Finite Group Databasebeta

← C24×#11761629 #31769715

Subgroups — C24×#11761629 #31769715

Order 17152.

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×#11761629C21120yesmaximal, direct product factor
2C24×D268C213yesmaximal, complemented
3C24×C4×D67C213yesmaximal, complemented
4C24×Q8×C67C211yesmaximal, characteristic, complemented
5C25×D67C2213yes—
6C22×#11761629C2212240yesdirect product factor
7C23×D268C221180yescomplemented
8C24×Dic67C2211yescharacteristic
9C23×C4×D67C221180yescomplemented
10C23×Q8×C67C22160yescomplemented
11C24×C4×C67C2213yes—
12C24×D67—248no—
13C24×D67C23190yes—
14C2×#11761629C2317680yesdirect product factor
15C22×D268C2311680yescomplemented
16C23×Dic67C23130yes—
17C22×C4×D67C2311680yescomplemented
18C25×C67C2311yescharacteristic
19C22×Q8×C67C231560yescomplemented
20C23×C4×C67C23190yes—
21C23×D67—2720no—
22C23×D67C241420yes—
23#11761629C2414096yesdirect product factor
24C2×D268C2412880yescomplemented
25C22×Dic67C241140yes—
26C2×C4×D67C2412880yescomplemented
27C24×C67#41116yescomplemented
28C24×C67C24115yes—
29C2×Q8×C67C241960yescomplemented
30C22×C4×C67C241420yes—
31C23×C67C2×#411120yescomplemented
32C23×C67C25135yes—
33C22×D67—21680no—
34C22×D67C251360yes—
35Q8×C67C251256yescomplemented
36C2×C4×C67C251360yes—
37C2×Dic67C251120yes—
38D268C251768yescomplemented
39C4×D67C251768yescomplemented
40C22×C67C22×#411140yescomplemented
41C22×C67C26115yes—
42C2×D67—2720no—
43C2×D67C26148yes—
44C4×C67C26148yes—
45Dic67C26116yes—
46C24×#41—671nomaximal, Sylow
47C2×C67C23×#41130yescomplemented
48C2×C67C2711yescharacteristic
49D67—248no—
50C23×#41—67120no—
51C24×Q8D6711yescharacteristic, complemented
52C24×D4—673no—
53C25×C4—673no—
54C67C24×#4111yescharacteristic, complemented, Sylow
55C26—673no—
56C22×#41—672240no—
57C23×Q8C2×D67160yescomplemented
58C23×D4—67180no—
59C24×C4—671no—
60C24×C4—67180no—
61C24×C4C2×D6713yes—
62C25—6790no—
63C25—13448no—
64C25C22×D6711yescharacteristic
65C2×#41—677680no—
66C22×Q8C22×D671560yescomplemented
67C22×D4—671680no—
68C23×C4—6730no—
69C23×C4—671680no—
70C23×C4C22×D67190yes—
71C24—67420no—
72C24—134720no—
73C24#11761629116yesdirect product factor
74C24C23×D67115yes—
75#41—674096no—
76C2×Q8C23×D671960yescomplemented
77C2×D4—672880no—
78C22×C4—67140no—
79C22×C4—672880no—
80C22×C4C23×D671420yes—
81C23—67360no—
82C23—1341680no—
83C23C2×#117616291120yesdirect product factor
84C23C24×D67135yes—
85Q8C24×D671256yescomplemented
86D4—67768no—
87C2×C4—67120no—
88C2×C4—67768no—
89C2×C4C24×D671360yes—
90C22—6748no—
91C22—134720no—
92C22C22×#117616291140yesdirect product factor
93C22C25×D67115yes—
94C4—6716no—
95C4C25×D67148yes—
96C2—13448no—
97C2C23×#11761629130yesdirect product factor
98C2C26×D6711yescharacteristic
subgroupG/H
centerC25C22×D67
commutator, cosocleC2×C67C27
FrattiniC2C26×D67
FittingC24×Q8×C67C2
socleC25×C67C23
hypercenterC24×Q8D67
hypocenterC67C24×#41
LCSG ▷ C2×C67 ▷ C67
UCSC1 ◁ C25 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C23×#11761629C2direct (N × H)
2. C24×D268C2semidirect
3. C24×C4×D67C2semidirect
4. C24×Q8×C67C2semidirect
6. C22×#11761629C22direct (N × H)
7. C23×D268C22semidirect
9. C23×C4×D67C22semidirect
10. C23×Q8×C67C22semidirect
14. C2×#11761629C23direct (N × H)
15. C22×D268C23semidirect
17. C22×C4×D67C23semidirect
19. C22×Q8×C67C23semidirect
23. #11761629C24direct (N × H)
24. C2×D268C24semidirect
26. C2×C4×D67C24semidirect
27. C24×C67#41semidirect
29. C2×Q8×C67C24semidirect
31. C23×C67C2×#41semidirect
35. Q8×C67C25semidirect
38. D268C25semidirect
39. C4×D67C25semidirect
40. C22×C67C22×#41semidirect
47. C2×C67C23×#41semidirect
51. C24×Q8D67semidirect
54. C67C24×#41semidirect
57. C23×Q8C2×D67semidirect
66. C22×Q8C22×D67semidirect
73. C24#11761629direct (N × H)
76. C2×Q8C23×D67semidirect
83. C23C2×#11761629direct (N × H)
85. Q8C24×D67semidirect
92. C22C22×#11761629direct (N × H)
97. C2C23×#11761629direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117616292
C24×D2682
C24×C4×D672
C24×Q8×C672
C24×#4167