Finite Group Databasebeta

← #10608303

Subgroups — #10608303

Order 640.

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

normal subgroups only

This group has 492 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×#4093C211yesmaximal, characteristic
2C2×#4090C211yesmaximal, characteristic, complemented
3C5×#391C211yesmaximal, characteristic, complemented
4#65283C212yesmaximal, complemented
5#65229C212yesmaximal, complemented
6#64962C212yesmaximal, complemented
7#64961C212yesmaximal
8#64959C212yesmaximal, complemented
9#64942C212yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C22×D20C2211yescharacteristic
12C22×Dic10C2211yescharacteristic
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C42×C5C2211yescharacteristic
15#4093—24no—
16#4090—24no—
17C2×#685C2212yes—
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×C4⋊Dic5C2212yes—
24C2×#683C2212yes—
25C2×#683C2212yes—
26C2×C4×Dic5C2211yescharacteristic
27C2×C4×Dic5C2211yescharacteristic
28C5×#95C2212yes—
29C5×#95C2212yes—
30#3963C2212yes—
31#3963C2212yes—
32#3963C2212yes—
33#2731—51nomaximal, Sylow
34C23×D5C2312yes—
35C2×C5×Q8—41no—
36C2×C5×Q8—41no—
37C2×C5×Q8D414yes—
38C22×C4×C5C2312yes—
39C22×C4×C5C2312yes—
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2311yescharacteristic
43C22×Dic5C2312yes—
44C22×Dic5C2312yes—
45C22×Dic5C2312yes—
46C2×D20—22no—
47C2×D20—41no—
48C2×D20—41no—
49C2×Dic10—22no—
50C2×Dic10—41no—
51C2×Dic10—41no—
52C5×C4⋊C4—41no—
53C42×C5—41no—
54#685—24no—
55#685—24no—
56#685—24no—
57#685—24no—
58#685—42no—
59#685—42no—
60C4⋊Dic5—24no—
61#683—24no—
62#683—24no—
63C4×Dic5—22no—
64C4×Dic5—22no—
65C2×C4⋊Q8—51no—
66C2×#124—51no—
67#396—52no—
68#394—52no—
69#393—52no—
70#391D511yescharacteristic, complemented
71#388—52no—
72#388—52no—
73#387—52no—
74C23×C5C2411yescharacteristic
75C22×D5—22no—
76C22×D5—22no—
77C22×D5—22no—
78C22×D5—22no—
79C22×D5—22no—
80C22×D5—22no—
81C22×D5—22no—
82C5×Q8—44no—
83C2×C4×C5—21no—
84C2×C4×C5—21no—
85C2×C4×C5—21no—
86C2×C4×C5—21no—
87C2×C4×C5—21no—
88C2×C4×C5—21no—
89C2×C4×C5—22no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5C2×D412yes—
96C2×C4×C5C2×D412yes—
97C2×C4×C5C2×D412yes—
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#4114yes—
107D20—44no—
108Dic10—44no—
109C22×Q8—51no—
110C22×Q8C2×D511yescharacteristic
111C22×D4—51no—
112C4⋊Q8—104no—
113#124—104no—
114C2×C4⋊C4—52no—
115C2×C4⋊C4—52no—
116C2×C4⋊C4—52no—
117C2×C4⋊C4C2×D511yescharacteristic
118C2×C22⋊C4—52no—
119C2×C22⋊C4—52no—
120C2×C22⋊C4—52no—
121C2×C22⋊C4—52no—
122C2×C22⋊C4—52no—
123C2×C22⋊C4—52no—
124C2×C42—51no—
125C2×C42—51no—
126C2×C42C2×D511yescharacteristic
127#95—52no—
128#95—52no—
129#95—52no—
130#95C2×D512yes—
131#95C2×D512yes—
132C22×C5C22×D411yescharacteristic
133C22×C5C2×#4112yes—
134C22×C5H(5,𝔽2)11yescharacteristic
135C22×C5H(5,𝔽2)11yescharacteristic
136C22×C52−1+412yes—
137C2×D5—42no—
138C2×D5—42no—
139C2×D5—42no—
140C2×D5—42no—
141C2×D5—42no—
142C2×D5—42no—
143C2×D5—42no—
144C4×C5—22no—
145C4×C5—22no—
146C4×C5—22no—
147C4×C5—42no—
148C4×C5—42no—
149Dic5—24no—
150Dic5—42no—
151Dic5—42no—
152C24—52no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—102no—
156C2×Q8—201no—
157C2×Q8—201no—
158C2×Q8D2014yes—
159C2×D4—102no—
160C2×D4—201no—
161C2×D4—201no—
162C22×C4—52no—
163C22×C4—52no—
164C22×C4—52no—
165C22×C4C22×D512yes—
166C22×C4C22×D512yes—
167C22×C4C22×D511yescharacteristic
168C22×C4C22×D511yescharacteristic
169C22×C4C22×D511yescharacteristic
170C4⋊C4—41no—
171C4⋊C4—104no—
172C4⋊C4—104no—
173C4⋊C4—104no—
174C22⋊C4—104no—
175C22⋊C4—104no—
176C22⋊C4—104no—
177C22⋊C4—104no—
178C22⋊C4—202no—
179C22⋊C4—202no—
180C42—41no—
181C42—102no—
182C42—102no—
183C2×C5#53411yescharacteristic
184C2×C5#54212yes—
185C2×C5#54812yes—
186C2×C5#55511yescharacteristic
187C2×C5#56311yescharacteristic
188D5—82no—
189C23—102no—
190C23—102no—
191C23—102no—
192C23—102no—
193C23—102no—
194C23—102no—
195C23—102no—
196C23C23×D511yescharacteristic
197Q8—44no—
198Q8—204no—
199D4—204no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—54no—
213C2×C4—102no—
214C2×C4—102no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—102no—
221C2×C4C2×D2012yes—
222C2×C4C2×D2012yes—
223C2×C4C2×D2012yes—
224C5#273111yescharacteristic, complemented, Sylow
225C22—202no—
226C22—202no—
227C22—202no—
228C22—202no—
229C22—202no—
230C22—202no—
231C22—202no—
232C22C22×D2011yescharacteristic
233C22#414411yescharacteristic
234C22#414411yescharacteristic
235C22#414712yes—
236C22D5×#4112yes—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—42no—
241C4—42no—
242C4—104no—
243C4—202no—
244C4—202no—
245C2—402no—
246C2#6582711yescharacteristic
247C2#6591012yes—
248C2#6597412yes—
249C2#6600411yescharacteristic
250C2#6606911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#391C2
prime radicalC22×D20C22
hypercenter#391D5
hypocenterC5#2731
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#4090C2semidirect
3. C5×#391C2semidirect
4. #65283C2semidirect
5. #65229C2semidirect
6. #64962C2semidirect
8. #64959C2semidirect
9. #64942C2semidirect
70. #391D5semidirect
224. C5#2731semidirect

Maximal subgroups

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

subgroupindex
C2×#40932
C2×#40902
C5×#3912
#652832
#652292
#649622
#649612
#649592
#649422
#27315