Finite Group Databasebeta

← #12008700

Subgroups — #12008700

Order 1280.

578 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 107 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×C5×#384C211yesmaximal, characteristic
2#10602345C212yesmaximal
3#10598126C214yesmaximal, complemented
4C22×C5×C4⋊C4—22no—
5C22×C5×C4⋊C4C411yescharacteristic
6C22×C5×C4⋊C4C2211yescharacteristic
7C22×C42×C5C411yescharacteristic
8C5×#384—22no—
9C5×#384C412yes—
10C5×#384C2212yes—
11#65496—22no—
12#64748—22no—
13#64748C2214yes—
14#11512—51nomaximal, Sylow
15C23×C4×C5—21no—
16C23×C4×C5—21no—
17C23×C4×C5C2×C411yescharacteristic
18C23×C4×C5D412yes—
19C2×C5×C4⋊C4—21no—
20C2×C5×C4⋊C4—24no—
21C2×C5×C4⋊C4—24no—
22C2×C5×C4⋊C4—41no—
23C2×C5×C4⋊C4—41no—
24C2×C5×C4⋊C4—41no—
25C2×C5×C4⋊C4—41no—
26C2×C5×C4⋊C4—48no—
27C2×C5×C4⋊C4C2×C412yes—
28C2×C5×C4⋊C4C2×C411yescharacteristic
29C2×C5×C4⋊C4D412yes—
30C2×C5×C4⋊C4C2311yescharacteristic
31C2×C42×C5—21no—
32C2×C42×C5—24no—
33C2×C42×C5C2×C412yes—
34C2×#681—42no—
35C22⋊#168—24no—
36C22⋊#168—42no—
37C2×#384D511yescharacteristic
38#1463—52no—
39#1325—54no—
40C24×C5C22⋊C411yescharacteristic
41C22×C4×C5—24no—
42C22×C4×C5—24no—
43C22×C4×C5—24no—
44C22×C4×C5—21no—
45C22×C4×C5—21no—
46C22×C4×C5—22no—
47C22×C4×C5—22no—
48C22×C4×C5—22no—
49C22×C4×C5—22no—
50C22×C4×C5—22no—
51C22×C4×C5—22no—
52C22×C4×C5—44no—
53C22×C4×C5—44no—
54C22×C4×C5—41no—
55C22×C4×C5—41no—
56C22×C4×C5C22⋊C411yescharacteristic
57C22×C4×C5C22⋊C412yes—
58C22×C4×C5C22×C411yescharacteristic
59C22×C4×C5C2×D412yes—
60C5×C4⋊C4—41no—
61C5×C4⋊C4—42no—
62C5×C4⋊C4—41no—
63C5×C4⋊C4—42no—
64C5×C4⋊C4—48no—
65C5×C4⋊C4—48no—
66C5×C4⋊C4—42no—
67C42×C5—24no—
68C42×C5—22no—
69C42×C5M1614yescomplemented
70#681—84no—
71C2×#168—44no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4Dic511yescharacteristic
74C22×C4⋊C4C2×D511yescharacteristic
75C22×C42Dic511yescharacteristic
76#407—102no—
77#384—22no—
78#384Dic512yes—
79#384C2×D512yes—
80#324—54no—
81#324—102no—
82C23×C5—22no—
83C23×C5—22no—
84C23×C5—21no—
85C23×C5—21no—
86C23×C5#9911yescharacteristic
87C23×C5#9911yescharacteristic
88C23×C5C2×C22⋊C411yescharacteristic
89C2×C4×C5—24no—
90C2×C4×C5—216no—
91C2×C4×C5—24no—
92C2×C4×C5—24no—
93C2×C4×C5—24no—
94C2×C4×C5—21no—
95C2×C4×C5—24no—
96C2×C4×C5—21no—
97C2×C4×C5—21no—
98C2×C4×C5—21no—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5—44no—
102C2×C4×C5—42no—
103C2×C4×C5—42no—
104C2×C4×C5—44no—
105C2×C4×C5—42no—
106C2×C4×C5—82no—
107C2×C4×C5—82no—
108C2×C4×C5—82no—
109C2×C4×C5#10112yes—
110C2×C4×C5C2×M1612yes—
111#168—84no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4C2×Dic511yescharacteristic
115C23×C4C22⋊D512yes—
116C2×M16—202no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—48no—
125C2×C4⋊C4C2×Dic512yes—
126C2×C4⋊C4C2×Dic511yescharacteristic
127C2×C4⋊C4C22⋊D512yes—
128C2×C4⋊C4C22×D511yescharacteristic
129C2×C42—21no—
130C2×C42—24no—
131C2×C42C2×Dic512yes—
132C22⋊C8—104no—
133C22⋊C8—202no—
134C22×C5—24no—
135C22×C5—24no—
136C22×C5—21no—
137C22×C5—21no—
138C22×C5—21no—
139C22×C5—21no—
140C22×C5—21no—
141C22×C5—21no—
142C22×C5#35312yes—
143C22×C5#35612yes—
144C22×C5#40711yescharacteristic
145C22×C5C2×#9911yescharacteristic
146C22×C5C2×#10111yescharacteristic
147C4×C5—28no—
148C4×C5—22no—
149C4×C5—22no—
150C4×C5—42no—
151C4×C5—42no—
152C4×C5—82no—
153C24C22⋊Dic511yescharacteristic
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4C22⋊Dic511yescharacteristic
170C22×C4C22⋊Dic512yes—
171C22×C4C22×Dic511yescharacteristic
172C22×C4C2×C22⋊D512yes—
173M16—404no—
174C2×C8—204no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—48no—
180C4⋊C4—48no—
181C4⋊C4—42no—
182C42—24no—
183C42—22no—
184C42#68114yescomplemented
185C2×C5—21no—
186C2×C5—22no—
187C2×C5—22no—
188C2×C5—21no—
189C2×C5#146311yescharacteristic
190C2×C5C2×#35311yescharacteristic
191C2×C5C2×#35611yescharacteristic
192C23—22no—
193C23—21no—
194C23—21no—
195C23—22no—
196C23#396611yescharacteristic
197C23#396611yescharacteristic
198C23C2×C22⋊Dic511yescharacteristic
199C2×C4—24no—
200C2×C4—216no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—21no—
205C2×C4—24no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4#396812yes—
220C2×C4C2×#68112yes—
221C8—404no—
222C5#1151211yescharacteristic, complemented, Sylow
223C22—24no—
224C22—24no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22#6476512yes—
232C22#6476812yes—
233C22#6549611yescharacteristic
234C22C2×#396611yescharacteristic
235C22C2×#396811yescharacteristic
236C4—28no—
237C4—22no—
238C4—22no—
239C4—42no—
240C4—42no—
241C4—82no—
242C2—21no—
243C2—22no—
244C2—21no—
245C2—22no—
246C2#1060234511yescharacteristic
247C2C2×#6476511yescharacteristic
248C2C2×#6476811yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC2×C5×#384C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC2×C5×C4⋊C4C23
hypercenterC2×#384D5
hypocenterC5#11512
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
3. #10598126C2semidirect
69. C42×C5M16semidirect
184. C42#681semidirect
222. C5#11512semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3842
#106023452
#105981262
#115125