Finite Group Databasebeta

← #72265383

Subgroups — #72265383

Order 10240.

At least 872265 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has at least 872265 conjugacy classes of subgroups — too many to draw. Only its 20 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#72265267C211yesmaximal, characteristic, complemented
2#72258201D511yescharacteristic, complemented
3#72227762C211yesmaximal, characteristic, complemented
4#72227629C211yesmaximal, characteristic, complemented
5#72210028C2211yescharacteristic
6H(9,𝔽2)C2×D511yescharacteristic
7#585(C24⋊C5)⋊C211yescharacteristic, complemented
8#585(C24⋊C5)⋊C211yescharacteristic, complemented
9#585(C24⋊C5)⋊C211yescharacteristic, complemented
10#585(C24⋊C5)⋊C211yescharacteristic, complemented
11C24×C4(C24⋊C5)⋊C211yescharacteristic
12C25C2×(C24⋊C5)⋊C211yescharacteristic
132−1+4C2×(C24⋊C5)⋊C211yescharacteristic
142−1+4C2×(C24⋊C5)⋊C211yescharacteristic
152−1+4C2×(C24⋊C5)⋊C211yescharacteristic
162−1+4C2×(C24⋊C5)⋊C211yescharacteristic
17C4#7220970111yescharacteristic
18C2C2×#7220970111yescharacteristic
subgroupG/H
center, hypercenterC4#72209701
commutator, cosocle, hypocenter#72210028C22
Frattini, socleC2C2×#72209701
Fitting#72258201D5
LCSG ▷ #72210028
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #72265267C2semidirect
2. #72258201D5semidirect
3. #72227762C2semidirect
4. #72227629C2semidirect
7. #585(C24⋊C5)⋊C2semidirect
8. #585(C24⋊C5)⋊C2semidirect
9. #585(C24⋊C5)⋊C2semidirect
10. #585(C24⋊C5)⋊C2semidirect

Maximal subgroups

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

subgroupindex
#722652672
#722630515
#722277622
#722276292
#1061938216
C4×(C24⋊C5)⋊C216