Finite Group Databasebeta

← #10608596

Subgroups — #10608596

Order 640.

780 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 161 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×#4092C212yesmaximal, complemented
2C2×D5×C22⋊C4C212yesmaximal
3C5×#393C211yesmaximal, characteristic, complemented
4#65244C211yesmaximal, characteristic
5#65238C211yesmaximal, characteristic, complemented
6#65237C212yesmaximal, complemented
7#64957C212yesmaximal, complemented
8#64957C212yesmaximal, complemented
9#64956C212yesmaximal, complemented
10C24×D5C2211yescharacteristic
11C22×C5×Q8C2211yescharacteristic, complemented
12C22×C4×D5C2212yes—
13C22×C4×D5C2212yes—
14C2×C5×C22⋊C4C2212yes—
15C2×C5×C22⋊C4C2211yescharacteristic
16C2×C22⋊Dic5C2212yes—
17C2×C22⋊Dic5C2211yescharacteristic
18#4092—28no—
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×C4⋊Dic5C2212yescomplemented
24C2×C4⋊Dic5C2211yescharacteristic, complemented
25C2×#683C2212yescomplemented
26C2×#683C2212yescomplemented
27C2×#683C2212yescomplemented
28D5×C22⋊C4—28no—
29C5×#95C2211yescharacteristic, complemented
30C5×#95C2212yescomplemented
31#3963C2212yescomplemented
32#3963C2211yescharacteristic, complemented
33#2437—51nomaximal, Sylow
34C24×C5C2311yescharacteristic
35C23×D5—24no—
36C23×D5—24no—
37C23×D5—41no—
38C23×D5—41no—
39C23×D5Q814yescomplemented
40C23×D5C2312yes—
41C2×C5×Q8—24no—
42C2×C5×Q8—41no—
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2311yescharacteristic
46C22×C4×C5C2312yes—
47C22×Dic5C2312yes—
48C22×Dic5C2312yes—
49C22×Dic5C2312yes—
50C2×C4×D5—24no—
51C2×C4×D5—24no—
52C2×C4×D5—24no—
53C2×C4×D5—24no—
54C2×C4×D5—42no—
55C2×C4×D5—42no—
56C5×C22⋊C4—24no—
57C5×C22⋊C4—41no—
58C22⋊Dic5—24no—
59C22⋊Dic5—41no—
60#685—24no—
61#685—24no—
62#685—24no—
63#685D418yes—
64C4⋊Dic5—24no—
65C4⋊Dic5—41no—
66#683—24no—
67#683—24no—
68#683—42no—
69C2×C22⋊Q8—52no—
70C22×C22⋊C4—52no—
71#399—51no—
72#396—52no—
73#393D511yescharacteristic, complemented
74#385—52no—
75#385—52no—
76#380—52no—
77#379—51no—
78C23×C5—22no—
79C23×C5—22no—
80C23×C5—21no—
81C23×C5—21no—
82C23×C5C2×Q812yes—
83C23×C5C2411yescharacteristic
84C22×D5—22no—
85C22×D5—22no—
86C22×D5—22no—
87C22×D5—22no—
88C22×D5—216no—
89C22×D5—44no—
90C22×D5—44no—
91C22×D5—48no—
92C22×D5—44no—
93C22×D5C2×D414yes—
94C22×D5C2×Q814yes—
95C22×D5#4114yes—
96C5×Q8—44no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—21no—
105C2×C4×C5—21no—
106C2×C4×C5C2×D414yes—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5C2×D414yes—
116C4×D5—48no—
117C4×D5—48no—
118C25—51no—
119C22×Q8C2×D511yescharacteristic, complemented
120C23×C4—52no—
121C23×C4—52no—
122C22⋊Q8—108no—
123C2×C4⋊C4—52no—
124C2×C4⋊C4—52no—
125C2×C4⋊C4—52no—
126C2×C4⋊C4—52no—
127C2×C4⋊C4—51no—
128C2×C22⋊C4—52no—
129C2×C22⋊C4—52no—
130C2×C22⋊C4—52no—
131C2×C22⋊C4—52no—
132C2×C22⋊C4—52no—
133C2×C22⋊C4—51no—
134C2×C22⋊C4—108no—
135C2×C22⋊C4C2×D512yes—
136C2×C22⋊C4C2×D511yescharacteristic
137#95—52no—
138#95—51no—
139#95C2×D511yescharacteristic, complemented
140#95C2×D512yescomplemented
141C22×C5—24no—
142C22×C5—22no—
143C22×C5—41no—
144C22×C5—42no—
145C22×C5—41no—
146C22×C5C22×D412yes—
147C22×C5C22×Q811yescharacteristic
148C22×C5C2×#4112yes—
149C22×C5H(5,𝔽2)11yescharacteristic
150C22×C5H(5,𝔽2)11yescharacteristic
151C2×D5—24no—
152C2×D5—24no—
153C2×D5—24no—
154C2×D5—24no—
155C2×D5—24no—
156C2×D5—24no—
157C2×D5—416no—
158C2×D5C22⋊Q818yescomplemented
159C4×C5—24no—
160C4×C5—41no—
161C4×C5—42no—
162C4×C5—41no—
163Dic5—24no—
164Dic5—42no—
165Dic5—42no—
166C24—52no—
167C24—54no—
168C24—104no—
169C24—104no—
170C24—201no—
171C24—201no—
172C24C22×D511yescharacteristic
173C2×Q8—24no—
174C2×Q8—41no—
175C22×C4—52no—
176C22×C4—52no—
177C22×C4—52no—
178C22×C4—104no—
179C22×C4—104no—
180C22×C4—104no—
181C22×C4—104no—
182C22×C4—202no—
183C22×C4—202no—
184C22×C4C22×D511yescharacteristic
185C22×C4C22×D512yes—
186C22×C4C22×D511yescharacteristic
187C22×C4C22×D512yes—
188C4⋊C4—104no—
189C4⋊C4—104no—
190C4⋊C4—104no—
191C4⋊C4—202no—
192C4⋊C4—201no—
193C22⋊C4—24no—
194C22⋊C4—41no—
195C22⋊C4—58no—
196C22⋊C4—104no—
197C22⋊C4—104no—
198C22⋊C4—104no—
199C22⋊C4—104no—
200C22⋊C4—201no—
201C2×C5—42no—
202C2×C5C2×C22⋊Q812yes—
203C2×C5#54311yescharacteristic
204C2×C5D4211yescharacteristic
205C2×C5#54612yes—
206C2×C5#55111yescharacteristic
207D5—28no—
208C23—22no—
209C23—21no—
210C23—22no—
211C23—21no—
212C23—54no—
213C23—54no—
214C23—54no—
215C23—1016no—
216C23—102no—
217C23—102no—
218C23—102no—
219C23—102no—
220C23—204no—
221C23—204no—
222C23—208no—
223C23—204no—
224C23Q8×D512yes—
225C23C23×D511yescharacteristic
226Q8—44no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—54no—
237C2×C4—102no—
238C2×C4—102no—
239C2×C4—102no—
240C2×C4—102no—
241C2×C4—102no—
242C2×C4—102no—
243C2×C4—102no—
244C2×C4—102no—
245C2×C4—208no—
246C2×C4—208no—
247C2×C4D4×D514yes—
248C5#243711yescharacteristic, complemented, Sylow
249C22—24no—
250C22—22no—
251C22—41no—
252C22—42no—
253C22—41no—
254C22—58no—
255C22—104no—
256C22—104no—
257C22—104no—
258C22—104no—
259C22—104no—
260C22—104no—
261C22—2016no—
262C22C2×D4×D512yes—
263C22#414411yescharacteristic
264C22C2×Q8×D511yescharacteristic
265C22D5×#4112yes—
266C22#414911yescharacteristic
267C4—24no—
268C4—41no—
269C4—42no—
270C4—41no—
271C4—104no—
272C4—202no—
273C4—202no—
274C2—42no—
275C2—108no—
276C2#6592411yescharacteristic
277C2D5×C22⋊Q812yes—
278C2#6596811yescharacteristic
279C2#6601012yes—
280C2#6601311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#393C2
prime radicalC24×D5C22
hypercenter#393D5
hypocenterC5#2437
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#4092C2semidirect
3. C5×#393C2semidirect
5. #65238C2semidirect
6. #65237C2semidirect
7. #64957C2semidirect
8. #64957C2semidirect
9. #64956C2semidirect
11. C22×C5×Q8C22semidirect
23. C2×C4⋊Dic5C22semidirect
24. C2×C4⋊Dic5C22semidirect
25. C2×#683C22semidirect
26. C2×#683C22semidirect
27. C2×#683C22semidirect
29. C5×#95C22semidirect
30. C5×#95C22semidirect
31. #3963C22semidirect
32. #3963C22semidirect
39. C23×D5Q8semidirect
73. #393D5semidirect
119. C22×Q8C2×D5semidirect
139. #95C2×D5semidirect
140. #95C2×D5semidirect
158. C2×D5C22⋊Q8semidirect
248. C5#2437semidirect

Maximal subgroups

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

subgroupindex
C2×#40922
C2×D5×C22⋊C42
C5×#3932
#652442
#652382
#652372
#649572
#649562
#24375