Finite Group Databasebeta

← #10609911

Subgroups — #10609911

Order 640.

532 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 137 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×#4039C211yesmaximal, characteristic
2C2×#4035C211yesmaximal, characteristic, complemented
3C5×#400C211yesmaximal, characteristic, complemented
4#65280C211yesmaximal, characteristic
5#65271C211yesmaximal, characteristic, complemented
6#64961C211yesmaximal, characteristic
7#64959C212yesmaximal, complemented
8#64958C212yesmaximal, complemented
9#64956C212yesmaximal, complemented
10#64956C212yesmaximal, complemented
11#64938C211yesmaximal, characteristic, complemented
12C22×D20C2211yescharacteristic
13C22×C4×D5C2212yes—
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×#685C2212yes—
17C2×#685C2212yes—
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×C4⋊Dic5C2211yescharacteristic
22C2×#683C2211yescharacteristic
23C2×#683C2211yescharacteristic
24C2×#683C2211yescharacteristic
25C2×#683C2211yescharacteristic
26C2×#683C2212yes—
27C2×C4×Dic5C2212yes—
28C2×C4×Dic5C2211yescharacteristic
29#4039—24no—
30#4035—24no—
31C5×#95C2211yescharacteristic
32C5×#95C2212yes—
33C5×#95C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2562—51nomaximal, Sylow
38C23×D5C2312yes—
39C22×C4×C5C2312yes—
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×Dic5C2312yes—
45C22×Dic5C2311yescharacteristic
46C22×Dic5C2311yescharacteristic
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2311yescharacteristic
49C2×D20—24no—
50C2×D20—22no—
51C2×C4×D5—24no—
52C2×C4×D5—24no—
53C2×C4×D5—24no—
54C5×C4⋊C4—22no—
55C5×C4⋊C4—22no—
56#685—24no—
57#685—24no—
58#685—24no—
59#685—42no—
60#685—42no—
61C4⋊Dic5—22no—
62#683—22no—
63#683—22no—
64#683—22no—
65#683—22no—
66#683—24no—
67C4×Dic5—42no—
68C4×Dic5D414yes—
69C2×#125—51no—
70C2×C4×D4—51no—
71#400—51no—
72#400D511yescharacteristic, complemented
73#394—51no—
74#394—51no—
75#388—52no—
76#388—52no—
77#380—52no—
78#380—52no—
79C4×C4⋊C4—51no—
80C23×C5C2411yescharacteristic
81C22×D5—22no—
82C22×D5—22no—
83C22×D5—22no—
84C22×D5—22no—
85C22×D5—22no—
86C22×D5—22no—
87C22×D5#4114yes—
88C2×C4×C5—22no—
89C2×C4×C5—21no—
90C2×C4×C5—21no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5—21no—
99C2×C4×C5—21no—
100C2×C4×C5C2×D412yes—
101C2×Dic5—21no—
102C2×Dic5—21no—
103C2×Dic5—21no—
104C2×Dic5—21no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—21no—
110C2×Dic5—21no—
111C2×Dic5C2×D412yes—
112C2×Dic5C2×D412yes—
113C2×Dic5#4114yes—
114D20—28no—
115C4×D5—48no—
116C22×D4—51no—
117C23×C4—52no—
118#125—104no—
119C4×D4—104no—
120C2×C4⋊C4—51no—
121C2×C4⋊C4—51no—
122C2×C4⋊C4—51no—
123C2×C4⋊C4—51no—
124C2×C4⋊C4—51no—
125C2×C4⋊C4—52no—
126C2×C4⋊C4C2×D511yescharacteristic
127C2×C4⋊C4C2×D511yescharacteristic
128C2×C22⋊C4—52no—
129C2×C22⋊C4—52no—
130C2×C22⋊C4—52no—
131C2×C22⋊C4—52no—
132C2×C22⋊C4—52no—
133C2×C42—51no—
134C2×C42—52no—
135#95—51no—
136#95—52no—
137#95—52no—
138#95C2×D511yescharacteristic
139#95C2×D512yes—
140#95C2×D512yes—
141C22×C5C22×D411yescharacteristic
142C22×C5C2×#4112yes—
143C22×C5C2×#4112yes—
144C22×C5H(5,𝔽2)11yescharacteristic
145C22×C52−1+411yescharacteristic
146C2×D5—24no—
147C2×D5—24no—
148C2×D5—24no—
149C2×D5—42no—
150C2×D5—42no—
151C2×D5—42no—
152C2×D5—42no—
153C4×C5—22no—
154C4×C5—41no—
155C4×C5—42no—
156C4×C5—41no—
157C4×C5—42no—
158Dic5—22no—
159Dic5—41no—
160Dic5—41no—
161Dic5—42no—
162Dic5#12414yes—
163C24—52no—
164C2×D4—104no—
165C2×D4—102no—
166C22×C4—51no—
167C22×C4—51no—
168C22×C4—52no—
169C22×C4—51no—
170C22×C4—51no—
171C22×C4—104no—
172C22×C4—104no—
173C22×C4—104no—
174C22×C4C22×D512yes—
175C22×C4C22×D512yes—
176C22×C4C22×D511yescharacteristic
177C22×C4C22×D511yescharacteristic
178C22×C4C22×D511yescharacteristic
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—102no—
182C4⋊C4—102no—
183C4⋊C4—102no—
184C4⋊C4—102no—
185C4⋊C4—104no—
186C4⋊C4—102no—
187C22⋊C4—104no—
188C22⋊C4—104no—
189C22⋊C4—104no—
190C22⋊C4—202no—
191C22⋊C4—202no—
192C42—54no—
193C42—202no—
194C2×C5#52511yescharacteristic
195C2×C5C2×#12411yescharacteristic
196C2×C5#53711yescharacteristic
197C2×C5#55212yes—
198C2×C5#55312yes—
199D5—44no—
200C23—54no—
201C23—102no—
202C23—102no—
203C23—102no—
204C23—102no—
205C23—102no—
206C23—102no—
207C23C23×D511yescharacteristic
208D4—108no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—52no—
222C2×C4—54no—
223C2×C4—52no—
224C2×C4—101no—
225C2×C4—101no—
226C2×C4—101no—
227C2×C4—101no—
228C2×C4—101no—
229C2×C4—102no—
230C2×C4—102no—
231C2×C4—102no—
232C2×C4—102no—
233C2×C4—101no—
234C2×C4—208no—
235C2×C4D4×D512yes—
236C5#256211yescharacteristic, complemented, Sylow
237C22—104no—
238C22—104no—
239C22—104no—
240C22—202no—
241C22—202no—
242C22—202no—
243C22—202no—
244C22C2×D4×D511yescharacteristic
245C22#414711yescharacteristic
246C22D5×#4112yes—
247C22D5×#4112yes—
248C22#414911yescharacteristic
249C4—22no—
250C4—41no—
251C4—41no—
252C4—42no—
253C4—42no—
254C4—54no—
255C4—102no—
256C4—201no—
257C4—201no—
258C4—202no—
259C2—204no—
260C2#6596311yescharacteristic
261C2#6599011yescharacteristic
262C2D5×#12411yescharacteristic
263C2#6602712yes—
264C2#6604312yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#400C2
prime radicalC22×D20C22
hypercenter#400D5
hypocenterC5#2562
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#4035C2semidirect
3. C5×#400C2semidirect
5. #65271C2semidirect
7. #64959C2semidirect
8. #64958C2semidirect
9. #64956C2semidirect
10. #64956C2semidirect
11. #64938C2semidirect
72. #400D5semidirect
236. C5#2562semidirect

Maximal subgroups

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

subgroupindex
C2×#40392
C2×#40352
C5×#4002
#652802
#652712
#649612
#649592
#649582
#649562
#649382
#25625