Finite Group Databasebeta

← #10606649

Subgroups — #10606649

Order 640.

688 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 137 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×#4083C211yesmaximal, characteristic
2C2×#4082C211yesmaximal, characteristic, complemented
3C5×#386C211yesmaximal, characteristic, complemented
4#65502C212yesmaximal, complemented
5#65250C212yesmaximal, complemented
6#65244C212yesmaximal, complemented
7#65242C212yesmaximal, complemented
8#64962C212yesmaximal, complemented
9#64959C212yesmaximal, complemented
10C24×D5C2211yescharacteristic
11C22×C5×D4C2211yescharacteristic
12C23×C4×C5C2211yescharacteristic, complemented
13C22×C22⋊D5C2212yes—
14C22×Dic10C2211yescharacteristic, complemented
15C2×C5×C22⋊C4C2212yes—
16C2×C5×C22⋊C4C2211yescharacteristic, complemented
17C2×C22⋊Dic5C2212yes—
18C2×C22⋊Dic5C2211yescharacteristic
19C2×C22⋊Dic5C2212yescomplemented
20#4083—24no—
21#4082—24no—
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×#685C2212yes—
26C2×#685C2212yes—
27C2×C4⋊Dic5C2212yescomplemented
28C2×C4×Dic5C2211yescharacteristic, complemented
29C5×#95C2212yescomplemented
30#3963C2212yescomplemented
31#3963C2212yescomplemented
32#3963C2212yescomplemented
33#2684—51nomaximal, Sylow
34C24×C5C2311yescharacteristic
35C24×C5C2311yescharacteristic
36C23×D5—22no—
37C23×D5—24no—
38C23×D5—41no—
39C23×D5—41no—
40C23×D5—42no—
41C23×D5C2312yes—
42C2×C5×D4—41no—
43C2×C5×D4—41no—
44C2×C5×D4D414yes—
45C22×C4×C5—41no—
46C22×C4×C5—42no—
47C22×C4×C5C2312yes—
48C22×C4×C5C2312yes—
49C22×C4×C5C2311yescharacteristic
50C2×C22⋊D5—24no—
51C2×C22⋊D5—24no—
52C2×C22⋊D5—42no—
53C22×Dic5C2312yes—
54C22×Dic5C2312yes—
55C22×Dic5C2312yes—
56C2×Dic10—22no—
57C2×Dic10—41no—
58C2×Dic10—41no—
59C5×C22⋊C4—42no—
60C5×C22⋊C4—41no—
61C22⋊Dic5—24no—
62C22⋊Dic5—22no—
63C22⋊Dic5—24no—
64#685—24no—
65#685—24no—
66#685—42no—
67#685—42no—
68#685—42no—
69C4⋊Dic5—42no—
70C4×Dic5—22no—
71C2×#124—51no—
72C2×#120—51no—
73#397—52no—
74#394—52no—
75#393—52no—
76#388—52no—
77#386—52no—
78#386D511yescharacteristic, complemented
79#379—52no—
80C23×C5—22no—
81C23×C5—22no—
82C23×C5—21no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5—21no—
86C23×C5—21no—
87C23×C5—21no—
88C23×C5C2×D412yes—
89C23×C5C2×D412yes—
90C23×C5C2411yescharacteristic
91C22×D5—22no—
92C22×D5—22no—
93C22×D5—22no—
94C22×D5—22no—
95C22×D5—22no—
96C22×D5—22no—
97C22×D5—44no—
98C22×D5—48no—
99C22×D5—48no—
100C22×D5—44no—
101C22×D5—44no—
102C22×D5#4114yes—
103C5×D4—44no—
104C2×C4×C5—21no—
105C2×C4×C5—21no—
106C2×C4×C5—22no—
107C2×C4×C5—22no—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—44no—
113C2×C4×C5C2×D412yes—
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—
124Dic10—44no—
125C25—51no—
126C22×Q8—51no—
127C22×D4—52no—
128C22×D4C2×D511yescharacteristic
129C23×C4C2×D511yescharacteristic, complemented
130#124—104no—
131#120—104no—
132C2×C4⋊C4—52no—
133C2×C22⋊C4—51no—
134C2×C22⋊C4—52no—
135C2×C22⋊C4—52no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4—52no—
141C2×C22⋊C4C2×D512yes—
142C2×C22⋊C4C2×D511yescharacteristic, complemented
143C2×C42—51no—
144#95—52no—
145#95—52no—
146#95—52no—
147#95C2×D512yescomplemented
148C22×C5—22no—
149C22×C5—22no—
150C22×C5—24no—
151C22×C5—24no—
152C22×C5—42no—
153C22×C5—42no—
154C22×C5—41no—
155C22×C5—41no—
156C22×C5—41no—
157C22×C5—41no—
158C22×C5C22×D411yescharacteristic
159C22×C5C2×#4112yes—
160C22×C5H(5,𝔽2)11yescharacteristic
161C22×C5H(5,𝔽2)11yescharacteristic
162C22×C5H(5,𝔽2)12yes—
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—22no—
172C4×C5—42no—
173C4×C5—42no—
174Dic5—42no—
175Dic5—42no—
176Dic5—42no—
177C24—52no—
178C24—104no—
179C24—102no—
180C24—202no—
181C24—201no—
182C24—201no—
183C24C22×D511yescharacteristic
184C24C22×D511yescharacteristic
185C2×Q8—102no—
186C2×Q8—201no—
187C2×Q8—201no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—104no—
191C2×D4—104no—
192C2×D4—202no—
193C2×D4D2014yes—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—52no—
197C22×C4—52no—
198C22×C4—52no—
199C22×C4C22×D512yes—
200C22×C4C22×D512yes—
201C22×C4C22×D511yescharacteristic
202C4⋊C4—202no—
203C22⋊C4—42no—
204C22⋊C4—41no—
205C22⋊C4—104no—
206C22⋊C4—102no—
207C22⋊C4—104no—
208C22⋊C4—104no—
209C22⋊C4—104no—
210C22⋊C4—202no—
211C22⋊C4—202no—
212C22⋊C4—202no—
213C42—102no—
214C2×C5—42no—
215C2×C5—42no—
216C2×C5#53411yescharacteristic
217C2×C5#53812yes—
218C2×C5#54612yes—
219C2×C5#55111yescharacteristic
220C2×C5H(3,𝔽4)11yescharacteristic
221D5—44no—
222C23—21no—
223C23—22no—
224C23—21no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—22no—
230C23—54no—
231C23—102no—
232C23—102no—
233C23—102no—
234C23—102no—
235C23—102no—
236C23—102no—
237C23—208no—
238C23—204no—
239C23—204no—
240C23—208no—
241C23—204no—
242C23C2×D2012yes—
243C23C2×D2012yes—
244C23C23×D511yescharacteristic
245Q8—204no—
246D4—44no—
247D4—208no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—44no—
257C2×C4—102no—
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×C4C2×D2012yes—
267C5#268411yescharacteristic, complemented, Sylow
268C22—22no—
269C22—22no—
270C22—24no—
271C22—24no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—104no—
279C22—104no—
280C22—104no—
281C22—2016no—
282C22—202no—
283C22—202no—
284C22—202no—
285C22—202no—
286C22C22×D2011yescharacteristic
287C22#414411yescharacteristic
288C22#414411yescharacteristic
289C22#414412yes—
290C22D5×#4112yes—
291C4—22no—
292C4—42no—
293C4—42no—
294C4—202no—
295C4—202no—
296C4—202no—
297C2—42no—
298C2—42no—
299C2—204no—
300C2#6582711yescharacteristic
301C2#6588812yes—
302C2#6592811yescharacteristic
303C2#6594712yes—
304C2#6599311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#386C2
prime radicalC2×#4083C2
hypercenter#386D5
hypocenterC5#2684
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#4082C2semidirect
3. C5×#386C2semidirect
4. #65502C2semidirect
5. #65250C2semidirect
6. #65244C2semidirect
7. #65242C2semidirect
8. #64962C2semidirect
9. #64959C2semidirect
12. C23×C4×C5C22semidirect
14. C22×Dic10C22semidirect
16. C2×C5×C22⋊C4C22semidirect
19. C2×C22⋊Dic5C22semidirect
27. C2×C4⋊Dic5C22semidirect
28. C2×C4×Dic5C22semidirect
29. C5×#95C22semidirect
30. #3963C22semidirect
31. #3963C22semidirect
32. #3963C22semidirect
78. #386D5semidirect
129. C23×C4C2×D5semidirect
142. C2×C22⋊C4C2×D5semidirect
147. #95C2×D5semidirect
267. C5#2684semidirect

Maximal subgroups

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

subgroupindex
C2×#40832
C2×#40822
C5×#3862
#655022
#652502
#652442
#652422
#649622
#649592
#26845