C17×D15—56no—
18C3×C5×D3×C17D516yescomplemented
19C3×C52×C17—21no—
20C3×C52×C17C2×D311yescharacteristic, complemented
21C3×C52×C17C2×D311yescharacteristic, complemented
22C2×C17×D15—156no—
23D3×D5×C17—156no—
24D3×C3⋊(C5⋊D5)C1711yesmaximal, characteristic, direct product factor, Hall
25C2×C52×C17—31no—
26C17×C5⋊D5—31no—
27C17×C5⋊D5—91no—
28C32×C5×C17C2×D516yescomplemented
29D32×C17—251noHall
30C2×C3×C5×C17—36no—
31C17×D15—56no—
32C17×D15—156no—
33C17×D15—156no—
34C17×D15—306no—
35C5×D3×C17D1516yescomplemented
36C3×D5×C17—156no—
37C3⋊(C3⋊(C5⋊D5))C2×C1711yescharacteristic, complemented
38C3×C3⋊(C5⋊D5)C2×C1711yescharacteristic, complemented
39C3×C52×D3C2×C1711yescharacteristic, complemented
40C52×C17D3211yescharacteristic, complemented, Hall
41C2×D5×C17—456no—
42C17×C3⋊D3—251no—
43C3×D3×C17—251no—
44C3×D3×C17C5⋊D511yescharacteristic, complemented
45C2×C3⋊(C5⋊D5)—31no—
46D3×C5⋊D5—31no—
47C3×C5×C17—26no—
48C3×C5×C17D3×D516yescomplemented
49C3×C5×C17C2×D1516yescomplemented
50C32×C52C22×C1711yescharacteristic, complemented, Hall
51C2×D3×C17—751no—
52C2×D3×C17—751no—
53D3×D15—56no—
54C2×C5×C17—36no—
55D5×C17—156no—
56D5×C17—456no—
57C32×C17C2×C5⋊D511yescharacteristic, complemented, Hall
58C2×C3×C52—31no—
59C3⋊(C5⋊D5)—31no—
60C3⋊(C5⋊D5)—31no—
61C3⋊(C5⋊D5)—61no—
62C3⋊(C5⋊D5)D3×C1711yescharacteristic, direct product factor
63C52×D3D3×C1711yescharacteristic, complemented
64C3×C5⋊D5—31no—
65C2×C3×C17—31no—
66C2×C3×C17—751no—
67D3×C17—251no—
68D3×C17—751no—
69D3×C17—751no—
70D3×C17—1501no—
71D3×C17C3⋊(C5⋊D5)11yescharacteristic, direct product factor
72C2×C5⋊D5—91noHall
73C3⋊D15—56no—
74C3×D15—56no—
75C3×C5×D3D5×C1716yescomplemented
76C5×C17D3×D1516yescomplemented
77C3×C52—21no—
78C3×C52C2×D3×C1711yescharacteristic, complemented
79C3×C52C2×D3×C1711yescharacteristic, complemented
80C22×C17—2251noHall
81C2×D15—156no—
82D3×D5—156no—
83C3×C17—21no—
84C3×C17D3×C5⋊D511yescharacteristic, complemented
85C3×C17C2×C3⋊(C5⋊D5)11yescharacteristic, complemented
86C2×C52—31no—
87C5⋊D5—31no—
88C5⋊D5—91no—
89C32×C5C2×D5×C1716yescomplemented
90D32—251noHall
91C2×C17—31no—
92C2×C17—751no—
93C2×C17—2251no—
94C2×C3×C5—36no—
95D15—56no—
96D15—156no—
97D15—156no—
98D15—306no—
99C3×D5—156no—
100C5×D3C17×D1516yescomplemented
101C52D32×C1711yescharacteristic, complemented, Sylow
102C2×D5—456no—
103C3⋊D3—251no—
104C3×D3—251no—
105C3×D3C17×C5⋊D511yescharacteristic, complemented
106C17D3×C3⋊(C5⋊D5)11yescharacteristic, direct product factor, Sylow
107C3×C5—26no—
108C3×C5D3×D5×C1716yescomplemented
109C3×C5C2×C17×D1516yescomplemented
110C2×D3—751no—
111C2×D3—751no—
112C2×C5—36no—
113D5—156no—
114D5—456no—
115C32C2×C17×C5⋊D511yescharacteristic, complemented, Sylow
116C2×C3—31no—
117C2×C3—751no—
118D3—251no—
119D3—751no—
120D3—751no—
121D3—1501no—
122D3C17×C3⋊(C5⋊D5)11yescharacteristic, direct product factor
123C5D3×C17×D1516yescomplemented
124C22—2251noSylow
125C3—21no—
126C3D3×C17×C5⋊D511yescharacteristic, complemented
127C3C2×C17×C3⋊(C5⋊D5)11yescharacteristic, complemented
128C2—31no—
129C2—751no—
130C2—2251no—
subgroupG/H
center, hypercenterC17D3×C3⋊(C5⋊D5)
commutator, cosocle, hypocenterC32×C52C22×C17
FrattiniC1G
Fitting, socleC32×C52×C17C22
LCSG ▷ C32×C52
UCSC1 ◁ C17

Complements

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

NH ≅ G/Ntype
1. C17×C3⋊(C3⋊(C5⋊D5))C2semidirect
2. C3×C17×C3⋊(C5⋊D5)C2semidirect
3. C3×C52×D3×C17C2semidirect
6. C32×C52×C17C22semidirect
11. C17×C3⋊(C5⋊D5)D3direct (N × H)
12. C52×D3×C17D3semidirect
18. C3×C5×D3×C17D5semidirect
20. C3×C52×C17C2×D3semidirect
21. C3×C52×C17C2×D3semidirect
24. D3×C3⋊(C5⋊D5)C17direct (N × H)
28. C32×C5×C17C2×D5semidirect
35. C5×D3×C17D15semidirect
37. C3⋊(C3⋊(C5⋊D5))C2×C17semidirect
38. C3×C3⋊(C5⋊D5)C2×C17semidirect
39. C3×C52×D3C2×C17semidirect
40. C52×C17D32semidirect
44. C3×D3×C17C5⋊D5semidirect
48. C3×C5×C17D3×D5semidirect
49. C3×C5×C17C2×D15semidirect
50. C32×C52C22×C17semidirect
57. C32×C17C2×C5⋊D5semidirect
62. C3⋊(C5⋊D5)D3×C17direct (N × H)
63. C52×D3D3×C17semidirect
71. D3×C17C3⋊(C5⋊D5)direct (N × H)
75. C3×C5×D3D5×C17semidirect
76. C5×C17D3×D15semidirect
78. C3×C52C2×D3×C17semidirect
79. C3×C52C2×D3×C17semidirect
84. C3×C17D3×C5⋊D5semidirect
85. C3×C17C2×C3⋊(C5⋊D5)semidirect
89. C32×C5C2×D5×C17semidirect
100. C5×D3C17×D15semidirect
101. C52D32×C17semidirect
105. C3×D3C17×C5⋊D5semidirect
106. C17D3×C3⋊(C5⋊D5)direct (N × H)
108. C3×C5D3×D5×C17semidirect
109. C3×C5C2×C17×D15semidirect
115. C32C2×C17×C5⋊D5semidirect
122. D3C17×C3⋊(C5⋊D5)direct (N × H)
123. C5D3×C17×D15semidirect
126. C3D3×C17×C5⋊D5semidirect
127. C3C2×C17×C3⋊(C5⋊D5)semidirect

Maximal subgroups

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

subgroupindex
C17×C3⋊(C3⋊(C5⋊D5))2
C3×C17×C3⋊(C5⋊D5)2
C3×C52×D3×C172
C2×C17×C3⋊(C5⋊D5)3
D3×C17×C5⋊D53
D3×C17×D155
D3×C3⋊(C5⋊D5)17