Finite Group Databasebeta

← #10608972

Subgroups — #10608972

Order 640.

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

normal subgroups only

This group has 444 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×#4092C211yesmaximal, characteristic
2C2×#4042C211yesmaximal, characteristic
3C5×#395C211yesmaximal, characteristic, complemented
4#65518C211yesmaximal, characteristic, complemented
5#65282C212yesmaximal
6#65275C211yesmaximal, characteristic, complemented
7#65268C212yesmaximal, complemented
8#64964C212yesmaximal
9#64953C211yesmaximal, characteristic
10#64948C211yesmaximal, characteristic, complemented
11#64947C212yesmaximal, complemented
12C22×C5×Q8C2211yescharacteristic
13C22×C4×D5C2211yescharacteristic
14C22×Dic10C2211yescharacteristic
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C4⋊C4C2212yes—
17#4092—24no—
18C2×#685C2211yescharacteristic
19C2×#685C2212yes—
20C2×#685C2211yescharacteristic
21C2×#685C2212yes—
22C2×C4⋊Dic5C2211yescharacteristic
23C2×#683C2211yescharacteristic
24C2×#683C2211yescharacteristic
25C2×#683C2212yes—
26C2×#683C2212yes—
27#4042—24no—
28C5×#95C2212yes—
29C5×#95C2211yescharacteristic
30#3963C2212yes—
31#3963C2212yes—
32#3963C2212yes—
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2211yescharacteristic
37#2817—51nomaximal, Sylow
38C23×D5C2311yescharacteristic
39C2×C5×Q8—22no—
40C2×C5×Q8—42no—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2311yescharacteristic
46C22×Dic5C2311yescharacteristic
47C22×Dic5C2312yes—
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2312yes—
50C22×Dic5C2311yescharacteristic
51C2×C4×D5—42no—
52C2×C4×D5D414yes—
53C2×Dic10—22no—
54C2×Dic10—42no—
55C5×C4⋊C4—22no—
56C5×C4⋊C4—42no—
57#685—22no—
58#685—22no—
59#685—24no—
60#685—42no—
61C4⋊Dic5—22no—
62#683—22no—
63#683—22no—
64#683—24no—
65#683—42no—
66C2×C22⋊Q8—51no—
67C2×C22⋊Q8—51no—
68#400—51no—
69#400—52no—
70#400—51no—
71#397—52no—
72#397—52no—
73#395—51no—
74#395—52no—
75#395D511yescharacteristic, complemented
76#381—51no—
77C23×C5C2411yescharacteristic
78C22×D5—22no—
79C22×D5—22no—
80C22×D5—21no—
81C22×D5—21no—
82C22×D5C2×D412yes—
83C5×Q8—44no—
84C2×C4×C5—22no—
85C2×C4×C5—21no—
86C2×C4×C5—22no—
87C2×C4×C5—21no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—21no—
93C2×C4×C5—22no—
94C2×C4×C5—21no—
95C2×C4×C5—22no—
96C2×C4×C5C2×D412yes—
97C2×Dic5—22no—
98C2×Dic5—21no—
99C2×Dic5—22no—
100C2×Dic5—21no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—21no—
104C2×Dic5—24no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—21no—
108C2×Dic5C2×D412yes—
109C4×D5—44no—
110Dic10—44no—
111C22×Q8—51no—
112C22×Q8C2×D511yescharacteristic
113C23×C4—51no—
114C22⋊Q8—104no—
115C22⋊Q8—104no—
116C2×C4⋊C4—52no—
117C2×C4⋊C4—51no—
118C2×C4⋊C4—51no—
119C2×C4⋊C4—51no—
120C2×C4⋊C4—52no—
121C2×C4⋊C4C2×D511yescharacteristic
122C2×C4⋊C4C2×D512yes—
123C2×C22⋊C4—51no—
124C2×C22⋊C4—52no—
125C2×C22⋊C4—51no—
126C2×C22⋊C4—52no—
127#95—52no—
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×C5C22×D411yescharacteristic
137C22×C5H(5,𝔽2)12yes—
138C22×C52−1+412yes—
139C22×C52−1+411yescharacteristic
140C22×C52−1+411yescharacteristic
141C2×D5—22no—
142C2×D5—22no—
143C2×D5—22no—
144C2×D5—42no—
145C2×D5—42no—
146C4×C5—22no—
147C4×C5—41no—
148C4×C5—42no—
149C4×C5—41no—
150C4×C5—42no—
151Dic5—22no—
152Dic5—41no—
153Dic5—42no—
154Dic5—41no—
155Dic5—42no—
156C24—51no—
157C2×Q8—22no—
158C2×Q8—42no—
159C2×Q8—102no—
160C2×Q8—202no—
161C22×C4—51no—
162C22×C4—54no—
163C22×C4—52no—
164C22×C4—52no—
165C22×C4—51no—
166C22×C4—51no—
167C22×C4—202no—
168C22×C4C22×D512yes—
169C22×C4C22×D511yescharacteristic
170C22×C4C22×D511yescharacteristic
171C22×C4C22×D512yes—
172C22×C4C22×D511yescharacteristic
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—102no—
176C4⋊C4—104no—
177C4⋊C4—102no—
178C4⋊C4—102no—
179C4⋊C4—202no—
180C22⋊C4—102no—
181C22⋊C4—102no—
182C22⋊C4—104no—
183C22⋊C4—202no—
184C2×C5#53411yescharacteristic
185C2×C5#53611yescharacteristic
186C2×C5#53611yescharacteristic
187C2×C5#56312yes—
188C2×C5#56312yes—
189D5—42no—
190C23—52no—
191C23—102no—
192C23—102no—
193C23—101no—
194C23—101no—
195C23C23×D511yescharacteristic
196Q8—44no—
197Q8—204no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—52no—
211C2×C4—102no—
212C2×C4—102no—
213C2×C4—101no—
214C2×C4—101no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4—104no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—101no—
221C2×C4—101no—
222C2×C4—204no—
223C2×C4D4×D512yes—
224C5#281711yescharacteristic, complemented, Sylow
225C22—102no—
226C22—102no—
227C22—102no—
228C22—202no—
229C22—202no—
230C22C2×D4×D511yescharacteristic
231C22#414412yes—
232C22#414712yes—
233C22#414711yescharacteristic
234C22#415011yescharacteristic
235C4—22no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—42no—
240C4—102no—
241C4—202no—
242C4—202no—
243C4—201no—
244C4—201no—
245C2—202no—
246C2#6592311yescharacteristic
247C2#6596211yescharacteristic
248C2#6598012yes—
249C2#6605811yescharacteristic
250C2#6606912yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#395C2
prime radicalC23×D5C23
hypercenter#395D5
hypocenterC5#2817
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
3. C5×#395C2semidirect
4. #65518C2semidirect
6. #65275C2semidirect
7. #65268C2semidirect
10. #64948C2semidirect
11. #64947C2semidirect
75. #395D5semidirect
224. C5#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#40922
C2×#40422
C5×#3952
#655182
#652822
#652752
#652682
#649642
#649532
#649482
#649472
#28175