Finite Group Databasebeta

← #10610091

Subgroups — #10610091

Order 640.

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

normal subgroups only

This group has 508 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×#4043C211yesmaximal, characteristic
2C2×D5×C4⋊C4C211yesmaximal, characteristic
3C5×#400C211yesmaximal, characteristic, complemented
4#65271C211yesmaximal, characteristic, complemented
5#65268C211yesmaximal, characteristic, complemented
6#64957C212yesmaximal
7#64956C211yesmaximal, characteristic
8#64953C212yesmaximal
9#64950C212yesmaximal, complemented
10#64946C212yesmaximal, complemented
11#64945C211yesmaximal, characteristic, complemented
12C22×C4×D5C2211yescharacteristic
13C22×C4×D5C2211yescharacteristic
14C22×C4×D5C2211yescharacteristic
15C22×Dic10C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×C4⋊Dic5C2211yescharacteristic
21C2×C4⋊Dic5C2212yes—
22C2×C4⋊Dic5C2212yes—
23C2×#683C2211yescharacteristic
24C2×#683C2211yescharacteristic
25C2×#683C2212yes—
26C2×#683C2212yes—
27#4043—24no—
28D5×C4⋊C4—24no—
29C5×#95C2211yescharacteristic
30C5×#95C2212yes—
31C5×#95C2212yes—
32#3963C2212yes—
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2211yescharacteristic
37#2627—51nomaximal, Sylow
38C23×D5C2311yescharacteristic
39C22×C4×C5C2311yescharacteristic
40C22×C4×C5C2312yes—
41C22×C4×C5C2312yes—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2311yescharacteristic
44C22×Dic5C2312yes—
45C22×Dic5C2311yescharacteristic
46C22×Dic5C2312yes—
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2311yescharacteristic
49C2×C4×D5—22no—
50C2×C4×D5—22no—
51C2×C4×D5—22no—
52C2×C4×D5—42no—
53C2×C4×D5—42no—
54C2×C4×D5D414yes—
55C2×C4×D5Q814yescomplemented
56C2×Dic10—22no—
57C2×Dic10—42no—
58C5×C4⋊C4—22no—
59C5×C4⋊C4—22no—
60#685—24no—
61#685—24no—
62C4⋊Dic5—22no—
63C4⋊Dic5—24no—
64C4⋊Dic5—42no—
65#683—22no—
66#683—22no—
67#683—42no—
68#683—42no—
69C2×C22⋊Q8—51no—
70C22×C4⋊C4—51no—
71#400—51no—
72#400D511yescharacteristic, complemented
73#398—52no—
74#398—52no—
75#395—51no—
76#395—51no—
77#385—52no—
78#381—52no—
79#380—51no—
80C23×C5C2411yescharacteristic
81C22×D5—22no—
82C22×D5—22no—
83C22×D5C2×D412yes—
84C22×D5C2×Q812yes—
85C22×D5#4112yes—
86C2×C4×C5—21no—
87C2×C4×C5—21no—
88C2×C4×C5—22no—
89C2×C4×C5—21no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—24no—
95C2×C4×C5—22no—
96C2×C4×C5C2×D412yes—
97C2×C4×C5C2×Q812yes—
98C2×Dic5—21no—
99C2×Dic5—21no—
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×Q812yes—
111C4×D5—28no—
112C4×D5—44no—
113C4×D5—44no—
114Dic10—44no—
115C22×Q8—51no—
116C23×C4—51no—
117C23×C4—51no—
118C23×C4—51no—
119C22⋊Q8—104no—
120C2×C4⋊C4—51no—
121C2×C4⋊C4—51no—
122C2×C4⋊C4—51no—
123C2×C4⋊C4—52no—
124C2×C4⋊C4—52no—
125C2×C4⋊C4—52no—
126C2×C4⋊C4—52no—
127C2×C4⋊C4—104no—
128C2×C4⋊C4C2×D511yescharacteristic
129C2×C4⋊C4C2×D511yescharacteristic
130C2×C22⋊C4—52no—
131C2×C22⋊C4—52no—
132#95—51no—
133#95—52no—
134#95—52no—
135#95—52no—
136#95—52no—
137#95C2×D511yescharacteristic
138#95C2×D512yes—
139#95C2×D512yes—
140C22×C5C22×D411yescharacteristic
141C22×C5C22×Q811yescharacteristic
142C22×C5C2×#4111yescharacteristic
143C22×C5H(5,𝔽2)12yes—
144C22×C52−1+412yes—
145C2×D5—22no—
146C2×D5—22no—
147C2×D5—24no—
148C2×D5—24no—
149C2×D5C22⋊Q814yes—
150C4×C5—22no—
151C4×C5—22no—
152C4×C5—41no—
153C4×C5—42no—
154C4×C5—42no—
155Dic5—22no—
156Dic5—22no—
157Dic5—42no—
158Dic5—41no—
159Dic5—42no—
160C24—51no—
161C2×Q8—102no—
162C2×Q8—202no—
163C22×C4—51no—
164C22×C4—51no—
165C22×C4—54no—
166C22×C4—54no—
167C22×C4—52no—
168C22×C4—52no—
169C22×C4—51no—
170C22×C4—102no—
171C22×C4—102no—
172C22×C4—102no—
173C22×C4—202no—
174C22×C4—202no—
175C22×C4C22×D512yes—
176C22×C4C22×D512yes—
177C22×C4C22×D511yescharacteristic
178C22×C4C22×D511yescharacteristic
179C22×C4C22×D511yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—102no—
183C4⋊C4—102no—
184C4⋊C4—104no—
185C4⋊C4—102no—
186C4⋊C4—202no—
187C4⋊C4—202no—
188C4⋊C4—202no—
189C22⋊C4—104no—
190C22⋊C4—104no—
191C2×C5C2×C22⋊Q811yescharacteristic
192C2×C5#53411yescharacteristic
193C2×C5#53611yescharacteristic
194C2×C5#53912yes—
195C2×C5#54412yes—
196D5—24no—
197C23—52no—
198C23—52no—
199C23—52no—
200C23—102no—
201C23—102no—
202C23C23×D511yescharacteristic
203Q8—204no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—52no—
215C2×C4—52no—
216C2×C4—108no—
217C2×C4—101no—
218C2×C4—101no—
219C2×C4—102no—
220C2×C4—102no—
221C2×C4—101no—
222C2×C4—102no—
223C2×C4—102no—
224C2×C4—102no—
225C2×C4—102no—
226C2×C4—102no—
227C2×C4—102no—
228C2×C4—204no—
229C2×C4—204no—
230C2×C4D4×D512yes—
231C2×C4Q8×D512yes—
232C5#262711yescharacteristic, complemented, Sylow
233C22—54no—
234C22—102no—
235C22—102no—
236C22—104no—
237C22—104no—
238C22C2×D4×D511yescharacteristic
239C22C2×Q8×D511yescharacteristic
240C22D5×#4111yescharacteristic
241C22#414912yes—
242C22#415012yes—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—42no—
247C4—42no—
248C4—102no—
249C4—102no—
250C4—201no—
251C4—202no—
252C4—202no—
253C2—104no—
254C2D5×C22⋊Q811yescharacteristic
255C2#6598711yescharacteristic
256C2#6600911yescharacteristic
257C2#6602312yes—
258C2#6604112yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#400C2
prime radicalC23×D5C23
hypercenter#400D5
hypocenterC5#2627
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. C5×#400C2semidirect
4. #65271C2semidirect
5. #65268C2semidirect
9. #64950C2semidirect
10. #64946C2semidirect
11. #64945C2semidirect
55. C2×C4×D5Q8semidirect
72. #400D5semidirect
232. C5#2627semidirect

Maximal subgroups

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

subgroupindex
C2×#40432
C2×D5×C4⋊C42
C5×#4002
#652712
#652682
#649572
#649562
#649532
#649502
#649462
#649452
#26275