Finite Group Databasebeta

← #10609344

Subgroups — #10609344

Order 640.

400 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C5×#397C211yesmaximal, characteristic
2#65273C212yesmaximal, complemented
3#65240C212yesmaximal
4#65236C211yesmaximal, characteristic
5#65235C212yesmaximal
6#65232C211yesmaximal, characteristic
7#64948C212yesmaximal, complemented
8#64940C212yesmaximal, complemented
9#64935C212yesmaximal, complemented
10C23×Dic5C2211yescharacteristic
11C22×Dic10C2211yescharacteristic
12C2×C5×C4⋊C4C2211yescharacteristic
13C2×C5×C22⋊C4C2212yes—
14C2×C5×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic5C2212yes—
16C2×C22⋊Dic5C2211yescharacteristic
17C2×C4⋊Dic5C2212yes—
18C2×#683C2212yes—
19C2×#683C2212yes—
20C2×C4×Dic5C2212yes—
21C5×#95C2212yes—
22C5×#95C2211yescharacteristic
23#3963C2212yes—
24#3963C2212yes—
25#3963C2212yes—
26#3963C2212yes—
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2211yescharacteristic
31#2764—51nomaximal, Sylow
32C24×C5C2311yescharacteristic
33C22×C4×C5C2311yescharacteristic
34C22×C4×C5C2312yes—
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2312yes—
37C22×Dic5—22no—
38C22×Dic5—41no—
39C22×Dic5—41no—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C22×Dic5C2312yes—
43C22×Dic5C2312yes—
44C2×Dic10—24no—
45C2×Dic10—41no—
46C5×C4⋊C4—41no—
47C5×C22⋊C4—24no—
48C5×C22⋊C4—41no—
49C22⋊Dic5—24no—
50C22⋊Dic5—41no—
51C4⋊Dic5—42no—
52#683—42no—
53#683—42no—
54C4×Dic5—24no—
55#400—52no—
56#398—52no—
57#397D511yescharacteristic
58#393—51no—
59#391—52no—
60#388—52no—
61#385—52no—
62#383—52no—
63#381—51no—
64C23×C5—22no—
65C23×C5—21no—
66C23×C5—21no—
67C23×C5—22no—
68C23×C5#4112yes—
69C23×C5C2411yescharacteristic
70C2×C4×C5—22no—
71C2×C4×C5—22no—
72C2×C4×C5—22no—
73C2×C4×C5—22no—
74C2×C4×C5—22no—
75C2×C4×C5—22no—
76C2×C4×C5—21no—
77C2×C4×C5—21no—
78C2×C4×C5—22no—
79C2×C4×C5—22no—
80C2×Dic5—22no—
81C2×Dic5—22no—
82C2×Dic5—22no—
83C2×Dic5—22no—
84C2×Dic5—22no—
85C2×Dic5—22no—
86C2×Dic5—22no—
87C2×Dic5—22no—
88C2×Dic5—22no—
89C2×Dic5—22no—
90C2×Dic5—22no—
91C2×Dic5—44no—
92C2×Dic5#4114yes—
93Dic10—44no—
94C22×Q8—51no—
95C23×C4—51no—
96C2×C4⋊C4—52no—
97C2×C4⋊C4—52no—
98C2×C4⋊C4—52no—
99C2×C4⋊C4C2×D511yescharacteristic
100C2×C22⋊C4—52no—
101C2×C22⋊C4—51no—
102C2×C22⋊C4C2×D512yes—
103C2×C22⋊C4C2×D511yescharacteristic
104C2×C42—52no—
105#95—52no—
106#95—52no—
107#95—52no—
108#95—52no—
109#95—52no—
110#95—52no—
111#95—52no—
112#95—51no—
113#95C2×D511yescharacteristic
114#95C2×D512yes—
115C22×C5—22no—
116C22×C5—24no—
117C22×C5—42no—
118C22×C5—41no—
119C22×C5—41no—
120C22×C5C2×#4111yescharacteristic
121C22×C5C2×#4112yes—
122C22×C5H(5,𝔽2)11yescharacteristic
123C22×C5H(5,𝔽2)11yescharacteristic
124C22×C52−1+412yes—
125C4×C5—42no—
126C4×C5—41no—
127C4×C5—42no—
128C4×C5—41no—
129Dic5—24no—
130Dic5—42no—
131Dic5—42no—
132Dic5—42no—
133C24C22×D511yescharacteristic
134C2×Q8—104no—
135C2×Q8—201no—
136C22×C4—52no—
137C22×C4—52no—
138C22×C4—52no—
139C22×C4—52no—
140C22×C4—102no—
141C22×C4—201no—
142C22×C4—201no—
143C22×C4C22×D511yescharacteristic
144C22×C4C22×D512yes—
145C22×C4C22×D511yescharacteristic
146C22×C4C22×D512yes—
147C4⋊C4—41no—
148C4⋊C4—202no—
149C4⋊C4—202no—
150C4⋊C4—202no—
151C22⋊C4—24no—
152C22⋊C4—41no—
153C22⋊C4—104no—
154C22⋊C4—201no—
155C42—104no—
156C2×C5—42no—
157C2×C5#53811yescharacteristic
158C2×C5#54212yes—
159C2×C5#55111yescharacteristic
160C2×C5#55212yes—
161C2×C5#56311yescharacteristic
162C23—21no—
163C23—22no—
164C23—21no—
165C23—22no—
166C23#71112yes—
167C23C23×D511yescharacteristic
168Q8—204no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—54no—
180C2×C4—102no—
181C2×C4—102no—
182C2×C4—102no—
183C2×C4—102no—
184C2×C4—102no—
185C2×C4—102no—
186C2×C4—102no—
187C2×C4—102no—
188C2×C4—102no—
189C2×C4—102no—
190C2×C4—102no—
191C2×C4—204no—
192C5#276411yescharacteristic, complemented, Sylow
193C22—22no—
194C22—24no—
195C22—41no—
196C22—41no—
197C22—42no—
198C22C2×#71111yescharacteristic
199C22#414411yescharacteristic
200C22D5×#4112yes—
201C22#414911yescharacteristic
202C22#415012yes—
203C4—42no—
204C4—42no—
205C4—41no—
206C4—41no—
207C4—104no—
208C4—202no—
209C4—202no—
210C4—202no—
211C2—42no—
212C2#6595111yescharacteristic
213C2#6598412yes—
214C2#6598511yescharacteristic
215C2#6601311yescharacteristic
216C2#6604412yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radicalC24×C5C23
hypercenter#397D5
hypocenterC5#2764
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #65273C2semidirect
7. #64948C2semidirect
8. #64940C2semidirect
9. #64935C2semidirect
192. C5#2764semidirect

Maximal subgroups

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

subgroupindex
C5×#3972
#652732
#652402
#652362
#652352
#652322
#649482
#649402
#649352
#27645