Finite Group Databasebeta

← #12008788

Subgroups — #12008788

Order 1280.

424 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2326C211yesmaximal, characteristic
2#10602418C212yesmaximal, complemented
3#10598131C212yesmaximal
4#10598126C212yesmaximal, complemented
5C2×C5×#124C411yescharacteristic
6C2×C4×C5×Q8C2211yescharacteristic
7C2×C4×C5×D4C411yescharacteristic
8C5×#391C411yescharacteristic
9C5×#390C2211yescharacteristic
10C5×#388—22no—
11C5×#385—22no—
12C5×#384C2211yescharacteristic
13C5×#376C411yescharacteristic
14#65215—22no—
15#64748—22no—
16#64748C2214yes—
17#11534—51nomaximal, Sylow
18C22×C5×Q8C2×C411yescharacteristic
19C22×C5×D4C2×C411yescharacteristic
20C23×C4×C5—21no—
21C5×#124—22no—
22C5×#124—22no—
23C4×C5×Q8—42no—
24C4×C5×D4—42no—
25C2×C5×C4⋊C4—22no—
26C2×C5×C4⋊C4C2×C411yescharacteristic
27C2×C5×C4⋊C4D412yes—
28C2×C5×C4⋊C4C2311yescharacteristic
29C2×C5×C22⋊C4—22no—
30C2×C5×C22⋊C4—21no—
31C2×C5×C22⋊C4—22no—
32C2×C42×C5C2×C411yescharacteristic
33C2×C42×C5C2×C411yescharacteristic
34C2×C42×C5C2×C411yescharacteristic
35C2×C42×C5D412yes—
36C5×#95—21no—
37C5×#95—21no—
38C5×#95—22no—
39C22⋊#168—24no—
40C4⋊#168—42no—
41#3935—42no—
42#2326D511yescharacteristic
43#1467—52no—
44#1325—52no—
45#1318—52no—
46C24×C5—21no—
47C2×C5×Q8—21no—
48C2×C5×Q8—21no—
49C2×C5×Q8—41no—
50C2×C5×Q8—41no—
51C2×C5×D4—21no—
52C2×C5×D4—41no—
53C2×C5×D4—41no—
54C2×C5×D4M1612yes—
55C22×C4×C5—21no—
56C22×C4×C5—21no—
57C22×C4×C5—21no—
58C22×C4×C5—42no—
59C22×C4×C5—82no—
60C22×C4×C5C22⋊C412yes—
61C22×C4×C5C22⋊C411yescharacteristic
62C22×C4×C5C22⋊C411yescharacteristic
63C22×C4×C5C22×C411yescharacteristic
64C22×C4×C5C2×D412yes—
65C5×C4⋊C4—42no—
66C5×C4⋊C4—41no—
67C5×C4⋊C4—41no—
68C5×C4⋊C4—82no—
69C5×C22⋊C4—28no—
70C5×C22⋊C4—42no—
71C5×C22⋊C4—82no—
72C42×C5—22no—
73C42×C5—22no—
74C42×C5—22no—
75C42×C5—21no—
76C42×C5—41no—
77C42×C5M1612yes—
78C2×#168—44no—
79C2×#124Dic511yescharacteristic
80C2×C4×Q8C2×D511yescharacteristic
81C2×C4×D4Dic511yescharacteristic
82#432—102no—
83#391Dic511yescharacteristic
84#390C2×D511yescharacteristic
85#388—22no—
86#385—22no—
87#384C2×D511yescharacteristic
88#376Dic511yescharacteristic
89#324—54no—
90#324—102no—
91C23×C5—22no—
92C23×C5—42no—
93C23×C5—42no—
94C23×C5—41no—
95C23×C5—41no—
96C23×C5C2×C22⋊C411yescharacteristic
97C5×Q8—42no—
98C5×Q8—42no—
99C5×D4—42no—
100C5×D4—42no—
101C2×C4×C5—21no—
102C2×C4×C5—21no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5—22no—
106C2×C4×C5—21no—
107C2×C4×C5—22no—
108C2×C4×C5—22no—
109C2×C4×C5—41no—
110C2×C4×C5—41no—
111C2×C4×C5—41no—
112C2×C4×C5—42no—
113C2×C4×C5—41no—
114C2×C4×C5—41no—
115C2×C4×C5—42no—
116C2×C4×C5—41no—
117C2×C4×C5—41no—
118C2×C4×C5—41no—
119C2×C4×C5—84no—
120C2×C4×C5#9912yes—
121C2×C4×C5C2×M1611yescharacteristic
122C2×C4×C5C2×M1611yescharacteristic
123#168—84no—
124C22×Q8C2×Dic511yescharacteristic
125C22×D4C2×Dic511yescharacteristic
126C23×C4—21no—
127#124—22no—
128#124—22no—
129C4×Q8—42no—
130C4×D4—42no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4C2×Dic511yescharacteristic
133C2×C4⋊C4C22⋊D512yes—
134C2×C4⋊C4C22×D511yescharacteristic
135C2×C22⋊C4—22no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—22no—
138C2×C42C2×Dic511yescharacteristic
139C2×C42C2×Dic511yescharacteristic
140C2×C42C2×Dic511yescharacteristic
141C2×C42C22⋊D512yes—
142C4⋊C8—202no—
143C22⋊C8—104no—
144#97—202no—
145#95—21no—
146#95—22no—
147#95—21no—
148C22×C5—21no—
149C22×C5—21no—
150C22×C5—44no—
151C22×C5—42no—
152C22×C5—82no—
153C22×C5—81no—
154C22×C5—81no—
155C22×C5#40711yescharacteristic
156C22×C5C2×#9911yescharacteristic
157C22×C5#41311yescharacteristic
158C4×C5—22no—
159C4×C5—21no—
160C4×C5—21no—
161C4×C5—42no—
162C4×C5—42no—
163C4×C5—41no—
164C4×C5—42no—
165C4×C5—81no—
166C4×C5—81no—
167C24—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4#68112yes—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—42no—
180C22×C4—82no—
181C22×C4C22⋊Dic512yes—
182C22×C4C22⋊Dic511yescharacteristic
183C22×C4C22⋊Dic511yescharacteristic
184C22×C4C22×Dic511yescharacteristic
185C22×C4C2×C22⋊D512yes—
186C2×C8—204no—
187C4⋊C4—42no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—82no—
191C22⋊C4—28no—
192C22⋊C4—42no—
193C22⋊C4—82no—
194C42—22no—
195C42—22no—
196C42—22no—
197C42—21no—
198C42—41no—
199C42#68112yes—
200C2×C5—21no—
201C2×C5—21no—
202C2×C5—82no—
203C2×C5#146511yescharacteristic
204C2×C5#212811yescharacteristic
205C2×C5#213311yescharacteristic
206C23—22no—
207C23—42no—
208C23—42no—
209C23—41no—
210C23—41no—
211C23C2×C22⋊Dic511yescharacteristic
212Q8—42no—
213Q8—42no—
214D4—42no—
215D4—42no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—84no—
235C2×C4#396612yes—
236C2×C4C2×#68111yescharacteristic
237C2×C4C2×#68111yescharacteristic
238C8—404no—
239C5#1153411yescharacteristic, complemented, Sylow
240C22—21no—
241C22—21no—
242C22—44no—
243C22—42no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22#6549611yescharacteristic
248C22C2×#396611yescharacteristic
249C22#6552611yescharacteristic
250C4—21no—
251C4—21no—
252C4—22no—
253C4—42no—
254C4—42no—
255C4—42no—
256C4—41no—
257C4—81no—
258C4—81no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2#1060239511yescharacteristic
263C2#1060265411yescharacteristic
264C2#1060266511yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2326C2
socleC22×C5C2×#99
prime radicalC22×C5×D4C2×C4
cosocleC2×C5×C4⋊C4C23
hypercenter#2326D5
hypocenterC5#11534
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #10602418C2semidirect
4. #10598126C2semidirect
239. C5#11534semidirect

Maximal subgroups

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

subgroupindex
C5×#23262
#106024182
#105981312
#105981262
#115345