Finite Group Databasebeta

← #10608735

Subgroups — #10608735

Order 640.

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

normal subgroups only

This group has 568 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×#4081C212yesmaximal
2C2×#4024C212yesmaximal, complemented
3C5×#394C211yesmaximal, characteristic
4#65268C211yesmaximal, characteristic, complemented
5#65265C211yesmaximal, characteristic, complemented
6#65241C212yesmaximal, complemented
7#65239C212yesmaximal, complemented
8#65233C212yesmaximal, complemented
9#64946C212yesmaximal, complemented
10C22×C5×D4C2211yescharacteristic
11C23×Dic5C2212yes—
12C22×Dic10C2211yescharacteristic
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C5×C22⋊C4C2212yes—
15C2×C5×C22⋊C4C2212yes—
16C2×C22⋊Dic5C2212yes—
17C2×C22⋊Dic5C2212yes—
18C2×C22⋊Dic5C2212yes—
19#4081—28no—
20C2×C4⋊Dic5C2212yes—
21C2×C4⋊Dic5C2211yescharacteristic
22C2×#683C2212yes—
23C2×#683C2212yes—
24C2×#683C2212yes—
25C2×#683C2212yes—
26C2×C4×Dic5C2211yescharacteristic
27#4024—28no—
28C5×#95C2211yescharacteristic
29#3963C2212yes—
30#3963C2212yes—
31#3963C2212yes—
32#3963C2211yescharacteristic
33#2680—51nomaximal, Sylow
34C24×C5C2312yes—
35C2×C5×D4—24no—
36C2×C5×D4—41no—
37C22×C4×C5C2311yescharacteristic
38C22×C4×C5C2312yes—
39C22×C4×C5C2311yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C22×Dic5—24no—
42C22×Dic5—24no—
43C22×Dic5—42no—
44C22×Dic5C2312yes—
45C22×Dic5C2312yes—
46C22×Dic5C2312yes—
47C22×Dic5C2312yes—
48C2×Dic10—24no—
49C2×Dic10—41no—
50C5×C4⋊C4—41no—
51C5×C22⋊C4—24no—
52C5×C22⋊C4—24no—
53C22⋊Dic5—24no—
54C22⋊Dic5—24no—
55C22⋊Dic5D418yes—
56C4⋊Dic5—24no—
57C4⋊Dic5—41no—
58#683—24no—
59#683—24no—
60#683—24no—
61#683—24no—
62C4×Dic5—22no—
63C2×#123—52no—
64C2×C22⋊Q8—52no—
65#398—52no—
66#397—52no—
67#395—51no—
68#394D511yescharacteristic
69#388—52no—
70#385—52no—
71#384—51no—
72C23×C5—22no—
73C23×C5—22no—
74C23×C5—22no—
75C23×C5—22no—
76C23×C5—22no—
77C23×C5—22no—
78C23×C5C2×D414yes—
79C23×C5C2411yescharacteristic
80C5×D4—44no—
81C2×C4×C5—21no—
82C2×C4×C5—21no—
83C2×C4×C5—22no—
84C2×C4×C5—22no—
85C2×C4×C5—22no—
86C2×C4×C5—22no—
87C2×C4×C5—21no—
88C2×C4×C5—21no—
89C2×C4×C5—22no—
90C2×C4×C5#4112yes—
91C2×Dic5—22no—
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—
101C2×Dic5—48no—
102C2×Dic5C2×D414yes—
103C2×Dic5C2×D414yes—
104Dic10—44no—
105C22×Q8—51no—
106C22×D4C2×D511yescharacteristic
107C23×C4—52no—
108#123—108no—
109C22⋊Q8—108no—
110C2×C4⋊C4—52no—
111C2×C4⋊C4—52no—
112C2×C4⋊C4—52no—
113C2×C4⋊C4—52no—
114C2×C4⋊C4—52no—
115C2×C4⋊C4—51no—
116C2×C4⋊C4C2×D511yescharacteristic
117C2×C22⋊C4—52no—
118C2×C22⋊C4—52no—
119C2×C22⋊C4—52no—
120C2×C22⋊C4C2×D512yes—
121C2×C22⋊C4C2×D512yes—
122C2×C42—51no—
123#95—52no—
124#95—52no—
125#95—52no—
126#95—51no—
127#95C2×D511yescharacteristic
128C22×C5—24no—
129C22×C5—24no—
130C22×C5—24no—
131C22×C5—42no—
132C22×C5—42no—
133C22×C5—42no—
134C22×C5—42no—
135C22×C5C22×D412yes—
136C22×C5C2×#4111yescharacteristic
137C22×C5H(5,𝔽2)11yescharacteristic
138C22×C52−1+411yescharacteristic
139C22×C52−1+412yes—
140C4×C5—22no—
141C4×C5—41no—
142C4×C5—41no—
143C4×C5—42no—
144Dic5—24no—
145Dic5—24no—
146Dic5—42no—
147Dic5—42no—
148C24C22×D512yes—
149C2×Q8—104no—
150C2×Q8—201no—
151C2×D4—24no—
152C2×D4—41no—
153C22×C4—52no—
154C22×C4—52no—
155C22×C4—52no—
156C22×C4—52no—
157C22×C4—104no—
158C22×C4—104no—
159C22×C4—202no—
160C22×C4C22×D512yes—
161C22×C4C22×D511yescharacteristic
162C22×C4C22×D511yescharacteristic
163C22×C4C22×D511yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—104no—
166C4⋊C4—104no—
167C4⋊C4—104no—
168C4⋊C4—104no—
169C4⋊C4—104no—
170C4⋊C4—201no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—58no—
174C22⋊C4—104no—
175C22⋊C4—104no—
176C42—102no—
177C2×C5—44no—
178C2×C5#53612yes—
179C2×C5#53911yescharacteristic
180C2×C5D4211yescharacteristic
181C2×C5#54712yes—
182C2×C5#56311yescharacteristic
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23D4×D514yes—
190C23C23×D511yescharacteristic
191Q8—204no—
192D4—44no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—54no—
203C2×C4—54no—
204C2×C4—102no—
205C2×C4—102no—
206C2×C4—102no—
207C2×C4—102no—
208C2×C4—102no—
209C2×C4—102no—
210C2×C4—102no—
211C2×C4—102no—
212C2×C4—102no—
213C2×C4—102no—
214C2×C4—208no—
215C2×C4#71112yes—
216C5#268011yescharacteristic, complemented, Sylow
217C22—24no—
218C22—24no—
219C22—24no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22C2×D4×D512yes—
225C22C2×#71111yescharacteristic
226C22#414411yescharacteristic
227C22#415011yescharacteristic
228C22#415012yes—
229C4—22no—
230C4—41no—
231C4—41no—
232C4—42no—
233C4—104no—
234C4—104no—
235C4—202no—
236C4—202no—
237C2—44no—
238C2#6592411yescharacteristic
239C2#6593611yescharacteristic
240C2#6594512yes—
241C2#6598212yes—
242C2#6600611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC22×C5×D4C22
hypercenter#394D5
hypocenterC5#2680
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#4024C2semidirect
4. #65268C2semidirect
5. #65265C2semidirect
6. #65241C2semidirect
7. #65239C2semidirect
8. #65233C2semidirect
9. #64946C2semidirect
216. C5#2680semidirect

Maximal subgroups

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

subgroupindex
C2×#40812
C2×#40242
C5×#3942
#652682
#652652
#652412
#652392
#652332
#649462
#26805