Finite Group Databasebeta

← C2×C3×C52×D51 #29506081

Subgroups — C2×C3×C52×D51 #29506081

Order 15300.

352 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 176 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×D51C212yesmaximal, direct product factor
2C2×C32×C52×C17C211yesmaximal, characteristic, complemented
3C2×C52×D51C311yesmaximal, characteristic, direct product factor
4C2×C3×C52×D17—31nomaximal
5C32×C52×C17C2211yescharacteristic, complemented, Hall
6C2×C3×C5×D51C516yesmaximal, direct product factor
7C52×D51C2×C312yesdirect product factor
8C3×C52×D17—32no—
9C2×C3×C52×C17—21no—
10C2×C3×C52×C17D311yescharacteristic, complemented
11C2×C3×C52×C17C2×C311yescharacteristic, complemented
12C2×C52×D17—31noHall
13C2×C32×C5×C17C2×C516yescomplemented
14C3×C5×D51C2×C5112yesdirect product factor
15C3×C52×C17—21no—
16C3×C52×C17C2×D311yescharacteristic, complemented
17C3×C52×C17C22×C311yescharacteristic, complemented
18C2×C5×D51C3×C516yesdirect product factor
19C2×C3×C5×D17—36no—
20C2×C3×C52×D3—171nomaximal, Hall
21C2×C52×C17C3×D311yescharacteristic, complemented
22C52×D17—32no—
23C32×C5×C17C22×C516yescomplemented
24C2×C3×D51C5211yescharacteristic, direct product factor, Hall
25C2×C3×C5×C17—26no—
26C2×C3×C5×C17C5×D316yescomplemented
27C2×C3×C5×C17C2×C3×C516yescomplemented
28C5×D51C2×C3×C5112yesdirect product factor
29C3×C5×D17—312no—
30C2×C32×C52D1711yescharacteristic, complemented
31C3×C52×D3—172no—
32C52×C17C2×C3×D311yescharacteristic, complemented, Hall
33C2×C5×D17—36no—
34C2×C32×C17C2×C5211yescharacteristic, complemented
35C3×D51C2×C5212yesdirect product factor
36C22×C3×C52—511no—
37C2×C52×D3—171no—
38C3×C5×C17—26no—
39C3×C5×C17C2×C5×D316yescomplemented
40C3×C5×C17C22×C3×C516yescomplemented
41C32×C52C2×D1711yescharacteristic, complemented, Hall
42C2×D51C3×C5211yescharacteristic, direct product factor
43C2×C3×D17—31no—
44C2×C3×C5×D3—176no—
45C2×C5×C17C3×C5×D316yescomplemented
46C5×D17—312no—
47C32×C17C22×C5211yescharacteristic, complemented, Hall
48C2×C3×C52—21no—
49C2×C3×C52—512no—
50C2×C3×C52C3×D1711yescharacteristic, complemented
51C2×C3×C52D5111yescharacteristic, direct product factor
52C52×D3—172no—
53C2×C3×C17—21no—
54C2×C3×C17C52×D311yescharacteristic, complemented
55C2×C3×C17C2×C3×C5211yescharacteristic, complemented
56D51C2×C3×C5212yesdirect product factor
57C3×D17—32no—
58C22×C52—511noHall
59C2×C32×C5C5×D1716yescomplemented
60C3×C5×D3—1712no—
61C5×C17C2×C3×C5×D316yescomplemented
62C3×C52—21no—
63C3×C52C2×C3×D1711yescharacteristic, complemented
64C3×C52C2×D5111yescharacteristic, direct product factor
65C2×D17—31noHall
66C22×C3×C5—516no—
67C2×C5×D3—176no—
68C3×C17—21no—
69C3×C17C2×C52×D311yescharacteristic, complemented
70C3×C17C22×C3×C5211yescharacteristic, complemented
71C2×C52—512no—
72C2×C52C3×D5111yescharacteristic, direct product factor
73C32×C5C2×C5×D1716yescomplemented
74C2×C3×D3—171noHall
75C2×C17C3×C52×D311yescharacteristic, complemented
76D17—32no—
77C2×C3×C5—26no—
78C2×C3×C5—5112no—
79C2×C3×C5C3×C5×D1716yescomplemented
80C2×C3×C5C5×D5116yesdirect product factor
81C5×D3—1712no—
82C52C2×C3×D5111yescharacteristic, direct product factor, Sylow
83C22×C5—516no—
84C2×C32C52×D1711yescharacteristic, complemented
85C3×D3—172no—
86C17C2×C3×C52×D311yescharacteristic, complemented, Sylow
87C3×C5—26no—
88C3×C5C2×C3×C5×D1716yescomplemented
89C3×C5C2×C5×D5116yesdirect product factor
90C22×C3—511no—
91C2×D3—171no—
92C2×C5—5112no—
93C2×C5C3×C5×D5116yesdirect product factor
94C32C2×C52×D1711yescharacteristic, complemented, Sylow
95C2×C3—21no—
96C2×C3—512no—
97C2×C3C3×C52×D1711yescharacteristic, complemented
98C2×C3C52×D5111yescharacteristic, direct product factor
99D3—172no—
100C5C2×C3×C5×D5116yesdirect product factor
101C22—511noSylow
102C3—21no—
103C3C2×C3×C52×D1711yescharacteristic, complemented
104C3C2×C52×D5111yescharacteristic, direct product factor
105C2—512no—
106C2C3×C52×D5111yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C52D51
commutator, cosocle, hypocenterC3×C17C22×C3×C52
FrattiniC1G
Fitting, socleC2×C32×C52×C17C2
LCSG ▷ C3×C17
UCSC1 ◁ C2×C3×C52

Complements

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

NH ≅ G/Ntype
1. C3×C52×D51C2direct (N × H)
2. C2×C32×C52×C17C2semidirect
3. C2×C52×D51C3direct (N × H)
5. C32×C52×C17C22semidirect
6. C2×C3×C5×D51C5direct (N × H)
7. C52×D51C2×C3direct (N × H)
10. C2×C3×C52×C17D3semidirect
11. C2×C3×C52×C17C2×C3semidirect
13. C2×C32×C5×C17C2×C5semidirect
14. C3×C5×D51C2×C5direct (N × H)
16. C3×C52×C17C2×D3semidirect
17. C3×C52×C17C22×C3semidirect
18. C2×C5×D51C3×C5direct (N × H)
21. C2×C52×C17C3×D3semidirect
23. C32×C5×C17C22×C5semidirect
24. C2×C3×D51C52direct (N × H)
26. C2×C3×C5×C17C5×D3semidirect
27. C2×C3×C5×C17C2×C3×C5semidirect
28. C5×D51C2×C3×C5direct (N × H)
30. C2×C32×C52D17semidirect
32. C52×C17C2×C3×D3semidirect
34. C2×C32×C17C2×C52semidirect
35. C3×D51C2×C52direct (N × H)
39. C3×C5×C17C2×C5×D3semidirect
40. C3×C5×C17C22×C3×C5semidirect
41. C32×C52C2×D17semidirect
42. C2×D51C3×C52direct (N × H)
45. C2×C5×C17C3×C5×D3semidirect
47. C32×C17C22×C52semidirect
50. C2×C3×C52C3×D17semidirect
51. C2×C3×C52D51direct (N × H)
54. C2×C3×C17C52×D3semidirect
55. C2×C3×C17C2×C3×C52semidirect
56. D51C2×C3×C52direct (N × H)
59. C2×C32×C5C5×D17semidirect
61. C5×C17C2×C3×C5×D3semidirect
63. C3×C52C2×C3×D17semidirect
64. C3×C52C2×D51direct (N × H)
69. C3×C17C2×C52×D3semidirect
70. C3×C17C22×C3×C52semidirect
72. C2×C52C3×D51direct (N × H)
73. C32×C5C2×C5×D17semidirect
75. C2×C17C3×C52×D3semidirect
79. C2×C3×C5C3×C5×D17semidirect
80. C2×C3×C5C5×D51direct (N × H)
82. C52C2×C3×D51direct (N × H)
84. C2×C32C52×D17semidirect
86. C17C2×C3×C52×D3semidirect
88. C3×C5C2×C3×C5×D17semidirect
89. C3×C5C2×C5×D51direct (N × H)
93. C2×C5C3×C5×D51direct (N × H)
94. C32C2×C52×D17semidirect
97. C2×C3C3×C52×D17semidirect
98. C2×C3C52×D51direct (N × H)
100. C5C2×C3×C5×D51direct (N × H)
103. C3C2×C3×C52×D17semidirect
104. C3C2×C52×D51direct (N × H)
106. C2C3×C52×D51direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C52×D512
C2×C32×C52×C172
C2×C52×D513
C2×C3×C52×D173
C2×C3×C5×D515
C2×C3×C52×D317