Finite Group Databasebeta

← #10609761

Subgroups — #10609761

Order 640.

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

normal subgroups only

This group has 708 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×#4040C212yesmaximal, complemented
2C2×D5×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#4030C211yesmaximal, characteristic
4C2×#4029C212yesmaximal, complemented
5C5×#399C211yesmaximal, characteristic, complemented
6#65281C211yesmaximal, characteristic, complemented
7#65265C212yesmaximal, complemented
8#65246C212yesmaximal, complemented
9#65238C211yesmaximal, characteristic, complemented
10#64966C211yesmaximal, characteristic, complemented
11#64955C211yesmaximal, characteristic, complemented
12C22×C22⋊D5C2211yescharacteristic
13C22×C22⋊D5C2211yescharacteristic
14C22×D20C2211yescharacteristic
15C22×C4×D5C2212yes—
16C22×C4×D5C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C4⋊C4C2212yes—
19C2×C5×C22⋊C4C2212yes—
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C22⋊Dic5C2212yes—
22C2×#685C2211yescharacteristic
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×#685C2212yes—
26C2×C4⋊Dic5C2212yes—
27C2×C4⋊Dic5C2211yescharacteristic
28C2×#683C2212yes—
29C2×#683C2211yescharacteristic
30C2×#683C2212yes—
31C2×#683C2212yes—
32C2×C4×Dic5C2212yes—
33#4040—28no—
34D5×C4⋊C4—24no—
35#4030—24no—
36#4029—28no—
37C5×#95C2211yescharacteristic
38#3963C2211yescharacteristic
39#2501—51nomaximal, Sylow
40C24×C5C2311yescharacteristic
41C23×D5C2311yescharacteristic
42C23×D5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2311yescharacteristic
46C22×C4×C5C2312yes—
47C2×C22⋊D5—24no—
48C2×C22⋊D5—22no—
49C2×C22⋊D5—22no—
50C2×C22⋊D5—24no—
51C22×Dic5C2312yes—
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2312yes—
54C22×Dic5C2311yescharacteristic
55C2×D20—22no—
56C2×D20—24no—
57C2×C4×D5—24no—
58C2×C4×D5—24no—
59C2×C4×D5—24no—
60C2×C4×D5—24no—
61C2×C4×D5D414yes—
62C5×C4⋊C4—22no—
63C5×C4⋊C4—24no—
64C5×C22⋊C4—24no—
65C5×C22⋊C4—22no—
66C22⋊Dic5—24no—
67#685—22no—
68#685—24no—
69#685—24no—
70#685D418yescomplemented
71C4⋊Dic5—24no—
72C4⋊Dic5—22no—
73#683—24no—
74#683—22no—
75#683—24no—
76#683—24no—
77C4×Dic5—24no—
78C2×#123—52no—
79C2×#123—52no—
80C2×#121—51no—
81C22×C4⋊C4—51no—
82#399—51no—
83#399—51no—
84#399—51no—
85#399D511yescharacteristic, complemented
86#390—52no—
87#384—52no—
88C4×C22⋊C4—51no—
89C23×C5—22no—
90C23×C5—21no—
91C23×C5—22no—
92C23×C5—21no—
93C23×C5—21no—
94C23×C5C2411yescharacteristic
95C22×D5—22no—
96C22×D5—21no—
97C22×D5—21no—
98C22×D5—22no—
99C22×D5—21no—
100C22×D5—22no—
101C22×D5—21no—
102C22×D5—21no—
103C22×D5C2×D414yes—
104C22×D5C2×D412yes—
105C2×C4×C5—22no—
106C2×C4×C5—22no—
107C2×C4×C5—22no—
108C2×C4×C5—22no—
109C2×C4×C5—21no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5C2×D414yes—
114C2×C4×C5C2×D412yes—
115C22⋊D5—44no—
116C22⋊D5—44no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—21no—
123C2×Dic5—22no—
124C2×Dic5C2×D414yes—
125C2×Dic5C2×D412yes—
126C2×Dic5#4114yes—
127D20—44no—
128C4×D5—28no—
129C4×D5—48no—
130C22×D4—51no—
131C22×D4—51no—
132C22×D4—51no—
133C23×C4—51no—
134C23×C4—52no—
135#123—108no—
136#123—108no—
137#121—104no—
138C2×C4⋊C4—52no—
139C2×C4⋊C4—51no—
140C2×C4⋊C4—51no—
141C2×C4⋊C4—52no—
142C2×C4⋊C4—52no—
143C2×C4⋊C4—52no—
144C2×C4⋊C4—104no—
145C2×C4⋊C4C2×D511yescharacteristic
146C2×C4⋊C4C2×D512yes—
147C2×C22⋊C4—52no—
148C2×C22⋊C4—52no—
149C2×C22⋊C4—51no—
150C2×C22⋊C4—52no—
151C2×C22⋊C4—52no—
152C2×C22⋊C4C2×D511yescharacteristic
153C2×C22⋊C4C2×D512yes—
154C2×C42—52no—
155#95—51no—
156#95C2×D511yescharacteristic
157C22×C5—41no—
158C22×C5—41no—
159C22×C5—41no—
160C22×C5—42no—
161C22×C5—42no—
162C22×C5C22×D411yescharacteristic
163C22×C5C22×D412yes—
164C22×C5C2×#4112yes—
165C22×C52−1+411yescharacteristic
166C22×C52−1+411yescharacteristic
167C2×D5—22no—
168C2×D5—22no—
169C2×D5—24no—
170C2×D5—24no—
171C2×D5—41no—
172C2×D5—41no—
173C2×D5—41no—
174C2×D5—42no—
175C2×D5—42no—
176C2×D5#12714yes—
177C4×C5—22no—
178C4×C5—24no—
179C4×C5—42no—
180C4×C5—41no—
181Dic5—24no—
182Dic5—42no—
183Dic5—41no—
184Dic5#12314yescomplemented
185C24—51no—
186C24—51no—
187C24C22×D511yescharacteristic
188C2×D4—102no—
189C2×D4—104no—
190C2×D4—102no—
191C2×D4—104no—
192C2×D4—102no—
193C2×D4—104no—
194C22×C4—51no—
195C22×C4—52no—
196C22×C4—54no—
197C22×C4—52no—
198C22×C4—51no—
199C22×C4—104no—
200C22×C4—104no—
201C22×C4—104no—
202C22×C4—104no—
203C22×C4C22×D512yes—
204C22×C4C22×D511yescharacteristic
205C22×C4C22×D511yescharacteristic
206C22×C4C22×D512yes—
207C4⋊C4—22no—
208C4⋊C4—24no—
209C4⋊C4—104no—
210C4⋊C4—102no—
211C4⋊C4—104no—
212C4⋊C4—104no—
213C4⋊C4—102no—
214C4⋊C4—104no—
215C22⋊C4—22no—
216C22⋊C4—24no—
217C22⋊C4—58no—
218C22⋊C4—104no—
219C22⋊C4—104no—
220C22⋊C4—102no—
221C22⋊C4—104no—
222C42—104no—
223C2×C5—81no—
224C2×C5C2×#12311yescharacteristic
225C2×C5C2×#12711yescharacteristic
226C2×C5#53611yescharacteristic
227C2×C5#54712yes—
228C2×C5#54712yes—
229D5—24no—
230D5—81no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—21no—
236C23—54no—
237C23—52no—
238C23—101no—
239C23—102no—
240C23—101no—
241C23—102no—
242C23—101no—
243C23—101no—
244C23—102no—
245C23—101no—
246C23C23×D511yescharacteristic
247D4—204no—
248D4—204no—
249D4—204no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—52no—
259C2×C4—54no—
260C2×C4—54no—
261C2×C4—108no—
262C2×C4—102no—
263C2×C4—102no—
264C2×C4—102no—
265C2×C4—102no—
266C2×C4—101no—
267C2×C4—102no—
268C2×C4—102no—
269C2×C4—208no—
270C2×C4D4×D514yes—
271C2×C4D4×D512yes—
272C5#250111yescharacteristic, complemented, Sylow
273C22—41no—
274C22—41no—
275C22—42no—
276C22—41no—
277C22—42no—
278C22—54no—
279C22—102no—
280C22—104no—
281C22—102no—
282C22—104no—
283C22—201no—
284C22—201no—
285C22—202no—
286C22—201no—
287C22—202no—
288C22C2×D4×D511yescharacteristic
289C22C2×D4×D512yes—
290C22#414711yescharacteristic
291C22D5×#4112yes—
292C22#415011yescharacteristic
293C4—22no—
294C4—24no—
295C4—42no—
296C4—41no—
297C4—54no—
298C4—104no—
299C4—202no—
300C4—201no—
301C2—81no—
302C2—104no—
303C2—401no—
304C2#6596211yescharacteristic
305C2#6596512yes—
306C2D5×#12311yescharacteristic
307C2#6598912yes—
308C2D5×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
prime radicalC2×#4030C2
hypercenter#399D5
hypocenterC5#2501
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#4040C2semidirect
2. C2×D5×C4⋊C4C2semidirect
4. C2×#4029C2semidirect
5. C5×#399C2semidirect
6. #65281C2semidirect
7. #65265C2semidirect
8. #65246C2semidirect
9. #65238C2semidirect
10. #64966C2semidirect
11. #64955C2semidirect
70. #685D4semidirect
85. #399D5semidirect
184. Dic5#123semidirect
272. C5#2501semidirect

Maximal subgroups

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

subgroupindex
C2×#40402
C2×D5×C4⋊C42
C2×#40302
C2×#40292
C5×#3992
#652812
#652652
#652462
#652382
#649662
#649552
#25015