Finite Group Databasebeta

← #12097537

Subgroups — #12097537

Order 1280.

906 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#405C211yesmaximal, characteristic
2#10613960C211yesmaximal, characteristic
3C2×#65289C211yesmaximal, characteristic
4#10602776C212yesmaximal
5#10602760C212yesmaximal
6#10600603C212yesmaximal, complemented
7#10600599C211yesmaximal, characteristic, complemented
8#10600598C211yesmaximal, characteristic, complemented
9#10600550C212yesmaximal, complemented
10#10600542C211yesmaximal, characteristic, complemented
11C4×#3947C211yesmaximal, characteristic, complemented
12C22×C5×M16C2211yescharacteristic
13C23×C5×C8C2211yescharacteristic
14C2×C5×#117C2211yescharacteristic
15C5×#405C418yescomplemented
16C2×C5×#97C2211yescharacteristic
17C2×C5×#97C2211yescharacteristic
18C2×C4×C5×C8C2211yescharacteristic
19C2×C4×C5×C8C2211yescharacteristic
20C4×C22⋊Dic5C2212yes—
21C2×C22⋊#168C2212yes—
22C2×#3947C2212yes—
23C2×#3947C2212yes—
24C2×#3947C2212yes—
25C2×C8×Dic5C2212yes—
26#65289C418yescomplemented
27#65270C414yes—
28#65240C414yes—
29C4×C4⋊Dic5C2212yes—
30#65221C2211yescharacteristic
31C42×Dic5C2211yescharacteristic
32C2×C4⋊#168C2212yes—
33C2×#3935C2211yescharacteristic
34C2×C4×#168C2211yescharacteristic
35#64770C2212yes—
36#64770C2212yes—
37#64770C2212yes—
38#64770C2212yes—
39#20475—51nomaximal, Sylow
40C23×C4×C5C2311yescharacteristic
41C2×C5×M16—24no—
42C2×C5×M16C2×C414yes—
43C22×C5×C8—24no—
44C22×C5×C8C2×C414yes—
45C22×C5×C8C2311yescharacteristic
46C22×C5×C8C2311yescharacteristic
47C22×C5×C8C2311yescharacteristic
48C22×C5×C8C2311yescharacteristic
49C5×#117C2×C414yes—
50C5×#117C2×C414yes—
51C2×C5×C4⋊C4C2×C412yes—
52C2×C5×C22⋊C4C2×C412yes—
53C2×C42×C5C2311yescharacteristic
54C2×C42×C5C2311yescharacteristic
55C2×C22⋊Dic5C2×C414yes—
56C2×C4⋊Dic5C2×C414yes—
57C2×C4×Dic5C2×C414yes—
58C2×C4×Dic5C2312yes—
59C2×C4×Dic5C2312yes—
60C22×#168C2312yes—
61C22×#168C2312yes—
62C5×#97C2×C414yes—
63C5×#97C2×C414yes—
64C4×C5×C8C2×C414yes—
65C4×C5×C8C2×C414yes—
66#3963C2×C414yes—
67#3963C2×C414yes—
68#3963C2×C414yes—
69C22⋊#168C2×C418yes—
70#3947—24no—
71#3947—24no—
72#3947—24no—
73C8×Dic5—24no—
74C4⋊#168C2×C418yes—
75#3935C2×C414yes—
76C4×#168C2×C414yes—
77C2×#433—51no—
78C2×#405D511yescharacteristic
79#2298—51no—
80#1771—52no—
81#1770—52no—
82#1753—52no—
83#1752—52no—
84#1746—51no—
85#1745—51no—
86#1726—51no—
87C4×#97—51no—
88C24×C5C22×C411yescharacteristic
89C22×C4×C5—22no—
90C22×C4×C5—22no—
91C22×C4×C5C22×C412yes—
92C22×C4×C5C22×C412yes—
93C22×C4×C5C22×C411yescharacteristic
94C22×C4×C5C22×C412yes—
95C22×C4×C5C22×C412yes—
96C22×C4×C5C2411yescharacteristic
97C22×Dic5C22×C414yes—
98C22×Dic5C22×C414yes—
99C5×M16—28no—
100C2×C5×C8—22no—
101C2×C5×C8—22no—
102C2×C5×C8—28no—
103C2×C5×C8C22×C412yes—
104C2×C5×C8C22×C412yes—
105C2×C5×C8C22×C412yes—
106C2×C5×C8C22×C412yes—
107C2×C5×C8#4114yes—
108C2×C5×C8#4114yes—
109C5×C4⋊C4C4218yes—
110C5×C22⋊C4C4218yes—
111C42×C5C22×C412yes—
112C42×C5C22×C412yes—
113C42×C5C22×C412yes—
114C42×C5C22×C412yes—
115C22⋊Dic5—44no—
116C4⋊Dic5—44no—
117C4×Dic5—22no—
118C4×Dic5—22no—
119C4×Dic5—22no—
120C4×Dic5—22no—
121C4×Dic5—44no—
122C2×#168—22no—
123C2×#168—22no—
124C2×#168—22no—
125C2×#168—22no—
126C2×#168C22×C414yes—
127C2×#168C22×C414yes—
128C22×M16C2×D511yescharacteristic
129C23×C8C2×D511yescharacteristic
130C2×#117C2×D511yescharacteristic
131#433—58no—
132C2×C4⋊C8—52no—
133C2×C22⋊C8—52no—
134#405Dic518yescomplemented
135C2×#97—51no—
136C2×#97—52no—
137C2×#97—52no—
138C2×#97—52no—
139C2×#97C2×D511yescharacteristic
140C2×#97C2×D511yescharacteristic
141C2×C4×C8—51no—
142C2×C4×C8—52no—
143C2×C4×C8C2×D511yescharacteristic
144C2×C4×C8C2×D511yescharacteristic
145#388—54no—
146#387—54no—
147C4×C4⋊C4—52no—
148C4×C22⋊C4—52no—
149#376—51no—
150C43—51no—
151#336—52no—
152#336—52no—
153#336—52no—
154#336—52no—
155C23×C5—22no—
156C23×C5—24no—
157C23×C5C2×C4212yes—
158C23×C5C23×C411yescharacteristic
159C2×C4×C5—22no—
160C2×C4×C5—24no—
161C2×C4×C5—24no—
162C2×C4×C5—22no—
163C2×C4×C5—22no—
164C2×C4×C5—24no—
165C2×C4×C5C2×C4214yes—
166C2×C4×C5C2×C4214yes—
167C2×C4×C5C2×C4212yes—
168C2×C4×C5C23×C411yescharacteristic
169C2×C4×C5C23×C411yescharacteristic
170C2×C4×C5C2×#4112yes—
171C2×C4×C5C2×#4112yes—
172C2×Dic5—24no—
173C2×Dic5—24no—
174C2×Dic5—24no—
175C2×Dic5—24no—
176C2×Dic5—24no—
177C2×Dic5—24no—
178C5×C8—22no—
179C5×C8—22no—
180C5×C8#11714yes—
181C5×C8#11714yes—
182#168—24no—
183#168—24no—
184C23×C4C22×D511yescharacteristic
185C2×M16—24no—
186C2×M16C2×Dic514yes—
187C22×C8—24no—
188C22×C8—52no—
189C22×C8—52no—
190C22×C8C2×Dic514yes—
191C22×C8C22×D511yescharacteristic
192C22×C8C22×D511yescharacteristic
193C22×C8C22×D511yescharacteristic
194C22×C8C22×D511yescharacteristic
195#117C4×D514yes—
196#117C2×Dic514yes—
197C2×C4⋊C4—54no—
198C2×C4⋊C4C4×D512yes—
199C2×C22⋊C4—54no—
200C2×C22⋊C4C4×D512yes—
201C2×C42—52no—
202C2×C42—52no—
203C2×C42—54no—
204C2×C42C22×D511yescharacteristic
205C2×C42C22×D511yescharacteristic
206C4⋊C8—58no—
207C22⋊C8—58no—
208#97—54no—
209#97—104no—
210#97—104no—
211#97—104no—
212#97C2×Dic514yes—
213#97C2×Dic514yes—
214C4×C8—54no—
215C4×C8—104no—
216C4×C8C2×Dic514yes—
217C4×C8C2×Dic514yes—
218#95—54no—
219#95—54no—
220#95—54no—
221C22×C5—24no—
222C22×C5—22no—
223C22×C5—44no—
224C22×C5C22×C4211yescharacteristic
225C22×C5C4×#4112yes—
226C22×C5#56814yes—
227C4×C5—24no—
228C4×C5—24no—
229C4×C5—22no—
230C4×C5C2×#11711yescharacteristic
231C4×C5C2×#11711yescharacteristic
232C4×C5C4×#4112yes—
233Dic5—44no—
234Dic5—44no—
235C24C2×C4×D511yescharacteristic
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—54no—
239C22×C4—54no—
240C22×C4C2×C4×D512yes—
241C22×C4C2×C4×D511yescharacteristic
242C22×C4C2×C4×D512yes—
243C22×C4C2×C4×D512yes—
244C22×C4C22×Dic512yes—
245C22×C4C23×D511yescharacteristic
246M16—28no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—28no—
250C2×C8—54no—
251C2×C8—54no—
252C2×C8—102no—
253C2×C8—102no—
254C2×C8—102no—
255C2×C8—102no—
256C2×C8#70914yes—
257C2×C8#70914yes—
258C2×C8C22×Dic512yes—
259C2×C8C22×Dic512yes—
260C2×C8C22×Dic512yes—
261C2×C8C22×Dic512yes—
262C4⋊C4—204no—
263C4⋊C4C4×Dic518yes—
264C22⋊C4—204no—
265C22⋊C4C4×Dic518yes—
266C42—102no—
267C42—102no—
268C42—102no—
269C42—102no—
270C42—204no—
271C42C2×C4×D512yes—
272C42C2×C4×D512yes—
273C42C22×Dic512yes—
274C42C22×Dic512yes—
275C2×C5—42no—
276C2×C5C4×#11711yescharacteristic
277C2×C5#287512yes—
278C2×C5#296714yes—
279C23—22no—
280C23—24no—
281C23C2×C4×Dic512yes—
282C23C22×C4×D511yescharacteristic
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—104no—
290C2×C4—104no—
291C2×C4—104no—
292C2×C4—104no—
293C2×C4—104no—
294C2×C4—104no—
295C2×C4C2×C4×Dic514yes—
296C2×C4C2×C4×Dic514yes—
297C2×C4C2×C4×Dic512yes—
298C2×C4C22×C4×D511yescharacteristic
299C2×C4C2×#70912yes—
300C2×C4C2×#70912yes—
301C2×C4C23×Dic511yescharacteristic
302C8—22no—
303C8—22no—
304C8—104no—
305C8—104no—
306C8#407214yes—
307C8#407214yes—
308C5#2047511yescharacteristic, complemented, Sylow
309C22—22no—
310C22—24no—
311C22—44no—
312C22C4×#70912yes—
313C22#6608414yes—
314C22C22×C4×Dic511yescharacteristic
315C4—22no—
316C4—24no—
317C4—24no—
318C4—204no—
319C4—204no—
320C4C4×#70912yes—
321C4C2×#407211yescharacteristic
322C4C2×#407211yescharacteristic
323C2—42no—
324C2#1061028814yes—
325C2C4×#407211yescharacteristic
326C2#1061551312yes—
subgroupG/H
center, FrattiniC22×C4C23×D5
commutatorC22×C5C22×C42
FittingC2×C5×#405C2
socleC23×C5C23×C4
prime radicalC24×C5C22×C4
cosocleC22×C4×C5C24
hypercenterC2×#405D5
hypocenterC5#20475
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
6. #10600603C2semidirect
7. #10600599C2semidirect
8. #10600598C2semidirect
9. #10600550C2semidirect
10. #10600542C2semidirect
11. C4×#3947C2semidirect
15. C5×#405C4semidirect
26. #65289C4semidirect
134. #405Dic5semidirect
308. C5#20475semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#4052
#106139602
C2×#652892
#106027762
#106027602
#106006032
#106005992
#106005982
#106005502
#106005422
C4×#39472
#204755