Finite Group Databasebeta

← #102×D20 #12114708

Subgroups — #102×D20 #12114708

Order 1280.

2014 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×D20C211yesmaximal, characteristic
2C2×D5×#102C212yesmaximal, complemented
3C2×C8×D20C211yesmaximal, characteristic, complemented
4C4⋊C4×D20C211yesmaximal, characteristic, complemented
5C4×C5×#102C211yesmaximal, characteristic, complemented
6#10602397C211yesmaximal, characteristic, complemented
7#10600991C212yesmaximal, complemented
8#10600565C211yesmaximal, characteristic, complemented
9#10600280C211yesmaximal, characteristic, complemented
10#10599676C212yesmaximal, complemented
11#10599673C212yesmaximal, complemented
12C22×D4×D5C2212yes—
13C23×D20—21no—
14C2×C4×C5×D4C2211yescharacteristic
15C2×#4076—21no—
16C22×C8×D5C2212yescomplemented
17D4×D20—24no—
18D4×D20C418yescomplemented
19C2×#4041C2212yes—
20C2×D5×C4⋊C4C2212yescomplemented
21C2×#4028—22no—
22C2×#4020C2211yescharacteristic
23C2×C4×D20—21no—
24C2×C4×D20C2211yescharacteristic
25C2×C5×#102C2211yescharacteristic
26C2×C5×#102C2211yescharacteristic
27C2×C4×C5×C8C2211yescharacteristic, complemented
28C4×C5×C4⋊C4C2211yescharacteristic, complemented
29C2×#3964C2212yes—
30C2×#3953C2212yescomplemented
31C2×#3953C2212yescomplemented
32C2×#3952C2212yescomplemented
33#65276C2212yescomplemented
34C2×#3941C2212yescomplemented
35C2×#3941C2212yescomplemented
36#65224C2211yescharacteristic, complemented
37#65217—21no—
38C2×C4⋊#168C2211yescharacteristic, complemented
39D5×#102—28no—
40C8×D20—24no—
41#64957—22no—
42C5×#340C2211yescharacteristic, complemented
43C5×#340C2211yescharacteristic, complemented
44#64749C2212yescomplemented
45D4×#102—51nomaximal, Sylow
46C24×D5—22no—
47C22×C5×D4C2311yescharacteristic
48C23×C4×C5—21no—
49C22×C22⋊D5—22no—
50C2×D4×D5—24no—
51C2×D4×D5—24no—
52C2×D4×D5—24no—
53C2×D4×D5—42no—
54C2×D4×D5—42no—
55C2×D4×D5C2×C418yescomplemented
56C22×D20—22no—
57C22×D20—21no—
58C22×D20—21no—
59C22×D20—28no—
60C22×D20—44no—
61C22×D20—44no—
62C22×D20D411yescharacteristic
63C22×D20C2312yes—
64C22×C4×D5—22no—
65C22×C4×D5C2312yes—
66C22×C5×C8C2311yescharacteristic
67C22×C5×C8C2311yescharacteristic
68C4×C5×D4—22no—
69C4×C5×D4C2×C414yescomplemented
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C4⋊C4C2311yescharacteristic
73C2×C5×C22⋊C4—21no—
74C2×C42×C5—21no—
75C2×C42×C5C2311yescharacteristic
76#4076—28no—
77C2×#685—22no—
78C2×#685—22no—
79C2×#685D412yes—
80C2×C4⋊Dic5—21no—
81C2×C4⋊Dic5—21no—
82C2×C4⋊Dic5D411yescharacteristic
83C2×C4⋊Dic5C2312yes—
84C2×#683—22no—
85C22×#168C2312yes—
86C2×C8×D5—24no—
87C2×C8×D5—24no—
88C2×C8×D5—42no—
89#4041—24no—
90#4041C2×C418yescomplemented
91D5×C4⋊C4—24no—
92D5×C4⋊C4—42no—
93#4028—216no—
94#4020—22no—
95#4020C2×C414yescomplemented
96C4×D20—44no—
97C4×D20C2×C414yes—
98C4×D20D414yes—
99C5×#102—22no—
100C5×#102D414yescomplemented
101C4×C5×C8—22no—
102C5×#95—21no—
103#3964—24no—
104#3963—22no—
105#3953—24no—
106#3953—42no—
107#3952—24no—
108#3941—24no—
109#3941—42no—
110C4⋊#168—22no—
111C2×D42—51no—
112C2×C8×D4—51no—
113C22×#102—52no—
114D4×C4⋊C4—51no—
115#1982—51no—
116#1957—51no—
117#1893—52no—
118#1874—52no—
119#1802—51no—
120#1786—52no—
121C4×#102D511yescharacteristic, complemented
122C24×C5—21no—
123C23×D5—22no—
124C23×D5—28no—
125C23×D5—44no—
126C23×D5—44no—
127C23×D5—44no—
128C23×D5—44no—
129C23×D5—82no—
130C23×D5—82no—
131C23×D5C2×D412yes—
132C2×C5×D4—21no—
133C2×C5×D4—21no—
134C2×C5×D4—21no—
135C2×C5×D4C22×C412yes—
136C2×C5×D4C2×D412yes—
137C2×C5×D4#4112yes—
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—24no—
141C22×C4×C5—21no—
142C22×C4×C5—42no—
143C22×C4×C5—42no—
144C22×C4×C5C2×D411yescharacteristic
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2411yescharacteristic
147C2×C22⋊D5—28no—
148C2×C22⋊D5—44no—
149C2×C22⋊D5—44no—
150C22×Dic5—22no—
151C22×Dic5C2×D412yes—
152D4×D5—216no—
153D4×D5—48no—
154D4×D5—48no—
155D4×D5—48no—
156C2×D20—22no—
157C2×D20—22no—
158C2×D20—22no—
159C2×D20—22no—
160C2×D20—28no—
161C2×D20—24no—
162C2×D20—22no—
163C2×D20—24no—
164C2×D20—44no—
165C2×D20—44no—
166C2×D20—42no—
167C2×D20—42no—
168C2×D20—432no—
169C2×D20—81no—
170C2×D20—81no—
171C2×D20C22⋊C414yes—
172C2×D20D814yes—
173C2×D20SD1614yes—
174C2×D20C22×C414yes—
175C2×C4×D5—22no—
176C2×C4×D5—22no—
177C2×C4×D5—22no—
178C2×C4×D5—22no—
179C2×C4×D5—44no—
180C2×C4×D5—44no—
181C2×C4×D5—82no—
182C2×C4×D5C22×C414yes—
183C2×C4×D5C2×D414yes—
184C2×C5×C8—21no—
185C2×C5×C8—21no—
186C2×C5×C8—21no—
187C2×C5×C8—21no—
188C2×C5×C8—21no—
189C2×C5×C8C2×D412yescomplemented
190C5×C4⋊C4—21no—
191C5×C4⋊C4—21no—
192C5×C4⋊C4—21no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4C22×C412yes—
195C5×C4⋊C4C2×D412yescomplemented
196C5×C22⋊C4—24no—
197C42×C5—42no—
198C42×C5C22×C412yes—
199C42×C5C2×D412yes—
200#685—28no—
201#685—44no—
202#685C22⋊C418yes—
203C4⋊Dic5—22no—
204C4⋊Dic5—22no—
205C4⋊Dic5—42no—
206C4⋊Dic5—81no—
207C4⋊Dic5C22⋊C414yes—
208#683—44no—
209C2×#168—22no—
210C2×#168—22no—
211C2×#168—22no—
212C8×D5—48no—
213C23×D4—52no—
214C23×D4—101no—
215C23×C8—52no—
216D42—58no—
217D42—104no—
218C2×#127—51no—
219C2×#121—52no—
220C2×#121—101no—
221C2×#120—102no—
222C2×C4×D4—51no—
223C2×C4×D4—101no—
224C2×C4×D4C2×D511yescharacteristic
225C22×C4⋊C4—52no—
226C8×D4—104no—
227C2×C4⋊C8—51no—
228C2×#102—52no—
229C2×#102—52no—
230C2×#102—52no—
231C2×#102—52no—
232C2×#102—52no—
233C2×#102—108no—
234C2×#102C2×D511yescharacteristic
235C2×#102C2×D511yescharacteristic
236C2×C22⋊C8—52no—
237C2×C4×C8C2×D511yescharacteristic, complemented
238#389—101no—
239#385—102no—
240#384—51no—
241#380—52no—
242C4×C4⋊C4C2×D511yescharacteristic, complemented
243#340—52no—
244#340C2×D511yescharacteristic, complemented
245#340C2×D511yescharacteristic, complemented
246C23×C5—22no—
247C23×C5—22no—
248C23×C5—22no—
249C23×C5—41no—
250C23×C5—41no—
251C23×C5—41no—
252C23×C5—41no—
253C23×C5C22×D411yescharacteristic
254C22×D5—22no—
255C22×D5—22no—
256C22×D5—22no—
257C22×D5—22no—
258C22×D5—24no—
259C22×D5—416no—
260C22×D5—42no—
261C22×D5—42no—
262C22×D5—42no—
263C22×D5—42no—
264C22×D5—42no—
265C22×D5—42no—
266C22×D5—416no—
267C22×D5—88no—
268C22×D5—88no—
269C22×D5—88no—
270C22×D5—88no—
271C22×D5—88no—
272C22×D5C2×C22⋊C414yes—
273C22×D5C2×D814yes—
274C22×D5C2×SD1614yes—
275C5×D4—22no—
276C5×D4—22no—
277C5×D4—22no—
278C5×D4C4×D414yescomplemented
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—22no—
282C2×C4×C5—22no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—48no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5—41no—
291C2×C4×C5C2×C22⋊C412yes—
292C2×C4×C5C2×C22⋊C412yes—
293C2×C4×C5C2×D812yes—
294C2×C4×C5C2×SD1612yes—
295C2×C4×C5C23×C411yescharacteristic
296C2×C4×C5C22×D411yescharacteristic
297C2×C4×C5C22×D411yescharacteristic
298C2×C4×C5C2×#4111yescharacteristic
299C2×C4×C5H(5,𝔽2)11yescharacteristic
300C2×C4×C5H(5,𝔽2)11yescharacteristic
301C22⋊D5—416no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—42no—
305C2×Dic5—42no—
306C2×Dic5—42no—
307C2×Dic5C2×C22⋊C414yes—
308D20—24no—
309D20—24no—
310D20—416no—
311D20—42no—
312D20—42no—
313D20—48no—
314D20—84no—
315D20#102116yesdirect product factor
316C4×D5—24no—
317C4×D5—24no—
318C4×D5—24no—
319C4×D5—24no—
320C4×D5—88no—
321C5×C8—22no—
322C5×C8—41no—
323#168—42no—
324C25—102no—
325C22×D4—58no—
326C22×D4—51no—
327C22×D4—52no—
328C22×D4—108no—
329C22×D4—102no—
330C22×D4—104no—
331C22×D4—102no—
332C22×D4—104no—
333C22×D4—104no—
334C22×D4—101no—
335C22×D4—101no—
336C22×D4—204no—
337C22×D4—204no—
338C22×D4—202no—
339C22×D4—202no—
340C22×D4C22×D511yescharacteristic
341C23×C4—21no—
342C23×C4—52no—
343C23×C4—102no—
344C22×C8—52no—
345C22×C8—104no—
346C22×C8—104no—
347C22×C8—202no—
348C22×C8C22×D511yescharacteristic
349C22×C8C22×D511yescharacteristic
350#127—54no—
351#127—102no—
352#121—58no—
353#121—104no—
354#121—108no—
355#120—1016no—
356C4×D4—22no—
357C4×D4—54no—
358C4×D4—54no—
359C4×D4—204no—
360C4×D4C4×D514yescomplemented
361C2×C4⋊C4—51no—
362C2×C4⋊C4—52no—
363C2×C4⋊C4—104no—
364C2×C4⋊C4—102no—
365C2×C4⋊C4—101no—
366C2×C4⋊C4—101no—
367C2×C4⋊C4—202no—
368C2×C4⋊C4C22×D511yescharacteristic
369C2×C4⋊C4C22×D511yescharacteristic
370C2×C4⋊C4C22×D511yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—52no—
373C2×C22⋊C4—102no—
374C2×C22⋊C4—102no—
375C2×C42—21no—
376C2×C42C22×D511yescharacteristic
377C4⋊C8—102no—
378#102—22no—
379#102—104no—
380#102—104no—
381#102—104no—
382#102—202no—
383#102—202no—
384#102D2014yesdirect product factor
385C22⋊C8—104no—
386C4×C8—22no—
387#95—21no—
388#95—102no—
389C22×C5—24no—
390C22×C5—42no—
391C22×C5—42no—
392C22×C5—42no—
393C22×C5—42no—
394C22×C5—42no—
395C22×C5—42no—
396C22×C5C22×C22⋊C411yescharacteristic
397C22×C5D4211yescharacteristic
398C22×C5#54611yescharacteristic
399C22×C5C22×D811yescharacteristic
400C22×C5C22×SD1611yescharacteristic
401C22×C5#57611yescharacteristic
402C22×C5#57711yescharacteristic
403C2×D5—24no—
404C2×D5—24no—
405C2×D5—24no—
406C2×D5—24no—
407C2×D5—24no—
408C2×D5—24no—
409C2×D5—44no—
410C2×D5—44no—
411C2×D5—44no—
412C2×D5—832no—
413C2×D5—82no—
414C2×D5—82no—
415C2×D5—82no—
416C2×D5—82no—
417C2×D5C2×#10218yescomplemented
418C4×C5—22no—
419C4×C5—42no—
420C4×C5—42no—
421C4×C5—81no—
422C4×C5C2×#10214yescomplemented
423C4×C5C2×C4×D411yescharacteristic
424C4×C5#51811yescharacteristic
425C4×C5D4211yescharacteristic
426C4×C5#54611yescharacteristic
427Dic5—24no—
428Dic5—82no—
429C24—21no—
430C24—52no—
431C24—108no—
432C24—102no—
433C24—204no—
434C24—204no—
435C24—204no—
436C24—204no—
437C24—402no—
438C24—402no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—54no—
443C2×D4—54no—
444C2×D4—54no—
445C2×D4—54no—
446C2×D4—108no—
447C2×D4—108no—
448C2×D4—1016no—
449C2×D4—102no—
450C2×D4—102no—
451C2×D4—102no—
452C2×D4—102no—
453C2×D4—102no—
454C2×D4—104no—
455C2×D4—104no—
456C2×D4—2032no—
457C2×D4—204no—
458C2×D4—204no—
459C2×D4—208no—
460C2×D4—208no—
461C2×D4—208no—
462C2×D4—204no—
463C2×D4—204no—
464C2×D4—202no—
465C2×D4—202no—
466C2×D4—401no—
467C2×D4—401no—
468C2×D4C2×C4×D512yes—
469C2×D4C2×D2012yes—
470C2×D4#70912yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—52no—
478C22×C4—54no—
479C22×C4—54no—
480C22×C4—102no—
481C22×C4—102no—
482C22×C4—102no—
483C22×C4—102no—
484C22×C4—102no—
485C22×C4—204no—
486C22×C4—204no—
487C22×C4—402no—
488C22×C4D4×D511yescharacteristic
489C22×C4D4×D511yescharacteristic
490C22×C4C23×D511yescharacteristic
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—102no—
497C2×C8—102no—
498C2×C8—102no—
499C2×C8—208no—
500C2×C8C2×D2012yescomplemented
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—54no—
506C4⋊C4—102no—
507C4⋊C4—102no—
508C4⋊C4—204no—
509C4⋊C4—202no—
510C4⋊C4—401no—
511C4⋊C4C2×C4×D512yes—
512C4⋊C4C2×D2012yescomplemented
513C22⋊C4—24no—
514C22⋊C4—58no—
515C22⋊C4—108no—
516C22⋊C4—204no—
517C42—42no—
518C42C2×C4×D512yes—
519C42D4×D512yes—
520C2×C5—44no—
521C2×C5D4×C22⋊C411yescharacteristic
522C2×C5C22×#10211yescharacteristic
523C2×C5#289711yescharacteristic
524C2×C5D4×D811yescharacteristic
525C2×C5D4×SD1611yescharacteristic
526C2×C5#329411yescharacteristic
527C2×C5#329511yescharacteristic
528D5—28no—
529D5—84no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—54no—
538C23—54no—
539C23—54no—
540C23—102no—
541C23—102no—
542C23—102no—
543C23—102no—
544C23—104no—
545C23—2016no—
546C23—2016no—
547C23—202no—
548C23—202no—
549C23—202no—
550C23—202no—
551C23—202no—
552C23—202no—
553C23—408no—
554C23—408no—
555C23—408no—
556C23—408no—
557C23—408no—
558C23C2×D4×D511yescharacteristic
559D4—22no—
560D4—22no—
561D4—22no—
562D4—516no—
563D4—104no—
564D4—104no—
565D4—2016no—
566D4—2016no—
567D4—208no—
568D4—202no—
569D4—202no—
570D4—404no—
571D4C4×D2014yescomplemented
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—48no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—54no—
585C2×C4—102no—
586C2×C4—102no—
587C2×C4—104no—
588C2×C4—104no—
589C2×C4—104no—
590C2×C4—104no—
591C2×C4—202no—
592C2×C4—202no—
593C2×C4—202no—
594C2×C4—408no—
595C2×C4D5×C22⋊C412yes—
596C2×C4D5×C22⋊C412yes—
597C2×C4D5×D812yes—
598C2×C4D5×SD1612yes—
599C2×C4C22×C4×D511yescharacteristic
600C2×C4C22×D2011yescharacteristic
601C2×C4C2×#70911yescharacteristic
602C2×C4C2×D4×D511yescharacteristic
603C2×C4#414911yescharacteristic
604C2×C4#414911yescharacteristic
605C8—22no—
606C8—41no—
607C8—202no—
608C5D4×#10211yescharacteristic, complemented, Sylow
609C22—24no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—58no—
617C22—104no—
618C22—104no—
619C22—104no—
620C22—104no—
621C22—104no—
622C22—104no—
623C22—204no—
624C22—204no—
625C22—204no—
626C22—4032no—
627C22—402no—
628C22—402no—
629C22—402no—
630C22—402no—
631C22C2×D5×C22⋊C411yescharacteristic
632C22D4×D2011yescharacteristic
633C22#6588411yescharacteristic
634C22C2×D5×D811yescharacteristic
635C22C2×D5×SD1611yescharacteristic
636C22#6610311yescharacteristic
637C22#6610411yescharacteristic
638C4—22no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—104no—
643C4—402no—
644C4D5×#10214yescomplemented
645C4C2×C4×D2011yescharacteristic
646C4#6585511yescharacteristic
647C4D4×D2011yescharacteristic
648C4#6588411yescharacteristic
649C2—44no—
650C2—108no—
651C2—404no—
652C2C22⋊C4×D2011yescharacteristic
653C2C2×D5×#10211yescharacteristic
654C2#1061056211yescharacteristic
655C2D8×D2011yescharacteristic
656C2#1061133911yescharacteristic
657C2SD16×D2011yescharacteristic
658C2#1061140711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC4×C5×#102C2
socleC23×C5C22×D4
prime radicalC2×D4×D20C2
cosocleC22×C4×C5C24
hypercenterC4×#102D5
hypocenterC5D4×#102
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×D5×#102C2semidirect
3. C2×C8×D20C2semidirect
4. C4⋊C4×D20C2semidirect
5. C4×C5×#102C2semidirect
6. #10602397C2semidirect
7. #10600991C2semidirect
8. #10600565C2semidirect
9. #10600280C2semidirect
10. #10599676C2semidirect
11. #10599673C2semidirect
16. C22×C8×D5C22semidirect
18. D4×D20C4semidirect
20. C2×D5×C4⋊C4C22semidirect
27. C2×C4×C5×C8C22semidirect
28. C4×C5×C4⋊C4C22semidirect
30. C2×#3953C22semidirect
31. C2×#3953C22semidirect
32. C2×#3952C22semidirect
33. #65276C22semidirect
34. C2×#3941C22semidirect
35. C2×#3941C22semidirect
36. #65224C22semidirect
38. C2×C4⋊#168C22semidirect
42. C5×#340C22semidirect
43. C5×#340C22semidirect
44. #64749C22semidirect
55. C2×D4×D5C2×C4semidirect
69. C4×C5×D4C2×C4semidirect
90. #4041C2×C4semidirect
95. #4020C2×C4semidirect
100. C5×#102D4semidirect
121. C4×#102D5semidirect
189. C2×C5×C8C2×D4semidirect
195. C5×C4⋊C4C2×D4semidirect
237. C2×C4×C8C2×D5semidirect
242. C4×C4⋊C4C2×D5semidirect
244. #340C2×D5semidirect
245. #340C2×D5semidirect
278. C5×D4C4×D4semidirect
315. D20#102direct (N × H)
360. C4×D4C4×D5semidirect
384. #102D20direct (N × H)
417. C2×D5C2×#102semidirect
422. C4×C5C2×#102semidirect
500. C2×C8C2×D20semidirect
512. C4⋊C4C2×D20semidirect
571. D4C4×D20semidirect
608. C5D4×#102semidirect
644. C4D5×#102semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D202
C2×D5×#1022
C2×C8×D202
C4⋊C4×D202
C4×C5×#1022
#106023972
#106009912
#106005652
#106002802
#105996762
#105996732
D4×#1025