Finite Group Databasebeta

← #27484232

Subgroups — #27484232

Order 13568.

776 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280549C211yesmaximal, characteristic, complemented
2#20277410C211yesmaximal, characteristic
3#20276281C211yesmaximal, characteristic, complemented
4#20276241C211yesmaximal, characteristic, complemented
5#20276056C211yesmaximal, characteristic, complemented
6#20275859C211yesmaximal, characteristic, complemented
7#20275310C211yesmaximal, characteristic, complemented
8#20275285C211yesmaximal, characteristic, complemented
9#20275154C211yesmaximal, characteristic
10#20274974C211yesmaximal, characteristic, complemented
11#20274964C211yesmaximal, characteristic, complemented
12#20274746C211yesmaximal, characteristic, complemented
13#20274744C211yesmaximal, characteristic, complemented
14#20274411C211yesmaximal, characteristic, complemented
15C53×#1893C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26#16810557—21no—
27#16810555—21no—
28#16810546—21no—
29#16810546—21no—
30#16810526—21no—
31#16810526C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810522C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810516—21no—
36#16810503—21no—
37#16810498—21no—
38#16810492C2211yescharacteristic
39#16810483C2211yescharacteristic
40#16810483C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810377C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×#121×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C23×Dic53—21no—
60C2×#11719917—22no—
61C2×#11719917—22no—
62C2×#11719917—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133208—41no—
66#13133208—41no—
67D4×Dic53—41no—
68D4×Dic53—41no—
69#13133176—41no—
70#13133176—41no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53D411yescharacteristic
75C2×C4⋊Dic53D411yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53D411yescharacteristic
90C2×C22⋊Dic53D411yescharacteristic
91C2×C4×Dic53—21no—
92C2×C4×Dic53C2311yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94#13133151—22no—
95#13133151—41no—
96#13133150—22no—
97#13133148—41no—
98#13133138—22no—
99#13133138—41no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106C22×D4×C53—21no—
107C22×D4×C53C2311yescharacteristic
108C23×C4×C53C2311yescharacteristic
109C22×C8×C53—22no—
110C22×C8×C53—22no—
111C22×C8×C53—41no—
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114#121×C53—22no—
115#121×C53Q814yes—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121#102×C53—22no—
122#102×C53—41no—
123#95×C53—21no—
124C24×C53—21no—
125C24×C53C2×D411yescharacteristic
126C2×D4×C53—24no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—81no—
133C2×D4×C53—81no—
134C2×D4×C53C2×Q812yes—
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×C53—21no—
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×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—44no—
163C2×C8×C53#4112yes—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53C2×Q812yes—
170C22⋊C4×C53—24no—
171C22⋊C4×C53—81no—
172C22⋊Dic53—41no—
173C22⋊Dic53—41no—
174C22⋊Dic53—42no—
175C22⋊Dic53—81no—
176C22⋊Dic53—81no—
177C22⋊Dic53—81no—
178C22⋊Dic53—81no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—41no—
188C4⋊Dic53—41no—
189C4⋊Dic53—42no—
190C4⋊Dic53—81no—
191C4⋊Dic53—81no—
192#11719919—42no—
193#11719919—81no—
194#11719919—81no—
195#11719919—81no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—81no—
201#11719917—44no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322#4112yes—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—22no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53C22⋊Q812yes—
222C23×C53C22×D411yescharacteristic
223D4×C53—22no—
224D4×C53—22no—
225D4×C53—41no—
226D4×C53—41no—
227D4×C53—84no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
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×C53C22⋊Q812yes—
244C2×C4×C53C22×Q811yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53H(5,𝔽2)11yescharacteristic
249C2×C4×C53H(5,𝔽2)11yescharacteristic
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—84no—
275C8×C53—22no—
276C8×C53—41no—
277#88322—22no—
278#88322—41no—
279#29169—531nomaximal, Sylow
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—42no—
288C22×C53—42no—
289C22×C53—42no—
290C22×C53—81no—
291C22×C53—81no—
292C22×C53—81no—
293C22×C53—81no—
294C22×C53C2×C22⋊Q811yescharacteristic
295C22×C53#53211yescharacteristic
296C22×C53#53411yescharacteristic
297C22×C53#57611yescharacteristic
298C22×C53#57611yescharacteristic
299C22×C53#57711yescharacteristic
300C22×C53#57811yescharacteristic
301C4×C53—22no—
302C4×C53—42no—
303C4×C53—81no—
304C4×C53—81no—
305C4×C53#55111yescharacteristic
306C4×C53#55111yescharacteristic
307C4×C53#55411yescharacteristic
308C4×C53#55411yescharacteristic
309Dic53—41no—
310Dic53—41no—
311Dic53—41no—
312Dic53—41no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316Dic53—81no—
317#2629—531no—
318#2324—531no—
319#2100—531no—
320#2093—531no—
321#2059—531no—
322#2025—531no—
323#1921—531no—
324#1918—531no—
325#1895—531no—
326#1893D5311yescharacteristic
327#1864—531no—
328#1862—531no—
329#1809—531no—
330#1808—531no—
331#1729—531no—
332C2×C53—42no—
333C2×C53—82no—
334C2×C53#256411yescharacteristic
335C2×C53#308111yescharacteristic
336C2×C53#308311yescharacteristic
337C2×C53#317611yescharacteristic
338C2×C53#317811yescharacteristic
339C2×C53#319811yescharacteristic
340C2×C53#320311yescharacteristic
341C22×M16—531no—
342C23×C8C2×D5311yescharacteristic
343C2×#125—531no—
344C2×#121C2×D5311yescharacteristic
345C2×C4×D4—531no—
346C2×#117—531no—
347C2×#107—531no—
348C2×#107—531no—
349C2×#107—531no—
350C2×#106—531no—
351C2×#102—531no—
352C2×#102—531no—
353C2×#102C2×D5311yescharacteristic
354C2×#102C2×D5311yescharacteristic
355#398—1061no—
356#398—1061no—
357#396—1061no—
358#396—1061no—
359#390—531no—
360#390—1061no—
361#389—531no—
362#389—531no—
363#389—531no—
364#388—1061no—
365#386—1061no—
366#385—1061no—
367#384—531no—
368#380—531no—
369#380—531no—
370#380C2×D5311yescharacteristic
371C4×C4⋊C4—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340C2×D5311yescharacteristic
379#340C2×D5311yescharacteristic
380#336—531no—
381#336—531no—
382#336—531no—
383#336—531no—
384C53#2916911yescharacteristic, complemented, Sylow
385C22×D4—21no—
386C22×D4C22×D5311yescharacteristic
387C23×C4—1061no—
388C23×C4C22×D5311yescharacteristic
389C2×M16—1062no—
390C2×M16—1062no—
391C2×M16—2121no—
392C22×C8—22no—
393C22×C8—22no—
394C22×C8—41no—
395C22×C8—531no—
396C22×C8—531no—
397C22×C8C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399#125—2121no—
400#125—2121no—
401#121—22no—
402#121Dic10614yes—
403C4×D4—2121no—
404C4×D4—2121no—
405#117—2121no—
406#117—2121no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—531no—
426C2×C22⋊C4—531no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C42—531no—
433C2×C42—531no—
434C2×C42—1061no—
435#107—1062no—
436#107—1062no—
437#107—2121no—
438#106—2121no—
439#102—22no—
440#102—41no—
441#102—1062no—
442#102—2121no—
443#95—21no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450C24—21no—
451C24D4×D5311yescharacteristic
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—81no—
459C2×D4—81no—
460C2×D4C2×Dic10612yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—531no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—4241no—
477C22×C4—4241no—
478C22×C4—2122no—
479C22×C4C2×Dic10612yes—
480C22×C4D4×D5311yescharacteristic
481C22×C4#1171994712yes—
482C22×C4C23×D5311yescharacteristic
483M16—2124no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—44no—
490C2×C8—532no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8#1171994712yes—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—4241no—
511C4⋊C4—4241no—
512C4⋊C4—4241no—
513C4⋊C4—4241no—
514C4⋊C4—4241no—
515C4⋊C4—2121no—
516C4⋊C4—2121no—
517C4⋊C4—2122no—
518C4⋊C4—2122no—
519C4⋊C4C2×Dic10612yes—
520C22⋊C4—24no—
521C22⋊C4—81no—
522C22⋊C4—4241no—
523C22⋊C4—4241no—
524C22⋊C4—4241no—
525C22⋊C4—4241no—
526C22⋊C4—2121no—
527C22⋊C4—2121no—
528C22⋊C4—2122no—
529C42—1061no—
530C42—1061no—
531C42—1061no—
532C42—1061no—
533C42—4241no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23#1313319412yes—
548C23C2×D4×D5311yescharacteristic
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
553D4—84no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—4244no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4#1313319412yes—
595C2×C4C22×Dic10611yescharacteristic
596C2×C4#41×D5311yescharacteristic
597C2×C4C2×#1171994711yescharacteristic
598C2×C4C2×#1171994711yescharacteristic
599C2×C4#1313326211yescharacteristic
600C2×C4#1313326311yescharacteristic
601C8—22no—
602C8—41no—
603C8—1062no—
604C8—2121no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C2×#1313319411yescharacteristic
620C22#1681116811yescharacteristic
621C22#1681118111yescharacteristic
622C22#1681142411yescharacteristic
623C22#1681142511yescharacteristic
624C22#1681142911yescharacteristic
625C22#1681143711yescharacteristic
626C4—22no—
627C4—42no—
628C4—81no—
629C4—81no—
630C4—4241no—
631C4—4241no—
632C4—4241no—
633C4—4241no—
634C4—2121no—
635C4—2121no—
636C4—2121no—
637C4—2121no—
638C4#1681129611yescharacteristic
639C4#1681130411yescharacteristic
640C4#1681133011yescharacteristic
641C4#1681133211yescharacteristic
642C2—42no—
643C2—82no—
644C2#2027989911yescharacteristic
645C2#2028454211yescharacteristic
646C2#2028456411yescharacteristic
647C2#2028535811yescharacteristic
648C2#2028537811yescharacteristic
649C2#2028557011yescharacteristic
650C2#2028561111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1893C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#1893D53
hypocenterC53#29169
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
1. #20280549C2semidirect
3. #20276281C2semidirect
4. #20276241C2semidirect
5. #20276056C2semidirect
6. #20275859C2semidirect
7. #20275310C2semidirect
8. #20275285C2semidirect
10. #20274974C2semidirect
11. #20274964C2semidirect
12. #20274746C2semidirect
13. #20274744C2semidirect
14. #20274411C2semidirect
384. C53#29169semidirect

Maximal subgroups

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

subgroupindex
#202805492
#202774102
#202762812
#202762412
#202760562
#202758592
#202753102
#202752852
#202751542
#202749742
#202749642
#202747462
#202747442
#202744112
C53×#18932
#2916953