Finite Group Databasebeta

← #10608505

Subgroups — #10608505

Order 640.

712 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 125 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
2C5×#392C211yesmaximal, characteristic, complemented
3#65512C212yesmaximal, complemented
4#65511C212yesmaximal, complemented
5#65250C212yesmaximal, complemented
6#65248C212yesmaximal, complemented
7#65244C211yesmaximal, characteristic, complemented
8#65242C211yesmaximal, characteristic, complemented
9#64964C212yesmaximal, complemented
10C24×D5C2211yescharacteristic
11C22×C5×D4C2212yes—
12C22×C5×D4C2211yescharacteristic, complemented
13C22×C22⋊D5C2212yes—
14C22×C22⋊D5C2212yes—
15C2×C5×C22⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2212yescomplemented
17C2×C22⋊Dic5C2212yes—
18C2×C22⋊Dic5C2211yescharacteristic
19C2×C22⋊Dic5C2212yescomplemented
20C2×C22⋊Dic5C2212yescomplemented
21#4083—28no—
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×#685C2212yes—
26C2×C4⋊Dic5C2211yescharacteristic, complemented
27C2×#683C2212yescomplemented
28C5×#95C2211yescharacteristic, complemented
29#3963C2212yescomplemented
30#3963C2212yescomplemented
31#3963C2211yescharacteristic, complemented
32#2812—51nomaximal, Sylow
33C24×C5C2312yes—
34C24×C5C2311yescharacteristic
35C23×D5—42no—
36C23×D5—42no—
37C23×D5—41no—
38C23×D5—41no—
39C23×D5D414yescomplemented
40C23×D5C2312yes—
41C2×C5×D4—24no—
42C2×C5×D4—42no—
43C2×C5×D4—42no—
44C2×C5×D4—42no—
45C2×C5×D4—41no—
46C22×C4×C5C2311yescharacteristic
47C22×C4×C5C2311yescharacteristic
48C22×C4×C5C2312yes—
49C2×C22⋊D5—24no—
50C2×C22⋊D5—24no—
51C2×C22⋊D5—42no—
52C2×C22⋊D5—42no—
53C2×C22⋊D5—42no—
54C2×C22⋊D5—42no—
55C22×Dic5C2312yes—
56C22×Dic5C2312yes—
57C22×Dic5C2312yes—
58C5×C22⋊C4—41no—
59C5×C22⋊C4—42no—
60C22⋊Dic5—24no—
61C22⋊Dic5—41no—
62C22⋊Dic5—42no—
63C22⋊Dic5—42no—
64#685—24no—
65#685—24no—
66#685—42no—
67#685—42no—
68C4⋊Dic5—41no—
69#683—42no—
70C2×#120—52no—
71#397—51no—
72#397—52no—
73#394—52no—
74#394—52no—
75#394—52no—
76#392—52no—
77#392D511yescharacteristic, complemented
78#379—51no—
79C23×C5—21no—
80C23×C5—21no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5C2×D412yes—
91C23×C5C2411yescharacteristic
92C22×D5—22no—
93C22×D5—22no—
94C22×D5—22no—
95C22×D5—22no—
96C22×D5—22no—
97C22×D5—22no—
98C22×D5—44no—
99C22×D5—48no—
100C22×D5—44no—
101C22×D5—48no—
102C22×D5—44no—
103C22×D5C2×D414yes—
104C5×D4—48no—
105C5×D4—44no—
106C2×C4×C5—22no—
107C2×C4×C5—22no—
108C2×C4×C5—21no—
109C2×C4×C5—22no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—21no—
113C22⋊D5—48no—
114C22⋊D5—48no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C25—51no—
125C22×D4—52no—
126C22×D4—52no—
127C22×D4C2×D512yes—
128C22×D4C2×D511yescharacteristic, complemented
129#120—108no—
130C2×C4⋊C4—52no—
131C2×C4⋊C4—51no—
132C2×C22⋊C4—52no—
133C2×C22⋊C4—52no—
134C2×C22⋊C4—52no—
135C2×C22⋊C4—51no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4C2×D511yescharacteristic
141C2×C22⋊C4C2×D512yescomplemented
142#95—52no—
143#95—51no—
144#95—52no—
145#95C2×D511yescharacteristic, complemented
146C22×C5—24no—
147C22×C5—22no—
148C22×C5—41no—
149C22×C5—42no—
150C22×C5—41no—
151C22×C5—42no—
152C22×C5—42no—
153C22×C5—42no—
154C22×C5—42no—
155C22×C5—42no—
156C22×C5—42no—
157C22×C5—42no—
158C22×C5C22×D411yescharacteristic
159C22×C5H(5,𝔽2)12yes—
160C22×C5H(5,𝔽2)11yescharacteristic
161C22×C5H(5,𝔽2)12yes—
162C22×C5H(5,𝔽2)11yescharacteristic
163C2×D5—24no—
164C2×D5—24no—
165C2×D5—24no—
166C2×D5—416no—
167C2×D5—42no—
168C2×D5—42no—
169C2×D5—42no—
170C2×D5—42no—
171C4×C5—42no—
172C4×C5—41no—
173C4×C5—41no—
174Dic5—42no—
175Dic5—42no—
176Dic5—42no—
177C24—52no—
178C24—54no—
179C24—201no—
180C24—202no—
181C24—202no—
182C24—201no—
183C24C22×D512yes—
184C24C22×D511yescharacteristic
185C2×D4—24no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—104no—
191C2×D4—104no—
192C2×D4—202no—
193C2×D4—202no—
194C2×D4—202no—
195C2×D4—202no—
196C22×C4—52no—
197C22×C4—52no—
198C22×C4—52no—
199C22×C4C22×D511yescharacteristic
200C22×C4C22×D512yes—
201C22×C4C22×D511yescharacteristic
202C4⋊C4—202no—
203C4⋊C4—201no—
204C22⋊C4—41no—
205C22⋊C4—42no—
206C22⋊C4—104no—
207C22⋊C4—104no—
208C22⋊C4—104no—
209C22⋊C4—202no—
210C22⋊C4—202no—
211C22⋊C4—202no—
212C22⋊C4—201no—
213C22⋊C4—202no—
214C2×C5—42no—
215C2×C5—82no—
216C2×C5#53412yes—
217C2×C5#53411yescharacteristic
218C2×C5#56011yescharacteristic
219C2×C5#56012yes—
220C2×C5H(3,𝔽4)11yescharacteristic
221D5—44no—
222C23—21no—
223C23—22no—
224C23—22no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—54no—
234C23—102no—
235C23—102no—
236C23—102no—
237C23—102no—
238C23—102no—
239C23—102no—
240C23—208no—
241C23—204no—
242C23—204no—
243C23—204no—
244C23—208no—
245C23D4×D512yes—
246C23C23×D511yescharacteristic
247D4—48no—
248D4—44no—
249D4—208no—
250D4—208no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—102no—
259C2×C4—102no—
260C2×C4—102no—
261C2×C4—102no—
262C2×C4—102no—
263C2×C4—102no—
264C2×C4—102no—
265C2×C4—102no—
266C2×C4—102no—
267C5#281211yescharacteristic, complemented, Sylow
268C22—24no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—104no—
281C22—104no—
282C22—104no—
283C22—2016no—
284C22—202no—
285C22—202no—
286C22—202no—
287C22—202no—
288C22C2×D4×D511yescharacteristic
289C22#414412yes—
290C22#414411yescharacteristic
291C22#414412yes—
292C22#414911yescharacteristic
293C4—41no—
294C4—42no—
295C4—41no—
296C4—202no—
297C4—202no—
298C4—202no—
299C2—42no—
300C2—82no—
301C2—204no—
302C2#6592312yes—
303C2#6592811yescharacteristic
304C2#6593911yescharacteristic
305C2#6595412yes—
306C2#6605411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#392C2
hypercenter#392D5
hypocenterC5#2812
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. C5×#392C2semidirect
3. #65512C2semidirect
4. #65511C2semidirect
5. #65250C2semidirect
6. #65248C2semidirect
7. #65244C2semidirect
8. #65242C2semidirect
9. #64964C2semidirect
12. C22×C5×D4C22semidirect
16. C2×C5×C22⋊C4C22semidirect
19. C2×C22⋊Dic5C22semidirect
20. C2×C22⋊Dic5C22semidirect
26. C2×C4⋊Dic5C22semidirect
27. C2×#683C22semidirect
28. C5×#95C22semidirect
29. #3963C22semidirect
30. #3963C22semidirect
31. #3963C22semidirect
39. C23×D5D4semidirect
77. #392D5semidirect
128. C22×D4C2×D5semidirect
141. C2×C22⋊C4C2×D5semidirect
145. #95C2×D5semidirect
267. C5#2812semidirect

Maximal subgroups

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

subgroupindex
C2×#40832
C5×#3922
#655122
#655112
#652502
#652482
#652442
#652422
#649642
#28125