Finite Group Databasebeta

← 2−1+6×C251 #44548182

Subgroups — 2−1+6×C251 #44548182

Order 32128.

5796 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#585×C251C2136yesmaximal, complemented
2C2×2−1+4×C251C2127yesmaximal, complemented
32−1+4×C251C221216yescomplemented
4H(5,𝔽2)×C251C221120yescomplemented
5C2×#41×C251C221270yescomplemented
6C22×Q8×C251C22145yescomplemented
7#41×C251C231720yes—
8C2×Q8×C251C231270yes—
9C2×D4×C251C231270yes—
10C22×C4×C251C231135yes—
11C23×C251C24145yes—
12Q8×C251C241120yes—
13D4×C251C241216yes—
14C2×C4×C251C241270yes—
15C22×C251—445no—
16C22×C251C25127yes—
17C4×C251C25136yes—
18C2×C251—227no—
19C2×C251C2611yescharacteristic
20C2512−1+611yescharacteristic, direct product factor, Sylow
212−1+6C25111yesmaximal, characteristic, direct product factor, Sylow
22#585C2×C251136yescomplemented
23C2×2−1+4C2×C251127yescomplemented
242−1+4C22×C2511216yescomplemented
25H(5,𝔽2)C22×C2511120yescomplemented
26C2×#41C22×C2511270yescomplemented
27C22×Q8C22×C251145yescomplemented
28#41C23×C2511720yes—
29C2×Q8C23×C2511270yes—
30C2×D4C23×C2511270yes—
31C22×C4C23×C2511135yes—
32C23C24×C251145yes—
33Q8C24×C2511120yes—
34D4C24×C2511216yes—
35C2×C4C24×C2511270yes—
36C22—445no—
37C22C25×C251127yes—
38C4C25×C251136yes—
39C2—227no—
40C2C26×C25111yescharacteristic
subgroupG/H
center, socleC2×C251C26
commutator, Frattini, cosocleC2C26×C251
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. #585×C251C2semidirect
2. C2×2−1+4×C251C2semidirect
3. 2−1+4×C251C22semidirect
4. H(5,𝔽2)×C251C22semidirect
5. C2×#41×C251C22semidirect
6. C22×Q8×C251C22semidirect
20. C2512−1+6direct (N × H)
21. 2−1+6C251direct (N × H)
22. #585C2×C251semidirect
23. C2×2−1+4C2×C251semidirect
24. 2−1+4C22×C251semidirect
25. H(5,𝔽2)C22×C251semidirect
26. C2×#41C22×C251semidirect
27. C22×Q8C22×C251semidirect

Maximal subgroups

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

subgroupindex
#585×C2512
C2×2−1+4×C2512
2−1+6251