Finite Group Databasebeta

← #10609547

Subgroups — #10609547

Order 640.

476 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 139 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×#4038C211yesmaximal, characteristic
2C2×#4036C211yesmaximal, characteristic, complemented
3C5×#398C211yesmaximal, characteristic, complemented
4#65282C211yesmaximal, characteristic
5#65278C212yesmaximal
6#65275C211yesmaximal, characteristic, complemented
7#65274C212yesmaximal, complemented
8#64966C211yesmaximal, characteristic
9#64959C212yesmaximal
10#64943C212yesmaximal, complemented
11#64937C211yesmaximal, characteristic, complemented
12C22×C4×D5C2211yescharacteristic
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C4⋊C4C2212yes—
16C2×#685C2211yescharacteristic
17C2×#685C2212yes—
18C2×#685C2211yescharacteristic
19C2×#685C2212yes—
20C2×C4⋊Dic5C2212yes—
21C2×C4⋊Dic5C2211yescharacteristic
22C2×C4⋊Dic5C2211yescharacteristic
23C2×C4⋊Dic5C2212yes—
24C2×#683C2212yes—
25C2×C4×Dic5C2211yescharacteristic
26C2×C4×Dic5C2212yes—
27C2×C4×Dic5C2211yescharacteristic
28#4038—24no—
29#4036—24no—
30C5×#95C2211yescharacteristic
31C5×#95C2212yes—
32#3963C2211yescharacteristic
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2611—51nomaximal, Sylow
38C23×D5C2311yescharacteristic
39C22×C4×C5C2312yes—
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×Dic5C2312yes—
45C22×Dic5C2311yescharacteristic
46C22×Dic5C2312yes—
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2311yescharacteristic
49C2×C4×D5—22no—
50C2×C4×D5—22no—
51C2×C4×D5—22no—
52C5×C4⋊C4—22no—
53C5×C4⋊C4—22no—
54C5×C4⋊C4—24no—
55#685—24no—
56#685—22no—
57#685—24no—
58#685—22no—
59C4⋊Dic5—22no—
60C4⋊Dic5—24no—
61C4⋊Dic5—24no—
62C4⋊Dic5—22no—
63#683—24no—
64C4×Dic5—24no—
65C4×Dic5—22no—
66C4×Dic5D414yes—
67C2×#125—51no—
68C2×#117—51no—
69#400—51no—
70#399—51no—
71#398D511yescharacteristic, complemented
72#397—51no—
73#389—52no—
74#388—52no—
75#387—52no—
76#385—52no—
77#376—51no—
78C23×C5C2411yescharacteristic
79C22×D5—21no—
80C22×D5—22no—
81C22×D5—21no—
82C22×D5—22no—
83C22×D5#4112yes—
84C2×C4×C5—21no—
85C2×C4×C5—22no—
86C2×C4×C5—21no—
87C2×C4×C5—22no—
88C2×C4×C5—22no—
89C2×C4×C5—21no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5C2×D412yes—
95C2×C4×C5#4112yes—
96C2×C4×C5#4114yes—
97C2×Dic5—22no—
98C2×Dic5—21no—
99C2×Dic5—21no—
100C2×Dic5—21no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5C2×D412yes—
109C2×Dic5C2×D412yes—
110C4×D5—28no—
111C23×C4—51no—
112#125—104no—
113#117—104no—
114C2×C4⋊C4—51no—
115C2×C4⋊C4—52no—
116C2×C4⋊C4—52no—
117C2×C4⋊C4—51no—
118C2×C4⋊C4—52no—
119C2×C4⋊C4C2×D511yescharacteristic
120C2×C4⋊C4C2×D511yescharacteristic
121C2×C4⋊C4C2×D512yes—
122C2×C22⋊C4—51no—
123C2×C22⋊C4—51no—
124C2×C22⋊C4—52no—
125C2×C22⋊C4—52no—
126C2×C42—51no—
127C2×C42—51no—
128C2×C42—52no—
129#95—52no—
130#95—51no—
131#95—52no—
132#95—52no—
133#95—52no—
134#95C2×D511yescharacteristic
135#95C2×D512yes—
136C22×C5C22×D411yescharacteristic
137C22×C5C2×#4111yescharacteristic
138C22×C5C2×#4112yes—
139C22×C5C2×#4111yescharacteristic
140C22×C52−1+412yes—
141C2×D5—22no—
142C2×D5—24no—
143C2×D5—41no—
144C2×D5—41no—
145C2×D5—42no—
146C4×C5—22no—
147C4×C5—22no—
148C4×C5—24no—
149C4×C5—42no—
150C4×C5—41no—
151Dic5—22no—
152Dic5—22no—
153Dic5—41no—
154Dic5—42no—
155Dic5—42no—
156C24—51no—
157C22×C4—51no—
158C22×C4—51no—
159C22×C4—52no—
160C22×C4—51no—
161C22×C4—52no—
162C22×C4—102no—
163C22×C4—102no—
164C22×C4—102no—
165C22×C4C22×D511yescharacteristic
166C22×C4C22×D512yes—
167C22×C4C22×D511yescharacteristic
168C22×C4C22×D511yescharacteristic
169C22×C4C22×D512yes—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—24no—
173C4⋊C4—102no—
174C4⋊C4—104no—
175C4⋊C4—104no—
176C4⋊C4—102no—
177C4⋊C4—104no—
178C22⋊C4—102no—
179C22⋊C4—104no—
180C22⋊C4—102no—
181C22⋊C4—104no—
182C42—54no—
183C42—102no—
184C42—104no—
185C2×C5#52511yescharacteristic
186C2×C5#53211yescharacteristic
187C2×C5#53611yescharacteristic
188C2×C5#55212yes—
189C2×C5#55212yes—
190D5—42no—
191C23—52no—
192C23—101no—
193C23—101no—
194C23—102no—
195C23—102no—
196C23C23×D511yescharacteristic
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—52no—
208C2×C4—52no—
209C2×C4—108no—
210C2×C4—101no—
211C2×C4—102no—
212C2×C4—101no—
213C2×C4—101no—
214C2×C4—102no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—102no—
221C2×C4D4×D512yes—
222C2×C4#71112yes—
223C2×C4#71314yes—
224C5#261111yescharacteristic, complemented, Sylow
225C22—102no—
226C22—104no—
227C22—201no—
228C22—202no—
229C22—201no—
230C22C2×D4×D511yescharacteristic
231C22C2×#71111yescharacteristic
232C22C2×#71312yes—
233C22D5×#4111yescharacteristic
234C22#415012yes—
235C4—22no—
236C4—24no—
237C4—22no—
238C4—42no—
239C4—41no—
240C4—102no—
241C4—102no—
242C4—201no—
243C4—202no—
244C4—202no—
245C2—202no—
246C2#6594011yescharacteristic
247C2#6597612yes—
248C2#6598211yescharacteristic
249C2#6602812yes—
250C2#6605911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#398C2
prime radicalC23×D5C23
hypercenter#398D5
hypocenterC5#2611
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#4036C2semidirect
3. C5×#398C2semidirect
6. #65275C2semidirect
7. #65274C2semidirect
10. #64943C2semidirect
11. #64937C2semidirect
71. #398D5semidirect
224. C5#2611semidirect

Maximal subgroups

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

subgroupindex
C2×#40382
C2×#40362
C5×#3982
#652822
#652782
#652752
#652742
#649662
#649592
#649432
#649372
#26115