Finite Group Databasebeta

← #12097585

Subgroups — #12097585

Order 1280.

1022 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64994C211yesmaximal, characteristic
2C2×#64996C211yesmaximal, characteristic
3C4×#4018C211yesmaximal, characteristic
4C5×#1745C211yesmaximal, characteristic, complemented
5#10600598C211yesmaximal, characteristic, complemented
6Dic5×#97C211yesmaximal, characteristic, complemented
7C4×#3935C211yesmaximal, characteristic, complemented
8#10599445C212yesmaximal, complemented
9#10599444C212yesmaximal, complemented
10#10599443C212yesmaximal
11#10599442C212yesmaximal
12C22×#676C2212yes—
13C2×#4018C412yes—
14C2×C42×D5C2211yescharacteristic
15C2×C5×#97C2211yescharacteristic
16C2×C5×#97C2211yescharacteristic
17C43×C5C2211yescharacteristic
18C2×#3952C2212yes—
19C2×#3947C2212yes—
20C2×#3946C2212yes—
21C2×C8×Dic5C2212yes—
22C2×C8×Dic5C2212yes—
23C4×#685C412yes—
24C4×#685C2211yescharacteristic
25C4×#685C2211yescharacteristic
26#65221C412yes—
27C4×#683C412yes—
28C4×#683C2211yescharacteristic
29C4×#683C2211yescharacteristic
30C42×Dic5C2211yescharacteristic
31C2×#3935C2212yes—
32C2×#3935C2211yescharacteristic
33C2×#3935C2211yescharacteristic
34C2×C4×#168C2212yes—
35#64996C418yescomplemented
36#64994C418yescomplemented
37C5×#336C2212yes—
38C5×#336C2212yes—
39#64770C2212yes—
40#64770C2212yes—
41#20480—51nomaximal, Sylow
42C22×C4×D5C2×C411yescharacteristic
43C22×C4×D5C2×C411yescharacteristic
44C22×C4×D5C2311yescharacteristic
45C22×C5×C8C2312yes—
46C22×C5×C8C2312yes—
47C2×C42×C5C2×C412yes—
48C2×C42×C5C2×C412yes—
49C2×C42×C5C2311yescharacteristic
50C2×C42×C5C2311yescharacteristic
51C2×C42×C5C2311yescharacteristic
52C2×#685C2×C412yes—
53C2×#685C2×C412yes—
54C2×#683C2×C412yes—
55C2×#683C2×C412yes—
56C2×C4×Dic5C2×C412yes—
57C2×C4×Dic5C2×C412yes—
58C2×C4×Dic5C2×C411yescharacteristic
59C2×C4×Dic5C2×C411yescharacteristic
60C2×C4×Dic5C2311yescharacteristic
61C2×C4×Dic5C2311yescharacteristic
62C2×C4×Dic5C2311yescharacteristic
63C22×#168C2312yes—
64C22×#168C2312yes—
65C2×#676—24no—
66C2×#676—24no—
67C2×#676C2×C418yes—
68#4018—28no—
69C42×D5C2×C414yes—
70C42×D5C2×C414yes—
71C5×#97C2×C414yes—
72C5×#97C2×C414yes—
73#3963C2×C412yes—
74#3963C2×C412yes—
75#3963C2×C412yes—
76#3963C2×C412yes—
77#3952C2×C418yes—
78#3947—24no—
79#3946C2×C418yes—
80C8×Dic5—24no—
81C8×Dic5C2×C418yes—
82#3935—24no—
83#3935C2×C414yes—
84#3935C2×C414yes—
85C4×#168—24no—
86C2×#432—51no—
87C2×C4×M16—51no—
88C4×#117—51no—
89#1770—52no—
90C8×C4⋊C4—52no—
91#1753—52no—
92#1753—52no—
93#1745—51no—
94#1745D511yescharacteristic, complemented
95C4×#97—51no—
96C4×#97—51no—
97C23×D5C22×C411yescharacteristic
98C22×C4×C5C22×C412yes—
99C22×C4×C5C22×C411yescharacteristic
100C22×C4×C5C22×C411yescharacteristic
101C22×C4×C5C22×C412yes—
102C22×C4×C5C2411yescharacteristic
103C22×Dic5C22×C412yes—
104C22×Dic5C22×C411yescharacteristic
105C22×Dic5C22×C411yescharacteristic
106C22×Dic5C22×C412yes—
107C22×Dic5C22×C411yescharacteristic
108C2×C4×D5—22no—
109C2×C4×D5—22no—
110C2×C4×D5—24no—
111C2×C4×D5—42no—
112C2×C4×D5C4214yes—
113C2×C4×D5C4214yes—
114C2×C4×D5C22×C412yes—
115C2×C4×D5C22×C412yes—
116C2×C5×C8—22no—
117C2×C5×C8—22no—
118C2×C5×C8—22no—
119C2×C5×C8—22no—
120C2×C5×C8C22×C414yes—
121C2×C5×C8C22×C414yes—
122C42×C5—24no—
123C42×C5—22no—
124C42×C5—22no—
125C42×C5—42no—
126C42×C5C22×C412yes—
127C42×C5C22×C412yes—
128#685—24no—
129#685—24no—
130#683—24no—
131#683M1618yescomplemented
132C4×Dic5—22no—
133C4×Dic5—22no—
134C4×Dic5—24no—
135C4×Dic5—42no—
136C4×Dic5C4214yes—
137C4×Dic5C4214yes—
138C4×Dic5C22×C412yes—
139C4×Dic5C22×C412yes—
140C2×#168—22no—
141C2×#168—22no—
142C2×#168C22×C414yes—
143C2×#168C22×C414yes—
144C2×#168#4114yes—
145C2×#168#4114yes—
146#676—216no—
147C22×M16—52no—
148C2×#117—52no—
149C22×C42—51no—
150#432—58no—
151C2×C4⋊C8—52no—
152C2×C22⋊C8—52no—
153C4×M16—58no—
154C2×#97—51no—
155C2×#97—51no—
156C2×#97—52no—
157C2×#97—52no—
158C2×#97C2×D511yescharacteristic
159C2×#97C2×D511yescharacteristic
160C2×C4×C8—52no—
161C2×C4×C8—52no—
162C2×C4×C8—52no—
163C4×C4⋊C4—52no—
164C4×C4⋊C4—51no—
165C4×C4⋊C4—51no—
166C4×C22⋊C4—52no—
167C4×C22⋊C4—51no—
168C4×C22⋊C4—51no—
169#376—52no—
170C43—51no—
171C43C2×D511yescharacteristic
172#336—52no—
173#336—52no—
174#336C2×D512yes—
175#336C2×D512yes—
176C23×C5C23×C411yescharacteristic
177C22×D5—21no—
178C22×D5—21no—
179C22×D5—22no—
180C22×D5—22no—
181C22×D5C2×C4212yes—
182C2×C4×C5—22no—
183C2×C4×C5—22no—
184C2×C4×C5—22no—
185C2×C4×C5—22no—
186C2×C4×C5—22no—
187C2×C4×C5—22no—
188C2×C4×C5—22no—
189C2×C4×C5C2×C4212yes—
190C2×C4×C5C2×C4212yes—
191C2×C4×C5C2×M1614yes—
192C2×C4×C5C23×C411yescharacteristic
193C2×C4×C5C23×C411yescharacteristic
194C2×C4×C5C2×#4112yes—
195C2×C4×C5C2×#4112yes—
196C2×Dic5—22no—
197C2×Dic5—21no—
198C2×Dic5—21no—
199C2×Dic5—22no—
200C2×Dic5—22no—
201C2×Dic5—22no—
202C2×Dic5—22no—
203C2×Dic5—22no—
204C2×Dic5C2×C4212yes—
205C2×Dic5C2×C4212yes—
206C2×Dic5C2×C4212yes—
207C2×Dic5C2×M1614yes—
208C2×Dic5C2×M1614yes—
209C4×D5—24no—
210C4×D5—24no—
211C4×D5—28no—
212C4×D5—44no—
213C5×C8—24no—
214C5×C8—24no—
215#168—24no—
216#168#11718yescomplemented
217C23×C4—51no—
218C23×C4—51no—
219C23×C4—51no—
220C2×M16—58no—
221C2×M16—104no—
222C2×M16—104no—
223C22×C8—52no—
224C22×C8—52no—
225C22×C8C22×D512yes—
226C22×C8C22×D512yes—
227#117—108no—
228C2×C4⋊C4—52no—
229C2×C4⋊C4—52no—
230C2×C22⋊C4—52no—
231C2×C22⋊C4—52no—
232C2×C42—51no—
233C2×C42—51no—
234C2×C42—51no—
235C2×C42—54no—
236C2×C42—54no—
237C2×C42—52no—
238C2×C42—51no—
239C2×C42—51no—
240C2×C42—52no—
241C2×C42C4×D512yes—
242C2×C42C4×D512yes—
243C2×C42C22×D511yescharacteristic
244C2×C42C22×D511yescharacteristic
245C2×C42C22×D511yescharacteristic
246C4⋊C8—58no—
247C22⋊C8—58no—
248#97—54no—
249#97—54no—
250#97—104no—
251#97—104no—
252#97C4×D514yes—
253#97C4×D514yes—
254C4×C8—58no—
255C4×C8—104no—
256C4×C8—104no—
257#95—52no—
258#95—52no—
259#95—52no—
260#95—52no—
261C22×C5C22×C4211yescharacteristic
262C22×C5C4×#4111yescharacteristic
263C22×C5C4×#4111yescharacteristic
264C22×C5C22×M1612yes—
265C22×C5#56812yes—
266C2×D5—22no—
267C2×D5—22no—
268C2×D5—22no—
269C2×D5—42no—
270C2×D5—42no—
271C4×C5—22no—
272C4×C5—22no—
273C4×C5—24no—
274C4×C5—42no—
275C4×C5C2×#11712yes—
276C4×C5C4×#4112yes—
277Dic5—22no—
278Dic5—24no—
279Dic5—22no—
280Dic5—42no—
281Dic5C4×M1614yes—
282C24—51no—
283C22×C4—51no—
284C22×C4—51no—
285C22×C4—51no—
286C22×C4—52no—
287C22×C4—52no—
288C22×C4—52no—
289C22×C4—54no—
290C22×C4—54no—
291C22×C4—52no—
292C22×C4—104no—
293C22×C4—102no—
294C22×C4—102no—
295C22×C4—202no—
296C22×C4C2×C4×D512yes—
297C22×C4C2×C4×D511yescharacteristic
298C22×C4C2×C4×D511yescharacteristic
299C22×C4C2×C4×D512yes—
300C22×C4C23×D511yescharacteristic
301M16—1016no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—54no—
307C2×C8—54no—
308C2×C8—54no—
309C2×C8—54no—
310C2×C8—102no—
311C2×C8—102no—
312C2×C8C2×C4×D514yes—
313C2×C8C2×C4×D514yes—
314C4⋊C4—58no—
315C4⋊C4—104no—
316C22⋊C4—104no—
317C22⋊C4—104no—
318C42—22no—
319C42—22no—
320C42—24no—
321C42—42no—
322C42—52no—
323C42—52no—
324C42—54no—
325C42—54no—
326C42—104no—
327C42—102no—
328C42—102no—
329C42—202no—
330C42C2×C4×D512yes—
331C42C2×C4×D512yes—
332C2×C5C4×#11711yescharacteristic
333C2×C5C2×C4×M1611yescharacteristic
334C2×C5#287311yescharacteristic
335C2×C5#297012yes—
336C2×C5#297112yes—
337D5—42no—
338C23—52no—
339C23—102no—
340C23—102no—
341C23—101no—
342C23—101no—
343C23C22×C4×D511yescharacteristic
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—52no—
352C2×C4—54no—
353C2×C4—52no—
354C2×C4—52no—
355C2×C4—54no—
356C2×C4—108no—
357C2×C4—102no—
358C2×C4—102no—
359C2×C4—102no—
360C2×C4—101no—
361C2×C4—101no—
362C2×C4—104no—
363C2×C4—104no—
364C2×C4—102no—
365C2×C4—102no—
366C2×C4—102no—
367C2×C4—204no—
368C2×C4C42×D512yes—
369C2×C4C42×D512yes—
370C2×C4D5×M1614yes—
371C2×C4C22×C4×D511yescharacteristic
372C2×C4C22×C4×D511yescharacteristic
373C2×C4D5×#4112yes—
374C2×C4D5×#4112yes—
375C8—24no—
376C8—24no—
377C8—58no—
378C8—104no—
379C5#2048011yescharacteristic, complemented, Sylow
380C22—102no—
381C22—102no—
382C22—102no—
383C22—202no—
384C22—202no—
385C22C2×C42×D511yescharacteristic
386C22#6585611yescharacteristic
387C22#6585611yescharacteristic
388C22C2×D5×M1612yes—
389C22#6607912yes—
390C4—22no—
391C4—22no—
392C4—24no—
393C4—42no—
394C4—54no—
395C4—102no—
396C4—102no—
397C4—104no—
398C4—202no—
399C4D5×#11712yes—
400C4#6585612yes—
401C2—202no—
402C2#1060511311yescharacteristic
403C2C4×D5×M1611yescharacteristic
404C2#1061024511yescharacteristic
405C2#1061105812yes—
406C2#1061106012yes—
subgroupG/H
center, FrattiniC22×C4C23×D5
commutatorC22×C5C22×C42
FittingC5×#1745C2
socleC23×C5C23×C4
prime radicalC23×D5C22×C4
cosocleC22×C4×C5C24
hypercenter#1745D5
hypocenterC5#20480
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ #1745

Complements

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

NH ≅ G/Ntype
4. C5×#1745C2semidirect
5. #10600598C2semidirect
6. Dic5×#97C2semidirect
7. C4×#3935C2semidirect
8. #10599445C2semidirect
9. #10599444C2semidirect
35. #64996C4semidirect
36. #64994C4semidirect
94. #1745D5semidirect
131. #683M16semidirect
216. #168#117semidirect
379. C5#20480semidirect

Maximal subgroups

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

subgroupindex
C2×#649942
C2×#649962
C4×#40182
C5×#17452
#106005982
Dic5×#972
C4×#39352
#105994452
#105994442
#105994432
#105994422
#204805