Finite Group Databasebeta

← C23×#13108075 #27185931

Subgroups — C23×#13108075 #27185931

Order 12928.

6520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13108075C2156yesmaximal, direct product factor
2C23×D404C213yesmaximal, complemented
3C23×C4×D101C213yesmaximal, complemented
4C23×Q8×C101C211yesmaximal, characteristic, complemented
5C24×D101C2213yes—
6C2×#13108075C221448yesdirect product factor
7C22×D404C22184yescomplemented
8C23×Dic101C2211yescharacteristic
9C22×C4×D101C22184yescomplemented
10C22×Q8×C101C22128yescomplemented
11C23×C4×C101C2213yes—
12C23×D101—224no—
13C23×D101C23142yes—
14#13108075C231512yesdirect product factor
15C2×D404C231336yescomplemented
16C22×Dic101C23114yes—
17C2×C4×D101C231336yescomplemented
18C24×C101C2311yescharacteristic
19C2×Q8×C101C231112yescomplemented
20C22×C4×C101C23142yes—
21C23×C101#4118yescomplemented
22C23×C101C2417yes—
23C22×D101—2168no—
24C22×D101C24184yes—
25Q8×C101C24164yescomplemented
26C2×C4×C101C24184yes—
27C2×Dic101C24128yes—
28D404C241192yescomplemented
29C4×D101C241192yescomplemented
30C22×C101C2×#41128yescomplemented
31C22×C101C2517yes—
32C2×D101—2168no—
33C2×D101C25124yes—
34C4×C101C25124yes—
35Dic101C2518yes—
36C2×C101C22×#41114yescomplemented
37C2×C101C2611yescharacteristic
38D101—224no—
39C23×#41—1011nomaximal, Sylow
40C101C23×#4111yescharacteristic, complemented, Sylow
41C22×#41—10156no—
42C23×Q8D10111yescharacteristic, complemented
43C23×D4—1013no—
44C24×C4—1013no—
45C25—1013no—
46C2×#41—101448no—
47C22×Q8C2×D101128yescomplemented
48C22×D4—10184no—
49C23×C4—1011no—
50C23×C4—10184no—
51C23×C4C2×D10113yes—
52C24—10142no—
53C24—20224no—
54C24C22×D10111yescharacteristic
55#41—101512no—
56C2×Q8C22×D1011112yescomplemented
57C2×D4—101336no—
58C22×C4—10114no—
59C22×C4—101336no—
60C22×C4C22×D101142yes—
61C23—10184no—
62C23—202168no—
63C23#1310807518yesdirect product factor
64C23C23×D10117yes—
65Q8C23×D101164yescomplemented
66D4—101192no—
67C2×C4—10128no—
68C2×C4—101192no—
69C2×C4C23×D101184yes—
70C22—10124no—
71C22—202168no—
72C22C2×#13108075128yesdirect product factor
73C22C24×D10117yes—
74C4—1018no—
75C4C24×D101124yes—
76C2—20224no—
77C2C22×#13108075114yesdirect product factor
78C2C25×D10111yescharacteristic
subgroupG/H
centerC24C22×D101
commutator, cosocleC2×C101C26
FrattiniC2C25×D101
FittingC23×Q8×C101C2
socleC24×C101C23
hypercenterC23×Q8D101
hypocenterC101C23×#41
LCSG ▷ C2×C101 ▷ C101
UCSC1 ◁ C24 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13108075C2direct (N × H)
2. C23×D404C2semidirect
3. C23×C4×D101C2semidirect
4. C23×Q8×C101C2semidirect
6. C2×#13108075C22direct (N × H)
7. C22×D404C22semidirect
9. C22×C4×D101C22semidirect
10. C22×Q8×C101C22semidirect
14. #13108075C23direct (N × H)
15. C2×D404C23semidirect
17. C2×C4×D101C23semidirect
19. C2×Q8×C101C23semidirect
21. C23×C101#41semidirect
25. Q8×C101C24semidirect
28. D404C24semidirect
29. C4×D101C24semidirect
30. C22×C101C2×#41semidirect
36. C2×C101C22×#41semidirect
40. C101C23×#41semidirect
42. C23×Q8D101semidirect
47. C22×Q8C2×D101semidirect
56. C2×Q8C22×D101semidirect
63. C23#13108075direct (N × H)
65. Q8C23×D101semidirect
72. C22C2×#13108075direct (N × H)
77. C2C22×#13108075direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131080752
C23×D4042
C23×C4×D1012
C23×Q8×C1012
C23×#41101