Finite Group Databasebeta

← #12009856

Subgroups — #12009856

Order 1280.

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

normal subgroups only

This group has 482 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×#2486C211yesmaximal, characteristic
2C4×#3966C211yesmaximal, characteristic
3#10602396C211yesmaximal, characteristic
4#10598140C214yesmaximal, complemented
5C2×C5×#123—21no—
6C2×C4×C5×D4C2211yescharacteristic
7C22×C5×C4⋊C4C411yescharacteristic, complemented
8C5×#398C412yescomplemented
9C5×#397C2212yes—
10C5×#389—21no—
11C5×#381—21no—
12C5×#380—21no—
13C5×#380C411yescharacteristic, complemented
14C2×#3966—21no—
15C4×C22⋊Dic5—21no—
16#65215—21no—
17#64746C2212yes—
18#64745—21no—
19#64745C2212yes—
20#11773—51nomaximal, Sylow
21C22×C5×D4D411yescharacteristic
22C23×C4×C5—21no—
23C23×C4×C5C2×C411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C5×#123—44no—
26C4×C5×D4—41no—
27C4×C5×D4—41no—
28C2×C5×C4⋊C4—21no—
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4—41no—
32C2×C5×C4⋊C4—41no—
33C2×C5×C4⋊C4C2×C412yes—
34C2×C5×C4⋊C4C2×C411yescharacteristic
35C2×C5×C22⋊C4—21no—
36C2×C5×C22⋊C4—22no—
37C2×C5×C22⋊C4D411yescharacteristic
38C2×C5×C22⋊C4C2311yescharacteristic
39C2×C42×C5D411yescharacteristic
40C2×C22⋊Dic5—22no—
41C2×C4×Dic5—41no—
42C5×#95—22no—
43C5×#95—22no—
44C5×#95C2×C412yes—
45#3966—24no—
46#3963—41no—
47C22⋊#168—22no—
48C4⋊#168—41no—
49#3935—41no—
50#2486D511yescharacteristic
51C4×#99—51no—
52#1466—51no—
53#1344—54no—
54C24×C5C22⋊C411yescharacteristic
55C24×C5C2×D411yescharacteristic
56C2×C5×D4—22no—
57C2×C5×D4—22no—
58C2×C5×D4—41no—
59C22×C4×C5—21no—
60C22×C4×C5—21no—
61C22×C4×C5—21no—
62C22×C4×C5—21no—
63C22×C4×C5—22no—
64C22×C4×C5—22no—
65C22×C4×C5—42no—
66C22×C4×C5—42no—
67C22×C4×C5—42no—
68C22×C4×C5—42no—
69C22×C4×C5—42no—
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C22⋊C411yescharacteristic
72C22×C4×C5C22⋊C411yescharacteristic
73C22×C4×C5C22×C411yescharacteristic
74C22×C4×C5C2×D411yescharacteristic
75C22×Dic5—42no—
76C5×C4⋊C4—41no—
77C5×C4⋊C4—42no—
78C5×C4⋊C4—42no—
79C5×C4⋊C4—44no—
80C5×C4⋊C4—41no—
81C5×C4⋊C4—41no—
82C5×C22⋊C4—41no—
83C5×C22⋊C4—41no—
84C5×C22⋊C4—42no—
85C5×C22⋊C4—44no—
86C42×C5—21no—
87C42×C5—21no—
88C22⋊Dic5—24no—
89C22⋊Dic5—42no—
90C4×Dic5—82no—
91C2×#168—42no—
92C2×#123—21no—
93C2×C4×D4C2×D511yescharacteristic
94C22×C4⋊C4Dic511yescharacteristic, complemented
95#432—101no—
96C2×#99—101no—
97#398Dic512yescomplemented
98#397C2×D512yes—
99#389—21no—
100#381—21no—
101#380—21no—
102#380Dic511yescharacteristic, complemented
103C4×C22⋊C4—101no—
104#342—52no—
105#323—52no—
106#323—101no—
107C23×C5—21no—
108C23×C5—21no—
109C23×C5—21no—
110C23×C5—21no—
111C23×C5—41no—
112C23×C5—41no—
113C23×C5—41no—
114C23×C5—41no—
115C23×C5#10412yes—
116C23×C5C2×C22⋊C411yescharacteristic
117C23×C5AGL(1,ℤ/8)12yes—
118C5×D4—44no—
119C2×C4×C5—22no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—42no—
127C2×C4×C5—42no—
128C2×C4×C5—42no—
129C2×C4×C5—41no—
130C2×C4×C5—41no—
131C2×C4×C5—41no—
132C2×C4×C5—42no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—42no—
137C2×C4×C5—42no—
138C2×C4×C5—81no—
139C2×C4×C5—84no—
140C2×C4×C5#9911yescharacteristic
141C2×C4×C5#9911yescharacteristic
142C2×Dic5—44no—
143C2×Dic5—82no—
144C2×Dic5—82no—
145#168—82no—
146C22×D4C22⋊D511yescharacteristic
147C23×C4—21no—
148C23×C4C2×Dic511yescharacteristic
149C23×C4C22⋊D511yescharacteristic
150#123—44no—
151C4×D4—41no—
152C4×D4—41no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4C2×Dic512yes—
159C2×C4⋊C4C2×Dic511yescharacteristic
160C2×C22⋊C4—21no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—102no—
163C2×C22⋊C4C22⋊D511yescharacteristic
164C2×C22⋊C4C22×D511yescharacteristic
165C2×C42—201no—
166C2×C42C22⋊D511yescharacteristic
167C4⋊C8—201no—
168#99—104no—
169C22⋊C8—102no—
170#97—201no—
171#95—22no—
172#95—22no—
173#95—201no—
174#95C2×Dic512yes—
175C22×C5—21no—
176C22×C5—21no—
177C22×C5—22no—
178C22×C5—22no—
179C22×C5—42no—
180C22×C5—42no—
181C22×C5—42no—
182C22×C5—41no—
183C22×C5—41no—
184C22×C5—41no—
185C22×C5—41no—
186C22×C5—81no—
187C22×C5C2×#9911yescharacteristic
188C22×C5#41811yescharacteristic
189C22×C5C2×#10411yescharacteristic
190C4×C5—21no—
191C4×C5—41no—
192C4×C5—41no—
193C4×C5—41no—
194C4×C5—41no—
195C4×C5—82no—
196C4×C5—82no—
197C4×C5#35212yes—
198Dic5—84no—
199C24C22⋊Dic511yescharacteristic
200C24C2×C22⋊D511yescharacteristic
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—41no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—202no—
216C22×C4C22⋊Dic511yescharacteristic
217C22×C4C22⋊Dic511yescharacteristic
218C22×C4C22⋊Dic511yescharacteristic
219C22×C4C22×Dic511yescharacteristic
220C22×C4C2×C22⋊D511yescharacteristic
221C2×C8—202no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—44no—
225C4⋊C4—42no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—44no—
232C22⋊C4—104no—
233C22⋊C4—202no—
234C42—21no—
235C42—21no—
236C42—402no—
237C2×C5—21no—
238C2×C5—21no—
239C2×C5—42no—
240C2×C5—42no—
241C2×C5#150711yescharacteristic
242C2×C5C2×#35211yescharacteristic
243C2×C5#212911yescharacteristic
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23#396912yes—
253C23#407812yes—
254C23C2×C22⋊Dic511yescharacteristic
255D4—44no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—81no—
276C2×C4—84no—
277C2×C4—204no—
278C2×C4—402no—
279C2×C4—402no—
280C2×C4#396611yescharacteristic
281C2×C4#396611yescharacteristic
282C8—402no—
283C5#1177311yescharacteristic, complemented, Sylow
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22#6550411yescharacteristic
297C22C2×#396611yescharacteristic
298C22C2×#396911yescharacteristic
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4—82no—
306C4—404no—
307C4#6475912yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#1060246611yescharacteristic
313C2C2×#6475911yescharacteristic
314C2#1060266411yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2486C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2486D5
hypocenterC5#11773
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
4. #10598140C2semidirect
7. C22×C5×C4⋊C4C4semidirect
8. C5×#398C4semidirect
13. C5×#380C4semidirect
94. C22×C4⋊C4Dic5semidirect
97. #398Dic5semidirect
102. #380Dic5semidirect
283. C5#11773semidirect

Maximal subgroups

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

subgroupindex
C5×#24862
C4×#39662
#106023962
#105981402
#117735