Finite Group Databasebeta

← #27425411

Subgroups — #27425411

Order 13568.

828 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810916C211yesmaximal, characteristic, complemented
2C2×#16810677C211yesmaximal, characteristic
3#20283065C211yesmaximal, characteristic
4C22⋊Q8×Dic53C211yesmaximal, characteristic
5#20276141C211yesmaximal, characteristic
6#20276138C211yesmaximal, characteristic
7#20275908C211yesmaximal, characteristic
8#20275661C211yesmaximal, characteristic, complemented
9#20275475C211yesmaximal, characteristic, complemented
10#20275459C211yesmaximal, characteristic, complemented
11#20275457C211yesmaximal, characteristic, complemented
12#20275407C211yesmaximal, characteristic, complemented
13#20275402C211yesmaximal, characteristic, complemented
14#20274718C211yesmaximal, characteristic, complemented
15C53×#2035C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133194—21no—
18C2×Q8×Dic53C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20#16810916—24no—
21#16810677—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31C2×#13133127C2211yescharacteristic
32#16810530C2211yescharacteristic
33#16810527—21no—
34#16810522C2211yescharacteristic
35#16810522C2211yescharacteristic
36#16810520—21no—
37#16810519C2211yescharacteristic
38#16810509—21no—
39#16810498—21no—
40#16810492C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810483C2211yescharacteristic
43#16810481—21no—
44C4⋊C4×Dic53—21no—
45C4⋊C4×Dic53C2211yescharacteristic
46C22⋊C4×Dic53—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54C2×C22⋊Q8×C53C2211yescharacteristic
55C2×#106×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×Dic106—21no—
62C23×Dic53D411yescharacteristic
63C23×Dic53D411yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133216—22no—
67#13133216—22no—
68#13133194—44no—
69Q8×Dic53—22no—
70Q8×Dic53—22no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C4×Dic53—22no—
85C2×C4×Dic53—22no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92#13133151—22no—
93#13133150—22no—
94#13133148—22no—
95#13133146—22no—
96#13133146D414yes—
97#13133141—22no—
98#13133141—22no—
99C22⋊#88322—22no—
100#13133127—22no—
101#13133127—22no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110C22×Q8×C53C2311yescharacteristic
111C23×C4×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22⋊Q8×C53—22no—
115C22⋊Q8×C53—22no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122#106×C53—22no—
123#103×C53—22no—
124#103×C53—22no—
125C22⋊C8×C53—22no—
126#95×C53—21no—
127C24×C53C2×D411yescharacteristic
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53#4112yes—
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×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—41no—
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160C2×Dic106—42no—
161C2×Dic106—42no—
162C2×Dic106—42no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53C2×D412yes—
169C4⋊C4×C53—24no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53#4112yes—
176C22⋊C4×C53—24no—
177C22⋊C4×C53—42no—
178C22⋊Dic53—42no—
179C22⋊Dic53—42no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—42no—
189C4⋊Dic53—42no—
190C4⋊Dic53—42no—
191#11719919—42no—
192#11719919—42no—
193#11719919—42no—
194#11719919—42no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—22no—
198C4×Dic53—22no—
199C4×Dic53—42no—
200C4×Dic53—42no—
201C4×Dic53C2×D412yes—
202C4×Dic53#4112yes—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322C2×D412yes—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53AGL(1,ℤ/8)12yes—
216C23×C53C22×D411yescharacteristic
217Q8×C53—22no—
218Q8×C53—22no—
219Q8×C53—41no—
220Q8×C53—41no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—21no—
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×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×D411yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C532−1+411yescharacteristic
245C2×C4×C532−1+411yescharacteristic
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—44no—
255C2×Dic53—44no—
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—41no—
268C2×Dic53#12112yes—
269C2×Dic53#12112yes—
270C2×Dic53#13712yes—
271C2×Dic53#13712yes—
272Dic106—84no—
273C8×C53—22no—
274C8×C53—41no—
275#88322—22no—
276#88322—41no—
277#24711—531nomaximal, Sylow
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53C2×#12111yescharacteristic
286C22×C53#52511yescharacteristic
287C22×C53#53611yescharacteristic
288C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C53C2×#13711yescharacteristic
290C22×C53C2×#13711yescharacteristic
291C22×C53#57511yescharacteristic
292C4×C53—42no—
293C4×C53—41no—
294C4×C53—42no—
295C4×C53—81no—
296C4×C53—81no—
297C4×C53#54711yescharacteristic
298C4×C53#54811yescharacteristic
299C4×C53#55211yescharacteristic
300C4×C53#55611yescharacteristic
301Dic53—22no—
302Dic53—41no—
303Dic53—41no—
304Dic53—81no—
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308Dic53#47014yes—
309C2×#477—531no—
310C2×#432—531no—
311#2857—531no—
312C4×C22⋊Q8—531no—
313#2075—531no—
314#2075—531no—
315#2035—531no—
316#2035D5311yescharacteristic
317#1987—531no—
318#1948—531no—
319#1944—531no—
320#1944—531no—
321#1935—531no—
322#1935—531no—
323#1803—531no—
324C2×C53—42no—
325C2×C53#241511yescharacteristic
326C2×C53C2×#47011yescharacteristic
327C2×C53#305511yescharacteristic
328C2×C53#311211yescharacteristic
329C2×C53#311611yescharacteristic
330C2×C53#323711yescharacteristic
331C2×C53#323811yescharacteristic
332C2×C4⋊Q8—531no—
333C2×C22⋊Q8—1061no—
334C2×C22⋊Q8C2×D5311yescharacteristic
335C2×C4×Q8—531no—
336C22×C42—531no—
337#477—1064no—
338#432—1064no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#106—531no—
342C2×#106C2×D5311yescharacteristic
343C2×C4⋊C8—531no—
344C2×C4⋊C8—531no—
345C2×#103—531no—
346C2×#103—531no—
347C2×#103C2×D5311yescharacteristic
348C2×#103C2×D5311yescharacteristic
349C2×C22⋊C8—531no—
350C2×C22⋊C8C2×D5311yescharacteristic
351C2×#97—531no—
352C2×#97—531no—
353#391—531no—
354#391—1061no—
355#389—531no—
356#389—531no—
357#389—531no—
358#389—1061no—
359#387—1061no—
360#385—1061no—
361#384—531no—
362#380—531no—
363#380—531no—
364#380—1061no—
365#380C2×D5311yescharacteristic
366C4×C4⋊C4—531no—
367C4×C4⋊C4—1061no—
368C4×C22⋊C4—1061no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340C2×D5311yescharacteristic
377C53#2471111yescharacteristic, complemented, Sylow
378C22×Q8—1061no—
379C22×Q8C22×D5311yescharacteristic
380C23×C4—531no—
381C23×C4—531no—
382C23×C4C22×D5311yescharacteristic
383C22×C8—531no—
384C22×C8—531no—
385C22×C8C22×D5311yescharacteristic
386C22×C8C22×D5311yescharacteristic
387C4⋊Q8—1062no—
388C4⋊Q8—1062no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8—2124no—
392C4×Q8—1062no—
393C4×Q8—1062no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—1061no—
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⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C42—531no—
414C2×C42—531no—
415C2×C42—531no—
416C2×C42—531no—
417C2×C42—1062no—
418C2×C42—1062no—
419C2×C42—1061no—
420C2×C42—1061no—
421#107—1062no—
422#107—1062no—
423#106—22no—
424#106—1062no—
425C4⋊C8—534no—
426C4⋊C8—1062no—
427#103—22no—
428#103—22no—
429#103—1062no—
430#103—1062no—
431C22⋊C8—22no—
432C22⋊C8—1062no—
433#97—1062no—
434#97—1062no—
435#95—21no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444C24D4×D5311yescharacteristic
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—2122no—
451C2×Q8—2122no—
452C2×Q8—2122no—
453C2×Q8#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—21no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—2121no—
478C22×C4D4×D5311yescharacteristic
479C22×C4C23×D5311yescharacteristic
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—
491C2×C8D4×D5312yes—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—2122no—
507C4⋊C4—2122no—
508C4⋊C4—2122no—
509C4⋊C4—2122no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4#1171994712yes—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—2122no—
517C22⋊C4—2122no—
518C42—532no—
519C42—532no—
520C42—1062no—
521C42—1062no—
522C42—1061no—
523C42—1061no—
524C42—2122no—
525C42—2122no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23#1313324412yes—
533C23C2×D4×D5311yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538Q8—4244no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
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—532no—
559C2×C4—532no—
560C2×C4—532no—
561C2×C4—532no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—2124no—
571C2×C4—2124no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
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×C4C2×D4×D5311yescharacteristic
585C2×C4#41×D5311yescharacteristic
586C2×C4C2×#1171994711yescharacteristic
587C2×C4C2×#1171994711yescharacteristic
588C2×C4#1313326411yescharacteristic
589C2×C4#1313326411yescharacteristic
590C8—22no—
591C8—41no—
592C8—1062no—
593C8—2121no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22#121×D5311yescharacteristic
602C22#1681112911yescharacteristic
603C22#1681119511yescharacteristic
604C22C2×#1313324411yescharacteristic
605C22#137×D5311yescharacteristic
606C22#137×D5311yescharacteristic
607C22#1681142011yescharacteristic
608C4—41no—
609C4—42no—
610C4—42no—
611C4—81no—
612C4—81no—
613C4—534no—
614C4—1062no—
615C4—4241no—
616C4—4241no—
617C4—4241no—
618C4—4241no—
619C4—2121no—
620C4—2121no—
621C4#1681127311yescharacteristic
622C4#1681127911yescharacteristic
623C4#1681131011yescharacteristic
624C4#1681134211yescharacteristic
625C2—42no—
626C2#2027813711yescharacteristic
627C2#470×D5311yescharacteristic
628C2#2028432711yescharacteristic
629C2#2028483211yescharacteristic
630C2#2028487011yescharacteristic
631C2#2028592611yescharacteristic
632C2#2028593911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2035C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2035D53
hypocenterC53#24711
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
1. C2×#16810916C2semidirect
8. #20275661C2semidirect
9. #20275475C2semidirect
10. #20275459C2semidirect
11. #20275457C2semidirect
12. #20275407C2semidirect
13. #20275402C2semidirect
14. #20274718C2semidirect
377. C53#24711semidirect

Maximal subgroups

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

subgroupindex
C2×#168109162
C2×#168106772
#202830652
C22⋊Q8×Dic532
#202761412
#202761382
#202759082
#202756612
#202754752
#202754592
#202754572
#202754072
#202754022
#202747182
C53×#20352
#2471153