Finite Group Databasebeta

← C2×C5×#11925895 #27130937

Subgroups — C2×C5×#11925895 #27130937

Order 12100.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#10597243C211yesmaximal, characteristic, complemented
2C5×#11925895C212yesmaximal, direct product factor
3C5×#10597243C2211yescharacteristic, complemented, Hall
4C2×C5×C11×D11C511yesmaximal, characteristic, complemented
5C2×#11925895C515yesmaximal, direct product factor
6C2×C5×C112C2×C511yescharacteristic, complemented
7C5×C11×D11C2×C512yescomplemented
8C2×#10597243C2×C515yescomplemented
9#11925895C2×C5110yesdirect product factor
10C22×C5×C11⋊C5—111nomaximal
11C2×C5×AGL(1,𝔽11)—111nomaximal
12C5×C112C22×C511yescharacteristic, complemented
13#10597243C22×C515yescomplemented
14C2×C5×C11⋊C5—112no—
15C2×C5×C11⋊C5—111no—
16C2×C5×C11⋊C5—111no—
17C2×C5×C11⋊C5—225no—
18C5×AGL(1,𝔽11)—112no—
19C2×C11×D11C5211yescharacteristic, complemented, Hall
20C5×C11⋊C5—111no—
21C5×C11⋊C5—111no—
22C5×C11⋊C5—225no—
23C2×C112C2×C5211yescharacteristic, complemented
24C11×D11C2×C5212yescomplemented
25C22×C5×C11—111no—
26C2×C5×D11C11⋊C511yescharacteristic, complemented
27C22×C11⋊C5—115no—
28C2×AGL(1,𝔽11)—115no—
29C112C22×C5211yescharacteristic, complemented, Sylow
30C2×C5×C11—25no—
31C2×C5×C11—112no—
32C2×C5×C11AGL(1,𝔽11)11yescharacteristic, complemented
33C2×C5×C11C2×C11⋊C511yescharacteristic, complemented
34C5×D11C2×C11⋊C512yescomplemented
35C2×C11⋊C5—1110no—
36C2×C11⋊C5—115no—
37C2×C11⋊C5—115no—
38C2×C11⋊C5—2225no—
39AGL(1,𝔽11)—1110no—
40C22×C52—1211noHall
41C5×C11—25no—
42C5×C11C2×AGL(1,𝔽11)11yescharacteristic, complemented
43C5×C11C22×C11⋊C511yescharacteristic, complemented
44C11⋊C5—115no—
45C11⋊C5—115no—
46C11⋊C5—2225no—
47C2×C52—1212no—
48C2×C52—1211no—
49C22×C11—111no—
50C2×D11C5×C11⋊C511yescharacteristic, complemented
51C52—1211noSylow
52C2×C11—25no—
53C2×C11—112no—
54C2×C11C5×AGL(1,𝔽11)11yescharacteristic, complemented
55C2×C11C2×C5×C11⋊C511yescharacteristic, complemented
56D11C2×C5×C11⋊C512yescomplemented
57C22×C5—111no—
58C22×C5—1215no—
59C11—25no—
60C11C2×C5×AGL(1,𝔽11)11yescharacteristic, complemented
61C11C22×C5×C11⋊C511yescharacteristic, complemented
62C2×C5—112no—
63C2×C5—12110no—
64C2×C5—1215no—
65C2×C5#1192589511yescharacteristic, direct product factor
66C5—1215no—
67C5C2×#1192589511yescharacteristic, direct product factor
68C22—111noSylow
69C2—112no—
70C2C5×#1192589511yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C5#11925895
commutator, cosocle, hypocenterC112C22×C52
FrattiniC1G
Fitting, socleC2×C5×C112C2×C5
LCSG ▷ C112
UCSC1 ◁ C2×C5

Complements

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

NH ≅ G/Ntype
1. C2×C5×#10597243C2semidirect
2. C5×#11925895C2direct (N × H)
3. C5×#10597243C22semidirect
4. C2×C5×C11×D11C5semidirect
5. C2×#11925895C5direct (N × H)
6. C2×C5×C112C2×C5semidirect
7. C5×C11×D11C2×C5semidirect
8. C2×#10597243C2×C5semidirect
9. #11925895C2×C5direct (N × H)
12. C5×C112C22×C5semidirect
13. #10597243C22×C5semidirect
19. C2×C11×D11C52semidirect
23. C2×C112C2×C52semidirect
24. C11×D11C2×C52semidirect
26. C2×C5×D11C11⋊C5semidirect
29. C112C22×C52semidirect
32. C2×C5×C11AGL(1,𝔽11)semidirect
33. C2×C5×C11C2×C11⋊C5semidirect
34. C5×D11C2×C11⋊C5semidirect
42. C5×C11C2×AGL(1,𝔽11)semidirect
43. C5×C11C22×C11⋊C5semidirect
50. C2×D11C5×C11⋊C5semidirect
54. C2×C11C5×AGL(1,𝔽11)semidirect
55. C2×C11C2×C5×C11⋊C5semidirect
56. D11C2×C5×C11⋊C5semidirect
60. C11C2×C5×AGL(1,𝔽11)semidirect
61. C11C22×C5×C11⋊C5semidirect
65. C2×C5#11925895direct (N × H)
67. C5C2×#11925895direct (N × H)
70. C2C5×#11925895direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#105972432
C5×#119258952
C2×C5×C11×D115
C2×#119258955
C22×C5×C11⋊C511
C2×C5×AGL(1,𝔽11)11