Finite Group Databasebeta

← #10609348

Subgroups — #10609348

Order 640.

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

normal subgroups only

This group has 392 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#65261C212yesmaximal, complemented
3#65240C212yesmaximal
4#65238C211yesmaximal, characteristic
5#65235C212yesmaximal
6#65232C211yesmaximal, characteristic
7#64949C212yesmaximal, complemented
8#64949C212yesmaximal, complemented
9#64942C212yesmaximal, complemented
10C23×Dic5C2211yescharacteristic
11C2×C5×C4⋊C4C2211yescharacteristic
12C2×C5×C22⋊C4C2212yes—
13C2×C5×C22⋊C4C2211yescharacteristic
14C2×C22⋊Dic5C2212yes—
15C2×C22⋊Dic5C2211yescharacteristic
16C2×C4⋊Dic5C2212yes—
17C2×C4⋊Dic5C2211yescharacteristic
18C2×#683C2212yes—
19C2×#683C2212yes—
20C2×#683C2212yes—
21C2×C4×Dic5C2212yes—
22C5×#95C2212yes—
23C5×#95C2211yescharacteristic
24#3963C2211yescharacteristic
25#3963C2212yes—
26#3963C2212yes—
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#2771—51nomaximal, Sylow
32C24×C5C2311yescharacteristic
33C22×C4×C5C2311yescharacteristic
34C22×C4×C5C2312yes—
35C22×C4×C5C2312yes—
36C22×C4×C5C2311yescharacteristic
37C22×Dic5—22no—
38C22×Dic5—41no—
39C22×Dic5—41no—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C22×Dic5C2312yes—
43C22×Dic5C2312yes—
44C5×C4⋊C4—41no—
45C5×C22⋊C4—24no—
46C5×C22⋊C4—41no—
47C22⋊Dic5—24no—
48C22⋊Dic5—41no—
49C4⋊Dic5—24no—
50C4⋊Dic5—41no—
51#683—24no—
52#683—42no—
53#683—42no—
54C4×Dic5—42no—
55#400—52no—
56#400—52no—
57#399—51no—
58#397D511yescharacteristic
59#389—52no—
60#388—52no—
61#387—52no—
62#385—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—
93C23×C4—51no—
94C2×C4⋊C4—52no—
95C2×C4⋊C4—52no—
96C2×C4⋊C4—52no—
97C2×C4⋊C4—52no—
98C2×C4⋊C4—51no—
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—51no—
110#95—52no—
111#95—52no—
112#95C2×D512yes—
113#95C2×D511yescharacteristic
114C22×C5—22no—
115C22×C5—24no—
116C22×C5—42no—
117C22×C5—41no—
118C22×C5—41no—
119C22×C5C2×#4111yescharacteristic
120C22×C5C2×#4112yes—
121C22×C5H(5,𝔽2)11yescharacteristic
122C22×C5H(5,𝔽2)11yescharacteristic
123C22×C52−1+412yes—
124C4×C5—42no—
125C4×C5—42no—
126C4×C5—41no—
127C4×C5—41no—
128Dic5—24no—
129Dic5—42no—
130Dic5—42no—
131Dic5—42no—
132C24C22×D511yescharacteristic
133C22×C4—52no—
134C22×C4—52no—
135C22×C4—52no—
136C22×C4—52no—
137C22×C4—102no—
138C22×C4—201no—
139C22×C4—201no—
140C22×C4C22×D512yes—
141C22×C4C22×D511yescharacteristic
142C22×C4C22×D511yescharacteristic
143C22×C4C22×D512yes—
144C4⋊C4—41no—
145C4⋊C4—104no—
146C4⋊C4—104no—
147C4⋊C4—202no—
148C4⋊C4—202no—
149C4⋊C4—201no—
150C22⋊C4—24no—
151C22⋊C4—41no—
152C22⋊C4—104no—
153C22⋊C4—201no—
154C42—202no—
155C2×C5—42no—
156C2×C5#53811yescharacteristic
157C2×C5#53912yes—
158C2×C5#55212yes—
159C2×C5#55911yescharacteristic
160C2×C5#56311yescharacteristic
161C23—21no—
162C23—22no—
163C23—21no—
164C23—22no—
165C23#71112yes—
166C23C23×D511yescharacteristic
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—54no—
178C2×C4—102no—
179C2×C4—102no—
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—204no—
190C5#277111yescharacteristic, complemented, Sylow
191C22—22no—
192C22—24no—
193C22—41no—
194C22—41no—
195C22—42no—
196C22C2×#71111yescharacteristic
197C22#414411yescharacteristic
198C22D5×#4112yes—
199C22#414911yescharacteristic
200C22#415012yes—
201C4—42no—
202C4—42no—
203C4—41no—
204C4—41no—
205C4—104no—
206C4—202no—
207C4—202no—
208C4—202no—
209C2—42no—
210C2#6595111yescharacteristic
211C2#6598412yes—
212C2#6599811yescharacteristic
213C2#6600611yescharacteristic
214C2#6604112yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radicalC24×C5C23
hypercenter#397D5
hypocenterC5#2771
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #65261C2semidirect
7. #64949C2semidirect
8. #64949C2semidirect
9. #64942C2semidirect
190. C5#2771semidirect

Maximal subgroups

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

subgroupindex
C5×#3972
#652612
#652402
#652382
#652352
#652322
#649492
#649422
#27715