Finite Group Databasebeta

← H(7,𝔽2)×C131 #30663777

Subgroups — H(7,𝔽2)×C131 #30663777

Order 16768.

5992 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#585×C131C2128yesmaximal, complemented
2C2×H(5,𝔽2)×C131C2135yesmaximal, complemented
32−1+4×C131C22156yescomplemented
4H(5,𝔽2)×C131C221280yescomplemented
5C2×#41×C131C221210yescomplemented
6C22×D4×C131C221105yescomplemented
7C24×C131C23130yescomplemented
8#41×C131C231560yescomplemented
9C2×Q8×C131C23170yescomplemented
10C2×D4×C131C231630yescomplemented
11C22×C4×C131C231105yescomplemented
12C23×C131—830no—
13C23×C131C241105yes—
14Q8×C131C24156yes—
15D4×C131C241280yes—
16C2×C4×C131C241210yes—
17C22×C131—4105no—
18C22×C131C25135yes—
19C4×C131C25128yes—
20C2×C131—235no—
21C2×C131C2611yescharacteristic
22C131H(7,𝔽2)11yescharacteristic, direct product factor, Sylow
23H(7,𝔽2)C13111yesmaximal, characteristic, direct product factor, Sylow
24#585C2×C131128yescomplemented
25C2×H(5,𝔽2)C2×C131135yescomplemented
262−1+4C22×C131156yescomplemented
27H(5,𝔽2)C22×C1311280yescomplemented
28C2×#41C22×C1311210yescomplemented
29C22×D4C22×C1311105yescomplemented
30C24C23×C131130yescomplemented
31#41C23×C1311560yescomplemented
32C2×Q8C23×C131170yescomplemented
33C2×D4C23×C1311630yescomplemented
34C22×C4C23×C1311105yescomplemented
35C23—830no—
36C23C24×C1311105yes—
37Q8C24×C131156yes—
38D4C24×C1311280yes—
39C2×C4C24×C1311210yes—
40C22—4105no—
41C22C25×C131135yes—
42C4C25×C131128yes—
43C2—235no—
44C2C26×C13111yescharacteristic
subgroupG/H
center, socleC2×C131C26
commutator, Frattini, cosocleC2C26×C131
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C131 ◁ G

Complements

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

NH ≅ G/Ntype
1. #585×C131C2semidirect
2. C2×H(5,𝔽2)×C131C2semidirect
3. 2−1+4×C131C22semidirect
4. H(5,𝔽2)×C131C22semidirect
5. C2×#41×C131C22semidirect
6. C22×D4×C131C22semidirect
7. C24×C131C23semidirect
8. #41×C131C23semidirect
9. C2×Q8×C131C23semidirect
10. C2×D4×C131C23semidirect
11. C22×C4×C131C23semidirect
22. C131H(7,𝔽2)direct (N × H)
23. H(7,𝔽2)C131direct (N × H)
24. #585C2×C131semidirect
25. C2×H(5,𝔽2)C2×C131semidirect
26. 2−1+4C22×C131semidirect
27. H(5,𝔽2)C22×C131semidirect
28. C2×#41C22×C131semidirect
29. C22×D4C22×C131semidirect
30. C24C23×C131semidirect
31. #41C23×C131semidirect
32. C2×Q8C23×C131semidirect
33. C2×D4C23×C131semidirect
34. C22×C4C23×C131semidirect

Maximal subgroups

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

subgroupindex
#585×C1312
C2×H(5,𝔽2)×C1312
H(7,𝔽2)131