Finite Group Databasebeta

← #27456882

Subgroups — #27456882

Order 13568.

762 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279137C211yesmaximal, characteristic, complemented
2#20277751C211yesmaximal, characteristic
3#20276120C211yesmaximal, characteristic, complemented
4#20276088C211yesmaximal, characteristic, complemented
5#20275905C211yesmaximal, characteristic
6#20275905C211yesmaximal, characteristic
7#20275479C211yesmaximal, characteristic, complemented
8#20275443C211yesmaximal, characteristic, complemented
9#20275308C211yesmaximal, characteristic, complemented
10#20275301C211yesmaximal, characteristic, complemented
11#20274965C211yesmaximal, characteristic, complemented
12#20274916C211yesmaximal, characteristic, complemented
13#20274811C211yesmaximal, characteristic, complemented
14C4×#13133139C211yesmaximal, characteristic, complemented
15C53×#1923C211yesmaximal, characteristic, complemented
16C2×#13133210—21no—
17C2×#13133191C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28#16810563—21no—
29#16810548—21no—
30#16810525C2211yescharacteristic
31#16810523—21no—
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810512—21no—
36#16810508—21no—
37#16810499—21no—
38#16810493C2211yescharacteristic
39#16810483C2211yescharacteristic
40#16810483C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42C4×C4⋊Dic53C2211yescharacteristic
43#16810470—21no—
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×C4⋊C8×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C22×C4×D53—21no—
62#13133210—44no—
63#13133191—22no—
64#13133191—22no—
65C4×D212—22no—
66C4×D212—41no—
67#13133176—22no—
68#13133176—41no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91#13133151—22no—
92#13133149—22no—
93#13133149—22no—
94#13133149—41no—
95#13133139—22no—
96#13133139—22no—
97#13133139—22no—
98#13133139—41no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C2×C22⋊C4×C53D411yescharacteristic
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115C4⋊C8×C53—22no—
116C22⋊C8×C53—22no—
117C22⋊C8×C53—41no—
118C4×C8×C53—41no—
119#95×C53D411yescharacteristic
120#95×C53D411yescharacteristic
121C24×C53C2×D411yescharacteristic
122C23×D53—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×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2411yescharacteristic
135C2×C22⋊D53—24no—
136C2×C22⋊D53—81no—
137C2×C22⋊D53—81no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212#4112yes—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53—81no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53#4112yes—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C22⋊Dic53—42no—
173C22⋊Dic53—42no—
174C22⋊Dic53—42no—
175#11719921—24no—
176#11719921—42no—
177#11719921—42no—
178#11719921—81no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—42no—
192C4⋊Dic53—42no—
193C4⋊Dic53#4112yes—
194C4⋊Dic53#4112yes—
195#11719919—42no—
196#11719919—42no—
197#11719919—81no—
198C4×Dic53—42no—
199C4×Dic53—42no—
200C4×Dic53—81no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53C22×D411yescharacteristic
209C22×D53—22no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53#13512yes—
234C2×C4×C53AGL(1,ℤ/8)12yes—
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53H(5,𝔽2)11yescharacteristic
240C2×C4×C532−1+411yescharacteristic
241C22⋊D53—84no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—41no—
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—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263D212—22no—
264D212—22no—
265D212—41no—
266D212—41no—
267C4×D53—84no—
268C8×C53—22no—
269C8×C53—41no—
270C8×C53—41no—
271C8×C53—41no—
272#27039—531nomaximal, Sylow
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53#52511yescharacteristic
281C22×C53#54611yescharacteristic
282C22×C53#54711yescharacteristic
283C22×C53C2×#13511yescharacteristic
284C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C53#57711yescharacteristic
286C22×C53#57711yescharacteristic
287C2×D53—42no—
288C2×D53—42no—
289C2×D53—42no—
290C2×D53—81no—
291C2×D53—81no—
292C2×D53—81no—
293C2×D53—81no—
294C4×C53—22no—
295C4×C53—22no—
296C4×C53—41no—
297C4×C53—41no—
298C4×C53—41no—
299C4×C53#52911yescharacteristic
300C4×C53#53911yescharacteristic
301C4×C53#55311yescharacteristic
302C4×C53#55611yescharacteristic
303Dic53—42no—
304Dic53—42no—
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308Dic53—81no—
309Dic53—81no—
310#2496—531no—
311#2363—531no—
312#2071—531no—
313#2065—531no—
314#2034—531no—
315#2034—531no—
316#1948—531no—
317#1941—531no—
318#1923D5311yescharacteristic, complemented
319#1921—531no—
320#1920—531no—
321#1862—531no—
322#1854—531no—
323#1825—531no—
324C4×#102—531no—
325C2×C53—81no—
326C2×C53#246611yescharacteristic
327C2×C53#310611yescharacteristic
328C2×C53#312311yescharacteristic
329C2×C53#330411yescharacteristic
330C2×C53#330611yescharacteristic
331C2×C53#333711yescharacteristic
332C2×C53#335411yescharacteristic
333D53—82no—
334C2×#126—1061no—
335C2×#121—531no—
336C2×C4×D4—531no—
337C2×#117—531no—
338C2×#107—531no—
339C2×#106—531no—
340C2×#106—531no—
341C2×#106—531no—
342C2×C4⋊C8C2×D5311yescharacteristic
343C2×#102—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×#102—531no—
347C2×C22⋊C8C2×D5311yescharacteristic
348C2×C22⋊C8C2×D5311yescharacteristic
349C2×C4×C8C2×D5311yescharacteristic
350#400—1061no—
351#397—1061no—
352#390—531no—
353#390—1061no—
354#389—531no—
355#389—531no—
356#389—531no—
357#388—1061no—
358#387—1061no—
359#386—1061no—
360#384—531no—
361#380—531no—
362#380—531no—
363C4×C4⋊C4—531no—
364C4×C4⋊C4—531no—
365C4×C22⋊C4—1061no—
366C4×C22⋊C4C2×D5311yescharacteristic
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#336C2×D5311yescharacteristic
376#336C2×D5311yescharacteristic
377C53#2703911yescharacteristic, complemented, Sylow
378C22×D4—531no—
379C22×D4—1061no—
380C23×C4—1061no—
381C23×C4C22×D5311yescharacteristic
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384C22×C8C22×D5311yescharacteristic
385C22×C8C22×D5311yescharacteristic
386#126—2124no—
387#121—1062no—
388#121—1062no—
389C4×D4—1062no—
390C4×D4—2121no—
391#117—1062no—
392#117—2121no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4D21211yescharacteristic
413C2×C22⋊C4D21211yescharacteristic
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42—1061no—
417C2×C42C22×D5311yescharacteristic
418C2×C42C22×D5311yescharacteristic
419#107—1062no—
420#106—1062no—
421#106—1062no—
422#106—2121no—
423C4⋊C8—22no—
424#102—1062no—
425#102—1062no—
426#102—1062no—
427#102—2121no—
428C22⋊C8—22no—
429C22⋊C8—41no—
430C4×C8—41no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95D21211yescharacteristic
439#95D21211yescharacteristic
440C24—1061no—
441C24C2×D21211yescharacteristic
442C2×D4—532no—
443C2×D4—1064no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—1061no—
448C2×D4—1061no—
449C2×D4—4241no—
450C2×D4—4241no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—4241no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4C2×D21211yescharacteristic
468C22×C4C2×D21211yescharacteristic
469C22×C4C2×D21211yescharacteristic
470C22×C4C2×D21211yescharacteristic
471C22×C4C2×D21211yescharacteristic
472C22×C4C23×D5311yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8#1171994712yes—
485C4⋊C4—532no—
486C4⋊C4—532no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—4241no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C4⋊C4—2122no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—1064no—
507C22⋊C4—4241no—
508C22⋊C4—2122no—
509C22⋊C4—2122no—
510C22⋊C4—2122no—
511C22⋊C4—2122no—
512C22⋊C4—2122no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—4241no—
518C42—2122no—
519C42—2122no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—1062no—
528C23—2121no—
529C23—2121no—
530C23—2121no—
531C23—2121no—
532C23—2121no—
533C23—2121no—
534C23C22×D21211yescharacteristic
535D4—1062no—
536D4—1062no—
537D4—4244no—
538D4—2121no—
539D4—2121no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—4244no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4#1313323912yes—
580C2×C4#1313324512yes—
581C2×C4#41×D5311yescharacteristic
582C2×C4#41×D5311yescharacteristic
583C2×C4#41×D5311yescharacteristic
584C2×C4C2×#1171994711yescharacteristic
585C2×C4#1313326211yescharacteristic
586C2×C4#1313326411yescharacteristic
587C8—22no—
588C8—41no—
589C8—41no—
590C8—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—4241no—
599C22—4241no—
600C22—4241no—
601C22—4241no—
602C22—2122no—
603C22—2122no—
604C22—2122no—
605C22#1681113511yescharacteristic
606C22#1681126811yescharacteristic
607C22#1681127711yescharacteristic
608C22C2×#1313323911yescharacteristic
609C22C2×#1313324511yescharacteristic
610C22#1681143511yescharacteristic
611C22#1681143511yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—4241no—
618C4—4241no—
619C4—4241no—
620C4—4241no—
621C4—4241no—
622C4—2122no—
623C4—2122no—
624C4#1681115211yescharacteristic
625C4#1681121611yescharacteristic
626C4#1681131611yescharacteristic
627C4#1681134011yescharacteristic
628C2—81no—
629C2—4242no—
630C2#2027874911yescharacteristic
631C2#2028477611yescharacteristic
632C2#2028496811yescharacteristic
633C2#2028665511yescharacteristic
634C2#2028667811yescharacteristic
635C2#2028709611yescharacteristic
636C2#2028731111yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1923C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1923D53
hypocenterC53#27039
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
1. #20279137C2semidirect
3. #20276120C2semidirect
4. #20276088C2semidirect
7. #20275479C2semidirect
8. #20275443C2semidirect
9. #20275308C2semidirect
10. #20275301C2semidirect
11. #20274965C2semidirect
12. #20274916C2semidirect
13. #20274811C2semidirect
14. C4×#13133139C2semidirect
15. C53×#1923C2semidirect
318. #1923D53semidirect
377. C53#27039semidirect

Maximal subgroups

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

subgroupindex
#202791372
#202777512
#202761202
#202760882
#202759052
#202754792
#202754432
#202753082
#202753012
#202749652
#202749162
#202748112
C4×#131331392
C53×#19232
#2703953