Finite Group Databasebeta

← C7×H(7,𝔽2) #11744576

Subgroups — C7×H(7,𝔽2) #11744576

Order 896.

5992 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#585C2128yesmaximal, complemented
2C2×C7×H(5,𝔽2)C2135yesmaximal, complemented
3C7×2−1+4C22156yescomplemented
4C7×H(5,𝔽2)C221280yescomplemented
5C2×C7×#41C221210yescomplemented
6C22×C7×D4C221105yescomplemented
7H(7,𝔽2)C711yesmaximal, characteristic, direct product factor, Sylow
8C24×C7C23130yescomplemented
9C7×#41C231560yescomplemented
10C2×C7×Q8C23170yescomplemented
11C2×C7×D4C231630yescomplemented
12C22×C4×C7C231105yescomplemented
13#585C2×C7128yescomplemented
14C2×H(5,𝔽2)C2×C7135yescomplemented
15C23×C7—830no—
16C23×C7C241105yes—
17C7×Q8C24156yes—
18C7×D4C241280yes—
19C2×C4×C7C241210yes—
202−1+4C22×C7156yescomplemented
21H(5,𝔽2)C22×C71280yescomplemented
22C2×#41C22×C71210yescomplemented
23C22×D4C22×C71105yescomplemented
24C22×C7—4105no—
25C22×C7C25135yes—
26C4×C7C25128yes—
27C24C23×C7130yescomplemented
28#41C23×C71560yescomplemented
29C2×Q8C23×C7170yescomplemented
30C2×D4C23×C71630yescomplemented
31C22×C4C23×C71105yescomplemented
32C2×C7—235no—
33C2×C7C2611yescharacteristic
34C23—830no—
35C23C24×C71105yes—
36Q8C24×C7156yes—
37D4C24×C71280yes—
38C2×C4C24×C71210yes—
39C7H(7,𝔽2)11yescharacteristic, direct product factor, Sylow
40C22—4105no—
41C22C25×C7135yes—
42C4C25×C7128yes—
43C2—235no—
44C2C26×C711yescharacteristic
subgroupG/H
center, socleC2×C7C26
commutator, Frattini, cosocleC2C26×C7
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#585C2semidirect
2. C2×C7×H(5,𝔽2)C2semidirect
3. C7×2−1+4C22semidirect
4. C7×H(5,𝔽2)C22semidirect
5. C2×C7×#41C22semidirect
6. C22×C7×D4C22semidirect
7. H(7,𝔽2)C7direct (N × H)
8. C24×C7C23semidirect
9. C7×#41C23semidirect
10. C2×C7×Q8C23semidirect
11. C2×C7×D4C23semidirect
12. C22×C4×C7C23semidirect
13. #585C2×C7semidirect
14. C2×H(5,𝔽2)C2×C7semidirect
20. 2−1+4C22×C7semidirect
21. H(5,𝔽2)C22×C7semidirect
22. C2×#41C22×C7semidirect
23. C22×D4C22×C7semidirect
27. C24C23×C7semidirect
28. #41C23×C7semidirect
29. C2×Q8C23×C7semidirect
30. C2×D4C23×C7semidirect
31. C22×C4C23×C7semidirect
39. C7H(7,𝔽2)direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#5852
C2×C7×H(5,𝔽2)2
H(7,𝔽2)7