Finite Group Databasebeta

← #27466252

Subgroups — #27466252

Order 13568.

732 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277399C211yesmaximal, characteristic
2#20277339C211yesmaximal, characteristic
3#20276288C211yesmaximal, characteristic, complemented
4#20276276C211yesmaximal, characteristic, complemented
5#20276265C211yesmaximal, characteristic, complemented
6#20276251C211yesmaximal, characteristic, complemented
7#20275482C211yesmaximal, characteristic, complemented
8#20275472C211yesmaximal, characteristic, complemented
9#20275028C211yesmaximal, characteristic
10#20274916C211yesmaximal, characteristic
11#20274903C211yesmaximal, characteristic
12#20274728C211yesmaximal, characteristic, complemented
13#20274718C211yesmaximal, characteristic, complemented
14#20274697C211yesmaximal, characteristic
15C53×#1872C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28#16810530C2211yescharacteristic
29#16810522—21no—
30#16810522C2211yescharacteristic
31#16810519—21no—
32#16810519—21no—
33#16810516—21no—
34#16810516—21no—
35#16810509—21no—
36#16810496—21no—
37#16810496—21no—
38#16810492C2211yescharacteristic
39#16810492C2211yescharacteristic
40#16810488C2211yescharacteristic
41#16810488C2211yescharacteristic
42#16810483C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C22⋊Q8×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61Q8×Dic53—22no—
62Q8×Dic53—22no—
63#13133176—22no—
64#13133176—22no—
65#13133176—41no—
66#13133176—41no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53D411yescharacteristic
70C2×C4⋊Dic53D411yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53D411yescharacteristic
82C2×C22⋊Dic53D411yescharacteristic
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133151—41no—
89#13133150—41no—
90#13133150—41no—
91#13133149—41no—
92#13133146—22no—
93#13133146—22no—
94#13133141—41no—
95#13133141—41no—
96C22⋊#88322—22no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106C22×Q8×C53C2311yescharacteristic
107C23×C4×C53—21no—
108C23×C4×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22⋊Q8×C53—22no—
112C22⋊Q8×C53Q814yes—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—41no—
116C2×C4⋊C4×C53—41no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121#103×C53—41no—
122#103×C53—41no—
123C22⋊C8×C53—22no—
124C24×C53C2×D411yescharacteristic
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53C2×Q812yes—
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—42no—
140C22×C4×C53—81no—
141C22×C4×C53—81no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×Q812yes—
144C22×C4×C53#4112yes—
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C4⋊C4×C53—24no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—42no—
167C4⋊C4×C53C2×Q812yes—
168C22⋊C4×C53—24no—
169C22⋊Dic53—22no—
170C22⋊Dic53—22no—
171C22⋊Dic53—42no—
172C22⋊Dic53—81no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—42no—
184C4⋊Dic53—81no—
185#11719919—81no—
186#11719919—81no—
187#11719919—81no—
188#11719919—81no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—42no—
192C4×Dic53—42no—
193C4×Dic53—81no—
194C4×Dic53#4112yes—
195C4×Dic53#4112yes—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53C22⋊Q812yes—
209C23×C53C22×D411yescharacteristic
210Q8×C53—22no—
211Q8×C53—22no—
212Q8×C53—41no—
213Q8×C53—41no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
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—84no—
233C2×C4×C53C22⋊Q812yes—
234C2×C4×C53C22×Q811yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53H(5,𝔽2)11yescharacteristic
239C2×C4×C53H(5,𝔽2)11yescharacteristic
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
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#12312yes—
263C2×Dic53#12312yes—
264C8×C53—41no—
265C8×C53—41no—
266#88322—41no—
267#88322—41no—
268#27690—531nomaximal, Sylow
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53C2×C22⋊Q811yescharacteristic
277C22×C53C2×#12311yescharacteristic
278C22×C53#53511yescharacteristic
279C22×C53#57611yescharacteristic
280C22×C53#57711yescharacteristic
281C22×C53#57811yescharacteristic
282C22×C53#57811yescharacteristic
283C4×C53—22no—
284C4×C53—42no—
285C4×C53—42no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53#55311yescharacteristic
289C4×C53#55311yescharacteristic
290C4×C53#55411yescharacteristic
291C4×C53#55411yescharacteristic
292Dic53—22no—
293Dic53—22no—
294Dic53—41no—
295Dic53—41no—
296Dic53—81no—
297Dic53—81no—
298Dic53—81no—
299Dic53—81no—
300#2323—531no—
301#2317—531no—
302#2101—531no—
303#2099—531no—
304#2097—531no—
305#2095—531no—
306#1949—531no—
307#1947—531no—
308#1873—531no—
309#1872D5311yescharacteristic
310#1854—531no—
311#1852—531no—
312#1805—531no—
313#1803—531no—
314#1798—531no—
315C2×C53—42no—
316C2×C53#248711yescharacteristic
317C2×C53#308211yescharacteristic
318C2×C53#308311yescharacteristic
319C2×C53#309711yescharacteristic
320C2×C53#309811yescharacteristic
321C2×C53#322311yescharacteristic
322C2×C53#322411yescharacteristic
323C2×C22⋊Q8C2×D5311yescharacteristic
324C2×C4×Q8—531no—
325C2×#117—531no—
326C2×#117—531no—
327C22×C4⋊C4C2×D5311yescharacteristic
328C2×#107—531no—
329C2×#107—531no—
330C2×#107—531no—
331C2×#106—531no—
332C2×C4⋊C8—531no—
333C2×C4⋊C8—531no—
334C2×#103—531no—
335C2×#103—531no—
336C2×#103C2×D5311yescharacteristic
337C2×#103C2×D5311yescharacteristic
338C2×C22⋊C8—531no—
339C2×C22⋊C8C2×D5311yescharacteristic
340#391—531no—
341#389—531no—
342#389—1061no—
343#389—1061no—
344#389—1061no—
345#388—1061no—
346#388—1061no—
347#387—1061no—
348#385—1061no—
349#385—1061no—
350#384—531no—
351#384—531no—
352#382—531no—
353#382—531no—
354#380—531no—
355C4×C4⋊C4—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340C2×D5311yescharacteristic
363#340C2×D5311yescharacteristic
364#336—531no—
365#336—531no—
366C53#2769011yescharacteristic, complemented, Sylow
367C22×Q8C22×D5311yescharacteristic
368C23×C4—21no—
369C23×C4C22×D5311yescharacteristic
370C22×C8—531no—
371C22×C8—531no—
372C22×C8C22×D5311yescharacteristic
373C22×C8C22×D5311yescharacteristic
374C22⋊Q8—22no—
375C22⋊Q8Dic10614yes—
376C4×Q8—1062no—
377C4×Q8—1062no—
378#117—1062no—
379#117—1062no—
380#117—2121no—
381#117—2121no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—41no—
385C2×C4⋊C4—41no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—531no—
403C2×C22⋊C4—531no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C42—531no—
407C2×C42—531no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42—1061no—
411#107—2121no—
412#107—2121no—
413#107—2121no—
414#106—2121no—
415C4⋊C8—1062no—
416C4⋊C8—1062no—
417#103—41no—
418#103—41no—
419#103—2121no—
420#103—2121no—
421C22⋊C8—22no—
422C22⋊C8—1062no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432C24D4×D5311yescharacteristic
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×Dic10612yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4C2×Dic10612yes—
459C22×C4D4×D5311yescharacteristic
460C22×C4#1171994712yes—
461C22×C4C23×D5311yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—1062no—
482C4⋊C4—1062no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—4241no—
492C4⋊C4—4241no—
493C4⋊C4—4241no—
494C4⋊C4—4241no—
495C4⋊C4—4241no—
496C4⋊C4—2122no—
497C4⋊C4C2×Dic10612yes—
498C22⋊C4—24no—
499C22⋊C4—1062no—
500C22⋊C4—1062no—
501C22⋊C4—4241no—
502C22⋊C4—2122no—
503C42—532no—
504C42—532no—
505C42—1061no—
506C42—1061no—
507C42—4241no—
508C42—2122no—
509C42—2122no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23#1313319412yes—
517C23C2×D4×D5311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—84no—
541C2×C4—532no—
542C2×C4—532no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4#1313319412yes—
566C2×C4C22×Dic10611yescharacteristic
567C2×C4#41×D5311yescharacteristic
568C2×C4#41×D5311yescharacteristic
569C2×C4C2×#1171994711yescharacteristic
570C2×C4#1313326211yescharacteristic
571C2×C4#1313326211yescharacteristic
572C8—41no—
573C8—41no—
574C8—2121no—
575C8—2121no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1313319411yescharacteristic
584C22#123×D5311yescharacteristic
585C22#1681118811yescharacteristic
586C22#1681142711yescharacteristic
587C22#1681142911yescharacteristic
588C22#1681143611yescharacteristic
589C22#1681143911yescharacteristic
590C4—22no—
591C4—42no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—1062no—
596C4—1062no—
597C4—4241no—
598C4—4241no—
599C4—4241no—
600C4—4241no—
601C4—2121no—
602C4—2121no—
603C4#1681131611yescharacteristic
604C4#1681131611yescharacteristic
605C4#1681133211yescharacteristic
606C4#1681133211yescharacteristic
607C2—42no—
608C2#2027902911yescharacteristic
609C2#2028455311yescharacteristic
610C2#2028457011yescharacteristic
611C2#2028467711yescharacteristic
612C2#2028469411yescharacteristic
613C2#2028579411yescharacteristic
614C2#2028581311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1872C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1872D53
hypocenterC53#27690
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
3. #20276288C2semidirect
4. #20276276C2semidirect
5. #20276265C2semidirect
6. #20276251C2semidirect
7. #20275482C2semidirect
8. #20275472C2semidirect
12. #20274728C2semidirect
13. #20274718C2semidirect
366. C53#27690semidirect

Maximal subgroups

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

subgroupindex
#202773992
#202773392
#202762882
#202762762
#202762652
#202762512
#202754822
#202754722
#202750282
#202749162
#202749032
#202747282
#202747182
#202746972
C53×#18722
#2769053