Finite Group Databasebeta

← #27447658

Subgroups — #27447658

Order 13568.

802 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#20278400C211yesmaximal, characteristic, complemented
2#20277753C211yesmaximal, characteristic
3#20276287C211yesmaximal, characteristic, complemented
4#20276218C211yesmaximal, characteristic, complemented
5#20276090C211yesmaximal, characteristic, complemented
6#20275850C211yesmaximal, characteristic, complemented
7#20275665C211yesmaximal, characteristic, complemented
8#20275305C211yesmaximal, characteristic, complemented
9#20275143C211yesmaximal, characteristic
10#20275037C211yesmaximal, characteristic
11#20274963C211yesmaximal, characteristic, complemented
12#20274914C211yesmaximal, characteristic, complemented
13#20274893C211yesmaximal, characteristic, complemented
14#20274724C211yesmaximal, characteristic, complemented
15C53×#1735C211yesmaximal, characteristic, complemented
16C2×#13133216C2211yescharacteristic
17C2×#13133191C2211yescharacteristic
18C2×#13133179C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133139C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28#16810546—21no—
29#16810535—21no—
30#16810529—21no—
31#16810524C2211yescharacteristic
32#16810523—21no—
33#16810522C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810519C2211yescharacteristic
36#16810514—21no—
37#16810507—21no—
38#16810499—21no—
39#16810488C2211yescharacteristic
40#16810483C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×#11719921—21no—
43C4×C22⋊Dic53C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C22×C4×D53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133216—22no—
66#13133216—41no—
67#13133191—22no—
68#13133191Q814yescomplemented
69#13133179—22no—
70#13133179—41no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53D411yescharacteristic
80C2×C22⋊Dic53D411yescharacteristic
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133150—22no—
89#13133148—41no—
90#13133146—41no—
91#13133146—41no—
92#13133139—41no—
93#13133139—41no—
94#13133137—22no—
95#13133137—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#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53—22no—
108C22×C8×C53—41no—
109C22×C8×C53—41no—
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112#117×C53—22no—
113#117×C53—22no—
114#117×C53—41no—
115#117×C53—41no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C42×C53—21no—
123C2×C42×C53—21no—
124C24×C53C2×D411yescharacteristic
125C23×D53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×Q812yes—
136C22×C4×C53#4112yes—
137C22×C4×C53C2411yescharacteristic
138C2×C22⋊D53—24no—
139C2×C22⋊D53—81no—
140C2×C22⋊D53—81no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C2×D212—21no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212C2×Q812yes—
154C2×C4×D53—42no—
155C2×C4×D53—42no—
156C2×C4×D53—81no—
157C2×Dic106—42no—
158C2×Dic106—42no—
159C2×Dic106—81no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—44no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53#4112yes—
174C4⋊C4×C53#4112yes—
175C22⋊C4×C53—42no—
176C22⋊C4×C53—42no—
177C42×C53—42no—
178C42×C53—42no—
179C22⋊Dic53—22no—
180C22⋊Dic53—41no—
181C22⋊Dic53—81no—
182#11719921—24no—
183#11719921—42no—
184#11719921—42no—
185#11719921—81no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—81no—
192C4⋊Dic53C2×Q812yes—
193#11719919—42no—
194#11719919—42no—
195#11719919—81no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—81no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53C22⋊Q812yes—
214C23×C53C22×D411yescharacteristic
215C22×D53—22no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—41no—
219C22×D53—41no—
220C22×D53—41no—
221C22×D53—41no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
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×C53—41no—
240C2×C4×C53C22⋊Q812yes—
241C2×C4×C53C22×Q811yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53H(5,𝔽2)11yescharacteristic
246C2×C4×C53H(5,𝔽2)11yescharacteristic
247C22⋊D53—84no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—22no—
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—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53#13512yes—
269D212—22no—
270D212—22no—
271D212—41no—
272D212—41no—
273C4×D53—84no—
274Dic106—84no—
275C8×C53—41no—
276C8×C53—41no—
277#88322—41no—
278#88322—41no—
279#26409—531nomaximal, Sylow
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53C2×C22⋊Q811yescharacteristic
288C22×C53#54611yescharacteristic
289C22×C53#54611yescharacteristic
290C22×C53C2×#13511yescharacteristic
291C22×C53#57511yescharacteristic
292C22×C53#57611yescharacteristic
293C22×C53#57711yescharacteristic
294C2×D53—42no—
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—81no—
298C2×D53—81no—
299C2×D53—81no—
300C2×D53—81no—
301C4×C53—22no—
302C4×C53—42no—
303C4×C53—42no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53#52911yescharacteristic
307C4×C53#53811yescharacteristic
308C4×C53#55411yescharacteristic
309C4×C53#55411yescharacteristic
310Dic53—22no—
311Dic53—42no—
312Dic53—41no—
313Dic53—41no—
314Dic53—41no—
315Dic53—81no—
316Dic53—81no—
317#2439—531no—
318#2363—531no—
319#2101—531no—
320#2088—531no—
321#2065—531no—
322#2023—531no—
323#1988—531no—
324#1921—531no—
325#1893—531no—
326#1875—531no—
327#1862—531no—
328#1854—531no—
329#1850—531no—
330#1804—531no—
331#1735D5311yescharacteristic, complemented
332C2×C53—42no—
333C2×C53#245011yescharacteristic
334C2×C53#307611yescharacteristic
335C2×C53#308111yescharacteristic
336C2×C53#329711yescharacteristic
337C2×C53#330411yescharacteristic
338C2×C53#331911yescharacteristic
339C2×C53#332111yescharacteristic
340D53—82no—
341C23×C8C2×D5311yescharacteristic
342C2×C4⋊Q8—531no—
343C2×#121—531no—
344C2×C4×D4—531no—
345C2×#117C2×D5311yescharacteristic
346C2×#117C2×D5311yescharacteristic
347C2×#107—531no—
348C2×#106—531no—
349C2×C4⋊C8—531no—
350C2×C4⋊C8—531no—
351C2×#102—531no—
352C2×#102—531no—
353C2×#102—531no—
354C2×#102—531no—
355C2×C22⋊C8—531no—
356#396—1061no—
357#393—1061no—
358#391—1061no—
359#390—531no—
360#390—1061no—
361#389—531no—
362#389—531no—
363#389—531no—
364#388—1061no—
365#387—1061no—
366#386—1061no—
367#382—531no—
368#380—531no—
369C4×C4⋊C4—531no—
370C4×C22⋊C4—531no—
371C4×C22⋊C4—1061no—
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#340C2×D5311yescharacteristic
381#340C2×D5311yescharacteristic
382#336—531no—
383#336—531no—
384C53#2640911yescharacteristic, complemented, Sylow
385C22×Q8—1061no—
386C22×D4—531no—
387C22×D4—1061no—
388C23×C4—1061no—
389C23×C4C22×D5311yescharacteristic
390C22×C8—22no—
391C22×C8—41no—
392C22×C8—41no—
393C22×C8—531no—
394C22×C8—531no—
395C22×C8C22×D5311yescharacteristic
396C22×C8C22×D5311yescharacteristic
397C4⋊Q8—1062no—
398C4⋊Q8—2121no—
399#121—534no—
400#121—1062no—
401C4×D4—1062no—
402C4×D4—2121no—
403#117—22no—
404#117—22no—
405#117—41no—
406#117—41no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—531no—
421C2×C22⋊C4—531no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—1061no—
432#107—1062no—
433#106—2121no—
434C4⋊C8—2121no—
435C4⋊C8—2121no—
436#102—1062no—
437#102—2121no—
438#102—2121no—
439#102—2121no—
440C22⋊C8—1062no—
441#95—531no—
442#95—531no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450C24—1061no—
451C24D4×D5311yescharacteristic
452C2×Q8—4241no—
453C2×Q8—2122no—
454C2×Q8—2122no—
455C2×D4—532no—
456C2×D4—1064no—
457C2×D4—1061no—
458C2×D4—1061no—
459C2×D4—1061no—
460C2×D4—1061no—
461C2×D4—1061no—
462C2×D4—4241no—
463C2×D4—4241no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—531no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—4241no—
480C22×C4—2122no—
481C22×C4—2122no—
482C22×C4#1171994512yes—
483C22×C4D4×D5311yescharacteristic
484C22×C4Q8×D5312yes—
485C22×C4C23×D5311yescharacteristic
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—44no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—532no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—4241no—
512C4⋊C4—4241no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C4⋊C4#1171994512yes—
516C4⋊C4#1171994512yes—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—1062no—
520C22⋊C4—1064no—
521C22⋊C4—4241no—
522C22⋊C4—4241no—
523C22⋊C4—2122no—
524C22⋊C4—2122no—
525C22⋊C4—2121no—
526C42—42no—
527C42—42no—
528C42—1061no—
529C42—1061no—
530C42—1061no—
531C42—1061no—
532C42—4241no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—1062no—
540C23—2121no—
541C23—2121no—
542C23—2121no—
543C23—2121no—
544C23—2121no—
545C23—2121no—
546C23#1313319512yes—
547C23C2×D4×D5311yescharacteristic
548Q8—4244no—
549D4—1062no—
550D4—1062no—
551D4—4244no—
552D4—2121no—
553D4—2121no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—532no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1062no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—4244no—
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#1313319512yes—
595C2×C4C2×Q8×D5311yescharacteristic
596C2×C4C2×#1171994511yescharacteristic
597C2×C4C2×#1171994511yescharacteristic
598C2×C4C2×#1171994511yescharacteristic
599C2×C4#1313326211yescharacteristic
600C2×C4#1313326211yescharacteristic
601C8—41no—
602C8—41no—
603C8—2121no—
604C8—2121no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—4241no—
613C22—4241no—
614C22—4241no—
615C22—4241no—
616C22—2122no—
617C22—2122no—
618C22—2122no—
619C22C2×#1313319511yescharacteristic
620C22#1681126311yescharacteristic
621C22#1681126311yescharacteristic
622C22#135×D5311yescharacteristic
623C22#1681142011yescharacteristic
624C22#1681142511yescharacteristic
625C22#1681143011yescharacteristic
626C4—22no—
627C4—42no—
628C4—42no—
629C4—81no—
630C4—81no—
631C4—1062no—
632C4—4241no—
633C4—4241no—
634C4—2122no—
635C4—2121no—
636C4—2121no—
637C4—2121no—
638C4#1681115711yescharacteristic
639C4#1681121111yescharacteristic
640C4#1681132811yescharacteristic
641C4#1681132811yescharacteristic
642C2—42no—
643C2—4242no—
644C2#2027853911yescharacteristic
645C2#2028450111yescharacteristic
646C2#2028454911yescharacteristic
647C2#2028656611yescharacteristic
648C2#2028665011yescharacteristic
649C2#2028686711yescharacteristic
650C2#2028689711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1735C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1735D53
hypocenterC53#26409
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
1. #20278400C2semidirect
3. #20276287C2semidirect
4. #20276218C2semidirect
5. #20276090C2semidirect
6. #20275850C2semidirect
7. #20275665C2semidirect
8. #20275305C2semidirect
11. #20274963C2semidirect
12. #20274914C2semidirect
13. #20274893C2semidirect
14. #20274724C2semidirect
15. C53×#1735C2semidirect
68. #13133191Q8semidirect
331. #1735D53semidirect
384. C53#26409semidirect

Maximal subgroups

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

subgroupindex
#202784002
#202777532
#202762872
#202762182
#202760902
#202758502
#202756652
#202753052
#202751432
#202750372
#202749632
#202749142
#202748932
#202747242
C53×#17352
#2640953