Finite Group Databasebeta

← #12004834

Subgroups — #12004834

Order 1280.

1204 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10617632C211yesmaximal, characteristic
2C2×#65428C211yesmaximal, characteristic
3C2×#65294C211yesmaximal, characteristic
4C2×C5×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
5C2×#64779C211yesmaximal, characteristic, complemented
6#10602385C211yesmaximal, characteristic, complemented
7#10602371C211yesmaximal, characteristic, complemented
8C22×#709C2211yescharacteristic
9C22×C5×M16C2211yescharacteristic
10C2×C5×#117C2211yescharacteristic
11C2×C5×#117C2211yescharacteristic
12C2×#4072C2211yescharacteristic
13C2×#4072C2211yescharacteristic
14C22×#681C2211yescharacteristic
15#65857—22no—
16#65856—22no—
17#65855C412yescomplemented
18D5×#117—22no—
19#65850C412yescomplemented
20#65838—22no—
21#65820—22no—
22C4×#709—22no—
23#65428—22no—
24#65428C414yescomplemented
25C2×#3957—22no—
26#65294—22no—
27#65294—22no—
28C2×#3937C414yescomplemented
29C5×H(3,ℤ/4)—22no—
30#64779—22no—
31#64746—22no—
32#64743—22no—
33#10783—51nomaximal, Sylow
34C23×C4×C5C2311yescharacteristic
35C22×C22⋊D5—21no—
36C2×#709—21no—
37C2×#709—21no—
38C2×#709—21no—
39C2×#709—21no—
40C2×#709—21no—
41C2×#709—21no—
42C2×#709C2×C412yes—
43C2×#709C2×C412yes—
44C2×#709D412yes—
45C2×#709D412yes—
46C2×#709D412yes—
47C2×#709Q812yes—
48C22×D20C2×C411yescharacteristic
49C22×C4×D5—21no—
50C22×Dic10C2×C411yescharacteristic
51C2×C5×M16—21no—
52C2×C5×M16—21no—
53C2×C5×M16—21no—
54C2×C5×M16—21no—
55C2×C5×M16—21no—
56C2×C5×M16C2×C412yescomplemented
57C22×C5×C8—21no—
58C5×#117—21no—
59C5×#117—21no—
60C5×#117—21no—
61C5×#117—21no—
62C5×#117—21no—
63C5×#117—21no—
64C5×#117C2×C412yes—
65C5×#117D412yes—
66C2×C5×C4⋊C4—21no—
67C2×C5×C4⋊C4—21no—
68C2×C5×C22⋊C4—21no—
69C2×C5×C22⋊C4—21no—
70C2×C42×C5C2×C412yescomplemented
71C2×C42×C5D412yes—
72C2×C22⋊Dic5—21no—
73C2×C22⋊Dic5D412yes—
74C4×C22⋊D5—28no—
75#4072—21no—
76#4072—21no—
77#4072—21no—
78#4072—21no—
79#4072—21no—
80#4072C2×C412yes—
81#4072C2×C412yescomplemented
82#4072D412yes—
83C2×C4⋊Dic5—21no—
84C2×C4⋊Dic5—21no—
85C2×C4×Dic5—21no—
86C2×C4×Dic5—21no—
87C2×#681—21no—
88C2×#681—21no—
89C2×#681—21no—
90C2×#681—21no—
91C2×#681—21no—
92C2×#681C2×C412yescomplemented
93C22×#168—21no—
94#4039—24no—
95#4038—24no—
96D5×C4⋊C4—24no—
97#4033—24no—
98#4027—28no—
99D5×C22⋊C4—24no—
100#4023—24no—
101C4×D20—24no—
102#4018—24no—
103C42×D5—24no—
104C4×Dic10—24no—
105#3957—28no—
106#3937—28no—
107#3428—51no—
108C2×#421—51no—
109C2×#421—51no—
110C2×#344—51no—
111#1740—51no—
112#1733—51no—
113C2×H(3,ℤ/4)D511yescharacteristic, complemented
114C24×C5C2×D411yescharacteristic
115C23×D5—21no—
116C22×C4×C5—22no—
117C22×C4×C5—22no—
118C22×C4×C5C22×C411yescharacteristic
119C22×C4×C5C22×C411yescharacteristic
120C22×C4×C5C22×C411yescharacteristic
121C22×C4×C5C2×D411yescharacteristic
122C22×C4×C5C2×D411yescharacteristic
123C22×C4×C5C2×D411yescharacteristic
124C22×C4×C5C2×D411yescharacteristic
125C22×C4×C5C2×D411yescharacteristic
126C22×C4×C5C2×D411yescharacteristic
127C22×C4×C5C2×Q811yescharacteristic
128C22×C4×C5#4111yescharacteristic
129C22×C4×C5#4111yescharacteristic
130C22×C4×C5#4111yescharacteristic
131C22×C4×C5#4111yescharacteristic
132C2×C22⋊D5—22no—
133C2×C22⋊D5—22no—
134C2×C22⋊D5—22no—
135C2×C22⋊D5—22no—
136C2×C22⋊D5—22no—
137C2×C22⋊D5—22no—
138C22×Dic5—21no—
139C22×Dic5—22no—
140C22×Dic5—41no—
141#709—22no—
142#709—22no—
143#709—22no—
144#709—22no—
145#709—22no—
146#709—22no—
147#709—22no—
148#709—22no—
149#709—22no—
150#709—22no—
151#709—22no—
152#709—22no—
153#709C22⋊C414yes—
154#709C22⋊C414yes—
155#709C4⋊C414yes—
156#709C4⋊C414yes—
157C2×D20—21no—
158C2×D20—21no—
159C2×D20—21no—
160C2×D20C4212yes—
161C2×D20C22⋊C412yes—
162C2×D20C4⋊C412yes—
163C2×C4×D5—22no—
164C2×C4×D5—22no—
165C2×C4×D5—22no—
166C2×C4×D5—22no—
167C2×C4×D5—22no—
168C2×C4×D5—22no—
169C2×C4×D5—28no—
170C2×Dic10—21no—
171C2×Dic10—21no—
172C2×Dic10—21no—
173C2×Dic10C4212yes—
174C2×Dic10C22⋊C412yes—
175C2×Dic10C4⋊C412yes—
176C5×M16—22no—
177C5×M16—22no—
178C5×M16—41no—
179C5×M16—41no—
180C2×C5×C8—22no—
181C2×C5×C8—41no—
182C2×C5×C8—41no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—41no—
186C5×C4⋊C4—41no—
187C5×C22⋊C4—22no—
188C5×C22⋊C4—22no—
189C5×C22⋊C4—41no—
190C5×C22⋊C4—41no—
191C42×C5—22no—
192C42×C5—22no—
193C42×C5—22no—
194C42×C5C22⋊C414yes—
195C22⋊Dic5—22no—
196C22⋊Dic5—22no—
197C22⋊Dic5—41no—
198C22⋊Dic5C22⋊C414yes—
199#685—28no—
200C4⋊Dic5—22no—
201C4⋊Dic5—22no—
202C4⋊Dic5—22no—
203C4⋊Dic5—41no—
204#683—28no—
205C4×Dic5—28no—
206C4×Dic5—22no—
207C4×Dic5—22no—
208C4×Dic5—22no—
209C4×Dic5—41no—
210#681—22no—
211#681—22no—
212#681—41no—
213#681—41no—
214C2×#168—22no—
215C2×#168—41no—
216C2×#168—41no—
217C22×#41—51no—
218C22×M16—51no—
219C22×M16C2×D511yescharacteristic
220#520—102no—
221#520—102no—
222#519—52no—
223#518—52no—
224#518—102no—
225C4×#41—102no—
226C4×#41—102no—
227C2×#117—51no—
228C2×#117—51no—
229C2×#117—102no—
230C2×#117C2×D511yescharacteristic
231C2×#117C2×D511yescharacteristic
232#421—54no—
233#421—102no—
234#421—102no—
235#421—102no—
236C2×#104—54no—
237C2×#104—102no—
238#344—102no—
239#342—102no—
240#339—102no—
241H(3,ℤ/4)—22no—
242C23×C5—81no—
243C23×C5C2×C22⋊C411yescharacteristic
244C23×C5C4×D411yescharacteristic
245C23×C5C4×D411yescharacteristic
246C23×C5#12011yescharacteristic
247C23×C5#12011yescharacteristic
248C23×C5#12111yescharacteristic
249C23×C5C22⋊Q811yescharacteristic
250C22×D5—22no—
251C22×D5—22no—
252C22×D5—22no—
253C22×D5—82no—
254C2×C4×C5—24no—
255C2×C4×C5—42no—
256C2×C4×C5—82no—
257C2×C4×C5—82no—
258C2×C4×C5C2×C4211yescharacteristic
259C2×C4×C5C2×C22⋊C411yescharacteristic
260C2×C4×C5C2×C22⋊C411yescharacteristic
261C2×C4×C5C2×C22⋊C411yescharacteristic
262C2×C4×C5C2×C22⋊C411yescharacteristic
263C2×C4×C5C2×C4⋊C411yescharacteristic
264C2×C4×C5C2×C4⋊C411yescharacteristic
265C2×C4×C5C2×C4⋊C411yescharacteristic
266C2×C4×C5#11711yescharacteristic
267C2×C4×C5#11711yescharacteristic
268C2×C4×C5C4×D411yescharacteristic
269C2×C4×C5C4×D411yescharacteristic
270C2×C4×C5C4×D411yescharacteristic
271C2×C4×C5C4×D411yescharacteristic
272C2×C4×C5C4×D411yescharacteristic
273C2×C4×C5C4×D411yescharacteristic
274C2×C4×C5#12011yescharacteristic
275C2×C4×C5#12011yescharacteristic
276C2×C4×C5#12111yescharacteristic
277C2×C4×C5#12111yescharacteristic
278C2×C4×C5#12111yescharacteristic
279C2×C4×C5C22⋊Q811yescharacteristic
280C2×C4×C5C22⋊Q811yescharacteristic
281C2×C4×C5C22⋊Q811yescharacteristic
282C2×C4×C5#12311yescharacteristic
283C2×C4×C5#12311yescharacteristic
284C2×C4×C5#12311yescharacteristic
285C2×C4×C5#12311yescharacteristic
286C22⋊D5—24no—
287C22⋊D5—24no—
288C22⋊D5—24no—
289C22⋊D5—24no—
290C2×Dic5—22no—
291C2×Dic5—22no—
292C2×Dic5—22no—
293C2×Dic5—24no—
294C2×Dic5—42no—
295C2×Dic5—82no—
296C2×Dic5—82no—
297D20—22no—
298D20—22no—
299D20—22no—
300D20#9514yes—
301C4×D5—24no—
302C4×D5—24no—
303C4×D5—24no—
304C4×D5—24no—
305C4×D5—88no—
306Dic10—22no—
307Dic10—22no—
308Dic10—22no—
309Dic10#9514yes—
310C5×C8—42no—
311#168—42no—
312C2×#41—52no—
313C2×#41—52no—
314C2×#41—52no—
315C2×#41—52no—
316C2×#41—52no—
317C2×#41—52no—
318C2×#41—101no—
319C2×#41—101no—
320C2×#41—101no—
321C2×#41—101no—
322C2×#41—101no—
323C2×#41—101no—
324C22×Q8—51no—
325C22×D4—51no—
326C22×D4—101no—
327C23×C4—101no—
328C23×C4C22×D511yescharacteristic
329C2×M16—21no—
330C2×M16—21no—
331C2×M16—21no—
332C2×M16—21no—
333C2×M16—21no—
334C2×M16—52no—
335C2×M16—101no—
336C2×M16—101no—
337C2×M16—101no—
338C2×M16—101no—
339C2×M16—101no—
340C2×M16C4×D512yescomplemented
341C22×C8—21no—
342C22×C8—101no—
343C4×Q8—104no—
344C4×Q8—104no—
345C4×D4—108no—
346C4×D4—104no—
347C4×D4—108no—
348C4×D4—104no—
349#117—21no—
350#117—21no—
351#117—21no—
352#117—21no—
353#117—21no—
354#117—21no—
355#117—52no—
356#117—52no—
357#117—52no—
358#117—104no—
359#117—104no—
360#117—104no—
361#117—101no—
362#117—101no—
363#117—101no—
364#117—101no—
365#117—101no—
366#117C4×D512yes—
367#117C22⋊D512yes—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—104no—
371C2×C4⋊C4—101no—
372C2×C4⋊C4—101no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—52no—
376C2×C22⋊C4—104no—
377C2×C22⋊C4—101no—
378C2×C42—104no—
379C2×C42—101no—
380C2×C42—101no—
381C2×C42C4×D512yescomplemented
382C2×C42D2012yes—
383#104—108no—
384#104—108no—
385C22×C5—41no—
386C22×C5—41no—
387C22×C5—41no—
388C22×C5—41no—
389C22×C5—41no—
390C22×C5—41no—
391C22×C5—41no—
392C22×C5C4×C22⋊C411yescharacteristic
393C22×C5#37911yescharacteristic
394C22×C5#38011yescharacteristic
395C22×C5#38511yescharacteristic
396C22×C5#38511yescharacteristic
397C22×C5#38611yescharacteristic
398C22×C5#38611yescharacteristic
399C2×D5—24no—
400C2×D5—44no—
401C2×D5—82no—
402C2×D5—82no—
403C4×C5—44no—
404C4×C5—82no—
405C4×C5C2×#9511yescharacteristic
406C4×C5C4×C22⋊C411yescharacteristic
407C4×C5#38011yescharacteristic
408C4×C5#38111yescharacteristic
409C4×C5#38511yescharacteristic
410C4×C5#38511yescharacteristic
411C4×C5#38611yescharacteristic
412C4×C5#38611yescharacteristic
413Dic5—24no—
414Dic5—44no—
415Dic5—82no—
416C24—101no—
417C24D4×D511yescharacteristic
418#41—54no—
419#41—54no—
420#41—54no—
421#41—54no—
422#41—102no—
423#41—102no—
424#41—102no—
425#41—102no—
426#41—102no—
427#41—102no—
428#41—102no—
429#41—102no—
430#41—102no—
431#41—102no—
432#41—102no—
433#41—102no—
434C2×Q8—52no—
435C2×Q8—52no—
436C2×Q8—52no—
437C2×Q8—101no—
438C2×Q8—101no—
439C2×Q8—101no—
440C2×D4—52no—
441C2×D4—52no—
442C2×D4—52no—
443C2×D4—102no—
444C2×D4—102no—
445C2×D4—102no—
446C2×D4—102no—
447C2×D4—102no—
448C2×D4—102no—
449C2×D4—101no—
450C2×D4—101no—
451C2×D4—101no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—108no—
455C22×C4—101no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4—102no—
459C22×C4—102no—
460C22×C4—102no—
461C22×C4—102no—
462C22×C4—102no—
463C22×C4—201no—
464C22×C4C2×C4×D511yescharacteristic
465C22×C4C2×C4×D511yescharacteristic
466C22×C4C2×C4×D511yescharacteristic
467C22×C4C2×D2011yescharacteristic
468C22×C4#70911yescharacteristic
469C22×C4#70911yescharacteristic
470C22×C4D4×D511yescharacteristic
471C22×C4D4×D511yescharacteristic
472C22×C4D4×D511yescharacteristic
473C22×C4D4×D511yescharacteristic
474C22×C4#71111yescharacteristic
475C22×C4Q8×D511yescharacteristic
476C22×C4#71311yescharacteristic
477C22×C4C2×C22⋊D511yescharacteristic
478M16—22no—
479M16—22no—
480M16—41no—
481M16—41no—
482M16—102no—
483M16—102no—
484M16—201no—
485M16—201no—
486C2×C8—22no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—102no—
490C2×C8—201no—
491C2×C8—201no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—108no—
497C4⋊C4—102no—
498C4⋊C4—102no—
499C4⋊C4—102no—
500C4⋊C4—201no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—54no—
506C22⋊C4—108no—
507C22⋊C4—102no—
508C22⋊C4—102no—
509C22⋊C4—201no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—108no—
514C42—102no—
515C42—102no—
516C42—102no—
517C42—201no—
518C42#68514yes—
519C2×C5—21no—
520C2×C5—21no—
521C2×C5—21no—
522C2×C5—21no—
523C2×C5—21no—
524C2×C5—21no—
525C2×C5—21no—
526C2×C5C22⋊#9511yescharacteristic
527D5—44no—
528C23—81no—
529C23—102no—
530C23—102no—
531C23—102no—
532C23—402no—
533C23D5×C22⋊C411yescharacteristic
534C23#402711yescharacteristic
535C23#402811yescharacteristic
536C23#403011yescharacteristic
537C23#403911yescharacteristic
538C23#404211yescharacteristic
539C23#408311yescharacteristic
540Q8—54no—
541Q8—102no—
542Q8—102no—
543Q8—102no—
544D4—54no—
545D4—104no—
546D4—104no—
547D4—104no—
548D4—104no—
549D4—102no—
550D4—102no—
551D4—102no—
552C2×C4—24no—
553C2×C4—42no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—102no—
557C2×C4—102no—
558C2×C4—102no—
559C2×C4—104no—
560C2×C4—104no—
561C2×C4—104no—
562C2×C4—104no—
563C2×C4—104no—
564C2×C4—202no—
565C2×C4—408no—
566C2×C4—402no—
567C2×C4—402no—
568C2×C4C42×D511yescharacteristic
569C2×C4#401811yescharacteristic
570C2×C4C4×D2011yescharacteristic
571C2×C4C4×D2011yescharacteristic
572C2×C4D5×C22⋊C411yescharacteristic
573C2×C4D5×C22⋊C411yescharacteristic
574C2×C4D5×C22⋊C411yescharacteristic
575C2×C4#402711yescharacteristic
576C2×C4#402811yescharacteristic
577C2×C4#402911yescharacteristic
578C2×C4#402911yescharacteristic
579C2×C4#403011yescharacteristic
580C2×C4D5×C4⋊C411yescharacteristic
581C2×C4D5×C4⋊C411yescharacteristic
582C2×C4D5×C4⋊C411yescharacteristic
583C2×C4#403811yescharacteristic
584C2×C4#403911yescharacteristic
585C2×C4#404011yescharacteristic
586C2×C4#404011yescharacteristic
587C2×C4#404111yescharacteristic
588C2×C4#404211yescharacteristic
589C2×C4#404311yescharacteristic
590C2×C4C2×#68511yescharacteristic
591C2×C4C4×C22⋊D511yescharacteristic
592C2×C4C4×C22⋊D511yescharacteristic
593C2×C4#408311yescharacteristic
594C2×C4#408411yescharacteristic
595C2×C4#409211yescharacteristic
596C8—42no—
597C8—202no—
598C5#1078311yescharacteristic, complemented, Sylow
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—104no—
607C22—204no—
608C22—402no—
609C22—402no—
610C22#6495411yescharacteristic
611C22#6495511yescharacteristic
612C22#6495611yescharacteristic
613C22#6495711yescharacteristic
614C22#6524411yescharacteristic
615C22#6524511yescharacteristic
616C22#6527811yescharacteristic
617C4—44no—
618C4—82no—
619C4—104no—
620C4—204no—
621C4—402no—
622C4D5×#9511yescharacteristic
623C4#6495311yescharacteristic
624C4#6495411yescharacteristic
625C4#6495711yescharacteristic
626C4C4×#68511yescharacteristic
627C4#6524511yescharacteristic
628C4#6527611yescharacteristic
629C4#6527811yescharacteristic
630C2—21no—
631C2—21no—
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—204no—
638C2#1060021811yescharacteristic
subgroupG/H
centerC4#64954
commutatorC2×C4×C5C2×C42
FrattiniC23×C4C22×D5
FittingC2×C5×H(3,ℤ/4)C2
socleC2×C5C22⋊#95
prime radicalC22×#709C22
cosocleC23×C4×C5C23
hypercenterC2×H(3,ℤ/4)D5
hypocenterC5#10783
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
4. C2×C5×H(3,ℤ/4)C2semidirect
5. C2×#64779C2semidirect
6. #10602385C2semidirect
7. #10602371C2semidirect
17. #65855C4semidirect
19. #65850C4semidirect
24. #65428C4semidirect
28. C2×#3937C4semidirect
56. C2×C5×M16C2×C4semidirect
70. C2×C42×C5C2×C4semidirect
81. #4072C2×C4semidirect
92. C2×#681C2×C4semidirect
113. C2×H(3,ℤ/4)D5semidirect
340. C2×M16C4×D5semidirect
381. C2×C42C4×D5semidirect
598. C5#10783semidirect

Maximal subgroups

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

subgroupindex
#106176322
C2×#654282
C2×#652942
C2×C5×H(3,ℤ/4)2
C2×#647792
#106023852
#106023712
#107835