Finite Group Databasebeta

← #10604590

Subgroups — #10604590

Order 640.

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

normal subgroups only

This group has 252 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#65689C211yesmaximal, characteristic
3C4×D5⋊C8C211yesmaximal, characteristic
4#64874C214yesmaximal
5C2×C4⋊AGL(1,𝔽5)C2212yes—
6C2×D5⋊C8C2212yes—
7C5×#125C411yescharacteristic, complemented
8D5×C4⋊C4—22no—
9D5×C4⋊C4C2212yes—
10#4036C412yescomplemented
11#4035—22no—
12C42×D5C2211yescharacteristic
13#4016C411yescharacteristic, complemented
14(C2×AGL(1,𝔽5))⋊C4—22no—
15C4×#170—22no—
16#1991—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—
23D5⋊C8—22no—
24D5⋊C8Q814yes—
25C5×C4⋊C4—22no—
26C5×C4⋊C4C2×C412yes—
27C42×C5C2×C411yescharacteristic
28C4⋊Dic5—22no—
29C4⋊Dic5C2×C412yes—
30#683—22no—
31#683—22no—
32#683—22no—
33C4×Dic5C2×C411yescharacteristic
34C4×Dic5D412yes—
35C2×#125—51no—
36C2×C4×C8—51no—
37#382—51no—
38#340—54no—
39C22×D5C2×D411yescharacteristic
40C2×AGL(1,𝔽5)—42no—
41C2×AGL(1,𝔽5)—44no—
42C2×C4×C5—22no—
43C2×C4×C5C22⋊C412yes—
44C2×C4×C5C22×C411yescharacteristic
45C2×Dic5—22no—
46C2×Dic5C22⋊C412yes—
47C2×Dic5C2×D411yescharacteristic
48C4×D5—24no—
49C4×D5—44no—
50C4×D5C2×Q812yes—
51C4×D5#4112yes—
52#170—24no—
53C22×C8—52no—
54#125—52no—
55#125—51no—
56#125—102no—
57#125AGL(1,𝔽5)11yescharacteristic, complemented
58C2×C4⋊C4—52no—
59C2×C4⋊C4—52no—
60C2×C4⋊C4—102no—
61C2×C42—51no—
62C4×C8—102no—
63#95—102no—
64C22×C5C2×C22⋊C411yescharacteristic
65C2×D5#12411yescharacteristic
66C2×D5C4⋊Q811yescharacteristic
67C2×D5#13514yes—
68AGL(1,𝔽5)—82no—
69C4×C5—22no—
70C4×C5—42no—
71C4×C5C4×Q812yes—
72Dic5—22no—
73Dic5—42no—
74Dic5C22⋊Q812yes—
75C22×C4—51no—
76C22×C4—52no—
77C22×C4—102no—
78C22×C4—102no—
79C2×C8—54no—
80C2×C8—102no—
81C2×C8—102no—
82C4⋊C4—22no—
83C4⋊C4—52no—
84C4⋊C4—102no—
85C4⋊C4—102no—
86C4⋊C4—102no—
87C4⋊C4—104no—
88C4⋊C4—102no—
89C4⋊C4C2×AGL(1,𝔽5)12yes—
90C42—52no—
91C42—51no—
92C42C2×AGL(1,𝔽5)11yescharacteristic
93C2×C5#39111yescharacteristic
94C2×C5#41612yes—
95D5#48812yes—
96D5#49912yes—
97C23—51no—
98C2×C4—22no—
99C2×C4—51no—
100C2×C4—52no—
101C2×C4—52no—
102C2×C4—52no—
103C2×C4—104no—
104C2×C4—102no—
105C2×C4—204no—
106C2×C4—202no—
107C2×C4—204no—
108C2×C4C22⋊AGL(1,𝔽5)12yes—
109C2×C4C22×AGL(1,𝔽5)11yescharacteristic
110C8—104no—
111C5#199111yescharacteristic, complemented, Sylow
112C22—54no—
113C22—51no—
114C22—51no—
115C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
116C4—22no—
117C4—42no—
118C4—52no—
119C4—102no—
120C4—202no—
121C4—402no—
122C4Q8×AGL(1,𝔽5)12yes—
123C2—52no—
124C2—52no—
125C2(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
126C2#6579412yes—
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#1991
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
7. C5×#125C4semidirect
10. #4036C4semidirect
13. #4016C4semidirect
57. #125AGL(1,𝔽5)semidirect
111. C5#1991semidirect

Maximal subgroups

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

subgroupindex
D5×#1252
#656892
C4×D5⋊C82
#648742
#19915