Finite Group Databasebeta

← #10609883

Subgroups — #10609883

Order 640.

512 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 141 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×#4042C211yesmaximal, characteristic
2C2×D5×C4⋊C4C211yesmaximal, characteristic
3C5×#400C211yesmaximal, characteristic, complemented
4#65275C211yesmaximal, characteristic, complemented
5#65271C211yesmaximal, characteristic, complemented
6#64955C211yesmaximal, characteristic
7#64953C212yesmaximal
8#64953C212yesmaximal
9#64948C211yesmaximal, characteristic, complemented
10#64939C212yesmaximal, complemented
11#64935C212yesmaximal, complemented
12C22×C4×D5C2212yes—
13C22×C4×D5C2211yescharacteristic
14C22×Dic10C2211yescharacteristic
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×#685C2211yescharacteristic
18C2×#685C2211yescharacteristic
19C2×#685C2212yes—
20C2×C4⋊Dic5C2211yescharacteristic
21C2×#683C2212yes—
22C2×#683C2211yescharacteristic
23C2×#683C2211yescharacteristic
24C2×C4×Dic5C2212yes—
25#4042—24no—
26D5×C4⋊C4—24no—
27C5×#95C2212yes—
28C5×#95C2212yes—
29C5×#95C2211yescharacteristic
30#3963C2212yes—
31#3963C2212yes—
32#3963C2212yes—
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2211yescharacteristic
37#2503—51nomaximal, Sylow
38C23×D5C2311yescharacteristic
39C22×C4×C5C2311yescharacteristic
40C22×C4×C5C2312yes—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×Dic5C2311yescharacteristic
45C22×Dic5C2312yes—
46C22×Dic5C2312yes—
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2311yescharacteristic
49C2×C4×D5—24no—
50C2×C4×D5—24no—
51C2×C4×D5—28no—
52C2×C4×D5D414yes—
53C2×Dic10—22no—
54C2×Dic10—24no—
55C5×C4⋊C4—22no—
56C5×C4⋊C4—22no—
57#685—22no—
58#685—22no—
59#685—24no—
60C4⋊Dic5—22no—
61#683—22no—
62#683—24no—
63#683—22no—
64C4×Dic5—24no—
65C2×C22⋊Q8—51no—
66C22×C4⋊C4—51no—
67#400—51no—
68#400—51no—
69#400—51no—
70#400D511yescharacteristic, complemented
71#391—52no—
72#382—52no—
73#381—52no—
74#381—52no—
75C4×C22⋊C4—51no—
76C23×C5C2411yescharacteristic
77C22×D5—22no—
78C22×D5—21no—
79C22×D5—21no—
80C22×D5C2×D412yes—
81C22×D5#4114yes—
82C2×C4×C5—22no—
83C2×C4×C5—21no—
84C2×C4×C5—22no—
85C2×C4×C5—21no—
86C2×C4×C5—22no—
87C2×C4×C5—22no—
88C2×C4×C5—22no—
89C2×C4×C5—24no—
90C2×C4×C5—21no—
91C2×C4×C5—21no—
92C2×C4×C5—22no—
93C2×C4×C5C2×D412yes—
94C2×Dic5—21no—
95C2×Dic5—22no—
96C2×Dic5—21no—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5—22no—
100C2×Dic5—24no—
101C2×Dic5—21no—
102C2×Dic5—22no—
103C2×Dic5—21no—
104C2×Dic5C2×D412yes—
105C2×Dic5#4114yes—
106C4×D5—28no—
107C4×D5—48no—
108Dic10—44no—
109C22×Q8—51no—
110C23×C4—51no—
111C23×C4—52no—
112C22⋊Q8—104no—
113C2×C4⋊C4—52no—
114C2×C4⋊C4—51no—
115C2×C4⋊C4—51no—
116C2×C4⋊C4—51no—
117C2×C4⋊C4—104no—
118C2×C4⋊C4C2×D511yescharacteristic
119C2×C4⋊C4C2×D511yescharacteristic
120C2×C22⋊C4—51no—
121C2×C22⋊C4—51no—
122C2×C22⋊C4—52no—
123C2×C42—52no—
124#95—51no—
125#95—52no—
126#95—52no—
127#95—52no—
128#95—52no—
129#95—52no—
130#95—52no—
131#95C2×D511yescharacteristic
132#95C2×D512yes—
133#95C2×D512yes—
134C22×C5C22×D411yescharacteristic
135C22×C5C2×#4112yes—
136C22×C5C2×#4112yes—
137C22×C52−1+411yescharacteristic
138C22×C52−1+411yescharacteristic
139C2×D5—22no—
140C2×D5—22no—
141C2×D5—24no—
142C2×D5—24no—
143C2×D5#12414yes—
144C4×C5—22no—
145C4×C5—41no—
146C4×C5—42no—
147C4×C5—41no—
148C4×C5—42no—
149Dic5—41no—
150Dic5—42no—
151Dic5—41no—
152Dic5—42no—
153Dic5#12314yes—
154C24—51no—
155C2×Q8—102no—
156C2×Q8—104no—
157C22×C4—51no—
158C22×C4—54no—
159C22×C4—52no—
160C22×C4—51no—
161C22×C4—51no—
162C22×C4—52no—
163C22×C4—104no—
164C22×C4—104no—
165C22×C4—108no—
166C22×C4C22×D512yes—
167C22×C4C22×D511yescharacteristic
168C22×C4C22×D511yescharacteristic
169C22×C4C22×D512yes—
170C22×C4C22×D511yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—104no—
174C4⋊C4—102no—
175C4⋊C4—102no—
176C4⋊C4—102no—
177C22⋊C4—102no—
178C22⋊C4—102no—
179C22⋊C4—104no—
180C42—104no—
181C2×C5C2×#12311yescharacteristic
182C2×C5C2×#12411yescharacteristic
183C2×C5#53611yescharacteristic
184C2×C5#55212yes—
185C2×C5#55212yes—
186D5—24no—
187C23—52no—
188C23—54no—
189C23—101no—
190C23—102no—
191C23—101no—
192C23C23×D511yescharacteristic
193Q8—204no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—52no—
206C2×C4—54no—
207C2×C4—108no—
208C2×C4—101no—
209C2×C4—101no—
210C2×C4—102no—
211C2×C4—101no—
212C2×C4—102no—
213C2×C4—102no—
214C2×C4—102no—
215C2×C4—104no—
216C2×C4—102no—
217C2×C4—101no—
218C2×C4—208no—
219C2×C4D4×D512yes—
220C5#250311yescharacteristic, complemented, Sylow
221C22—54no—
222C22—102no—
223C22—102no—
224C22—104no—
225C22—104no—
226C22C2×D4×D511yescharacteristic
227C22#414711yescharacteristic
228C22D5×#4112yes—
229C22D5×#4112yes—
230C22#415011yescharacteristic
231C4—22no—
232C4—41no—
233C4—41no—
234C4—42no—
235C4—42no—
236C4—54no—
237C4—201no—
238C4—201no—
239C4—202no—
240C4—202no—
241C2—104no—
242C2#6596211yescharacteristic
243C2D5×#12311yescharacteristic
244C2D5×#12411yescharacteristic
245C2#6602712yes—
246C2#6604212yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#400C2
prime radicalC23×D5C23
hypercenter#400D5
hypocenterC5#2503
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. C5×#400C2semidirect
4. #65275C2semidirect
5. #65271C2semidirect
9. #64948C2semidirect
10. #64939C2semidirect
11. #64935C2semidirect
70. #400D5semidirect
220. C5#2503semidirect

Maximal subgroups

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

subgroupindex
C2×#40422
C2×D5×C4⋊C42
C5×#4002
#652752
#652712
#649552
#649532
#649482
#649392
#649352
#25035