Finite Group Databasebeta

← #27315225

Subgroups — #27315225

Order 13568.

1486 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810485C211yesmaximal, characteristic
2C2×#16810482C211yesmaximal, characteristic
3C2×#16810480C211yesmaximal, characteristic
4#20272980C211yesmaximal, characteristic, complemented
5#20272979C212yesmaximal, complemented
6#20272979C211yesmaximal, characteristic, complemented
7#20272974C211yesmaximal, characteristic, complemented
8#20272973C212yesmaximal, complemented
9#20272961C212yesmaximal
10#20272959C212yesmaximal
11C53×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×C4⋊Dic53C2212yes—
15C22×#11719919C2211yescharacteristic
16C22×#11719919C2211yescharacteristic
17C22×#11719919C2212yes—
18C22×#11719921C2211yescharacteristic
19C22×#11719921C2212yes—
20C22×#11719921C2211yescharacteristic
21C22×#11719921C2212yes—
22#16810485C418yescomplemented
23#16810482—24no—
24#16810480—24no—
25C2×#13133124C2212yes—
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C2×#95×C53C2211yescharacteristic
35C2×#95×C53C2212yes—
36C2×#95×C53C2212yes—
37C2×#95×C53C2211yescharacteristic
38C24×D53C2311yescharacteristic
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2312yes—
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C22×C4×D53—42no—
45C22×C4×D53—42no—
46C22×C4×D53C2×C414yes—
47C22×C4×D53D414yes—
48C22×C4×D53Q814yes—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—42no—
51C2×C4⋊Dic53—42no—
52C2×C4⋊Dic53—42no—
53C2×C4⋊Dic53—42no—
54C2×#11719919—22no—
55C2×#11719919—22no—
56C2×#11719919—42no—
57C2×#11719919—42no—
58C2×#11719919—42no—
59C2×#11719919—42no—
60C2×#11719919—42no—
61C2×#11719921—24no—
62C2×#11719921—22no—
63C2×#11719921—22no—
64C2×#11719921—42no—
65C2×#11719921—42no—
66C2×#11719921—42no—
67C2×#11719921—42no—
68C2×#11719921—42no—
69C2×#11719921—42no—
70C2×#11719921C2×C418yes—
71#13133124—22no—
72#13133124—24no—
73#13133124—42no—
74#13133124—42no—
75#13133124—42no—
76#13133124—42no—
77#13133124C2×C414yes—
78#13133124C2×C418yes—
79C23×C4×C53C2312yes—
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2312yes—
83C23×C4×C53C2311yescharacteristic
84C2×C4⋊C4×C53—22no—
85C2×C4⋊C4×C53—42no—
86#95×C53—22no—
87#95×C53—42no—
88#95×C53—42no—
89#95×C53C2×C414yes—
90C24×C53C2411yescharacteristic
91C23×D53—21no—
92C23×D53—22no—
93C23×D53—21no—
94C23×D53—22no—
95C23×D53—22no—
96C23×D53—21no—
97C23×D53—22no—
98C23×D53—21no—
99C23×D53C22×C412yes—
100C23×D53C2×D412yes—
101C23×D53C2×Q812yes—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53C22×C414yes—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×Q812yes—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—21no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53C22×C414yes—
157C22×Dic53C22×C412yes—
158C22×Dic53C2×D412yes—
159C22×Dic53C2×Q812yes—
160C2×C4×D53—48no—
161C2×C4×D53—44no—
162C2×C4×D53—44no—
163C2×C4×D53—48no—
164C2×C4×D53C4⋊C4116yescomplemented
165C4⋊C4×C53—44no—
166#11719921—48no—
167#11719921—48no—
168#11719921—44no—
169#11719921—44no—
170C4⋊Dic53—44no—
171C4⋊Dic53—48no—
172#11719919—44no—
173#11719919—44no—
174#11719919—48no—
175C23×C53C23×C411yescharacteristic
176C23×C53C22×D411yescharacteristic
177C23×C53C22×Q811yescharacteristic
178C23×C53H(5,𝔽2)12yes—
179C23×C53H(5,𝔽2)11yescharacteristic
180C23×C53H(5,𝔽2)12yes—
181C23×C53H(5,𝔽2)12yes—
182C23×C53H(5,𝔽2)11yescharacteristic
183C23×C532−1+411yescharacteristic
184C23×C532−1+412yes—
185C23×C532−1+411yescharacteristic
186C22×D53—24no—
187C22×D53—22no—
188C22×D53—24no—
189C22×D53—22no—
190C22×D53—22no—
191C22×D53—22no—
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—22no—
195C22×D53—22no—
196C22×D53—24no—
197C22×D53—22no—
198C22×D53—22no—
199C22×D53—24no—
200C22×D53—42no—
201C22×D53—42no—
202C22×D53—42no—
203C22×D53—42no—
204C22×D53—42no—
205C22×D53—42no—
206C22×D53—42no—
207C22×D53—42no—
208C22×D53C2×C4⋊C414yes—
209C2×C4×C53—24no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—24no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—24no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—42no—
223C2×C4×C53—41no—
224C2×C4×C53—42no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—42no—
228C2×C4×C53—41no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53C2×C4⋊C414yes—
238C2×Dic53—24no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—42no—
252C2×Dic53—41no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53C2×C4⋊C414yes—
267C4×D53—416no—
268#14674—531nomaximal, Sylow
269C22×C53C22×C4⋊C411yescharacteristic
270C22×C53#51812yes—
271C22×C53#52011yescharacteristic
272C22×C53#52012yes—
273C22×C53#52011yescharacteristic
274C22×C53#53411yescharacteristic
275C22×C53#53411yescharacteristic
276C22×C53#53511yescharacteristic
277C22×C53#53511yescharacteristic
278C22×C53#53611yescharacteristic
279C22×C53#53611yescharacteristic
280C22×C53#54312yes—
281C22×C53#54411yescharacteristic
282C22×C53#54411yescharacteristic
283C22×C53#54412yes—
284C22×C53#56012yes—
285C22×C53#56012yes—
286C22×C53#56212yes—
287C22×C53#56212yes—
288C22×C53#56212yes—
289C22×C53#56212yes—
290C22×C53#56312yes—
291C22×C53#56312yes—
292C2×D53—24no—
293C2×D53—24no—
294C2×D53—24no—
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—42no—
298C2×D53—44no—
299C2×D53—44no—
300C2×D53—44no—
301C2×D53—44no—
302C2×D53—42no—
303C4×C53—24no—
304C4×C53—44no—
305C4×C53—81no—
306C4×C53—82no—
307C4×C53—81no—
308Dic53—24no—
309Dic53—44no—
310Dic53—81no—
311Dic53—81no—
312Dic53—82no—
313C2×#381—531no—
314C2×#380—531no—
315C2×#380—531no—
316#1444—532no—
317#1444—531no—
318#1444—531no—
319#1442—532no—
320#1442—531no—
321#1442D5311yescharacteristic, complemented
322#1439—532no—
323#1439—532no—
324C2×C53#231511yescharacteristic
325C2×C53#231711yescharacteristic
326C2×C53#231711yescharacteristic
327C2×C53#237612yes—
328C2×C53#237612yes—
329C2×C53#281511yescharacteristic
330C2×C53#281611yescharacteristic
331C2×C53#281611yescharacteristic
332C2×C53#281711yescharacteristic
333C2×C53#284112yes—
334C2×C53#284112yes—
335D53—44no—
336C24×C4—531no—
337C22×C4⋊C4—531no—
338C22×C4⋊C4—531no—
339C22×C4⋊C4—531no—
340C22×C4⋊C4—532no—
341C22×C4⋊C4—532no—
342C22×C4⋊C4C2×D5311yescharacteristic
343C22×C22⋊C4—532no—
344C22×C22⋊C4—531no—
345C22×C22⋊C4—532no—
346C22×C22⋊C4—531no—
347#381—538no—
348#380—1064no—
349#380—1064no—
350C2×#95—532no—
351C2×#95—531no—
352C2×#95—532no—
353C2×#95—531no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95C2×D5311yescharacteristic
359C2×#95C2×D5312yes—
360C2×#95C2×D5312yes—
361C2×#95C2×D5311yescharacteristic
362C53#1467411yescharacteristic, complemented, Sylow
363C25—531no—
364C23×C4—531no—
365C23×C4—534no—
366C23×C4—534no—
367C23×C4—534no—
368C23×C4—532no—
369C23×C4—532no—
370C23×C4—531no—
371C23×C4—531no—
372C23×C4—2122no—
373C23×C4—2122no—
374C23×C4C22×D5311yescharacteristic
375C23×C4C22×D5311yescharacteristic
376C23×C4C22×D5312yes—
377C23×C4C22×D5312yes—
378C23×C4C22×D5311yescharacteristic
379C2×C4⋊C4—22no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4—1062no—
384C2×C4⋊C4—2122no—
385C2×C4⋊C4—2122no—
386C2×C4⋊C4—2122no—
387C2×C4⋊C4—2122no—
388C2×C4⋊C4—2122no—
389C2×C4⋊C4—2122no—
390C2×C4⋊C4—2122no—
391C2×C4⋊C4—2122no—
392C2×C4⋊C4—2122no—
393C2×C22⋊C4—538no—
394C2×C22⋊C4—1062no—
395C2×C22⋊C4—1062no—
396C2×C22⋊C4—1064no—
397C2×C22⋊C4—2122no—
398C2×C22⋊C4—2122no—
399C2×C22⋊C4—2122no—
400C2×C22⋊C4—2122no—
401C2×C22⋊C4—2122no—
402C2×C22⋊C4—2122no—
403#95—22no—
404#95—42no—
405#95—42no—
406#95—538no—
407#95—534no—
408#95—1064no—
409#95—1062no—
410#95—2122no—
411#95—2122no—
412#95—2122no—
413#95—2122no—
414#95C4×D5314yes—
415C24—532no—
416C24—532no—
417C24—532no—
418C24—1061no—
419C24—1062no—
420C24—1062no—
421C24—1061no—
422C24—1062no—
423C24—1061no—
424C24—1062no—
425C24—1061no—
426C24C23×D5311yescharacteristic
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—532no—
453C22×C4—532no—
454C22×C4—532no—
455C22×C4—5316no—
456C22×C4—534no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1061no—
466C22×C4—1062no—
467C22×C4—1061no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1061no—
471C22×C4—1062no—
472C22×C4—1061no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—2128no—
483C22×C4—2124no—
484C22×C4—2124no—
485C22×C4—2128no—
486C22×C4C2×C4×D5314yes—
487C22×C4C2×C4×D5312yes—
488C22×C4D4×D5312yes—
489C22×C4Q8×D5312yes—
490C4⋊C4—44no—
491C4⋊C4—2124no—
492C4⋊C4—2124no—
493C4⋊C4—2124no—
494C4⋊C4—2128no—
495C4⋊C4—2128no—
496C22⋊C4—2128no—
497C22⋊C4—2128no—
498C22⋊C4—2124no—
499C22⋊C4—2124no—
500C23—534no—
501C23—1064no—
502C23—1062no—
503C23—1064no—
504C23—1062no—
505C23—1062no—
506C23—1062no—
507C23—1062no—
508C23—1062no—
509C23—1062no—
510C23—1062no—
511C23—1064no—
512C23—1062no—
513C23—1062no—
514C23—1064no—
515C23—2122no—
516C23—2122no—
517C23—2122no—
518C23—2122no—
519C23—2122no—
520C23—2122no—
521C23—2122no—
522C23—2122no—
523C23C22×C4×D5311yescharacteristic
524C23C2×D4×D5311yescharacteristic
525C23C2×Q8×D5311yescharacteristic
526C23#1313326212yes—
527C23#1313326211yescharacteristic
528C23#1313326312yes—
529C23#1313326312yes—
530C23#1313326311yescharacteristic
531C23#1313326411yescharacteristic
532C23#1313326412yes—
533C23#1313326511yescharacteristic
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—534no—
563C2×C4—1064no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1064no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—21216no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2122no—
587C2×C4—2121no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4C4⋊C4×D5314yes—
593C22—1064no—
594C22—1064no—
595C22—1064no—
596C22—2124no—
597C22—2124no—
598C22—2124no—
599C22—2124no—
600C22—2122no—
601C22—2122no—
602C22—2122no—
603C22—2122no—
604C22C2×C4⋊C4×D5311yescharacteristic
605C22#1681109512yes—
606C22#1681110211yescharacteristic
607C22#1681110312yes—
608C22#1681110511yescharacteristic
609C22#1681118311yescharacteristic
610C22#1681118411yescharacteristic
611C22#1681118911yescharacteristic
612C22#1681119111yescharacteristic
613C22#1681119311yescharacteristic
614C22#1681119611yescharacteristic
615C22#1681124812yes—
616C22#1681125011yescharacteristic
617C22#1681125311yescharacteristic
618C22#1681125312yes—
619C22#1681136312yes—
620C22#1681136412yes—
621C22#1681136712yes—
622C22#1681136812yes—
623C22#1681136912yes—
624C22#1681137012yes—
625C22#1681137512yes—
626C22#1681137712yes—
627C4—24no—
628C4—44no—
629C4—81no—
630C4—81no—
631C4—82no—
632C4—1064no—
633C4—4241no—
634C4—4242no—
635C4—4241no—
636C4—2124no—
637C2—2124no—
638C2#2027732611yescharacteristic
639C2#2027733711yescharacteristic
640C2#2027734011yescharacteristic
641C2#2027785712yes—
642C2#2027786212yes—
643C2#2028273811yescharacteristic
644C2#2028274111yescharacteristic
645C2#2028274611yescharacteristic
646C2#2028275111yescharacteristic
647C2#2028297312yes—
648C2#2028297412yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1442C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1442D53
hypocenterC53#14674
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
4. #20272980C2semidirect
5. #20272979C2semidirect
6. #20272979C2semidirect
7. #20272974C2semidirect
8. #20272973C2semidirect
11. C53×#1442C2semidirect
22. #16810485C4semidirect
164. C2×C4×D53C4⋊C4semidirect
321. #1442D53semidirect
362. C53#14674semidirect

Maximal subgroups

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

subgroupindex
C2×#168104852
C2×#168104822
C2×#168104802
#202729802
#202729792
#202729742
#202729732
#202729612
#202729592
C53×#14422
#1467453