Finite Group Databasebeta

← #10609789

Subgroups — #10609789

Order 640.

498 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 133 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×#4043C211yesmaximal, characteristic, complemented
2C2×#4029C211yesmaximal, characteristic
3C5×#399C211yesmaximal, characteristic, complemented
4#65282C212yesmaximal, complemented
5#65274C212yesmaximal, complemented
6#65271C211yesmaximal, characteristic, complemented
7#65250C212yesmaximal
8#65240C212yesmaximal, complemented
9#65236C211yesmaximal, characteristic, complemented
10#64955C211yesmaximal, characteristic, complemented
11#64945C211yesmaximal, characteristic, complemented
12C22×C22⋊D5C2211yescharacteristic
13C22×C4×D5C2211yescharacteristic
14C22×Dic10C2211yescharacteristic
15C2×C5×C4⋊C4C2212yes—
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2212yes—
19C2×C22⋊Dic5C2211yescharacteristic
20C2×C22⋊Dic5C2212yes—
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×C4⋊Dic5C2211yescharacteristic
25C2×C4⋊Dic5C2212yes—
26C2×C4⋊Dic5C2212yes—
27C2×#683C2211yescharacteristic
28C2×C4×Dic5C2212yes—
29#4043—24no—
30#4029—24no—
31C5×#95C2211yescharacteristic
32#3963C2212yes—
33#3963C2212yes—
34#3963C2212yes—
35#3963C2211yescharacteristic
36#3963C2212yes—
37#2663—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2312yes—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C2×C22⋊D5—22no—
45C2×C22⋊D5—42no—
46C22×Dic5C2312yes—
47C22×Dic5C2312yes—
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2311yescharacteristic
51C2×C4×D5—42no—
52C2×C4×D5D414yes—
53C2×Dic10—22no—
54C2×Dic10—42no—
55C5×C4⋊C4—22no—
56C5×C4⋊C4—24no—
57C5×C22⋊C4—22no—
58C5×C22⋊C4—24no—
59C22⋊Dic5—22no—
60C22⋊Dic5—42no—
61#685—22no—
62#685—24no—
63#685—24no—
64C4⋊Dic5—22no—
65C4⋊Dic5—24no—
66C4⋊Dic5—42no—
67#683—22no—
68C4×Dic5—42no—
69C2×#123—51no—
70C2×C22⋊Q8—51no—
71#400—51no—
72#399D511yescharacteristic, complemented
73#397—52no—
74#395—51no—
75#394—52no—
76#393—51no—
77#389—52no—
78#388—52no—
79C4×C22⋊C4—51no—
80C23×C5—21no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5—21no—
84C23×C5—21no—
85C23×C5C2411yescharacteristic
86C22×D5—22no—
87C22×D5—22no—
88C22×D5—21no—
89C22×D5—21no—
90C22×D5C2×D412yes—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—21no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5C2×D412yes—
100C2×C4×C5#4114yes—
101C22⋊D5—44no—
102C2×Dic5—22no—
103C2×Dic5—21no—
104C2×Dic5—21no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—21no—
112C2×Dic5—22no—
113C2×Dic5—21no—
114C2×Dic5C2×D412yes—
115C4×D5—44no—
116Dic10—44no—
117C22×Q8—51no—
118C22×D4—51no—
119C23×C4—51no—
120#123—104no—
121C22⋊Q8—104no—
122C2×C4⋊C4—52no—
123C2×C4⋊C4—51no—
124C2×C4⋊C4—51no—
125C2×C4⋊C4—52no—
126C2×C4⋊C4C2×D511yescharacteristic
127C2×C4⋊C4C2×D512yes—
128C2×C22⋊C4—51no—
129C2×C22⋊C4—52no—
130C2×C22⋊C4—51no—
131C2×C22⋊C4—52no—
132C2×C22⋊C4—52no—
133C2×C22⋊C4C2×D511yescharacteristic
134C2×C22⋊C4C2×D512yes—
135C2×C42—52no—
136#95—51no—
137#95—52no—
138#95—52no—
139#95—52no—
140#95—52no—
141#95C2×D511yescharacteristic
142C22×C5—41no—
143C22×C5—41no—
144C22×C5—41no—
145C22×C5—42no—
146C22×C5—42no—
147C22×C5C22×D411yescharacteristic
148C22×C5C2×#4112yes—
149C22×C5H(5,𝔽2)12yes—
150C22×C52−1+411yescharacteristic
151C22×C52−1+411yescharacteristic
152C2×D5—22no—
153C2×D5—22no—
154C2×D5—22no—
155C2×D5—42no—
156C2×D5—42no—
157C4×C5—22no—
158C4×C5—24no—
159C4×C5—41no—
160C4×C5—42no—
161Dic5—22no—
162Dic5—42no—
163Dic5—41no—
164Dic5—41no—
165Dic5—42no—
166C24—51no—
167C24C22×D511yescharacteristic
168C2×Q8—102no—
169C2×Q8—202no—
170C2×D4—102no—
171C2×D4—202no—
172C22×C4—51no—
173C22×C4—54no—
174C22×C4—52no—
175C22×C4—51no—
176C22×C4—52no—
177C22×C4—51no—
178C22×C4—202no—
179C22×C4C22×D511yescharacteristic
180C22×C4C22×D512yes—
181C22×C4C22×D511yescharacteristic
182C22×C4C22×D512yes—
183C4⋊C4—22no—
184C4⋊C4—24no—
185C4⋊C4—104no—
186C4⋊C4—102no—
187C4⋊C4—102no—
188C4⋊C4—202no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—104no—
192C22⋊C4—102no—
193C22⋊C4—102no—
194C22⋊C4—104no—
195C22⋊C4—202no—
196C42—202no—
197C2×C5—81no—
198C2×C5#53211yescharacteristic
199C2×C5#53411yescharacteristic
200C2×C5#53611yescharacteristic
201C2×C5#53912yes—
202C2×C5#54212yes—
203D5—42no—
204C23—21no—
205C23—22no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—52no—
210C23—102no—
211C23—101no—
212C23—102no—
213C23—101no—
214C23C23×D511yescharacteristic
215Q8—204no—
216D4—204no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—52no—
226C2×C4—102no—
227C2×C4—101no—
228C2×C4—102no—
229C2×C4—101no—
230C2×C4—102no—
231C2×C4—101no—
232C2×C4—102no—
233C2×C4—102no—
234C2×C4—102no—
235C2×C4—102no—
236C2×C4—102no—
237C2×C4—101no—
238C2×C4—204no—
239C2×C4D4×D512yes—
240C2×C4#71114yes—
241C5#266311yescharacteristic, complemented, Sylow
242C22—41no—
243C22—41no—
244C22—42no—
245C22—41no—
246C22—42no—
247C22—102no—
248C22—102no—
249C22—102no—
250C22—202no—
251C22—202no—
252C22C2×D4×D511yescharacteristic
253C22C2×#71112yes—
254C22#414711yescharacteristic
255C22#414912yes—
256C22#415011yescharacteristic
257C4—22no—
258C4—24no—
259C4—41no—
260C4—42no—
261C4—102no—
262C4—201no—
263C4—202no—
264C4—201no—
265C4—202no—
266C2—81no—
267C2—202no—
268C2#6596211yescharacteristic
269C2#6597712yes—
270C2#6598711yescharacteristic
271C2#6599912yes—
272C2#6605011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
prime radicalC22×C22⋊D5C22
hypercenter#399D5
hypocenterC5#2663
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#4043C2semidirect
3. C5×#399C2semidirect
4. #65282C2semidirect
5. #65274C2semidirect
6. #65271C2semidirect
8. #65240C2semidirect
9. #65236C2semidirect
10. #64955C2semidirect
11. #64945C2semidirect
72. #399D5semidirect
241. C5#2663semidirect

Maximal subgroups

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

subgroupindex
C2×#40432
C2×#40292
C5×#3992
#652822
#652742
#652712
#652502
#652402
#652362
#649552
#649452
#26635