Finite Group Databasebeta

← #10609322

Subgroups — #10609322

Order 640.

448 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 129 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×#4025C212yesmaximal
2C5×#397C211yesmaximal, characteristic
3#65261C212yesmaximal, complemented
4#65242C211yesmaximal, characteristic
5#65235C211yesmaximal, characteristic
6#65231C212yesmaximal
7#64946C212yesmaximal, complemented
8#64946C212yesmaximal, complemented
9#64944C212yesmaximal, 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×#683C2212yes—
22C2×#683C2212yes—
23C2×C4×Dic5C2212yes—
24#4025—28no—
25C5×#95C2212yes—
26C5×#95C2211yescharacteristic
27#3963C2211yescharacteristic
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#3963C2212yes—
32#2722—51nomaximal, Sylow
33C24×C5C2311yescharacteristic
34C22×C4×C5C2311yescharacteristic
35C22×C4×C5C2312yes—
36C22×C4×C5C2311yescharacteristic
37C22×C4×C5C2312yes—
38C22×Dic5—24no—
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⋊Dic5D414yes—
49C4⋊Dic5—24no—
50C4⋊Dic5—41no—
51#683—24no—
52#683—24no—
53#683—24no—
54#683—24no—
55#683—42no—
56C4×Dic5—24no—
57C2×#126—52no—
58#398—52no—
59#398—52no—
60#397—51no—
61#397D511yescharacteristic
62#389—52no—
63#387—52no—
64#385—51no—
65#380—52no—
66C23×C5—22no—
67C23×C5—21no—
68C23×C5—21no—
69C23×C5—22no—
70C23×C5C2×D412yes—
71C23×C5C2411yescharacteristic
72C2×C4×C5—22no—
73C2×C4×C5—22no—
74C2×C4×C5—22no—
75C2×C4×C5—22no—
76C2×C4×C5—22no—
77C2×C4×C5—21no—
78C2×C4×C5—21no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5—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—22no—
92C2×Dic5—44no—
93C2×Dic5C2×D414yes—
94C2×Dic5#4114yes—
95C23×C4—51no—
96#126—108no—
97C2×C4⋊C4—52no—
98C2×C4⋊C4—52no—
99C2×C4⋊C4—52no—
100C2×C4⋊C4—52no—
101C2×C4⋊C4—52no—
102C2×C4⋊C4—52no—
103C2×C4⋊C4—51no—
104C2×C4⋊C4C2×D511yescharacteristic
105C2×C22⋊C4—51no—
106C2×C22⋊C4—52no—
107C2×C22⋊C4C2×D512yes—
108C2×C22⋊C4C2×D511yescharacteristic
109C2×C42—52no—
110#95—52no—
111#95—52no—
112#95—52no—
113#95—51no—
114#95—52no—
115#95C2×D512yes—
116#95C2×D511yescharacteristic
117C22×C5—22no—
118C22×C5—24no—
119C22×C5—42no—
120C22×C5—41no—
121C22×C5—41no—
122C22×C5C22×D411yescharacteristic
123C22×C5C2×#4112yes—
124C22×C5H(5,𝔽2)11yescharacteristic
125C22×C52−1+411yescharacteristic
126C22×C52−1+412yes—
127C4×C5—42no—
128C4×C5—42no—
129C4×C5—41no—
130C4×C5—41no—
131Dic5—24no—
132Dic5—24no—
133Dic5—42no—
134Dic5—42no—
135C24C22×D511yescharacteristic
136C22×C4—52no—
137C22×C4—52no—
138C22×C4—52no—
139C22×C4—52no—
140C22×C4—104no—
141C22×C4—201no—
142C22×C4C22×D511yescharacteristic
143C22×C4C22×D511yescharacteristic
144C22×C4C22×D512yes—
145C22×C4C22×D512yes—
146C4⋊C4—41no—
147C4⋊C4—104no—
148C4⋊C4—104no—
149C4⋊C4—104no—
150C4⋊C4—104no—
151C4⋊C4—104no—
152C4⋊C4—202no—
153C4⋊C4—201no—
154C22⋊C4—24no—
155C22⋊C4—41no—
156C22⋊C4—54no—
157C22⋊C4—104no—
158C42—104no—
159C2×C5—42no—
160C2×C5#53711yescharacteristic
161C2×C5#54112yes—
162C2×C5#54712yes—
163C2×C5#55111yescharacteristic
164C2×C5#56311yescharacteristic
165C23—21no—
166C23—22no—
167C23—21no—
168C23—22no—
169C23D4×D512yes—
170C23C23×D511yescharacteristic
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—54no—
182C2×C4—54no—
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—102no—
192C2×C4—102no—
193C2×C4—204no—
194C5#272211yescharacteristic, complemented, Sylow
195C22—22no—
196C22—24no—
197C22—41no—
198C22—41no—
199C22—42no—
200C22C2×D4×D511yescharacteristic
201C22#414411yescharacteristic
202C22D5×#4112yes—
203C22#415011yescharacteristic
204C22#415012yes—
205C4—42no—
206C4—42no—
207C4—41no—
208C4—41no—
209C4—104no—
210C4—104no—
211C4—202no—
212C4—202no—
213C2—42no—
214C2#6593111yescharacteristic
215C2#6598912yes—
216C2#6599311yescharacteristic
217C2#6600611yescharacteristic
218C2#6603512yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radicalC24×C5C23
hypercenter#397D5
hypocenterC5#2722
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
3. #65261C2semidirect
7. #64946C2semidirect
8. #64946C2semidirect
9. #64944C2semidirect
194. C5#2722semidirect

Maximal subgroups

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

subgroupindex
C2×#40252
C5×#3972
#652612
#652422
#652352
#652312
#649462
#649442
#27225