Finite Group Databasebeta

← #10604538

Subgroups — #10604538

Order 640.

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

normal subgroups only

This group has 242 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
2C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3#65684C211yesmaximal, characteristic, complemented
4#64926C212yesmaximal, complemented
5#64920C212yesmaximal
6C2×C4⋊AGL(1,𝔽5)—21no—
7C2×C4×AGL(1,𝔽5)—21no—
8C5×#124C411yescharacteristic, complemented
9#4092C2211yescharacteristic
10Q8×Dic5C411yescharacteristic, complemented
11#4084C2211yescharacteristic
12D4×Dic5C411yescharacteristic, complemented
13#4029—21no—
14#4027—21no—
15#4025—21no—
16#4023—21no—
17C42×D5C2211yescharacteristic
18#4016C411yescharacteristic, complemented
19(C2×AGL(1,𝔽5))⋊C4C2212yes—
20#4004—21no—
21#4003C2212yes—
22#4002—21no—
23#1509—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—
34C4×AGL(1,𝔽5)—42no—
35C5×C22⋊C4—21no—
36C5×C22⋊C4—21no—
37C42×C5C2×C411yescharacteristic
38C22⋊Dic5—21no—
39C22⋊Dic5—21no—
40#685—21no—
41C4⋊Dic5—21no—
42C4⋊Dic5C2×C411yescharacteristic
43C4⋊Dic5C2×C411yescharacteristic
44#683—21no—
45#683—21no—
46#683—21no—
47C4×Dic5—21no—
48C4×Dic5C2×C411yescharacteristic
49C4×Dic5D411yescharacteristic
50C4×Dic5D411yescharacteristic
51#529—51no—
52#432—51no—
53C4×C4⋊C4—51no—
54#328—52no—
55#327—52no—
56C23×C5—21no—
57C22×D5C2×D411yescharacteristic
58C2×AGL(1,𝔽5)—24no—
59C2×AGL(1,𝔽5)—42no—
60C2×AGL(1,𝔽5)—42no—
61C5×Q8—41no—
62C5×D4—41no—
63C2×C4×C5—21no—
64C2×C4×C5C22⋊C411yescharacteristic
65C2×C4×C5C22⋊C411yescharacteristic
66C2×C4×C5C22×C411yescharacteristic
67C22⋊D5—41no—
68C22⋊D5—41no—
69C2×Dic5—21no—
70C2×Dic5—21no—
71C2×Dic5—42no—
72C2×Dic5C22⋊C411yescharacteristic
73C2×Dic5C22⋊C411yescharacteristic
74C2×Dic5C2×D411yescharacteristic
75C4×D5—21no—
76C4×D5—21no—
77C4×D5—22no—
78C4×D5—41no—
79#170—42no—
80#126—101no—
81#125—51no—
82#124AGL(1,𝔽5)11yescharacteristic, complemented
83#123—101no—
84C22⋊Q8—51no—
85#121—51no—
86C4×Q8—51no—
87C4×D4—51no—
88C4×D4—101no—
89#117—101no—
90C2×C4⋊C4—101no—
91C2×C42—51no—
92C2×C42—101no—
93C4⋊C8—101no—
94C22⋊C8—52no—
95#97—101no—
96#95—52no—
97C22×C5—41no—
98C22×C5—41no—
99C22×C5—41no—
100C22×C5C2×C22⋊C411yescharacteristic
101C2×D5—21no—
102C2×D5—21no—
103C2×D5AGL(1,ℤ/8)11yescharacteristic
104C2×D5#13711yescharacteristic
105AGL(1,𝔽5)—44no—
106C4×C5—21no—
107C4×C5—21no—
108C4×C5—21no—
109C4×C5—41no—
110Dic5—21no—
111Dic5—21no—
112Dic5—41no—
113Dic5—41no—
114Dic5#10412yes—
115C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
116C2×D4—101no—
117C2×D4C2×AGL(1,𝔽5)11yescharacteristic
118C22×C4—51no—
119C22×C4—51no—
120C22×C4—51no—
121C22×C4—101no—
122C22×C4—102no—
123C2×C8—102no—
124C4⋊C4—51no—
125C4⋊C4—51no—
126C4⋊C4—101no—
127C4⋊C4—101no—
128C4⋊C4—104no—
129C4⋊C4—101no—
130C4⋊C4—101no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—101no—
134C22⋊C4—101no—
135C22⋊C4—101no—
136C42—51no—
137C42—51no—
138C42—51no—
139C42—101no—
140C42—202no—
141C42C2×AGL(1,𝔽5)11yescharacteristic
142C2×C5—81no—
143C2×C5#41011yescharacteristic
144C2×C5#41711yescharacteristic
145C2×C5C2×#10411yescharacteristic
146D5—21no—
147D5—21no—
148C23—21no—
149C23—51no—
150Q8—41no—
151D4—41no—
152D4—201no—
153D4—201no—
154C2×C4—21no—
155C2×C4—51no—
156C2×C4—51no—
157C2×C4—51no—
158C2×C4—102no—
159C2×C4—101no—
160C2×C4—101no—
161C2×C4—104no—
162C2×C4—101no—
163C2×C4—101no—
164C2×C4—201no—
165C2×C4—202no—
166C2×C4—202no—
167C2×C4—202no—
168C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
169C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
170C2×C4C22×AGL(1,𝔽5)11yescharacteristic
171C8—202no—
172C5#150911yescharacteristic, complemented, Sylow
173C22—41no—
174C22—41no—
175C22—41no—
176C22—51no—
177C22—51no—
178C22—101no—
179C22—101no—
180C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—41no—
185C4—52no—
186C4—101no—
187C4—101no—
188C4—204no—
189C4—201no—
190C4—201no—
191C2—81no—
192C2—101no—
193C2—101no—
194C2#6577111yescharacteristic
195C2D5⋊#10411yescharacteristic
196C2#6579311yescharacteristic
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#1509
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. C4×C4⋊AGL(1,𝔽5)C2semidirect
3. #65684C2semidirect
4. #64926C2semidirect
8. C5×#124C4semidirect
10. Q8×Dic5C4semidirect
12. D4×Dic5C4semidirect
18. #4016C4semidirect
82. #124AGL(1,𝔽5)semidirect
172. C5#1509semidirect

Maximal subgroups

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

subgroupindex
#660142
C4×C4⋊AGL(1,𝔽5)2
#656842
#649262
#649202
#15095