Finite Group Databasebeta

← C23×#41×C251 #44548177

Subgroups — C23×#41×C251 #44548177

Order 32128.

6520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#41×C251C2156yesmaximal, direct product factor
2C23×Q8×C251C211yesmaximal, characteristic, complemented
3C23×D4×C251C213yesmaximal, complemented
4C24×C4×C251C213yesmaximal, complemented
5C25×C251C2213yes—
6C2×#41×C251C221448yesdirect product factor
7C22×Q8×C251C22128yescomplemented
8C22×D4×C251C22184yescomplemented
9C23×C4×C251C2213yes—
10C23×C4×C251C2211yescharacteristic
11C23×C4×C251C22184yescomplemented
12C24×C251—224no—
13C24×C251C23142yes—
14C24×C251C2311yescharacteristic
15#41×C251C231512yesdirect product factor
16C2×Q8×C251C231112yescomplemented
17C2×D4×C251C231336yescomplemented
18C22×C4×C251C23142yes—
19C22×C4×C251C23114yes—
20C22×C4×C251C231336yescomplemented
21C23×C251—2168no—
22C23×C251#4118yesdirect product factor
23C23×C251C24184yes—
24C23×C251C2417yes—
25Q8×C251C24164yescomplemented
26D4×C251C241192yescomplemented
27C2×C4×C251C24128yes—
28C2×C4×C251C24184yes—
29C2×C4×C251C241192yescomplemented
30C22×C251—2168no—
31C22×C251C2×#41128yesdirect product factor
32C22×C251C25124yes—
33C22×C251C2517yes—
34C4×C251C2518yes—
35C4×C251C25124yes—
36C2×C251—224no—
37C2×C251C22×#41114yesdirect product factor
38C2×C251C2611yescharacteristic
39C251C23×#4111yescharacteristic, direct product factor, Sylow
40C23×#41C25111yesmaximal, characteristic, direct product factor, Sylow
41C22×#41C2×C251156yesdirect product factor
42C23×Q8C2×C25111yescharacteristic, complemented
43C23×D4C2×C25113yescomplemented
44C24×C4C2×C25113yescomplemented
45C25C22×C25113yes—
46C2×#41C22×C2511448yesdirect product factor
47C22×Q8C22×C251128yescomplemented
48C22×D4C22×C251184yescomplemented
49C23×C4C22×C25111yescharacteristic
50C23×C4C22×C25113yes—
51C23×C4C22×C251184yescomplemented
52C24—224no—
53C24C23×C251142yes—
54C24C23×C25111yescharacteristic
55#41C23×C2511512yesdirect product factor
56C2×Q8C23×C2511112yescomplemented
57C2×D4C23×C2511336yescomplemented
58C22×C4C23×C251114yes—
59C22×C4C23×C251142yes—
60C22×C4C23×C2511336yescomplemented
61C23—2168no—
62C23#41×C25118yesdirect product factor
63C23C24×C251184yes—
64C23C24×C25117yes—
65Q8C24×C251164yescomplemented
66D4C24×C2511192yescomplemented
67C2×C4C24×C251128yes—
68C2×C4C24×C251184yes—
69C2×C4C24×C2511192yescomplemented
70C22—2168no—
71C22C2×#41×C251128yesdirect product factor
72C22C25×C251124yes—
73C22C25×C25117yes—
74C4C25×C25118yes—
75C4C25×C251124yes—
76C2—224no—
77C2C22×#41×C251114yesdirect product factor
78C2C26×C25111yescharacteristic
subgroupG/H
centerC23×C4×C251C22
commutator, Frattini, cosocleC2C26×C251
socleC24×C251C23
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C23×C4×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×#41×C251C2direct (N × H)
2. C23×Q8×C251C2semidirect
3. C23×D4×C251C2semidirect
4. C24×C4×C251C2semidirect
6. C2×#41×C251C22direct (N × H)
7. C22×Q8×C251C22semidirect
8. C22×D4×C251C22semidirect
11. C23×C4×C251C22semidirect
15. #41×C251C23direct (N × H)
16. C2×Q8×C251C23semidirect
17. C2×D4×C251C23semidirect
20. C22×C4×C251C23semidirect
22. C23×C251#41direct (N × H)
25. Q8×C251C24semidirect
26. D4×C251C24semidirect
29. C2×C4×C251C24semidirect
31. C22×C251C2×#41direct (N × H)
37. C2×C251C22×#41direct (N × H)
39. C251C23×#41direct (N × H)
40. C23×#41C251direct (N × H)
41. C22×#41C2×C251direct (N × H)
42. C23×Q8C2×C251semidirect
43. C23×D4C2×C251semidirect
44. C24×C4C2×C251semidirect
46. C2×#41C22×C251direct (N × H)
47. C22×Q8C22×C251semidirect
48. C22×D4C22×C251semidirect
51. C23×C4C22×C251semidirect
55. #41C23×C251direct (N × H)
56. C2×Q8C23×C251semidirect
57. C2×D4C23×C251semidirect
60. C22×C4C23×C251semidirect
62. C23#41×C251direct (N × H)
65. Q8C24×C251semidirect
66. D4C24×C251semidirect
69. C2×C4C24×C251semidirect
71. C22C2×#41×C251direct (N × H)
77. C2C22×#41×C251direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#41×C2512
C23×Q8×C2512
C23×D4×C2512
C24×C4×C2512
C23×#41251