Finite Group Databasebeta

← H(7,𝔽2)×C223 #43252609

Subgroups — H(7,𝔽2)×C223 #43252609

Order 28544.

5992 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#585×C223C2128yesmaximal, complemented
2C2×H(5,𝔽2)×C223C2135yesmaximal, complemented
32−1+4×C223C22156yescomplemented
4H(5,𝔽2)×C223C221280yescomplemented
5C2×#41×C223C221210yescomplemented
6C22×D4×C223C221105yescomplemented
7C24×C223C23130yescomplemented
8#41×C223C231560yescomplemented
9C2×Q8×C223C23170yescomplemented
10C2×D4×C223C231630yescomplemented
11C22×C4×C223C231105yescomplemented
12C23×C223—830no—
13C23×C223C241105yes—
14Q8×C223C24156yes—
15D4×C223C241280yes—
16C2×C4×C223C241210yes—
17C22×C223—4105no—
18C22×C223C25135yes—
19C4×C223C25128yes—
20C2×C223—235no—
21C2×C223C2611yescharacteristic
22C223H(7,𝔽2)11yescharacteristic, direct product factor, Sylow
23H(7,𝔽2)C22311yesmaximal, characteristic, direct product factor, Sylow
24#585C2×C223128yescomplemented
25C2×H(5,𝔽2)C2×C223135yescomplemented
262−1+4C22×C223156yescomplemented
27H(5,𝔽2)C22×C2231280yescomplemented
28C2×#41C22×C2231210yescomplemented
29C22×D4C22×C2231105yescomplemented
30C24C23×C223130yescomplemented
31#41C23×C2231560yescomplemented
32C2×Q8C23×C223170yescomplemented
33C2×D4C23×C2231630yescomplemented
34C22×C4C23×C2231105yescomplemented
35C23—830no—
36C23C24×C2231105yes—
37Q8C24×C223156yes—
38D4C24×C2231280yes—
39C2×C4C24×C2231210yes—
40C22—4105no—
41C22C25×C223135yes—
42C4C25×C223128yes—
43C2—235no—
44C2C26×C22311yescharacteristic
subgroupG/H
center, socleC2×C223C26
commutator, Frattini, cosocleC2C26×C223
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C223 ◁ G

Complements

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

NH ≅ G/Ntype
1. #585×C223C2semidirect
2. C2×H(5,𝔽2)×C223C2semidirect
3. 2−1+4×C223C22semidirect
4. H(5,𝔽2)×C223C22semidirect
5. C2×#41×C223C22semidirect
6. C22×D4×C223C22semidirect
7. C24×C223C23semidirect
8. #41×C223C23semidirect
9. C2×Q8×C223C23semidirect
10. C2×D4×C223C23semidirect
11. C22×C4×C223C23semidirect
22. C223H(7,𝔽2)direct (N × H)
23. H(7,𝔽2)C223direct (N × H)
24. #585C2×C223semidirect
25. C2×H(5,𝔽2)C2×C223semidirect
26. 2−1+4C22×C223semidirect
27. H(5,𝔽2)C22×C223semidirect
28. C2×#41C22×C223semidirect
29. C22×D4C22×C223semidirect
30. C24C23×C223semidirect
31. #41C23×C223semidirect
32. C2×Q8C23×C223semidirect
33. C2×D4C23×C223semidirect
34. C22×C4C23×C223semidirect

Maximal subgroups

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

subgroupindex
#585×C2232
C2×H(5,𝔽2)×C2232
H(7,𝔽2)223