Finite Group Databasebeta

← #27465869

Subgroups — #27465869

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×#16810951C211yesmaximal, characteristic
2#20277399C211yesmaximal, characteristic
3#20277335C211yesmaximal, characteristic
4#20276199C211yesmaximal, characteristic, complemented
5#20276194C211yesmaximal, characteristic, complemented
6#20276136C211yesmaximal, characteristic, complemented
7#20276126C211yesmaximal, characteristic, complemented
8#20275480C211yesmaximal, characteristic, complemented
9#20275475C211yesmaximal, characteristic, complemented
10#20274810C211yesmaximal, characteristic
11#20274806C211yesmaximal, characteristic
12#20274728C211yesmaximal, characteristic, complemented
13#20274718C211yesmaximal, characteristic, complemented
14#20274697C211yesmaximal, characteristic
15C2×C53×#485C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19#16810951—24no—
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27#16810530C2211yescharacteristic
28#16810522—21no—
29#16810522C2211yescharacteristic
30#16810521—21no—
31#16810521—21no—
32#16810516—21no—
33#16810516—21no—
34#16810509—21no—
35#16810495—21no—
36#16810495—21no—
37#16810492C2211yescharacteristic
38#16810492C2211yescharacteristic
39#16810488C2211yescharacteristic
40#16810488C2211yescharacteristic
41#16810483C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×C22⋊Q8×C53C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C53×#485—24no—
56C2×#107×C53C2211yescharacteristic
57C2×#106×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×C22⋊C8×C53C2211yescharacteristic
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64Q8×Dic53—22no—
65Q8×Dic53—22no—
66#13133176—22no—
67#13133176—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53D411yescharacteristic
74C2×C4⋊Dic53D411yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53D411yescharacteristic
85C2×C22⋊Dic53D411yescharacteristic
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53C2311yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90#13133148—22no—
91#13133148—22no—
92#13133146—22no—
93#13133146—22no—
94#13133141—22no—
95#13133141—22no—
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#13133124—21no—
105#13133124—21no—
106C22×Q8×C53C2311yescharacteristic
107C23×C4×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22⋊Q8×C53—22no—
111C22⋊Q8×C53D414yes—
112#117×C53—22no—
113#117×C53—22no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C42×C53—21no—
121#107×C53—22no—
122#106×C53—22no—
123#103×C53—22no—
124#103×C53—22no—
125C22⋊C8×C53—22no—
126C24×C53C2×D411yescharacteristic
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53C2×D412yes—
133C22×C4×C53—21no—
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×C53C2×D411yescharacteristic
142C22×C4×C53C2×D412yes—
143C22×C4×C53#4112yes—
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—42no—
150C22×Dic53—81no—
151C22×Dic53—81no—
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
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—24no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53C2×D412yes—
169C22⋊C4×C53—24no—
170C22⋊C4×C53—42no—
171C42×C53—42no—
172C22⋊Dic53—22no—
173C22⋊Dic53—22no—
174C22⋊Dic53—81no—
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—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—42no—
186C4⋊Dic53—42no—
187C4⋊Dic53—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
212Q8×C53—22no—
213Q8×C53—22no—
214Q8×C53—41no—
215Q8×C53—41no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
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—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53#12112yes—
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C532−1+411yescharacteristic
240C2×C4×C532−1+411yescharacteristic
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
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—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—84no—
264C2×Dic53#12312yes—
265C2×Dic53#12312yes—
266C8×C53—41no—
267C8×C53—41no—
268#88322—41no—
269#88322—41no—
270#27657—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#57711yescharacteristic
282C22×C53#57711yescharacteristic
283C22×C53#57711yescharacteristic
284C22×C53#57811yescharacteristic
285C4×C53—22no—
286C4×C53—42no—
287C4×C53—42no—
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—81no—
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302C2×#485D5311yescharacteristic
303C2×#483—531no—
304#2323—531no—
305#2317—531no—
306#2085—531no—
307#2084—531no—
308#2074—531no—
309#2072—531no—
310#1949—531no—
311#1948—531no—
312#1825—531no—
313#1824—531no—
314#1805—531no—
315#1803—531no—
316#1798—531no—
317C2×C53—42no—
318C2×C53#248611yescharacteristic
319C2×C53#304311yescharacteristic
320C2×C53#304611yescharacteristic
321C2×C53#309711yescharacteristic
322C2×C53#309911yescharacteristic
323C2×C53#324211yescharacteristic
324C2×C53#324411yescharacteristic
325C2×C22⋊Q8C2×D5311yescharacteristic
326C2×C4×Q8—531no—
327C2×#117—531no—
328C2×#117C2×D5311yescharacteristic
329C22×C4⋊C4—531no—
330#485—24no—
331#483—1064no—
332C2×#107C2×D5311yescharacteristic
333C2×#106—531no—
334C2×#106—531no—
335C2×#106C2×D5311yescharacteristic
336C2×C4⋊C8—531no—
337C2×C4⋊C8—531no—
338C2×#103—531no—
339C2×#103—531no—
340C2×#103C2×D5311yescharacteristic
341C2×#103C2×D5311yescharacteristic
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#380—531no—
359C4×C4⋊C4—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#336—531no—
369#336—531no—
370C53#2765711yescharacteristic, complemented, Sylow
371C22×Q8C22×D5311yescharacteristic
372C23×C4—1061no—
373C23×C4C22×D5311yescharacteristic
374C22×C8—531no—
375C22×C8—531no—
376C22×C8C22×D5311yescharacteristic
377C22×C8C22×D5311yescharacteristic
378C22⋊Q8—22no—
379C22⋊Q8C22⋊D5314yes—
380C4×Q8—1062no—
381C4×Q8—1062no—
382#117—22no—
383#117—22no—
384#117—1062no—
385#117—1062no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
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⋊C4—1061no—
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—22no—
416#106—22no—
417#106—1062no—
418#106—1062no—
419C4⋊C8—1062no—
420C4⋊C8—1062no—
421#103—22no—
422#103—22no—
423#103—1062no—
424#103—1062no—
425C22⋊C8—22no—
426C22⋊C8—1062no—
427#95—1061no—
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—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C22⋊D5312yes—
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×C4D4×D5311yescharacteristic
463C22×C4#1171994712yes—
464C22×C4C2×C22⋊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—24no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
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—1061no—
494C4⋊C4—4241no—
495C4⋊C4—4241no—
496C4⋊C4—4241no—
497C4⋊C4—4241no—
498C4⋊C4—4241no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4C2×C22⋊D5312yes—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—1062no—
505C22⋊C4—1062no—
506C22⋊C4—4241no—
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#1313318912yes—
521C23C2×D4×D5311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
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—1061no—
551C2×C4—1061no—
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—2121no—
569C2×C4#1313318912yes—
570C2×C4C22×C22⋊D5311yescharacteristic
571C2×C4#41×D5311yescharacteristic
572C2×C4#41×D5311yescharacteristic
573C2×C4C2×#1171994711yescharacteristic
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×#1313318911yescharacteristic
588C22#123×D5311yescharacteristic
589C22#1681119511yescharacteristic
590C22#1681143011yescharacteristic
591C22#1681143211yescharacteristic
592C22#1681143211yescharacteristic
593C22#1681143711yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—1062no—
600C4—1062no—
601C4—4241no—
602C4—4241no—
603C4—4241no—
604C4—4241no—
605C4—2121no—
606C4—2121no—
607C4#1681127211yescharacteristic
608C4#1681127211yescharacteristic
609C4#1681131011yescharacteristic
610C4#1681131011yescharacteristic
611C2—42no—
612C2#2027902011yescharacteristic
613C2#2028423811yescharacteristic
614C2#2028426311yescharacteristic
615C2#2028467911yescharacteristic
616C2#2028470111yescharacteristic
617C2#2028598111yescharacteristic
618C2#2028601911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#485C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#485D53
hypocenterC53#27657
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#485

Complements

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

NH ≅ G/Ntype
4. #20276199C2semidirect
5. #20276194C2semidirect
6. #20276136C2semidirect
7. #20276126C2semidirect
8. #20275480C2semidirect
9. #20275475C2semidirect
12. #20274728C2semidirect
13. #20274718C2semidirect
370. C53#27657semidirect

Maximal subgroups

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

subgroupindex
C2×#168109512
#202773992
#202773352
#202761992
#202761942
#202761362
#202761262
#202754802
#202754752
#202748102
#202748062
#202747282
#202747182
#202746972
C2×C53×#4852
#2765753