Finite Group Databasebeta

← #10608800

Subgroups — #10608800

Order 640.

448 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 125 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×#394C211yesmaximal, characteristic
2#65509C212yesmaximal
3#65275C211yesmaximal, characteristic, complemented
4#65266C211yesmaximal, characteristic, complemented
5#65241C212yesmaximal, complemented
6#65239C212yesmaximal, complemented
7#65239C212yesmaximal, complemented
8#65233C212yesmaximal, complemented
9#64948C212yesmaximal, complemented
10C22×C5×D4C2211yescharacteristic
11C23×Dic5C2212yes—
12C2×C5×C4⋊C4C2211yescharacteristic
13C2×C5×C22⋊C4C2212yes—
14C2×C5×C22⋊C4C2212yes—
15C2×C22⋊Dic5C2212yes—
16C2×C22⋊Dic5C2212yes—
17C2×C22⋊Dic5C2212yes—
18C2×C4⋊Dic5C2212yes—
19C2×C4⋊Dic5C2211yescharacteristic
20C2×C4⋊Dic5C2211yescharacteristic
21C2×#683C2212yes—
22C2×C4×Dic5C2211yescharacteristic
23C5×#95C2211yescharacteristic
24#3963C2212yes—
25#3963C2212yes—
26#3963C2212yes—
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2211yescharacteristic
31#2782—51nomaximal, Sylow
32C24×C5C2312yes—
33C2×C5×D4—24no—
34C2×C5×D4—41no—
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2312yes—
37C22×C4×C5C2311yescharacteristic
38C22×C4×C5C2311yescharacteristic
39C22×Dic5—42no—
40C22×Dic5—42no—
41C22×Dic5—42no—
42C22×Dic5C2312yes—
43C22×Dic5C2312yes—
44C22×Dic5C2312yes—
45C22×Dic5C2312yes—
46C5×C4⋊C4—41no—
47C5×C22⋊C4—24no—
48C5×C22⋊C4—24no—
49C22⋊Dic5—24no—
50C22⋊Dic5—42no—
51C22⋊Dic5—42no—
52C4⋊Dic5—41no—
53C4⋊Dic5—42no—
54C4⋊Dic5—41no—
55#683—42no—
56C4×Dic5—22no—
57#400—51no—
58#400—52no—
59#397—52no—
60#394D511yescharacteristic
61#388—52no—
62#386—52no—
63#385—52no—
64#385—52no—
65#382—51no—
66C23×C5—22no—
67C23×C5—22no—
68C23×C5—22no—
69C23×C5—22no—
70C23×C5—22no—
71C23×C5—22no—
72C23×C5#4114yes—
73C23×C5C2411yescharacteristic
74C5×D4—44no—
75C2×C4×C5—21no—
76C2×C4×C5—21no—
77C2×C4×C5—22no—
78C2×C4×C5—22no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—21no—
83C2×C4×C5—21no—
84C2×C4×C5#4112yes—
85C2×Dic5—22no—
86C2×Dic5—22no—
87C2×Dic5—22no—
88C2×Dic5—22no—
89C2×Dic5—22no—
90C2×Dic5—22no—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—22no—
96C2×Dic5—22no—
97C2×Dic5—48no—
98C22×D4C2×D511yescharacteristic
99C23×C4—52no—
100C2×C4⋊C4—51no—
101C2×C4⋊C4—51no—
102C2×C4⋊C4—52no—
103C2×C4⋊C4—52no—
104C2×C4⋊C4C2×D511yescharacteristic
105C2×C22⋊C4—52no—
106C2×C22⋊C4—52no—
107C2×C22⋊C4—52no—
108C2×C22⋊C4C2×D512yes—
109C2×C22⋊C4C2×D512yes—
110C2×C42—51no—
111#95—52no—
112#95—52no—
113#95—52no—
114#95—52no—
115#95—52no—
116#95—52no—
117#95—51no—
118#95C2×D511yescharacteristic
119C22×C5—24no—
120C22×C5—24no—
121C22×C5—24no—
122C22×C5—42no—
123C22×C5—42no—
124C22×C5—42no—
125C22×C5—42no—
126C22×C5C2×#4112yes—
127C22×C5C2×#4111yescharacteristic
128C22×C5H(5,𝔽2)11yescharacteristic
129C22×C5H(5,𝔽2)12yes—
130C22×C52−1+411yescharacteristic
131C4×C5—22no—
132C4×C5—41no—
133C4×C5—41no—
134C4×C5—42no—
135Dic5—42no—
136Dic5—42no—
137Dic5—42no—
138Dic5—42no—
139C24C22×D512yes—
140C2×D4—24no—
141C2×D4—41no—
142C22×C4—52no—
143C22×C4—52no—
144C22×C4—52no—
145C22×C4—52no—
146C22×C4—202no—
147C22×C4—202no—
148C22×C4—202no—
149C22×C4C22×D511yescharacteristic
150C22×C4C22×D511yescharacteristic
151C22×C4C22×D512yes—
152C22×C4C22×D511yescharacteristic
153C4⋊C4—41no—
154C4⋊C4—202no—
155C4⋊C4—201no—
156C4⋊C4—201no—
157C4⋊C4—202no—
158C22⋊C4—24no—
159C22⋊C4—24no—
160C22⋊C4—104no—
161C22⋊C4—202no—
162C22⋊C4—202no—
163C42—102no—
164C2×C5—44no—
165C2×C5#53911yescharacteristic
166C2×C5#53912yes—
167C2×C5#55111yescharacteristic
168C2×C5#55312yes—
169C2×C5H(3,𝔽4)11yescharacteristic
170C23—22no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23#71114yes—
177C23C23×D511yescharacteristic
178D4—44no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—102no—
189C2×C4—102no—
190C2×C4—102no—
191C2×C4—102no—
192C2×C4—102no—
193C2×C4—102no—
194C2×C4—102no—
195C2×C4—102no—
196C2×C4—102no—
197C2×C4—102no—
198C2×C4—102no—
199C2×C4—102no—
200C2×C4—208no—
201C2×C4#71112yes—
202C5#278211yescharacteristic, complemented, Sylow
203C22—24no—
204C22—24no—
205C22—24no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22C2×#71112yes—
211C22C2×#71111yescharacteristic
212C22#414411yescharacteristic
213C22#414912yes—
214C22#415011yescharacteristic
215C4—22no—
216C4—41no—
217C4—41no—
218C4—42no—
219C4—202no—
220C4—202no—
221C4—202no—
222C4—202no—
223C2—44no—
224C2#6592111yescharacteristic
225C2#6593611yescharacteristic
226C2#6595212yes—
227C2#6599912yes—
228C2#6601711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC22×C5×D4C22
hypercenter#394D5
hypocenterC5#2782
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
3. #65275C2semidirect
4. #65266C2semidirect
5. #65241C2semidirect
6. #65239C2semidirect
7. #65239C2semidirect
8. #65233C2semidirect
9. #64948C2semidirect
202. C5#2782semidirect

Maximal subgroups

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

subgroupindex
C5×#3942
#655092
#652752
#652662
#652412
#652392
#652332
#649482
#27825