Finite Group Databasebeta

← #10608535

Subgroups — #10608535

Order 640.

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

normal subgroups only

This group has 520 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×Q8×Dic5C211yesmaximal, characteristic, complemented
2C2×#4031C211yesmaximal, characteristic
3C5×#393C211yesmaximal, characteristic, complemented
4#65520C211yesmaximal, characteristic, complemented
5#65246C212yesmaximal
6#65242C211yesmaximal, characteristic, complemented
7#65240C212yesmaximal, complemented
8#64963C211yesmaximal, characteristic, complemented
9#64959C212yesmaximal, complemented
10#64938C211yesmaximal, characteristic, complemented
11#64935C212yesmaximal, complemented
12C22×C5×Q8C2211yescharacteristic
13C22×C22⋊D5C2211yescharacteristic
14C22×Dic10C2211yescharacteristic
15C2×C5×C22⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2212yes—
17C2×C22⋊Dic5C2211yescharacteristic
18C2×C22⋊Dic5C2212yes—
19Q8×Dic5—24no—
20C2×#685C2211yescharacteristic
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×C4⋊Dic5C2212yes—
25C2×C4⋊Dic5C2211yescharacteristic
26C2×#683C2212yes—
27C2×C4×Dic5C2212yes—
28C2×C4×Dic5C2212yes—
29C2×C4×Dic5C2211yescharacteristic
30#4031—24no—
31C5×#95C2212yes—
32C5×#95C2211yescharacteristic
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2211yescharacteristic
37#2592—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C2×C5×Q8—22no—
41C2×C5×Q8—24no—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2312yes—
46C2×C22⋊D5—22no—
47C2×C22⋊D5—24no—
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2312yes—
50C22×Dic5C2312yes—
51C22×Dic5C2311yescharacteristic
52C22×Dic5C2311yescharacteristic
53C2×Dic10—22no—
54C2×Dic10—24no—
55C5×C22⋊C4—22no—
56C5×C22⋊C4—24no—
57C22⋊Dic5—22no—
58C22⋊Dic5—42no—
59#685—22no—
60#685—24no—
61#685—22no—
62#685—42no—
63C4⋊Dic5—22no—
64C4⋊Dic5—24no—
65#683—24no—
66C4×Dic5—24no—
67C4×Dic5—42no—
68C4×Dic5D414yes—
69C2×#124—51no—
70C2×C4×Q8—51no—
71#397—51no—
72#397—51no—
73#393—51no—
74#393D511yescharacteristic, complemented
75#391—52no—
76#390—52no—
77#388—52no—
78#388—52no—
79C4×C4⋊C4—51no—
80C23×C5—21no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5—21no—
84C23×C5—21no—
85C23×C5C2411yescharacteristic
86C22×D5—21no—
87C22×D5—22no—
88C22×D5—22no—
89C22×D5—21no—
90C22×D5—21no—
91C5×Q8—28no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—21no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5C2×D412yes—
101C2×C4×C5#4114yes—
102C22⋊D5—44no—
103C2×Dic5—21no—
104C2×Dic5—21no—
105C2×Dic5—22no—
106C2×Dic5—21no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5C2×D412yes—
114C2×Dic5C2×D412yes—
115C2×Dic5#4114yes—
116Dic10—44no—
117C22×Q8—51no—
118C22×Q8C2×D511yescharacteristic
119C22×D4—51no—
120#124—104no—
121C4×Q8—104no—
122C2×C4⋊C4—52no—
123C2×C4⋊C4—51no—
124C2×C4⋊C4—52no—
125C2×C22⋊C4—51no—
126C2×C22⋊C4—51no—
127C2×C22⋊C4—51no—
128C2×C22⋊C4—52no—
129C2×C22⋊C4—52no—
130C2×C22⋊C4—52no—
131C2×C22⋊C4C2×D511yescharacteristic
132C2×C22⋊C4C2×D512yes—
133C2×C42—51no—
134C2×C42—52no—
135C2×C42—52no—
136#95—51no—
137#95—52no—
138#95—52no—
139#95—52no—
140#95C2×D511yescharacteristic
141#95C2×D512yes—
142C22×C5—41no—
143C22×C5—41no—
144C22×C5—41no—
145C22×C5—42no—
146C22×C5—42no—
147C22×C5C22×D411yescharacteristic
148C22×C5C2×#4112yes—
149C22×C5C2×#4112yes—
150C22×C5H(5,𝔽2)11yescharacteristic
151C22×C52−1+411yescharacteristic
152C2×D5—41no—
153C2×D5—41no—
154C2×D5—41no—
155C2×D5—42no—
156C2×D5—42no—
157C4×C5—22no—
158C4×C5—24no—
159C4×C5—41no—
160C4×C5—42no—
161Dic5—22no—
162Dic5—41no—
163Dic5—42no—
164Dic5—42no—
165Dic5#12414yes—
166C24—51no—
167C24C22×D511yescharacteristic
168C2×Q8—22no—
169C2×Q8—24no—
170C2×Q8—102no—
171C2×Q8—104no—
172C2×D4—102no—
173C2×D4—104no—
174C22×C4—51no—
175C22×C4—51no—
176C22×C4—51no—
177C22×C4—52no—
178C22×C4—52no—
179C22×C4C22×D511yescharacteristic
180C22×C4C22×D512yes—
181C22×C4C22×D511yescharacteristic
182C22×C4C22×D512yes—
183C4⋊C4—104no—
184C4⋊C4—104no—
185C4⋊C4—102no—
186C22⋊C4—22no—
187C22⋊C4—24no—
188C22⋊C4—102no—
189C22⋊C4—102no—
190C22⋊C4—102no—
191C22⋊C4—104no—
192C22⋊C4—202no—
193C22⋊C4—202no—
194C42—54no—
195C42—104no—
196C42—202no—
197C2×C5—81no—
198C2×C5#52511yescharacteristic
199C2×C5C2×#12411yescharacteristic
200C2×C5#53711yescharacteristic
201C2×C5#55612yes—
202C2×C5#55912yes—
203D5—81no—
204C23—21no—
205C23—22no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—101no—
210C23—102no—
211C23—101no—
212C23—102no—
213C23—101no—
214C23C23×D511yescharacteristic
215Q8—28no—
216Q8—204no—
217D4—204no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—52no—
227C2×C4—54no—
228C2×C4—52no—
229C2×C4—101no—
230C2×C4—101no—
231C2×C4—102no—
232C2×C4—101no—
233C2×C4—102no—
234C2×C4—102no—
235C2×C4—102no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4—102no—
239C2×C4D4×D512yes—
240C2×C4#71114yes—
241C5#259211yescharacteristic, complemented, Sylow
242C22—41no—
243C22—41no—
244C22—42no—
245C22—41no—
246C22—42no—
247C22—201no—
248C22—201no—
249C22—202no—
250C22—201no—
251C22—202no—
252C22C2×D4×D511yescharacteristic
253C22C2×#71112yes—
254C22#414711yescharacteristic
255C22D5×#4112yes—
256C22#414911yescharacteristic
257C4—22no—
258C4—24no—
259C4—41no—
260C4—42no—
261C4—54no—
262C4—102no—
263C4—202no—
264C4—201no—
265C4—202no—
266C2—81no—
267C2—401no—
268C2#6592011yescharacteristic
269C2#6596311yescharacteristic
270C2#6597512yes—
271C2D5×#12411yescharacteristic
272C2#6601512yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#393C2
prime radicalC22×C22⋊D5C22
hypercenter#393D5
hypocenterC5#2592
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic5C2semidirect
3. C5×#393C2semidirect
4. #65520C2semidirect
6. #65242C2semidirect
7. #65240C2semidirect
8. #64963C2semidirect
9. #64959C2semidirect
10. #64938C2semidirect
11. #64935C2semidirect
74. #393D5semidirect
241. C5#2592semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic52
C2×#40312
C5×#3932
#655202
#652462
#652422
#652402
#649632
#649592
#649382
#649352
#25925