Finite Group Databasebeta

← C23×2−1+6 #15671214

Subgroups — C23×2−1+6 #15671214

Order 2944.

5796 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#585C2136yesmaximal, complemented
2C2×C23×2−1+4C2127yesmaximal, complemented
3C23×2−1+4C221216yescomplemented
4C23×H(5,𝔽2)C221120yescomplemented
5C2×#41×C23C221270yescomplemented
6C22×Q8×C23C22145yescomplemented
7#41×C23C231720yes—
8C2×Q8×C23C231270yes—
9C2×D4×C23C231270yes—
10C22×C4×C23C231135yes—
11C23×C23C24145yes—
12Q8×C23C241120yes—
13D4×C23C241216yes—
14C2×C4×C23C241270yes—
152−1+6C2311yesmaximal, characteristic, direct product factor, Sylow
16C22×C23—445no—
17C22×C23C25127yes—
18C4×C23C25136yes—
19#585C2×C23136yescomplemented
20C2×2−1+4C2×C23127yescomplemented
21C2×C23—227no—
22C2×C23C2611yescharacteristic
232−1+4C22×C231216yescomplemented
24H(5,𝔽2)C22×C231120yescomplemented
25C2×#41C22×C231270yescomplemented
26C22×Q8C22×C23145yescomplemented
27C232−1+611yescharacteristic, direct product factor, Sylow
28#41C23×C231720yes—
29C2×Q8C23×C231270yes—
30C2×D4C23×C231270yes—
31C22×C4C23×C231135yes—
32C23C24×C23145yes—
33Q8C24×C231120yes—
34D4C24×C231216yes—
35C2×C4C24×C231270yes—
36C22—445no—
37C22C25×C23127yes—
38C4C25×C23136yes—
39C2—227no—
40C2C26×C2311yescharacteristic
subgroupG/H
center, socleC2×C23C26
commutator, Frattini, cosocleC2C26×C23
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#585C2semidirect
2. C2×C23×2−1+4C2semidirect
3. C23×2−1+4C22semidirect
4. C23×H(5,𝔽2)C22semidirect
5. C2×#41×C23C22semidirect
6. C22×Q8×C23C22semidirect
15. 2−1+6C23direct (N × H)
19. #585C2×C23semidirect
20. C2×2−1+4C2×C23semidirect
23. 2−1+4C22×C23semidirect
24. H(5,𝔽2)C22×C23semidirect
25. C2×#41C22×C23semidirect
26. C22×Q8C22×C23semidirect
27. C232−1+6direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5852
C2×C23×2−1+42
2−1+623