Finite Group Databasebeta

← #10608686

Subgroups — #10608686

Order 640.

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

normal subgroups only

This group has 532 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×D4×Dic5C211yesmaximal, characteristic
2C2×#4035C211yesmaximal, characteristic, complemented
3C5×#394C211yesmaximal, characteristic
4#65511C211yesmaximal, characteristic
5#65275C211yesmaximal, characteristic, complemented
6#65240C212yesmaximal, complemented
7#65233C212yesmaximal, complemented
8#65231C212yesmaximal, complemented
9#65231C212yesmaximal, complemented
10#64948C211yesmaximal, characteristic, complemented
11#64938C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C23×Dic5C2212yes—
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C22⋊C4C2212yes—
16C2×C5×C22⋊C4C2212yes—
17C2×C22⋊Dic5C2212yes—
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2212yes—
20D4×Dic5—24no—
21C2×C4⋊Dic5C2211yescharacteristic
22C2×C4⋊Dic5C2211yescharacteristic
23C2×#683C2211yescharacteristic
24C2×#683C2211yescharacteristic
25C2×#683C2211yescharacteristic
26C2×#683C2211yescharacteristic
27C2×#683C2212yes—
28C2×C4×Dic5C2212yes—
29C2×C4×Dic5C2211yescharacteristic
30#4035—24no—
31C5×#95C2211yescharacteristic
32#3963C2211yescharacteristic
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2562—51nomaximal, Sylow
38C24×C5C2312yes—
39C2×C5×D4—24no—
40C2×C5×D4—22no—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C22×Dic5—24no—
46C22×Dic5—24no—
47C22×Dic5—24no—
48C22×Dic5C2312yes—
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2311yescharacteristic
51C22×Dic5C2312yes—
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2311yescharacteristic
54C5×C4⋊C4—22no—
55C5×C22⋊C4—24no—
56C5×C22⋊C4—42no—
57C22⋊Dic5—24no—
58C22⋊Dic5—24no—
59C22⋊Dic5—42no—
60C4⋊Dic5—22no—
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#400—51no—
73#394—51no—
74#394D511yescharacteristic
75#388—52no—
76#388—52no—
77#380—52no—
78#380—52no—
79C4×C4⋊C4—51no—
80C23×C5—22no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5#4114yes—
87C23×C5C2411yescharacteristic
88C5×D4—28no—
89C2×C4×C5—22no—
90C2×C4×C5—21no—
91C2×C4×C5—21no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5—21no—
97C2×C4×C5—21no—
98C2×C4×C5C2×D412yes—
99C2×Dic5—21no—
100C2×Dic5—21no—
101C2×Dic5—21no—
102C2×Dic5—21no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—21no—
111C2×Dic5—21no—
112C2×Dic5—48no—
113C2×Dic5C2×D412yes—
114C2×Dic5C2×D412yes—
115C2×Dic5#4114yes—
116C22×D4C2×D511yescharacteristic
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—51no—
126C2×C4⋊C4—52no—
127C2×C4⋊C4C2×D511yescharacteristic
128C2×C22⋊C4—52no—
129C2×C22⋊C4—52no—
130C2×C22⋊C4—52no—
131C2×C22⋊C4C2×D512yes—
132C2×C22⋊C4C2×D512yes—
133C2×C42—51no—
134C2×C42—52no—
135#95—51no—
136#95—52no—
137#95—52no—
138#95—52no—
139#95—52no—
140#95C2×D511yescharacteristic
141C22×C5—24no—
142C22×C5—24no—
143C22×C5—24no—
144C22×C5—42no—
145C22×C5—42no—
146C22×C5—42no—
147C22×C5—42no—
148C22×C5C22×D411yescharacteristic
149C22×C5C2×#4112yes—
150C22×C5C2×#4112yes—
151C22×C5H(5,𝔽2)11yescharacteristic
152C22×C52−1+411yescharacteristic
153C4×C5—22no—
154C4×C5—41no—
155C4×C5—41no—
156C4×C5—42no—
157Dic5—22no—
158Dic5—41no—
159Dic5—42no—
160Dic5—41no—
161Dic5—42no—
162Dic5#12414yes—
163C24C22×D512yes—
164C2×D4—24no—
165C2×D4—22no—
166C22×C4—51no—
167C22×C4—51no—
168C22×C4—52no—
169C22×C4—52no—
170C22×C4—51no—
171C22×C4—51no—
172C22×C4—104no—
173C22×C4—104no—
174C22×C4—104no—
175C22×C4C22×D512yes—
176C22×C4C22×D511yescharacteristic
177C22×C4C22×D511yescharacteristic
178C22×C4C22×D511yescharacteristic
179C4⋊C4—22no—
180C4⋊C4—102no—
181C4⋊C4—102no—
182C4⋊C4—102no—
183C4⋊C4—102no—
184C4⋊C4—104no—
185C4⋊C4—102no—
186C4⋊C4—102no—
187C22⋊C4—24no—
188C22⋊C4—42no—
189C22⋊C4—104no—
190C22⋊C4—104no—
191C22⋊C4—202no—
192C42—54no—
193C42—202no—
194C2×C5—44no—
195C2×C5#52511yescharacteristic
196C2×C5C2×#12411yescharacteristic
197C2×C5#53711yescharacteristic
198C2×C5#55212yes—
199C2×C5#55312yes—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23#71114yes—
207C23C23×D511yescharacteristic
208D4—28no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—52no—
219C2×C4—54no—
220C2×C4—52no—
221C2×C4—101no—
222C2×C4—101no—
223C2×C4—102no—
224C2×C4—101no—
225C2×C4—101no—
226C2×C4—102no—
227C2×C4—102no—
228C2×C4—102no—
229C2×C4—101no—
230C2×C4—102no—
231C2×C4—102no—
232C2×C4—102no—
233C2×C4—101no—
234C2×C4—208no—
235C2×C4D4×D512yes—
236C5#256211yescharacteristic, complemented, Sylow
237C22—24no—
238C22—24no—
239C22—24no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22C2×D4×D511yescharacteristic
245C22C2×#71112yes—
246C22#414411yescharacteristic
247C22D5×#4112yes—
248C22#415011yescharacteristic
249C4—22no—
250C4—41no—
251C4—42no—
252C4—41no—
253C4—54no—
254C4—102no—
255C4—201no—
256C4—202no—
257C4—201no—
258C4—202no—
259C2—44no—
260C2#6592011yescharacteristic
261C2#6593512yes—
262C2#6594211yescharacteristic
263C2#6598412yes—
264C2D5×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC22×C5×D4C22
hypercenter#394D5
hypocenterC5#2562
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#4035C2semidirect
5. #65275C2semidirect
6. #65240C2semidirect
7. #65233C2semidirect
8. #65231C2semidirect
9. #65231C2semidirect
10. #64948C2semidirect
11. #64938C2semidirect
236. C5#2562semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic52
C2×#40352
C5×#3942
#655112
#652752
#652402
#652332
#652312
#649482
#649382
#25625