Finite Group Databasebeta

← #10608498

Subgroups — #10608498

Order 640.

796 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 141 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×#4083C212yesmaximal, complemented
2C2×#4031C212yesmaximal, complemented
3C5×#392C211yesmaximal, characteristic, complemented
4#65512C212yesmaximal, complemented
5#65510C212yesmaximal, complemented
6#65250C212yesmaximal, complemented
7#65244C211yesmaximal, characteristic, complemented
8#65236C211yesmaximal, characteristic, complemented
9#64959C212yesmaximal, complemented
10C24×D5C2211yescharacteristic
11C22×C5×D4C2212yes—
12C22×C5×D4C2211yescharacteristic, complemented
13C22×C22⋊D5C2212yes—
14C22×C22⋊D5C2212yes—
15C22×Dic10C2211yescharacteristic, complemented
16C2×C5×C22⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yescomplemented
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2211yescharacteristic
20C2×C22⋊Dic5C2212yescomplemented
21C2×C22⋊Dic5C2212yescomplemented
22#4083—28no—
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×#685C2212yes—
26C2×#685C2212yes—
27C2×C4⋊Dic5C2212yescomplemented
28C2×C4×Dic5C2212yescomplemented
29#4031—28no—
30C5×#95C2211yescharacteristic, complemented
31#3963C2212yescomplemented
32#3963C2211yescharacteristic, complemented
33#2670—51nomaximal, Sylow
34C24×C5C2312yes—
35C24×C5C2311yescharacteristic
36C23×D5—22no—
37C23×D5—24no—
38C23×D5—42no—
39C23×D5—41no—
40C23×D5—41no—
41C23×D5C2312yes—
42C2×C5×D4—24no—
43C2×C5×D4—24no—
44C2×C5×D4—42no—
45C2×C5×D4—42no—
46C2×C5×D4—41no—
47C22×C4×C5C2311yescharacteristic
48C22×C4×C5C2311yescharacteristic
49C22×C4×C5C2312yes—
50C2×C22⋊D5—24no—
51C2×C22⋊D5—24no—
52C2×C22⋊D5—24no—
53C2×C22⋊D5—42no—
54C2×C22⋊D5—42no—
55C2×C22⋊D5—42no—
56C22×Dic5C2312yes—
57C22×Dic5C2312yes—
58C22×Dic5C2312yes—
59C2×Dic10—24no—
60C2×Dic10—41no—
61C5×C22⋊C4—24no—
62C5×C22⋊C4—41no—
63C22⋊Dic5—24no—
64C22⋊Dic5—24no—
65C22⋊Dic5—41no—
66C22⋊Dic5—42no—
67#685—24no—
68#685—24no—
69#685—42no—
70#685D418yescomplemented
71C4⋊Dic5—42no—
72C4×Dic5—24no—
73C2×#124—52no—
74C2×#120—52no—
75#394—52no—
76#393—51no—
77#392—52no—
78#392D511yescharacteristic, complemented
79#390—52no—
80#388—52no—
81#379—51no—
82C23×C5—22no—
83C23×C5—21no—
84C23×C5—21no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5—22no—
91C23×C5—22no—
92C23×C5—22no—
93C23×C5#4112yes—
94C23×C5C2411yescharacteristic
95C22×D5—22no—
96C22×D5—22no—
97C22×D5—22no—
98C22×D5—22no—
99C22×D5—22no—
100C22×D5—22no—
101C22×D5—44no—
102C22×D5—48no—
103C22×D5—48no—
104C22×D5—44no—
105C22×D5—44no—
106C22×D5C2×D414yes—
107C5×D4—48no—
108C5×D4—44no—
109C2×C4×C5—22no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5—21no—
114C2×C4×C5—21no—
115C2×C4×C5C2×D414yes—
116C22⋊D5—48no—
117C22⋊D5—48no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C2×Dic5—22no—
125C2×Dic5—22no—
126C2×Dic5C2×D414yes—
127Dic10—44no—
128C25—51no—
129C22×Q8—51no—
130C22×D4—52no—
131C22×D4—52no—
132C22×D4C2×D512yes—
133C22×D4C2×D511yescharacteristic, complemented
134#124—108no—
135#120—108no—
136C2×C4⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4—52no—
141C2×C22⋊C4—52no—
142C2×C22⋊C4—51no—
143C2×C22⋊C4—52no—
144C2×C22⋊C4—52no—
145C2×C22⋊C4C2×D511yescharacteristic
146C2×C22⋊C4C2×D512yescomplemented
147C2×C42—52no—
148#95—52no—
149#95—51no—
150#95C2×D511yescharacteristic, complemented
151C22×C5—22no—
152C22×C5—24no—
153C22×C5—42no—
154C22×C5—41no—
155C22×C5—41no—
156C22×C5—42no—
157C22×C5—42no—
158C22×C5—42no—
159C22×C5—42no—
160C22×C5—42no—
161C22×C5—42no—
162C22×C5—42no—
163C22×C5C22×D412yes—
164C22×C5C2×#4111yescharacteristic
165C22×C5H(5,𝔽2)12yes—
166C22×C5H(5,𝔽2)11yescharacteristic
167C22×C5H(5,𝔽2)11yescharacteristic
168C2×D5—24no—
169C2×D5—24no—
170C2×D5—24no—
171C2×D5—416no—
172C2×D5—42no—
173C2×D5—42no—
174C2×D5—42no—
175C2×D5—42no—
176C4×C5—24no—
177C4×C5—41no—
178C4×C5—41no—
179Dic5—24no—
180Dic5—42no—
181Dic5—42no—
182C24—52no—
183C24—104no—
184C24—102no—
185C24—201no—
186C24—202no—
187C24—201no—
188C24C22×D512yes—
189C24C22×D511yescharacteristic
190C2×Q8—104no—
191C2×Q8—201no—
192C2×D4—24no—
193C2×D4—24no—
194C2×D4—42no—
195C2×D4—41no—
196C2×D4—42no—
197C2×D4—104no—
198C2×D4—104no—
199C2×D4—104no—
200C2×D4—202no—
201C2×D4—202no—
202C2×D4—202no—
203C22×C4—52no—
204C22×C4—52no—
205C22×C4—52no—
206C22×C4C22×D511yescharacteristic
207C22×C4C22×D511yescharacteristic
208C22×C4C22×D512yes—
209C4⋊C4—202no—
210C22⋊C4—24no—
211C22⋊C4—41no—
212C22⋊C4—58no—
213C22⋊C4—104no—
214C22⋊C4—104no—
215C22⋊C4—104no—
216C22⋊C4—104no—
217C22⋊C4—202no—
218C22⋊C4—201no—
219C22⋊C4—202no—
220C42—104no—
221C2×C5—42no—
222C2×C5—82no—
223C2×C5#53512yes—
224C2×C5#53811yescharacteristic
225C2×C5D4211yescharacteristic
226C2×C5#54612yes—
227C2×C5H(3,𝔽4)11yescharacteristic
228D5—44no—
229C23—22no—
230C23—21no—
231C23—22no—
232C23—21no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—54no—
241C23—102no—
242C23—102no—
243C23—102no—
244C23—102no—
245C23—102no—
246C23—102no—
247C23—208no—
248C23—204no—
249C23—204no—
250C23—208no—
251C23—204no—
252C23#71112yes—
253C23C23×D511yescharacteristic
254Q8—204no—
255D4—48no—
256D4—44no—
257D4—208no—
258D4—208no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—54no—
266C2×C4—102no—
267C2×C4—102no—
268C2×C4—102no—
269C2×C4—102no—
270C2×C4—102no—
271C2×C4—102no—
272C2×C4—102no—
273C2×C4—102no—
274C2×C4D4×D514yes—
275C5#267011yescharacteristic, complemented, Sylow
276C22—24no—
277C22—22no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—104no—
289C22—104no—
290C22—104no—
291C22—2016no—
292C22—202no—
293C22—202no—
294C22—202no—
295C22—202no—
296C22C2×D4×D512yes—
297C22C2×#71111yescharacteristic
298C22#414411yescharacteristic
299C22#414412yes—
300C22#414911yescharacteristic
301C4—24no—
302C4—41no—
303C4—41no—
304C4—104no—
305C4—202no—
306C4—202no—
307C2—42no—
308C2—82no—
309C2—204no—
310C2#6592512yes—
311C2#6592811yescharacteristic
312C2#6594112yes—
313C2#6595111yescharacteristic
314C2#6605111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#392C2
hypercenter#392D5
hypocenterC5#2670
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#4083C2semidirect
2. C2×#4031C2semidirect
3. C5×#392C2semidirect
4. #65512C2semidirect
5. #65510C2semidirect
6. #65250C2semidirect
7. #65244C2semidirect
8. #65236C2semidirect
9. #64959C2semidirect
12. C22×C5×D4C22semidirect
15. C22×Dic10C22semidirect
17. C2×C5×C22⋊C4C22semidirect
20. C2×C22⋊Dic5C22semidirect
21. C2×C22⋊Dic5C22semidirect
27. C2×C4⋊Dic5C22semidirect
28. C2×C4×Dic5C22semidirect
30. C5×#95C22semidirect
31. #3963C22semidirect
32. #3963C22semidirect
70. #685D4semidirect
78. #392D5semidirect
133. C22×D4C2×D5semidirect
146. C2×C22⋊C4C2×D5semidirect
150. #95C2×D5semidirect
275. C5#2670semidirect

Maximal subgroups

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

subgroupindex
C2×#40832
C2×#40312
C5×#3922
#655122
#655102
#652502
#652442
#652362
#649592
#26705