Finite Group Databasebeta

← #10608602

Subgroups — #10608602

Order 640.

668 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 143 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×#4092C211yesmaximal, characteristic, complemented
2C2×D5×C22⋊C4C211yesmaximal, characteristic
3C5×#393C211yesmaximal, characteristic, complemented
4#65520C211yesmaximal, characteristic, complemented
5#65244C212yesmaximal
6#65238C211yesmaximal, characteristic, complemented
7#65237C212yesmaximal, complemented
8#64966C211yesmaximal, characteristic, complemented
9#64965C212yesmaximal, complemented
10#64957C212yesmaximal, complemented
11#64956C211yesmaximal, characteristic, complemented
12C24×D5C2211yescharacteristic
13C22×C5×Q8C2211yescharacteristic, complemented
14C22×C4×D5C2211yescharacteristic
15C22×C4×D5C2211yescharacteristic
16C2×C5×C22⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2211yescharacteristic
19C2×C22⋊Dic5C2212yes—
20#4092—24no—
21C2×#685C2211yescharacteristic
22C2×#685C2211yescharacteristic
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×#685C2212yes—
26C2×#685C2212yes—
27C2×C4⋊Dic5C2211yescharacteristic, complemented
28C2×C4⋊Dic5C2212yescomplemented
29C2×#683C2212yescomplemented
30C2×#683C2212yescomplemented
31C2×#683C2212yescomplemented
32D5×C22⋊C4—24no—
33C5×#95C2211yescharacteristic, complemented
34C5×#95C2212yescomplemented
35#3963C2211yescharacteristic, complemented
36#3963C2212yescomplemented
37#2626—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5—22no—
40C23×D5—22no—
41C23×D5—42no—
42C23×D5—42no—
43C23×D5Q814yescomplemented
44C23×D5C2311yescharacteristic
45C23×D5C2311yescharacteristic
46C2×C5×Q8—22no—
47C2×C5×Q8—42no—
48C22×C4×C5C2311yescharacteristic
49C22×C4×C5C2312yes—
50C22×C4×C5C2312yes—
51C22×C4×C5C2311yescharacteristic
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2312yes—
54C22×Dic5C2312yes—
55C22×Dic5C2311yescharacteristic
56C2×C4×D5—22no—
57C2×C4×D5—42no—
58C2×C4×D5—42no—
59C2×C4×D5D414yes—
60C5×C22⋊C4—22no—
61C5×C22⋊C4—42no—
62C22⋊Dic5—22no—
63C22⋊Dic5—42no—
64#685—24no—
65#685—24no—
66#685—22no—
67#685—22no—
68#685—42no—
69#685—42no—
70C4⋊Dic5—22no—
71C4⋊Dic5—42no—
72#683—24no—
73#683—42no—
74#683—42no—
75C2×C22⋊Q8—51no—
76C22×C22⋊C4—51no—
77#399—51no—
78#399—51no—
79#396—52no—
80#396—52no—
81#393—51no—
82#393D511yescharacteristic, complemented
83#385—52no—
84#380—51no—
85#379—52no—
86C23×C5—21no—
87C23×C5—21no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5C2×Q812yes—
91C23×C5C2411yescharacteristic
92C22×D5—22no—
93C22×D5—22no—
94C22×D5—21no—
95C22×D5—21no—
96C22×D5—22no—
97C22×D5—22no—
98C22×D5—28no—
99C22×D5—48no—
100C22×D5—44no—
101C22×D5—44no—
102C22×D5—48no—
103C22×D5C2×D412yes—
104C22×D5C2×Q812yes—
105C22×D5C2×Q812yes—
106C22×D5#4112yes—
107C5×Q8—44no—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5—21no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—22no—
116C2×C4×C5—22no—
117C2×C4×C5C2×D412yes—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5—21no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C2×Dic5—22no—
125C2×Dic5—22no—
126C2×Dic5—22no—
127C2×Dic5C2×D412yes—
128C4×D5—44no—
129C4×D5—44no—
130C25—51no—
131C22×Q8C2×D511yescharacteristic, complemented
132C23×C4—51no—
133C23×C4—51no—
134C22⋊Q8—104no—
135C2×C4⋊C4—52no—
136C2×C4⋊C4—51no—
137C2×C4⋊C4—52no—
138C2×C4⋊C4—52no—
139C2×C4⋊C4—52no—
140C2×C22⋊C4—51no—
141C2×C22⋊C4—51no—
142C2×C22⋊C4—51no—
143C2×C22⋊C4—52no—
144C2×C22⋊C4—52no—
145C2×C22⋊C4—52no—
146C2×C22⋊C4—52no—
147C2×C22⋊C4—52no—
148C2×C22⋊C4—104no—
149C2×C22⋊C4C2×D511yescharacteristic
150C2×C22⋊C4C2×D512yes—
151#95—51no—
152#95—52no—
153#95C2×D511yescharacteristic, complemented
154#95C2×D512yescomplemented
155C22×C5—22no—
156C22×C5—24no—
157C22×C5—41no—
158C22×C5—41no—
159C22×C5—42no—
160C22×C5C22×D411yescharacteristic
161C22×C5C22×Q811yescharacteristic
162C22×C5C2×#4111yescharacteristic
163C22×C5H(5,𝔽2)12yes—
164C22×C5H(5,𝔽2)12yes—
165C2×D5—22no—
166C2×D5—22no—
167C2×D5—24no—
168C2×D5—24no—
169C2×D5—22no—
170C2×D5—24no—
171C2×D5—416no—
172C2×D5—41no—
173C2×D5—41no—
174C2×D5—42no—
175C2×D5C22⋊Q814yes—
176C4×C5—22no—
177C4×C5—42no—
178C4×C5—41no—
179C4×C5—42no—
180Dic5—22no—
181Dic5—41no—
182Dic5—42no—
183Dic5—42no—
184C24—51no—
185C24—51no—
186C24—54no—
187C24—102no—
188C24—102no—
189C24—202no—
190C24—202no—
191C24C22×D511yescharacteristic
192C2×Q8—22no—
193C2×Q8—42no—
194C22×C4—51no—
195C22×C4—54no—
196C22×C4—52no—
197C22×C4—51no—
198C22×C4—52no—
199C22×C4—102no—
200C22×C4—202no—
201C22×C4—202no—
202C22×C4C22×D512yes—
203C22×C4C22×D512yes—
204C22×C4C22×D511yescharacteristic
205C22×C4C22×D511yescharacteristic
206C4⋊C4—104no—
207C4⋊C4—102no—
208C4⋊C4—202no—
209C4⋊C4—202no—
210C4⋊C4—202no—
211C22⋊C4—22no—
212C22⋊C4—42no—
213C22⋊C4—102no—
214C22⋊C4—104no—
215C22⋊C4—104no—
216C22⋊C4—102no—
217C22⋊C4—102no—
218C22⋊C4—202no—
219C22⋊C4—202no—
220C22⋊C4—202no—
221C2×C5—42no—
222C2×C5C2×C22⋊Q811yescharacteristic
223C2×C5#53411yescharacteristic
224C2×C5#53511yescharacteristic
225C2×C5#53812yes—
226C2×C5#54312yes—
227D5—24no—
228D5—42no—
229C23—21no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—52no—
234C23—52no—
235C23—52no—
236C23—52no—
237C23—108no—
238C23—102no—
239C23—102no—
240C23—101no—
241C23—102no—
242C23—101no—
243C23—102no—
244C23—208no—
245C23—204no—
246C23—204no—
247C23—208no—
248C23Q8×D512yes—
249C23C23×D511yescharacteristic
250Q8—44no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—52no—
261C2×C4—102no—
262C2×C4—101no—
263C2×C4—102no—
264C2×C4—102no—
265C2×C4—102no—
266C2×C4—102no—
267C2×C4—102no—
268C2×C4—102no—
269C2×C4—102no—
270C2×C4—204no—
271C2×C4—204no—
272C2×C4D4×D512yes—
273C5#262611yescharacteristic, complemented, Sylow
274C22—22no—
275C22—24no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—54no—
280C22—102no—
281C22—104no—
282C22—102no—
283C22—104no—
284C22—102no—
285C22—104no—
286C22—2016no—
287C22—201no—
288C22—201no—
289C22—202no—
290C22C2×D4×D511yescharacteristic
291C22#414412yes—
292C22C2×Q8×D511yescharacteristic
293C22D5×#4111yescharacteristic
294C22#414912yes—
295C4—22no—
296C4—42no—
297C4—41no—
298C4—42no—
299C4—102no—
300C4—201no—
301C4—202no—
302C4—202no—
303C2—42no—
304C2—104no—
305C2—202no—
306C2#6592311yescharacteristic
307C2D5×C22⋊Q811yescharacteristic
308C2#6596812yes—
309C2#6600811yescharacteristic
310C2#6601212yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#393C2
prime radicalC24×D5C22
hypercenter#393D5
hypocenterC5#2626
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
4. #65520C2semidirect
6. #65238C2semidirect
7. #65237C2semidirect
8. #64966C2semidirect
9. #64965C2semidirect
10. #64957C2semidirect
11. #64956C2semidirect
13. C22×C5×Q8C22semidirect
27. C2×C4⋊Dic5C22semidirect
28. C2×C4⋊Dic5C22semidirect
29. C2×#683C22semidirect
30. C2×#683C22semidirect
31. C2×#683C22semidirect
33. C5×#95C22semidirect
34. C5×#95C22semidirect
35. #3963C22semidirect
36. #3963C22semidirect
43. C23×D5Q8semidirect
82. #393D5semidirect
131. C22×Q8C2×D5semidirect
153. #95C2×D5semidirect
154. #95C2×D5semidirect
273. C5#2626semidirect

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
#655202
#652442
#652382
#652372
#649662
#649652
#649572
#649562
#26265