Finite Group Databasebeta

← #12008708

Subgroups — #12008708

Order 1280.

434 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C5×#2317C211yesmaximal, characteristic
2#10602345C212yesmaximal
3#10598126C212yesmaximal, complemented
4#10598123C212yesmaximal, complemented
5C2×C5×#117C411yescharacteristic
6C2×C5×#117C411yescharacteristic
7C22×C5×C4⋊C4C2211yescharacteristic
8C5×#389—22no—
9C5×#385—22no—
10C5×#384C411yescharacteristic
11C5×#384C2211yescharacteristic
12C5×#382C411yescharacteristic
13C5×#382C2211yescharacteristic
14#65496—22no—
15#64748—22no—
16#64748C2214yes—
17#11514—51nomaximal, Sylow
18C23×C4×C5C2×C411yescharacteristic
19C23×C4×C5D412yes—
20C5×#117—22no—
21C5×#117—22no—
22C5×#117—42no—
23C2×C5×C4⋊C4—21no—
24C2×C5×C4⋊C4—22no—
25C2×C5×C4⋊C4—22no—
26C2×C5×C4⋊C4—42no—
27C2×C5×C4⋊C4C2×C411yescharacteristic
28C2×C5×C4⋊C4C2×C411yescharacteristic
29C2×C5×C4⋊C4C2×C411yescharacteristic
30C2×C5×C4⋊C4D412yes—
31C2×C5×C4⋊C4C2311yescharacteristic
32C2×C5×C22⋊C4—21no—
33C2×C5×C22⋊C4—21no—
34C2×C42×C5—21no—
35C2×C42×C5C2×C411yescharacteristic
36C2×C42×C5C2×C411yescharacteristic
37C2×#681—42no—
38C5×#95—22no—
39C5×#95—22no—
40C22⋊#168—24no—
41C22⋊#168—42no—
42#2317D511yescharacteristic
43#1463—52no—
44#1325—52no—
45#1324—52no—
46C24×C5C22⋊C411yescharacteristic
47C22×C4×C5—22no—
48C22×C4×C5—22no—
49C22×C4×C5—21no—
50C22×C4×C5—21no—
51C22×C4×C5—22no—
52C22×C4×C5—21no—
53C22×C4×C5—21no—
54C22×C4×C5—22no—
55C22×C4×C5—21no—
56C22×C4×C5—21no—
57C22×C4×C5—82no—
58C22×C4×C5C22⋊C411yescharacteristic
59C22×C4×C5C22⋊C412yes—
60C22×C4×C5C22×C411yescharacteristic
61C22×C4×C5C2×D412yes—
62C5×C4⋊C4—24no—
63C5×C4⋊C4—41no—
64C5×C4⋊C4—42no—
65C5×C4⋊C4—41no—
66C5×C4⋊C4—41no—
67C5×C4⋊C4—41no—
68C5×C4⋊C4—82no—
69C5×C4⋊C4—82no—
70C5×C22⋊C4—24no—
71C5×C22⋊C4—42no—
72C42×C5—21no—
73C42×C5—21no—
74C42×C5—42no—
75C42×C5M1612yescomplemented
76C42×C5M1612yescomplemented
77#681—84no—
78C2×#168—44no—
79C2×#117Dic511yescharacteristic
80C2×#117Dic511yescharacteristic
81C22×C4⋊C4C2×D511yescharacteristic
82#407—102no—
83#389—22no—
84#385—22no—
85#384Dic511yescharacteristic
86#384C2×D511yescharacteristic
87#382Dic511yescharacteristic
88#382C2×D511yescharacteristic
89#324—54no—
90#324—102no—
91C23×C5—21no—
92C23×C5—21no—
93C23×C5—41no—
94C23×C5—41no—
95C23×C5#9912yes—
96C23×C5C2×C22⋊C411yescharacteristic
97C2×C4×C5—24no—
98C2×C4×C5—24no—
99C2×C4×C5—21no—
100C2×C4×C5—21no—
101C2×C4×C5—21no—
102C2×C4×C5—21no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5—22no—
106C2×C4×C5—22no—
107C2×C4×C5—44no—
108C2×C4×C5—41no—
109C2×C4×C5—41no—
110C2×C4×C5—41no—
111C2×C4×C5—41no—
112C2×C4×C5—42no—
113C2×C4×C5—42no—
114C2×C4×C5—42no—
115C2×C4×C5—42no—
116C2×C4×C5—41no—
117C2×C4×C5—41no—
118C2×C4×C5—82no—
119C2×C4×C5#10112yes—
120C2×C4×C5C2×M1611yescharacteristic
121C2×C4×C5C2×M1611yescharacteristic
122#168—84no—
123C23×C4C2×Dic511yescharacteristic
124C23×C4C22⋊D512yes—
125C2×M16—202no—
126#117—22no—
127#117—22no—
128#117—42no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×Dic511yescharacteristic
134C2×C4⋊C4C2×Dic511yescharacteristic
135C2×C4⋊C4C2×Dic511yescharacteristic
136C2×C4⋊C4C22⋊D512yes—
137C2×C4⋊C4C22×D511yescharacteristic
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C42—21no—
141C2×C42C2×Dic511yescharacteristic
142C2×C42C2×Dic511yescharacteristic
143C22⋊C8—104no—
144C22⋊C8—202no—
145#95—22no—
146#95—22no—
147C22×C5—21no—
148C22×C5—21no—
149C22×C5—22no—
150C22×C5—22no—
151C22×C5—22no—
152C22×C5—42no—
153C22×C5—42no—
154C22×C5#40711yescharacteristic
155C22×C5C2×#9911yescharacteristic
156C22×C5C2×#10111yescharacteristic
157C4×C5—22no—
158C4×C5—21no—
159C4×C5—21no—
160C4×C5—42no—
161C4×C5—42no—
162C4×C5—42no—
163C4×C5—42no—
164C4×C5—81no—
165C4×C5—81no—
166C24C22⋊Dic511yescharacteristic
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—82no—
178C22×C4C22⋊Dic511yescharacteristic
179C22×C4C22⋊Dic512yes—
180C22×C4C22×Dic511yescharacteristic
181C22×C4C2×C22⋊D512yes—
182M16—404no—
183C2×C8—204no—
184C4⋊C4—24no—
185C4⋊C4—41no—
186C4⋊C4—42no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C22⋊C4—24no—
193C22⋊C4—42no—
194C42—21no—
195C42—21no—
196C42—42no—
197C42#68112yescomplemented
198C42#68112yescomplemented
199C2×C5—21no—
200C2×C5—21no—
201C2×C5—42no—
202C2×C5#146311yescharacteristic
203C2×C5#212711yescharacteristic
204C2×C5#213311yescharacteristic
205C23—21no—
206C23—21no—
207C23—41no—
208C23—41no—
209C23#396612yes—
210C23C2×C22⋊Dic511yescharacteristic
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—44no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—82no—
233C2×C4#396812yes—
234C2×C4C2×#68111yescharacteristic
235C2×C4C2×#68111yescharacteristic
236C8—404no—
237C5#1151411yescharacteristic, complemented, Sylow
238C22—21no—
239C22—21no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22#6549611yescharacteristic
246C22C2×#396611yescharacteristic
247C22C2×#396811yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—81no—
256C4—81no—
257C2—21no—
258C2—21no—
259C2—42no—
260C2#1060234511yescharacteristic
261C2#1060265311yescharacteristic
262C2#1060266511yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2317C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC2×C5×C4⋊C4C23
hypercenter#2317D5
hypocenterC5#11514
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
3. #10598126C2semidirect
4. #10598123C2semidirect
75. C42×C5M16semidirect
76. C42×C5M16semidirect
197. C42#681semidirect
198. C42#681semidirect
237. C5#11514semidirect

Maximal subgroups

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

subgroupindex
C5×#23172
#106023452
#105981262
#105981232
#115145