Finite Group Databasebeta

← C24×#11761627 #31769713

Subgroups — C24×#11761627 #31769713

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×#11761627C21120yesmaximal, direct product factor
2C24×Dic134C211yesmaximal, characteristic, complemented
3C24×C22⋊D67C212yesmaximal, complemented
4C25×Dic67C212yesmaximal, complemented
5C24×C4×D67C211yesmaximal, characteristic, complemented
6C24×D4×C67C211yesmaximal, characteristic, complemented
7C25×D67C2211yescharacteristic
8C22×#11761627C2212240yesdirect product factor
9C23×Dic134C22160yescomplemented
10C23×C22⋊D67C221120yescomplemented
11C24×Dic67C2211yescharacteristic
12C24×Dic67C2212yes—
13C24×Dic67C221120yescomplemented
14C23×C4×D67C22160yescomplemented
15C26×C67C2212yes—
16C23×D4×C67C22160yescomplemented
17C24×C4×C67C2211yescharacteristic
18C24×D67—216no—
19C24×D67C23130yes—
20C2×#11761627C2317680yesdirect product factor
21C22×Dic134C231560yescomplemented
22C22×C22⋊D67C2311120yescomplemented
23C23×Dic67C23130yes—
24C23×Dic67C23160yes—
25C23×Dic67C2311120yescomplemented
26C22×C4×D67C231560yescomplemented
27C25×C67—232no—
28C25×C67C23160yes—
29C25×C67C2311yescharacteristic
30C22×D4×C67C231560yescomplemented
31C23×C4×C67C23130yes—
32C23×D67—2240no—
33C23×D67C241140yes—
34#11761627C2414096yesdirect product factor
35C2×Dic134C241960yescomplemented
36C2×C22⋊D67C2411920yescomplemented
37C22×Dic67C241140yes—
38C22×Dic67C241280yes—
39C22×Dic67C2411920yescomplemented
40C2×C4×D67C241960yescomplemented
41C24×C67—2480no—
42C24×C67#41116yescomplemented
43C24×C67C241280yes—
44C24×C67C24115yes—
45C2×D4×C67C241960yescomplemented
46C22×C4×C67C241140yes—
47C23×C67—21120no—
48C23×C67C2×#411120yescomplemented
49C23×C67C251240yes—
50C23×C67C25135yes—
51C22×D67—2560no—
52C22×D67C251120yes—
53D4×C67C251256yescomplemented
54C2×C4×C67C251120yes—
55C22⋊D67C251512yescomplemented
56C2×Dic67C251120yes—
57C2×Dic67C251240yes—
58C2×Dic67C251512yescomplemented
59C4×D67C251256yescomplemented
60Dic134C251256yescomplemented
61C22×C67—2480no—
62C22×C67C22×#411140yescomplemented
63C22×C67C26132yes—
64C22×C67C26115yes—
65C2×D67—2240no—
66C2×D67C26116yes—
67C4×C67C26116yes—
68Dic67C26132yes—
69Dic67C26116yes—
70C24×#41—671nomaximal, Sylow
71C2×C67—232no—
72C2×C67C23×#41130yescomplemented
73C2×C67C2711yescharacteristic
74D67—216no—
75C23×#41—67120no—
76C24×Q8—671no—
77C24×D4—672no—
78C24×D4D6711yescharacteristic, complemented
79C25×C4—672no—
80C25×C4—671no—
81C67C24×#4111yescharacteristic, complemented, Sylow
82C26—671no—
83C26C2×D6712yes—
84C22×#41—672240no—
85C23×Q8—6760no—
86C23×D4—67120no—
87C23×D4C2×D67160yescomplemented
88C24×C4—67120no—
89C24×C4—671no—
90C24×C4—672no—
91C24×C4—6760no—
92C24×C4C2×D6711yescharacteristic
93C25—232no—
94C25—6730no—
95C25—13416no—
96C25C22×D67160yes—
97C25C22×D6711yescharacteristic
98C2×#41—677680no—
99C22×Q8—67560no—
100C22×D4—671120no—
101C22×D4C22×D671560yescomplemented
102C23×C4—6730no—
103C23×C4—6760no—
104C23×C4—671120no—
105C23×C4—67560no—
106C23×C4C22×D67130yes—
107C24—2480no—
108C24—67140no—
109C24—134240no—
110C24#11761627116yesdirect product factor
111C24C23×D671280yes—
112C24C23×D67115yes—
113#41—674096no—
114C2×Q8—67960no—
115C2×D4—671920no—
116C2×D4C23×D671960yescomplemented
117C22×C4—67140no—
118C22×C4—67280no—
119C22×C4—671920no—
120C22×C4—67960no—
121C22×C4C23×D671140yes—
122C23—21120no—
123C23—67120no—
124C23—134560no—
125C23C2×#117616271120yesdirect product factor
126C23C24×D671240yes—
127C23C24×D67135yes—
128Q8—67256no—
129D4—67512no—
130D4C24×D671256yescomplemented
131C2×C4—67120no—
132C2×C4—67240no—
133C2×C4—67512no—
134C2×C4—67256no—
135C2×C4C24×D671120yes—
136C22—2480no—
137C22—6716no—
138C22—134240no—
139C22C22×#117616271140yesdirect product factor
140C22C25×D67132yes—
141C22C25×D67115yes—
142C4—6716no—
143C4—6732no—
144C4C25×D67116yes—
145C2—232no—
146C2—13416no—
147C2C23×#11761627130yesdirect product factor
148C2C26×D6711yescharacteristic
subgroupG/H
centerC25C22×D67
commutator, cosocleC2×C67C27
FrattiniC2C26×D67
FittingC24×D4×C67C2
socleC25×C67C23
hypercenterC24×D4D67
hypocenterC67C24×#41
LCSG ▷ C2×C67 ▷ C67
UCSC1 ◁ C25 ◁ C24×D4

Complements

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

NH ≅ G/Ntype
1. C23×#11761627C2direct (N × H)
2. C24×Dic134C2semidirect
3. C24×C22⋊D67C2semidirect
4. C25×Dic67C2semidirect
5. C24×C4×D67C2semidirect
6. C24×D4×C67C2semidirect
8. C22×#11761627C22direct (N × H)
9. C23×Dic134C22semidirect
10. C23×C22⋊D67C22semidirect
13. C24×Dic67C22semidirect
14. C23×C4×D67C22semidirect
16. C23×D4×C67C22semidirect
20. C2×#11761627C23direct (N × H)
21. C22×Dic134C23semidirect
22. C22×C22⋊D67C23semidirect
25. C23×Dic67C23semidirect
26. C22×C4×D67C23semidirect
30. C22×D4×C67C23semidirect
34. #11761627C24direct (N × H)
35. C2×Dic134C24semidirect
36. C2×C22⋊D67C24semidirect
39. C22×Dic67C24semidirect
40. C2×C4×D67C24semidirect
42. C24×C67#41semidirect
45. C2×D4×C67C24semidirect
48. C23×C67C2×#41semidirect
53. D4×C67C25semidirect
55. C22⋊D67C25semidirect
58. C2×Dic67C25semidirect
59. C4×D67C25semidirect
60. Dic134C25semidirect
62. C22×C67C22×#41semidirect
72. C2×C67C23×#41semidirect
78. C24×D4D67semidirect
81. C67C24×#41semidirect
87. C23×D4C2×D67semidirect
101. C22×D4C22×D67semidirect
110. C24#11761627direct (N × H)
116. C2×D4C23×D67semidirect
125. C23C2×#11761627direct (N × H)
130. D4C24×D67semidirect
139. C22C22×#11761627direct (N × H)
147. C2C23×#11761627direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117616272
C24×Dic1342
C24×C22⋊D672
C25×Dic672
C24×C4×D672
C24×D4×C672
C24×#4167