Finite Group Databasebeta

← #12015566

Subgroups — #12015566

Order 1280.

638 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 115 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×#391C211yesmaximal, characteristic
2#10602926C212yesmaximal
3#10598135C214yesmaximal, complemented
4C23×C5×Q8C2211yescharacteristic
5C22×C5×C4⋊C4C411yescharacteristic
6C22×C42×C5C411yescharacteristic
7C5×#391—22no—
8C5×#391C412yes—
9C5×#391C2212yes—
10C2×C5×#95—22no—
11C2×#3968—22no—
12C2×#3968C2214yes—
13#65496—22no—
14#12860—51nomaximal, Sylow
15C22×C5×Q8—24no—
16C22×C5×Q8—41no—
17C22×C5×Q8—41no—
18C22×C5×Q8—42no—
19C22×C5×Q8C2×C411yescharacteristic
20C22×C5×Q8D412yes—
21C22×C5×Q8C2311yescharacteristic
22C23×C4×C5—21no—
23C23×C4×C5—21no—
24C23×C4×C5C2×C411yescharacteristic
25C23×C4×C5D412yes—
26C2×C5×C4⋊C4—21no—
27C2×C5×C4⋊C4—41no—
28C2×C5×C4⋊C4—41no—
29C2×C5×C4⋊C4C2×C412yes—
30C2×C42×C5—21no—
31C2×C42×C5—42no—
32C2×C42×C5C2×C412yes—
33C2×#681—24no—
34C5×#95—24no—
35C5×#95—24no—
36#3968—28no—
37#3968—44no—
38C22⋊#168—42no—
39C22⋊#168—42no—
40C2×#391D511yescharacteristic
41#1784—52no—
42#1394—54no—
43C24×C5C22⋊C411yescharacteristic
44C2×C5×Q8—24no—
45C2×C5×Q8—22no—
46C2×C5×Q8—24no—
47C2×C5×Q8—41no—
48C2×C5×Q8—41no—
49C2×C5×Q8—42no—
50C2×C5×Q8—41no—
51C2×C5×Q8—41no—
52C2×C5×Q8—48no—
53C2×C5×Q8—48no—
54C2×C5×Q8#4114yes—
55C22×C4×C5—21no—
56C22×C4×C5—21no—
57C22×C4×C5—22no—
58C22×C4×C5—22no—
59C22×C4×C5—24no—
60C22×C4×C5—22no—
61C22×C4×C5—22no—
62C22×C4×C5—22no—
63C22×C4×C5—24no—
64C22×C4×C5—22no—
65C22×C4×C5—42no—
66C22×C4×C5—42no—
67C22×C4×C5—42no—
68C22×C4×C5—44no—
69C22×C4×C5—41no—
70C22×C4×C5—41no—
71C22×C4×C5C22⋊C412yes—
72C22×C4×C5C22⋊C411yescharacteristic
73C22×C4×C5C22×C411yescharacteristic
74C22×C4×C5C2×D412yes—
75C5×C4⋊C4—42no—
76C5×C4⋊C4—42no—
77C42×C5—42no—
78C42×C5—42no—
79#681—44no—
80#681—44no—
81C2×#168—44no—
82C23×Q8C2×D511yescharacteristic
83C22×C4⋊C4Dic511yescharacteristic
84C22×C42Dic511yescharacteristic
85C2×#101—54no—
86C2×#101—102no—
87#407—102no—
88#391—22no—
89#391Dic512yes—
90#391C2×D512yes—
91C2×#95—22no—
92C23×C5—22no—
93C23×C5—22no—
94C23×C5—21no—
95C23×C5—21no—
96C23×C5#9911yescharacteristic
97C23×C5#9911yescharacteristic
98C23×C5C2×C22⋊C411yescharacteristic
99C5×Q8—44no—
100C5×Q8—48no—
101C5×Q8—44no—
102C2×C4×C5—24no—
103C2×C4×C5—24no—
104C2×C4×C5—21no—
105C2×C4×C5—22no—
106C2×C4×C5—22no—
107C2×C4×C5—21no—
108C2×C4×C5—21no—
109C2×C4×C5—21no—
110C2×C4×C5—24no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—44no—
115C2×C4×C5—44no—
116C2×C4×C5—48no—
117C2×C4×C5—42no—
118C2×C4×C5—42no—
119C2×C4×C5—44no—
120C2×C4×C5—42no—
121C2×C4×C5—82no—
122C2×C4×C5—82no—
123C2×C4×C5#9912yes—
124C2×C4×C5#11712yes—
125C2×C4×C5#12314yes—
126#168—84no—
127C22×Q8—24no—
128C22×Q8—41no—
129C22×Q8—41no—
130C22×Q8—42no—
131C22×Q8C2×Dic511yescharacteristic
132C22×Q8C22⋊D512yes—
133C22×Q8C22×D511yescharacteristic
134C23×C4—21no—
135C23×C4—21no—
136C23×C4C2×Dic511yescharacteristic
137C23×C4C22⋊D512yes—
138C2×M16—104no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4C2×Dic512yes—
143C2×C42—21no—
144C2×C42—42no—
145C2×C42C2×Dic512yes—
146#101—108no—
147#101—204no—
148C22⋊C8—202no—
149C22⋊C8—202no—
150#95—24no—
151#95—24no—
152C22×C5—24no—
153C22×C5—24no—
154C22×C5—21no—
155C22×C5—21no—
156C22×C5—21no—
157C22×C5—21no—
158C22×C5—21no—
159C22×C5—21no—
160C22×C5#35512yes—
161C22×C5#35612yes—
162C22×C5#38111yescharacteristic
163C22×C5C2×#9911yescharacteristic
164C22×C5C2×#9911yescharacteristic
165C4×C5—24no—
166C4×C5—22no—
167C4×C5—22no—
168C4×C5—44no—
169C4×C5—42no—
170C4×C5—82no—
171C24C22⋊Dic511yescharacteristic
172C2×Q8—24no—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—42no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—48no—
181C2×Q8—48no—
182C2×Q8#70914yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4C22⋊Dic512yes—
200C22×C4C22⋊Dic511yescharacteristic
201C22×C4C22×Dic511yescharacteristic
202C22×C4C2×C22⋊D512yes—
203M16—204no—
204M16—204no—
205C2×C8—204no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C42—42no—
209C42—42no—
210C2×C5—21no—
211C2×C5—22no—
212C2×C5—22no—
213C2×C5—21no—
214C2×C5#181911yescharacteristic
215C2×C5C2×#35511yescharacteristic
216C2×C5C2×#35611yescharacteristic
217C23—22no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23#396611yescharacteristic
222C23#396611yescharacteristic
223C23C2×C22⋊Dic511yescharacteristic
224Q8—44no—
225Q8—48no—
226Q8—44no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—48no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4#396612yes—
249C2×C4#407212yes—
250C2×C4#407514yes—
251C8—404no—
252C5#1286011yescharacteristic, complemented, Sylow
253C22—24no—
254C22—24no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22#6476212yes—
262C22#6476812yes—
263C22#6550011yescharacteristic
264C22C2×#396611yescharacteristic
265C22C2×#396611yescharacteristic
266C4—24no—
267C4—22no—
268C4—22no—
269C4—44no—
270C4—42no—
271C4—82no—
272C2—21no—
273C2—22no—
274C2—21no—
275C2—22no—
276C2#1060233511yescharacteristic
277C2C2×#6476211yescharacteristic
278C2C2×#6476811yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC22×Q8C22×D5
FittingC2×C5×#391C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC22×C5×Q8C23
hypercenterC2×#391D5
hypocenterC5#12860
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. #10598135C2semidirect
252. C5#12860semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3912
#106029262
#105981352
#128605