Finite Group Databasebeta

← #27465496

Subgroups — #27465496

Order 13568.

770 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277243C211yesmaximal, characteristic
2#20277117C211yesmaximal, characteristic
3#20276152C211yesmaximal, characteristic
4#20275912C211yesmaximal, characteristic
5#20275911C211yesmaximal, characteristic
6#20275477C211yesmaximal, characteristic, complemented
7#20275475C211yesmaximal, characteristic, complemented
8#20275348C211yesmaximal, characteristic, complemented
9#20275347C211yesmaximal, characteristic, complemented
10#20275299C211yesmaximal, characteristic, complemented
11#20275298C211yesmaximal, characteristic, complemented
12#20274719C211yesmaximal, characteristic, complemented
13#20274719C211yesmaximal, characteristic, complemented
14#20274693C211yesmaximal, characteristic
15C53×#2077C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×#13133182C2211yescharacteristic
18C2×#13133173—21no—
19C22×C4×Dic53C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30#16810528C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810505—21no—
33#16810495—21no—
34#16810492C2211yescharacteristic
35#16810492C2211yescharacteristic
36#16810487—21no—
37#16810483C2211yescharacteristic
38#16810481—21no—
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42#16810474—21no—
43#16810474—21no—
44C22⋊C4×Dic53—21no—
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#106×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C53×#380C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C23×Dic53D411yescharacteristic
61C23×Dic53D411yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133196—22no—
65#13133196—22no—
66#13133182—22no—
67#13133182—22no—
68#13133173—44no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—22no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133149—22no—
92#13133148—22no—
93#13133148—41no—
94#13133146—22no—
95#13133146—22no—
96#13133142—22no—
97#13133142—41no—
98#13133140—22no—
99#13133140—41no—
100C22⋊#88322—41no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C23×C4×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117#106×C53—41no—
118C22⋊C8×C53—41no—
119#95×C53—21no—
120#95×C53—21no—
121C24×C53C2×D411yescharacteristic
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2411yescharacteristic
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—41no—
142C22×Dic53—41no—
143C22×Dic53SD1614yes—
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106#4112yes—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—42no—
170C22⋊Dic53—24no—
171C22⋊Dic53—42no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53#4112yes—
178#11719919—24no—
179#11719919—42no—
180#11719919—42no—
181#11719919—42no—
182#11719919—42no—
183#11719919—42no—
184#11719919—42no—
185C4×Dic53—21no—
186C4×Dic53—21no—
187C4×Dic53—22no—
188C4×Dic53—22no—
189C4×Dic53—42no—
190C4×Dic53—42no—
191C4×Dic53—42no—
192C4×Dic53—42no—
193C4×Dic53#4112yes—
194C4×Dic53#4112yes—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53C2×SD1612yes—
208C23×C53C22×D411yescharacteristic
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53H(5,𝔽2)11yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—28no—
239C2×Dic53—21no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—44no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53#12312yes—
256C2×Dic53#12312yes—
257C2×Dic53C2×SD1612yes—
258C2×Dic53C2×SD1612yes—
259C2×Dic53#13712yes—
260Dic106—22no—
261Dic106—22no—
262Dic106—41no—
263Dic106—41no—
264C8×C53—41no—
265C8×C53—41no—
266#88322—41no—
267#88322—41no—
268#27628—531nomaximal, Sylow
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53C2×#12311yescharacteristic
277C22×C53#52511yescharacteristic
278C22×C53#53711yescharacteristic
279C22×C53C22×SD1611yescharacteristic
280C22×C53C2×#13711yescharacteristic
281C22×C53#57511yescharacteristic
282C22×C53#57511yescharacteristic
283C4×C53—41no—
284C4×C53—81no—
285C4×C53—81no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53#55211yescharacteristic
289C4×C53#55311yescharacteristic
290C4×C53#55511yescharacteristic
291C4×C53#55611yescharacteristic
292Dic53—22no—
293Dic53—22no—
294Dic53—41no—
295Dic53—42no—
296Dic53—42no—
297Dic53—81no—
298Dic53—81no—
299Dic53#48314yes—
300C4×C22⋊Q8—531no—
301#2285—531no—
302#2077—531no—
303#2077D5311yescharacteristic
304#2035—531no—
305#2035—531no—
306#1948—531no—
307#1948—531no—
308#1927—531no—
309#1927—531no—
310#1920—531no—
311#1920—531no—
312#1803—531no—
313#1803—531no—
314#1797—531no—
315C2×C53—42no—
316C2×C53#248511yescharacteristic
317C2×C53C2×#48311yescharacteristic
318C2×C53#309311yescharacteristic
319C2×C53#310311yescharacteristic
320C2×C53#311611yescharacteristic
321C2×C53#324611yescharacteristic
322C2×C53#325211yescharacteristic
323C2×C22⋊Q8—531no—
324C2×C4×Q8—531no—
325C2×#117—1061no—
326C22×C42—531no—
327C2×#106—531no—
328C2×#106—531no—
329C2×#106—531no—
330C2×#106C2×D5311yescharacteristic
331C2×C4⋊C8—531no—
332C2×C4⋊C8—531no—
333C2×#103—531no—
334C2×#103—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×C22⋊C8—531no—
338C2×C22⋊C8C2×D5311yescharacteristic
339#391—531no—
340#389—531no—
341#387—1061no—
342#385—1061no—
343#384—531no—
344#384—531no—
345#382—1061no—
346#380—531no—
347#380—1061no—
348#380C2×D5311yescharacteristic
349#380C2×D5311yescharacteristic
350C4×C4⋊C4—531no—
351C4×C4⋊C4—531no—
352C4×C4⋊C4—531no—
353C4×C4⋊C4—1061no—
354C4×C4⋊C4—1061no—
355C4×C22⋊C4—1061no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340C2×D5311yescharacteristic
362#340C2×D5311yescharacteristic
363#340C2×D5311yescharacteristic
364#336—531no—
365#336—531no—
366C53#2762811yescharacteristic, complemented, Sylow
367C22×Q8—531no—
368C23×C4—531no—
369C23×C4—531no—
370C23×C4C22×D5311yescharacteristic
371C22×C8—531no—
372C22×C8—531no—
373C22×C8C22×D5311yescharacteristic
374C22×C8C22×D5311yescharacteristic
375C22⋊Q8—1062no—
376C22⋊Q8—1062no—
377C4×Q8—1062no—
378C4×Q8—1062no—
379#117—2124no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—1062no—
403C2×C42—1062no—
404C2×C42—1061no—
405C2×C42—1061no—
406C2×C42—1061no—
407C2×C42—1061no—
408#106—41no—
409#106—1062no—
410#106—1062no—
411#106—2121no—
412C4⋊C8—1062no—
413C4⋊C8—1062no—
414#103—1062no—
415#103—1062no—
416#103—2121no—
417#103—2121no—
418C22⋊C8—41no—
419C22⋊C8—2121no—
420#95—21no—
421#95—21no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429C24D4×D5311yescharacteristic
430C2×Q8—532no—
431C2×Q8—1061no—
432C2×Q8—1061no—
433C2×Q8—1061no—
434C2×Q8—1061no—
435C2×Q8—1061no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—531no—
447C22×C4—534no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—2121no—
459C22×C4—2121no—
460C22×C4D4×D5311yescharacteristic
461C22×C4C23×D5311yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—532no—
483C4⋊C4—1064no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—2122no—
490C4⋊C4—2122no—
491C4⋊C4—2122no—
492C4⋊C4—2122no—
493C4⋊C4—2122no—
494C4⋊C4—2122no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—1064no—
498C22⋊C4—2122no—
499C42—532no—
500C42—532no—
501C42—1062no—
502C42—1062no—
503C42—1061no—
504C42—1061no—
505C42—2122no—
506C42—2122no—
507C42—2122no—
508C42—2122no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23SD16×D5312yes—
516C23C2×D4×D5311yescharacteristic
517Q8—1062no—
518Q8—1062no—
519Q8—2121no—
520Q8—2121no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—532no—
541C2×C4—532no—
542C2×C4—532no—
543C2×C4—532no—
544C2×C4—532no—
545C2×C4—1068no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—2124no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4#41×D5311yescharacteristic
567C2×C4#41×D5311yescharacteristic
568C2×C4#41×D5311yescharacteristic
569C2×C4#41×D5311yescharacteristic
570C2×C4#1313326211yescharacteristic
571C2×C4#1313326411yescharacteristic
572C8—41no—
573C8—41no—
574C8—2121no—
575C8—2121no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22#123×D5311yescharacteristic
584C22#1681113111yescharacteristic
585C22#1681120011yescharacteristic
586C22C2×SD16×D5311yescharacteristic
587C22#137×D5311yescharacteristic
588C22#1681141911yescharacteristic
589C22#1681142011yescharacteristic
590C4—41no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4—534no—
596C4—1062no—
597C4—1062no—
598C4—4241no—
599C4—4241no—
600C4—2122no—
601C4—2122no—
602C4—2121no—
603C4#1681130911yescharacteristic
604C4#1681131811yescharacteristic
605C4#1681133511yescharacteristic
606C4#1681134011yescharacteristic
607C2—42no—
608C2#2027900711yescharacteristic
609C2#483×D5311yescharacteristic
610C2#2028464311yescharacteristic
611C2#2028474311yescharacteristic
612C2#2028487211yescharacteristic
613C2#2028602611yescharacteristic
614C2#2028608111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2077C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2077D53
hypocenterC53#27628
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2077

Complements

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

NH ≅ G/Ntype
6. #20275477C2semidirect
7. #20275475C2semidirect
8. #20275348C2semidirect
9. #20275347C2semidirect
10. #20275299C2semidirect
11. #20275298C2semidirect
12. #20274719C2semidirect
13. #20274719C2semidirect
366. C53#27628semidirect

Maximal subgroups

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

subgroupindex
#202772432
#202771172
#202761522
#202759122
#202759112
#202754772
#202754752
#202753482
#202753472
#202752992
#202752982
#202747192
#202746932
C53×#20772
#2762853