Finite Group Databasebeta

← #12117449

Subgroups — #12117449

Order 1280.

1042 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10614003C211yesmaximal, characteristic
2C2×#65284C211yesmaximal, characteristic, complemented
3C2×D5×#106C211yesmaximal, characteristic, complemented
4C2×#65017C211yesmaximal, characteristic
5#10605741C211yesmaximal, characteristic
6C5×#1854C211yesmaximal, characteristic, complemented
7#10601425C211yesmaximal, characteristic, complemented
8#10601401C211yesmaximal, characteristic, complemented
9#10601356C211yesmaximal, characteristic, complemented
10#10600721C211yesmaximal, characteristic, complemented
11#10600306C211yesmaximal, characteristic, complemented
12#10599668C211yesmaximal, characteristic, complemented
13#10599662C211yesmaximal, characteristic, complemented
14#10599590C211yesmaximal, characteristic, complemented
15#10599579C211yesmaximal, characteristic, complemented
16C2×C5×#117C2211yescharacteristic
17C2×C4×C22⋊D5C2211yescharacteristic
18C22×C4⋊Dic5C2211yescharacteristic
19C22×#681C2211yescharacteristic
20C22×C8×D5C2211yescharacteristic
21C2×D5×C4⋊C4C2211yescharacteristic
22C2×D5×C4⋊C4C2211yescharacteristic
23C2×#4027—21no—
24C2×C5×#107C2211yescharacteristic
25C2×C5×#106C2211yescharacteristic
26C2×C5×C22⋊C8C2211yescharacteristic
27C5×#380C2211yescharacteristic
28#65501C2211yescharacteristic
29C2×#3952C2211yescharacteristic
30C2×#3949C2211yescharacteristic
31C2×#3949C2211yescharacteristic
32C2×#3947C2211yescharacteristic
33C2×#3946C2211yescharacteristic
34#65284—24no—
35#65276C2211yescharacteristic
36#65276C2211yescharacteristic
37#65266C2211yescharacteristic
38#65265C2211yescharacteristic
39C4⋊C4×Dic5C2211yescharacteristic
40C4⋊C4×Dic5C2211yescharacteristic
41C2×#3940C2211yescharacteristic
42C2×#3940C2211yescharacteristic
43C2×#3940C2211yescharacteristic
44C2×#3939C2211yescharacteristic
45C2×#3939C2211yescharacteristic
46C2×#3939C2211yescharacteristic
47#65245—21no—
48#65235—21no—
49#65234—21no—
50C4×#685—21no—
51#65222—21no—
52D5×#106—24no—
53#65017C418yescomplemented
54#64957—21no—
55#64936—21no—
56C5×#340C2211yescharacteristic
57C5×#340C2211yescharacteristic
58#64771C2211yescharacteristic
59#64771C2211yescharacteristic
60#64749C2211yescharacteristic
61#64749C2211yescharacteristic
62#22659—51nomaximal, Sylow
63C23×C4×C5C2311yescharacteristic
64C22×C22⋊D5D411yescharacteristic
65C23×Dic5—21no—
66C22×C4×D5—21no—
67C22×C4×D5—21no—
68C22×C4×D5C2311yescharacteristic
69C22×C5×C8C2311yescharacteristic
70C22×C5×C8C2311yescharacteristic
71C5×#117—22no—
72C5×#117—41no—
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C4⋊C4C2311yescharacteristic
76C2×C5×C4⋊C4C2311yescharacteristic
77C2×C5×C22⋊C4—21no—
78C2×C5×C22⋊C4—21no—
79C2×C42×C5—21no—
80C2×C22⋊Dic5—21no—
81C2×C22⋊Dic5D411yescharacteristic
82C4×C22⋊D5C2×C414yes—
83C4×C22⋊D5Q814yes—
84C2×#685—21no—
85C2×#685—21no—
86C2×#685D411yescharacteristic
87C2×C4⋊Dic5—22no—
88C2×C4⋊Dic5—21no—
89C2×C4⋊Dic5—41no—
90C2×C4⋊Dic5C2311yescharacteristic
91C2×C4⋊Dic5C2311yescharacteristic
92C2×C4⋊Dic5C2311yescharacteristic
93C2×C4⋊Dic5C2311yescharacteristic
94C2×#683—21no—
95C2×#683—21no—
96C2×#683—21no—
97C2×#683—21no—
98C2×#683D411yescharacteristic
99C2×C4×Dic5—21no—
100C2×C4×Dic5—21no—
101C2×C4×Dic5C2311yescharacteristic
102C2×#681—22no—
103C2×#681—22no—
104C2×#681C2×C414yescomplemented
105C22×#168C2311yescharacteristic
106C22×#168C2311yescharacteristic
107C2×C8×D5—22no—
108C2×C8×D5—22no—
109C2×C8×D5C2×C414yescomplemented
110D5×C4⋊C4—22no—
111D5×C4⋊C4—22no—
112D5×C4⋊C4—41no—
113D5×C4⋊C4—41no—
114#4027—44no—
115C5×#107—41no—
116C5×#106—22no—
117C5×C22⋊C8C2×C414yescomplemented
118C5×#95—21no—
119#3963—21no—
120#3963—21no—
121#3963—21no—
122#3963—21no—
123#3963—21no—
124#3963—21no—
125#3963—21no—
126#3952C2×C414yescomplemented
127#3949—22no—
128#3949—41no—
129#3947C2×C414yescomplemented
130#3946C2×C414yescomplemented
131#3940—22no—
132#3940—22no—
133#3940D414yescomplemented
134#3939—22no—
135#3939—22no—
136#3939—22no—
137C2×#435—51no—
138C2×#428—51no—
139C22×#106—51no—
140#2352—51no—
141D4×C4⋊C4—51no—
142#1949—51no—
143#1948—51no—
144#1854D511yescharacteristic, complemented
145#1853—51no—
146#1853—51no—
147#1852—51no—
148#1847—51no—
149#1809—51no—
150#1808—51no—
151#1776—51no—
152C24×C5C2×D411yescharacteristic
153C23×D5C2×D411yescharacteristic
154C22×C4×C5—21no—
155C22×C4×C5—21no—
156C22×C4×C5—21no—
157C22×C4×C5—21no—
158C22×C4×C5—21no—
159C22×C4×C5—21no—
160C22×C4×C5—21no—
161C22×C4×C5—21no—
162C22×C4×C5C22×C412yes—
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2×Q812yes—
165C22×C4×C5C2411yescharacteristic
166C2×C22⋊D5—22no—
167C2×C22⋊D5—22no—
168C2×C22⋊D5C4⋊C414yes—
169C22×Dic5—21no—
170C22×Dic5—21no—
171C22×Dic5—21no—
172C22×Dic5—21no—
173C22×Dic5—42no—
174C22×Dic5—42no—
175C22×Dic5—81no—
176C22×Dic5C2×D411yescharacteristic
177C22×Dic5C2×D411yescharacteristic
178C22×Dic5C2×D411yescharacteristic
179C2×C4×D5—21no—
180C2×C4×D5—21no—
181C2×C4×D5—21no—
182C2×C4×D5—21no—
183C2×C4×D5—42no—
184C2×C4×D5—42no—
185C2×C4×D5—42no—
186C2×C4×D5—42no—
187C2×C4×D5—81no—
188C2×C4×D5—81no—
189C2×C4×D5C22×C412yes—
190C2×C4×D5C2×Q812yes—
191C2×C5×C8—21no—
192C2×C5×C8—21no—
193C2×C5×C8—21no—
194C2×C5×C8—21no—
195C2×C5×C8C22×C412yes—
196C2×C5×C8C22×C412yes—
197C5×C4⋊C4—21no—
198C5×C4⋊C4—21no—
199C5×C4⋊C4—21no—
200C5×C4⋊C4—21no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4—21no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4C2×D412yes—
205C5×C22⋊C4—42no—
206C5×C22⋊C4—42no—
207C42×C5—42no—
208C22⋊Dic5—81no—
209C22⋊Dic5C4⋊C414yes—
210#685—42no—
211#685—42no—
212#685C4⋊C414yes—
213C4⋊Dic5—21no—
214C4⋊Dic5—21no—
215C4⋊Dic5—21no—
216C4⋊Dic5—21no—
217C4⋊Dic5—21no—
218C4⋊Dic5—21no—
219C4⋊Dic5—21no—
220C4⋊Dic5—42no—
221C4⋊Dic5C2×D412yes—
222#683—42no—
223#683—42no—
224#683—42no—
225#683—81no—
226#683C4⋊C414yes—
227C4×Dic5—42no—
228C4×Dic5—81no—
229C4×Dic5C22×C412yes—
230C4×Dic5C2×Q812yes—
231#681—28no—
232C2×#168—21no—
233C2×#168—21no—
234C2×#168C22×C412yes—
235C2×#168C22×C412yes—
236C2×#168C2×D412yes—
237C2×#168#4112yes—
238C8×D5—28no—
239C22×M16—51no—
240C23×C8—51no—
241C2×C4×D4—51no—
242C2×C4×D4—101no—
243C2×#117C2×D511yescharacteristic
244C22×C4⋊C4—51no—
245C22×C4⋊C4—51no—
246C22×C4⋊C4—51no—
247#435—58no—
248#428—104no—
249C2×#107—51no—
250C2×#107—51no—
251C2×#107—51no—
252C2×#107C2×D511yescharacteristic
253C2×#106—51no—
254C2×#106—51no—
255C2×#106—51no—
256C2×#106—51no—
257C2×#106—51no—
258C2×#106—104no—
259C2×#106C2×D511yescharacteristic
260C2×C4⋊C8—51no—
261C2×C22⋊C8—51no—
262C2×C22⋊C8C2×D511yescharacteristic
263C2×#97—51no—
264#389—101no—
265#388—101no—
266#387—101no—
267#386—101no—
268#385—101no—
269#385—101no—
270#384—51no—
271#382—51no—
272#380—51no—
273#380—51no—
274#380—51no—
275#380C2×D511yescharacteristic
276C4×C4⋊C4—51no—
277C4×C4⋊C4—51no—
278C4×C22⋊C4—101no—
279#340—51no—
280#340—51no—
281#340—51no—
282#340—51no—
283#340C2×D511yescharacteristic
284#340C2×D511yescharacteristic
285C23×C5—21no—
286C23×C5—21no—
287C23×C5—21no—
288C23×C5—21no—
289C23×C5—21no—
290C23×C5—21no—
291C23×C5C2×C4⋊C412yes—
292C23×C5C22×D411yescharacteristic
293C22×D5—21no—
294C22×D5—21no—
295C22×D5—21no—
296C22×D5—21no—
297C22×D5C2×C4⋊C412yes—
298C22×D5AGL(1,ℤ/8)12yes—
299C22×D5#13712yes—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—22no—
303C2×C4×C5—22no—
304C2×C4×C5—22no—
305C2×C4×C5—22no—
306C2×C4×C5—22no—
307C2×C4×C5—41no—
308C2×C4×C5—41no—
309C2×C4×C5—41no—
310C2×C4×C5—41no—
311C2×C4×C5—41no—
312C2×C4×C5—41no—
313C2×C4×C5—41no—
314C2×C4×C5—41no—
315C2×C4×C5—41no—
316C2×C4×C5—41no—
317C2×C4×C5—41no—
318C2×C4×C5C2×C4⋊C412yes—
319C2×C4×C5C23×C411yescharacteristic
320C2×C4×C5C22×D411yescharacteristic
321C2×C4×C5C22×Q811yescharacteristic
322C2×C4×C5C2×#4111yescharacteristic
323C2×C4×C5H(5,𝔽2)11yescharacteristic
324C2×C4×C52−1+411yescharacteristic
325C22⋊D5—28no—
326C2×Dic5—21no—
327C2×Dic5—21no—
328C2×Dic5—21no—
329C2×Dic5—21no—
330C2×Dic5—21no—
331C2×Dic5—21no—
332C2×Dic5—22no—
333C2×Dic5—41no—
334C2×Dic5—41no—
335C2×Dic5—41no—
336C2×Dic5—41no—
337C2×Dic5—41no—
338C2×Dic5—41no—
339C2×Dic5—41no—
340C2×Dic5—41no—
341C2×Dic5—41no—
342C2×Dic5—41no—
343C2×Dic5—41no—
344C2×Dic5—84no—
345C2×Dic5C2×C4⋊C412yes—
346C2×Dic5C2×C4⋊C412yes—
347C2×Dic5C2×C4⋊C412yes—
348C4×D5—22no—
349C4×D5—22no—
350C4×D5—22no—
351C4×D5—22no—
352C4×D5—84no—
353C4×D5—84no—
354C5×C8—22no—
355C5×C8—22no—
356#168—22no—
357#168C4×D414yescomplemented
358C22×D4—51no—
359C23×C4—51no—
360C23×C4—101no—
361C23×C4—101no—
362C23×C4—101no—
363C23×C4C22×D511yescharacteristic
364C2×M16—54no—
365C2×M16—102no—
366C2×M16—102no—
367C22×C8—51no—
368C22×C8—51no—
369C22×C8—54no—
370C22×C8—102no—
371C22×C8—102no—
372C22×C8C22×D511yescharacteristic
373C22×C8C22×D511yescharacteristic
374C4×D4—54no—
375C4×D4—54no—
376C4×D4—204no—
377#117—22no—
378#117—41no—
379C2×C4⋊C4—51no—
380C2×C4⋊C4—51no—
381C2×C4⋊C4—51no—
382C2×C4⋊C4—51no—
383C2×C4⋊C4—51no—
384C2×C4⋊C4—102no—
385C2×C4⋊C4—102no—
386C2×C4⋊C4—102no—
387C2×C4⋊C4—101no—
388C2×C4⋊C4—101no—
389C2×C4⋊C4—101no—
390C2×C4⋊C4—101no—
391C2×C4⋊C4—101no—
392C2×C4⋊C4—201no—
393C2×C4⋊C4—201no—
394C2×C4⋊C4—201no—
395C2×C4⋊C4C22×D511yescharacteristic
396C2×C4⋊C4C22×D511yescharacteristic
397C2×C4⋊C4C22×D511yescharacteristic
398C2×C4⋊C4C22×D511yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—51no—
402C2×C22⋊C4—51no—
403C2×C22⋊C4—101no—
404C2×C22⋊C4—101no—
405C2×C22⋊C4—101no—
406C2×C42—21no—
407C2×C42—51no—
408C2×C42—101no—
409C2×C42—101no—
410#107—41no—
411#107—102no—
412#107—102no—
413#107—102no—
414#106—22no—
415#106—54no—
416#106—102no—
417#106—102no—
418#106—102no—
419#106—201no—
420C4⋊C8—54no—
421C22⋊C8—54no—
422C22⋊C8C4×D514yescomplemented
423#97—54no—
424#95—21no—
425#95—101no—
426#95—101no—
427#95—101no—
428#95—101no—
429#95—101no—
430#95—101no—
431#95—101no—
432C22×C5—22no—
433C22×C5—22no—
434C22×C5—22no—
435C22×C5—41no—
436C22×C5—41no—
437C22×C5—41no—
438C22×C5—41no—
439C22×C5C22×C4⋊C411yescharacteristic
440C22×C5D4211yescharacteristic
441C22×C5#54711yescharacteristic
442C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
443C22×C5C2×#13711yescharacteristic
444C22×C5#57711yescharacteristic
445C22×C5#57711yescharacteristic
446C2×D5—22no—
447C2×D5—22no—
448C2×D5—22no—
449C2×D5—22no—
450C2×D5—22no—
451C2×D5—22no—
452C2×D5#42814yescomplemented
453C4×C5—22no—
454C4×C5—42no—
455C4×C5—81no—
456C4×C5—81no—
457C4×C5—81no—
458C4×C5C2×C4×D411yescharacteristic
459C4×C5#52011yescharacteristic
460C4×C5D4×Q811yescharacteristic
461C4×C5#55411yescharacteristic
462Dic5—22no—
463Dic5—22no—
464Dic5—22no—
465Dic5—42no—
466Dic5—81no—
467Dic5—81no—
468Dic5—81no—
469C24—51no—
470C24D4×D511yescharacteristic
471C2×D4—54no—
472C2×D4—102no—
473C2×D4—102no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—51no—
483C22×C4—51no—
484C22×C4—51no—
485C22×C4—52no—
486C22×C4—52no—
487C22×C4—101no—
488C22×C4—101no—
489C22×C4—101no—
490C22×C4—101no—
491C22×C4—101no—
492C22×C4—101no—
493C22×C4—101no—
494C22×C4—101no—
495C22×C4—202no—
496C22×C4—202no—
497C22×C4—202no—
498C22×C4—202no—
499C22×C4—202no—
500C22×C4—202no—
501C22×C4—401no—
502C22×C4—401no—
503C22×C4—401no—
504C22×C4C2×C4×D512yes—
505C22×C4D4×D511yescharacteristic
506C22×C4Q8×D512yes—
507C22×C4C23×D511yescharacteristic
508M16—108no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—52no—
514C2×C8—52no—
515C2×C8—52no—
516C2×C8—52no—
517C2×C8—108no—
518C2×C8—101no—
519C2×C8—101no—
520C2×C8C2×C4×D512yes—
521C2×C8C2×C4×D512yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—54no—
530C4⋊C4—52no—
531C4⋊C4—101no—
532C4⋊C4—101no—
533C4⋊C4—101no—
534C4⋊C4—101no—
535C4⋊C4—101no—
536C4⋊C4—101no—
537C4⋊C4—101no—
538C4⋊C4—202no—
539C4⋊C4—202no—
540C4⋊C4—202no—
541C4⋊C4—202no—
542C4⋊C4—401no—
543C4⋊C4D4×D512yes—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—54no—
547C22⋊C4—54no—
548C22⋊C4—202no—
549C22⋊C4—202no—
550C22⋊C4—401no—
551C42—42no—
552C42—52no—
553C42—52no—
554C42—202no—
555C42—401no—
556C2×C5—42no—
557C2×C5D4×C4⋊C411yescharacteristic
558C2×C5C2×#42811yescharacteristic
559C2×C5#291211yescharacteristic
560C2×C5#326511yescharacteristic
561C2×C5#326611yescharacteristic
562C2×C5#334411yescharacteristic
563C2×C5#334511yescharacteristic
564D5—24no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—52no—
572C23—52no—
573C23—52no—
574C23—101no—
575C23—101no—
576C23—101no—
577C23—101no—
578C23D5×C4⋊C412yes—
579C23C2×D4×D511yescharacteristic
580D4—108no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—52no—
600C2×C4—52no—
601C2×C4—52no—
602C2×C4—101no—
603C2×C4—101no—
604C2×C4—101no—
605C2×C4—101no—
606C2×C4—101no—
607C2×C4—101no—
608C2×C4—102no—
609C2×C4—102no—
610C2×C4—102no—
611C2×C4—102no—
612C2×C4—102no—
613C2×C4—201no—
614C2×C4—201no—
615C2×C4—201no—
616C2×C4—201no—
617C2×C4—201no—
618C2×C4—201no—
619C2×C4—201no—
620C2×C4—201no—
621C2×C4—201no—
622C2×C4—201no—
623C2×C4—201no—
624C2×C4—404no—
625C2×C4—404no—
626C2×C4—404no—
627C2×C4D5×C4⋊C412yes—
628C2×C4C22×C4×D511yescharacteristic
629C2×C4C2×D4×D511yescharacteristic
630C2×C4#414411yescharacteristic
631C2×C4C2×Q8×D511yescharacteristic
632C2×C4D5×#4111yescharacteristic
633C2×C4#415011yescharacteristic
634C8—22no—
635C8—22no—
636C8—54no—
637C8—102no—
638C5#2265911yescharacteristic, complemented, Sylow
639C22—22no—
640C22—22no—
641C22—22no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—54no—
647C22—102no—
648C22—102no—
649C22—102no—
650C22—102no—
651C22—102no—
652C22—102no—
653C22C2×D5×C4⋊C411yescharacteristic
654C22#6592411yescharacteristic
655C22#6598911yescharacteristic
656C22#6610411yescharacteristic
657C22#6610411yescharacteristic
658C22D5×AGL(1,ℤ/8)11yescharacteristic
659C22D5×#13711yescharacteristic
660C4—22no—
661C4—42no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—102no—
666C4—102no—
667C4—102no—
668C4—202no—
669C4—401no—
670C4—401no—
671C4—401no—
672C4C4×D4×D511yescharacteristic
673C4#6588011yescharacteristic
674C4#6596611yescharacteristic
675C4#6597011yescharacteristic
676C2—42no—
677C2—104no—
678C2#1060612411yescharacteristic
679C2#1061090911yescharacteristic
680C2D5×#42811yescharacteristic
681C2#1061185211yescharacteristic
682C2#1061197211yescharacteristic
683C2#1061312211yescharacteristic
684C2#1061326511yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1854C2
socleC23×C5C22×D4
prime radicalC22×C22⋊D5D4
cosocleC22×C4×C5C24
hypercenter#1854D5
hypocenterC5#22659
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
2. C2×#65284C2semidirect
3. C2×D5×#106C2semidirect
6. C5×#1854C2semidirect
7. #10601425C2semidirect
8. #10601401C2semidirect
9. #10601356C2semidirect
10. #10600721C2semidirect
11. #10600306C2semidirect
12. #10599668C2semidirect
13. #10599662C2semidirect
14. #10599590C2semidirect
15. #10599579C2semidirect
53. #65017C4semidirect
104. C2×#681C2×C4semidirect
109. C2×C8×D5C2×C4semidirect
117. C5×C22⋊C8C2×C4semidirect
126. #3952C2×C4semidirect
129. #3947C2×C4semidirect
130. #3946C2×C4semidirect
133. #3940D4semidirect
144. #1854D5semidirect
357. #168C4×D4semidirect
422. C22⋊C8C4×D5semidirect
452. C2×D5#428semidirect
638. C5#22659semidirect

Maximal subgroups

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

subgroupindex
#106140032
C2×#652842
C2×D5×#1062
C2×#650172
#106057412
C5×#18542
#106014252
#106014012
#106013562
#106007212
#106003062
#105996682
#105996622
#105995902
#105995792
#226595