Finite Group Databasebeta

← #27456790

Subgroups — #27456790

Order 13568.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277729C211yesmaximal, characteristic
2#20276129C211yesmaximal, characteristic, complemented
3#20276100C211yesmaximal, characteristic, complemented
4#20275904C211yesmaximal, characteristic
5#20275897C211yesmaximal, characteristic
6#20275481C211yesmaximal, characteristic, complemented
7#20275445C211yesmaximal, characteristic, complemented
8#20275323C211yesmaximal, characteristic, complemented
9#20275304C211yesmaximal, characteristic, complemented
10#20275293C211yesmaximal, characteristic, complemented
11#20274961C211yesmaximal, characteristic, complemented
12#20274902C211yesmaximal, characteristic, complemented
13#20274803C211yesmaximal, characteristic, complemented
14C4×#13133137C211yesmaximal, characteristic, complemented
15C53×#2494C211yesmaximal, characteristic, complemented
16C2×#13133191C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×#13133177—21no—
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31C2×C4×#88322C2211yescharacteristic
32#16810525C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810515—21no—
35#16810500—21no—
36#16810497—21no—
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C53×#401—21no—
49C53×#397—21no—
50C53×#389C2211yescharacteristic
51C53×#389C2211yescharacteristic
52C53×#388—21no—
53C53×#387—21no—
54C53×#382C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C23×Dic53—21no—
62C22×C4×D53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133191—22no—
68#13133191—22no—
69C4×D212—22no—
70C4×D212—41no—
71#13133177—44no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53—41no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×C4×Dic53—21no—
84#13133150—22no—
85#13133150—22no—
86#13133150—41no—
87#13133148—22no—
88C4⋊#88322—22no—
89#13133137—22no—
90#13133137—22no—
91#13133137—22no—
92#13133137—41no—
93C22⋊#88322—22no—
94C22⋊#88322—41no—
95C4×#88322—41no—
96#13133124—21no—
97#13133124—21no—
98C23×C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C22⋊C4×C53D411yescharacteristic
109C2×C42×C53—21no—
110C2×C42×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112#95×C53—21no—
113#95×C53—21no—
114#95×C53—21no—
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
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×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C2×C22⋊D53—24no—
140C2×C22⋊D53—42no—
141C2×C22⋊D53—42no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—42no—
146C22×Dic53—42no—
147C22×Dic53—81no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212#4112yes—
154C2×C4×D53—42no—
155C2×C4×D53—42no—
156C2×C4×D53—81no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×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—42no—
166C4⋊C4×C53—81no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—42no—
170C22⋊C4×C53—81no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—81no—
176#11719921—24no—
177#11719921—42no—
178#11719921—42no—
179#11719921—81no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—42no—
185C4⋊Dic53#4112yes—
186C4⋊Dic53#4112yes—
187#11719919—42no—
188C4×Dic53—42no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322#4112yes—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53#13512yes—
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—21no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53AGL(1,ℤ/8)12yes—
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×C4×C532−1+411yescharacteristic
253C22⋊D53—48no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—84no—
264D212—22no—
265D212—22no—
266D212—41no—
267D212—41no—
268C4×D53—84no—
269#88322—22no—
270#88322—41no—
271#88322—41no—
272#88322—41no—
273#27033—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53#52511yescharacteristic
282C22×C53#54611yescharacteristic
283C22×C53#54711yescharacteristic
284C22×C53C2×#13511yescharacteristic
285C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C53#57611yescharacteristic
287C22×C53#57811yescharacteristic
288C2×D53—42no—
289C2×D53—42no—
290C2×D53—42no—
291C2×D53—81no—
292C2×D53—81no—
293C2×D53—81no—
294C2×D53—81no—
295C4×C53—22no—
296C4×C53—41no—
297C4×C53—41no—
298C4×C53—41no—
299C4×C53—41no—
300C4×C53—81no—
301C4×C53—81no—
302C4×C53—81no—
303C4×C53—81no—
304C4×C53#52911yescharacteristic
305C4×C53#53911yescharacteristic
306C4×C53#55311yescharacteristic
307C4×C53#55611yescharacteristic
308Dic53—42no—
309Dic53—42no—
310Dic53—81no—
311#2494D5311yescharacteristic, complemented
312#2361—531no—
313#2073—531no—
314#2067—531no—
315#2034—531no—
316#2033—531no—
317#1949—531no—
318#1942—531no—
319#1923—531no—
320#1921—531no—
321#1919—531no—
322#1862—531no—
323#1852—531no—
324#1824—531no—
325C4×#102—531no—
326C2×C53—42no—
327C2×C53#246611yescharacteristic
328C2×C53#310511yescharacteristic
329C2×C53#312411yescharacteristic
330C2×C53#330011yescharacteristic
331C2×C53#331811yescharacteristic
332C2×C53#333911yescharacteristic
333C2×C53#335311yescharacteristic
334D53—82no—
335C2×#121—531no—
336C2×C4×D4—531no—
337C2×C4×D4—1061no—
338C22×C4⋊C4—531no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#107—531no—
342C2×#106—531no—
343C2×C4⋊C8—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×#102—531no—
347C2×#102—531no—
348C2×C22⋊C8—531no—
349C2×C22⋊C8—531no—
350C2×C4×C8—531no—
351#401—21no—
352#397—21no—
353#390—531no—
354#389—531no—
355#389C2×D5311yescharacteristic
356#389C2×D5311yescharacteristic
357#388—21no—
358#388—1061no—
359#387—21no—
360#386—1061no—
361#385—1061no—
362#382C2×D5311yescharacteristic
363#380C2×D5311yescharacteristic
364#380C2×D5311yescharacteristic
365C4×C4⋊C4—531no—
366C4×C4⋊C4C2×D5311yescharacteristic
367C4×C22⋊C4C2×D5311yescharacteristic
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#336—531no—
377#336—531no—
378C53#2703311yescharacteristic, complemented, Sylow
379C22×D4—531no—
380C22×D4—1061no—
381C23×C4—1061no—
382C23×C4—1061no—
383C23×C4C22×D5311yescharacteristic
384C22×C8—531no—
385C22×C8—531no—
386C22×C8—531no—
387C22×C8—531no—
388#121—1062no—
389#121—1062no—
390C4×D4—1062no—
391C4×D4—2121no—
392C4×D4—2124no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—2121no—
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4C22⋊D5311yescharacteristic
413C2×C22⋊C4C22⋊D5311yescharacteristic
414C2×C42—21no—
415C2×C42—1061no—
416C2×C42C22×D5311yescharacteristic
417C2×C42C22×D5311yescharacteristic
418#107—1062no—
419#107—1062no—
420#107—2121no—
421#106—1062no—
422C4⋊C8—1062no—
423#102—1062no—
424#102—1062no—
425#102—1062no—
426#102—2121no—
427C22⋊C8—1062no—
428C22⋊C8—2121no—
429C4×C8—2121no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—1061no—
438#95—1061no—
439#95C22⋊D5311yescharacteristic
440#95C22⋊D5311yescharacteristic
441C24—1061no—
442C24C2×C22⋊D5311yescharacteristic
443C2×D4—532no—
444C2×D4—1064no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—1061no—
448C2×D4—1061no—
449C2×D4—1061no—
450C2×D4—2122no—
451C2×D4—2122no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—4241no—
466C22×C4—4241no—
467C22×C4—2122no—
468C22×C4—2122no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C23×D5311yescharacteristic
477C2×C8—532no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—532no—
500C4⋊C4—532no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—42no—
510C22⋊C4—81no—
511C22⋊C4—1064no—
512C22⋊C4—4241no—
513C22⋊C4—2122no—
514C22⋊C4—2122no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—81no—
520C42—2122no—
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—
534C23#1313323712yes—
535C23C22×C22⋊D5311yescharacteristic
536D4—1062no—
537D4—1062no—
538D4—2128no—
539D4—2121no—
540D4—2121no—
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—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—4244no—
574C2×C4—4244no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4#1313324212yes—
583C2×C4#41×D5311yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4#41×D5311yescharacteristic
586C2×C4#41×D5311yescharacteristic
587C2×C4#1313326311yescharacteristic
588C2×C4#1313326411yescharacteristic
589C8—1062no—
590C8—2121no—
591C8—2121no—
592C8—2121no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—4241no—
601C22—4241no—
602C22—4241no—
603C22—4241no—
604C22—2122no—
605C22—2122no—
606C22—2122no—
607C22#1681113011yescharacteristic
608C22#1681126111yescharacteristic
609C22#1681127211yescharacteristic
610C22C2×#1313323711yescharacteristic
611C22C2×#1313324211yescharacteristic
612C22#1681142611yescharacteristic
613C22#1681143811yescharacteristic
614C4—22no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—4241no—
624C4—2122no—
625C4—2122no—
626C4#1681115611yescharacteristic
627C4#1681121411yescharacteristic
628C4#1681132411yescharacteristic
629C4#1681134011yescharacteristic
630C2—42no—
631C2—4242no—
632C2#2027874711yescharacteristic
633C2#2028475811yescharacteristic
634C2#2028497111yescharacteristic
635C2#2028659611yescharacteristic
636C2#2028685011yescharacteristic
637C2#2028711111yescharacteristic
638C2#2028729311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2494C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2494D53
hypocenterC53#27033
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2494

Complements

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

NH ≅ G/Ntype
2. #20276129C2semidirect
3. #20276100C2semidirect
6. #20275481C2semidirect
7. #20275445C2semidirect
8. #20275323C2semidirect
9. #20275304C2semidirect
10. #20275293C2semidirect
11. #20274961C2semidirect
12. #20274902C2semidirect
13. #20274803C2semidirect
14. C4×#13133137C2semidirect
15. C53×#2494C2semidirect
311. #2494D53semidirect
378. C53#27033semidirect

Maximal subgroups

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

subgroupindex
#202777292
#202761292
#202761002
#202759042
#202758972
#202754812
#202754452
#202753232
#202753042
#202752932
#202749612
#202749022
#202748032
C4×#131331372
C53×#24942
#2703353