Finite Group Databasebeta

← #10604592

Subgroups — #10604592

Order 640.

260 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 72 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#66022C211yesmaximal, characteristic
2C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3C4×#700C211yesmaximal, characteristic
4#64875C212yesmaximal
5#64875C212yesmaximal
6C2×C4⋊AGL(1,𝔽5)—21no—
7C2×C4×AGL(1,𝔽5)C2212yes—
8C2×#700C2212yes—
9C5×#125C411yescharacteristic, complemented
10#4042—21no—
11#4042—21no—
12#4038C2211yescharacteristic
13#4038C2211yescharacteristic
14#4034C411yescharacteristic, complemented
15#4034C411yescharacteristic, complemented
16#4033—21no—
17#4033—21no—
18C42×D5C2211yescharacteristic
19#4016C411yescharacteristic, complemented
20(C2×AGL(1,𝔽5))⋊C4—21no—
21#4002—21no—
22C4×#170—21no—
23#1986—51nomaximal, Sylow
24C22×AGL(1,𝔽5)—22no—
25C2×C4×D5D411yescharacteristic
26C2×C4×D5D411yescharacteristic
27C2×C4×D5C2311yescharacteristic
28C2×Dic10—21no—
29C2×Dic10—21no—
30C2×#170—22no—
31C4⋊AGL(1,𝔽5)—24no—
32C4×AGL(1,𝔽5)—24no—
33#700—22no—
34#700Q814yescomplemented
35C5×C4⋊C4—21no—
36C5×C4⋊C4—21no—
37C5×C4⋊C4C2×C411yescharacteristic
38C5×C4⋊C4C2×C411yescharacteristic
39C42×C5C2×C411yescharacteristic
40#685—21no—
41#685—21no—
42C4⋊Dic5C2×C411yescharacteristic
43C4⋊Dic5C2×C411yescharacteristic
44#683—21no—
45#683—21no—
46#683—21no—
47#683—21no—
48C4×Dic5—21no—
49C4×Dic5—21no—
50C4×Dic5C2×C411yescharacteristic
51C4×Dic5D411yescharacteristic
52C4×Dic5D411yescharacteristic
53#533—51no—
54C4×M16—51no—
55C4×C4⋊C4—51no—
56#339—52no—
57#339—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—41no—
74C4×D5C2×Q812yes—
75C4×D5#4112yes—
76Dic10—42no—
77Dic10—42no—
78#170—24no—
79C2×M16—52no—
80C4⋊Q8—51no—
81C4⋊Q8—51no—
82#125—51no—
83#125AGL(1,𝔽5)11yescharacteristic, complemented
84C22⋊Q8—101no—
85C22⋊Q8—101no—
86C4×Q8—101no—
87C4×Q8—101no—
88#117—51no—
89#117—51no—
90C2×C4⋊C4—101no—
91C2×C42—51no—
92C2×C42—52no—
93#97—101no—
94C4×C8—101no—
95#95—101no—
96C22×C5C2×C22⋊C411yescharacteristic
97C2×D5—21no—
98C2×D5—21no—
99C2×D5#12411yescharacteristic
100C2×D5C4⋊Q811yescharacteristic
101AGL(1,𝔽5)—44no—
102C4×C5—21no—
103C4×C5—21no—
104C4×C5—41no—
105C4×C5—41no—
106C4×C5C4×Q812yes—
107Dic5—21no—
108Dic5—41no—
109Dic5—41no—
110Dic5#10412yes—
111Dic5C22⋊Q812yes—
112C2×Q8—101no—
113C2×Q8—101no—
114C22×C4—51no—
115C22×C4—51no—
116C22×C4—51no—
117C22×C4—102no—
118M16—54no—
119M16—102no—
120C2×C8—102no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—51no—
124C4⋊C4—51no—
125C4⋊C4—101no—
126C4⋊C4—101no—
127C4⋊C4—104no—
128C4⋊C4—101no—
129C4⋊C4—101no—
130C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
131C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
132C22⋊C4—101no—
133C22⋊C4—101no—
134C42—51no—
135C42—51no—
136C42—51no—
137C42—101no—
138C42—101no—
139C42—104no—
140C42C2×AGL(1,𝔽5)11yescharacteristic
141C2×C5#39111yescharacteristic
142C2×C5C2×#10411yescharacteristic
143C2×C5#42111yescharacteristic
144D5—21no—
145D5—21no—
146C23—51no—
147Q8—202no—
148Q8—202no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—51no—
152C2×C4—51no—
153C2×C4—52no—
154C2×C4—52no—
155C2×C4—51no—
156C2×C4—102no—
157C2×C4—104no—
158C2×C4—101no—
159C2×C4—101no—
160C2×C4—201no—
161C2×C4—202no—
162C2×C4—202no—
163C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
164C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
165C2×C4C22×AGL(1,𝔽5)11yescharacteristic
166C8—104no—
167C5#198611yescharacteristic, complemented, Sylow
168C22—51no—
169C22—51no—
170C22—101no—
171C22—101no—
172C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—52no—
178C4—52no—
179C4—101no—
180C4—204no—
181C4—201no—
182C4—201no—
183C4Q8×AGL(1,𝔽5)12yes—
184C2—101no—
185C2—101no—
186C2(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
187C2D5⋊#10411yescharacteristic
188C2#6579511yescharacteristic
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#1986
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. #4034C4semidirect
15. #4034C4semidirect
19. #4016C4semidirect
34. #700Q8semidirect
83. #125AGL(1,𝔽5)semidirect
167. C5#1986semidirect

Maximal subgroups

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

subgroupindex
#660222
C4×C4⋊AGL(1,𝔽5)2
C4×#7002
#648752
#19865