Finite Group Databasebeta

← #27456865

Subgroups — #27456865

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277751C211yesmaximal, 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#20275300C211yesmaximal, characteristic, complemented
10#20275297C211yesmaximal, characteristic, complemented
11#20274975C211yesmaximal, characteristic, complemented
12#20274912C211yesmaximal, characteristic, complemented
13#20274807C211yesmaximal, characteristic, complemented
14C4×#13133137C211yesmaximal, characteristic, complemented
15C53×#2495C211yesmaximal, characteristic, complemented
16C2×#13133191C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133137C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31#16810525C2211yescharacteristic
32#16810523—21no—
33#16810521C2211yescharacteristic
34#16810512—21no—
35#16810499—21no—
36C4×C4⋊Dic53C2211yescharacteristic
37#16810470—21no—
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C53×#400—21no—
49C53×#397—21no—
50C53×#389C2211yescharacteristic
51C53×#389C2211yescharacteristic
52C53×#388—21no—
53C53×#383—21no—
54C53×#382C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C22×C4×D53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133191—22no—
67#13133191—22no—
68C4×D212—22no—
69C4×D212—41no—
70#13133176—22no—
71#13133176—41no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×C22⋊Dic53—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83#13133150—22no—
84#13133150—41no—
85#13133148—22no—
86#13133148—22no—
87C4⋊#88322—22no—
88#13133137—22no—
89#13133137—22no—
90#13133137—22no—
91#13133137—41no—
92C22⋊#88322—22no—
93C22⋊#88322—41no—
94C4×#88322—41no—
95#13133124—21no—
96#13133124—21no—
97C23×C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53D411yescharacteristic
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C42×C53—21no—
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111#95×C53—21no—
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×C53D411yescharacteristic
119#95×C53D411yescharacteristic
120C24×C53C2×D411yescharacteristic
121C23×D53—21no—
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×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2411yescharacteristic
138C2×C22⋊D53—24no—
139C2×C22⋊D53—81no—
140C2×C22⋊D53—81no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212#4112yes—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—81no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—81no—
162C4⋊C4×C53—81no—
163C22⋊C4×C53—22no—
164C22⋊C4×C53—41no—
165C22⋊C4×C53—42no—
166C22⋊C4×C53—81no—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—42no—
172C22⋊Dic53—42no—
173#11719921—24no—
174#11719921—42no—
175#11719921—42no—
176#11719921—81no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53#4112yes—
182C4⋊Dic53#4112yes—
183#11719919—81no—
184C4×Dic53—42no—
185C4×Dic53—81no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
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#4112yes—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53C22×D411yescharacteristic
206C22×D53—22no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
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—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
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#13512yes—
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53H(5,𝔽2)11yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C22⋊D53—84no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260D212—22no—
261D212—22no—
262D212—41no—
263D212—41no—
264C4×D53—84no—
265#88322—22no—
266#88322—41no—
267#88322—41no—
268#88322—41no—
269#27038—531nomaximal, Sylow
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53#52511yescharacteristic
278C22×C53#54611yescharacteristic
279C22×C53#54711yescharacteristic
280C22×C53C2×#13511yescharacteristic
281C22×C53#57511yescharacteristic
282C22×C53#57611yescharacteristic
283C22×C53#57711yescharacteristic
284C2×D53—42no—
285C2×D53—42no—
286C2×D53—42no—
287C2×D53—81no—
288C2×D53—81no—
289C2×D53—81no—
290C2×D53—81no—
291C4×C53—22no—
292C4×C53—41no—
293C4×C53—41no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—81no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53—81no—
300C4×C53#52911yescharacteristic
301C4×C53#53911yescharacteristic
302C4×C53#55311yescharacteristic
303C4×C53#55611yescharacteristic
304Dic53—42no—
305Dic53—42no—
306Dic53—81no—
307#2495D5311yescharacteristic, complemented
308#2363—531no—
309#2073—531no—
310#2067—531no—
311#2034—531no—
312#2033—531no—
313#1949—531no—
314#1942—531no—
315#1923—531no—
316#1920—531no—
317#1920—531no—
318#1864—531no—
319#1854—531no—
320#1825—531no—
321C4×#102—531no—
322C2×C53—81no—
323C2×C53#246611yescharacteristic
324C2×C53#310211yescharacteristic
325C2×C53#311811yescharacteristic
326C2×C53#329611yescharacteristic
327C2×C53#330911yescharacteristic
328C2×C53#334111yescharacteristic
329C2×C53#335111yescharacteristic
330D53—82no—
331C2×#121—531no—
332C2×C4×D4—531no—
333C2×#117—531no—
334C2×#107—531no—
335C2×#107—531no—
336C2×#106—531no—
337C2×#106—531no—
338C2×C4⋊C8—531no—
339C2×#102—531no—
340C2×#102—531no—
341C2×#102—531no—
342C2×#102—531no—
343C2×C22⋊C8—531no—
344C2×C22⋊C8—531no—
345C2×C4×C8—531no—
346#400—21no—
347#397—21no—
348#390—531no—
349#390—1061no—
350#389—531no—
351#389C2×D5311yescharacteristic
352#389C2×D5311yescharacteristic
353#388—21no—
354#388—1061no—
355#386—1061no—
356#383—21no—
357#382C2×D5311yescharacteristic
358#380C2×D5311yescharacteristic
359#380C2×D5311yescharacteristic
360C4×C4⋊C4—531no—
361C4×C4⋊C4C2×D5311yescharacteristic
362C4×C22⋊C4—1061no—
363C4×C22⋊C4C2×D5311yescharacteristic
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#336—531no—
373#336—531no—
374C53#2703811yescharacteristic, complemented, Sylow
375C22×D4—531no—
376C22×D4—1061no—
377C23×C4—1061no—
378C23×C4C22×D5311yescharacteristic
379C22×C8—531no—
380C22×C8—531no—
381C22×C8—531no—
382C22×C8—531no—
383#121—1062no—
384#121—1062no—
385C4×D4—1062no—
386C4×D4—2121no—
387#117—1062no—
388#117—2121no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C22⋊C4C22⋊D5311yescharacteristic
407C2×C22⋊C4C22⋊D5311yescharacteristic
408C2×C42—21no—
409C2×C42—1061no—
410C2×C42—1061no—
411C2×C42C22×D5311yescharacteristic
412C2×C42C22×D5311yescharacteristic
413#107—1062no—
414#107—2121no—
415#106—1062no—
416#106—1062no—
417C4⋊C8—1062no—
418#102—1062no—
419#102—1062no—
420#102—1062no—
421#102—2121no—
422C22⋊C8—1062no—
423C22⋊C8—2121no—
424C4×C8—2121no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—1061no—
433#95—1061no—
434#95C22⋊D5311yescharacteristic
435#95C22⋊D5311yescharacteristic
436C24—1061no—
437C24C2×C22⋊D5311yescharacteristic
438C2×D4—532no—
439C2×D4—1064no—
440C2×D4—1061no—
441C2×D4—1061no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—4241no—
446C2×D4—4241no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
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—4241no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4C2×C22⋊D5311yescharacteristic
464C22×C4C2×C22⋊D5311yescharacteristic
465C22×C4C2×C22⋊D5311yescharacteristic
466C22×C4C2×C22⋊D5311yescharacteristic
467C22×C4C2×C22⋊D5311yescharacteristic
468C22×C4C23×D5311yescharacteristic
469C2×C8—532no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—532no—
492C4⋊C4—532no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—4241no—
498C22⋊C4—22no—
499C22⋊C4—41no—
500C22⋊C4—42no—
501C22⋊C4—81no—
502C22⋊C4—1064no—
503C22⋊C4—4241no—
504C22⋊C4—2122no—
505C22⋊C4—2122no—
506C22⋊C4—2122no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42—4241no—
513C42—2122no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—1062no—
522C23—2121no—
523C23—2121no—
524C23—2121no—
525C23—2121no—
526C23—2121no—
527C23—2121no—
528C23C22×C22⋊D5311yescharacteristic
529D4—1062no—
530D4—1062no—
531D4—4244no—
532D4—2121no—
533D4—2121no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
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—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
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—1062no—
565C2×C4—1062no—
566C2×C4—4244no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4#1313323712yes—
575C2×C4#41×D5311yescharacteristic
576C2×C4#41×D5311yescharacteristic
577C2×C4#41×D5311yescharacteristic
578C2×C4#41×D5311yescharacteristic
579C2×C4#1313326311yescharacteristic
580C2×C4#1313326411yescharacteristic
581C8—1062no—
582C8—2121no—
583C8—2121no—
584C8—2121no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—4241no—
593C22—4241no—
594C22—4241no—
595C22—4241no—
596C22—2122no—
597C22—2122no—
598C22—2122no—
599C22#1681113011yescharacteristic
600C22#1681126111yescharacteristic
601C22#1681127211yescharacteristic
602C22C2×#1313323711yescharacteristic
603C22#1681142111yescharacteristic
604C22#1681142611yescharacteristic
605C22#1681143111yescharacteristic
606C4—22no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—4241no—
616C4—2122no—
617C4—2122no—
618C4#1681115611yescharacteristic
619C4#1681121411yescharacteristic
620C4#1681132411yescharacteristic
621C4#1681134011yescharacteristic
622C2—81no—
623C2—4242no—
624C2#2027874711yescharacteristic
625C2#2028472711yescharacteristic
626C2#2028489311yescharacteristic
627C2#2028654411yescharacteristic
628C2#2028671511yescharacteristic
629C2#2028713311yescharacteristic
630C2#2028727511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2495C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2495D53
hypocenterC53#27038
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2495

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. #20275300C2semidirect
10. #20275297C2semidirect
11. #20274975C2semidirect
12. #20274912C2semidirect
13. #20274807C2semidirect
14. C4×#13133137C2semidirect
15. C53×#2495C2semidirect
307. #2495D53semidirect
374. C53#27038semidirect

Maximal subgroups

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

subgroupindex
#202777512
#202761292
#202761002
#202759042
#202758972
#202754812
#202754452
#202753232
#202753002
#202752972
#202749752
#202749122
#202748072
C4×#131331372
C53×#24952
#2703853