Finite Group Databasebeta

← #10604547

Subgroups — #10604547

Order 640.

270 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 52 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#66008C211yesmaximal, characteristic
2#65785C212yesmaximal, complemented
3#64922C212yesmaximal, complemented
4#64853C212yesmaximal, complemented
5C2×#713C2211yescharacteristic
6C2×D4×D5C411yescharacteristic, complemented
7#4130—22no—
8C5×#124C411yescharacteristic, complemented
9#4093C411yescharacteristic, complemented
10#4086C2211yescharacteristic
11#4030—22no—
12#4028—22no—
13#4020C2211yescharacteristic
14#4018C411yescharacteristic, complemented
15#3999—22no—
16#3999C2214yescomplemented
17#2128—51nomaximal, Sylow
18C23×D5—21no—
19C2×C5×Q8C2×C411yescharacteristic
20C2×C5×D4C2×C411yescharacteristic
21C2×C22⋊D5—22no—
22C2×C22⋊D5—21no—
23#713—42no—
24D4×D5—42no—
25C2×D20—22no—
26C2×D20C2×C411yescharacteristic
27C2×D20D412yes—
28C2×D20C2311yescharacteristic
29C2×C4×D5C2×C411yescharacteristic
30C2×C4×D5D412yes—
31C22⋊AGL(1,𝔽5)—24no—
32C4⋊AGL(1,𝔽5)—42no—
33C4×AGL(1,𝔽5)—42no—
34C5×C22⋊C4—22no—
35C42×C5C2×C411yescharacteristic
36#685—21no—
37#685—22no—
38#683—21no—
39C4×Dic5C2×C411yescharacteristic
40#535—51no—
41#410—52no—
42#353—52no—
43#353—52no—
44C23×C5—21no—
45C22×D5—21no—
46C22×D5—42no—
47C22×D5—82no—
48C22×D5C22⋊C411yescharacteristic
49C22×D5C2×D412yes—
50C2×AGL(1,𝔽5)—44no—
51C5×Q8—41no—
52C5×D4—41no—
53C2×C4×C5—21no—
54C2×C4×C5C22⋊C412yes—
55C2×C4×C5C22×C411yescharacteristic
56C22⋊D5—42no—
57C22⋊D5—82no—
58C2×Dic5—21no—
59C2×Dic5C22⋊C411yescharacteristic
60D20—42no—
61D20—41no—
62D20—41no—
63D20—82no—
64C4×D5—22no—
65C4×D5—22no—
66C4×D5—22no—
67C2×#41—51no—
68C22×D4—51no—
69#127—51no—
70#127—51no—
71#124—51no—
72#124AGL(1,𝔽5)11yescharacteristic, complemented
73#121—102no—
74#120—102no—
75#117—51no—
76#117—102no—
77#99—54no—
78#99—102no—
79C22×C5—42no—
80C22×C5—41no—
81C22×C5C2×C22⋊C411yescharacteristic
82C2×D5—22no—
83C2×D5—21no—
84C2×D5—42no—
85C2×D5—41no—
86C2×D5—42no—
87C2×D5—41no—
88C2×D5—84no—
89AGL(1,𝔽5)—84no—
90C4×C5—21no—
91C4×C5—22no—
92C4×C5—41no—
93Dic5—41no—
94Dic5#9912yes—
95C24—101no—
96#41—202no—
97C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
98C2×D4—52no—
99C2×D4—51no—
100C2×D4—51no—
101C2×D4—102no—
102C2×D4—101no—
103C2×D4—102no—
104C2×D4—202no—
105C2×D4C2×AGL(1,𝔽5)11yescharacteristic
106C22×C4—52no—
107C22×C4—51no—
108C4⋊C4—101no—
109C4⋊C4—202no—
110C22⋊C4—22no—
111C22⋊C4—101no—
112C22⋊C4—102no—
113C22⋊C4—104no—
114C42—51no—
115C42—202no—
116C42C2×AGL(1,𝔽5)11yescharacteristic
117C2×C5—21no—
118C2×C5—81no—
119C2×C5C2×#9911yescharacteristic
120D5—42no—
121D5—41no—
122D5—81no—
123C23—21no—
124C23—52no—
125C23—51no—
126C23—101no—
127C23—202no—
128C23—402no—
129Q8—41no—
130D4—41no—
131D4—202no—
132D4—201no—
133D4—201no—
134D4—202no—
135D4—402no—
136D4—402no—
137C2×C4—21no—
138C2×C4—51no—
139C2×C4—102no—
140C2×C4—102no—
141C2×C4—102no—
142C2×C4—101no—
143C2×C4—204no—
144C2×C4C22⋊AGL(1,𝔽5)12yes—
145C2×C4C22×AGL(1,𝔽5)11yescharacteristic
146C5#212811yescharacteristic, complemented, Sylow
147C22—42no—
148C22—41no—
149C22—102no—
150C22—101no—
151C22—202no—
152C22—201no—
153C22—202no—
154C22—201no—
155C22—404no—
156C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
157C4—22no—
158C4—21no—
159C4—41no—
160C4—52no—
161C4—201no—
162C4—404no—
163C2—21no—
164C2—81no—
165C2—202no—
166C2—201no—
167C2—401no—
168C2D5⋊#9911yescharacteristic
subgroupG/H
centerC2D5⋊#99
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×#124C4
socleC2×C5C2×#99
prime radical#66008C2
cosocleC2×D20C23
hypercenter#124AGL(1,𝔽5)
hypocenterC5#2128
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #65785C2semidirect
3. #64922C2semidirect
4. #64853C2semidirect
6. C2×D4×D5C4semidirect
8. C5×#124C4semidirect
9. #4093C4semidirect
14. #4018C4semidirect
16. #3999C22semidirect
72. #124AGL(1,𝔽5)semidirect
146. C5#2128semidirect

Maximal subgroups

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

subgroupindex
#660082
#657852
#649222
#648532
#21285