Finite Group Databasebeta

← C24×#709 #13013183

Subgroups — C24×#709 #13013183

Order 1280.

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
1C25×C4×C5C211yesmaximal, characteristic, complemented
2C24×C22⋊D5C212yesmaximal, complemented
3C23×#709C21120yesmaximal, direct product factor
4C24×D20C211yesmaximal, characteristic, complemented
5C24×C4×D5C212yesmaximal, complemented
6C24×Dic10C211yesmaximal, characteristic, complemented
7C26×C5C2211yescharacteristic
8C25×D5C2212yes—
9C24×C4×C5C2211yescharacteristic
10C24×C4×C5C2211yescharacteristic
11C24×C4×C5C22160yescomplemented
12C23×C22⋊D5C221120yescomplemented
13C24×Dic5C2212yes—
14C22×#709C2212240yesdirect product factor
15C23×D20C22160yescomplemented
16C23×C4×D5C221120yescomplemented
17C23×Dic10C22160yescomplemented
18C24×#41—51nomaximal, Sylow
19C25×C5—216no—
20C25×C5C23130yes—
21C25×C5C2311yescharacteristic
22C24×D5—232no—
23C24×D5C23160yes—
24C23×C4×C5C23130yes—
25C23×C4×C5C23130yes—
26C23×C4×C5C231560yescomplemented
27C22×C22⋊D5C2311120yescomplemented
28C23×Dic5C23160yes—
29C2×#709C2317680yesdirect product factor
30C22×D20C231560yescomplemented
31C22×C4×D5C2311120yescomplemented
32C22×Dic10C231560yescomplemented
33C23×#41—5120no—
34C24×Q8—51no—
35C24×D4—52no—
36C24×D4—51no—
37C25×C4—52no—
38C25×C4D511yescharacteristic, complemented
39C24×C5—2240no—
40C24×C5#41116yescomplemented
41C24×C5C241140yes—
42C24×C5C24115yes—
43C23×D5—2480no—
44C23×D5C241280yes—
45C22×C4×C5C241140yes—
46C22×C4×C5C241140yes—
47C22×C4×C5C241960yescomplemented
48C2×C22⋊D5C2411920yescomplemented
49C22×Dic5C241280yes—
50#709C2414096yesdirect product factor
51C2×D20C241960yescomplemented
52C2×C4×D5C2411920yescomplemented
53C2×Dic10C241960yescomplemented
54C26—52no—
55C26C2×D511yescharacteristic
56C22×#41—52240no—
57C23×Q8—560no—
58C23×D4—5120no—
59C23×D4—560no—
60C24×C4—52no—
61C24×C4—5120no—
62C24×C4C2×D511yescharacteristic
63C24×C4C2×D511yescharacteristic
64C24×C4C2×D5160yescomplemented
65C23×C5—2560no—
66C23×C5C2×#411120yescomplemented
67C23×C5C251120yes—
68C23×C5C25135yes—
69C22×D5—21120no—
70C22×D5C251240yes—
71C2×C4×C5C251120yes—
72C2×C4×C5C251120yes—
73C2×C4×C5C251256yescomplemented
74C22⋊D5C251512yescomplemented
75C2×Dic5C251240yes—
76D20C251256yescomplemented
77C4×D5C251512yescomplemented
78Dic10C251256yescomplemented
79C25—216no—
80C25—560no—
81C25—1032no—
82C25C22×D5130yes—
83C25C22×D511yescharacteristic
84C2×#41—57680no—
85C22×Q8—5560no—
86C22×D4—51120no—
87C22×D4—5560no—
88C23×C4—560no—
89C23×C4—51120no—
90C23×C4C22×D5130yes—
91C23×C4C22×D5130yes—
92C23×C4C22×D51560yescomplemented
93C22×C5—2240no—
94C22×C5C22×#411140yescomplemented
95C22×C5C26116yes—
96C22×C5C26115yes—
97C2×D5—2480no—
98C2×D5C26132yes—
99C4×C5C26116yes—
100C4×C5C26116yes—
101Dic5C26132yes—
102C24—2240no—
103C24—5280no—
104C24—10480no—
105C24#709116yesdirect product factor
106C24C23×D51140yes—
107C24C23×D5115yes—
108#41—54096no—
109C2×Q8—5960no—
110C2×D4—51920no—
111C2×D4—5960no—
112C22×C4—5280no—
113C22×C4—51920no—
114C22×C4C23×D51140yes—
115C22×C4C23×D51140yes—
116C22×C4C23×D51960yescomplemented
117C2×C5—216no—
118C2×C5C23×#41130yescomplemented
119C2×C5C2711yescharacteristic
120D5—232no—
121C23—2560no—
122C23—5240no—
123C23—101120no—
124C23C2×#7091120yesdirect product factor
125C23C24×D51120yes—
126C23C24×D5135yes—
127Q8—5256no—
128D4—5512no—
129D4—5256no—
130C2×C4—5240no—
131C2×C4—5512no—
132C2×C4C24×D51120yes—
133C2×C4C24×D51120yes—
134C2×C4C24×D51256yescomplemented
135C5C24×#4111yescharacteristic, complemented, Sylow
136C22—2240no—
137C22—532no—
138C22—10480no—
139C22C22×#7091140yesdirect product factor
140C22C25×D5116yes—
141C22C25×D5115yes—
142C4—532no—
143C4C25×D5116yes—
144C4C25×D5116yes—
145C2—216no—
146C2—1032no—
147C2C23×#709130yesdirect product factor
148C2C26×D511yescharacteristic
subgroupG/H
centerC24×C4C2×D5
commutator, cosocleC2×C5C27
FrattiniC2C26×D5
FittingC25×C4×C5C2
socleC25×C5C23
hypercenterC25×C4D5
hypocenterC5C24×#41
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C24×C4 ◁ C25×C4

Complements

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

NH ≅ G/Ntype
1. C25×C4×C5C2semidirect
2. C24×C22⋊D5C2semidirect
3. C23×#709C2direct (N × H)
4. C24×D20C2semidirect
5. C24×C4×D5C2semidirect
6. C24×Dic10C2semidirect
11. C24×C4×C5C22semidirect
12. C23×C22⋊D5C22semidirect
14. C22×#709C22direct (N × H)
15. C23×D20C22semidirect
16. C23×C4×D5C22semidirect
17. C23×Dic10C22semidirect
26. C23×C4×C5C23semidirect
27. C22×C22⋊D5C23semidirect
29. C2×#709C23direct (N × H)
30. C22×D20C23semidirect
31. C22×C4×D5C23semidirect
32. C22×Dic10C23semidirect
38. C25×C4D5semidirect
40. C24×C5#41semidirect
47. C22×C4×C5C24semidirect
48. C2×C22⋊D5C24semidirect
50. #709C24direct (N × H)
51. C2×D20C24semidirect
52. C2×C4×D5C24semidirect
53. C2×Dic10C24semidirect
64. C24×C4C2×D5semidirect
66. C23×C5C2×#41semidirect
73. C2×C4×C5C25semidirect
74. C22⋊D5C25semidirect
76. D20C25semidirect
77. C4×D5C25semidirect
78. Dic10C25semidirect
92. C23×C4C22×D5semidirect
94. C22×C5C22×#41semidirect
105. C24#709direct (N × H)
116. C22×C4C23×D5semidirect
118. C2×C5C23×#41semidirect
124. C23C2×#709direct (N × H)
134. C2×C4C24×D5semidirect
135. C5C24×#41semidirect
139. C22C22×#709direct (N × H)
147. C2C23×#709direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C4×C52
C24×C22⋊D52
C23×#7092
C24×D202
C24×C4×D52
C24×Dic102
C24×#415