Finite Group Databasebeta

← C83×2−1+6 #24905280

Subgroups — C83×2−1+6 #24905280

Order 10624.

5796 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#585×C83C2136yesmaximal, complemented
2C2×2−1+4×C83C2127yesmaximal, complemented
32−1+4×C83C221216yescomplemented
4H(5,𝔽2)×C83C221120yescomplemented
5C2×#41×C83C221270yescomplemented
6C22×Q8×C83C22145yescomplemented
7#41×C83C231720yes—
8C2×Q8×C83C231270yes—
9C2×D4×C83C231270yes—
10C22×C4×C83C231135yes—
11C23×C83C24145yes—
12Q8×C83C241120yes—
13D4×C83C241216yes—
14C2×C4×C83C241270yes—
15C22×C83—445no—
16C22×C83C25127yes—
17C4×C83C25136yes—
18C2×C83—227no—
19C2×C83C2611yescharacteristic
202−1+6C8311yesmaximal, characteristic, direct product factor, Sylow
21C832−1+611yescharacteristic, direct product factor, Sylow
22#585C2×C83136yescomplemented
23C2×2−1+4C2×C83127yescomplemented
242−1+4C22×C831216yescomplemented
25H(5,𝔽2)C22×C831120yescomplemented
26C2×#41C22×C831270yescomplemented
27C22×Q8C22×C83145yescomplemented
28#41C23×C831720yes—
29C2×Q8C23×C831270yes—
30C2×D4C23×C831270yes—
31C22×C4C23×C831135yes—
32C23C24×C83145yes—
33Q8C24×C831120yes—
34D4C24×C831216yes—
35C2×C4C24×C831270yes—
36C22—445no—
37C22C25×C83127yes—
38C4C25×C83136yes—
39C2—227no—
40C2C26×C8311yescharacteristic
subgroupG/H
center, socleC2×C83C26
commutator, Frattini, cosocleC2C26×C83
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. #585×C83C2semidirect
2. C2×2−1+4×C83C2semidirect
3. 2−1+4×C83C22semidirect
4. H(5,𝔽2)×C83C22semidirect
5. C2×#41×C83C22semidirect
6. C22×Q8×C83C22semidirect
20. 2−1+6C83direct (N × H)
21. C832−1+6direct (N × H)
22. #585C2×C83semidirect
23. C2×2−1+4C2×C83semidirect
24. 2−1+4C22×C83semidirect
25. H(5,𝔽2)C22×C83semidirect
26. C2×#41C22×C83semidirect
27. C22×Q8C22×C83semidirect

Maximal subgroups

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

subgroupindex
#585×C832
C2×2−1+4×C832
2−1+683