Finite Group Databasebeta

← #10606556

Subgroups — #10606556

Order 640.

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

normal subgroups only

This group has 512 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×#4082C211yesmaximal, characteristic
2C2×#4081C211yesmaximal, characteristic
3C5×#386C211yesmaximal, characteristic
4#65501C212yesmaximal, complemented
5#65242C212yesmaximal, complemented
6#65242C212yesmaximal, complemented
7#65239C212yesmaximal, complemented
8#64945C212yesmaximal, complemented
9#64942C212yesmaximal, complemented
10C22×C5×D4C2211yescharacteristic
11C23×C4×C5C2211yescharacteristic
12C23×Dic5C2211yescharacteristic
13C22×Dic10C2211yescharacteristic
14C2×C5×C22⋊C4C2212yes—
15C2×C5×C22⋊C4C2211yescharacteristic
16C2×C22⋊Dic5C2211yescharacteristic
17C2×C22⋊Dic5C2212yes—
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2212yes—
20#4082—24no—
21#4081—24no—
22C2×C4⋊Dic5C2212yes—
23C2×C4⋊Dic5C2212yes—
24C2×C4⋊Dic5C2212yes—
25C2×#683C2212yes—
26C2×C4×Dic5C2211yescharacteristic
27C5×#95C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#3963C2212yes—
32#3963C2212yes—
33#2717—51nomaximal, Sylow
34C24×C5C2311yescharacteristic
35C24×C5C2311yescharacteristic
36C2×C5×D4—41no—
37C2×C5×D4—41no—
38C2×C5×D4D414yes—
39C22×C4×C5—24no—
40C22×C4×C5—41no—
41C22×C4×C5C2312yes—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×Dic5—22no—
45C22×Dic5—41no—
46C22×Dic5—41no—
47C22×Dic5C2312yes—
48C22×Dic5C2312yes—
49C22×Dic5C2312yes—
50C22×Dic5C2312yes—
51C2×Dic10—22no—
52C2×Dic10—41no—
53C2×Dic10—41no—
54C5×C22⋊C4—24no—
55C5×C22⋊C4—41no—
56C22⋊Dic5—22no—
57C22⋊Dic5—24no—
58C22⋊Dic5—24no—
59C22⋊Dic5—24no—
60C4⋊Dic5—24no—
61C4⋊Dic5—42no—
62C4⋊Dic5—42no—
63#683—24no—
64C4×Dic5—22no—
65C2×#124—51no—
66C2×#123—51no—
67#397—52no—
68#397—52no—
69#395—52no—
70#387—52no—
71#386D511yescharacteristic
72#385—52no—
73#380—52no—
74C23×C5—22no—
75C23×C5—22no—
76C23×C5—21no—
77C23×C5—21no—
78C23×C5—21no—
79C23×C5—21no—
80C23×C5—22no—
81C23×C5—22no—
82C23×C5C2×D412yes—
83C23×C5C2×D412yes—
84C23×C5C2411yescharacteristic
85C5×D4—44no—
86C2×C4×C5—22no—
87C2×C4×C5—22no—
88C2×C4×C5—21no—
89C2×C4×C5—21no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—44no—
94C2×C4×C5C2×D412yes—
95C2×C4×C5#4114yes—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5—22no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—44no—
109Dic10—44no—
110C22×Q8—51no—
111C22×D4C2×D511yescharacteristic
112C23×C4—51no—
113C23×C4C2×D511yescharacteristic
114#124—104no—
115#123—104no—
116C2×C4⋊C4—52no—
117C2×C4⋊C4—52no—
118C2×C4⋊C4—52no—
119C2×C4⋊C4—52no—
120C2×C22⋊C4—52no—
121C2×C22⋊C4—51no—
122C2×C22⋊C4—52no—
123C2×C22⋊C4—52no—
124C2×C22⋊C4C2×D512yes—
125C2×C22⋊C4C2×D511yescharacteristic
126C2×C42—51no—
127#95—52no—
128#95—52no—
129#95—52no—
130#95—52no—
131#95—52no—
132#95C2×D512yes—
133C22×C5—22no—
134C22×C5—22no—
135C22×C5—24no—
136C22×C5—24no—
137C22×C5—42no—
138C22×C5—42no—
139C22×C5—41no—
140C22×C5—41no—
141C22×C5—41no—
142C22×C5—41no—
143C22×C5C22×D411yescharacteristic
144C22×C5C2×#4112yes—
145C22×C5H(5,𝔽2)11yescharacteristic
146C22×C5H(5,𝔽2)11yescharacteristic
147C22×C52−1+412yes—
148C4×C5—22no—
149C4×C5—24no—
150C4×C5—42no—
151Dic5—42no—
152Dic5—42no—
153Dic5—42no—
154Dic5—42no—
155C24C22×D511yescharacteristic
156C24C22×D511yescharacteristic
157C2×Q8—102no—
158C2×Q8—201no—
159C2×Q8—201no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4D2014yes—
163C22×C4—24no—
164C22×C4—41no—
165C22×C4—52no—
166C22×C4—52no—
167C22×C4—52no—
168C22×C4—52no—
169C22×C4—102no—
170C22×C4—201no—
171C22×C4—201no—
172C22×C4C22×D512yes—
173C22×C4C22×D511yescharacteristic
174C22×C4C22×D512yes—
175C4⋊C4—104no—
176C4⋊C4—104no—
177C4⋊C4—202no—
178C4⋊C4—202no—
179C22⋊C4—24no—
180C22⋊C4—41no—
181C22⋊C4—102no—
182C22⋊C4—104no—
183C22⋊C4—104no—
184C22⋊C4—104no—
185C42—102no—
186C2×C5—42no—
187C2×C5—42no—
188C2×C5#53411yescharacteristic
189C2×C5#54212yes—
190C2×C5#54712yes—
191C2×C5#55111yescharacteristic
192C2×C5#56311yescharacteristic
193C23—22no—
194C23—22no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—22no—
201C23C2×D2012yes—
202C23C2×D2012yes—
203C23C23×D511yescharacteristic
204Q8—204no—
205D4—44no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—44no—
214C2×C4—102no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—102no—
221C2×C4—102no—
222C2×C4—102no—
223C2×C4—102no—
224C2×C4—102no—
225C2×C4—102no—
226C2×C4—204no—
227C2×C4C2×D2012yes—
228C2×C4#71114yes—
229C5#271711yescharacteristic, complemented, Sylow
230C22—22no—
231C22—22no—
232C22—24no—
233C22—24no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—42no—
238C22—41no—
239C22—41no—
240C22C22×D2011yescharacteristic
241C22C2×#71112yes—
242C22#414411yescharacteristic
243C22#414411yescharacteristic
244C22#415012yes—
245C4—22no—
246C4—24no—
247C4—42no—
248C4—202no—
249C4—202no—
250C4—202no—
251C4—202no—
252C2—42no—
253C2—42no—
254C2#6582711yescharacteristic
255C2#6588912yes—
256C2#6592111yescharacteristic
257C2#6593712yes—
258C2#6598511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#386C2
prime radicalC22×C5×D4C22
hypercenter#386D5
hypocenterC5#2717
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
4. #65501C2semidirect
5. #65242C2semidirect
6. #65242C2semidirect
7. #65239C2semidirect
8. #64945C2semidirect
9. #64942C2semidirect
229. C5#2717semidirect

Maximal subgroups

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

subgroupindex
C2×#40822
C2×#40812
C5×#3862
#655012
#652422
#652392
#649452
#649422
#27175