Finite Group Databasebeta

← #27465857

Subgroups — #27465857

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×#16810956C211yesmaximal, characteristic
2C2×#16810961C211yesmaximal, characteristic
3#20277338C211yesmaximal, characteristic
4#20276196C211yesmaximal, characteristic, complemented
5#20276186C211yesmaximal, characteristic, complemented
6#20276133C211yesmaximal, characteristic, complemented
7#20276123C211yesmaximal, characteristic, complemented
8#20275483C211yesmaximal, characteristic, complemented
9#20275473C211yesmaximal, characteristic, complemented
10#20274807C211yesmaximal, characteristic
11#20274799C211yesmaximal, characteristic
12#20274730C211yesmaximal, characteristic, complemented
13#20274720C211yesmaximal, characteristic, complemented
14#20274699C211yesmaximal, characteristic
15C53×#2323C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18#16810961—24no—
19#16810956—24no—
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31C2×C22⋊#88322C2211yescharacteristic
32#16810522—21no—
33#16810522—21no—
34#16810498—21no—
35#16810498—21no—
36#16810493C2211yescharacteristic
37#16810493C2211yescharacteristic
38#16810489C2211yescharacteristic
39#16810489C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#389—21no—
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#388—21no—
58C53×#387—21no—
59C53×#380C2211yescharacteristic
60C4×C4⋊C4×C53C2211yescharacteristic
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133176—22no—
67#13133176—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C22⋊Dic53—21no—
80C2×C4×Dic53—21no—
81#13133150—22no—
82#13133150—22no—
83#13133150—22no—
84#13133148—22no—
85C4⋊#88322—22no—
86C4⋊#88322—22no—
87#13133141—22no—
88#13133141—22no—
89#13133141—22no—
90#13133141—22no—
91C22⋊#88322—22no—
92C22⋊#88322—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53C2311yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C22⋊Q8×C53—22no—
100C22⋊Q8×C53D414yes—
101C4×Q8×C53—22no—
102C4×Q8×C53—22no—
103#117×C53—22no—
104#117×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C42×C53—21no—
118C2×C42×C53—21no—
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
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×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53#4112yes—
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—42no—
154C22×Dic53—81no—
155C22×Dic53—81no—
156C4⋊C4×C53—24no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—42no—
165C4⋊C4×C53—81no—
166C4⋊C4×C53C2×D412yes—
167C22⋊C4×C53—24no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—81no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—42no—
174C42×C53—81no—
175C42×C53#4112yes—
176C42×C53#4112yes—
177C22⋊Dic53—42no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—42no—
187C4⋊Dic53—81no—
188C4⋊Dic53—81no—
189C4⋊Dic53—81no—
190C4⋊Dic53—81no—
191C4×Dic53—42no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
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—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
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×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53#12112yes—
245C2×C4×C53#12312yes—
246C2×C4×C53#12312yes—
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C532−1+411yescharacteristic
252C2×C4×C532−1+411yescharacteristic
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—41no—
264C2×Dic53—41no—
265C2×Dic53—84no—
266#88322—41no—
267#88322—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—22no—
287C4×C53—22no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53—42no—
291C4×C53—42no—
292C4×C53—81no—
293C4×C53—81no—
294C4×C53#54711yescharacteristic
295C4×C53#54711yescharacteristic
296C4×C53#55211yescharacteristic
297C4×C53#55211yescharacteristic
298Dic53—81no—
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302C2×#484—531no—
303C2×#485—531no—
304#2323D5311yescharacteristic
305#2317—531no—
306#2085—531no—
307#2083—531no—
308#2074—531no—
309#2072—531no—
310#1949—531no—
311#1947—531no—
312#1825—531no—
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⋊Q8C2×D5311yescharacteristic
326C2×C4×Q8C2×D5311yescharacteristic
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⋊C8—531no—
344#391C2×D5311yescharacteristic
345#389—21no—
346#389—1061no—
347#389—1061no—
348#389C2×D5311yescharacteristic
349#388—21no—
350#388—21no—
351#387—21no—
352#385—1061no—
353#385—1061no—
354#384—531no—
355#384—531no—
356#382—531no—
357#382—531no—
358#380C2×D5311yescharacteristic
359C4×C4⋊C4C2×D5311yescharacteristic
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#2765611yescharacteristic, complemented, Sylow
371C22×Q8C22×D5311yescharacteristic
372C23×C4—1061no—
373C23×C4C22×D5311yescharacteristic
374C22×C8—531no—
375C22×C8—531no—
376C22×C8—531no—
377C22×C8—531no—
378C22⋊Q8—22no—
379C22⋊Q8D21214yes—
380C4×Q8—22no—
381C4×Q8—22no—
382#117—22no—
383#117—22no—
384#117—1062no—
385#117—1062no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
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⋊C4C22⋊D5311yescharacteristic
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—1061no—
408C2×C22⋊C4C22⋊D5311yescharacteristic
409C2×C22⋊C4C22⋊D5311yescharacteristic
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—1061no—
413C2×C42C22×D5311yescharacteristic
414C2×C42C22×D5311yescharacteristic
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—1062no—
426C22⋊C8—1062no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436C24C2×C22⋊D5311yescharacteristic
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×D21212yes—
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—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—4241no—
456C22×C4—4241no—
457C22×C4—2122no—
458C22×C4C2×D21212yes—
459C22×C4#1171994512yes—
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C2×C22⋊D5311yescharacteristic
464C22×C4C2×C22⋊D5311yescharacteristic
465C22×C4C23×D5311yescharacteristic
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—81no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—4241no—
497C4⋊C4—4241no—
498C4⋊C4—4241no—
499C4⋊C4—4241no—
500C4⋊C4—2122no—
501C4⋊C4C2×D21212yes—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—81no—
506C22⋊C4—2122no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—81no—
511C42—2122no—
512C42#1171994712yes—
513C42#1171994712yes—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23#1313319112yes—
521C23C22×C22⋊D5311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
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—4244no—
555C2×C4—2121no—
556C2×C4—2121no—
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#1313319112yes—
568C2×C4#1313319812yes—
569C2×C4#1313319812yes—
570C2×C4C22×D21211yescharacteristic
571C2×C4C2×#1171994711yescharacteristic
572C2×C4C2×#1171994711yescharacteristic
573C2×C4C2×#1171994511yescharacteristic
574C2×C4#1313326411yescharacteristic
575C2×C4#1313326411yescharacteristic
576C8—2121no—
577C8—2121no—
578C8—2121no—
579C8—2121no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1313319111yescharacteristic
588C22C2×#1313319811yescharacteristic
589C22#1681119411yescharacteristic
590C22#1681142611yescharacteristic
591C22#1681143111yescharacteristic
592C22#1681143811yescharacteristic
593C22#1681143811yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—42no—
600C4—42no—
601C4—81no—
602C4—81no—
603C4—4241no—
604C4—4241no—
605C4—4241no—
606C4—4241no—
607C4#1681127711yescharacteristic
608C4#1681127711yescharacteristic
609C4#1681131311yescharacteristic
610C4#1681131311yescharacteristic
611C2—42no—
612C2#2027902311yescharacteristic
613C2#2028424511yescharacteristic
614C2#2028425411yescharacteristic
615C2#2028467811yescharacteristic
616C2#2028468911yescharacteristic
617C2#2028597711yescharacteristic
618C2#2028599211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2323C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2323D53
hypocenterC53#27656
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
4. #20276196C2semidirect
5. #20276186C2semidirect
6. #20276133C2semidirect
7. #20276123C2semidirect
8. #20275483C2semidirect
9. #20275473C2semidirect
12. #20274730C2semidirect
13. #20274720C2semidirect
370. C53#27656semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
C2×#168109612
#202773382
#202761962
#202761862
#202761332
#202761232
#202754832
#202754732
#202748072
#202747992
#202747302
#202747202
#202746992
C53×#23232
#2765653