Finite Group Databasebeta

← #10609549

Subgroups — #10609549

Order 640.

528 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 143 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×#4042C212yesmaximal
2C2×#4038C211yesmaximal, characteristic
3C2×#4034C211yesmaximal, characteristic, complemented
4C5×#398C211yesmaximal, characteristic, complemented
5#65282C211yesmaximal, characteristic
6#65273C212yesmaximal, complemented
7#65272C211yesmaximal, characteristic, complemented
8#64964C212yesmaximal
9#64963C211yesmaximal, characteristic
10#64939C211yesmaximal, characteristic, complemented
11#64935C212yesmaximal, complemented
12C22×C4×D5C2211yescharacteristic
13C22×Dic10C2212yes—
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C4⋊C4C2212yes—
17C2×#685C2211yescharacteristic
18C2×#685C2212yes—
19C2×#685C2211yescharacteristic
20C2×#685C2212yes—
21C2×C4⋊Dic5C2211yescharacteristic
22C2×C4⋊Dic5C2211yescharacteristic
23C2×#683C2212yes—
24C2×#683C2212yes—
25C2×#683C2212yes—
26C2×C4×Dic5C2211yescharacteristic
27C2×C4×Dic5C2211yescharacteristic
28#4042—28no—
29#4038—24no—
30#4034—24no—
31C5×#95C2212yes—
32C5×#95C2211yescharacteristic
33#3963C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#3963C2212yes—
38#2621—51nomaximal, Sylow
39C23×D5C2311yescharacteristic
40C22×C4×C5C2312yes—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×Dic5C2312yes—
46C22×Dic5C2311yescharacteristic
47C22×Dic5C2312yes—
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2311yescharacteristic
50C2×C4×D5—22no—
51C2×C4×D5—22no—
52C2×C4×D5D414yes—
53C2×Dic10—24no—
54C2×Dic10—24no—
55C2×Dic10—24no—
56C5×C4⋊C4—22no—
57C5×C4⋊C4—22no—
58C5×C4⋊C4—24no—
59#685—24no—
60#685—24no—
61#685—22no—
62#685—22no—
63C4⋊Dic5—22no—
64C4⋊Dic5—22no—
65#683—24no—
66#683—24no—
67#683—24no—
68C4×Dic5—22no—
69C4×Dic5D414yes—
70C2×C4⋊Q8—51no—
71C2×C22⋊Q8—52no—
72C2×#117—51no—
73#398—51no—
74#398—52no—
75#398D511yescharacteristic, complemented
76#397—51no—
77#397—51no—
78#397—52no—
79#391—52no—
80#382—51no—
81C23×C5C2411yescharacteristic
82C22×D5—21no—
83C22×D5—21no—
84C22×D5—22no—
85C22×D5—22no—
86C22×D5C2×D412yes—
87C2×C4×C5—21no—
88C2×C4×C5—21no—
89C2×C4×C5—22no—
90C2×C4×C5—21no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5C2×D412yes—
99C2×C4×C5C2×D412yes—
100C2×Dic5—22no—
101C2×Dic5—21no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5C2×D412yes—
110C2×Dic5C2×D412yes—
111C2×Dic5C2×D412yes—
112C2×Dic5#4112yes—
113C4×D5—28no—
114Dic10—48no—
115C22×Q8—52no—
116C23×C4—51no—
117C4⋊Q8—104no—
118C22⋊Q8—108no—
119#117—104no—
120C2×C4⋊C4—51no—
121C2×C4⋊C4—52no—
122C2×C4⋊C4—52no—
123C2×C4⋊C4—52no—
124C2×C4⋊C4—51no—
125C2×C4⋊C4C2×D511yescharacteristic
126C2×C4⋊C4C2×D512yes—
127C2×C4⋊C4C2×D511yescharacteristic
128C2×C22⋊C4—51no—
129C2×C22⋊C4—51no—
130C2×C22⋊C4—52no—
131C2×C22⋊C4—52no—
132C2×C42—51no—
133C2×C42—51no—
134#95—52no—
135#95—51no—
136#95—52no—
137#95—52no—
138#95—52no—
139#95C2×D511yescharacteristic
140#95C2×D512yes—
141C22×C5C22×D411yescharacteristic
142C22×C5C22×D411yescharacteristic
143C22×C5C2×#4111yescharacteristic
144C22×C52−1+412yes—
145C22×C52−1+412yes—
146C2×D5—22no—
147C2×D5—24no—
148C2×D5—41no—
149C2×D5—41no—
150C2×D5—42no—
151C4×C5—22no—
152C4×C5—22no—
153C4×C5—42no—
154C4×C5—42no—
155C4×C5—41no—
156Dic5—22no—
157Dic5—42no—
158Dic5—41no—
159Dic5—42no—
160Dic5#12114yes—
161C24—51no—
162C2×Q8—104no—
163C2×Q8—104no—
164C2×Q8—104no—
165C22×C4—51no—
166C22×C4—51no—
167C22×C4—54no—
168C22×C4—51no—
169C22×C4—52no—
170C22×C4—52no—
171C22×C4—102no—
172C22×C4—102no—
173C22×C4C22×D511yescharacteristic
174C22×C4C22×D512yes—
175C22×C4C22×D512yes—
176C22×C4C22×D511yescharacteristic
177C22×C4C22×D511yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4—102no—
182C4⋊C4—104no—
183C4⋊C4—104no—
184C4⋊C4—104no—
185C4⋊C4—102no—
186C22⋊C4—102no—
187C22⋊C4—104no—
188C22⋊C4—104no—
189C22⋊C4—102no—
190C42—54no—
191C42—102no—
192C2×C5C2×#12111yescharacteristic
193C2×C5#53611yescharacteristic
194C2×C5#53612yes—
195C2×C5#53611yescharacteristic
196C2×C5#54112yes—
197D5—42no—
198C23—52no—
199C23—101no—
200C23—102no—
201C23—101no—
202C23—102no—
203C23C23×D511yescharacteristic
204Q8—208no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—52no—
217C2×C4—52no—
218C2×C4—52no—
219C2×C4—52no—
220C2×C4—108no—
221C2×C4—102no—
222C2×C4—101no—
223C2×C4—102no—
224C2×C4—102no—
225C2×C4—102no—
226C2×C4—102no—
227C2×C4—102no—
228C2×C4—102no—
229C2×C4—102no—
230C2×C4D4×D512yes—
231C2×C4D4×D512yes—
232C5#262111yescharacteristic, complemented, Sylow
233C22—102no—
234C22—104no—
235C22—201no—
236C22—201no—
237C22—202no—
238C22C2×D4×D511yescharacteristic
239C22C2×D4×D511yescharacteristic
240C22#414712yes—
241C22D5×#4111yescharacteristic
242C22#415012yes—
243C4—22no—
244C4—22no—
245C4—42no—
246C4—42no—
247C4—41no—
248C4—54no—
249C4—102no—
250C4—201no—
251C4—202no—
252C4—202no—
253C2—202no—
254C2D5×#12111yescharacteristic
255C2#6596212yes—
256C2#6598211yescharacteristic
257C2#6603012yes—
258C2#6605811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#398C2
prime radicalC23×D5C23
hypercenter#398D5
hypocenterC5#2621
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. C2×#4034C2semidirect
4. C5×#398C2semidirect
6. #65273C2semidirect
7. #65272C2semidirect
10. #64939C2semidirect
11. #64935C2semidirect
75. #398D5semidirect
232. C5#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#40422
C2×#40382
C2×#40342
C5×#3982
#652822
#652732
#652722
#649642
#649632
#649392
#649352
#26215