Finite Group Databasebeta

← #27465856

Subgroups — #27465856

Order 13568.

772 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
1C2×#16810959C211yesmaximal, characteristic
2C2×#16810964C211yesmaximal, characteristic
3#20277393C211yesmaximal, characteristic
4#20277335C211yesmaximal, characteristic
5#20276202C211yesmaximal, characteristic, complemented
6#20276190C211yesmaximal, characteristic, complemented
7#20276137C211yesmaximal, characteristic, complemented
8#20276127C211yesmaximal, characteristic, complemented
9#20275486C211yesmaximal, characteristic, complemented
10#20275472C211yesmaximal, characteristic, complemented
11#20274799C211yesmaximal, characteristic
12#20274729C211yesmaximal, characteristic, complemented
13#20274719C211yesmaximal, characteristic, complemented
14#20274697C211yesmaximal, characteristic
15C53×#1825C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×#13133182C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20#16810964—24no—
21#16810959—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133151C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133146C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×C22⋊#88322C2211yescharacteristic
33#16810528C2211yescharacteristic
34#16810521—21no—
35#16810521—21no—
36#16810520—21no—
37#16810519C2211yescharacteristic
38#16810517—21no—
39#16810511—21no—
40#16810510—21no—
41#16810495—21no—
42#16810495—21no—
43#16810492C2211yescharacteristic
44#16810492C2211yescharacteristic
45#16810488C2211yescharacteristic
46#16810488C2211yescharacteristic
47C4⋊C4×Dic53C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C23×Dic53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133196—22no—
66#13133196D414yes—
67#13133182—22no—
68#13133182—22no—
69#13133176—22no—
70#13133176—22no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53D411yescharacteristic
75C2×C4⋊Dic53D411yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53D411yescharacteristic
88C2×C22⋊Dic53D411yescharacteristic
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133151—22no—
94#13133151—22no—
95#13133150—22no—
96#13133148—22no—
97#13133146—22no—
98#13133146—22no—
99#13133142—22no—
100#13133142—22no—
101#13133140—22no—
102#13133140—22no—
103C22⋊#88322—22no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112C23×C4×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115#117×C53—22no—
116#117×C53—22no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C42×C53—21no—
124C22⋊C8×C53—22no—
125#95×C53—21no—
126C24×C53C2×D411yescharacteristic
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×C53—21no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D412yes—
137C22×C4×C53#4112yes—
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—42no—
144C22×Dic53—81no—
145C22×Dic53—81no—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106C2×D412yes—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C22⋊C4×C53—42no—
171C22⋊C4×C53—81no—
172C42×C53—42no—
173C22⋊Dic53—24no—
174C22⋊Dic53—22no—
175C22⋊Dic53—22no—
176C4⋊Dic53—21no—
177C4⋊Dic53—22no—
178C4⋊Dic53—22no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—42no—
184C4⋊Dic53C2×D412yes—
185#11719919—24no—
186#11719919—42no—
187#11719919—81no—
188#11719919—81no—
189#11719919—81no—
190#11719919—81no—
191#11719919—81no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—42no—
195C4×Dic53—81no—
196C4×Dic53#4112yes—
197C4×Dic53#4112yes—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53#12112yes—
211C23×C53C22×D411yescharacteristic
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
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#12112yes—
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C2×C4×C532−1+411yescharacteristic
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—41no—
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—84no—
260C2×Dic53#12312yes—
261C2×Dic53#12312yes—
262Dic106—22no—
263Dic106—22no—
264Dic106—41no—
265Dic106—41no—
266C8×C53—41no—
267C8×C53—41no—
268#88322—41no—
269#88322—41no—
270#27656—531nomaximal, Sylow
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53C2×#12111yescharacteristic
279C22×C53C2×#12311yescharacteristic
280C22×C53#53611yescharacteristic
281C22×C53#57611yescharacteristic
282C22×C53#57711yescharacteristic
283C22×C53#57811yescharacteristic
284C22×C53#57811yescharacteristic
285C4×C53—22no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53#54711yescharacteristic
291C4×C53#54711yescharacteristic
292C4×C53#55211yescharacteristic
293C4×C53#55211yescharacteristic
294Dic53—22no—
295Dic53—22no—
296Dic53—41no—
297Dic53—41no—
298Dic53—42no—
299Dic53—42no—
300Dic53—81no—
301Dic53—81no—
302C2×#484—531no—
303C2×#485—531no—
304#2323—531no—
305#2317—531no—
306#2085—531no—
307#2083—531no—
308#2074—531no—
309#2072—531no—
310#1949—531no—
311#1947—531no—
312#1825D5311yescharacteristic
313#1823—531no—
314#1805—531no—
315#1803—531no—
316#1798—531no—
317C2×C53—42no—
318C2×C53#248611yescharacteristic
319C2×C53#304411yescharacteristic
320C2×C53#304511yescharacteristic
321C2×C53#309711yescharacteristic
322C2×C53#309811yescharacteristic
323C2×C53#324211yescharacteristic
324C2×C53#324311yescharacteristic
325C2×C22⋊Q8—531no—
326C2×C4×Q8—531no—
327C2×#117—531no—
328C2×#117C2×D5311yescharacteristic
329C22×C4⋊C4—531no—
330#485—1064no—
331#484—1064no—
332C2×#107—531no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#106—531no—
336C2×C4⋊C8—531no—
337C2×C4⋊C8—531no—
338C2×#103—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×C22⋊C8—531no—
343C2×C22⋊C8C2×D5311yescharacteristic
344#391—531no—
345#389—531no—
346#389—1061no—
347#389—1061no—
348#389—1061no—
349#388—1061no—
350#388—1061no—
351#387—1061no—
352#385—1061no—
353#385—1061no—
354#384—531no—
355#384—531no—
356#382—531no—
357#382—531no—
358#380C2×D5311yescharacteristic
359C4×C4⋊C4—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340C2×D5311yescharacteristic
365#340C2×D5311yescharacteristic
366#340C2×D5311yescharacteristic
367#340C2×D5311yescharacteristic
368#336—531no—
369#336—531no—
370C53#2765611yescharacteristic, complemented, Sylow
371C22×Q8—531no—
372C23×C4—1061no—
373C23×C4C22×D5311yescharacteristic
374C22×C8—531no—
375C22×C8—531no—
376C22×C8C22×D5311yescharacteristic
377C22×C8C22×D5311yescharacteristic
378C22⋊Q8—534no—
379C22⋊Q8—1062no—
380C4×Q8—1062no—
381C4×Q8—1062no—
382#117—22no—
383#117—22no—
384#117—1062no—
385#117—1062no—
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—1062no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—531no—
408C2×C22⋊C4—531no—
409C2×C22⋊C4—1061no—
410C2×C42—21no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—1061no—
414C2×C42—1061no—
415#107—1062no—
416#107—1062no—
417#107—1062no—
418#106—1062no—
419C4⋊C8—1062no—
420C4⋊C8—1062no—
421#103—1062no—
422#103—1062no—
423#103—1062no—
424#103—1062no—
425C22⋊C8—22no—
426C22⋊C8—1062no—
427#95—21no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436C24D4×D5311yescharacteristic
437C2×Q8—532no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C2×Q8—1061no—
441C2×Q8—1061no—
442C2×Q8—1061no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
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—4241no—
461C22×C4—2122no—
462C22×C4#1171994512yes—
463C22×C4D4×D5311yescharacteristic
464C22×C4D4×D5312yes—
465C22×C4C23×D5311yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—532no—
487C4⋊C4—1062no—
488C4⋊C4—1062no—
489C4⋊C4—1064no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—4241no—
496C4⋊C4—4241no—
497C4⋊C4—4241no—
498C4⋊C4—4241no—
499C4⋊C4—4241no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—1062no—
505C22⋊C4—1062no—
506C22⋊C4—1064no—
507C42—42no—
508C42—532no—
509C42—532no—
510C42—1061no—
511C42—1061no—
512C42—4241no—
513C42—2122no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23#1313319012yes—
521C23C2×D4×D5311yescharacteristic
522Q8—1062no—
523Q8—1062no—
524Q8—2121no—
525Q8—2121no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
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—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—532no—
545C2×C4—532no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—4244no—
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—2121no—
569C2×C4#1313319012yes—
570C2×C4C2×D4×D5311yescharacteristic
571C2×C4#41×D5311yescharacteristic
572C2×C4#41×D5311yescharacteristic
573C2×C4C2×#1171994511yescharacteristic
574C2×C4#1313326411yescharacteristic
575C2×C4#1313326411yescharacteristic
576C8—41no—
577C8—41no—
578C8—2121no—
579C8—2121no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1313319011yescharacteristic
588C22#123×D5311yescharacteristic
589C22#1681119511yescharacteristic
590C22#1681142711yescharacteristic
591C22#1681143211yescharacteristic
592C22#1681143711yescharacteristic
593C22#1681143711yescharacteristic
594C4—22no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—1062no—
600C4—1062no—
601C4—4241no—
602C4—4241no—
603C4—2121no—
604C4—2121no—
605C4—2122no—
606C4—2122no—
607C4#1681127111yescharacteristic
608C4#1681127111yescharacteristic
609C4#1681130611yescharacteristic
610C4#1681130611yescharacteristic
611C2—42no—
612C2#2027902211yescharacteristic
613C2#2028424811yescharacteristic
614C2#2028425311yescharacteristic
615C2#2028467911yescharacteristic
616C2#2028469011yescharacteristic
617C2#2028598111yescharacteristic
618C2#2028599611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1825C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1825D53
hypocenterC53#27656
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
5. #20276202C2semidirect
6. #20276190C2semidirect
7. #20276137C2semidirect
8. #20276127C2semidirect
9. #20275486C2semidirect
10. #20275472C2semidirect
12. #20274729C2semidirect
13. #20274719C2semidirect
370. C53#27656semidirect

Maximal subgroups

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

subgroupindex
C2×#168109592
C2×#168109642
#202773932
#202773352
#202762022
#202761902
#202761372
#202761272
#202754862
#202754722
#202747992
#202747292
#202747192
#202746972
C53×#18252
#2765653