Finite Group Databasebeta

← #10604918

Subgroups — #10604918

Order 640.

270 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 54 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
1#66139C211yesmaximal, characteristic
2#65745C212yesmaximal, complemented
3#64922C214yesmaximal, complemented
4C22×C5×D4C411yescharacteristic, complemented
5C2×#709C2211yescharacteristic
6#4130—22no—
7#4099—22no—
8#4086C2212yes—
9#4084—22no—
10#4082C412yescomplemented
11#4072C411yescharacteristic, complemented
12#4011—22no—
13#3999C2214yescomplemented
14#2128—51nomaximal, Sylow
15C24×C5—21no—
16C2×C5×D4—21no—
17C2×C5×D4—42no—
18C2×C5×D4C2×C412yes—
19C22×C4×C5C2×C411yescharacteristic
20C2×C22⋊D5—24no—
21C2×C22⋊D5D412yes—
22#709—42no—
23C2×D20C2311yescharacteristic
24C2×C4×D5D412yes—
25C2×Dic10C2×C411yescharacteristic
26C22⋊AGL(1,𝔽5)—24no—
27C4⋊AGL(1,𝔽5)—42no—
28C4×AGL(1,𝔽5)—42no—
29C22⋊Dic5—21no—
30C22⋊Dic5—24no—
31C4⋊Dic5—21no—
32C4×Dic5C2×C412yes—
33#535—51no—
34#410—52no—
35#353—54no—
36C23×C5—22no—
37C23×C5—42no—
38C23×C5—82no—
39C23×C5C22⋊C411yescharacteristic
40C22×D5C2×D412yes—
41C2×AGL(1,𝔽5)—44no—
42C5×D4—42no—
43C5×D4—42no—
44C2×C4×C5—22no—
45C2×C4×C5C22⋊C411yescharacteristic
46C2×C4×C5C22×C411yescharacteristic
47C22⋊D5—42no—
48C22⋊D5—84no—
49C2×Dic5—22no—
50C2×Dic5C22⋊C412yes—
51D20—41no—
52C4×D5—22no—
53C4×D5—22no—
54Dic10—41no—
55C2×#41—51no—
56C22×D4AGL(1,𝔽5)11yescharacteristic, complemented
57#127—52no—
58#124—52no—
59#121—102no—
60#120—102no—
61#117—51no—
62#117—102no—
63#99—54no—
64#99—102no—
65C22×C5—21no—
66C22×C5—41no—
67C22×C5—44no—
68C22×C5—42no—
69C22×C5—84no—
70C22×C5C2×C22⋊C411yescharacteristic
71C2×D5—22no—
72C2×D5—42no—
73AGL(1,𝔽5)—84no—
74C4×C5—21no—
75C4×C5#9912yes—
76Dic5—22no—
77Dic5—42no—
78C24—21no—
79#41—202no—
80C2×Q8—51no—
81C2×D4—21no—
82C2×D4—42no—
83C2×D4—52no—
84C2×D4—51no—
85C2×D4—104no—
86C2×D4C2×AGL(1,𝔽5)12yes—
87C22×C4—52no—
88C22×C4C2×AGL(1,𝔽5)11yescharacteristic
89C4⋊C4—101no—
90C4⋊C4—202no—
91C22⋊C4—104no—
92C22⋊C4—101no—
93C22⋊C4—104no—
94C42—52no—
95C42—202no—
96C2×C5—21no—
97C2×C5—41no—
98C2×C5—82no—
99C2×C5C2×#9911yescharacteristic
100D5—42no—
101C23—22no—
102C23—42no—
103C23—52no—
104C23—82no—
105C23C22⋊AGL(1,𝔽5)11yescharacteristic
106Q8—201no—
107D4—42no—
108D4—42no—
109D4—201no—
110D4—202no—
111D4—404no—
112C2×C4—22no—
113C2×C4—52no—
114C2×C4—102no—
115C2×C4—102no—
116C2×C4—102no—
117C2×C4—204no—
118C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
119C2×C4C22×AGL(1,𝔽5)11yescharacteristic
120C5#212811yescharacteristic, complemented, Sylow
121C22—21no—
122C22—41no—
123C22—44no—
124C22—42no—
125C22—84no—
126C22—102no—
127C22—202no—
128C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
129C4—21no—
130C4—102no—
131C4—202no—
132C4—404no—
133C4#401112yes—
134C2—21no—
135C2—41no—
136C2—82no—
137C2—202no—
138C2C2×#401111yescharacteristic
subgroupG/H
centerC2C2×#4011
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC22×C5×D4C4
socleC2×C5C2×#99
prime radical#66139C2
cosocleC2×D20C23
hypercenterC22×D4AGL(1,𝔽5)
hypocenterC5#2128
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. #65745C2semidirect
3. #64922C2semidirect
4. C22×C5×D4C4semidirect
10. #4082C4semidirect
11. #4072C4semidirect
13. #3999C22semidirect
56. C22×D4AGL(1,𝔽5)semidirect
120. C5#2128semidirect

Maximal subgroups

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

subgroupindex
#661392
#657452
#649222
#21285