Finite Group Databasebeta

← #10605861

Subgroups — #10605861

Order 640.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#685C211yesmaximal, characteristic
2C2×D5×C22⋊C4C211yesmaximal, characteristic
3C5×#381C211yesmaximal, characteristic, complemented
4#65500C211yesmaximal, characteristic, complemented
5#65244C211yesmaximal, characteristic
6#65239C211yesmaximal, characteristic, complemented
7#65235C211yesmaximal, characteristic, complemented
8#64964C212yesmaximal, complemented
9#64963C212yesmaximal, complemented
10#64957C212yesmaximal, complemented
11#64953C212yesmaximal, complemented
12C24×D5C2211yescharacteristic
13C23×C4×C5C2211yescharacteristic
14C23×Dic5C2211yescharacteristic
15C22×C4×D5C2211yescharacteristic
16C22×C4×D5C2211yescharacteristic
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C22⋊Dic5C2211yescharacteristic
20C2×C22⋊Dic5C2211yescharacteristic
21C2×#685—24no—
22C2×#685C2211yescharacteristic
23C2×#685C2211yescharacteristic
24C2×#685C2212yes—
25C2×#685C2212yes—
26C2×#685C2212yes—
27C2×#685C2212yes—
28C2×C4⋊Dic5C2212yescomplemented
29C2×#683C2212yescomplemented
30D5×C22⋊C4—24no—
31C5×#95C2212yescomplemented
32C5×#95C2212yescomplemented
33#3963C2212yescomplemented
34#3963C2212yescomplemented
35#3963C2212yescomplemented
36#3963C2212yescomplemented
37#2483—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5—22no—
40C23×D5—22no—
41C23×D5—24no—
42C23×D5—42no—
43C23×D5D414yes—
44C23×D5C2311yescharacteristic
45C23×D5C2311yescharacteristic
46C22×C4×C5—22no—
47C22×C4×C5—22no—
48C22×C4×C5—22no—
49C22×C4×C5C2311yescharacteristic
50C22×C4×C5C2311yescharacteristic
51C22×C4×C5C2312yes—
52C22×C4×C5C2312yes—
53C22×Dic5—22no—
54C22×Dic5—42no—
55C22×Dic5C2311yescharacteristic
56C22×Dic5C2311yescharacteristic
57C22×Dic5C2312yes—
58C22×Dic5C2312yes—
59C2×C4×D5—22no—
60C2×C4×D5—22no—
61C2×C4×D5—24no—
62C2×C4×D5—42no—
63C5×C22⋊C4—22no—
64C5×C22⋊C4—22no—
65C22⋊Dic5—22no—
66C22⋊Dic5—22no—
67#685—22no—
68#685—22no—
69#685—24no—
70#685—24no—
71#685—24no—
72#685—42no—
73C4⋊Dic5—42no—
74#683—42no—
75C22×C22⋊C4—51no—
76C22×C22⋊C4—51no—
77#397—52no—
78#397—52no—
79#385—51no—
80#385—51no—
81#385—52no—
82#381—51no—
83#381—52no—
84#381D511yescharacteristic, complemented
85#379—51no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5C2×D412yes—
89C23×C5#4112yes—
90C23×C5#4112yes—
91C23×C5C2411yescharacteristic
92C22×D5—21no—
93C22×D5—21no—
94C22×D5—22no—
95C22×D5—28no—
96C22×D5—21no—
97C22×D5—21no—
98C22×D5—22no—
99C22×D5—22no—
100C22×D5—28no—
101C22×D5—44no—
102C22×D5—48no—
103C22×D5—48no—
104C22×D5C2×D412yes—
105C22×D5C2×D412yes—
106C22×D5#4114yes—
107C2×C4×C5—22no—
108C2×C4×C5—21no—
109C2×C4×C5—21no—
110C2×C4×C5—22no—
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×C5—44no—
118C2×Dic5—22no—
119C2×Dic5—21no—
120C2×Dic5—21no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C2×Dic5—22no—
125C2×Dic5—22no—
126C2×Dic5—22no—
127C2×Dic5—22no—
128C2×Dic5—44no—
129C4×D5—44no—
130C4×D5—44no—
131C25—51no—
132C23×C4—51no—
133C23×C4—51no—
134C23×C4—51no—
135C23×C4C2×D511yescharacteristic
136C2×C4⋊C4—52no—
137C2×C4⋊C4—52no—
138C2×C22⋊C4—51no—
139C2×C22⋊C4—51no—
140C2×C22⋊C4—52no—
141C2×C22⋊C4—51no—
142C2×C22⋊C4—52no—
143C2×C22⋊C4—51no—
144C2×C22⋊C4—52no—
145C2×C22⋊C4—52no—
146C2×C22⋊C4—104no—
147C2×C22⋊C4—104no—
148C2×C22⋊C4C2×D511yescharacteristic
149C2×C22⋊C4C2×D511yescharacteristic
150#95—52no—
151#95—52no—
152#95—52no—
153#95—52no—
154#95C2×D512yescomplemented
155#95C2×D512yescomplemented
156C22×C5—22no—
157C22×C5—22no—
158C22×C5—24no—
159C22×C5—24no—
160C22×C5#12314yes—
161C22×C5C22×D411yescharacteristic
162C22×C5C2×#4111yescharacteristic
163C22×C5C2×#4111yescharacteristic
164C22×C5C2×#4112yes—
165C22×C5H(5,𝔽2)12yes—
166C2×D5—22no—
167C2×D5—22no—
168C2×D5—24no—
169C2×D5—24no—
170C2×D5—22no—
171C2×D5—22no—
172C2×D5—22no—
173C2×D5—416no—
174C2×D5—42no—
175C2×D5—42no—
176C2×D5#12314yes—
177C4×C5—41no—
178C4×C5—42no—
179C4×C5—41no—
180C4×C5—42no—
181Dic5—41no—
182Dic5—41no—
183Dic5—42no—
184Dic5—42no—
185C24—51no—
186C24—51no—
187C24—54no—
188C24—104no—
189C24—102no—
190C24—102no—
191C24—202no—
192C24C22×D511yescharacteristic
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—51no—
197C22×C4—52no—
198C22×C4—51no—
199C22×C4—52no—
200C22×C4—102no—
201C22×C4—102no—
202C22×C4—104no—
203C22×C4—102no—
204C22×C4—202no—
205C22×C4—202no—
206C22×C4C22×D512yes—
207C22×C4C22×D511yescharacteristic
208C22×C4C22×D512yes—
209C22×C4C22×D511yescharacteristic
210C4⋊C4—202no—
211C4⋊C4—202no—
212C22⋊C4—22no—
213C22⋊C4—22no—
214C22⋊C4—102no—
215C22⋊C4—102no—
216C22⋊C4—104no—
217C22⋊C4—104no—
218C22⋊C4—104no—
219C22⋊C4—102no—
220C22⋊C4—102no—
221C22⋊C4—202no—
222C2×C5—24no—
223C2×C5C2×#12311yescharacteristic
224C2×C5C2×#12311yescharacteristic
225C2×C5#53411yescharacteristic
226C2×C5#55112yes—
227C2×C5#55112yes—
228D5—24no—
229D5—42no—
230C23—22no—
231C23—22no—
232C23—52no—
233C23—54no—
234C23—52no—
235C23—108no—
236C23—101no—
237C23—101no—
238C23—102no—
239C23—101no—
240C23—101no—
241C23—102no—
242C23—102no—
243C23—108no—
244C23—204no—
245C23—208no—
246C23—208no—
247C23#70912yes—
248C23D4×D512yes—
249C23#71112yes—
250C23C23×D511yescharacteristic
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—44no—
262C2×C4—102no—
263C2×C4—101no—
264C2×C4—102no—
265C2×C4—101no—
266C2×C4—102no—
267C2×C4—102no—
268C2×C4—102no—
269C2×C4—102no—
270C2×C4—102no—
271C2×C4—102no—
272C2×C4—204no—
273C2×C4—204no—
274C2×C4—204no—
275C5#248311yescharacteristic, complemented, Sylow
276C22—24no—
277C22—22no—
278C22—22no—
279C22—24no—
280C22—54no—
281C22—102no—
282C22—104no—
283C22—102no—
284C22—104no—
285C22—102no—
286C22—102no—
287C22—102no—
288C22—2016no—
289C22—202no—
290C22—202no—
291C22#402914yes—
292C22C2×#70911yescharacteristic
293C22C2×D4×D511yescharacteristic
294C22C2×#71111yescharacteristic
295C22D5×#4112yes—
296C22#414912yes—
297C4—42no—
298C4—41no—
299C4—41no—
300C4—42no—
301C4—202no—
302C4—201no—
303C4—201no—
304C4—202no—
305C2—24no—
306C2—104no—
307C2—202no—
308C2C2×#402911yescharacteristic
309C2#6586112yes—
310C2D5×#12311yescharacteristic
311C2#6598711yescharacteristic
312C2#6599212yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#381C2
prime radicalC24×D5C22
hypercenter#381D5
hypocenterC5#2483
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. C5×#381C2semidirect
4. #65500C2semidirect
6. #65239C2semidirect
7. #65235C2semidirect
8. #64964C2semidirect
9. #64963C2semidirect
10. #64957C2semidirect
11. #64953C2semidirect
28. C2×C4⋊Dic5C22semidirect
29. C2×#683C22semidirect
31. C5×#95C22semidirect
32. C5×#95C22semidirect
33. #3963C22semidirect
34. #3963C22semidirect
35. #3963C22semidirect
36. #3963C22semidirect
84. #381D5semidirect
154. #95C2×D5semidirect
155. #95C2×D5semidirect
275. C5#2483semidirect

Maximal subgroups

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

subgroupindex
C22×#6852
C2×D5×C22⋊C42
C5×#3812
#655002
#652442
#652392
#652352
#649642
#649632
#649572
#649532
#24835