Finite Group Databasebeta

← C23×#13095512 #27156535

Subgroups — C23×#13095512 #27156535

Order 12416.

6520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13095512C2156yesmaximal, direct product factor
2C23×D388C213yesmaximal, complemented
3C23×C4×D97C213yesmaximal, complemented
4C23×Q8×C97C211yesmaximal, characteristic, complemented
5C24×D97C2213yes—
6C2×#13095512C221448yesdirect product factor
7C22×D388C22184yescomplemented
8C23×Dic97C2211yescharacteristic
9C22×C4×D97C22184yescomplemented
10C22×Q8×C97C22128yescomplemented
11C23×C4×C97C2213yes—
12C23×D97—224no—
13C23×D97C23142yes—
14#13095512C231512yesdirect product factor
15C2×D388C231336yescomplemented
16C22×Dic97C23114yes—
17C2×C4×D97C231336yescomplemented
18C24×C97C2311yescharacteristic
19C2×Q8×C97C231112yescomplemented
20C22×C4×C97C23142yes—
21C23×C97#4118yescomplemented
22C23×C97C2417yes—
23C22×D97—2168no—
24C22×D97C24184yes—
25Q8×C97C24164yescomplemented
26C2×C4×C97C24184yes—
27C2×Dic97C24128yes—
28D388C241192yescomplemented
29C4×D97C241192yescomplemented
30C22×C97C2×#41128yescomplemented
31C22×C97C2517yes—
32C2×D97—2168no—
33C2×D97C25124yes—
34C4×C97C25124yes—
35Dic97C2518yes—
36C2×C97C22×#41114yescomplemented
37C2×C97C2611yescharacteristic
38D97—224no—
39C23×#41—971nomaximal, Sylow
40C97C23×#4111yescharacteristic, complemented, Sylow
41C22×#41—9756no—
42C23×Q8D9711yescharacteristic, complemented
43C23×D4—973no—
44C24×C4—973no—
45C25—973no—
46C2×#41—97448no—
47C22×Q8C2×D97128yescomplemented
48C22×D4—9784no—
49C23×C4—971no—
50C23×C4—9784no—
51C23×C4C2×D9713yes—
52C24—9742no—
53C24—19424no—
54C24C22×D9711yescharacteristic
55#41—97512no—
56C2×Q8C22×D971112yescomplemented
57C2×D4—97336no—
58C22×C4—9714no—
59C22×C4—97336no—
60C22×C4C22×D97142yes—
61C23—9784no—
62C23—194168no—
63C23#1309551218yesdirect product factor
64C23C23×D9717yes—
65Q8C23×D97164yescomplemented
66D4—97192no—
67C2×C4—9728no—
68C2×C4—97192no—
69C2×C4C23×D97184yes—
70C22—9724no—
71C22—194168no—
72C22C2×#13095512128yesdirect product factor
73C22C24×D9717yes—
74C4—978no—
75C4C24×D97124yes—
76C2—19424no—
77C2C22×#13095512114yesdirect product factor
78C2C25×D9711yescharacteristic
subgroupG/H
centerC24C22×D97
commutator, cosocleC2×C97C26
FrattiniC2C25×D97
FittingC23×Q8×C97C2
socleC24×C97C23
hypercenterC23×Q8D97
hypocenterC97C23×#41
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C24 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13095512C2direct (N × H)
2. C23×D388C2semidirect
3. C23×C4×D97C2semidirect
4. C23×Q8×C97C2semidirect
6. C2×#13095512C22direct (N × H)
7. C22×D388C22semidirect
9. C22×C4×D97C22semidirect
10. C22×Q8×C97C22semidirect
14. #13095512C23direct (N × H)
15. C2×D388C23semidirect
17. C2×C4×D97C23semidirect
19. C2×Q8×C97C23semidirect
21. C23×C97#41semidirect
25. Q8×C97C24semidirect
28. D388C24semidirect
29. C4×D97C24semidirect
30. C22×C97C2×#41semidirect
36. C2×C97C22×#41semidirect
40. C97C23×#41semidirect
42. C23×Q8D97semidirect
47. C22×Q8C2×D97semidirect
56. C2×Q8C22×D97semidirect
63. C23#13095512direct (N × H)
65. Q8C23×D97semidirect
72. C22C2×#13095512direct (N × H)
77. C2C22×#13095512direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955122
C23×D3882
C23×C4×D972
C23×Q8×C972
C23×#4197