Finite Group Databasebeta

← #10604283

Subgroups — #10604283

Order 640.

344 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C2×D5×C4⋊C4C211yesmaximal, characteristic
2#65765C211yesmaximal, characteristic
3D5⋊(C22⋊C8)C211yesmaximal, characteristic
4#64874C214yesmaximal, complemented
5C22×C4×D5—21no—
6C22×C4×D5C2211yescharacteristic
7C2×C22⋊AGL(1,𝔽5)—21no—
8C2×C4⋊AGL(1,𝔽5)C2212yes—
9C2×D5⋊C8C2212yes—
10C2×C5×C4⋊C4C411yescharacteristic, complemented
11C2×C4⋊Dic5C411yescharacteristic, complemented
12C2×#683—21no—
13D5×C4⋊C4—21no—
14D5×C4⋊C4—21no—
15D5×C4⋊C4C412yescomplemented
16D5×C4⋊C4C2212yes—
17C22⋊#170—21no—
18(C2×AGL(1,𝔽5))⋊C4—21no—
19#4003—21no—
20#1824—51nomaximal, Sylow
21C23×D5D411yescharacteristic
22C22×AGL(1,𝔽5)—22no—
23C22×C4×C5—21no—
24C22×C4×C5C2×C411yescharacteristic
25C22×Dic5—21no—
26C22×Dic5D411yescharacteristic
27C2×C4×D5—24no—
28C2×C4×D5—22no—
29C2×C4×D5—22no—
30C2×C4×D5—44no—
31C2×C4×D5C2×C411yescharacteristic
32C2×C4×D5D411yescharacteristic
33C2×C4×D5D411yescharacteristic
34C2×C4×D5C2311yescharacteristic
35C22⋊AGL(1,𝔽5)—42no—
36C2×#170—22no—
37C4⋊AGL(1,𝔽5)—24no—
38D5⋊C8—24no—
39C5×C4⋊C4—21no—
40C5×C4⋊C4C2×C412yes—
41C4⋊Dic5—21no—
42C4⋊Dic5C2×C412yes—
43#683—22no—
44#683—22no—
45C22×C4⋊C4—51no—
46C2×C22⋊C8—51no—
47#380—51no—
48#340—54no—
49C23×C5C22⋊C411yescharacteristic
50C22×D5—22no—
51C22×D5—22no—
52C22×D5C22⋊C411yescharacteristic
53C22×D5SD1612yes—
54C22×D5Q1612yes—
55C22×D5C2×D411yescharacteristic
56C2×AGL(1,𝔽5)—42no—
57C2×AGL(1,𝔽5)—42no—
58C2×AGL(1,𝔽5)—42no—
59C2×C4×C5—22no—
60C2×C4×C5—22no—
61C2×C4×C5—42no—
62C2×C4×C5C22⋊C411yescharacteristic
63C2×C4×C5C22×C411yescharacteristic
64C2×Dic5—22no—
65C2×Dic5—22no—
66C2×Dic5—42no—
67C2×Dic5C22⋊C411yescharacteristic
68C2×Dic5C2×D411yescharacteristic
69C4×D5—22no—
70C4×D5—22no—
71C4×D5—22no—
72C4×D5—44no—
73C4×D5—44no—
74C4×D5#4114yes—
75#170—42no—
76C23×C4—51no—
77C23×C4—101no—
78C22×C8—52no—
79C2×C4⋊C4—51no—
80C2×C4⋊C4—52no—
81C2×C4⋊C4—52no—
82C2×C4⋊C4—52no—
83C2×C4⋊C4—101no—
84C2×C4⋊C4—101no—
85C2×C4⋊C4—101no—
86C2×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
87C2×C22⋊C4—101no—
88C22⋊C8—101no—
89C22⋊C8—101no—
90#95—101no—
91C22×C5—21no—
92C22×C5—21no—
93C22×C5#10312yes—
94C22×C5C2×C22⋊C411yescharacteristic
95C2×D5—21no—
96C2×D5—21no—
97C2×D5—24no—
98C2×D5—24no—
99C2×D5#10312yes—
100C2×D5#12312yes—
101C2×D5C2×SD1611yescharacteristic
102C2×D5C2×Q1611yescharacteristic
103C2×D5AGL(1,ℤ/8)12yes—
104AGL(1,𝔽5)—82no—
105C4×C5—21no—
106C4×C5—42no—
107C4×C5#11712yes—
108Dic5—21no—
109Dic5—42no—
110Dic5#12312yes—
111C24—51no—
112C22×C4—21no—
113C22×C4—51no—
114C22×C4—51no—
115C22×C4—51no—
116C22×C4—51no—
117C22×C4—51no—
118C22×C4—104no—
119C22×C4—102no—
120C22×C4—102no—
121C22×C4—101no—
122C22×C4—102no—
123C22×C4—204no—
124C22×C4C2×AGL(1,𝔽5)11yescharacteristic
125C2×C8—102no—
126C2×C8—104no—
127C4⋊C4—21no—
128C4⋊C4—52no—
129C4⋊C4—102no—
130C4⋊C4—102no—
131C4⋊C4—104no—
132C4⋊C4—101no—
133C4⋊C4C2×AGL(1,𝔽5)12yes—
134C22⋊C4—202no—
135C2×C5—22no—
136C2×C5#38111yescharacteristic
137C2×C5C2×#10311yescharacteristic
138C2×C5#41811yescharacteristic
139D5—22no—
140D5#48112yes—
141D5#48412yes—
142C23—51no—
143C23—51no—
144C23—52no—
145C23—52no—
146C23—102no—
147C23—102no—
148C23C22⋊AGL(1,𝔽5)11yescharacteristic
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—51no—
153C2×C4—54no—
154C2×C4—51no—
155C2×C4—102no—
156C2×C4—102no—
157C2×C4—102no—
158C2×C4—102no—
159C2×C4—102no—
160C2×C4—204no—
161C2×C4—204no—
162C2×C4—202no—
163C2×C4—202no—
164C2×C4—202no—
165C2×C4—202no—
166C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
167C2×C4C22×AGL(1,𝔽5)11yescharacteristic
168C8—202no—
169C5#182411yescharacteristic, complemented, Sylow
170C22—21no—
171C22—21no—
172C22—52no—
173C22—52no—
174C22—51no—
175C22—51no—
176C22—52no—
177C22—104no—
178C22—104no—
179C22—101no—
180C22—101no—
181C22#400912yes—
182C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
183C4—21no—
184C4—42no—
185C4—52no—
186C4—101no—
187C4—202no—
188C4—402no—
189C4#413012yes—
190C2—22no—
191C2—52no—
192C2—52no—
193C2—102no—
194C2#6576411yescharacteristic
195C2#6576711yescharacteristic
196C2C2×#400911yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC2×C5×C4⋊C4C4
socleC22×C5C2×C22⋊C4
prime radicalC23×D5D4
cosocleC2×C4×D5C23
hypercenterC2×C4⋊C4AGL(1,𝔽5)
hypocenterC5#1824
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #64874C2semidirect
10. C2×C5×C4⋊C4C4semidirect
11. C2×C4⋊Dic5C4semidirect
15. D5×C4⋊C4C4semidirect
86. C2×C4⋊C4AGL(1,𝔽5)semidirect
169. C5#1824semidirect

Maximal subgroups

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

subgroupindex
C2×D5×C4⋊C42
#657652
D5⋊(C22⋊C8)2
#648742
#18245