Finite Group Databasebeta

← #12108981

Subgroups — #12108981

Order 1280.

1222 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65912C211yesmaximal, characteristic
2C2×#65122C211yesmaximal, characteristic
3D5×#384C211yesmaximal, characteristic
4C5×#1803C211yesmaximal, characteristic, complemented
5#10602419C211yesmaximal, characteristic, complemented
6#10601278C211yesmaximal, characteristic, complemented
7#10601061C212yesmaximal
8#10601018C212yesmaximal, complemented
9#10600381C211yesmaximal, characteristic, complemented
10#10599671C212yesmaximal
11#10599596C212yesmaximal, complemented
12C2×C4×C5×Q8C2211yescharacteristic
13C2×#4092C2211yescharacteristic
14C2×#4090C2211yescharacteristic
15#65912—24no—
16#65912C418yescomplemented
17C2×#4043C2211yescharacteristic
18C2×#4042—21no—
19C2×D5×C4⋊C4—21no—
20C2×D5×C4⋊C4—21no—
21C2×D5×C4⋊C4C2212yes—
22C2×#4035—21no—
23C2×#4034C2211yescharacteristic
24C2×#4018—21no—
25C2×C42×D5C2211yescharacteristic
26C2×C4×Dic10—21no—
27C2×C4×Dic10C2211yescharacteristic
28C2×C5×C4⋊C8C2211yescharacteristic
29C2×C5×#103C2212yes—
30C5×#384C2211yescharacteristic
31C2×#3967C2212yes—
32C2×#3952C2212yes—
33C2×#3948C2212yes—
34C2×C8×Dic5C2212yes—
35#65265—21no—
36#65265—21no—
37#65265C2212yes—
38C2×#3942C2212yes—
39#65224C2211yescharacteristic
40C2×C4⋊#168C2211yescharacteristic
41#65122—24no—
42C5×#340C2212yes—
43#64771C2212yes—
44#64749C2212yes—
45#64749C2212yes—
46#21860—51nomaximal, Sylow
47C22×C5×Q8C2311yescharacteristic
48C22×C4×D5—22no—
49C22×C4×D5D411yescharacteristic
50C22×C4×D5D411yescharacteristic
51C22×C4×D5C2311yescharacteristic
52C22×Dic10—21no—
53C22×Dic10C2311yescharacteristic
54C22×C5×C8C2312yes—
55C4×C5×Q8—22no—
56C4×C5×Q8C2×C414yescomplemented
57C2×C5×C4⋊C4—21no—
58C2×C5×C4⋊C4—21no—
59C2×C5×C4⋊C4—21no—
60C2×C5×C4⋊C4C2312yes—
61C2×C5×C4⋊C4C2311yescharacteristic
62C2×C42×C5—21no—
63C2×C42×C5C2311yescharacteristic
64#4092—22no—
65#4092C2×C414yes—
66#4090—22no—
67#4090C2×C414yescomplemented
68C2×#685—21no—
69C2×#685—21no—
70C2×C4⋊Dic5—21no—
71C2×C4⋊Dic5—21no—
72C2×C4⋊Dic5—21no—
73C2×C4⋊Dic5C2312yes—
74C2×C4⋊Dic5C2311yescharacteristic
75C2×#683—21no—
76C2×#683—21no—
77C2×#683—21no—
78C2×#683—21no—
79C2×#683—21no—
80C2×#683—22no—
81C2×#683—22no—
82C2×#683—22no—
83C2×C4×Dic5—21no—
84C2×C4×Dic5D411yescharacteristic
85C2×C4×Dic5D411yescharacteristic
86C2×C4×Dic5C2311yescharacteristic
87C22×#168C2312yes—
88#4043—22no—
89#4043C2×C414yes—
90#4042—28no—
91D5×C4⋊C4—42no—
92D5×C4⋊C4—42no—
93D5×C4⋊C4—44no—
94D5×C4⋊C4—44no—
95#4035—28no—
96#4034—22no—
97#4034C2×C414yescomplemented
98#4018—28no—
99C42×D5C2×C414yes—
100C42×D5D414yes—
101C4×Dic10—22no—
102C4×Dic10—28no—
103C4×Dic10C2×C414yescomplemented
104C5×C4⋊C8—22no—
105C5×#103—42no—
106#3967—42no—
107#3952—24no—
108#3948—42no—
109C8×Dic5—24no—
110#3942—42no—
111C4⋊#168—22no—
112C2×#533—51no—
113C2×#431—51no—
114C2×#384—51no—
115#1990—52no—
116#1956—52no—
117#1872—52no—
118#1872—52no—
119#1803—51no—
120#1803—51no—
121#1803—51no—
122#1803D511yescharacteristic, complemented
123C23×D5C2×D411yescharacteristic
124C2×C5×Q8—21no—
125C2×C5×Q8—21no—
126C2×C5×Q8—21no—
127C2×C5×Q8—21no—
128C2×C5×Q8—21no—
129C2×C5×Q8C22×C412yes—
130C22×C4×C5—21no—
131C22×C4×C5—21no—
132C22×C4×C5—22no—
133C22×C4×C5C2×D411yescharacteristic
134C22×C4×C5C2×D411yescharacteristic
135C22×C4×C5C2411yescharacteristic
136C22×Dic5—21no—
137C22×Dic5—21no—
138C22×Dic5—22no—
139C22×Dic5C2×D411yescharacteristic
140C22×Dic5C2×D411yescharacteristic
141C22×Dic5C2×D411yescharacteristic
142C2×C4×D5—21no—
143C2×C4×D5—21no—
144C2×C4×D5—21no—
145C2×C4×D5—21no—
146C2×C4×D5—22no—
147C2×C4×D5—22no—
148C2×C4×D5—44no—
149C2×C4×D5—44no—
150C2×C4×D5—44no—
151C2×C4×D5C22⋊C414yes—
152C2×C4×D5C22⋊C414yes—
153C2×C4×D5SD1614yes—
154C2×C4×D5Q1614yes—
155C2×C4×D5C22×C412yes—
156C2×C4×D5C2×D412yes—
157C2×Dic10—21no—
158C2×Dic10—21no—
159C2×Dic10—21no—
160C2×Dic10—21no—
161C2×Dic10—21no—
162C2×Dic10—24no—
163C2×Dic10—42no—
164C2×Dic10—42no—
165C2×Dic10C22×C412yes—
166C2×C5×C8—22no—
167C2×C5×C8—22no—
168C2×C5×C8—22no—
169C5×C4⋊C4—24no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—22no—
172C5×C4⋊C4—22no—
173C5×C4⋊C4—42no—
174C5×C4⋊C4—42no—
175C5×C4⋊C4C22×C412yes—
176C42×C5—24no—
177C42×C5C22×C412yes—
178C42×C5C2×D412yes—
179#685—24no—
180#685—24no—
181C4⋊Dic5—21no—
182C4⋊Dic5—24no—
183C4⋊Dic5—22no—
184C4⋊Dic5—22no—
185C4⋊Dic5—42no—
186C4⋊Dic5—42no—
187C4⋊Dic5C22×C412yes—
188#683—24no—
189#683—24no—
190#683—24no—
191#683—24no—
192#683—24no—
193#683—44no—
194#683—44no—
195#683—44no—
196C4×Dic5—24no—
197C4×Dic5C22⋊C414yes—
198C4×Dic5C22⋊C414yes—
199C4×Dic5C22×C412yes—
200C4×Dic5C2×D412yes—
201C2×#168—22no—
202C2×#168—22no—
203C2×#168—22no—
204#533—58no—
205#533—104no—
206C2×C4⋊Q8—51no—
207C2×C4⋊Q8—51no—
208C2×#125—101no—
209C2×C22⋊Q8—51no—
210C2×C22⋊Q8—51no—
211C2×C22⋊Q8—101no—
212C2×C4×Q8—51no—
213C2×C4×Q8—101no—
214C2×C4×Q8C2×D511yescharacteristic
215C2×#117—101no—
216C22×C4⋊C4—52no—
217C22×C4⋊C4—101no—
218C22×C4⋊C4—101no—
219C22×C42—51no—
220#431—104no—
221C2×C4⋊C8—51no—
222C2×C4⋊C8C2×D511yescharacteristic
223C2×#103—52no—
224C2×#103—52no—
225C2×#103—52no—
226C2×#103C2×D512yes—
227C2×C22⋊C8—52no—
228C2×C4×C8—52no—
229#384—52no—
230#384—51no—
231#384—101no—
232#384—101no—
233#384C2×D511yescharacteristic
234#340—52no—
235#340—52no—
236#340—52no—
237#340C2×D512yes—
238C23×C5C22×D411yescharacteristic
239C22×D5—21no—
240C22×D5—21no—
241C22×D5—21no—
242C22×D5—21no—
243C22×D5C2×C22⋊C412yes—
244C22×D5C2×SD1612yes—
245C22×D5C2×Q1612yes—
246C5×Q8—22no—
247C5×Q8—22no—
248C5×Q8—41no—
249C5×Q8—41no—
250C2×C4×C5—21no—
251C2×C4×C5—21no—
252C2×C4×C5—22no—
253C2×C4×C5—22no—
254C2×C4×C5—41no—
255C2×C4×C5—41no—
256C2×C4×C5—41no—
257C2×C4×C5—41no—
258C2×C4×C5—42no—
259C2×C4×C5—42no—
260C2×C4×C5—42no—
261C2×C4×C5C2×C22⋊C412yes—
262C2×C4×C5C2×C22⋊C412yes—
263C2×C4×C5C2×SD1612yes—
264C2×C4×C5C2×Q1612yes—
265C2×C4×C5C23×C411yescharacteristic
266C2×C4×C5C22×D411yescharacteristic
267C2×C4×C5H(5,𝔽2)11yescharacteristic
268C2×C4×C5H(5,𝔽2)11yescharacteristic
269C2×C4×C52−1+411yescharacteristic
270C2×C4×C52−1+411yescharacteristic
271C2×Dic5—21no—
272C2×Dic5—21no—
273C2×Dic5—22no—
274C2×Dic5—22no—
275C2×Dic5—41no—
276C2×Dic5—41no—
277C2×Dic5—41no—
278C2×Dic5—41no—
279C2×Dic5—42no—
280C2×Dic5—42no—
281C2×Dic5—42no—
282C2×Dic5C2×C22⋊C412yes—
283C2×Dic5C2×C22⋊C412yes—
284C2×Dic5C2×C22⋊C412yes—
285C2×Dic5C2×SD1612yes—
286C2×Dic5C2×Q1612yes—
287C2×Dic5AGL(1,ℤ/8)12yes—
288C2×Dic5#13712yes—
289C4×D5—22no—
290C4×D5—22no—
291C4×D5—22no—
292C4×D5—22no—
293C4×D5—28no—
294C4×D5—88no—
295C4×D5#103116yescomplemented
296Dic10—22no—
297Dic10—22no—
298Dic10—41no—
299Dic10—41no—
300Dic10—48no—
301C5×C8—42no—
302#168—42no—
303C22×Q8—51no—
304C22×Q8—101no—
305C22×Q8C22×D511yescharacteristic
306C23×C4—51no—
307C23×C4—51no—
308C23×C4—51no—
309C23×C4—102no—
310C22×C8—52no—
311C22×C8C22×D512yes—
312C4⋊Q8—54no—
313C4⋊Q8—54no—
314C4⋊Q8—102no—
315C4⋊Q8—102no—
316#125—108no—
317C22⋊Q8—54no—
318C22⋊Q8—54no—
319C22⋊Q8—102no—
320C22⋊Q8—108no—
321C22⋊Q8—102no—
322C4×Q8—22no—
323C4×Q8—54no—
324C4×Q8—108no—
325C4×Q8—102no—
326C4×Q8C4×D514yescomplemented
327#117—108no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—51no—
332C2×C4⋊C4—52no—
333C2×C4⋊C4—101no—
334C2×C4⋊C4—101no—
335C2×C4⋊C4—101no—
336C2×C4⋊C4—101no—
337C2×C4⋊C4—101no—
338C2×C4⋊C4—101no—
339C2×C4⋊C4—102no—
340C2×C4⋊C4—102no—
341C2×C4⋊C4—102no—
342C2×C4⋊C4—101no—
343C2×C4⋊C4—101no—
344C2×C4⋊C4—202no—
345C2×C4⋊C4—202no—
346C2×C4⋊C4—204no—
347C2×C4⋊C4—204no—
348C2×C4⋊C4C22×D512yes—
349C2×C4⋊C4C22×D511yescharacteristic
350C2×C22⋊C4—101no—
351C2×C22⋊C4—101no—
352C2×C42—21no—
353C2×C42—51no—
354C2×C42—51no—
355C2×C42—51no—
356C2×C42—54no—
357C2×C42—54no—
358C2×C42—101no—
359C2×C42C22×D511yescharacteristic
360C4⋊C8—22no—
361C4⋊C8—102no—
362#103—42no—
363#103—202no—
364#103—202no—
365#103—202no—
366C22⋊C8—104no—
367C4×C8—104no—
368C22×C5C22×C22⋊C411yescharacteristic
369C22×C5#53511yescharacteristic
370C22×C5#53611yescharacteristic
371C22×C5C22×SD1611yescharacteristic
372C22×C5C22×Q1611yescharacteristic
373C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
374C22×C5C2×#13711yescharacteristic
375C2×D5—22no—
376C2×D5—22no—
377C2×D5—22no—
378C2×D5—22no—
379C2×D5—22no—
380C2×D5—22no—
381C2×D5C2×#10314yes—
382C4×C5—22no—
383C4×C5—42no—
384C4×C5—42no—
385C4×C5—82no—
386C4×C5C2×#10314yes—
387C4×C5#52011yescharacteristic
388C4×C5#52011yescharacteristic
389C4×C5#53711yescharacteristic
390C4×C5#53711yescharacteristic
391Dic5—22no—
392Dic5—42no—
393Dic5—42no—
394Dic5—82no—
395Dic5C2×#10314yes—
396Dic5#41714yes—
397C24—51no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—52no—
404C2×Q8—104no—
405C2×Q8—101no—
406C2×Q8—101no—
407C2×Q8—101no—
408C2×Q8—101no—
409C2×Q8—101no—
410C2×Q8—202no—
411C2×Q8—202no—
412C2×Q8C2×C4×D512yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—51no—
417C22×C4—51no—
418C22×C4—54no—
419C22×C4—54no—
420C22×C4—54no—
421C22×C4—51no—
422C22×C4—52no—
423C22×C4—52no—
424C22×C4—54no—
425C22×C4—102no—
426C22×C4—102no—
427C22×C4—101no—
428C22×C4—101no—
429C22×C4—101no—
430C22×C4—101no—
431C22×C4—101no—
432C22×C4—101no—
433C22×C4—102no—
434C22×C4—204no—
435C22×C4—204no—
436C22×C4—204no—
437C22×C4D4×D511yescharacteristic
438C22×C4D4×D511yescharacteristic
439C22×C4C23×D511yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—102no—
444C2×C8—102no—
445C2×C8—102no—
446C4⋊C4—21no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—52no—
453C4⋊C4—104no—
454C4⋊C4—104no—
455C4⋊C4—104no—
456C4⋊C4—104no—
457C4⋊C4—104no—
458C4⋊C4—101no—
459C4⋊C4—104no—
460C4⋊C4—102no—
461C4⋊C4—102no—
462C4⋊C4—202no—
463C4⋊C4—204no—
464C4⋊C4—204no—
465C4⋊C4—204no—
466C4⋊C4—202no—
467C4⋊C4C2×C4×D512yes—
468C22⋊C4—104no—
469C22⋊C4—104no—
470C42—24no—
471C42—52no—
472C42—52no—
473C42—54no—
474C42—54no—
475C42—104no—
476C42C2×C4×D512yes—
477C42D4×D512yes—
478C2×C5#231111yescharacteristic
479C2×C5C22×#10311yescharacteristic
480C2×C5C2×#41711yescharacteristic
481C2×C5#320811yescharacteristic
482C2×C5#320911yescharacteristic
483C2×C5#322911yescharacteristic
484C2×C5#323011yescharacteristic
485D5—24no—
486C23—52no—
487C23—52no—
488C23—52no—
489C23—101no—
490C23—101no—
491C23—101no—
492C23—101no—
493C23C2×D4×D511yescharacteristic
494Q8—22no—
495Q8—22no—
496Q8—41no—
497Q8—41no—
498Q8—102no—
499Q8—102no—
500Q8—208no—
501Q8—201no—
502Q8—201no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—52no—
515C2×C4—52no—
516C2×C4—52no—
517C2×C4—52no—
518C2×C4—52no—
519C2×C4—52no—
520C2×C4—516no—
521C2×C4—52no—
522C2×C4—108no—
523C2×C4—101no—
524C2×C4—101no—
525C2×C4—102no—
526C2×C4—102no—
527C2×C4—102no—
528C2×C4—102no—
529C2×C4—102no—
530C2×C4—102no—
531C2×C4—201no—
532C2×C4—201no—
533C2×C4—201no—
534C2×C4—201no—
535C2×C4—202no—
536C2×C4—202no—
537C2×C4—202no—
538C2×C4—408no—
539C2×C4D5×C22⋊C412yes—
540C2×C4D5×C22⋊C412yes—
541C2×C4D5×SD1612yes—
542C2×C4D5×Q1612yes—
543C2×C4C22×C4×D511yescharacteristic
544C2×C4C2×D4×D511yescharacteristic
545C2×C4#414411yescharacteristic
546C2×C4#414711yescharacteristic
547C2×C4#414911yescharacteristic
548C2×C4#415011yescharacteristic
549C8—42no—
550C8—202no—
551C5#2186011yescharacteristic, complemented, Sylow
552C22—54no—
553C22—102no—
554C22—102no—
555C22—102no—
556C22—102no—
557C22—102no—
558C22—102no—
559C22C2×D5×C22⋊C411yescharacteristic
560C22#6594111yescharacteristic
561C22#6596211yescharacteristic
562C22C2×D5×SD1611yescharacteristic
563C22C2×D5×Q1611yescharacteristic
564C22D5×AGL(1,ℤ/8)11yescharacteristic
565C22D5×#13711yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—54no—
570C4—54no—
571C4—82no—
572C4—102no—
573C4—202no—
574C4—202no—
575C4—402no—
576C4D5×#10314yes—
577C4#6583811yescharacteristic
578C4#6585711yescharacteristic
579C4#6594211yescharacteristic
580C4#6596311yescharacteristic
581C2—104no—
582C2#1060573411yescharacteristic
583C2C2×D5×#10311yescharacteristic
584C2D5×#41711yescharacteristic
585C2#1061211811yescharacteristic
586C2#1061221111yescharacteristic
587C2#1061274611yescharacteristic
588C2#1061284311yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1803C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1803D5
hypocenterC5#21860
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. C5×#1803C2semidirect
5. #10602419C2semidirect
6. #10601278C2semidirect
8. #10601018C2semidirect
9. #10600381C2semidirect
11. #10599596C2semidirect
16. #65912C4semidirect
56. C4×C5×Q8C2×C4semidirect
67. #4090C2×C4semidirect
97. #4034C2×C4semidirect
103. C4×Dic10C2×C4semidirect
122. #1803D5semidirect
295. C4×D5#103semidirect
326. C4×Q8C4×D5semidirect
551. C5#21860semidirect

Maximal subgroups

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

subgroupindex
C2×#659122
C2×#651222
D5×#3842
C5×#18032
#106024192
#106012782
#106010612
#106010182
#106003812
#105996712
#105995962
#218605