Finite Group Databasebeta

← C23×H(7,𝔽2) #15671213

Subgroups — C23×H(7,𝔽2) #15671213

Order 2944.

5992 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#585C2128yesmaximal, complemented
2C2×C23×H(5,𝔽2)C2135yesmaximal, complemented
3C23×2−1+4C22156yescomplemented
4C23×H(5,𝔽2)C221280yescomplemented
5C2×#41×C23C221210yescomplemented
6C22×D4×C23C221105yescomplemented
7C24×C23C23130yescomplemented
8#41×C23C231560yescomplemented
9C2×Q8×C23C23170yescomplemented
10C2×D4×C23C231630yescomplemented
11C22×C4×C23C231105yescomplemented
12C23×C23—830no—
13C23×C23C241105yes—
14Q8×C23C24156yes—
15D4×C23C241280yes—
16C2×C4×C23C241210yes—
17H(7,𝔽2)C2311yesmaximal, characteristic, direct product factor, Sylow
18C22×C23—4105no—
19C22×C23C25135yes—
20C4×C23C25128yes—
21#585C2×C23128yescomplemented
22C2×H(5,𝔽2)C2×C23135yescomplemented
23C2×C23—235no—
24C2×C23C2611yescharacteristic
252−1+4C22×C23156yescomplemented
26H(5,𝔽2)C22×C231280yescomplemented
27C2×#41C22×C231210yescomplemented
28C22×D4C22×C231105yescomplemented
29C23H(7,𝔽2)11yescharacteristic, direct product factor, Sylow
30C24C23×C23130yescomplemented
31#41C23×C231560yescomplemented
32C2×Q8C23×C23170yescomplemented
33C2×D4C23×C231630yescomplemented
34C22×C4C23×C231105yescomplemented
35C23—830no—
36C23C24×C231105yes—
37Q8C24×C23156yes—
38D4C24×C231280yes—
39C2×C4C24×C231210yes—
40C22—4105no—
41C22C25×C23135yes—
42C4C25×C23128yes—
43C2—235no—
44C2C26×C2311yescharacteristic
subgroupG/H
center, socleC2×C23C26
commutator, Frattini, cosocleC2C26×C23
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#585C2semidirect
2. C2×C23×H(5,𝔽2)C2semidirect
3. C23×2−1+4C22semidirect
4. C23×H(5,𝔽2)C22semidirect
5. C2×#41×C23C22semidirect
6. C22×D4×C23C22semidirect
7. C24×C23C23semidirect
8. #41×C23C23semidirect
9. C2×Q8×C23C23semidirect
10. C2×D4×C23C23semidirect
11. C22×C4×C23C23semidirect
17. H(7,𝔽2)C23direct (N × H)
21. #585C2×C23semidirect
22. C2×H(5,𝔽2)C2×C23semidirect
25. 2−1+4C22×C23semidirect
26. H(5,𝔽2)C22×C23semidirect
27. C2×#41C22×C23semidirect
28. C22×D4C22×C23semidirect
29. C23H(7,𝔽2)direct (N × H)
30. C24C23×C23semidirect
31. #41C23×C23semidirect
32. C2×Q8C23×C23semidirect
33. C2×D4C23×C23semidirect
34. C22×C4C23×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#5852
C2×C23×H(5,𝔽2)2
H(7,𝔽2)23