Finite Group Databasebeta

← C24×#11765866 #32911336

Subgroups — C24×#11765866 #32911336

Order 18176.

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×#11765866C21120yesmaximal, direct product factor
2C24×D284C213yesmaximal, complemented
3C24×C4×D71C213yesmaximal, complemented
4C24×Q8×C71C211yesmaximal, characteristic, complemented
5C25×D71C2213yes—
6C22×#11765866C2212240yesdirect product factor
7C23×D284C221180yescomplemented
8C24×Dic71C2211yescharacteristic
9C23×C4×D71C221180yescomplemented
10C23×Q8×C71C22160yescomplemented
11C24×C4×C71C2213yes—
12C24×D71—248no—
13C24×D71C23190yes—
14C2×#11765866C2317680yesdirect product factor
15C22×D284C2311680yescomplemented
16C23×Dic71C23130yes—
17C22×C4×D71C2311680yescomplemented
18C25×C71C2311yescharacteristic
19C22×Q8×C71C231560yescomplemented
20C23×C4×C71C23190yes—
21C23×D71—2720no—
22C23×D71C241420yes—
23#11765866C2414096yesdirect product factor
24C2×D284C2412880yescomplemented
25C22×Dic71C241140yes—
26C2×C4×D71C2412880yescomplemented
27C24×C71#41116yescomplemented
28C24×C71C24115yes—
29C2×Q8×C71C241960yescomplemented
30C22×C4×C71C241420yes—
31C23×C71C2×#411120yescomplemented
32C23×C71C25135yes—
33C22×D71—21680no—
34C22×D71C251360yes—
35Q8×C71C251256yescomplemented
36C2×C4×C71C251360yes—
37C2×Dic71C251120yes—
38D284C251768yescomplemented
39C4×D71C251768yescomplemented
40C22×C71C22×#411140yescomplemented
41C22×C71C26115yes—
42C2×D71—2720no—
43C2×D71C26148yes—
44C4×C71C26148yes—
45Dic71C26116yes—
46C24×#41—711nomaximal, Sylow
47C2×C71C23×#41130yescomplemented
48C2×C71C2711yescharacteristic
49D71—248no—
50C23×#41—71120no—
51C24×Q8D7111yescharacteristic, complemented
52C24×D4—713no—
53C25×C4—713no—
54C71C24×#4111yescharacteristic, complemented, Sylow
55C26—713no—
56C22×#41—712240no—
57C23×Q8C2×D71160yescomplemented
58C23×D4—71180no—
59C24×C4—711no—
60C24×C4—71180no—
61C24×C4C2×D7113yes—
62C25—7190no—
63C25—14248no—
64C25C22×D7111yescharacteristic
65C2×#41—717680no—
66C22×Q8C22×D711560yescomplemented
67C22×D4—711680no—
68C23×C4—7130no—
69C23×C4—711680no—
70C23×C4C22×D71190yes—
71C24—71420no—
72C24—142720no—
73C24#11765866116yesdirect product factor
74C24C23×D71115yes—
75#41—714096no—
76C2×Q8C23×D711960yescomplemented
77C2×D4—712880no—
78C22×C4—71140no—
79C22×C4—712880no—
80C22×C4C23×D711420yes—
81C23—71360no—
82C23—1421680no—
83C23C2×#117658661120yesdirect product factor
84C23C24×D71135yes—
85Q8C24×D711256yescomplemented
86D4—71768no—
87C2×C4—71120no—
88C2×C4—71768no—
89C2×C4C24×D711360yes—
90C22—7148no—
91C22—142720no—
92C22C22×#117658661140yesdirect product factor
93C22C25×D71115yes—
94C4—7116no—
95C4C25×D71148yes—
96C2—14248no—
97C2C23×#11765866130yesdirect product factor
98C2C26×D7111yescharacteristic
subgroupG/H
centerC25C22×D71
commutator, cosocleC2×C71C27
FrattiniC2C26×D71
FittingC24×Q8×C71C2
socleC25×C71C23
hypercenterC24×Q8D71
hypocenterC71C24×#41
LCSG ▷ C2×C71 ▷ C71
UCSC1 ◁ C25 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C23×#11765866C2direct (N × H)
2. C24×D284C2semidirect
3. C24×C4×D71C2semidirect
4. C24×Q8×C71C2semidirect
6. C22×#11765866C22direct (N × H)
7. C23×D284C22semidirect
9. C23×C4×D71C22semidirect
10. C23×Q8×C71C22semidirect
14. C2×#11765866C23direct (N × H)
15. C22×D284C23semidirect
17. C22×C4×D71C23semidirect
19. C22×Q8×C71C23semidirect
23. #11765866C24direct (N × H)
24. C2×D284C24semidirect
26. C2×C4×D71C24semidirect
27. C24×C71#41semidirect
29. C2×Q8×C71C24semidirect
31. C23×C71C2×#41semidirect
35. Q8×C71C25semidirect
38. D284C25semidirect
39. C4×D71C25semidirect
40. C22×C71C22×#41semidirect
47. C2×C71C23×#41semidirect
51. C24×Q8D71semidirect
54. C71C24×#41semidirect
57. C23×Q8C2×D71semidirect
66. C22×Q8C22×D71semidirect
73. C24#11765866direct (N × H)
76. C2×Q8C23×D71semidirect
83. C23C2×#11765866direct (N × H)
85. Q8C24×D71semidirect
92. C22C22×#11765866direct (N × H)
97. C2C23×#11765866direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117658662
C24×D2842
C24×C4×D712
C24×Q8×C712
C24×#4171