Finite Group Databasebeta

← #10609851

Subgroups — #10609851

Order 640.

552 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C5×#399C211yesmaximal, characteristic, complemented
2#65283C212yesmaximal, complemented
3#65279C212yesmaximal, complemented
4#65279C212yesmaximal, complemented
5#65244C212yesmaximal
6#65244C211yesmaximal, characteristic
7#65242C211yesmaximal, characteristic, complemented
8#65233C212yesmaximal, complemented
9#64964C212yesmaximal, complemented
10C24×D5C2211yescharacteristic
11C2×C5×C4⋊C4C2212yescomplemented
12C2×C5×C4⋊C4C2211yescharacteristic, complemented
13C2×C5×C22⋊C4C2212yes—
14C2×C5×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic5C2212yes—
16C2×C22⋊Dic5C2211yescharacteristic
17C2×#685C2212yes—
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×C4⋊Dic5C2211yescharacteristic, complemented
24C2×#683C2212yescomplemented
25C2×C4×Dic5C2212yescomplemented
26C5×#95C2211yescharacteristic, complemented
27#3963C2212yescomplemented
28#3963C2212yescomplemented
29#3963C2212yescomplemented
30#3963C2211yescharacteristic, complemented
31#2735—51nomaximal, Sylow
32C24×C5C2311yescharacteristic
33C23×D5—22no—
34C23×D5—24no—
35C23×D5—41no—
36C23×D5—42no—
37C23×D5—41no—
38C23×D5C2312yes—
39C22×C4×C5C2312yes—
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×Dic5C2312yes—
44C22×Dic5C2312yes—
45C22×Dic5C2312yes—
46C5×C4⋊C4—42no—
47C5×C4⋊C4—41no—
48C5×C22⋊C4—42no—
49C5×C22⋊C4—41no—
50C22⋊Dic5—24no—
51C22⋊Dic5—41no—
52#685—24no—
53#685—24no—
54#685—42no—
55#685—42no—
56#685—42no—
57#685—42no—
58C4⋊Dic5—41no—
59#683—42no—
60C4×Dic5—42no—
61#399D511yescharacteristic, complemented
62#397—51no—
63#397—52no—
64#396—52no—
65#388—52no—
66#388—52no—
67#388—52no—
68#379—52no—
69#379—51no—
70C23×C5—22no—
71C23×C5—21no—
72C23×C5—21no—
73C23×C5—22no—
74C23×C5#4112yes—
75C23×C5C2411yescharacteristic
76C22×D5—22no—
77C22×D5—22no—
78C22×D5—22no—
79C22×D5—22no—
80C22×D5—22no—
81C22×D5—22no—
82C22×D5—44no—
83C22×D5—48no—
84C22×D5—48no—
85C22×D5—44no—
86C22×D5—44no—
87C22×D5#4114yes—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—21no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—21no—
98C2×Dic5—22no—
99C2×Dic5—22no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C25—51no—
108C2×C4⋊C4—52no—
109C2×C4⋊C4—51no—
110C2×C4⋊C4C2×D511yescharacteristic, complemented
111C2×C4⋊C4C2×D512yescomplemented
112C2×C22⋊C4—52no—
113C2×C22⋊C4—52no—
114C2×C22⋊C4—52no—
115C2×C22⋊C4—51no—
116C2×C22⋊C4—52no—
117C2×C22⋊C4—52no—
118C2×C22⋊C4—52no—
119C2×C22⋊C4—52no—
120C2×C22⋊C4C2×D512yes—
121C2×C22⋊C4C2×D511yescharacteristic
122C2×C42—52no—
123#95—52no—
124#95—51no—
125#95—52no—
126#95—52no—
127#95C2×D511yescharacteristic, complemented
128C22×C5—22no—
129C22×C5—24no—
130C22×C5—42no—
131C22×C5—41no—
132C22×C5—41no—
133C22×C5C2×#4111yescharacteristic
134C22×C5C2×#4112yes—
135C22×C5H(5,𝔽2)12yes—
136C22×C5H(5,𝔽2)11yescharacteristic
137C22×C5H(5,𝔽2)11yescharacteristic
138C2×D5—24no—
139C2×D5—24no—
140C2×D5—24no—
141C2×D5—416no—
142C2×D5—42no—
143C2×D5—42no—
144C2×D5—42no—
145C2×D5—42no—
146C4×C5—42no—
147C4×C5—41no—
148C4×C5—42no—
149C4×C5—41no—
150Dic5—42no—
151Dic5—42no—
152Dic5—42no—
153C24—52no—
154C24—104no—
155C24—102no—
156C24—202no—
157C24—201no—
158C24—201no—
159C24C22×D511yescharacteristic
160C22×C4—52no—
161C22×C4—52no—
162C22×C4—52no—
163C22×C4C22×D511yescharacteristic
164C22×C4C22×D512yes—
165C22×C4C22×D512yes—
166C22×C4C22×D511yescharacteristic
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—202no—
170C4⋊C4—201no—
171C22⋊C4—42no—
172C22⋊C4—41no—
173C22⋊C4—104no—
174C22⋊C4—104no—
175C22⋊C4—104no—
176C22⋊C4—202no—
177C22⋊C4—202no—
178C22⋊C4—202no—
179C22⋊C4—202no—
180C22⋊C4—201no—
181C42—202no—
182C2×C5—42no—
183C2×C5#53811yescharacteristic
184C2×C5#53812yes—
185C2×C5#55112yes—
186C2×C5#55111yescharacteristic
187C2×C5#56011yescharacteristic
188D5—44no—
189C23—22no—
190C23—21no—
191C23—22no—
192C23—21no—
193C23—54no—
194C23—102no—
195C23—102no—
196C23—102no—
197C23—102no—
198C23—102no—
199C23—102no—
200C23—208no—
201C23—204no—
202C23—204no—
203C23—208no—
204C23—204no—
205C23#71312yes—
206C23C23×D511yescharacteristic
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—102no—
221C2×C4—102no—
222C2×C4—102no—
223C2×C4—102no—
224C2×C4—102no—
225C2×C4—102no—
226C5#273511yescharacteristic, complemented, Sylow
227C22—24no—
228C22—22no—
229C22—41no—
230C22—42no—
231C22—41no—
232C22—104no—
233C22—104no—
234C22—104no—
235C22—2016no—
236C22—202no—
237C22—202no—
238C22—202no—
239C22—202no—
240C22#414412yes—
241C22#414411yescharacteristic
242C22C2×#71311yescharacteristic
243C22D5×#4112yes—
244C22#414911yescharacteristic
245C4—42no—
246C4—41no—
247C4—42no—
248C4—41no—
249C4—202no—
250C4—202no—
251C4—202no—
252C2—42no—
253C2—204no—
254C2#6597112yes—
255C2#6597811yescharacteristic
256C2#6599112yes—
257C2#6599311yescharacteristic
258C2#6605411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
prime radicalC24×D5C22
hypercenter#399D5
hypocenterC5#2735
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C5×#399C2semidirect
2. #65283C2semidirect
3. #65279C2semidirect
4. #65279C2semidirect
7. #65242C2semidirect
8. #65233C2semidirect
9. #64964C2semidirect
11. C2×C5×C4⋊C4C22semidirect
12. C2×C5×C4⋊C4C22semidirect
23. C2×C4⋊Dic5C22semidirect
24. C2×#683C22semidirect
25. C2×C4×Dic5C22semidirect
26. C5×#95C22semidirect
27. #3963C22semidirect
28. #3963C22semidirect
29. #3963C22semidirect
30. #3963C22semidirect
61. #399D5semidirect
110. C2×C4⋊C4C2×D5semidirect
111. C2×C4⋊C4C2×D5semidirect
127. #95C2×D5semidirect
226. C5#2735semidirect

Maximal subgroups

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

subgroupindex
C5×#3992
#652832
#652792
#652442
#652422
#652332
#649642
#27355