Finite Group Databasebeta

← #12022061

Subgroups — #12022061

Order 1280.

1862 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#685C211yesmaximal, characteristic
2C2×#64956C211yesmaximal, characteristic
3C2×D5×#95C211yesmaximal, characteristic
4C5×#1434C211yesmaximal, characteristic, complemented
5#10602327C211yesmaximal, characteristic, complemented
6#10600218C212yesmaximal
7#10600218C212yesmaximal
8#10600207C212yesmaximal, complemented
9#10600206C212yesmaximal, complemented
10#10600196C211yesmaximal, characteristic, complemented
11#10600196C211yesmaximal, characteristic, complemented
12C23×C4×D5C2212yes—
13C23×C4×D5C2211yescharacteristic
14C22×C42×C5C2211yescharacteristic
15C22×#685C2212yes—
16C22×#685C2211yescharacteristic
17C22×#685C2211yescharacteristic
18C22×#683C2212yes—
19C22×C4×Dic5C2211yescharacteristic
20C2×C5×#95C2211yescharacteristic
21C2×C5×#95C2211yescharacteristic
22C2×C5×#95C2212yes—
23C2×C5×#95C2212yes—
24C2×#3963C2212yes—
25C2×#3963C2212yes—
26C2×#3963C2211yescharacteristic
27C2×#3963C2211yescharacteristic
28C2×#3963C2212yes—
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2211yescharacteristic
34C2×#3963C2211yescharacteristic
35C4×#685C418yescomplemented
36#64956—24no—
37D5×#95—24no—
38#13966—51nomaximal, Sylow
39C24×D5C2311yescharacteristic
40C23×C4×C5C2312yes—
41C23×C4×C5C2311yescharacteristic
42C23×C4×C5C2311yescharacteristic
43C23×C4×C5C2312yes—
44C23×C4×C5C2311yescharacteristic
45C23×Dic5C2312yes—
46C23×Dic5C2311yescharacteristic
47C23×Dic5C2311yescharacteristic
48C23×Dic5C2312yes—
49C23×Dic5C2311yescharacteristic
50C22×C4×D5—24no—
51C22×C4×D5—24no—
52C22×C4×D5—24no—
53C22×C4×D5—24no—
54C22×C4×D5—24no—
55C22×C4×D5—42no—
56C22×C4×D5—42no—
57C22×C4×D5—42no—
58C22×C4×D5—42no—
59C22×C4×D5C2×C414yes—
60C22×C4×D5D414yes—
61C22×C4×D5Q814yes—
62C2×C42×C5—42no—
63C2×C42×C5C2×C414yescomplemented
64C2×#685—22no—
65C2×#685—22no—
66C2×#685—22no—
67C2×#685—22no—
68C2×#685—24no—
69C2×#685—24no—
70C2×#685—24no—
71C2×#685C2×C414yes—
72C2×#685C2×C414yes—
73C2×#683—24no—
74C2×#683—42no—
75C2×#683—42no—
76C2×C4×Dic5—42no—
77C2×C4×Dic5C2×C414yescomplemented
78C5×#95—22no—
79C5×#95—22no—
80C5×#95—24no—
81C5×#95—42no—
82#3963—22no—
83#3963—22no—
84#3963—24no—
85#3963—24no—
86#3963—24no—
87#3963—24no—
88#3963—42no—
89#3963—42no—
90#3963—42no—
91#3963C2×C414yescomplemented
92#3963C2×C414yescomplemented
93C2×#380—51no—
94C2×C4×C22⋊C4—51no—
95C22×#95—51no—
96#1442—52no—
97#1442—52no—
98#1434—51no—
99#1434—51no—
100#1434—51no—
101#1434D511yescharacteristic, complemented
102C22⋊#95—52no—
103C22⋊#95—52no—
104C24×C5C2411yescharacteristic
105C23×D5—22no—
106C23×D5—22no—
107C23×D5—22no—
108C23×D5—22no—
109C23×D5C22×C412yes—
110C23×D5C2×D412yes—
111C23×D5C2×Q812yes—
112C23×D5#4112yes—
113C23×D5#4112yes—
114C23×D5#4112yes—
115C23×D5#4112yes—
116C22×C4×C5—22no—
117C22×C4×C5—22no—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5—21no—
125C22×C4×C5—21no—
126C22×C4×C5—21no—
127C22×C4×C5—21no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—22no—
132C22×C4×C5—22no—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—22no—
138C22×C4×C5—22no—
139C22×C4×C5C22×C412yes—
140C22×C4×C5C22×C412yes—
141C22×C4×C5C22×C412yes—
142C22×C4×C5C2×D412yes—
143C22×C4×C5C2×Q812yes—
144C22×C4×C5#4114yes—
145C22×Dic5—22no—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—22no—
150C22×Dic5—22no—
151C22×Dic5—22no—
152C22×Dic5—22no—
153C22×Dic5—21no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5—22no—
164C22×Dic5—22no—
165C22×Dic5—22no—
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5C22×C412yes—
169C22×Dic5C22×C412yes—
170C22×Dic5C22×C412yes—
171C22×Dic5C2×D412yes—
172C22×Dic5C2×Q812yes—
173C22×Dic5#4114yes—
174C2×C4×D5—216no—
175C2×C4×D5—44no—
176C2×C4×D5—44no—
177C2×C4×D5—48no—
178C2×C4×D5—48no—
179C2×C4×D5—48no—
180C2×C4×D5—48no—
181C2×C4×D5—48no—
182C2×C4×D5—48no—
183C2×C4×D5—48no—
184C2×C4×D5—48no—
185C2×C4×D5C4⋊C4116yes—
186C42×C5—44no—
187#685—28no—
188#685—28no—
189#685—48no—
190#683—48no—
191C4×Dic5—44no—
192C24×C4—51no—
193C24×C4—52no—
194C22×C4⋊C4—52no—
195C22×C22⋊C4—51no—
196C22×C22⋊C4—51no—
197C22×C22⋊C4—52no—
198C22×C42—51no—
199C22×C42C2×D511yescharacteristic
200#380—104no—
201C4×C22⋊C4—58no—
202C2×#95—51no—
203C2×#95—51no—
204C2×#95—52no—
205C2×#95—52no—
206C2×#95—52no—
207C2×#95—51no—
208C2×#95—51no—
209C2×#95—52no—
210C2×#95—52no—
211C2×#95—52no—
212C2×#95—52no—
213C2×#95—104no—
214C2×#95C2×D512yes—
215C2×#95C2×D511yescharacteristic
216C2×#95C2×D511yescharacteristic
217C2×#95C2×D512yes—
218C23×C5C23×C411yescharacteristic
219C23×C5C22×D411yescharacteristic
220C23×C5C22×Q811yescharacteristic
221C23×C5C2×#4112yes—
222C23×C5C2×#4111yescharacteristic
223C23×C5C2×#4111yescharacteristic
224C23×C5C2×#4111yescharacteristic
225C23×C5C2×#4111yescharacteristic
226C23×C5C2×#4112yes—
227C23×C5H(5,𝔽2)12yes—
228C23×C5H(5,𝔽2)12yes—
229C22×D5—22no—
230C22×D5—22no—
231C22×D5—22no—
232C22×D5—22no—
233C22×D5—24no—
234C22×D5—24no—
235C22×D5—24no—
236C22×D5—24no—
237C22×D5—24no—
238C22×D5—24no—
239C22×D5—24no—
240C22×D5—24no—
241C22×D5—24no—
242C22×D5—24no—
243C22×D5—24no—
244C22×D5—24no—
245C22×D5C2×C4⋊C414yes—
246C22×D5#11714yes—
247C22×D5#11714yes—
248C22×D5#12414yes—
249C22×D5#12414yes—
250C22×D5#12514yes—
251C22×D5#12514yes—
252C2×C4×C5—22no—
253C2×C4×C5—22no—
254C2×C4×C5—24no—
255C2×C4×C5—24no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—24no—
263C2×C4×C5—24no—
264C2×C4×C5—24no—
265C2×C4×C5—42no—
266C2×C4×C5—42no—
267C2×C4×C5—42no—
268C2×C4×C5—42no—
269C2×C4×C5—42no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5—42no—
277C2×C4×C5—42no—
278C2×C4×C5—42no—
279C2×C4×C5—42no—
280C2×C4×C5C2×C4⋊C414yes—
281C2×Dic5—22no—
282C2×Dic5—22no—
283C2×Dic5—24no—
284C2×Dic5—24no—
285C2×Dic5—22no—
286C2×Dic5—22no—
287C2×Dic5—22no—
288C2×Dic5—22no—
289C2×Dic5—22no—
290C2×Dic5—22no—
291C2×Dic5—24no—
292C2×Dic5—24no—
293C2×Dic5—24no—
294C2×Dic5—42no—
295C2×Dic5—42no—
296C2×Dic5—42no—
297C2×Dic5—42no—
298C2×Dic5—42no—
299C2×Dic5—42no—
300C2×Dic5—42no—
301C2×Dic5—42no—
302C2×Dic5—42no—
303C2×Dic5—42no—
304C2×Dic5—42no—
305C2×Dic5—42no—
306C2×Dic5—42no—
307C2×Dic5—42no—
308C2×Dic5—42no—
309C2×Dic5C2×C4⋊C414yes—
310C4×D5—416no—
311C4×D5—432no—
312C25—51no—
313C23×C4—51no—
314C23×C4—54no—
315C23×C4—54no—
316C23×C4—54no—
317C23×C4—51no—
318C23×C4—51no—
319C23×C4—52no—
320C23×C4—52no—
321C23×C4—104no—
322C23×C4—104no—
323C23×C4—104no—
324C23×C4—104no—
325C23×C4—104no—
326C23×C4—202no—
327C23×C4—202no—
328C23×C4—202no—
329C23×C4—202no—
330C23×C4C22×D512yes—
331C23×C4C22×D511yescharacteristic
332C23×C4C22×D511yescharacteristic
333C23×C4C22×D512yes—
334C23×C4C22×D511yescharacteristic
335C2×C4⋊C4—104no—
336C2×C4⋊C4—202no—
337C2×C4⋊C4—202no—
338C2×C22⋊C4—54no—
339C2×C22⋊C4—54no—
340C2×C22⋊C4—102no—
341C2×C22⋊C4—102no—
342C2×C22⋊C4—102no—
343C2×C22⋊C4—102no—
344C2×C22⋊C4—104no—
345C2×C22⋊C4—104no—
346C2×C22⋊C4—104no—
347C2×C42—42no—
348C2×C42—54no—
349C2×C42—202no—
350C2×C42C4×D514yescomplemented
351#95—24no—
352#95—22no—
353#95—22no—
354#95—42no—
355#95—54no—
356#95—54no—
357#95—104no—
358#95—104no—
359#95—104no—
360#95—104no—
361#95—102no—
362#95—102no—
363#95—202no—
364#95—202no—
365#95—202no—
366C22×C5C22×C4⋊C411yescharacteristic
367C22×C5C2×#11711yescharacteristic
368C22×C5C2×#11711yescharacteristic
369C22×C5C4×#4111yescharacteristic
370C22×C5C4×#4111yescharacteristic
371C22×C5#51811yescharacteristic
372C22×C5#51811yescharacteristic
373C22×C5#52512yes—
374C22×C5C2×#12411yescharacteristic
375C22×C5C2×#12411yescharacteristic
376C22×C5C2×#12511yescharacteristic
377C22×C5C2×#12511yescharacteristic
378C22×C5#53312yes—
379C22×C5#53412yes—
380C22×C5#54312yes—
381C22×C5#55112yes—
382C22×C5#55112yes—
383C22×C5#55112yes—
384C22×C5#55112yes—
385C22×C5#55312yes—
386C22×C5#55312yes—
387C22×C5#55312yes—
388C22×C5#55312yes—
389C2×D5—28no—
390C2×D5—28no—
391C2×D5—28no—
392C2×D5—24no—
393C2×D5—28no—
394C2×D5—24no—
395C2×D5—24no—
396C2×D5—24no—
397C2×D5—24no—
398C2×D5—24no—
399C2×D5#38218yes—
400C4×C5—24no—
401C4×C5—44no—
402C4×C5—42no—
403C4×C5—42no—
404C4×C5—82no—
405Dic5—24no—
406Dic5—44no—
407Dic5—42no—
408Dic5—42no—
409Dic5—82no—
410C24—52no—
411C24—52no—
412C24—52no—
413C24—52no—
414C24—52no—
415C24—52no—
416C24—52no—
417C24—102no—
418C24—102no—
419C24—102no—
420C24—102no—
421C24C23×D511yescharacteristic
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—52no—
446C22×C4—52no—
447C22×C4—52no—
448C22×C4—516no—
449C22×C4—52no—
450C22×C4—52no—
451C22×C4—54no—
452C22×C4—1016no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4—102no—
459C22×C4—102no—
460C22×C4—101no—
461C22×C4—101no—
462C22×C4—101no—
463C22×C4—101no—
464C22×C4—102no—
465C22×C4—102no—
466C22×C4—102no—
467C22×C4—102no—
468C22×C4—102no—
469C22×C4—102no—
470C22×C4—102no—
471C22×C4—102no—
472C22×C4—102no—
473C22×C4—102no—
474C22×C4—102no—
475C22×C4—102no—
476C22×C4—204no—
477C22×C4—204no—
478C22×C4—208no—
479C22×C4—208no—
480C22×C4—208no—
481C22×C4—208no—
482C22×C4—208no—
483C22×C4—208no—
484C22×C4—208no—
485C22×C4—208no—
486C22×C4C2×C4×D512yes—
487C22×C4C2×C4×D512yes—
488C22×C4C2×C4×D512yes—
489C22×C4#70914yes—
490C22×C4D4×D512yes—
491C22×C4Q8×D512yes—
492C4⋊C4—208no—
493C22⋊C4—108no—
494C22⋊C4—108no—
495C22⋊C4—208no—
496C42—44no—
497C42—204no—
498C2×C5C2×#38211yescharacteristic
499C2×C5#229611yescharacteristic
500C2×C5#231511yescharacteristic
501C2×C5#235311yescharacteristic
502C2×C5#235311yescharacteristic
503C2×C5#235311yescharacteristic
504C2×C5#235311yescharacteristic
505C2×C5#256112yes—
506C2×C5#256912yes—
507C2×C5#256912yes—
508C2×C5#259112yes—
509D5—28no—
510C23—54no—
511C23—54no—
512C23—54no—
513C23—54no—
514C23—54no—
515C23—54no—
516C23—54no—
517C23—102no—
518C23—102no—
519C23—102no—
520C23—102no—
521C23—104no—
522C23—104no—
523C23—104no—
524C23—104no—
525C23—104no—
526C23—104no—
527C23—104no—
528C23—104no—
529C23—104no—
530C23—104no—
531C23—104no—
532C23—104no—
533C23C22×C4×D511yescharacteristic
534C23C2×#70912yes—
535C23C2×D4×D511yescharacteristic
536C23#414412yes—
537C23C2×Q8×D511yescharacteristic
538C23D5×#4111yescharacteristic
539C23D5×#4111yescharacteristic
540C23D5×#4111yescharacteristic
541C23D5×#4111yescharacteristic
542C23D5×#4112yes—
543C23#414912yes—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—54no—
573C2×C4—102no—
574C2×C4—102no—
575C2×C4—104no—
576C2×C4—104no—
577C2×C4—102no—
578C2×C4—102no—
579C2×C4—102no—
580C2×C4—102no—
581C2×C4—102no—
582C2×C4—102no—
583C2×C4—104no—
584C2×C4—104no—
585C2×C4—104no—
586C2×C4—2016no—
587C2×C4—2032no—
588C2×C4—202no—
589C2×C4—202no—
590C2×C4—202no—
591C2×C4—202no—
592C2×C4—202no—
593C2×C4—202no—
594C2×C4—202no—
595C2×C4—202no—
596C2×C4—202no—
597C2×C4—202no—
598C2×C4—202no—
599C2×C4—202no—
600C2×C4—202no—
601C2×C4—202no—
602C2×C4—202no—
603C2×C4D5×C4⋊C414yes—
604C5#1396611yescharacteristic, complemented, Sylow
605C22—58no—
606C22—108no—
607C22—108no—
608C22—104no—
609C22—104no—
610C22—104no—
611C22—104no—
612C22—104no—
613C22—108no—
614C22—108no—
615C22—104no—
616C22C4×#70911yescharacteristic
617C22#6582011yescharacteristic
618C22C2×D5×C4⋊C411yescharacteristic
619C22D5×#11711yescharacteristic
620C22D5×#11711yescharacteristic
621C22#6585511yescharacteristic
622C22#6585611yescharacteristic
623C22#6586112yes—
624C22#6586412yes—
625C22#6588112yes—
626C22#6589012yes—
627C22#6589212yes—
628C22#6591212yes—
629C22#6593912yes—
630C22#6596812yes—
631C22#6599312yes—
632C22#6599512yes—
633C22D5×#12411yescharacteristic
634C22D5×#12411yescharacteristic
635C22D5×#12511yescharacteristic
636C22D5×#12511yescharacteristic
637C22#6603912yes—
638C22#6604312yes—
639C4—24no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—82no—
644C4—104no—
645C4—204no—
646C4—202no—
647C4—202no—
648C4—402no—
649C2—108no—
650C2#1060514811yescharacteristic
651C2#1060530711yescharacteristic
652C2#1060548611yescharacteristic
653C2#1060573611yescharacteristic
654C2#1060585411yescharacteristic
655C2D5×#38211yescharacteristic
656C2#1060602811yescharacteristic
657C2#1060709212yes—
658C2#1060709312yes—
659C2#1060762112yes—
660C2#1060762512yes—
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC5×#1434C2
socle, cosocleC24×C5C24
prime radicalC24×D5C23
hypercenter#1434D5
hypocenterC5#13966
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. C5×#1434C2semidirect
5. #10602327C2semidirect
8. #10600207C2semidirect
9. #10600206C2semidirect
10. #10600196C2semidirect
11. #10600196C2semidirect
35. C4×#685C4semidirect
63. C2×C42×C5C2×C4semidirect
77. C2×C4×Dic5C2×C4semidirect
91. #3963C2×C4semidirect
92. #3963C2×C4semidirect
101. #1434D5semidirect
350. C2×C42C4×D5semidirect
604. C5#13966semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#6852
C2×#649562
C2×D5×#952
C5×#14342
#106023272
#106002182
#106002072
#106002062
#106001962
#139665