Finite Group Databasebeta

← H(11,𝔽2) #72258202

Subgroups — H(11,𝔽2) #72258202

Order 2048.

229974158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72258201C21496yesmaximal, complemented
2C2×H(9,𝔽2)C21527yesmaximal, complemented
32−1+8C22119840yescomplemented
4H(9,𝔽2)C22167456yescomplemented
5C2×#63103C22163240yescomplemented
6C22×H(7,𝔽2)C22123715yescomplemented
7#63103C2312698240yescomplemented
8C2×2−1+6C231590240yescomplemented
9C2×H(7,𝔽2)C2312276640yescomplemented
10C22×#585C231664020yescomplemented
11C23×H(5,𝔽2)C231118575yescomplemented
122−1+6C2417555072yescomplemented
13H(7,𝔽2)C24116189440yescomplemented
14C2×#585C24121248640yescomplemented
15C22×2−1+4C2411328040yescomplemented
16C22×H(5,𝔽2)C2416640200yescomplemented
17C23×#41C241711450yescomplemented
18C24×D4C24171145yescomplemented
19C26C2514590yescomplemented
20#585C25145330432yescomplemented
21C2×2−1+4C25112749184yescomplemented
22C2×H(5,𝔽2)C25135414400yescomplemented
23C22×#41C25113280400yescomplemented
24C23×Q8C251237150yescomplemented
25C23×D4C2512134350yescomplemented
26C24×C4C25171145yescomplemented
27C25—324590no—
28C25C26171145yes—
292−1+4C2617555072yes—
30H(5,𝔽2)C26116189440yes—
31C2×#41C26121248640yes—
32C22×Q8C2611328040yes—
33C22×D4C2616640200yes—
34C23×C4C261711450yes—
35C24—1671145no—
36C24C271118575yes—
37#41C2712698240yes—
38C2×Q8C271590240yes—
39C2×D4C2712276640yes—
40C22×C4C271664020yes—
41C23—8118575no—
42C23C28123715yes—
43Q8C28119840yes—
44D4C28167456yes—
45C2×C4C28163240yes—
46C22—423715no—
47C22C291527yes—
48C4C291496yes—
49C2—2527no—
50C2C21011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC2C210
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ G
LECSG ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72258201C2semidirect
2. C2×H(9,𝔽2)C2semidirect
3. 2−1+8C22semidirect
4. H(9,𝔽2)C22semidirect
5. C2×#63103C22semidirect
6. C22×H(7,𝔽2)C22semidirect
7. #63103C23semidirect
8. C2×2−1+6C23semidirect
9. C2×H(7,𝔽2)C23semidirect
10. C22×#585C23semidirect
11. C23×H(5,𝔽2)C23semidirect
12. 2−1+6C24semidirect
13. H(7,𝔽2)C24semidirect
14. C2×#585C24semidirect
15. C22×2−1+4C24semidirect
16. C22×H(5,𝔽2)C24semidirect
17. C23×#41C24semidirect
18. C24×D4C24semidirect
19. C26C25semidirect
20. #585C25semidirect
21. C2×2−1+4C25semidirect
22. C2×H(5,𝔽2)C25semidirect
23. C22×#41C25semidirect
24. C23×Q8C25semidirect
25. C23×D4C25semidirect
26. C24×C4C25semidirect

Maximal subgroups

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

subgroupindex
#722582012
C2×H(9,𝔽2)2