Finite Group Databasebeta

← #27456862

Subgroups — #27456862

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#20279128C211yesmaximal, characteristic, complemented
2#20277757C211yesmaximal, characteristic
3#20276131C211yesmaximal, characteristic, complemented
4#20276099C211yesmaximal, characteristic, complemented
5#20275894C211yesmaximal, characteristic
6#20275484C211yesmaximal, characteristic, complemented
7#20275447C211yesmaximal, characteristic, complemented
8#20275318C211yesmaximal, characteristic, complemented
9#20275303C211yesmaximal, characteristic, complemented
10#20275302C211yesmaximal, characteristic, complemented
11#20274974C211yesmaximal, characteristic, complemented
12#20274912C211yesmaximal, characteristic, complemented
13#20274811C211yesmaximal, characteristic, complemented
14#102×Dic53C211yesmaximal, characteristic, complemented
15C53×#2034C211yesmaximal, characteristic
16C2×D4×Dic53C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C8×Dic53C2211yescharacteristic
26#16810563—21no—
27#16810551—21no—
28#16810526—21no—
29#16810526C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810516—21no—
34#16810516—21no—
35#16810503—21no—
36#16810490—21no—
37#16810488C2211yescharacteristic
38#16810483C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C22⋊C4×Dic53—21no—
42C4×C22⋊Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#121×C53C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62D4×Dic53—22no—
63D4×Dic53—41no—
64#13133176—22no—
65#13133176—41no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53D411yescharacteristic
79C2×C22⋊Dic53D411yescharacteristic
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53C2311yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85#13133150—22no—
86#13133150—41no—
87#13133148—22no—
88#13133146—22no—
89#13133138—22no—
90#13133138—22no—
91C22⋊#88322—22no—
92C8×Dic53—41no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124D411yescharacteristic
102#13133124D411yescharacteristic
103C22×D4×C53—21no—
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108#121×C53—22no—
109#121×C53—22no—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115#106×C53—22no—
116#102×C53—22no—
117#102×C53—41no—
118C22⋊C8×C53—41no—
119#95×C53—21no—
120C24×C53—21no—
121C24×C53C2×D411yescharacteristic
122C2×D4×C53—24no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—81no—
129C2×D4×C53—81no—
130C2×D4×C53#4112yes—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—81no—
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×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#4112yes—
165C22⋊C4×C53—24no—
166C22⋊C4×C53—42no—
167C22⋊Dic53—22no—
168C22⋊Dic53—41no—
169C22⋊Dic53—42no—
170C22⋊Dic53—42no—
171C22⋊Dic53—42no—
172C22⋊Dic53—81no—
173C22⋊Dic53—81no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—81no—
182C4⋊Dic53—81no—
183C4⋊Dic53#4112yes—
184#11719919—81no—
185C4×Dic53—21no—
186C4×Dic53—21no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—42no—
190C4×Dic53—42no—
191C4×Dic53—81no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322#4112yes—
198C23×C53—21no—
199C23×C53—22no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53C22×D411yescharacteristic
213D4×C53—22no—
214D4×C53—22no—
215D4×C53—41no—
216D4×C53—41no—
217D4×C53—84no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
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×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53H(5,𝔽2)11yescharacteristic
239C2×C4×C532−1+411yescharacteristic
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—22no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
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—
263C2×Dic53—84no—
264C2×Dic53#13512yes—
265C8×C53—41no—
266C8×C53—41no—
267#88322—22no—
268#88322—41no—
269#27038—531nomaximal, Sylow
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—42no—
274C22×C53—41no—
275C22×C53—42no—
276C22×C53—41no—
277C22×C53—42no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53—81no—
283C22×C53—81no—
284C22×C53#52511yescharacteristic
285C22×C53#54611yescharacteristic
286C22×C53#54711yescharacteristic
287C22×C53C2×#13511yescharacteristic
288C22×C53#57511yescharacteristic
289C22×C53#57611yescharacteristic
290C22×C53#57711yescharacteristic
291C4×C53—42no—
292C4×C53—41no—
293C4×C53—81no—
294C4×C53—81no—
295C4×C53#52911yescharacteristic
296C4×C53#53911yescharacteristic
297C4×C53#55311yescharacteristic
298C4×C53#55611yescharacteristic
299Dic53—22no—
300Dic53—41no—
301Dic53—41no—
302Dic53—42no—
303Dic53—41no—
304Dic53—81no—
305Dic53—81no—
306Dic53—81no—
307#2495—531no—
308#2363—531no—
309#2073—531no—
310#2067—531no—
311#2034D5311yescharacteristic
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—82no—
324C2×C53#246611yescharacteristic
325C2×C53#310211yescharacteristic
326C2×C53#311811yescharacteristic
327C2×C53#329611yescharacteristic
328C2×C53#330911yescharacteristic
329C2×C53#334111yescharacteristic
330C2×C53#335111yescharacteristic
331C2×#121C2×D5311yescharacteristic
332C2×C4×D4—531no—
333C2×#117—531no—
334C2×#107—531no—
335C2×#107—531no—
336C2×#106—531no—
337C2×#106C2×D5311yescharacteristic
338C2×C4⋊C8—531no—
339C2×#102—531no—
340C2×#102—531no—
341C2×#102C2×D5311yescharacteristic
342C2×#102C2×D5311yescharacteristic
343C2×C22⋊C8—531no—
344C2×C22⋊C8C2×D5311yescharacteristic
345C2×C4×C8—531no—
346#400—1061no—
347#397—1061no—
348#390—531no—
349#390—1061no—
350#389—531no—
351#389—531no—
352#389—531no—
353#388—1061no—
354#388—1061no—
355#386—1061no—
356#383—1061no—
357#382—531no—
358#380—531no—
359#380C2×D5311yescharacteristic
360C4×C4⋊C4—531no—
361C4×C4⋊C4—531no—
362C4×C22⋊C4—531no—
363C4×C22⋊C4—1061no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340C2×D5311yescharacteristic
372#336—531no—
373#336—531no—
374C53#2703811yescharacteristic, complemented, Sylow
375C22×D4—21no—
376C22×D4C22×D5311yescharacteristic
377C23×C4—1061no—
378C23×C4C22×D5311yescharacteristic
379C22×C8—531no—
380C22×C8—531no—
381C22×C8C22×D5311yescharacteristic
382C22×C8C22×D5311yescharacteristic
383#121—22no—
384#121—22no—
385C4×D4—1062no—
386C4×D4—2121no—
387#117—1062no—
388#117—2121no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—531no—
402C2×C22⋊C4—531no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C42—531no—
409C2×C42—531no—
410C2×C42—1061no—
411C2×C42—1061no—
412C2×C42—1061no—
413#107—1062no—
414#107—2121no—
415#106—22no—
416#106—1062no—
417C4⋊C8—1062no—
418#102—22no—
419#102—41no—
420#102—1062no—
421#102—1062no—
422C22⋊C8—41no—
423C22⋊C8—1062no—
424C4×C8—2121no—
425#95—21no—
426#95—531no—
427#95—531no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436C24—21no—
437C24D4×D5311yescharacteristic
438C2×D4—24no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—81no—
445C2×D4—81no—
446C2×D4#1171994712yes—
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—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—4241no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4D4×D5311yescharacteristic
468C22×C4C23×D5311yescharacteristic
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—532no—
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—532no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—4241no—
495C4⋊C4—4241no—
496C4⋊C4—4241no—
497C4⋊C4#1171994712yes—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—1062no—
501C22⋊C4—4241no—
502C22⋊C4—4241no—
503C22⋊C4—2121no—
504C22⋊C4—2122no—
505C22⋊C4—2122no—
506C22⋊C4—2122no—
507C42—1061no—
508C42—1061no—
509C42—1061no—
510C42—1061no—
511C42—4241no—
512C42—2122no—
513C42—2122no—
514C23—21no—
515C23—22no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C2×D4×D5311yescharacteristic
529D4—22no—
530D4—22no—
531D4—41no—
532D4—41no—
533D4—84no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—532no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1062no—
563C2×C4—4244no—
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#41×D5311yescharacteristic
576C2×C4#41×D5311yescharacteristic
577C2×C4C2×#1171994711yescharacteristic
578C2×C4C2×#1171994711yescharacteristic
579C2×C4#1313326211yescharacteristic
580C2×C4#1313326411yescharacteristic
581C8—41no—
582C8—41no—
583C8—1062no—
584C8—2121no—
585C22—41no—
586C22—41no—
587C22—42no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—42no—
592C22—41no—
593C22—42no—
594C22—41no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22#1681112911yescharacteristic
600C22#1681126211yescharacteristic
601C22#1681127111yescharacteristic
602C22#135×D5311yescharacteristic
603C22#1681141911yescharacteristic
604C22#1681142711yescharacteristic
605C22#1681142911yescharacteristic
606C4—41no—
607C4—42no—
608C4—81no—
609C4—81no—
610C4—1062no—
611C4—4241no—
612C4—4241no—
613C4—4241no—
614C4—2121no—
615C4—2122no—
616C4—2121no—
617C4—2121no—
618C4#1681115011yescharacteristic
619C4#1681121811yescharacteristic
620C4#1681132011yescharacteristic
621C4#1681134211yescharacteristic
622C2—81no—
623C2—82no—
624C2#2027874411yescharacteristic
625C2#2028472611yescharacteristic
626C2#2028490011yescharacteristic
627C2#2028654511yescharacteristic
628C2#2028671611yescharacteristic
629C2#2028713211yescharacteristic
630C2#2028727911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2034C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2034D53
hypocenterC53#27038
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2034

Complements

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

NH ≅ G/Ntype
1. #20279128C2semidirect
3. #20276131C2semidirect
4. #20276099C2semidirect
6. #20275484C2semidirect
7. #20275447C2semidirect
8. #20275318C2semidirect
9. #20275303C2semidirect
10. #20275302C2semidirect
11. #20274974C2semidirect
12. #20274912C2semidirect
13. #20274811C2semidirect
14. #102×Dic53C2semidirect
374. C53#27038semidirect

Maximal subgroups

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

subgroupindex
#202791282
#202777572
#202761312
#202760992
#202758942
#202754842
#202754472
#202753182
#202753032
#202753022
#202749742
#202749122
#202748112
#102×Dic532
C53×#20342
#2703853