Finite Group Databasebeta

← #12098884

Subgroups — #12098884

Order 1280.

1590 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#676C211yesmaximal, characteristic
2C2×#65170C212yesmaximal, complemented
3D5×#380C211yesmaximal, characteristic
4#10605593C211yesmaximal, characteristic
5C5×#1808C211yesmaximal, characteristic, complemented
6#10602763C211yesmaximal, characteristic, complemented
7#10601424C212yesmaximal, complemented
8#10600305C211yesmaximal, characteristic, complemented
9#10600304C211yesmaximal, characteristic, complemented
10#10599663C212yesmaximal, complemented
11#10599660C212yesmaximal, complemented
12C23×C4×D5C2211yescharacteristic
13C23×C5×C8C2211yescharacteristic
14C23×#168C2211yescharacteristic
15C22×#676—28no—
16C22×#676C418yescomplemented
17C22×#676C2211yescharacteristic
18C22×#676C2211yescharacteristic
19C22×#676C2211yescharacteristic
20C22×#676C2211yescharacteristic
21C2×#4038C2212yes—
22C2×D5×C4⋊C4C2212yes—
23C2×D5×C22⋊C4—21no—
24C2×C5×#107C2212yescomplemented
25C5×#380C2211yescharacteristic
26C5×#380C2211yescharacteristic
27#65501C2211yescharacteristic
28#65501C2211yescharacteristic
29C2×#3949C2212yescomplemented
30#65276C2212yes—
31#65276C2212yes—
32C2×#3940C2212yescomplemented
33C2×#3939C2212yescomplemented
34#65244—21no—
35#65231—21no—
36C22⋊C4×Dic5—21no—
37#65170—28no—
38#64956—21no—
39#64955—21no—
40#64953—21no—
41D5×#95—21no—
42C5×#340C2212yescomplemented
43#64771C2212yescomplemented
44#64749C2212yescomplemented
45#64749C2212yescomplemented
46#20661—51nomaximal, Sylow
47C24×D5D411yescharacteristic
48C23×C4×C5C2311yescharacteristic
49C23×Dic5D411yescharacteristic
50C22×C4×D5—24no—
51C22×C4×D5—24no—
52C22×C4×D5—22no—
53C22×C4×D5C2×C414yes—
54C22×C4×D5D411yescharacteristic
55C22×C4×D5D411yescharacteristic
56C22×C4×D5Q814yes—
57C22×C4×D5C2311yescharacteristic
58C22×C4×D5C2311yescharacteristic
59C22×C5×C8—24no—
60C22×C5×C8C2×C414yescomplemented
61C22×C5×C8C2311yescharacteristic
62C22×C5×C8C2311yescharacteristic
63C2×C5×C4⋊C4C2312yes—
64C2×C5×C4⋊C4C2312yes—
65C2×C5×C22⋊C4—21no—
66C2×C5×C22⋊C4—21no—
67C2×C22⋊Dic5—21no—
68C2×C22⋊Dic5—21no—
69C2×#685—22no—
70C2×#685—22no—
71C2×#685—22no—
72C2×C4⋊Dic5C2312yes—
73C2×C4⋊Dic5C2312yes—
74C2×#683—22no—
75C2×C4×Dic5—22no—
76C22×#168—24no—
77C22×#168C2×C414yescomplemented
78C22×#168C2311yescharacteristic
79C22×#168C2311yescharacteristic
80C2×#676—24no—
81C2×#676—24no—
82C2×#676—24no—
83C2×#676—232no—
84C2×#676C2×C414yes—
85C2×#676C2×C414yes—
86C2×#676C2×C414yescomplemented
87C2×#676C2×C414yescomplemented
88C2×#676D418yes—
89#4038—24no—
90#4038—24no—
91D5×C4⋊C4—24no—
92D5×C4⋊C4—24no—
93D5×C22⋊C4—44no—
94C5×#107—24no—
95C5×#95—21no—
96C5×#95—21no—
97#3963—21no—
98#3963—21no—
99#3963—22no—
100#3963—22no—
101#3949—24no—
102#3940—24no—
103#3939—24no—
104C23×M16—51no—
105C2×#428—52no—
106#2280—51no—
107C2×#380—51no—
108#1809—52no—
109#1809—52no—
110#1808—51no—
111#1808D511yescharacteristic, complemented
112#1807—51no—
113#1807—51no—
114#1736—52no—
115C24×C5C2×D411yescharacteristic
116C23×D5—22no—
117C23×D5—22no—
118C23×D5—24no—
119C23×D5—42no—
120C23×D5C4⋊C414yes—
121C23×D5C2×D411yescharacteristic
122C23×D5C2×D411yescharacteristic
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5C22×C412yes—
128C22×C4×C5C2×D411yescharacteristic
129C22×C4×C5C2×Q812yes—
130C22×C4×C5C2411yescharacteristic
131C22×Dic5—24no—
132C22×Dic5—22no—
133C22×Dic5—22no—
134C22×Dic5C4⋊C414yes—
135C22×Dic5C2×D411yescharacteristic
136C22×Dic5C2×D411yescharacteristic
137C2×C4×D5—24no—
138C2×C4×D5—22no—
139C2×C4×D5—22no—
140C2×C4×D5—24no—
141C2×C4×D5—28no—
142C2×C4×D5—28no—
143C2×C4×D5—216no—
144C2×C4×D5—44no—
145C2×C4×D5—44no—
146C2×C4×D5—44no—
147C2×C4×D5C4⋊C414yes—
148C2×C4×D5C4⋊C414yes—
149C2×C4×D5C22×C412yes—
150C2×C4×D5C22×C412yes—
151C2×C4×D5C2×D414yes—
152C2×C4×D5C2×Q812yes—
153C2×C4×D5C2×Q812yes—
154C2×C4×D5#4114yes—
155C2×C5×C8—22no—
156C2×C5×C8—28no—
157C2×C5×C8C22×C412yes—
158C2×C5×C8C22×C412yes—
159C2×C5×C8C2×D414yes—
160C5×C4⋊C4—22no—
161C5×C4⋊C4—22no—
162C5×C4⋊C4—22no—
163C5×C4⋊C4—22no—
164C5×C22⋊C4—42no—
165C5×C22⋊C4—42no—
166C22⋊Dic5—42no—
167C22⋊Dic5—42no—
168#685—44no—
169#685—44no—
170#685—44no—
171C4⋊Dic5—22no—
172C4⋊Dic5—22no—
173C4⋊Dic5—22no—
174C4⋊Dic5—22no—
175#683—44no—
176C4×Dic5—44no—
177C2×#168—22no—
178C2×#168—28no—
179C2×#168C22×C412yes—
180C2×#168C22×C412yes—
181C2×#168C2×D414yes—
182#676—28no—
183#676—28no—
184#676—28no—
185#676C22⋊C4116yescomplemented
186C24×C4—51no—
187C22×M16—58no—
188C22×M16—51no—
189C22×M16—51no—
190C22×M16—51no—
191C22×M16—51no—
192C22×M16—108no—
193C23×C8—51no—
194C23×C8C2×D511yescharacteristic
195C2×#117—52no—
196C22×C4⋊C4—52no—
197C22×C22⋊C4—101no—
198#428—108no—
199C2×#107—52no—
200C2×#107C2×D512yescomplemented
201C2×#106—52no—
202C2×#106—52no—
203#381—101no—
204#380—51no—
205#380—51no—
206#380—52no—
207#380—52no—
208#380—101no—
209#380—101no—
210#380C2×D511yescharacteristic
211#380C2×D511yescharacteristic
212#379—101no—
213C4×C22⋊C4—101no—
214C4×C22⋊C4—101no—
215C2×#95—101no—
216#340—52no—
217#340—52no—
218#340—52no—
219#340C2×D512yescomplemented
220C23×C5—22no—
221C23×C5—22no—
222C23×C5C2×C4⋊C412yes—
223C23×C5AGL(1,ℤ/8)12yes—
224C23×C5#13712yes—
225C23×C5C22×D411yescharacteristic
226C22×D5—21no—
227C22×D5—21no—
228C22×D5—22no—
229C22×D5—22no—
230C22×D5—21no—
231C22×D5—21no—
232C22×D5—22no—
233C22×D5—28no—
234C22×D5—28no—
235C22×D5—44no—
236C22×D5—48no—
237C22×D5—48no—
238C22×D5C2×C4⋊C412yes—
239C22×D5C2×C4⋊C412yes—
240C22×D5#12114yes—
241C2×C4×C5—22no—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5—24no—
245C2×C4×C5—42no—
246C2×C4×C5—42no—
247C2×C4×C5—42no—
248C2×C4×C5—42no—
249C2×C4×C5—42no—
250C2×C4×C5—42no—
251C2×C4×C5C2×C4⋊C412yes—
252C2×C4×C5C23×C411yescharacteristic
253C2×C4×C5C22×D412yes—
254C2×C4×C5C22×Q811yescharacteristic
255C2×C4×C5C2×#4112yes—
256C2×Dic5—22no—
257C2×Dic5—28no—
258C2×Dic5—42no—
259C2×Dic5—42no—
260C2×Dic5—42no—
261C2×Dic5—42no—
262C2×Dic5—42no—
263C2×Dic5—42no—
264C2×Dic5C2×C4⋊C412yes—
265C2×Dic5C2×C4⋊C412yes—
266C2×Dic5#12114yes—
267C4×D5—28no—
268C4×D5—22no—
269C4×D5—22no—
270C4×D5—24no—
271C4×D5—28no—
272C4×D5—88no—
273C4×D5C2×C22⋊C414yes—
274C4×D5#11714yes—
275C4×D5C22⋊Q818yes—
276C5×C8—22no—
277C5×C8C2×C22⋊C414yescomplemented
278#168—22no—
279#168C2×C22⋊C414yescomplemented
280C25—51no—
281C23×C4—51no—
282C23×C4—51no—
283C23×C4—54no—
284C23×C4—51no—
285C23×C4—51no—
286C23×C4—54no—
287C23×C4—51no—
288C23×C4—104no—
289C23×C4—104no—
290C23×C4—102no—
291C23×C4C22×D511yescharacteristic
292C2×M16—54no—
293C2×M16—54no—
294C2×M16—54no—
295C2×M16—54no—
296C2×M16—58no—
297C2×M16—104no—
298C2×M16—1032no—
299C2×M16—104no—
300C2×M16—104no—
301C22×C8—24no—
302C22×C8—51no—
303C22×C8—54no—
304C22×C8—51no—
305C22×C8—104no—
306C22×C8C4×D514yescomplemented
307C22×C8C22×D511yescharacteristic
308C22×C8C22×D511yescharacteristic
309#117—104no—
310#117—104no—
311C2×C4⋊C4—52no—
312C2×C4⋊C4—52no—
313C2×C4⋊C4—104no—
314C2×C4⋊C4—104no—
315C2×C4⋊C4—102no—
316C2×C4⋊C4C22×D512yes—
317C2×C4⋊C4C22×D512yes—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—102no—
321C2×C22⋊C4—102no—
322C2×C22⋊C4—102no—
323C2×C22⋊C4—101no—
324C2×C22⋊C4—101no—
325C2×C22⋊C4—204no—
326C2×C42—102no—
327#107—24no—
328#107—104no—
329#106—104no—
330#106—104no—
331#95—21no—
332#95—21no—
333#95—102no—
334#95—102no—
335#95—101no—
336#95—101no—
337C22×C5—22no—
338C22×C5—22no—
339C22×C5—24no—
340C22×C5—24no—
341C22×C5#42814yescomplemented
342C22×C5C22×C4⋊C411yescharacteristic
343C22×C5C2×#12112yes—
344C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
345C22×C5C2×#13711yescharacteristic
346C22×C5#57512yes—
347C2×D5—22no—
348C2×D5—22no—
349C2×D5—24no—
350C2×D5—24no—
351C2×D5—22no—
352C2×D5—22no—
353C2×D5—22no—
354C2×D5—416no—
355C2×D5—42no—
356C2×D5—42no—
357C2×D5#38014yes—
358C4×C5—22no—
359C4×C5—82no—
360C4×C5—82no—
361C4×C5C22×C22⋊C411yescharacteristic
362C4×C5C2×#11711yescharacteristic
363C4×C5C2×C22⋊Q812yes—
364Dic5—22no—
365Dic5—82no—
366Dic5—82no—
367Dic5#38014yes—
368C24—51no—
369C24—51no—
370C24—54no—
371C24—104no—
372C24—102no—
373C24—102no—
374C24—202no—
375C24D4×D511yescharacteristic
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—51no—
381C22×C4—52no—
382C22×C4—54no—
383C22×C4—52no—
384C22×C4—54no—
385C22×C4—51no—
386C22×C4—54no—
387C22×C4—52no—
388C22×C4—52no—
389C22×C4—54no—
390C22×C4—54no—
391C22×C4—108no—
392C22×C4—104no—
393C22×C4—108no—
394C22×C4—1016no—
395C22×C4—104no—
396C22×C4—104no—
397C22×C4—102no—
398C22×C4—102no—
399C22×C4—102no—
400C22×C4—102no—
401C22×C4—204no—
402C22×C4—204no—
403C22×C4—204no—
404C22×C4C2×C4×D512yes—
405C22×C4D4×D511yescharacteristic
406C22×C4Q8×D512yes—
407C22×C4C23×D511yescharacteristic
408M16—516no—
409M16—108no—
410M16—108no—
411M16—108no—
412C2×C8—22no—
413C2×C8—28no—
414C2×C8—52no—
415C2×C8—54no—
416C2×C8—52no—
417C2×C8—108no—
418C2×C8—102no—
419C2×C8C2×C4×D512yes—
420C2×C8C2×C4×D512yes—
421C2×C8D4×D514yes—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—102no—
427C4⋊C4—102no—
428C4⋊C4—102no—
429C4⋊C4—102no—
430C4⋊C4—204no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—204no—
434C22⋊C4—204no—
435C22⋊C4—204no—
436C22⋊C4—202no—
437C22⋊C4—202no—
438C42—204no—
439C2×C5—24no—
440C2×C5C2×#38011yescharacteristic
441C2×C5C2×#42811yescharacteristic
442C2×C5#291111yescharacteristic
443C2×C5#305512yes—
444C2×C5#305612yes—
445D5—24no—
446D5—42no—
447C23—22no—
448C23—22no—
449C23—52no—
450C23—54no—
451C23—52no—
452C23—108no—
453C23—101no—
454C23—101no—
455C23—102no—
456C23—101no—
457C23—101no—
458C23—102no—
459C23—102no—
460C23—108no—
461C23—204no—
462C23—208no—
463C23—208no—
464C23D5×C4⋊C412yes—
465C23#405712yes—
466C23#406412yes—
467C23C2×D4×D511yescharacteristic
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—52no—
479C2×C4—54no—
480C2×C4—54no—
481C2×C4—54no—
482C2×C4—58no—
483C2×C4—52no—
484C2×C4—108no—
485C2×C4—108no—
486C2×C4—108no—
487C2×C4—102no—
488C2×C4—102no—
489C2×C4—102no—
490C2×C4—104no—
491C2×C4—202no—
492C2×C4—202no—
493C2×C4—202no—
494C2×C4—202no—
495C2×C4—202no—
496C2×C4—202no—
497C2×C4—408no—
498C2×C4D5×C4⋊C412yes—
499C2×C4C22×C4×D511yescharacteristic
500C2×C4C2×D4×D512yes—
501C2×C4C2×Q8×D511yescharacteristic
502C2×C4D5×#4112yes—
503C8—22no—
504C8—54no—
505C8—102no—
506C8D5×C22⋊C414yescomplemented
507C5#2066111yescharacteristic, complemented, Sylow
508C22—24no—
509C22—22no—
510C22—22no—
511C22—24no—
512C22—54no—
513C22—104no—
514C22—102no—
515C22—102no—
516C22—104no—
517C22—102no—
518C22—102no—
519C22—102no—
520C22—2016no—
521C22—202no—
522C22—202no—
523C22#6517014yescomplemented
524C22C2×D5×C4⋊C411yescharacteristic
525C22D5×#12112yes—
526C22C2×#405711yescharacteristic
527C22C2×#406411yescharacteristic
528C22#6610212yes—
529C4—22no—
530C4—54no—
531C4—82no—
532C4—82no—
533C4—102no—
534C4—402no—
535C4—402no—
536C4C2×D5×C22⋊C411yescharacteristic
537C4D5×#11711yescharacteristic
538C4D5×C22⋊Q812yes—
539C2—24no—
540C2—104no—
541C2—202no—
542C2D5×#38011yescharacteristic
543C2C2×#6517011yescharacteristic
544C2#1061090811yescharacteristic
545C2#1061244612yes—
546C2#1061249012yes—
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1808C2
socleC23×C5C22×D4
prime radicalC24×D5D4
cosocleC22×C4×C5C24
hypercenter#1808D5
hypocenterC5#20661
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
2. C2×#65170C2semidirect
5. C5×#1808C2semidirect
6. #10602763C2semidirect
7. #10601424C2semidirect
8. #10600305C2semidirect
9. #10600304C2semidirect
10. #10599663C2semidirect
11. #10599660C2semidirect
16. C22×#676C4semidirect
24. C2×C5×#107C22semidirect
29. C2×#3949C22semidirect
32. C2×#3940C22semidirect
33. C2×#3939C22semidirect
42. C5×#340C22semidirect
43. #64771C22semidirect
44. #64749C22semidirect
45. #64749C22semidirect
60. C22×C5×C8C2×C4semidirect
77. C22×#168C2×C4semidirect
86. C2×#676C2×C4semidirect
87. C2×#676C2×C4semidirect
111. #1808D5semidirect
185. #676C22⋊C4semidirect
200. C2×#107C2×D5semidirect
219. #340C2×D5semidirect
277. C5×C8C2×C22⋊C4semidirect
279. #168C2×C22⋊C4semidirect
306. C22×C8C4×D5semidirect
341. C22×C5#428semidirect
506. C8D5×C22⋊C4semidirect
507. C5#20661semidirect
523. C22#65170semidirect

Maximal subgroups

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

subgroupindex
C23×#6762
C2×#651702
D5×#3802
#106055932
C5×#18082
#106027632
#106014242
#106003052
#106003042
#105996632
#105996602
#206615