Finite Group Databasebeta

← #10608687

Subgroups — #10608687

Order 640.

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

normal subgroups only

This group has 476 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
1C5×#394C211yesmaximal, characteristic
2#65509C212yesmaximal
3#65275C211yesmaximal, characteristic, complemented
4#65263C211yesmaximal, characteristic, complemented
5#65240C212yesmaximal, complemented
6#65236C212yesmaximal, complemented
7#65232C212yesmaximal, complemented
8C22⋊C4×Dic5C212yesmaximal, complemented
9#64940C212yesmaximal, complemented
10C22×C5×D4C2211yescharacteristic
11C23×Dic5C2212yes—
12C22×Dic10C2211yescharacteristic
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C5×C22⋊C4C2212yes—
15C2×C5×C22⋊C4C2212yes—
16C2×C22⋊Dic5C2212yes—
17C2×C22⋊Dic5C2212yes—
18C2×C22⋊Dic5C2212yes—
19C2×C4⋊Dic5C2211yescharacteristic
20C2×C4×Dic5C2212yes—
21C2×C4×Dic5C2211yescharacteristic
22C5×#95C2211yescharacteristic
23#3963C2212yes—
24#3963C2212yes—
25#3963C2212yes—
26#3963C2212yes—
27#3963C2212yes—
28#3963C2212yes—
29#3963C2211yescharacteristic
30#3963C2212yes—
31#2572—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—24no—
40C22×Dic5—24no—
41C22×Dic5—42no—
42C22×Dic5C2312yes—
43C22×Dic5C2312yes—
44C22×Dic5C2312yes—
45C22×Dic5C2312yes—
46C2×Dic10—22no—
47C2×Dic10—41no—
48C2×Dic10—41no—
49C5×C4⋊C4—41no—
50C5×C22⋊C4—24no—
51C5×C22⋊C4—24no—
52C22⋊Dic5—24no—
53C22⋊Dic5—42no—
54C22⋊Dic5—42no—
55C4⋊Dic5—41no—
56C4×Dic5—24no—
57C4×Dic5—41no—
58#400—51no—
59#394D511yescharacteristic
60#393—52no—
61#391—51no—
62#388—52no—
63#386—52no—
64#383—52no—
65#381—52no—
66C4×C22⋊C4—52no—
67C23×C5—22no—
68C23×C5—22no—
69C23×C5—22no—
70C23×C5—22no—
71C23×C5—22no—
72C23×C5—22no—
73C23×C5#4114yes—
74C23×C5C2411yescharacteristic
75C5×D4—44no—
76C2×C4×C5—21no—
77C2×C4×C5—21no—
78C2×C4×C5—22no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—21no—
84C2×C4×C5—21no—
85C2×C4×C5#4112yes—
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—
98C2×Dic5#4114yes—
99Dic10—44no—
100C22×Q8—51no—
101C22×D4C2×D511yescharacteristic
102C23×C4—52no—
103C2×C4⋊C4—51no—
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—52no—
111C2×C42—51no—
112#95—52no—
113#95—52no—
114#95—51no—
115#95—52no—
116#95—52no—
117#95—52no—
118#95—52no—
119#95—52no—
120#95C2×D511yescharacteristic
121C22×C5—24no—
122C22×C5—24no—
123C22×C5—24no—
124C22×C5—42no—
125C22×C5—42no—
126C22×C5—42no—
127C22×C5—42no—
128C22×C5C2×#4111yescharacteristic
129C22×C5C2×#4112yes—
130C22×C5C2×#4112yes—
131C22×C5H(5,𝔽2)11yescharacteristic
132C22×C5H(5,𝔽2)11yescharacteristic
133C4×C5—22no—
134C4×C5—42no—
135C4×C5—41no—
136C4×C5—41no—
137Dic5—24no—
138Dic5—42no—
139Dic5—42no—
140Dic5—42no—
141C24C22×D512yes—
142C2×Q8—102no—
143C2×Q8—201no—
144C2×Q8—201no—
145C2×D4—24no—
146C2×D4—41no—
147C22×C4—52no—
148C22×C4—52no—
149C22×C4—52no—
150C22×C4—52no—
151C22×C4—104no—
152C22×C4—104no—
153C22×C4—202no—
154C22×C4C22×D511yescharacteristic
155C22×C4C22×D512yes—
156C22×C4C22×D511yescharacteristic
157C22×C4C22×D511yescharacteristic
158C4⋊C4—41no—
159C4⋊C4—201no—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—104no—
163C22⋊C4—202no—
164C22⋊C4—202no—
165C42—104no—
166C42—201no—
167C2×C5—44no—
168C2×C5#52912yes—
169C2×C5#53811yescharacteristic
170C2×C5#55111yescharacteristic
171C2×C5#55112yes—
172C2×C5#55511yescharacteristic
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23#71114yes—
180C23C23×D511yescharacteristic
181Q8—204no—
182D4—44no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—54no—
193C2×C4—102no—
194C2×C4—102no—
195C2×C4—102no—
196C2×C4—102no—
197C2×C4—102no—
198C2×C4—102no—
199C2×C4—102no—
200C2×C4—102no—
201C2×C4—102no—
202C2×C4—102no—
203C2×C4—102no—
204C2×C4—208no—
205C2×C4#71112yes—
206C5#257211yescharacteristic, complemented, Sylow
207C22—24no—
208C22—24no—
209C22—24no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22C2×#71112yes—
215C22C2×#71111yescharacteristic
216C22#414411yescharacteristic
217C22D5×#4112yes—
218C22#414911yescharacteristic
219C4—22no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—104no—
224C4—202no—
225C4—202no—
226C4—202no—
227C2—44no—
228C2#6592111yescharacteristic
229C2#6593412yes—
230C2#6595111yescharacteristic
231C2#6599212yes—
232C2#6600411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC22×C5×D4C22
hypercenter#394D5
hypocenterC5#2572
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
3. #65275C2semidirect
4. #65263C2semidirect
5. #65240C2semidirect
6. #65236C2semidirect
7. #65232C2semidirect
8. C22⋊C4×Dic5C2semidirect
9. #64940C2semidirect
206. C5#2572semidirect

Maximal subgroups

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

subgroupindex
C5×#3942
#655092
#652752
#652632
#652402
#652362
#652322
C22⋊C4×Dic52
#649402
#25725