Finite Group Databasebeta

← #27449513

Subgroups — #27449513

Order 13568.

752 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279751C211yesmaximal, characteristic, complemented
2#20277790C211yesmaximal, characteristic
3#20276256C211yesmaximal, characteristic, complemented
4#20276246C211yesmaximal, characteristic
5#20276092C211yesmaximal, characteristic, complemented
6#20276085C211yesmaximal, characteristic
7#20275648C211yesmaximal, characteristic
8#20275476C211yesmaximal, characteristic, complemented
9#20275438C211yesmaximal, characteristic, complemented
10#20274971C211yesmaximal, characteristic, complemented
11#20274964C211yesmaximal, characteristic
12#20274718C211yesmaximal, characteristic, complemented
13#20274713C211yesmaximal, characteristic
14#20274612C211yesmaximal, characteristic, complemented
15C53×#1959C211yesmaximal, characteristic
16C2×#13133208—21no—
17C2×Q8×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30#16810556—21no—
31#16810556—21no—
32#16810530C2211yescharacteristic
33#16810526C2211yescharacteristic
34#16810522C2211yescharacteristic
35#16810522C2211yescharacteristic
36#16810519—21no—
37#16810516—21no—
38#16810516—21no—
39#16810498—21no—
40#16810492C2211yescharacteristic
41#16810492C2211yescharacteristic
42C4×C4⋊Dic53C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45C22⋊C4×Dic53—21no—
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#124×C53C2211yescharacteristic
53C2×#103×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×#97×C53C2211yescharacteristic
58C53×#384C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133208—44no—
63Q8×Dic53—22no—
64Q8×Dic53—41no—
65D4×Dic53—22no—
66D4×Dic53—41no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53D411yescharacteristic
71C2×C4⋊Dic53D411yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133151—22no—
89#13133149—22no—
90#13133148—41no—
91#13133148—41no—
92C4⋊#88322—22no—
93#13133146—41no—
94#13133146—41no—
95#13133141—41no—
96#13133141—41no—
97#13133138—41no—
98#13133138—41no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124D411yescharacteristic
106#13133124D411yescharacteristic
107C22×Q8×C53C2311yescharacteristic
108C22×D4×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111#124×C53—22no—
112#124×C53Q814yes—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C42×C53C2311yescharacteristic
120#103×C53—22no—
121#103×C53—41no—
122#102×C53—22no—
123#102×C53—41no—
124#97×C53—22no—
125C24×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53C2×Q812yes—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53C2×Q812yes—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53—81no—
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53#4112yes—
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—24no—
168C22⋊C4×C53—24no—
169C42×C53C2×Q812yes—
170C42×C53#4112yes—
171C22⋊Dic53—42no—
172C22⋊Dic53—42no—
173C22⋊Dic53—81no—
174C4⋊Dic53—21no—
175C4⋊Dic53—22no—
176C4⋊Dic53—22no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—42no—
184C4⋊Dic53—42no—
185C4⋊Dic53—42no—
186C4⋊Dic53#4112yes—
187#11719919—42no—
188#11719919—42no—
189#11719919—81no—
190#11719919—81no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53—81no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C23×C53—22no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53C22×D411yescharacteristic
212Q8×C53—22no—
213Q8×C53—22no—
214Q8×C53—41no—
215Q8×C53—41no—
216D4×C53—22no—
217D4×C53—22no—
218D4×C53—41no—
219D4×C53—41no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
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×C53C22⋊Q812yes—
234C2×C4×C53C22⋊Q812yes—
235C2×C4×C53C22×Q811yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53H(5,𝔽2)11yescharacteristic
240C2×C4×C53H(5,𝔽2)11yescharacteristic
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—22no—
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×Dic53AGL(1,ℤ/8)12yes—
265C2×Dic53#13712yes—
266C8×C53—22no—
267C8×C53—41no—
268#88322—41no—
269#88322—41no—
270#26544—531nomaximal, Sylow
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—42no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53—81no—
277C22×C53—81no—
278C22×C53C2×C22⋊Q811yescharacteristic
279C22×C53#54611yescharacteristic
280C22×C53#55011yescharacteristic
281C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C53C2×#13711yescharacteristic
283C22×C53#57711yescharacteristic
284C22×C53#57711yescharacteristic
285C4×C53—22no—
286C4×C53—22no—
287C4×C53—42no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53#52911yescharacteristic
291C4×C53#54011yescharacteristic
292C4×C53#55411yescharacteristic
293C4×C53#55411yescharacteristic
294Dic53—22no—
295Dic53—22no—
296Dic53—41no—
297Dic53—42no—
298Dic53—41no—
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302#2552—531no—
303#2367—531no—
304#2096—531no—
305#2094—531no—
306#2066—531no—
307#2065—531no—
308#1983—531no—
309#1959D5311yescharacteristic
310#1948—531no—
311#1940—531no—
312#1863—531no—
313#1862—531no—
314#1803—531no—
315#1802—531no—
316#1776—531no—
317C2×C53—82no—
318C2×C53#245411yescharacteristic
319C2×C53#307911yescharacteristic
320C2×C53#308411yescharacteristic
321C2×C53#330611yescharacteristic
322C2×C53#331611yescharacteristic
323C2×C53#339711yescharacteristic
324C2×C53#340011yescharacteristic
325C2×#125—1061no—
326C2×#124C2×D5311yescharacteristic
327C2×C4×Q8—531no—
328C2×C4×D4—531no—
329C2×#107—531no—
330C2×#106—531no—
331C2×#106—531no—
332C2×#106—531no—
333C2×C4⋊C8—531no—
334C2×C4⋊C8—531no—
335C2×C4⋊C8—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#103C2×D5311yescharacteristic
339C2×#103C2×D5311yescharacteristic
340C2×#102—531no—
341C2×#102—531no—
342C2×#102C2×D5311yescharacteristic
343C2×#102C2×D5311yescharacteristic
344C2×#97C2×D5311yescharacteristic
345#398—1061no—
346#398—1061no—
347#391—531no—
348#390—531no—
349#389—531no—
350#389—531no—
351#389—1061no—
352#388—1061no—
353#388—1061no—
354#385—1061no—
355#384—531no—
356#384—531no—
357#384C2×D5311yescharacteristic
358C4×C4⋊C4—531no—
359C4×C4⋊C4—531no—
360C4×C4⋊C4—531no—
361C4×C22⋊C4—1061no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#336—531no—
367#336—531no—
368C53#2654411yescharacteristic, complemented, Sylow
369C22×Q8C22×D5311yescharacteristic
370C22×D4C22×D5311yescharacteristic
371C23×C4—1061no—
372C22×C8—531no—
373C22×C8—531no—
374C22×C8C22×D5311yescharacteristic
375C22×C8C22×D5311yescharacteristic
376#125—2124no—
377#124—22no—
378#124Dic10614yes—
379C4×Q8—1062no—
380C4×Q8—2121no—
381C4×D4—1062no—
382C4×D4—2121no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—1061no—
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×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C42—531no—
406C2×C42—531no—
407C2×C42—1061no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42C22×D5311yescharacteristic
411#107—1062no—
412#106—1062no—
413#106—2121no—
414#106—2121no—
415C4⋊C8—1062no—
416C4⋊C8—2121no—
417C4⋊C8—2121no—
418#103—22no—
419#103—41no—
420#103—2121no—
421#103—2121no—
422#102—22no—
423#102—41no—
424#102—2121no—
425#102—2121no—
426#97—22no—
427#95—531no—
428#95—531no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435C24—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8C2×Dic10612yes—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4C2×Dic10612yes—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—4241no—
460C22×C4—2122no—
461C22×C4—2122no—
462C22×C4D4×D5311yescharacteristic
463C22×C4D4×D5311yescharacteristic
464C22×C4C23×D5311yescharacteristic
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8#1171994712yes—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—532no—
484C4⋊C4—1062no—
485C4⋊C4—1062no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—4241no—
494C4⋊C4—4241no—
495C4⋊C4—2122no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—4241no—
503C22⋊C4—2122no—
504C22⋊C4—2122no—
505C42—1061no—
506C42—1061no—
507C42—1061no—
508C42—1061no—
509C42—4241no—
510C42—2122no—
511C42—2122no—
512C42C2×Dic10612yes—
513C42#1171994712yes—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D5311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—532no—
544C2×C4—532no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1062no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—4244no—
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—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4#1313319412yes—
569C2×C4#1313319412yes—
570C2×C4C22×Dic10611yescharacteristic
571C2×C4#41×D5311yescharacteristic
572C2×C4C2×#1171994711yescharacteristic
573C2×C4C2×#1171994711yescharacteristic
574C2×C4#1313326211yescharacteristic
575C2×C4#1313326311yescharacteristic
576C8—22no—
577C8—41no—
578C8—2121no—
579C8—2121no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C2×#1313319411yescharacteristic
588C22#1681126611yescharacteristic
589C22#1681129311yescharacteristic
590C22AGL(1,ℤ/8)×D5311yescharacteristic
591C22#137×D5311yescharacteristic
592C22#1681143211yescharacteristic
593C22#1681143311yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—1062no—
600C4—1062no—
601C4—4241no—
602C4—4241no—
603C4—4241no—
604C4—2121no—
605C4—2121no—
606C4—2122no—
607C4#1681115411yescharacteristic
608C4#1681122611yescharacteristic
609C4#1681133011yescharacteristic
610C4#1681133211yescharacteristic
611C2—82no—
612C2#2027859011yescharacteristic
613C2#2028452611yescharacteristic
614C2#2028458111yescharacteristic
615C2#2028667511yescharacteristic
616C2#2028682511yescharacteristic
617C2#2028773211yescharacteristic
618C2#2028776511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1959C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#1959D53
hypocenterC53#26544
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
1. #20279751C2semidirect
3. #20276256C2semidirect
5. #20276092C2semidirect
8. #20275476C2semidirect
9. #20275438C2semidirect
10. #20274971C2semidirect
12. #20274718C2semidirect
14. #20274612C2semidirect
368. C53#26544semidirect

Maximal subgroups

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

subgroupindex
#202797512
#202777902
#202762562
#202762462
#202760922
#202760852
#202756482
#202754762
#202754382
#202749712
#202749642
#202747182
#202747132
#202746122
C53×#19592
#2654453