Finite Group Databasebeta

← #27456772

Subgroups — #27456772

Order 13568.

792 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279150C211yesmaximal, characteristic, complemented
2#20277735C211yesmaximal, characteristic
3#20276121C211yesmaximal, characteristic, complemented
4#20276085C211yesmaximal, characteristic, complemented
5#20275901C211yesmaximal, characteristic
6#20275475C211yesmaximal, characteristic, complemented
7#20275442C211yesmaximal, characteristic, complemented
8#20275318C211yesmaximal, characteristic, complemented
9#20275303C211yesmaximal, characteristic, complemented
10#20275295C211yesmaximal, characteristic, complemented
11#20274974C211yesmaximal, characteristic, complemented
12#20274907C211yesmaximal, characteristic, complemented
13#20274802C211yesmaximal, characteristic, complemented
14#102×Dic53C211yesmaximal, characteristic, complemented
15C53×#2034C211yesmaximal, characteristic
16C2×D4×Dic53—21no—
17C2×D4×Dic53C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C8×Dic53C2211yescharacteristic
27#16810564—21no—
28#16810546—21no—
29#16810526C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810517—21no—
34#16810516—21no—
35#16810508—21no—
36#16810503—21no—
37#16810498—21no—
38#16810492C2211yescharacteristic
39#16810483C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
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—
60C23×Dic53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63D4×Dic53—22no—
64D4×Dic53—41no—
65D4×Dic53—44no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—22no—
70C2×C4⋊Dic53—41no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53D411yescharacteristic
83C2×C22⋊Dic53D411yescharacteristic
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133148—22no—
89#13133148—22no—
90#13133148—41no—
91#13133146—22no—
92#13133138—22no—
93#13133138—22no—
94C22⋊#88322—22no—
95C8×Dic53—41no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124D411yescharacteristic
103#13133124D411yescharacteristic
104C22×D4×C53—21no—
105C22×D4×C53C2311yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109#121×C53—22no—
110#121×C53—22no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116#106×C53—22no—
117#102×C53—22no—
118#102×C53—41no—
119C22⋊C8×C53—41no—
120#95×C53—21no—
121C24×C53—21no—
122C24×C53C2×D411yescharacteristic
123C2×D4×C53—24no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—42no—
130C2×D4×C53—42no—
131C2×D4×C53#4112yes—
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×C53—21no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53—81no—
152C22×Dic53—81no—
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53#4112yes—
169C22⋊C4×C53—24no—
170C22⋊C4×C53—81no—
171C22⋊Dic53—41no—
172C22⋊Dic53—42no—
173C22⋊Dic53—42no—
174C22⋊Dic53—81no—
175C22⋊Dic53—81no—
176C22⋊Dic53SD1614yes—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—42no—
185C4⋊Dic53—42no—
186C4⋊Dic53—42no—
187C4⋊Dic53#4112yes—
188#11719919—81no—
189#11719919—81no—
190#11719919—81no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—42no—
196C4×Dic53—81no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322#4112yes—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—22no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53C2×SD1612yes—
217C23×C53C22×D411yescharacteristic
218D4×C53—22no—
219D4×C53—22no—
220D4×C53—48no—
221D4×C53—41no—
222D4×C53—41no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
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×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53H(5,𝔽2)11yescharacteristic
244C2×C4×C532−1+411yescharacteristic
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—22no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—84no—
268C2×Dic53—84no—
269C2×Dic53C2×SD1612yes—
270C2×Dic53C2×SD1612yes—
271C8×C53—41no—
272C8×C53—41no—
273#88322—22no—
274#88322—41no—
275#27032—531nomaximal, Sylow
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—81no—
287C22×C53—81no—
288C22×C53—81no—
289C22×C53—81no—
290C22×C53#52511yescharacteristic
291C22×C53#54611yescharacteristic
292C22×C53#54711yescharacteristic
293C22×C53C22×SD1611yescharacteristic
294C22×C53#57511yescharacteristic
295C22×C53#57611yescharacteristic
296C22×C53#57711yescharacteristic
297C4×C53—42no—
298C4×C53—41no—
299C4×C53—81no—
300C4×C53—81no—
301C4×C53#52911yescharacteristic
302C4×C53#53911yescharacteristic
303C4×C53#55311yescharacteristic
304C4×C53#55611yescharacteristic
305Dic53—22no—
306Dic53—22no—
307Dic53—41no—
308Dic53—42no—
309Dic53—41no—
310Dic53—81no—
311Dic53—81no—
312Dic53—81no—
313#2497—531no—
314#2361—531no—
315#2071—531no—
316#2065—531no—
317#2034—531no—
318#2034D5311yescharacteristic
319#1948—531no—
320#1941—531no—
321#1923—531no—
322#1920—531no—
323#1919—531no—
324#1864—531no—
325#1853—531no—
326#1823—531no—
327C4×#102—531no—
328C2×C53—42no—
329C2×C53—82no—
330C2×C53#246611yescharacteristic
331C2×C53#310111yescharacteristic
332C2×C53#311911yescharacteristic
333C2×C53#329511yescharacteristic
334C2×C53#332111yescharacteristic
335C2×C53#333511yescharacteristic
336C2×C53#335511yescharacteristic
337C2×#121C2×D5311yescharacteristic
338C2×C4×D4—531no—
339C2×C4×D4—1061no—
340C22×C4⋊C4—531no—
341C2×#106—531no—
342C2×#106—531no—
343C2×#106—531no—
344C2×#106C2×D5311yescharacteristic
345C2×C4⋊C8—531no—
346C2×#102—531no—
347C2×#102—531no—
348C2×#102C2×D5311yescharacteristic
349C2×#102C2×D5311yescharacteristic
350C2×C22⋊C8—531no—
351C2×C22⋊C8C2×D5311yescharacteristic
352C2×C4×C8—531no—
353#400—1061no—
354#396—1061no—
355#390—531no—
356#389—531no—
357#389—531no—
358#389—531no—
359#388—1061no—
360#388—1061no—
361#387—1061no—
362#386—1061no—
363#385—1061no—
364#384—531no—
365#380—531no—
366#380C2×D5311yescharacteristic
367C4×C4⋊C4—531no—
368C4×C4⋊C4—531no—
369C4×C22⋊C4—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340C2×D5311yescharacteristic
378#336—531no—
379#336—531no—
380C53#2703211yescharacteristic, complemented, Sylow
381C22×D4—21no—
382C22×D4C22×D5311yescharacteristic
383C23×C4—1061no—
384C23×C4—1061no—
385C23×C4C22×D5311yescharacteristic
386C22×C8—531no—
387C22×C8—531no—
388C22×C8C22×D5311yescharacteristic
389C22×C8C22×D5311yescharacteristic
390#121—22no—
391#121—22no—
392C4×D4—1062no—
393C4×D4—2121no—
394C4×D4—2124no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—2121no—
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—531no—
413C2×C22⋊C4—531no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C42—531no—
419C2×C42—531no—
420C2×C42—1061no—
421C2×C42—1061no—
422#106—22no—
423#106—1062no—
424#106—1062no—
425#106—2121no—
426C4⋊C8—1062no—
427#102—22no—
428#102—41no—
429#102—1062no—
430#102—1062no—
431C22⋊C8—41no—
432C22⋊C8—1062no—
433C4×C8—2121no—
434#95—21no—
435#95—531no—
436#95—531no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443C24—21no—
444C24D4×D5311yescharacteristic
445C2×D4—24no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4#1171994712yes—
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—21no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—4241no—
472C22×C4—4241no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4—2122no—
476C22×C4—2122no—
477C22×C4D4×D5311yescharacteristic
478C22×C4C23×D5311yescharacteristic
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—532no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—532no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—4241no—
505C4⋊C4—4241no—
506C4⋊C4—4241no—
507C4⋊C4—2122no—
508C4⋊C4—2122no—
509C4⋊C4—2122no—
510C4⋊C4#1171994712yes—
511C22⋊C4—24no—
512C22⋊C4—81no—
513C22⋊C4—534no—
514C22⋊C4—4241no—
515C22⋊C4—4241no—
516C22⋊C4—2122no—
517C22⋊C4—2121no—
518C22⋊C4—2122no—
519C42—1061no—
520C42—1061no—
521C42—1061no—
522C42—1061no—
523C42—4241no—
524C42—2122no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23SD16×D5312yes—
539C23C2×D4×D5311yescharacteristic
540D4—22no—
541D4—22no—
542D4—48no—
543D4—41no—
544D4—41no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
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—532no—
562C2×C4—532no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1062no—
574C2×C4—4244no—
575C2×C4—4244no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4#41×D5311yescharacteristic
588C2×C4#41×D5311yescharacteristic
589C2×C4C2×#1171994711yescharacteristic
590C2×C4C2×#1171994711yescharacteristic
591C2×C4#1313326211yescharacteristic
592C2×C4#1313326411yescharacteristic
593C8—41no—
594C8—41no—
595C8—1062no—
596C8—2121no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#1681112911yescharacteristic
612C22#1681126211yescharacteristic
613C22#1681127111yescharacteristic
614C22C2×SD16×D5311yescharacteristic
615C22#1681141911yescharacteristic
616C22#1681142511yescharacteristic
617C22#1681143311yescharacteristic
618C4—42no—
619C4—41no—
620C4—81no—
621C4—81no—
622C4—1062no—
623C4—1062no—
624C4—4241no—
625C4—4241no—
626C4—4241no—
627C4—2122no—
628C4—2121no—
629C4—2121no—
630C4#1681115011yescharacteristic
631C4#1681121811yescharacteristic
632C4#1681132011yescharacteristic
633C4#1681134211yescharacteristic
634C2—42no—
635C2—82no—
636C2#2027874411yescharacteristic
637C2#2028471811yescharacteristic
638C2#2028490711yescharacteristic
639C2#2028653411yescharacteristic
640C2#2028689611yescharacteristic
641C2#2028707011yescharacteristic
642C2#2028731411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2034C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2034D53
hypocenterC53#27032
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. #20279150C2semidirect
3. #20276121C2semidirect
4. #20276085C2semidirect
6. #20275475C2semidirect
7. #20275442C2semidirect
8. #20275318C2semidirect
9. #20275303C2semidirect
10. #20275295C2semidirect
11. #20274974C2semidirect
12. #20274907C2semidirect
13. #20274802C2semidirect
14. #102×Dic53C2semidirect
380. C53#27032semidirect

Maximal subgroups

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

subgroupindex
#202791502
#202777352
#202761212
#202760852
#202759012
#202754752
#202754422
#202753182
#202753032
#202752952
#202749742
#202749072
#202748022
#102×Dic532
C53×#20342
#2703253