Finite Group Databasebeta

← #10604337

Subgroups — #10604337

Order 640.

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

normal subgroups only

This group has 316 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
1C2×D5×C4⋊C4C211yesmaximal, characteristic
2#65755C212yesmaximal
3#64919C214yesmaximal, complemented
4C22×C4×D5C412yes—
5C22×C4×D5C2211yescharacteristic
6C2×#704C2214yes—
7C2×C5×C4⋊C4C411yescharacteristic
8C2×C4⋊Dic5C411yescharacteristic
9C2×#683C2212yes—
10D5×C4⋊C4—22no—
11D5×C4⋊C4—22no—
12#4003—22no—
13#4003—22no—
14#1821—51nomaximal, Sylow
15C23×D5C2×C411yescharacteristic
16C22×C4×C5C2×C412yes—
17C22×C4×C5C2×C411yescharacteristic
18C22×Dic5C2×C412yes—
19C22×Dic5C2311yescharacteristic
20C2×C4×D5—24no—
21C2×C4×D5—22no—
22C2×C4×D5—22no—
23C2×C4×D5—44no—
24C2×C4×D5D412yes—
25C2×C4×D5D412yes—
26#704—28no—
27C2×#170—24no—
28C5×C4⋊C4—22no—
29C4⋊Dic5—22no—
30#683—24no—
31C22×C4⋊C4—51no—
32#407—52no—
33#343—54no—
34C23×C5C22×C411yescharacteristic
35C22×D5—22no—
36C22×D5—22no—
37C22×D5—22no—
38C22×D5C22⋊C411yescharacteristic
39C22×D5C22⋊C411yescharacteristic
40C2×C4×C5—22no—
41C2×C4×C5—22no—
42C2×C4×C5—42no—
43C2×C4×C5C22⋊C412yes—
44C2×Dic5—22no—
45C2×Dic5—42no—
46C2×Dic5C2×D412yes—
47C2×Dic5#4114yes—
48C4×D5—24no—
49C4×D5—24no—
50C4×D5—44no—
51C4×D5—44no—
52#170—44no—
53C23×C4—51no—
54C23×C4—52no—
55C2×M16—54no—
56C2×C4⋊C4—52no—
57C2×C4⋊C4—51no—
58C2×C4⋊C4—102no—
59C2×C4⋊C4—102no—
60C2×C4⋊C4AGL(1,𝔽5)11yescharacteristic
61C22⋊C8—102no—
62C22⋊C8—102no—
63C22×C5—21no—
64C22×C5—21no—
65C22×C5C2×C22⋊C411yescharacteristic
66C22×C5#11712yes—
67C2×D5—28no—
68C2×D5—21no—
69C2×D5—21no—
70C2×D5—21no—
71C2×D5—21no—
72C2×D5#10012yes—
73C2×D5#10112yes—
74C4×C5—22no—
75C4×C5—42no—
76Dic5—42no—
77Dic5#12314yes—
78C24—51no—
79C22×C4—51no—
80C22×C4—52no—
81C22×C4—52no—
82C22×C4—52no—
83C22×C4—104no—
84C22×C4—102no—
85C22×C4—102no—
86C22×C4—204no—
87C22×C4C2×AGL(1,𝔽5)12yes—
88C22×C4C2×AGL(1,𝔽5)11yescharacteristic
89M16—108no—
90C2×C8—104no—
91C4⋊C4—22no—
92C4⋊C4—104no—
93C4⋊C4—102no—
94C2×C5—22no—
95C2×C5#38111yescharacteristic
96C2×C5C2×#10011yescharacteristic
97C2×C5C2×#10111yescharacteristic
98D5—22no—
99D5—22no—
100C23—51no—
101C23—51no—
102C23—102no—
103C23—102no—
104C23—102no—
105C23C22×AGL(1,𝔽5)11yescharacteristic
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—42no—
109C2×C4—54no—
110C2×C4—52no—
111C2×C4—104no—
112C2×C4—104no—
113C2×C4—102no—
114C2×C4—204no—
115C2×C4—204no—
116C2×C4—202no—
117C2×C4C22⋊AGL(1,𝔽5)12yes—
118C8—204no—
119C5#182111yescharacteristic, complemented, Sylow
120C22—21no—
121C22—21no—
122C22—52no—
123C22—52no—
124C22—108no—
125C22—101no—
126C22—101no—
127C22—101no—
128C22—101no—
129C22#413012yes—
130C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
131C4—22no—
132C4—42no—
133C4—54no—
134C4—202no—
135C2—22no—
136C2—102no—
137C2—102no—
138C2#6576411yescharacteristic
139C2D5⋊#10011yescharacteristic
140C2D5⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C4
FrattiniC23C22×AGL(1,𝔽5)
FittingC2×C5×C4⋊C4C4
socleC22×C5C2×C22⋊C4
prime radicalC23×D5C2×C4
cosocleC22×Dic5C23
hypercenterC2×C4⋊C4AGL(1,𝔽5)
hypocenterC5#1821
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #64919C2semidirect
119. C5#1821semidirect

Maximal subgroups

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

subgroupindex
C2×D5×C4⋊C42
#657552
#649192
#18215