Finite Group Databasebeta

← #10604591

Subgroups — #10604591

Order 640.

260 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 78 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
1D5×#125C211yesmaximal, characteristic
2#65688C211yesmaximal, characteristic
3#65676C211yesmaximal, characteristic
4#64874C214yesmaximal
5C2×C4⋊AGL(1,𝔽5)—22no—
6C2×C4⋊AGL(1,𝔽5)C2212yes—
7C2×D5⋊C8C2212yes—
8C5×#125C411yescharacteristic, complemented
9D5×C4⋊C4—22no—
10D5×C4⋊C4C2212yes—
11#4036C412yescomplemented
12#4035—22no—
13C42×D5C2211yescharacteristic
14#4016C411yescharacteristic, complemented
15#4002—22no—
16#1992—51nomaximal, Sylow
17C22×AGL(1,𝔽5)—22no—
18C2×C4×D5—22no—
19C2×C4×D5D412yes—
20C2×C4×D5C2311yescharacteristic
21C2×#170—22no—
22C4⋊AGL(1,𝔽5)—24no—
23C4⋊AGL(1,𝔽5)—44no—
24D5⋊C8—22no—
25D5⋊C8Q814yes—
26C5×C4⋊C4—22no—
27C5×C4⋊C4C2×C412yes—
28C42×C5C2×C411yescharacteristic
29C4⋊Dic5—22no—
30C4⋊Dic5C2×C412yes—
31#683—22no—
32#683—22no—
33#683—22no—
34C4×Dic5C2×C411yescharacteristic
35C4×Dic5D412yes—
36C2×#125—51no—
37C2×#97—51no—
38#384—51no—
39#340—54no—
40C22×D5C2×D411yescharacteristic
41C2×AGL(1,𝔽5)—42no—
42C2×AGL(1,𝔽5)—44no—
43C2×C4×C5—22no—
44C2×C4×C5C22⋊C412yes—
45C2×C4×C5C22×C411yescharacteristic
46C2×Dic5—22no—
47C2×Dic5C22⋊C412yes—
48C2×Dic5C2×D411yescharacteristic
49C4×D5—22no—
50C4×D5—22no—
51C4×D5—42no—
52C4×D5—42no—
53C4×D5C2×Q812yes—
54C4×D5#4112yes—
55#170—24no—
56C22×C8—52no—
57#125—52no—
58#125—51no—
59#125—102no—
60#125AGL(1,𝔽5)11yescharacteristic, complemented
61C2×C4⋊C4—52no—
62C2×C4⋊C4—52no—
63C2×C4⋊C4—102no—
64C2×C4⋊C4—102no—
65C2×C42—51no—
66#97—102no—
67C22×C5C2×C22⋊C411yescharacteristic
68C2×D5#12411yescharacteristic
69C2×D5C4⋊Q811yescharacteristic
70C2×D5AGL(1,ℤ/8)12yes—
71C2×D5#13712yes—
72AGL(1,𝔽5)—82no—
73C4×C5—22no—
74C4×C5—42no—
75C4×C5C4×Q812yes—
76Dic5—22no—
77Dic5—42no—
78Dic5C22⋊Q812yes—
79C22×C4—51no—
80C22×C4—52no—
81C22×C4—102no—
82C22×C4—102no—
83C2×C8—54no—
84C2×C8—102no—
85C2×C8—102no—
86C4⋊C4—22no—
87C4⋊C4—52no—
88C4⋊C4—102no—
89C4⋊C4—102no—
90C4⋊C4—102no—
91C4⋊C4—104no—
92C4⋊C4—102no—
93C4⋊C4—204no—
94C4⋊C4C2×AGL(1,𝔽5)12yes—
95C42—52no—
96C42—51no—
97C42C2×AGL(1,𝔽5)11yescharacteristic
98C2×C5#39111yescharacteristic
99C2×C5#41712yes—
100D5#49011yescharacteristic
101D5#49111yescharacteristic
102D5#50112yes—
103C23—51no—
104C2×C4—22no—
105C2×C4—51no—
106C2×C4—52no—
107C2×C4—52no—
108C2×C4—52no—
109C2×C4—102no—
110C2×C4—102no—
111C2×C4—102no—
112C2×C4—202no—
113C2×C4—202no—
114C2×C4—202no—
115C2×C4—204no—
116C2×C4C22⋊AGL(1,𝔽5)12yes—
117C2×C4C22×AGL(1,𝔽5)11yescharacteristic
118C8—104no—
119C5#199211yescharacteristic, complemented, Sylow
120C22—52no—
121C22—52no—
122C22—51no—
123C22—51no—
124C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
125C4—22no—
126C4—42no—
127C4—52no—
128C4—102no—
129C4—202no—
130C4—402no—
131C4Q8×AGL(1,𝔽5)12yes—
132C2—51no—
133C2—51no—
134C2—52no—
135C2(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
136C2#6579312yes—
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×#125C4
socleC22×C5C2×C22⋊C4
prime radicalC22×D5C2×D4
cosocleC2×C4×D5C23
hypercenter#125AGL(1,𝔽5)
hypocenterC5#1992
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
8. C5×#125C4semidirect
11. #4036C4semidirect
14. #4016C4semidirect
60. #125AGL(1,𝔽5)semidirect
119. C5#1992semidirect

Maximal subgroups

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

subgroupindex
D5×#1252
#656882
#656762
#648742
#19925