Finite Group Databasebeta

← #10604589

Subgroups — #10604589

Order 640.

260 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 78 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
1D5×#125C211yesmaximal, characteristic
2C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3D5⋊(C4⋊C8)C211yesmaximal, characteristic
4#64874C212yesmaximal
5#64874C212yesmaximal
6C2×C4⋊AGL(1,𝔽5)C2212yes—
7C2×C4×AGL(1,𝔽5)—21no—
8C2×D5⋊C8C2212yes—
9C5×#125C411yescharacteristic, complemented
10D5×C4⋊C4—21no—
11D5×C4⋊C4—21no—
12D5×C4⋊C4C2211yescharacteristic
13D5×C4⋊C4C2211yescharacteristic
14#4036C411yescharacteristic, complemented
15#4036C411yescharacteristic, complemented
16#4035—21no—
17#4035—21no—
18C42×D5C2211yescharacteristic
19#4016C411yescharacteristic, complemented
20(C2×AGL(1,𝔽5))⋊C4—21no—
21#4004—21no—
22C4⋊#170—21no—
23#1989—51nomaximal, Sylow
24C22×AGL(1,𝔽5)—22no—
25C2×C4×D5—21no—
26C2×C4×D5—21no—
27C2×C4×D5D411yescharacteristic
28C2×C4×D5D411yescharacteristic
29C2×C4×D5C2311yescharacteristic
30C2×#170—22no—
31C4⋊AGL(1,𝔽5)—22no—
32C4⋊AGL(1,𝔽5)Q814yes—
33C4×AGL(1,𝔽5)—42no—
34D5⋊C8—24no—
35C5×C4⋊C4—21no—
36C5×C4⋊C4—21no—
37C5×C4⋊C4C2×C411yescharacteristic
38C5×C4⋊C4C2×C411yescharacteristic
39C42×C5C2×C411yescharacteristic
40C4⋊Dic5—21no—
41C4⋊Dic5—21no—
42C4⋊Dic5C2×C411yescharacteristic
43C4⋊Dic5C2×C411yescharacteristic
44#683—21no—
45#683—21no—
46#683—21no—
47#683—21no—
48#683—21no—
49#683—21no—
50C4×Dic5C2×C411yescharacteristic
51C4×Dic5D411yescharacteristic
52C4×Dic5D411yescharacteristic
53C2×#125—51no—
54C2×C4⋊C8—51no—
55C4×C4⋊C4—51no—
56#340—52no—
57#340—52no—
58C22×D5C2×D411yescharacteristic
59C2×AGL(1,𝔽5)—24no—
60C2×AGL(1,𝔽5)—42no—
61C2×AGL(1,𝔽5)—42no—
62C2×C4×C5—21no—
63C2×C4×C5—21no—
64C2×C4×C5C22⋊C411yescharacteristic
65C2×C4×C5C22⋊C411yescharacteristic
66C2×C4×C5C22×C411yescharacteristic
67C2×Dic5—21no—
68C2×Dic5—21no—
69C2×Dic5C22⋊C411yescharacteristic
70C2×Dic5C22⋊C411yescharacteristic
71C2×Dic5C2×D411yescharacteristic
72C4×D5—22no—
73C4×D5—21no—
74C4×D5—21no—
75C4×D5—42no—
76C4×D5—41no—
77C4×D5—41no—
78C4×D5C2×Q812yes—
79C4×D5#4112yes—
80#170—42no—
81C22×C8—52no—
82#125—51no—
83#125—51no—
84#125—51no—
85#125—101no—
86#125—101no—
87#125AGL(1,𝔽5)11yescharacteristic, complemented
88C2×C4⋊C4—52no—
89C2×C4⋊C4—51no—
90C2×C4⋊C4—51no—
91C2×C4⋊C4—101no—
92C2×C4⋊C4—101no—
93C2×C42—51no—
94C2×C42—101no—
95C4⋊C8—101no—
96C4⋊C8—101no—
97#95—101no—
98C22×C5C2×C22⋊C411yescharacteristic
99C2×D5C22⋊Q812yes—
100C2×D5#13512yes—
101C2×D5AGL(1,ℤ/8)11yescharacteristic
102C2×D5#13711yescharacteristic
103AGL(1,𝔽5)—44no—
104C4×C5—21no—
105C4×C5—21no—
106C4×C5—41no—
107C4×C5—41no—
108C4×C5C4×Q812yes—
109Dic5—21no—
110Dic5—21no—
111Dic5—41no—
112Dic5—41no—
113Dic5#12411yescharacteristic
114Dic5C4⋊Q811yescharacteristic
115C22×C4—51no—
116C22×C4—51no—
117C22×C4—51no—
118C22×C4—101no—
119C22×C4—101no—
120C22×C4—102no—
121C2×C8—102no—
122C2×C8—104no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—54no—
126C4⋊C4—51no—
127C4⋊C4—51no—
128C4⋊C4—101no—
129C4⋊C4—101no—
130C4⋊C4—101no—
131C4⋊C4—101no—
132C4⋊C4—101no—
133C4⋊C4—101no—
134C4⋊C4—102no—
135C4⋊C4—101no—
136C4⋊C4—101no—
137C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
138C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
139C42—51no—
140C42—51no—
141C42—51no—
142C42—202no—
143C42C2×AGL(1,𝔽5)11yescharacteristic
144C2×C5#39111yescharacteristic
145C2×C5#41611yescharacteristic
146C2×C5#41711yescharacteristic
147D5#47812yes—
148D5#47912yes—
149C23—51no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—51no—
153C2×C4—52no—
154C2×C4—52no—
155C2×C4—51no—
156C2×C4—51no—
157C2×C4—101no—
158C2×C4—101no—
159C2×C4—102no—
160C2×C4—104no—
161C2×C4—101no—
162C2×C4—101no—
163C2×C4—201no—
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#198911yescharacteristic, complemented, Sylow
173C22—51no—
174C22—51no—
175C22—52no—
176C22—52no—
177C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
178C4—21no—
179C4—21no—
180C4—41no—
181C4—41no—
182C4—51no—
183C4—51no—
184C4—101no—
185C4—101no—
186C4—204no—
187C4—201no—
188C4—201no—
189C4Q8×AGL(1,𝔽5)12yes—
190C2—52no—
191C2—52no—
192C2(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
193C2#6579311yescharacteristic
194C2#6579411yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×#125C4
socleC22×C5C2×C22⋊C4
prime radicalC22×D5C2×D4
cosocleC2×C4×D5C23
hypercenter#125AGL(1,𝔽5)
hypocenterC5#1989
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
9. C5×#125C4semidirect
14. #4036C4semidirect
15. #4036C4semidirect
19. #4016C4semidirect
87. #125AGL(1,𝔽5)semidirect
172. C5#1989semidirect

Maximal subgroups

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

subgroupindex
D5×#1252
C4×C4⋊AGL(1,𝔽5)2
D5⋊(C4⋊C8)2
#648742
#19895