Finite Group Databasebeta

← C2×C3×C52×D3×C31 #42124985

Subgroups — C2×C3×C52×D3×C31 #42124985

Order 27900.

352 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 224 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×D3×C31C212yesmaximal, direct product factor
2C2×C32×C52×C31C211yesmaximal, characteristic, complemented
3C2×C52×D3×C31C311yesmaximal, characteristic, direct product factor
4C22×C3×C52×C31—31nomaximal
5C32×C52×C31C2211yescharacteristic, complemented, Hall
6C2×C3×C5×D3×C31C516yesmaximal, direct product factor
7C52×D3×C31C2×C312yesdirect product factor
8C2×C3×C52×C31—21no—
9C2×C3×C52×C31—32no—
10C2×C3×C52×C31D311yescharacteristic, direct product factor
11C2×C3×C52×C31C2×C311yescharacteristic, complemented
12C22×C52×C31—31noHall
13C3×C5×D3×C31C2×C5112yesdirect product factor
14C2×C32×C5×C31C2×C516yescomplemented
15C3×C52×C31—21no—
16C3×C52×C31C2×D311yescharacteristic, direct product factor
17C3×C52×C31C22×C311yescharacteristic, complemented
18C22×C3×C5×C31—36no—
19C2×C5×D3×C31C3×C516yesdirect product factor
20C2×C52×C31—32no—
21C2×C52×C31C3×D311yescharacteristic, direct product factor
22C32×C5×C31C22×C516yescomplemented
23C2×C3×D3×C31C5211yescharacteristic, direct product factor, Hall
24C2×C3×C5×C31—26no—
25C2×C3×C5×C31—312no—
26C2×C3×C5×C31C5×D316yesdirect product factor
27C2×C3×C5×C31C2×C3×C516yescomplemented
28C5×D3×C31C2×C3×C5112yesdirect product factor
29C2×C3×C52×D3C3111yesmaximal, characteristic, direct product factor, Hall
30C52×C31C2×C3×D311yescharacteristic, direct product factor, Hall
31C22×C5×C31—36no—
32C2×C32×C31C2×C5211yescharacteristic, complemented
33C3×D3×C31C2×C5212yesdirect product factor
34C3×C5×C31—26no—
35C3×C5×C31C2×C5×D316yesdirect product factor
36C3×C5×C31C22×C3×C516yescomplemented
37C2×C32×C52C2×C3111yescharacteristic, complemented
38C3×C52×D3C2×C3112yesdirect product factor
39C22×C3×C31—31no—
40C2×D3×C31C3×C5211yescharacteristic, direct product factor
41C2×C5×C31—312no—
42C2×C5×C31C3×C5×D316yesdirect product factor
43C22×C3×C52—31no—
44C2×C52×D3C3×C3111yescharacteristic, direct product factor
45C32×C31C22×C5211yescharacteristic, complemented, Hall
46C32×C52C22×C3111yescharacteristic, complemented, Hall
47C2×C3×C31—21no—
48C2×C3×C31—32no—
49C2×C3×C31C52×D311yescharacteristic, direct product factor
50C2×C3×C31C2×C3×C5211yescharacteristic, complemented
51D3×C31C2×C3×C5212yesdirect product factor
52C2×C3×C5×D3C5×C3116yesdirect product factor
53C5×C31C2×C3×C5×D316yesdirect product factor
54C2×C3×C52—21no—
55C2×C3×C52—32no—
56C2×C3×C52D3×C3111yescharacteristic, direct product factor
57C2×C3×C52C2×C3×C3111yescharacteristic, complemented
58C52×D3C2×C3×C3112yesdirect product factor
59C22×C31—31noHall
60C22×C52—31noHall
61C3×C31—21no—
62C3×C31C2×C52×D311yescharacteristic, direct product factor
63C3×C31C22×C3×C5211yescharacteristic, complemented
64C2×C32×C5C2×C5×C3116yescomplemented
65C3×C5×D3C2×C5×C31112yesdirect product factor
66C3×C52—21no—
67C3×C52C2×D3×C3111yescharacteristic, direct product factor
68C3×C52C22×C3×C3111yescharacteristic, complemented
69C2×C31—32no—
70C2×C31C3×C52×D311yescharacteristic, direct product factor
71C22×C3×C5—36no—
72C2×C5×D3C3×C5×C3116yesdirect product factor
73C2×C52—32no—
74C2×C52C3×D3×C3111yescharacteristic, direct product factor
75C32×C5C22×C5×C3116yescomplemented
76C2×C3×D3C52×C3111yescharacteristic, direct product factor, Hall
77C31C2×C3×C52×D311yescharacteristic, direct product factor, Sylow
78C2×C3×C5—26no—
79C2×C3×C5—312no—
80C2×C3×C5C5×D3×C3116yesdirect product factor
81C2×C3×C5C2×C3×C5×C3116yescomplemented
82C5×D3C2×C3×C5×C31112yesdirect product factor
83C52C2×C3×D3×C3111yescharacteristic, direct product factor, Sylow
84C22×C5—36no—
85C2×C32C2×C52×C3111yescharacteristic, complemented
86C3×D3C2×C52×C3112yesdirect product factor
87C3×C5—26no—
88C3×C5C2×C5×D3×C3116yesdirect product factor
89C3×C5C22×C3×C5×C3116yescomplemented
90C22×C3—31no—
91C2×D3C3×C52×C3111yescharacteristic, direct product factor
92C2×C5—312no—
93C2×C5C3×C5×D3×C3116yesdirect product factor
94C32C22×C52×C3111yescharacteristic, complemented, Sylow
95C2×C3—21no—
96C2×C3—32no—
97C2×C3C2×C3×C52×C3111yescharacteristic, complemented
98C2×C3C52×D3×C3111yescharacteristic, direct product factor
99D3C2×C3×C52×C3112yesdirect product factor
100C5C2×C3×C5×D3×C3116yesdirect product factor
101C22—31noSylow
102C3—21no—
103C3C22×C3×C52×C3111yescharacteristic, complemented
104C3C2×C52×D3×C3111yescharacteristic, direct product factor
105C2—32no—
106C2C3×C52×D3×C3111yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C52×C31D3
commutator, cosocle, hypocenterC3C22×C3×C52×C31
FrattiniC1G
Fitting, socleC2×C32×C52×C31C2
LCSG ▷ C3
UCSC1 ◁ C2×C3×C52×C31

Complements

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

NH ≅ G/Ntype
1. C3×C52×D3×C31C2direct (N × H)
2. C2×C32×C52×C31C2semidirect
3. C2×C52×D3×C31C3direct (N × H)
5. C32×C52×C31C22semidirect
6. C2×C3×C5×D3×C31C5direct (N × H)
7. C52×D3×C31C2×C3direct (N × H)
10. C2×C3×C52×C31D3direct (N × H)
11. C2×C3×C52×C31C2×C3semidirect
13. C3×C5×D3×C31C2×C5direct (N × H)
14. C2×C32×C5×C31C2×C5semidirect
16. C3×C52×C31C2×D3direct (N × H)
17. C3×C52×C31C22×C3semidirect
19. C2×C5×D3×C31C3×C5direct (N × H)
21. C2×C52×C31C3×D3direct (N × H)
22. C32×C5×C31C22×C5semidirect
23. C2×C3×D3×C31C52direct (N × H)
26. C2×C3×C5×C31C5×D3direct (N × H)
27. C2×C3×C5×C31C2×C3×C5semidirect
28. C5×D3×C31C2×C3×C5direct (N × H)
29. C2×C3×C52×D3C31direct (N × H)
30. C52×C31C2×C3×D3direct (N × H)
32. C2×C32×C31C2×C52semidirect
33. C3×D3×C31C2×C52direct (N × H)
35. C3×C5×C31C2×C5×D3direct (N × H)
36. C3×C5×C31C22×C3×C5semidirect
37. C2×C32×C52C2×C31semidirect
38. C3×C52×D3C2×C31direct (N × H)
40. C2×D3×C31C3×C52direct (N × H)
42. C2×C5×C31C3×C5×D3direct (N × H)
44. C2×C52×D3C3×C31direct (N × H)
45. C32×C31C22×C52semidirect
46. C32×C52C22×C31semidirect
49. C2×C3×C31C52×D3direct (N × H)
50. C2×C3×C31C2×C3×C52semidirect
51. D3×C31C2×C3×C52direct (N × H)
52. C2×C3×C5×D3C5×C31direct (N × H)
53. C5×C31C2×C3×C5×D3direct (N × H)
56. C2×C3×C52D3×C31direct (N × H)
57. C2×C3×C52C2×C3×C31semidirect
58. C52×D3C2×C3×C31direct (N × H)
62. C3×C31C2×C52×D3direct (N × H)
63. C3×C31C22×C3×C52semidirect
64. C2×C32×C5C2×C5×C31semidirect
65. C3×C5×D3C2×C5×C31direct (N × H)
67. C3×C52C2×D3×C31direct (N × H)
68. C3×C52C22×C3×C31semidirect
70. C2×C31C3×C52×D3direct (N × H)
72. C2×C5×D3C3×C5×C31direct (N × H)
74. C2×C52C3×D3×C31direct (N × H)
75. C32×C5C22×C5×C31semidirect
76. C2×C3×D3C52×C31direct (N × H)
77. C31C2×C3×C52×D3direct (N × H)
80. C2×C3×C5C5×D3×C31direct (N × H)
81. C2×C3×C5C2×C3×C5×C31semidirect
82. C5×D3C2×C3×C5×C31direct (N × H)
83. C52C2×C3×D3×C31direct (N × H)
85. C2×C32C2×C52×C31semidirect
86. C3×D3C2×C52×C31direct (N × H)
88. C3×C5C2×C5×D3×C31direct (N × H)
89. C3×C5C22×C3×C5×C31semidirect
91. C2×D3C3×C52×C31direct (N × H)
93. C2×C5C3×C5×D3×C31direct (N × H)
94. C32C22×C52×C31semidirect
97. C2×C3C2×C3×C52×C31semidirect
98. C2×C3C52×D3×C31direct (N × H)
99. D3C2×C3×C52×C31direct (N × H)
100. C5C2×C3×C5×D3×C31direct (N × H)
103. C3C22×C3×C52×C31semidirect
104. C3C2×C52×D3×C31direct (N × H)
106. C2C3×C52×D3×C31direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C52×D3×C312
C2×C32×C52×C312
C2×C52×D3×C313
C22×C3×C52×C313
C2×C3×C5×D3×C315
C2×C3×C52×D331