Finite Group Databasebeta

← D51⋊(C3⋊(C5⋊D5)) #29506032

Subgroups — D51⋊(C3⋊(C5⋊D5)) #29506032

Order 15300.

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
1C17×C3⋊(C3⋊(C5⋊D5))C211yesmaximal, characteristic, complemented
2C3×C3⋊(C5⋊D85)C211yesmaximal, characteristic, complemented
3C3×C52×D51C211yesmaximal, characteristic, complemented
4D51⋊(C5⋊D5)—31nomaximal
5D17×C3⋊(C5⋊D5)—31nomaximal
6C32×C52×C17C2211yescharacteristic, complemented, Hall
7D51⋊D15—56nomaximal
8C3⋊(C5⋊D85)D311yescharacteristic, complemented
9C17×C3⋊(C5⋊D5)—31no—
10C17×C3⋊(C5⋊D5)—31no—
11C17×C3⋊(C5⋊D5)—61no—
12C52×D51D311yescharacteristic, complemented
13C3×C5⋊D85—31no—
14C3×C52×D17—31no—
15D17×C5⋊D5—91noHall
16C17×C3⋊D15—56no—
17C3×D255—56no—
18C3×C5×D51D516yescomplemented
19C3×C52×C17—21no—
20C3×C52×C17C2×D311yescharacteristic, complemented
21C3×C52×C17C2×D311yescharacteristic, complemented
22D15⋊D17—156no—
23D15×D17—156no—
24D3×C3⋊(C5⋊D5)—171nomaximal, Hall
25C5⋊D85—31no—
26C17×C5⋊D5—91no—
27C52×D17—31no—
28C32×C5×C17C2×D516yescomplemented
29(C3⋊D3)⋊D17—251noHall
30D255—56no—
31C17×D15—156no—
32C17×D15—156no—
33C17×D15—306no—
34C5×D51D1516yescomplemented
35C3×D85—156no—
36C3×C5×D17—36no—
37C3⋊(C3⋊(C5⋊D5))D1711yescharacteristic, complemented
38C3×C3⋊(C5⋊D5)—171no—
39C3×C52×D3—171no—
40C52×C17D3211yescharacteristic, complemented, Hall
41D5×D17—456no—
42C17×C3⋊D3—251no—
43C3×D51—251no—
44C3×D51C5⋊D511yescharacteristic, complemented
45C2×C3⋊(C5⋊D5)—511no—
46D3×C5⋊D5—511no—
47C3×C5×C17—26no—
48C3×C5×C17D3×D516yescomplemented
49C3×C5×C17C2×D1516yescomplemented
50C32×C52C2×D1711yescharacteristic, complemented, Hall
51D3×D17—751no—
52D3×D17—751no—
53D3×D15—856no—
54D85—156no—
55C5×D17—36no—
56D5×C17—456no—
57C32×C17C2×C5⋊D511yescharacteristic, complemented, Hall
58C2×C3×C52—511no—
59C3⋊(C5⋊D5)—31no—
60C3⋊(C5⋊D5)—31no—
61C3⋊(C5⋊D5)—61no—
62C3⋊(C5⋊D5)—171no—
63C52×D3—171no—
64C3×C5⋊D5—511no—
65D51—251no—
66D51C3⋊(C5⋊D5)11yescharacteristic, complemented
67C3×D17—31no—
68C3×D17—751no—
69D3×C17—751no—
70D3×C17—751no—
71D3×C17—1501no—
72C2×C5⋊D5—1531noHall
73C3⋊D15—56no—
74C3×D15—856no—
75C3×C5×D3—176no—
76C5×C17D3×D1516yescomplemented
77C3×C52—21no—
78C3×C52D3×D1711yescharacteristic, complemented
79C3×C52D3×D1711yescharacteristic, complemented
80C2×D17—2251noHall
81C2×D15—2556no—
82D3×D5—2556no—
83C3×C17—21no—
84C3×C17D3×C5⋊D511yescharacteristic, complemented
85C3×C17C2×C3⋊(C5⋊D5)11yescharacteristic, complemented
86C2×C52—511no—
87C5⋊D5—91no—
88C5⋊D5—511no—
89C32×C5D5×D1716yescomplemented
90D32—4251noHall
91C2×C17—2251no—
92D17—31no—
93D17—751no—
94C2×C3×C5—516no—
95D15—156no—
96D15—156no—
97D15—306no—
98D15—856no—
99C3×D5—2556no—
100C5×D3—176no—
101C52(C3⋊D3)⋊D1711yescharacteristic, complemented, Sylow
102C2×D5—7656no—
103C3⋊D3—251no—
104C3×D3—171no—
105C3×D3—4251no—
106C17D3×C3⋊(C5⋊D5)11yescharacteristic, complemented, Sylow
107C3×C5—26no—
108C3×C5D15×D1716yescomplemented
109C3×C5D15⋊D1716yescomplemented
110C2×D3—12751no—
111C2×D3—12751no—
112C2×C5—516no—
113D5—456no—
114D5—2556no—
115C32D17×C5⋊D511yescharacteristic, complemented, Sylow
116C2×C3—511no—
117C2×C3—12751no—
118D3—171no—
119D3—751no—
120D3—751no—
121D3—1501no—
122D3—4251no—
123C5D51⋊D1516yescomplemented
124C22—38251noSylow
125C3—21no—
126C3D17×C3⋊(C5⋊D5)11yescharacteristic, complemented
127C3D51⋊(C5⋊D5)11yescharacteristic, complemented
128C2—511no—
129C2—2251no—
130C2—12751no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C52×C17C22
LCSG ▷ C32×C52×C17
UCSC1

Complements

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

NH ≅ G/Ntype
1. C17×C3⋊(C3⋊(C5⋊D5))C2semidirect
2. C3×C3⋊(C5⋊D85)C2semidirect
3. C3×C52×D51C2semidirect
6. C32×C52×C17C22semidirect
8. C3⋊(C5⋊D85)D3semidirect
12. C52×D51D3semidirect
18. C3×C5×D51D5semidirect
20. C3×C52×C17C2×D3semidirect
21. C3×C52×C17C2×D3semidirect
28. C32×C5×C17C2×D5semidirect
34. C5×D51D15semidirect
37. C3⋊(C3⋊(C5⋊D5))D17semidirect
40. C52×C17D32semidirect
44. C3×D51C5⋊D5semidirect
48. C3×C5×C17D3×D5semidirect
49. C3×C5×C17C2×D15semidirect
50. C32×C52C2×D17semidirect
57. C32×C17C2×C5⋊D5semidirect
66. D51C3⋊(C5⋊D5)semidirect
76. C5×C17D3×D15semidirect
78. C3×C52D3×D17semidirect
79. C3×C52D3×D17semidirect
84. C3×C17D3×C5⋊D5semidirect
85. C3×C17C2×C3⋊(C5⋊D5)semidirect
89. C32×C5D5×D17semidirect
101. C52(C3⋊D3)⋊D17semidirect
106. C17D3×C3⋊(C5⋊D5)semidirect
108. C3×C5D15×D17semidirect
109. C3×C5D15⋊D17semidirect
115. C32D17×C5⋊D5semidirect
123. C5D51⋊D15semidirect
126. C3D17×C3⋊(C5⋊D5)semidirect
127. C3D51⋊(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×C3⋊(C5⋊D85)2
C3×C52×D512
D51⋊(C5⋊D5)3
D17×C3⋊(C5⋊D5)3
D51⋊D155
D3×C3⋊(C5⋊D5)17