Finite Group Databasebeta

← #10604588

Subgroups — #10604588

Order 640.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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#66022C211yesmaximal, characteristic
2#65686C211yesmaximal, characteristic
3#65681C211yesmaximal, characteristic
4#64875C214yesmaximal
5C2×C4×AGL(1,𝔽5)C2212yes—
6C2×#700C2212yes—
7C5×#125C411yescharacteristic, complemented
8#4042—22no—
9#4038C2212yes—
10#4034C412yescomplemented
11#4033—22no—
12C42×D5C2211yescharacteristic
13#4016C411yescharacteristic, complemented
14(C2×AGL(1,𝔽5))⋊C4—22no—
15C4⋊#170—22no—
16#1995—51nomaximal, Sylow
17C22×AGL(1,𝔽5)—22no—
18C2×C4×D5D412yes—
19C2×C4×D5C2311yescharacteristic
20C2×Dic10—22no—
21C2×#170—22no—
22C4×AGL(1,𝔽5)—22no—
23C4×AGL(1,𝔽5)Q814yescomplemented
24#700—24no—
25C5×C4⋊C4—22no—
26C5×C4⋊C4C2×C412yes—
27C42×C5C2×C411yescharacteristic
28#685—22no—
29C4⋊Dic5C2×C412yes—
30#683—22no—
31#683—22no—
32C4×Dic5—22no—
33C4×Dic5C2×C411yescharacteristic
34C4×Dic5D412yes—
35#533—51no—
36#423—51no—
37#376—51no—
38#339—54no—
39C22×D5C2×D411yescharacteristic
40C2×AGL(1,𝔽5)—24no—
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—42no—
49C4×D5C2×Q812yes—
50C4×D5#4112yes—
51Dic10—44no—
52#170—42no—
53C2×M16—52no—
54C4⋊Q8—52no—
55#125—51no—
56#125AGL(1,𝔽5)11yescharacteristic, complemented
57C22⋊Q8—102no—
58C4×Q8—102no—
59#117—52no—
60C2×C42—51no—
61C2×C42—52no—
62C4⋊C8—102no—
63#95—102no—
64C22×C5C2×C22⋊C411yescharacteristic
65C2×D5—22no—
66C2×D5C22⋊Q812yes—
67AGL(1,𝔽5)—44no—
68C4×C5—22no—
69C4×C5—42no—
70C4×C5C4×Q812yes—
71Dic5—22no—
72Dic5—42no—
73Dic5#12411yescharacteristic
74Dic5C4⋊Q811yescharacteristic
75C2×Q8—102no—
76C22×C4—51no—
77C22×C4—52no—
78C22×C4—102no—
79M16—104no—
80C2×C8—102no—
81C4⋊C4—22no—
82C4⋊C4—52no—
83C4⋊C4—102no—
84C4⋊C4—102no—
85C4⋊C4C2×AGL(1,𝔽5)12yes—
86C22⋊C4—102no—
87C42—52no—
88C42—51no—
89C42—54no—
90C42—102no—
91C42—102no—
92C42C2×AGL(1,𝔽5)11yescharacteristic
93C2×C5#39111yescharacteristic
94C2×C5#42112yes—
95D5—22no—
96C23—51no—
97Q8—204no—
98C2×C4—22no—
99C2×C4—51no—
100C2×C4—52no—
101C2×C4—52no—
102C2×C4—52no—
103C2×C4—104no—
104C2×C4—102no—
105C2×C4—202no—
106C2×C4—204no—
107C2×C4C22⋊AGL(1,𝔽5)12yes—
108C2×C4C22×AGL(1,𝔽5)11yescharacteristic
109C8—202no—
110C5#199511yescharacteristic, complemented, Sylow
111C22—52no—
112C22—102no—
113C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
114C4—22no—
115C4—42no—
116C4—51no—
117C4—51no—
118C4—102no—
119C4—204no—
120C4—202no—
121C4Q8×AGL(1,𝔽5)12yes—
122C2—102no—
123C2(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
124C2#6579512yes—
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#1995
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. #4034C4semidirect
13. #4016C4semidirect
23. C4×AGL(1,𝔽5)Q8semidirect
56. #125AGL(1,𝔽5)semidirect
110. C5#1995semidirect

Maximal subgroups

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

subgroupindex
#660222
#656862
#656812
#648752
#19955