Finite Group Databasebeta

← C24×#11719949 #28297976

Subgroups — C24×#11719949 #28297976

Order 13568.

65470 conjugacy classes of subgroups; only the 32651 normal subgroups are shown.

normal subgroups only

This group has 65470 conjugacy classes of subgroups — too many to draw. Only its 32651 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
1C23×#11719949C21120yesmaximal, direct product factor
2C24×D212C213yesmaximal, complemented
3C24×C4×D53C213yesmaximal, complemented
4C24×Q8×C53C211yesmaximal, characteristic, complemented
5C25×D53C2213yes—
6C22×#11719949C2212240yesdirect product factor
7C23×D212C221180yescomplemented
8C24×Dic53C2211yescharacteristic
9C23×C4×D53C221180yescomplemented
10C23×Q8×C53C22160yescomplemented
11C24×C4×C53C2213yes—
12C24×D53—248no—
13C24×D53C23190yes—
14C2×#11719949C2317680yesdirect product factor
15C22×D212C2311680yescomplemented
16C23×Dic53C23130yes—
17C22×C4×D53C2311680yescomplemented
18C25×C53C2311yescharacteristic
19C22×Q8×C53C231560yescomplemented
20C23×C4×C53C23190yes—
21C24×C53#41116yescomplemented
22C24×C53C24115yes—
23C23×D53—2720no—
24C23×D53C241420yes—
25C2×Q8×C53C241960yescomplemented
26C22×C4×C53C241420yes—
27C22×Dic53C241140yes—
28#11719949C2414096yesdirect product factor
29C2×D212C2412880yescomplemented
30C2×C4×D53C2412880yescomplemented
31C23×C53C2×#411120yescomplemented
32C23×C53C25135yes—
33C22×D53—21680no—
34C22×D53C251360yes—
35Q8×C53C251256yescomplemented
36C2×C4×C53C251360yes—
37C2×Dic53C251120yes—
38D212C251768yescomplemented
39C4×D53C251768yescomplemented
40C24×#41—531nomaximal, Sylow
41C22×C53C22×#411140yescomplemented
42C22×C53C26115yes—
43C2×D53—2720no—
44C2×D53C26148yes—
45C4×C53C26148yes—
46Dic53C26116yes—
47C23×#41—53120no—
48C24×Q8D5311yescharacteristic, complemented
49C24×D4—533no—
50C25×C4—533no—
51C2×C53C23×#41130yescomplemented
52C2×C53C2711yescharacteristic
53D53—248no—
54C26—533no—
55C22×#41—532240no—
56C23×Q8C2×D53160yescomplemented
57C23×D4—53180no—
58C24×C4—531no—
59C24×C4—53180no—
60C24×C4C2×D5313yes—
61C53C24×#4111yescharacteristic, complemented, Sylow
62C25—5390no—
63C25—10648no—
64C25C22×D5311yescharacteristic
65C2×#41—537680no—
66C22×Q8C22×D531560yescomplemented
67C22×D4—531680no—
68C23×C4—5330no—
69C23×C4—531680no—
70C23×C4C22×D53190yes—
71C24—53420no—
72C24—106720no—
73C24#11719949116yesdirect product factor
74C24C23×D53115yes—
75#41—534096no—
76C2×Q8C23×D531960yescomplemented
77C2×D4—532880no—
78C22×C4—53140no—
79C22×C4—532880no—
80C22×C4C23×D531420yes—
81C23—53360no—
82C23—1061680no—
83C23C2×#117199491120yesdirect product factor
84C23C24×D53135yes—
85Q8C24×D531256yescomplemented
86D4—53768no—
87C2×C4—53120no—
88C2×C4—53768no—
89C2×C4C24×D531360yes—
90C22—5348no—
91C22—106720no—
92C22C22×#117199491140yesdirect product factor
93C22C25×D53115yes—
94C4—5316no—
95C4C25×D53148yes—
96C2—10648no—
97C2C23×#11719949130yesdirect product factor
98C2C26×D5311yescharacteristic
subgroupG/H
centerC25C22×D53
commutator, cosocleC2×C53C27
FrattiniC2C26×D53
FittingC24×Q8×C53C2
socleC25×C53C23
hypercenterC24×Q8D53
hypocenterC53C24×#41
LCSG ▷ C2×C53 ▷ C53
UCSC1 ◁ C25 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C23×#11719949C2direct (N × H)
2. C24×D212C2semidirect
3. C24×C4×D53C2semidirect
4. C24×Q8×C53C2semidirect
6. C22×#11719949C22direct (N × H)
7. C23×D212C22semidirect
9. C23×C4×D53C22semidirect
10. C23×Q8×C53C22semidirect
14. C2×#11719949C23direct (N × H)
15. C22×D212C23semidirect
17. C22×C4×D53C23semidirect
19. C22×Q8×C53C23semidirect
21. C24×C53#41semidirect
25. C2×Q8×C53C24semidirect
28. #11719949C24direct (N × H)
29. C2×D212C24semidirect
30. C2×C4×D53C24semidirect
31. C23×C53C2×#41semidirect
35. Q8×C53C25semidirect
38. D212C25semidirect
39. C4×D53C25semidirect
41. C22×C53C22×#41semidirect
48. C24×Q8D53semidirect
51. C2×C53C23×#41semidirect
56. C23×Q8C2×D53semidirect
61. C53C24×#41semidirect
66. C22×Q8C22×D53semidirect
73. C24#11719949direct (N × H)
76. C2×Q8C23×D53semidirect
83. C23C2×#11719949direct (N × H)
85. Q8C24×D53semidirect
92. C22C22×#11719949direct (N × H)
97. C2C23×#11719949direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199492
C24×D2122
C24×C4×D532
C24×Q8×C532
C24×#4153