Finite Group Databasebeta

← #10608385

Subgroups — #10608385

Order 640.

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

normal subgroups only

This group has 512 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×#4092C212yesmaximal, complemented
2C5×#391C211yesmaximal, characteristic, complemented
3#65280C211yesmaximal, characteristic
4#65272C211yesmaximal, characteristic, complemented
5#65229C212yesmaximal, complemented
6#64963C212yesmaximal, complemented
7#64957C212yesmaximal, complemented
8#64954C212yesmaximal
9#64948C212yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C22×D20C2211yescharacteristic
12C22×C4×D5C2212yes—
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C42×C5C2211yescharacteristic
15#4092—28no—
16C2×#685C2212yes—
17C2×#685C2212yes—
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×C4⋊Dic5C2212yes—
22C2×C4⋊Dic5C2211yescharacteristic
23C2×#683C2212yes—
24C2×#683C2212yes—
25C5×#95C2212yes—
26C5×#95C2212yes—
27#3963C2211yescharacteristic
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#3963C2211yescharacteristic
32#2728—51nomaximal, Sylow
33C23×D5C2312yes—
34C2×C5×Q8—41no—
35C2×C5×Q8—41no—
36C2×C5×Q8D414yescomplemented
37C22×C4×C5C2312yes—
38C22×C4×C5C2311yescharacteristic
39C22×C4×C5C2311yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2312yes—
42C22×Dic5C2312yes—
43C22×Dic5C2312yes—
44C22×Dic5C2312yes—
45C2×D20—24no—
46C2×D20—41no—
47C2×C4×D5—24no—
48C2×C4×D5—42no—
49C2×C4×D5—42no—
50C5×C4⋊C4—41no—
51C42×C5—41no—
52#685—24no—
53#685—24no—
54#685—24no—
55#685—24no—
56#685—42no—
57C4⋊Dic5—24no—
58C4⋊Dic5—41no—
59#683—24no—
60#683—24no—
61C2×C22⋊Q8—52no—
62#400—52no—
63#398—51no—
64#397—52no—
65#394—51no—
66#391D511yescharacteristic, complemented
67#388—52no—
68#386—52no—
69#385—52no—
70C23×C5C2411yescharacteristic
71C22×D5—22no—
72C22×D5—22no—
73C22×D5—22no—
74C22×D5—22no—
75C22×D5—22no—
76C22×D5—22no—
77C22×D5#4114yes—
78C5×Q8—44no—
79C2×C4×C5—21no—
80C2×C4×C5—21no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—22no—
85C2×C4×C5—22no—
86C2×C4×C5—21no—
87C2×C4×C5—22no—
88C2×C4×C5—21no—
89C2×C4×C5—22no—
90C2×C4×C5C2×D412yes—
91C2×C4×C5C2×D414yes—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—22no—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5—22no—
100C2×Dic5—22no—
101D20—44no—
102C4×D5—48no—
103C22×Q8C2×D511yescharacteristic
104C22×D4—51no—
105C23×C4—52no—
106C22⋊Q8—108no—
107C2×C4⋊C4—52no—
108C2×C4⋊C4—52no—
109C2×C4⋊C4—52no—
110C2×C4⋊C4—51no—
111C2×C4⋊C4C2×D511yescharacteristic
112C2×C22⋊C4—52no—
113C2×C22⋊C4—52no—
114C2×C22⋊C4—52no—
115C2×C22⋊C4—52no—
116C2×C22⋊C4—52no—
117C2×C42C2×D511yescharacteristic
118#95—52no—
119#95—52no—
120#95—52no—
121#95—51no—
122#95—51no—
123#95C2×D512yes—
124#95C2×D512yes—
125C22×C5C22×D411yescharacteristic
126C22×C5C2×#4112yes—
127C22×C5H(5,𝔽2)11yescharacteristic
128C22×C5H(5,𝔽2)11yescharacteristic
129C22×C52−1+412yes—
130C2×D5—24no—
131C2×D5—24no—
132C2×D5—24no—
133C2×D5—42no—
134C2×D5—42no—
135C2×D5—42no—
136C2×D5—42no—
137C4×C5—24no—
138C4×C5—22no—
139C4×C5—42no—
140C4×C5—41no—
141C4×C5—41no—
142Dic5—42no—
143Dic5—42no—
144Dic5—42no—
145C24—52no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8D2014yescomplemented
149C2×D4—104no—
150C2×D4—201no—
151C22×C4—52no—
152C22×C4—52no—
153C22×C4—52no—
154C22×C4—104no—
155C22×C4—202no—
156C22×C4—202no—
157C22×C4C22×D511yescharacteristic
158C22×C4C22×D512yes—
159C22×C4C22×D511yescharacteristic
160C22×C4C22×D512yes—
161C22×C4C22×D511yescharacteristic
162C4⋊C4—41no—
163C4⋊C4—104no—
164C4⋊C4—104no—
165C4⋊C4—104no—
166C4⋊C4—201no—
167C22⋊C4—104no—
168C22⋊C4—104no—
169C22⋊C4—104no—
170C22⋊C4—104no—
171C22⋊C4—202no—
172C42—41no—
173C2×C5#53411yescharacteristic
174C2×C5#53912yes—
175C2×C5#54812yes—
176C2×C5#55111yescharacteristic
177C2×C5#56311yescharacteristic
178D5—44no—
179C23—54no—
180C23—102no—
181C23—102no—
182C23—102no—
183C23—102no—
184C23—102no—
185C23—102no—
186C23C23×D511yescharacteristic
187Q8—44no—
188D4—204no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—102no—
201C2×C4—102no—
202C2×C4—102no—
203C2×C4—102no—
204C2×C4—102no—
205C2×C4—102no—
206C2×C4—102no—
207C2×C4—102no—
208C2×C4—102no—
209C2×C4—208no—
210C2×C4C2×D2012yes—
211C2×C4C2×D2014yes—
212C5#272811yescharacteristic, complemented, Sylow
213C22—104no—
214C22—104no—
215C22—104no—
216C22—202no—
217C22—202no—
218C22—202no—
219C22—202no—
220C22C22×D2011yescharacteristic
221C22#414411yescharacteristic
222C22#414411yescharacteristic
223C22#414712yes—
224C22D5×#4112yes—
225C4—24no—
226C4—22no—
227C4—42no—
228C4—41no—
229C4—41no—
230C4—202no—
231C4—202no—
232C4—202no—
233C2—204no—
234C2#6582711yescharacteristic
235C2#6591012yes—
236C2#6597212yes—
237C2#6601311yescharacteristic
238C2#6606911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#391C2
prime radicalC22×D20C22
hypercenter#391D5
hypocenterC5#2728
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#4092C2semidirect
2. C5×#391C2semidirect
4. #65272C2semidirect
5. #65229C2semidirect
6. #64963C2semidirect
7. #64957C2semidirect
9. #64948C2semidirect
36. C2×C5×Q8D4semidirect
66. #391D5semidirect
148. C2×Q8D20semidirect
212. C5#2728semidirect

Maximal subgroups

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

subgroupindex
C2×#40922
C5×#3912
#652802
#652722
#652292
#649632
#649572
#649542
#649482
#27285