Finite Group Databasebeta

← #10604307

Subgroups — #10604307

Order 640.

300 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C2×#4038C211yesmaximal, characteristic
2#65764C212yesmaximal
3#64917C214yesmaximal
4C22×C4×D5C2211yescharacteristic
5C2×C22⋊AGL(1,𝔽5)C2214yes—
6C2×C5×C4⋊C4C411yescharacteristic, complemented
7C2×#685C2212yes—
8C2×C4⋊Dic5C411yescharacteristic, complemented
9C2×C4×Dic5C412yescomplemented
10#4038—24no—
11(C2×AGL(1,𝔽5))⋊C4—24no—
12#1820—51nomaximal, Sylow
13C23×D5C2311yescharacteristic
14C22×AGL(1,𝔽5)—24no—
15C22×C4×C5C2×C412yes—
16C22×C4×C5C2×C411yescharacteristic
17C22×Dic5C2×C411yescharacteristic
18C22×Dic5C2×C412yes—
19C2×C4×D5—24no—
20C2×C4×D5D414yes—
21C22⋊AGL(1,𝔽5)—28no—
22C5×C4⋊C4—22no—
23#685—24no—
24C4⋊Dic5—22no—
25C4×Dic5—24no—
26C2×#117—51no—
27#381—52no—
28#342—54no—
29C23×C5C22×C411yescharacteristic
30C22×D5—41no—
31C22×D5—41no—
32C22×D5C2×D412yes—
33C22×D5#4114yes—
34C2×AGL(1,𝔽5)—44no—
35C2×AGL(1,𝔽5)—44no—
36C2×AGL(1,𝔽5)—44no—
37C2×C4×C5—22no—
38C2×C4×C5—22no—
39C2×C4×C5—42no—
40C2×C4×C5C22⋊C412yes—
41C2×Dic5—22no—
42C2×Dic5—22no—
43C2×Dic5—42no—
44C2×Dic5C22⋊C412yes—
45C4×D5—28no—
46C23×C4—51no—
47#117—104no—
48C2×C4⋊C4—51no—
49C2×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
50C2×C22⋊C4—52no—
51C2×C22⋊C4—54no—
52C2×C42—52no—
53#95—104no—
54C22×C5—21no—
55C22×C5—21no—
56C22×C5C2×C22⋊C411yescharacteristic
57C22×C5#11712yes—
58C2×D5—22no—
59C2×D5—22no—
60C2×D5—44no—
61C2×D5#12314yes—
62AGL(1,𝔽5)—84no—
63C4×C5—22no—
64C4×C5—42no—
65Dic5—22no—
66Dic5—42no—
67C24—51no—
68C22×C4—51no—
69C22×C4—54no—
70C22×C4—52no—
71C22×C4—104no—
72C22×C4—104no—
73C22×C4C2×AGL(1,𝔽5)12yes—
74C22×C4C2×AGL(1,𝔽5)11yescharacteristic
75C4⋊C4—22no—
76C4⋊C4—102no—
77C22⋊C4—104no—
78C22⋊C4—108no—
79C42—104no—
80C2×C5—22no—
81C2×C5#38111yescharacteristic
82C2×C5#41011yescharacteristic
83C2×C5#41011yescharacteristic
84D5—24no—
85C23—52no—
86C23—54no—
87C23—201no—
88C23—201no—
89C23C22×AGL(1,𝔽5)11yescharacteristic
90C2×C4—22no—
91C2×C4—22no—
92C2×C4—42no—
93C2×C4—52no—
94C2×C4—108no—
95C2×C4—102no—
96C2×C4—102no—
97C2×C4—202no—
98C2×C4—204no—
99C2×C4—204no—
100C2×C4—204no—
101C2×C4C22⋊AGL(1,𝔽5)12yes—
102C5#182011yescharacteristic, complemented, Sylow
103C22—21no—
104C22—21no—
105C22—54no—
106C22—102no—
107C22—102no—
108C22—204no—
109C22#413012yes—
110C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
111C4—22no—
112C4—42no—
113C4—102no—
114C4—202no—
115C4—404no—
116C2—22no—
117C2—104no—
118C2#6576411yescharacteristic
119C2#6577111yescharacteristic
120C2#6578511yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C4
FrattiniC23C22×AGL(1,𝔽5)
FittingC2×C5×C4⋊C4C4
socleC22×C5C2×C22⋊C4
prime radical, cosocleC23×D5C23
hypercenterC2×C4⋊C4AGL(1,𝔽5)
hypocenterC5#1820
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. C2×C5×C4⋊C4C4semidirect
8. C2×C4⋊Dic5C4semidirect
9. C2×C4×Dic5C4semidirect
49. C2×C4⋊C4AGL(1,𝔽5)semidirect
102. C5#1820semidirect

Maximal subgroups

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

subgroupindex
C2×#40382
#657642
#649172
#18205