Finite Group Databasebeta

← #10604537

Subgroups — #10604537

Order 640.

238 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 58 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#66014C211yesmaximal, characteristic
2#65686C211yesmaximal, characteristic, complemented
3#65678C211yesmaximal, characteristic, complemented
4#64926C212yesmaximal, complemented
5#64920C212yesmaximal
6C2×C4×AGL(1,𝔽5)—21no—
7C5×#124C411yescharacteristic, complemented
8#4092C2211yescharacteristic
9Q8×Dic5C411yescharacteristic, complemented
10#4084C2211yescharacteristic
11D4×Dic5C411yescharacteristic, complemented
12#4029—21no—
13#4027—21no—
14#4025—21no—
15#4023—21no—
16C42×D5C2211yescharacteristic
17#4016C411yescharacteristic, complemented
18(C2×AGL(1,𝔽5))⋊C4—21no—
19(C2×AGL(1,𝔽5))⋊C4C2212yes—
20#4004—21no—
21#4003C2212yes—
22C4×#170—21no—
23#1500—51nomaximal, Sylow
24C22×AGL(1,𝔽5)—22no—
25C2×C5×Q8C2×C411yescharacteristic
26C2×C5×D4C2×C411yescharacteristic
27C2×C22⋊D5—21no—
28C22×Dic5—21no—
29C2×C4×D5D411yescharacteristic
30C2×C4×D5D411yescharacteristic
31C2×C4×D5C2311yescharacteristic
32C2×#170—22no—
33C4×AGL(1,𝔽5)—24no—
34C5×C22⋊C4—21no—
35C5×C22⋊C4—21no—
36C42×C5C2×C411yescharacteristic
37C22⋊Dic5—21no—
38C22⋊Dic5—21no—
39#685—21no—
40C4⋊Dic5—21no—
41C4⋊Dic5C2×C411yescharacteristic
42C4⋊Dic5C2×C411yescharacteristic
43#683—21no—
44#683—21no—
45#683—21no—
46C4×Dic5—21no—
47C4×Dic5C2×C411yescharacteristic
48C4×Dic5D411yescharacteristic
49C4×Dic5D411yescharacteristic
50#529—51no—
51#431—51no—
52#376—51no—
53#328—52no—
54#327—52no—
55C23×C5—21no—
56C22×D5C2×D411yescharacteristic
57C2×AGL(1,𝔽5)—24no—
58C2×AGL(1,𝔽5)—42no—
59C2×AGL(1,𝔽5)—42no—
60C5×Q8—41no—
61C5×D4—41no—
62C2×C4×C5—21no—
63C2×C4×C5C22⋊C411yescharacteristic
64C2×C4×C5C22⋊C411yescharacteristic
65C2×C4×C5C22×C411yescharacteristic
66C22⋊D5—42no—
67C2×Dic5—21no—
68C2×Dic5—21no—
69C2×Dic5—42no—
70C2×Dic5C22⋊C411yescharacteristic
71C2×Dic5C22⋊C411yescharacteristic
72C2×Dic5C2×D411yescharacteristic
73C4×D5—22no—
74C4×D5—22no—
75C4×D5—41no—
76#170—42no—
77#126—101no—
78#125—51no—
79#124AGL(1,𝔽5)11yescharacteristic, complemented
80#123—101no—
81C22⋊Q8—51no—
82#121—51no—
83C4×Q8—51no—
84C4×D4—51no—
85C4×D4—101no—
86#117—101no—
87C2×C42—51no—
88C2×C42—101no—
89C4⋊C8—101no—
90C22⋊C8—52no—
91C4×C8—101no—
92#95—52no—
93#95—101no—
94C22×C5—41no—
95C22×C5—41no—
96C22×C5—41no—
97C22×C5C2×C22⋊C411yescharacteristic
98C2×D5—21no—
99C2×D5—21no—
100C2×D5#13512yes—
101AGL(1,𝔽5)—44no—
102C4×C5—21no—
103C4×C5—21no—
104C4×C5—21no—
105C4×C5—41no—
106Dic5—21no—
107Dic5—21no—
108Dic5—41no—
109Dic5—41no—
110Dic5#10412yes—
111C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
112C2×D4—101no—
113C2×D4C2×AGL(1,𝔽5)11yescharacteristic
114C22×C4—51no—
115C22×C4—51no—
116C22×C4—51no—
117C22×C4—101no—
118C22×C4—102no—
119C2×C8—102no—
120C4⋊C4—51no—
121C4⋊C4—51no—
122C4⋊C4—101no—
123C4⋊C4—101no—
124C4⋊C4—101no—
125C4⋊C4—101no—
126C22⋊C4—21no—
127C22⋊C4—21no—
128C22⋊C4—101no—
129C22⋊C4—101no—
130C22⋊C4—101no—
131C42—51no—
132C42—51no—
133C42—51no—
134C42—101no—
135C42—104no—
136C42C2×AGL(1,𝔽5)11yescharacteristic
137C2×C5—81no—
138C2×C5#41011yescharacteristic
139C2×C5#41611yescharacteristic
140C2×C5C2×#10411yescharacteristic
141D5—22no—
142C23—21no—
143C23—51no—
144Q8—41no—
145D4—41no—
146D4—202no—
147C2×C4—21no—
148C2×C4—51no—
149C2×C4—51no—
150C2×C4—51no—
151C2×C4—102no—
152C2×C4—102no—
153C2×C4—104no—
154C2×C4—101no—
155C2×C4—101no—
156C2×C4—201no—
157C2×C4—202no—
158C2×C4—202no—
159C2×C4—202no—
160C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
161C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
162C2×C4C22×AGL(1,𝔽5)11yescharacteristic
163C8—202no—
164C5#150011yescharacteristic, complemented, Sylow
165C22—41no—
166C22—41no—
167C22—41no—
168C22—52no—
169C22—101no—
170C22—101no—
171C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—52no—
177C4—101no—
178C4—101no—
179C4—204no—
180C4—201no—
181C4—201no—
182C2—81no—
183C2—102no—
184C2#6577111yescharacteristic
185C2D5⋊#10411yescharacteristic
186C2#6579411yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×#124C4
socleC22×C5C2×C22⋊C4
prime radical#4084C22
cosocleC2×C4×D5C23
hypercenter#124AGL(1,𝔽5)
hypocenterC5#1500
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #65686C2semidirect
3. #65678C2semidirect
4. #64926C2semidirect
7. C5×#124C4semidirect
9. Q8×Dic5C4semidirect
11. D4×Dic5C4semidirect
17. #4016C4semidirect
79. #124AGL(1,𝔽5)semidirect
164. C5#1500semidirect

Maximal subgroups

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

subgroupindex
#660142
#656862
#656782
#649262
#649202
#15005