Finite Group Databasebeta

← #10608585

Subgroups — #10608585

Order 640.

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

normal subgroups only

This group has 430 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×#4090C211yesmaximal, characteristic, complemented
2C2×#4025C211yesmaximal, characteristic
3C5×#393C211yesmaximal, characteristic
4#65518C211yesmaximal, characteristic, complemented
5#65241C212yesmaximal
6#65238C212yesmaximal
7#65236C211yesmaximal, characteristic
8#64949C212yesmaximal, complemented
9#64946C212yesmaximal, complemented
10#64945C211yesmaximal, characteristic, complemented
11#64940C211yesmaximal, characteristic, complemented
12C22×C5×Q8C2211yescharacteristic
13C22×Dic10C2211yescharacteristic
14C2×C5×C22⋊C4C2211yescharacteristic
15C2×C5×C22⋊C4C2212yes—
16C2×C22⋊Dic5C2211yescharacteristic
17C2×C22⋊Dic5C2211yescharacteristic
18C2×C22⋊Dic5C2212yes—
19#4090—24no—
20C2×C4⋊Dic5C2211yescharacteristic
21C2×C4⋊Dic5C2212yes—
22C2×#683C2211yescharacteristic
23C2×#683C2211yescharacteristic
24C2×#683C2212yes—
25C2×#683C2212yes—
26C2×#683C2212yes—
27C2×C4×Dic5C2211yescharacteristic
28#4025—24no—
29C5×#95C2211yescharacteristic
30C5×#95C2212yes—
31#3963C2212yes—
32#3963C2212yes—
33#3963C2211yescharacteristic
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2821—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C2×C5×Q8—22no—
40C2×C5×Q8—42no—
41C22×C4×C5C2312yes—
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×Dic5C2312yes—
46C22×Dic5C2312yes—
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2311yescharacteristic
51C2×Dic10—22no—
52C2×Dic10—42no—
53C5×C22⋊C4—22no—
54C5×C22⋊C4—42no—
55C22⋊Dic5—22no—
56C22⋊Dic5—22no—
57C22⋊Dic5—42no—
58C4⋊Dic5—22no—
59C4⋊Dic5—42no—
60#683—22no—
61#683—22no—
62#683—24no—
63#683—24no—
64#683—42no—
65C4×Dic5D414yes—
66C2×C4⋊Q8—51no—
67C2×#126—51no—
68#400—52no—
69#399—52no—
70#398—52no—
71#397—52no—
72#395—51no—
73#395—51no—
74#393—51no—
75#393D511yescharacteristic
76#383—51no—
77C23×C5—21no—
78C23×C5—22no—
79C23×C5—21no—
80C23×C5—21no—
81C23×C5—22no—
82C23×C5C2411yescharacteristic
83C5×Q8—44no—
84C2×C4×C5—22no—
85C2×C4×C5—22no—
86C2×C4×C5—21no—
87C2×C4×C5—22no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5C2×D412yes—
94C2×Dic5—21no—
95C2×Dic5—22no—
96C2×Dic5—21no—
97C2×Dic5—21no—
98C2×Dic5—22no—
99C2×Dic5—21no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—21no—
105C2×Dic5—22no—
106C2×Dic5—21no—
107C2×Dic5—22no—
108C2×Dic5C2×D412yes—
109C2×Dic5C2×D412yes—
110Dic10—44no—
111C22×Q8—51no—
112C22×Q8C2×D511yescharacteristic
113C4⋊Q8—104no—
114#126—104no—
115C2×C4⋊C4—51no—
116C2×C4⋊C4—51no—
117C2×C4⋊C4—51no—
118C2×C4⋊C4—52no—
119C2×C4⋊C4—52no—
120C2×C4⋊C4—52no—
121C2×C4⋊C4—52no—
122C2×C22⋊C4—51no—
123C2×C22⋊C4—51no—
124C2×C22⋊C4—52no—
125C2×C22⋊C4C2×D511yescharacteristic
126C2×C22⋊C4C2×D512yes—
127C2×C42—51no—
128#95—51no—
129#95—52no—
130#95—52no—
131#95—52no—
132#95—52no—
133#95—52no—
134#95C2×D511yescharacteristic
135#95C2×D512yes—
136C22×C5—41no—
137C22×C5—41no—
138C22×C5—41no—
139C22×C5—42no—
140C22×C5—42no—
141C22×C5C22×D411yescharacteristic
142C22×C5H(5,𝔽2)12yes—
143C22×C52−1+411yescharacteristic
144C22×C52−1+411yescharacteristic
145C22×C52−1+412yes—
146C4×C5—22no—
147C4×C5—42no—
148C4×C5—41no—
149C4×C5—42no—
150Dic5—22no—
151Dic5—22no—
152Dic5—42no—
153Dic5—41no—
154Dic5—41no—
155Dic5—42no—
156C24C22×D511yescharacteristic
157C2×Q8—22no—
158C2×Q8—42no—
159C2×Q8—102no—
160C2×Q8—202no—
161C22×C4—51no—
162C22×C4—51no—
163C22×C4—52no—
164C22×C4—51no—
165C22×C4—52no—
166C22×C4—51no—
167C22×C4C22×D512yes—
168C22×C4C22×D511yescharacteristic
169C22×C4C22×D512yes—
170C22×C4C22×D511yescharacteristic
171C4⋊C4—102no—
172C4⋊C4—102no—
173C4⋊C4—104no—
174C4⋊C4—104no—
175C4⋊C4—102no—
176C4⋊C4—202no—
177C4⋊C4—202no—
178C22⋊C4—22no—
179C22⋊C4—42no—
180C22⋊C4—102no—
181C22⋊C4—102no—
182C22⋊C4—202no—
183C42—54no—
184C2×C5—81no—
185C2×C5#53511yescharacteristic
186C2×C5#53611yescharacteristic
187C2×C5#53611yescharacteristic
188C2×C5#56312yes—
189C2×C5#56312yes—
190C23—21no—
191C23—22no—
192C23—21no—
193C23—21no—
194C23—22no—
195C23C23×D511yescharacteristic
196Q8—44no—
197Q8—204no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—52no—
208C2×C4—52no—
209C2×C4—101no—
210C2×C4—101no—
211C2×C4—102no—
212C2×C4—101no—
213C2×C4—101no—
214C2×C4—102no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—101no—
220C2×C4—102no—
221C2×C4—101no—
222C2×C4—102no—
223C2×C4D4×D512yes—
224C5#282111yescharacteristic, complemented, Sylow
225C22—41no—
226C22—41no—
227C22—42no—
228C22—42no—
229C22—41no—
230C22C2×D4×D511yescharacteristic
231C22#414412yes—
232C22#414711yescharacteristic
233C22#415011yescharacteristic
234C22#415012yes—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—41no—
239C4—102no—
240C4—102no—
241C4—202no—
242C4—201no—
243C4—201no—
244C4—202no—
245C2—81no—
246C2#6592611yescharacteristic
247C2#6596211yescharacteristic
248C2#6598012yes—
249C2#6600612yes—
250C2#6600911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#393C2
prime radicalC24×C5C23
hypercenter#393D5
hypocenterC5#2821
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#4090C2semidirect
4. #65518C2semidirect
8. #64949C2semidirect
9. #64946C2semidirect
10. #64945C2semidirect
11. #64940C2semidirect
224. C5#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#40902
C2×#40252
C5×#3932
#655182
#652412
#652382
#652362
#649492
#649462
#649452
#649402
#28215