Finite Group Databasebeta

← #72246783

Subgroups — #72246783

Order 6144.

At least 7677 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has at least 7677 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72245113C211yesmaximal, characteristic, complemented
2#72244982C211yesmaximal, characteristic, complemented
3#72238589D311yescharacteristic, complemented
4C2×#72209772C211yesmaximal, characteristic
5C2×#11710730C2211yescharacteristic
6#72209772C412yescomplemented
7#11710730C2×C411yescharacteristic, complemented
8C2×#62817C2×D311yescharacteristic
9#62817C4×D311yescharacteristic, complemented
10#62652S411yescharacteristic, complemented
11#20609S411yescharacteristic, complemented
12#14679S411yescharacteristic, complemented
13C27C2×S411yescharacteristic
14C22×H(5,𝔽2)C2×S411yescharacteristic
15C22×H(5,𝔽2)C2×S411yescharacteristic
16C26C4×S411yescharacteristic, complemented
17C2×H(5,𝔽2)C4×S411yescharacteristic
18C2×H(5,𝔽2)C4×S411yescharacteristic, complemented
19C22×C22⋊C4#102011yescharacteristic
20C25#1020⋊C212yes—
21C25C2×#102011yescharacteristic
22C23×C4#593411yescharacteristic
23C23×C4#593411yescharacteristic
24C24#1020⋊C412yescomplemented
25C24C4×#102011yescharacteristic
26C24C2×#593411yescharacteristic
27C24#8751911yescharacteristic
28C23C4×#593411yescharacteristic
29C23#87588⋊C211yescharacteristic
30C22#7220965311yescharacteristic
31C22#7223498111yescharacteristic
32C22#7224127611yescharacteristic
33C2#7220983211yescharacteristic
34C2#7223344711yescharacteristic
35C2#7224628211yescharacteristic
subgroupG/H
center, socleC22#72209653
commutator, hypocenter#11710730C2×C4
FrattiniC25C2×#1020
Fitting#72238589D3
prime radicalC2×#72209772C2
cosocleC2×#11710730C22
hypercenterC23#87588⋊C2
LCSG ▷ #11710730
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. #72245113C2semidirect
2. #72244982C2semidirect
3. #72238589D3semidirect
6. #72209772C4semidirect
7. #11710730C2×C4semidirect
9. #62817C4×D3semidirect
10. #62652S4semidirect
11. #20609S4semidirect
12. #14679S4semidirect
16. C26C4×S4semidirect
18. C2×H(5,𝔽2)C4×S4semidirect
24. C24#1020⋊C4semidirect

Maximal subgroups

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

subgroupindex
#722467583
#722467094
#722466974
#722451132
#722449822
#722437774
C2×#722097722