Finite Group Databasebeta

← #10609414

Subgroups — #10609414

Order 640.

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

normal subgroups only

This group has 672 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×#4043C211yesmaximal, characteristic, complemented
2C2×D5×C22⋊C4C211yesmaximal, characteristic
3C5×#397C211yesmaximal, characteristic, complemented
4#65282C211yesmaximal, characteristic, complemented
5#65244C212yesmaximal
6#65240C212yesmaximal, complemented
7#65236C211yesmaximal, characteristic, complemented
8#64962C211yesmaximal, characteristic, complemented
9#64959C212yesmaximal, complemented
10#64955C211yesmaximal, characteristic, complemented
11#64953C212yesmaximal, complemented
12C24×D5C2211yescharacteristic
13C22×C4×D5C2211yescharacteristic
14C22×C4×D5C2211yescharacteristic
15C22×Dic10C2211yescharacteristic, complemented
16C2×C5×C4⋊C4C2211yescharacteristic, complemented
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2212yes—
19C2×C22⋊Dic5C2211yescharacteristic
20C2×C22⋊Dic5C2212yes—
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#685C2211yescharacteristic
24C2×#685C2212yes—
25C2×#685C2212yes—
26C2×#685C2212yes—
27C2×C4⋊Dic5C2212yescomplemented
28C2×C4×Dic5C2212yescomplemented
29#4043—24no—
30D5×C22⋊C4—24no—
31C5×#95C2212yescomplemented
32C5×#95C2211yescharacteristic, complemented
33#3963C2212yescomplemented
34#3963C2211yescharacteristic, complemented
35#3963C2212yescomplemented
36#3963C2212yescomplemented
37#2561—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5—22no—
40C23×D5—22no—
41C23×D5—22no—
42C23×D5—24no—
43C23×D5—42no—
44C23×D5C2311yescharacteristic
45C23×D5C2311yescharacteristic
46C22×C4×C5C2312yes—
47C22×C4×C5C2311yescharacteristic
48C22×C4×C5C2312yes—
49C22×C4×C5C2311yescharacteristic
50C22×Dic5C2312yes—
51C22×Dic5C2311yescharacteristic
52C22×Dic5C2312yes—
53C22×Dic5C2311yescharacteristic
54C2×C4×D5—22no—
55C2×C4×D5—24no—
56C2×C4×D5—42no—
57C2×C4×D5D414yes—
58C2×Dic10—22no—
59C2×Dic10—42no—
60C5×C4⋊C4—22no—
61C5×C22⋊C4—24no—
62C5×C22⋊C4—22no—
63C22⋊Dic5—22no—
64C22⋊Dic5—42no—
65#685—24no—
66#685—24no—
67#685—24no—
68#685—22no—
69#685—22no—
70#685—42no—
71C4⋊Dic5—24no—
72C4×Dic5—42no—
73C2×C22⋊Q8—51no—
74C22×C22⋊C4—51no—
75#397—51no—
76#397D511yescharacteristic, complemented
77#393—51no—
78#393—51no—
79#388—52no—
80#388—52no—
81#381—52no—
82#379—52no—
83C4×C22⋊C4—51no—
84C23×C5—21no—
85C23×C5—21no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5#4112yes—
89C23×C5C2411yescharacteristic
90C22×D5—21no—
91C22×D5—21no—
92C22×D5—22no—
93C22×D5—28no—
94C22×D5—21no—
95C22×D5—21no—
96C22×D5—22no—
97C22×D5—22no—
98C22×D5—48no—
99C22×D5—48no—
100C22×D5—44no—
101C22×D5—44no—
102C22×D5C2×D412yes—
103C22×D5#4112yes—
104C22×D5#4114yes—
105C2×C4×C5—22no—
106C2×C4×C5—22no—
107C2×C4×C5—21no—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5C2×D412yes—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—21no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C2×Dic5C2×D412yes—
125C4×D5—44no—
126C4×D5—44no—
127Dic10—44no—
128C25—51no—
129C22×Q8—51no—
130C23×C4—51no—
131C23×C4—51no—
132C22⋊Q8—104no—
133C2×C4⋊C4—52no—
134C2×C4⋊C4C2×D511yescharacteristic, complemented
135C2×C22⋊C4—51no—
136C2×C22⋊C4—51no—
137C2×C22⋊C4—51no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4—52no—
141C2×C22⋊C4—52no—
142C2×C22⋊C4—52no—
143C2×C22⋊C4—104no—
144C2×C22⋊C4C2×D511yescharacteristic
145C2×C22⋊C4C2×D512yes—
146C2×C42—52no—
147#95—51no—
148#95—52no—
149#95—52no—
150#95—52no—
151#95C2×D511yescharacteristic, complemented
152#95C2×D512yescomplemented
153C22×C5—22no—
154C22×C5—24no—
155C22×C5—41no—
156C22×C5—41no—
157C22×C5—42no—
158C22×C5C22×D411yescharacteristic
159C22×C5C2×#4111yescharacteristic
160C22×C5C2×#4111yescharacteristic
161C22×C5C2×#4112yes—
162C22×C5H(5,𝔽2)12yes—
163C2×D5—22no—
164C2×D5—22no—
165C2×D5—24no—
166C2×D5—24no—
167C2×D5—22no—
168C2×D5—24no—
169C2×D5—416no—
170C2×D5—41no—
171C2×D5—41no—
172C2×D5—42no—
173C2×D5#12414yes—
174C4×C5—22no—
175C4×C5—42no—
176C4×C5—41no—
177C4×C5—42no—
178Dic5—22no—
179Dic5—41no—
180Dic5—42no—
181Dic5—42no—
182C24—51no—
183C24—51no—
184C24—102no—
185C24—102no—
186C24—104no—
187C24—102no—
188C24—202no—
189C24C22×D511yescharacteristic
190C2×Q8—102no—
191C2×Q8—202no—
192C22×C4—51no—
193C22×C4—54no—
194C22×C4—51no—
195C22×C4—52no—
196C22×C4—52no—
197C22×C4—102no—
198C22×C4—104no—
199C22×C4—202no—
200C22×C4C22×D511yescharacteristic
201C22×C4C22×D512yes—
202C22×C4C22×D512yes—
203C22×C4C22×D511yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—104no—
206C22⋊C4—24no—
207C22⋊C4—22no—
208C22⋊C4—102no—
209C22⋊C4—104no—
210C22⋊C4—104no—
211C22⋊C4—104no—
212C22⋊C4—102no—
213C22⋊C4—102no—
214C22⋊C4—202no—
215C22⋊C4—202no—
216C42—202no—
217C2×C5—42no—
218C2×C5#52511yescharacteristic
219C2×C5C2×#12411yescharacteristic
220C2×C5#53411yescharacteristic
221C2×C5#55112yes—
222C2×C5#55112yes—
223D5—24no—
224D5—42no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—52no—
230C23—54no—
231C23—52no—
232C23—108no—
233C23—101no—
234C23—101no—
235C23—102no—
236C23—101no—
237C23—101no—
238C23—102no—
239C23—102no—
240C23—208no—
241C23—204no—
242C23—208no—
243C23—204no—
244C23#71112yes—
245C23C23×D511yescharacteristic
246Q8—204no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—52no—
257C2×C4—102no—
258C2×C4—101no—
259C2×C4—102no—
260C2×C4—102no—
261C2×C4—102no—
262C2×C4—102no—
263C2×C4—102no—
264C2×C4—102no—
265C2×C4—102no—
266C2×C4—204no—
267C2×C4—204no—
268C2×C4D4×D512yes—
269C5#256111yescharacteristic, complemented, Sylow
270C22—22no—
271C22—24no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—54no—
276C22—102no—
277C22—102no—
278C22—104no—
279C22—104no—
280C22—102no—
281C22—104no—
282C22—2016no—
283C22—201no—
284C22—202no—
285C22—201no—
286C22C2×D4×D511yescharacteristic
287C22C2×#71111yescharacteristic
288C22D5×#4111yescharacteristic
289C22D5×#4112yes—
290C22#414912yes—
291C4—22no—
292C4—42no—
293C4—42no—
294C4—41no—
295C4—102no—
296C4—202no—
297C4—201no—
298C4—202no—
299C2—42no—
300C2—104no—
301C2—202no—
302C2#6594011yescharacteristic
303C2#6598711yescharacteristic
304C2#6599212yes—
305C2D5×#12411yescharacteristic
306C2#6603912yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radicalC24×D5C22
hypercenter#397D5
hypocenterC5#2561
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#4043C2semidirect
3. C5×#397C2semidirect
4. #65282C2semidirect
6. #65240C2semidirect
7. #65236C2semidirect
8. #64962C2semidirect
9. #64959C2semidirect
10. #64955C2semidirect
11. #64953C2semidirect
15. C22×Dic10C22semidirect
16. C2×C5×C4⋊C4C22semidirect
27. C2×C4⋊Dic5C22semidirect
28. C2×C4×Dic5C22semidirect
31. C5×#95C22semidirect
32. C5×#95C22semidirect
33. #3963C22semidirect
34. #3963C22semidirect
35. #3963C22semidirect
36. #3963C22semidirect
76. #397D5semidirect
134. C2×C4⋊C4C2×D5semidirect
151. #95C2×D5semidirect
152. #95C2×D5semidirect
269. C5#2561semidirect

Maximal subgroups

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

subgroupindex
C2×#40432
C2×D5×C22⋊C42
C5×#3972
#652822
#652442
#652402
#652362
#649622
#649592
#649552
#649532
#25615