Finite Group Databasebeta

← #10604355

Subgroups — #10604355

Order 640.

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

normal subgroups only

This group has 238 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#65860C211yesmaximal, characteristic
2#65745C212yesmaximal
3#64854C214yesmaximal, complemented
4C2×#709C2211yescharacteristic
5C22×Dic10C411yescharacteristic, complemented
6#4130—22no—
7C5×#117C411yescharacteristic, complemented
8#4043—22no—
9#4032—22no—
10#4021C2212yes—
11#4015C412yescomplemented
12#4011—22no—
13#3999C2214yes—
14#2129—51nomaximal, Sylow
15C22×C4×C5C2×C411yescharacteristic
16C2×C22⋊D5D412yes—
17C22×Dic5—21no—
18#709—42no—
19C2×D20C2311yescharacteristic
20C2×C4×D5D412yes—
21C2×Dic10—21no—
22C2×Dic10—42no—
23C2×Dic10C2×C412yes—
24C2×Dic10C2×C411yescharacteristic
25C22⋊AGL(1,𝔽5)—24no—
26C4⋊AGL(1,𝔽5)—42no—
27C4×AGL(1,𝔽5)—42no—
28C5×C4⋊C4—21no—
29C5×C22⋊C4—21no—
30C42×C5C2×C412yes—
31#685—24no—
32C4⋊Dic5—24no—
33#536—51no—
34#410—52no—
35#354—54no—
36C23×C5C22⋊C411yescharacteristic
37C22×D5C2×D412yes—
38C2×AGL(1,𝔽5)—44no—
39C2×C4×C5—22no—
40C2×C4×C5—22no—
41C2×C4×C5C22⋊C411yescharacteristic
42C2×C4×C5C22×C411yescharacteristic
43C22⋊D5—42no—
44C2×Dic5—22no—
45C2×Dic5—42no—
46C2×Dic5C22⋊C412yes—
47D20—41no—
48C4×D5—22no—
49C4×D5—22no—
50Dic10—41no—
51Dic10—42no—
52Dic10—42no—
53C2×#41—51no—
54C22×Q8—51no—
55C4⋊Q8—52no—
56#124—52no—
57#123—102no—
58C22⋊Q8—102no—
59#117—102no—
60#117AGL(1,𝔽5)11yescharacteristic, complemented
61#99—54no—
62#99—102no—
63C22×C5—21no—
64C22×C5—41no—
65C22×C5C2×C22⋊C411yescharacteristic
66C2×D5—22no—
67C2×D5—42no—
68AGL(1,𝔽5)—84no—
69C4×C5—21no—
70C4×C5—42no—
71C4×C5#9912yes—
72Dic5—22no—
73Dic5—42no—
74#41—202no—
75C2×Q8—51no—
76C2×Q8—52no—
77C2×Q8—101no—
78C2×Q8—202no—
79C2×D4—52no—
80C2×D4—51no—
81C22×C4—52no—
82C22×C4—101no—
83C22×C4C2×AGL(1,𝔽5)11yescharacteristic
84C4⋊C4—21no—
85C4⋊C4—104no—
86C4⋊C4—202no—
87C22⋊C4—21no—
88C22⋊C4—104no—
89C22⋊C4—104no—
90C42—202no—
91C42C2×AGL(1,𝔽5)12yes—
92C2×C5—21no—
93C2×C5—41no—
94C2×C5C2×#9911yescharacteristic
95D5—42no—
96C23—52no—
97C23C22⋊AGL(1,𝔽5)11yescharacteristic
98Q8—201no—
99Q8—202no—
100Q8—202no—
101D4—201no—
102D4—202no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—52no—
106C2×C4—102no—
107C2×C4—102no—
108C2×C4—102no—
109C2×C4—202no—
110C2×C4—204no—
111C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
112C2×C4C22×AGL(1,𝔽5)11yescharacteristic
113C5#212911yescharacteristic, complemented, Sylow
114C22—21no—
115C22—41no—
116C22—102no—
117C22—202no—
118C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
119C4—21no—
120C4—42no—
121C4—102no—
122C4—202no—
123C4—404no—
124C4#399912yes—
125C2—21no—
126C2—41no—
127C2—202no—
128C2C2×#399911yescharacteristic
subgroupG/H
centerC2C2×#3999
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×#117C4
socleC2×C5C2×#99
prime radicalC2×#709C22
cosocleC2×D20C23
hypercenter#117AGL(1,𝔽5)
hypocenterC5#2129
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #64854C2semidirect
5. C22×Dic10C4semidirect
7. C5×#117C4semidirect
11. #4015C4semidirect
60. #117AGL(1,𝔽5)semidirect
113. C5#2129semidirect

Maximal subgroups

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

subgroupindex
#658602
#657452
#648542
#21295