Finite Group Databasebeta

← C24×#6127 #16776802

Subgroups — C24×#6127 #16776802

Order 3328.

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×#6127C21120yesmaximal, direct product factor
2C24×Dic26C211yesmaximal, characteristic, complemented
3C24×D52C211yesmaximal, characteristic, complemented
4C24×C22⋊D13C212yesmaximal, complemented
5C24×C4×D13C212yesmaximal, complemented
6C25×C4×C13C211yesmaximal, characteristic, complemented
7C26×C13C2211yescharacteristic
8C25×D13C2212yes—
9C24×C4×C13C2211yescharacteristic
10C24×C4×C13C2211yescharacteristic
11C24×C4×C13C22160yescomplemented
12C23×C22⋊D13C221120yescomplemented
13C24×Dic13C2212yes—
14C22×#6127C2212240yesdirect product factor
15C23×D52C22160yescomplemented
16C23×C4×D13C221120yescomplemented
17C23×Dic26C22160yescomplemented
18C25×C13—216no—
19C25×C13C23130yes—
20C25×C13C2311yescharacteristic
21C24×D13—232no—
22C24×D13C23160yes—
23C23×C4×C13C23130yes—
24C23×C4×C13C23130yes—
25C23×C4×C13C231560yescomplemented
26C22×C22⋊D13C2311120yescomplemented
27C23×Dic13C23160yes—
28C2×#6127C2317680yesdirect product factor
29C22×D52C231560yescomplemented
30C22×C4×D13C2311120yescomplemented
31C22×Dic26C231560yescomplemented
32C24×#41—131nomaximal, Sylow
33C24×C13—2240no—
34C24×C13#41116yescomplemented
35C24×C13C241140yes—
36C24×C13C24115yes—
37C23×D13—2480no—
38C23×D13C241280yes—
39C22×C4×C13C241140yes—
40C22×C4×C13C241140yes—
41C22×C4×C13C241960yescomplemented
42C2×C22⋊D13C2411920yescomplemented
43C22×Dic13C241280yes—
44#6127C2414096yesdirect product factor
45C2×D52C241960yescomplemented
46C2×C4×D13C2411920yescomplemented
47C2×Dic26C241960yescomplemented
48C23×#41—13120no—
49C24×Q8—131no—
50C24×D4—132no—
51C24×D4—131no—
52C25×C4—132no—
53C25×C4D1311yescharacteristic, complemented
54C23×C13—2560no—
55C23×C13C2×#411120yescomplemented
56C23×C13C251120yes—
57C23×C13C25135yes—
58C22×D13—21120no—
59C22×D13C251240yes—
60C2×C4×C13C251120yes—
61C2×C4×C13C251120yes—
62C2×C4×C13C251256yescomplemented
63C22⋊D13C251512yescomplemented
64C2×Dic13C251240yes—
65D52C251256yescomplemented
66C4×D13C251512yescomplemented
67Dic26C251256yescomplemented
68C26—132no—
69C26C2×D1311yescharacteristic
70C22×#41—132240no—
71C23×Q8—1360no—
72C23×D4—13120no—
73C23×D4—1360no—
74C24×C4—132no—
75C24×C4—13120no—
76C24×C4C2×D1311yescharacteristic
77C24×C4C2×D1311yescharacteristic
78C24×C4C2×D13160yescomplemented
79C22×C13—2240no—
80C22×C13C22×#411140yescomplemented
81C22×C13C26116yes—
82C22×C13C26115yes—
83C2×D13—2480no—
84C2×D13C26132yes—
85C4×C13C26116yes—
86C4×C13C26116yes—
87Dic13C26132yes—
88C25—216no—
89C25—1360no—
90C25—2632no—
91C25C22×D13130yes—
92C25C22×D1311yescharacteristic
93C2×#41—137680no—
94C22×Q8—13560no—
95C22×D4—131120no—
96C22×D4—13560no—
97C23×C4—1360no—
98C23×C4—131120no—
99C23×C4C22×D13130yes—
100C23×C4C22×D13130yes—
101C23×C4C22×D131560yescomplemented
102C2×C13—216no—
103C2×C13C23×#41130yescomplemented
104C2×C13C2711yescharacteristic
105D13—232no—
106C24—2240no—
107C24—13280no—
108C24—26480no—
109C24#6127116yesdirect product factor
110C24C23×D131140yes—
111C24C23×D13115yes—
112#41—134096no—
113C2×Q8—13960no—
114C2×D4—131920no—
115C2×D4—13960no—
116C22×C4—13280no—
117C22×C4—131920no—
118C22×C4C23×D131140yes—
119C22×C4C23×D131140yes—
120C22×C4C23×D131960yescomplemented
121C13C24×#4111yescharacteristic, complemented, Sylow
122C23—2560no—
123C23—13240no—
124C23—261120no—
125C23C2×#61271120yesdirect product factor
126C23C24×D131120yes—
127C23C24×D13135yes—
128Q8—13256no—
129D4—13512no—
130D4—13256no—
131C2×C4—13240no—
132C2×C4—13512no—
133C2×C4C24×D131120yes—
134C2×C4C24×D131120yes—
135C2×C4C24×D131256yescomplemented
136C22—2240no—
137C22—1332no—
138C22—26480no—
139C22C22×#61271140yesdirect product factor
140C22C25×D13116yes—
141C22C25×D13115yes—
142C4—1332no—
143C4C25×D13116yes—
144C4C25×D13116yes—
145C2—216no—
146C2—2632no—
147C2C23×#6127130yesdirect product factor
148C2C26×D1311yescharacteristic
subgroupG/H
centerC24×C4C2×D13
commutator, cosocleC2×C13C27
FrattiniC2C26×D13
FittingC25×C4×C13C2
socleC25×C13C23
hypercenterC25×C4D13
hypocenterC13C24×#41
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C24×C4 ◁ C25×C4

Complements

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

NH ≅ G/Ntype
1. C23×#6127C2direct (N × H)
2. C24×Dic26C2semidirect
3. C24×D52C2semidirect
4. C24×C22⋊D13C2semidirect
5. C24×C4×D13C2semidirect
6. C25×C4×C13C2semidirect
11. C24×C4×C13C22semidirect
12. C23×C22⋊D13C22semidirect
14. C22×#6127C22direct (N × H)
15. C23×D52C22semidirect
16. C23×C4×D13C22semidirect
17. C23×Dic26C22semidirect
25. C23×C4×C13C23semidirect
26. C22×C22⋊D13C23semidirect
28. C2×#6127C23direct (N × H)
29. C22×D52C23semidirect
30. C22×C4×D13C23semidirect
31. C22×Dic26C23semidirect
34. C24×C13#41semidirect
41. C22×C4×C13C24semidirect
42. C2×C22⋊D13C24semidirect
44. #6127C24direct (N × H)
45. C2×D52C24semidirect
46. C2×C4×D13C24semidirect
47. C2×Dic26C24semidirect
53. C25×C4D13semidirect
55. C23×C13C2×#41semidirect
62. C2×C4×C13C25semidirect
63. C22⋊D13C25semidirect
65. D52C25semidirect
66. C4×D13C25semidirect
67. Dic26C25semidirect
78. C24×C4C2×D13semidirect
80. C22×C13C22×#41semidirect
101. C23×C4C22×D13semidirect
103. C2×C13C23×#41semidirect
109. C24#6127direct (N × H)
120. C22×C4C23×D13semidirect
121. C13C24×#41semidirect
125. C23C2×#6127direct (N × H)
135. C2×C4C24×D13semidirect
139. C22C22×#6127direct (N × H)
147. C2C23×#6127direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#61272
C24×Dic262
C24×D522
C24×C22⋊D132
C24×C4×D132
C25×C4×C132
C24×#4113