Finite Group Databasebeta

← C26×C3×C5×C25 #72201293

Subgroups — C26×C3×C5×C25 #72201293

Order 24000.

79100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C25×C3×C5×C25C2163yesmaximal, direct product factor
2C26×C5×C25C311yesmaximal, characteristic, direct product factor, Hall
3C24×C3×C5×C25C221651yesdirect product factor
4C26×C3×C52C511yesmaximal, characteristic
5C26×C3×C25C515yesmaximal, direct product factor
6C25×C5×C25C2×C3163yesdirect product factor
7C23×C3×C5×C25C2311395yesdirect product factor
8C25×C3×C52C2×C5163yes—
9C25×C3×C25C2×C51315yesdirect product factor
10C24×C5×C25C22×C31651yesdirect product factor
11C26×C52C3×C511yescharacteristic
12C26×C25C3×C515yesdirect product factor
13C22×C3×C5×C25C241651yesdirect product factor
14C24×C3×C52C22×C51651yes—
15C24×C3×C25C22×C513255yesdirect product factor
16C23×C5×C25C23×C311395yesdirect product factor
17C26×C3×C5C2515yesdirect product factor
18C26×C3×C5C5211yescharacteristic
19C25×C52C2×C3×C5163yes—
20C25×C25C2×C3×C51315yesdirect product factor
21C2×C3×C5×C25C25163yesdirect product factor
22C23×C3×C52C23×C511395yes—
23C23×C3×C25C23×C516975yesdirect product factor
24C22×C5×C25C24×C31651yesdirect product factor
25C25×C3×C5C2×C251315yesdirect product factor
26C25×C3×C5C2×C52163yes—
27C24×C52C22×C3×C51651yes—
28C24×C25C22×C3×C513255yesdirect product factor
29C3×C5×C25C2611yescharacteristic, direct product factor, Hall
30C26×C5C3×C2515yesdirect product factor
31C26×C5C3×C5211yescharacteristic
32C22×C3×C52C24×C51651yes—
33C22×C3×C25C24×C513255yesdirect product factor
34C2×C5×C25C25×C3163yesdirect product factor
35C24×C3×C5C22×C2513255yesdirect product factor
36C24×C3×C5C22×C521651yes—
37C23×C52C23×C3×C511395yes—
38C23×C25C23×C3×C516975yesdirect product factor
39C26×C3C5×C2511yescharacteristic, direct product factor, Hall
40C25×C5C2×C3×C251315yesdirect product factor
41C25×C5C2×C3×C52163yes—
42C2×C3×C52C25×C5163yes—
43C2×C3×C25C25×C51315yesdirect product factor
44C5×C25C26×C311yescharacteristic, direct product factor, Sylow
45C23×C3×C5C23×C2516975yesdirect product factor
46C23×C3×C5C23×C5211395yes—
47C22×C52C24×C3×C51651yes—
48C22×C25C24×C3×C513255yesdirect product factor
49C25×C3C2×C5×C25163yesdirect product factor
50C24×C5C22×C3×C2513255yesdirect product factor
51C24×C5C22×C3×C521651yes—
52C3×C52C26×C511yescharacteristic
53C3×C25C26×C515yesdirect product factor
54C26C3×C5×C2511yescharacteristic, direct product factor, Sylow
55C22×C3×C5C24×C2513255yesdirect product factor
56C22×C3×C5C24×C521651yes—
57C2×C52C25×C3×C5163yes—
58C2×C25C25×C3×C51315yesdirect product factor
59C24×C3C22×C5×C251651yesdirect product factor
60C23×C5C23×C3×C2516975yesdirect product factor
61C23×C5C23×C3×C5211395yes—
62C25C2×C3×C5×C25163yesdirect product factor
63C2×C3×C5C25×C251315yesdirect product factor
64C2×C3×C5C25×C52163yes—
65C52C26×C3×C511yescharacteristic
66C25C26×C3×C515yesdirect product factor
67C23×C3C23×C5×C2511395yesdirect product factor
68C22×C5C24×C3×C2513255yesdirect product factor
69C22×C5C24×C3×C521651yes—
70C24C22×C3×C5×C251651yesdirect product factor
71C3×C5C26×C2515yesdirect product factor
72C3×C5C26×C5211yescharacteristic
73C22×C3C24×C5×C251651yesdirect product factor
74C2×C5C25×C3×C251315yesdirect product factor
75C2×C5C25×C3×C52163yes—
76C23C23×C3×C5×C2511395yesdirect product factor
77C2×C3C25×C5×C25163yesdirect product factor
78C5C26×C3×C2515yesdirect product factor
79C5C26×C3×C5211yescharacteristic
80C22C24×C3×C5×C251651yesdirect product factor
81C3C26×C5×C2511yescharacteristic, direct product factor, Sylow
82C2C25×C3×C5×C25163yesdirect product factor
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC5C26×C3×C52
socle, prime radicalC26×C3×C52C5
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
1. C25×C3×C5×C25C2direct (N × H)
2. C26×C5×C25C3direct (N × H)
3. C24×C3×C5×C25C22direct (N × H)
5. C26×C3×C25C5direct (N × H)
6. C25×C5×C25C2×C3direct (N × H)
7. C23×C3×C5×C25C23direct (N × H)
9. C25×C3×C25C2×C5direct (N × H)
10. C24×C5×C25C22×C3direct (N × H)
12. C26×C25C3×C5direct (N × H)
13. C22×C3×C5×C25C24direct (N × H)
15. C24×C3×C25C22×C5direct (N × H)
16. C23×C5×C25C23×C3direct (N × H)
17. C26×C3×C5C25direct (N × H)
20. C25×C25C2×C3×C5direct (N × H)
21. C2×C3×C5×C25C25direct (N × H)
23. C23×C3×C25C23×C5direct (N × H)
24. C22×C5×C25C24×C3direct (N × H)
25. C25×C3×C5C2×C25direct (N × H)
28. C24×C25C22×C3×C5direct (N × H)
29. C3×C5×C25C26direct (N × H)
30. C26×C5C3×C25direct (N × H)
33. C22×C3×C25C24×C5direct (N × H)
34. C2×C5×C25C25×C3direct (N × H)
35. C24×C3×C5C22×C25direct (N × H)
38. C23×C25C23×C3×C5direct (N × H)
39. C26×C3C5×C25direct (N × H)
40. C25×C5C2×C3×C25direct (N × H)
43. C2×C3×C25C25×C5direct (N × H)
44. C5×C25C26×C3direct (N × H)
45. C23×C3×C5C23×C25direct (N × H)
48. C22×C25C24×C3×C5direct (N × H)
49. C25×C3C2×C5×C25direct (N × H)
50. C24×C5C22×C3×C25direct (N × H)
53. C3×C25C26×C5direct (N × H)
54. C26C3×C5×C25direct (N × H)
55. C22×C3×C5C24×C25direct (N × H)
58. C2×C25C25×C3×C5direct (N × H)
59. C24×C3C22×C5×C25direct (N × H)
60. C23×C5C23×C3×C25direct (N × H)
62. C25C2×C3×C5×C25direct (N × H)
63. C2×C3×C5C25×C25direct (N × H)
66. C25C26×C3×C5direct (N × H)
67. C23×C3C23×C5×C25direct (N × H)
68. C22×C5C24×C3×C25direct (N × H)
70. C24C22×C3×C5×C25direct (N × H)
71. C3×C5C26×C25direct (N × H)
73. C22×C3C24×C5×C25direct (N × H)
74. C2×C5C25×C3×C25direct (N × H)
76. C23C23×C3×C5×C25direct (N × H)
77. C2×C3C25×C5×C25direct (N × H)
78. C5C26×C3×C25direct (N × H)
80. C22C24×C3×C5×C25direct (N × H)
81. C3C26×C5×C25direct (N × H)
82. C2C25×C3×C5×C25direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C3×C5×C252
C26×C5×C253
C26×C3×C525
C26×C3×C255