Finite Group Databasebeta

← C22×H(5,𝔽2)×C263 #46780560

Subgroups — C22×H(5,𝔽2)×C263 #46780560

Order 33664.

6560 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×H(5,𝔽2)×C263C2148yesmaximal, direct product factor
2C22×#41×C263C216yesmaximal, complemented
3C23×D4×C263C219yesmaximal, complemented
4C25×C263C2216yescomplemented
5H(5,𝔽2)×C263C221256yesdirect product factor
6C2×#41×C263C221144yescomplemented
7C22×Q8×C263C2212yescomplemented
8C22×D4×C263C22118yescomplemented
9C22×D4×C263C221216yescomplemented
10C23×C4×C263C2219yescomplemented
11C24×C263—424no—
12C24×C263C2319yes—
13C24×C263C23172yescomplemented
14#41×C263C231384yescomplemented
15C2×Q8×C263C23124yescomplemented
16C2×D4×C263C231216yescomplemented
17C2×D4×C263C231576yescomplemented
18C22×C4×C263C2316yes—
19C22×C4×C263C231108yescomplemented
20C23×C263—236no—
21C23×C263—4144no—
22C23×C263C24154yes—
23C23×C263C2411yescharacteristic
24C23×C263C24196yescomplemented
25Q8×C263C24132yescomplemented
26D4×C263C241288yescomplemented
27C2×C4×C263C24136yes—
28C2×C4×C263C241144yescomplemented
29C22×C263—2108no—
30C22×C263—496no—
31C22×C263H(5,𝔽2)14yesdirect product factor
32C22×C263C25136yes—
33C22×C263C2513yes—
34C4×C263C25124yes—
35C2×C263—236no—
36C2×C263C2×H(5,𝔽2)16yesdirect product factor
37C2×C263C2611yescharacteristic
38C263C22×H(5,𝔽2)11yescharacteristic, direct product factor, Sylow
39C22×H(5,𝔽2)C26311yesmaximal, characteristic, direct product factor, Sylow
40C2×H(5,𝔽2)C2×C263148yesdirect product factor
41C22×#41C2×C26316yescomplemented
42C23×D4C2×C26319yescomplemented
43C25C22×C26316yescomplemented
44H(5,𝔽2)C22×C2631256yesdirect product factor
45C2×#41C22×C2631144yescomplemented
46C22×Q8C22×C26312yescomplemented
47C22×D4C22×C263118yescomplemented
48C22×D4C22×C2631216yescomplemented
49C23×C4C22×C26319yescomplemented
50C24—424no—
51C24C23×C26319yes—
52C24C23×C263172yescomplemented
53#41C23×C2631384yescomplemented
54C2×Q8C23×C263124yescomplemented
55C2×D4C23×C2631216yescomplemented
56C2×D4C23×C2631576yescomplemented
57C22×C4C23×C26316yes—
58C22×C4C23×C2631108yescomplemented
59C23—236no—
60C23—4144no—
61C23C24×C263154yes—
62C23C24×C26311yescharacteristic
63C23C24×C263196yescomplemented
64Q8C24×C263132yescomplemented
65D4C24×C2631288yescomplemented
66C2×C4C24×C263136yes—
67C2×C4C24×C2631144yescomplemented
68C22—2108no—
69C22—496no—
70C22H(5,𝔽2)×C26314yesdirect product factor
71C22C25×C263136yes—
72C22C25×C26313yes—
73C4C25×C263124yes—
74C2—236no—
75C2C2×H(5,𝔽2)×C26316yesdirect product factor
76C2C26×C26311yescharacteristic
subgroupG/H
center, socleC23×C263C24
commutator, Frattini, cosocleC2C26×C263
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C23×C263 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×H(5,𝔽2)×C263C2direct (N × H)
2. C22×#41×C263C2semidirect
3. C23×D4×C263C2semidirect
4. C25×C263C22semidirect
5. H(5,𝔽2)×C263C22direct (N × H)
6. C2×#41×C263C22semidirect
7. C22×Q8×C263C22semidirect
8. C22×D4×C263C22semidirect
9. C22×D4×C263C22semidirect
10. C23×C4×C263C22semidirect
13. C24×C263C23semidirect
14. #41×C263C23semidirect
15. C2×Q8×C263C23semidirect
16. C2×D4×C263C23semidirect
17. C2×D4×C263C23semidirect
19. C22×C4×C263C23semidirect
24. C23×C263C24semidirect
25. Q8×C263C24semidirect
26. D4×C263C24semidirect
28. C2×C4×C263C24semidirect
31. C22×C263H(5,𝔽2)direct (N × H)
36. C2×C263C2×H(5,𝔽2)direct (N × H)
38. C263C22×H(5,𝔽2)direct (N × H)
39. C22×H(5,𝔽2)C263direct (N × H)
40. C2×H(5,𝔽2)C2×C263direct (N × H)
41. C22×#41C2×C263semidirect
42. C23×D4C2×C263semidirect
43. C25C22×C263semidirect
44. H(5,𝔽2)C22×C263direct (N × H)
45. C2×#41C22×C263semidirect
46. C22×Q8C22×C263semidirect
47. C22×D4C22×C263semidirect
48. C22×D4C22×C263semidirect
49. C23×C4C22×C263semidirect
52. C24C23×C263semidirect
53. #41C23×C263semidirect
54. C2×Q8C23×C263semidirect
55. C2×D4C23×C263semidirect
56. C2×D4C23×C263semidirect
58. C22×C4C23×C263semidirect
63. C23C24×C263semidirect
64. Q8C24×C263semidirect
65. D4C24×C263semidirect
67. C2×C4C24×C263semidirect
70. C22H(5,𝔽2)×C263direct (N × H)
75. C2C2×H(5,𝔽2)×C263direct (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)×C2632
C22×#41×C2632
C23×D4×C2632
C22×H(5,𝔽2)263