Finite Group Databasebeta

← C22×H(5,𝔽2)×C251 #44548178

Subgroups — C22×H(5,𝔽2)×C251 #44548178

Order 32128.

6560 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×H(5,𝔽2)×C251C2148yesmaximal, direct product factor
2C22×#41×C251C216yesmaximal, complemented
3C23×D4×C251C219yesmaximal, complemented
4C25×C251C2216yescomplemented
5H(5,𝔽2)×C251C221256yesdirect product factor
6C2×#41×C251C221144yescomplemented
7C22×Q8×C251C2212yescomplemented
8C22×D4×C251C22118yescomplemented
9C22×D4×C251C221216yescomplemented
10C23×C4×C251C2219yescomplemented
11C24×C251—424no—
12C24×C251C2319yes—
13C24×C251C23172yescomplemented
14#41×C251C231384yescomplemented
15C2×Q8×C251C23124yescomplemented
16C2×D4×C251C231216yescomplemented
17C2×D4×C251C231576yescomplemented
18C22×C4×C251C2316yes—
19C22×C4×C251C231108yescomplemented
20C23×C251—236no—
21C23×C251—4144no—
22C23×C251C24154yes—
23C23×C251C2411yescharacteristic
24C23×C251C24196yescomplemented
25Q8×C251C24132yescomplemented
26D4×C251C241288yescomplemented
27C2×C4×C251C24136yes—
28C2×C4×C251C241144yescomplemented
29C22×C251—2108no—
30C22×C251—496no—
31C22×C251H(5,𝔽2)14yesdirect product factor
32C22×C251C25136yes—
33C22×C251C2513yes—
34C4×C251C25124yes—
35C2×C251—236no—
36C2×C251C2×H(5,𝔽2)16yesdirect product factor
37C2×C251C2611yescharacteristic
38C251C22×H(5,𝔽2)11yescharacteristic, direct product factor, Sylow
39C22×H(5,𝔽2)C25111yesmaximal, characteristic, direct product factor, Sylow
40C2×H(5,𝔽2)C2×C251148yesdirect product factor
41C22×#41C2×C25116yescomplemented
42C23×D4C2×C25119yescomplemented
43C25C22×C25116yescomplemented
44H(5,𝔽2)C22×C2511256yesdirect product factor
45C2×#41C22×C2511144yescomplemented
46C22×Q8C22×C25112yescomplemented
47C22×D4C22×C251118yescomplemented
48C22×D4C22×C2511216yescomplemented
49C23×C4C22×C25119yescomplemented
50C24—424no—
51C24C23×C25119yes—
52C24C23×C251172yescomplemented
53#41C23×C2511384yescomplemented
54C2×Q8C23×C251124yescomplemented
55C2×D4C23×C2511216yescomplemented
56C2×D4C23×C2511576yescomplemented
57C22×C4C23×C25116yes—
58C22×C4C23×C2511108yescomplemented
59C23—236no—
60C23—4144no—
61C23C24×C251154yes—
62C23C24×C25111yescharacteristic
63C23C24×C251196yescomplemented
64Q8C24×C251132yescomplemented
65D4C24×C2511288yescomplemented
66C2×C4C24×C251136yes—
67C2×C4C24×C2511144yescomplemented
68C22—2108no—
69C22—496no—
70C22H(5,𝔽2)×C25114yesdirect product factor
71C22C25×C251136yes—
72C22C25×C25113yes—
73C4C25×C251124yes—
74C2—236no—
75C2C2×H(5,𝔽2)×C25116yesdirect product factor
76C2C26×C25111yescharacteristic
subgroupG/H
center, socleC23×C251C24
commutator, Frattini, cosocleC2C26×C251
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C23×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×H(5,𝔽2)×C251C2direct (N × H)
2. C22×#41×C251C2semidirect
3. C23×D4×C251C2semidirect
4. C25×C251C22semidirect
5. H(5,𝔽2)×C251C22direct (N × H)
6. C2×#41×C251C22semidirect
7. C22×Q8×C251C22semidirect
8. C22×D4×C251C22semidirect
9. C22×D4×C251C22semidirect
10. C23×C4×C251C22semidirect
13. C24×C251C23semidirect
14. #41×C251C23semidirect
15. C2×Q8×C251C23semidirect
16. C2×D4×C251C23semidirect
17. C2×D4×C251C23semidirect
19. C22×C4×C251C23semidirect
24. C23×C251C24semidirect
25. Q8×C251C24semidirect
26. D4×C251C24semidirect
28. C2×C4×C251C24semidirect
31. C22×C251H(5,𝔽2)direct (N × H)
36. C2×C251C2×H(5,𝔽2)direct (N × H)
38. C251C22×H(5,𝔽2)direct (N × H)
39. C22×H(5,𝔽2)C251direct (N × H)
40. C2×H(5,𝔽2)C2×C251direct (N × H)
41. C22×#41C2×C251semidirect
42. C23×D4C2×C251semidirect
43. C25C22×C251semidirect
44. H(5,𝔽2)C22×C251direct (N × H)
45. C2×#41C22×C251semidirect
46. C22×Q8C22×C251semidirect
47. C22×D4C22×C251semidirect
48. C22×D4C22×C251semidirect
49. C23×C4C22×C251semidirect
52. C24C23×C251semidirect
53. #41C23×C251semidirect
54. C2×Q8C23×C251semidirect
55. C2×D4C23×C251semidirect
56. C2×D4C23×C251semidirect
58. C22×C4C23×C251semidirect
63. C23C24×C251semidirect
64. Q8C24×C251semidirect
65. D4C24×C251semidirect
67. C2×C4C24×C251semidirect
70. C22H(5,𝔽2)×C251direct (N × H)
75. C2C2×H(5,𝔽2)×C251direct (N × H)

Maximal subgroups

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

subgroupindex
C2×H(5,𝔽2)×C2512
C22×#41×C2512
C23×D4×C2512
C22×H(5,𝔽2)251