Finite Group Databasebeta

← #72258201

Subgroups — #72258201

Order 1024.

8302839 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
12−1+8C21120yesmaximal, complemented
2H(9,𝔽2)C21136yesmaximal, complemented
3C2×#63103C21255yesmaximal, complemented
4#63103C2215440yescomplemented
5#63103C22116320yescomplemented
6C2×2−1+6C2217140yescomplemented
7C2×H(7,𝔽2)C2219180yescomplemented
8C22×#585C2215355yescomplemented
92−1+6C231152320yescomplemented
10H(7,𝔽2)C231195840yescomplemented
11C2×#585C23185680yescomplemented
12C2×#585C231257040yescomplemented
13C22×2−1+4C23132130yescomplemented
14C22×H(5,𝔽2)C23153550yescomplemented
15C23×#41C23111475yescomplemented
16#585C24191392yescomplemented
17#585C2411370880yescomplemented
18C2×2−1+4C241514080yescomplemented
19C2×H(5,𝔽2)C241856800yescomplemented
20C22×#41C241107100yescomplemented
21C22×#41C241321300yescomplemented
22C23×Q8C24111475yescomplemented
23C23×D4C24134425yescomplemented
24C24×C4C2412295yescomplemented
25C25C2512295yes—
262−1+4C251548352yes—
27H(5,𝔽2)C251913920yes—
28C2×#41C25185680yes—
29C2×#41C2511285200yes—
30C22×Q8C251107100yes—
31C22×D4C251321300yes—
32C23×C4C25111475yes—
33C23×C4C25134425yes—
34C24—162295no—
35C24C26111475yes—
36#41C2615440yes—
37#41C261342720yes—
38C2×Q8C26185680yes—
39C2×D4C261257040yes—
40C22×C4C2615355yes—
41C22×C4C26180325yes—
42C23—811475no—
43C23C2715355yes—
44Q8C2715440yes—
45D4C27116320yes—
46C2×C4C27116065yes—
47C2×C4C271255yes—
48C22—45355no—
49C22C281255yes—
50C4C281255yes—
51C4C2811yescharacteristic
52C2—2255no—
53C2C2911yescharacteristic
subgroupG/H
center, socleC4C28
commutator, Frattini, cosocleC2C29
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ G
LECSG ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. 2−1+8C2semidirect
2. H(9,𝔽2)C2semidirect
3. C2×#63103C2semidirect
4. #63103C22semidirect
5. #63103C22semidirect
6. C2×2−1+6C22semidirect
7. C2×H(7,𝔽2)C22semidirect
8. C22×#585C22semidirect
9. 2−1+6C23semidirect
10. H(7,𝔽2)C23semidirect
11. C2×#585C23semidirect
12. C2×#585C23semidirect
13. C22×2−1+4C23semidirect
14. C22×H(5,𝔽2)C23semidirect
15. C23×#41C23semidirect
16. #585C24semidirect
17. #585C24semidirect
18. C2×2−1+4C24semidirect
19. C2×H(5,𝔽2)C24semidirect
20. C22×#41C24semidirect
21. C22×#41C24semidirect
22. C23×Q8C24semidirect
23. C23×D4C24semidirect
24. C24×C4C24semidirect

Maximal subgroups

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

subgroupindex
2−1+82
H(9,𝔽2)2
C2×#631032