Finite Group Databasebeta

← C89×2−1+6 #26018069

Subgroups — C89×2−1+6 #26018069

Order 11392.

5796 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#585×C89C2136yesmaximal, complemented
2C2×2−1+4×C89C2127yesmaximal, complemented
32−1+4×C89C221216yescomplemented
4H(5,𝔽2)×C89C221120yescomplemented
5C2×#41×C89C221270yescomplemented
6C22×Q8×C89C22145yescomplemented
7#41×C89C231720yes—
8C2×Q8×C89C231270yes—
9C2×D4×C89C231270yes—
10C22×C4×C89C231135yes—
11C23×C89C24145yes—
12Q8×C89C241120yes—
13D4×C89C241216yes—
14C2×C4×C89C241270yes—
15C22×C89—445no—
16C22×C89C25127yes—
17C4×C89C25136yes—
18C2×C89—227no—
19C2×C89C2611yescharacteristic
202−1+6C8911yesmaximal, characteristic, direct product factor, Sylow
21C892−1+611yescharacteristic, direct product factor, Sylow
22#585C2×C89136yescomplemented
23C2×2−1+4C2×C89127yescomplemented
242−1+4C22×C891216yescomplemented
25H(5,𝔽2)C22×C891120yescomplemented
26C2×#41C22×C891270yescomplemented
27C22×Q8C22×C89145yescomplemented
28#41C23×C891720yes—
29C2×Q8C23×C891270yes—
30C2×D4C23×C891270yes—
31C22×C4C23×C891135yes—
32C23C24×C89145yes—
33Q8C24×C891120yes—
34D4C24×C891216yes—
35C2×C4C24×C891270yes—
36C22—445no—
37C22C25×C89127yes—
38C4C25×C89136yes—
39C2—227no—
40C2C26×C8911yescharacteristic
subgroupG/H
center, socleC2×C89C26
commutator, Frattini, cosocleC2C26×C89
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. #585×C89C2semidirect
2. C2×2−1+4×C89C2semidirect
3. 2−1+4×C89C22semidirect
4. H(5,𝔽2)×C89C22semidirect
5. C2×#41×C89C22semidirect
6. C22×Q8×C89C22semidirect
20. 2−1+6C89direct (N × H)
21. C892−1+6direct (N × H)
22. #585C2×C89semidirect
23. C2×2−1+4C2×C89semidirect
24. 2−1+4C22×C89semidirect
25. H(5,𝔽2)C22×C89semidirect
26. C2×#41C22×C89semidirect
27. C22×Q8C22×C89semidirect

Maximal subgroups

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

subgroupindex
#585×C892
C2×2−1+4×C892
2−1+689