Finite Group Databasebeta

← #12008792

Subgroups — #12008792

Order 1280.

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

normal subgroups only

This group has 440 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#10602395C212yesmaximal
3#10598134C212yesmaximal, complemented
4#10598126C212yesmaximal, complemented
5C2×C5×#124C411yescharacteristic
6C2×C4×C5×Q8C411yescharacteristic
7C2×C4×C5×D4C2211yescharacteristic
8C5×#391C2211yescharacteristic
9C5×#390C411yescharacteristic
10C5×#388—22no—
11C5×#385—22no—
12C5×#384C2211yescharacteristic
13C5×#376C411yescharacteristic
14#65496—22no—
15#64748C2214yes—
16#64745—22no—
17#11535—51nomaximal, Sylow
18C22×C5×Q8C2×C411yescharacteristic
19C22×C5×D4C2×C411yescharacteristic
20C23×C4×C5D412yes—
21C5×#124—22no—
22C5×#124—22no—
23C4×C5×Q8—42no—
24C4×C5×D4—42no—
25C2×C5×C4⋊C4—21no—
26C2×C5×C4⋊C4—22no—
27C2×C5×C4⋊C4C2×C411yescharacteristic
28C2×C5×C4⋊C4C2311yescharacteristic
29C2×C5×C22⋊C4—22no—
30C2×C5×C22⋊C4—22no—
31C2×C5×C22⋊C4D412yes—
32C2×C42×C5—21no—
33C2×C42×C5C2×C411yescharacteristic
34C2×C42×C5C2×C411yescharacteristic
35C2×C42×C5C2×C411yescharacteristic
36C2×#681—42no—
37C5×#95—21no—
38C5×#95—21no—
39C5×#95—22no—
40C22⋊#168—24no—
41C22⋊#168—42no—
42#2326D511yescharacteristic
43#1465—52no—
44#1325—52no—
45#1319—52no—
46C24×C5C22⋊C412yes—
47C2×C5×Q8—21no—
48C2×C5×Q8—41no—
49C2×C5×Q8—41no—
50C2×C5×Q8M1612yescomplemented
51C2×C5×D4—21no—
52C2×C5×D4—21no—
53C2×C5×D4—41no—
54C2×C5×D4—41no—
55C22×C4×C5—21no—
56C22×C4×C5—21no—
57C22×C4×C5—22no—
58C22×C4×C5—22no—
59C22×C4×C5—21no—
60C22×C4×C5—21no—
61C22×C4×C5—82no—
62C22×C4×C5C22⋊C411yescharacteristic
63C22×C4×C5C22⋊C411yescharacteristic
64C22×C4×C5C22×C411yescharacteristic
65C22×C4×C5C2×D412yes—
66C5×C4⋊C4—41no—
67C5×C4⋊C4—41no—
68C5×C4⋊C4—42no—
69C5×C4⋊C4—82no—
70C5×C22⋊C4—28no—
71C5×C22⋊C4—42no—
72C5×C22⋊C4—82no—
73C42×C5—22no—
74C42×C5—21no—
75C42×C5—41no—
76C42×C5—42no—
77C42×C5M1612yescomplemented
78#681—84no—
79C2×#168—44no—
80C2×#124Dic511yescharacteristic
81C2×C4×Q8Dic511yescharacteristic
82C2×C4×D4C2×D511yescharacteristic
83#407—102no—
84#391C2×D511yescharacteristic
85#390Dic511yescharacteristic
86#388—22no—
87#385—22no—
88#384C2×D511yescharacteristic
89#376Dic511yescharacteristic
90#324—54no—
91#323—102no—
92C23×C5—22no—
93C23×C5—22no—
94C23×C5—22no—
95C23×C5—42no—
96C23×C5—42no—
97C23×C5C2×C22⋊C411yescharacteristic
98C5×Q8—42no—
99C5×Q8—42no—
100C5×D4—42no—
101C5×D4—42no—
102C2×C4×C5—21no—
103C2×C4×C5—21no—
104C2×C4×C5—22no—
105C2×C4×C5—22no—
106C2×C4×C5—22no—
107C2×C4×C5—21no—
108C2×C4×C5—41no—
109C2×C4×C5—44no—
110C2×C4×C5—41no—
111C2×C4×C5—41no—
112C2×C4×C5—41no—
113C2×C4×C5—41no—
114C2×C4×C5—42no—
115C2×C4×C5—41no—
116C2×C4×C5—41no—
117C2×C4×C5—42no—
118C2×C4×C5—41no—
119C2×C4×C5—41no—
120C2×C4×C5—41no—
121C2×C4×C5—82no—
122C2×C4×C5#9912yes—
123C2×C4×C5C2×M1611yescharacteristic
124C2×C4×C5C2×M1611yescharacteristic
125#168—84no—
126C22×Q8C2×Dic511yescharacteristic
127C22×D4C2×Dic511yescharacteristic
128C23×C4C22⋊D512yes—
129C2×M16—202no—
130#124—22no—
131#124—22no—
132C4×Q8—42no—
133C4×D4—42no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4C2×Dic511yescharacteristic
137C2×C4⋊C4C22×D511yescharacteristic
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4C22⋊D512yes—
141C2×C42—21no—
142C2×C42C2×Dic511yescharacteristic
143C2×C42C2×Dic511yescharacteristic
144C2×C42C2×Dic511yescharacteristic
145C22⋊C8—104no—
146C22⋊C8—202no—
147#95—21no—
148#95—21no—
149#95—22no—
150C22×C5—21no—
151C22×C5—21no—
152C22×C5—44no—
153C22×C5—42no—
154C22×C5—42no—
155C22×C5—42no—
156C22×C5—82no—
157C22×C5#40711yescharacteristic
158C22×C5C2×#9911yescharacteristic
159C22×C5#41311yescharacteristic
160C4×C5—22no—
161C4×C5—21no—
162C4×C5—21no—
163C4×C5—42no—
164C4×C5—42no—
165C4×C5—41no—
166C4×C5—42no—
167C4×C5—81no—
168C4×C5—81no—
169C24C22⋊Dic512yes—
170C2×Q8—21no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8#68112yescomplemented
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—41no—
177C2×D4—41no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—82no—
185C22×C4C22⋊Dic511yescharacteristic
186C22×C4C22⋊Dic511yescharacteristic
187C22×C4C22×Dic511yescharacteristic
188C22×C4C2×C22⋊D512yes—
189M16—404no—
190C2×C8—204no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—82no—
195C22⋊C4—28no—
196C22⋊C4—42no—
197C22⋊C4—82no—
198C42—22no—
199C42—21no—
200C42—41no—
201C42—42no—
202C42#68112yescomplemented
203C2×C5—21no—
204C2×C5—21no—
205C2×C5—82no—
206C2×C5#146511yescharacteristic
207C2×C5#212711yescharacteristic
208C2×C5#213511yescharacteristic
209C23—22no—
210C23—22no—
211C23—22no—
212C23—42no—
213C23—42no—
214C23C2×C22⋊Dic511yescharacteristic
215Q8—42no—
216Q8—42no—
217D4—42no—
218D4—42no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—44no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—82no—
239C2×C4#396612yes—
240C2×C4C2×#68111yescharacteristic
241C2×C4C2×#68111yescharacteristic
242C8—404no—
243C5#1153511yescharacteristic, complemented, Sylow
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22#6549611yescharacteristic
252C22C2×#396611yescharacteristic
253C22#6552611yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#1060239511yescharacteristic
267C2#1060265311yescharacteristic
268C2#1060266611yescharacteristic
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#11535
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2326

Complements

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

NH ≅ G/Ntype
3. #10598134C2semidirect
4. #10598126C2semidirect
50. C2×C5×Q8M16semidirect
77. C42×C5M16semidirect
173. C2×Q8#681semidirect
202. C42#681semidirect
243. C5#11535semidirect

Maximal subgroups

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

subgroupindex
C5×#23262
#106023952
#105981342
#105981262
#115355