Finite Group Databasebeta

← #27484014

Subgroups — #27484014

Order 13568.

734 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
1#20277426C211yesmaximal, characteristic
2#20275863C211yesmaximal, characteristic, complemented
3#20275858C211yesmaximal, characteristic
4#20275853C211yesmaximal, characteristic, complemented
5#20275848C211yesmaximal, characteristic
6#20275529C211yesmaximal, characteristic
7#20275524C211yesmaximal, characteristic
8#20274848C211yesmaximal, characteristic, complemented
9#20274842C211yesmaximal, characteristic, complemented
10#20274727C211yesmaximal, characteristic, complemented
11#20274722C211yesmaximal, characteristic
12#20274717C211yesmaximal, characteristic, complemented
13#20274712C211yesmaximal, characteristic
14#20274572C211yesmaximal, characteristic, complemented
15C53×#2660C211yesmaximal, characteristic, complemented
16C2×#13133205C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133140C2211yescharacteristic
22C2×#13133140C2211yescharacteristic
23C2×#13133140C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133128C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31#16810529C2211yescharacteristic
32#16810525C2211yescharacteristic
33#16810515—21no—
34#16810515—21no—
35#16810497—21no—
36#16810497—21no—
37#16810493C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C4⋊Q8×C53C2211yescharacteristic
49C2×#125×C53C2211yescharacteristic
50C53×#398—21no—
51C53×#398—21no—
52C53×#395—21no—
53C53×#395—21no—
54C53×#384C2211yescharacteristic
55C53×#384C2211yescharacteristic
56C53×#382C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C53×#376C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133205—22no—
67#13133205Q814yescomplemented
68C4×Dic106—22no—
69C4×Dic106—22no—
70C4×D212—22no—
71C4×D212—22no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×C4×Dic53—21no—
83C4⋊#88322—22no—
84C4⋊#88322—22no—
85#13133140—41no—
86#13133140—41no—
87#13133140—41no—
88#13133140—41no—
89#13133137—41no—
90#13133137—41no—
91#13133137—41no—
92#13133137—41no—
93#13133128—22no—
94C4×#88322—22no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53—21no—
98C4⋊Q8×C53—41no—
99C4⋊Q8×C53—41no—
100#125×C53—41no—
101#125×C53—41no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115C2×C42×C53C2311yescharacteristic
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119#95×C53—21no—
120#95×C53D411yescharacteristic
121#95×C53D411yescharacteristic
122#95×C53D411yescharacteristic
123#95×C53D411yescharacteristic
124C23×D53—21no—
125C2×Q8×C53—42no—
126C2×Q8×C53—81no—
127C2×Q8×C53—81no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212C2×Q812yes—
148C2×C4×D53—42no—
149C2×C4×D53—81no—
150C2×C4×D53—81no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106C2×Q812yes—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—42no—
166C4⋊C4×C53—42no—
167C4⋊C4×C53—42no—
168C4⋊C4×C53—81no—
169C4⋊C4×C53—81no—
170C4⋊C4×C53—81no—
171C4⋊C4×C53—81no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53C2×Q812yes—
175C42×C53#4112yes—
176C42×C53#4112yes—
177C42×C53#4112yes—
178#11719921—24no—
179#11719921—24no—
180#11719921—42no—
181#11719921—81no—
182#11719921—81no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—81no—
188C4⋊Dic53—81no—
189#11719919—42no—
190C4×Dic53—42no—
191C2×#88322—21no—
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—
203C23×C53C22×D411yescharacteristic
204C22×D53—22no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211Q8×C53—84no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
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—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
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×C53C22⋊Q812yes—
239C2×C4×C53C22⋊Q812yes—
240C2×C4×C53#12412yes—
241C2×C4×C53#12412yes—
242C2×C4×C53C22×Q811yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53H(5,𝔽2)11yescharacteristic
247C2×C4×C53H(5,𝔽2)11yescharacteristic
248C2×Dic53—22no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257D212—22no—
258D212—22no—
259D212—41no—
260D212—41no—
261C4×D53—84no—
262Dic106—22no—
263Dic106—22no—
264Dic106—41no—
265Dic106—41no—
266#88322—41no—
267#88322—41no—
268#88322—41no—
269#88322—41no—
270#29151—531nomaximal, Sylow
271C22×C53C2×C22⋊Q811yescharacteristic
272C22×C53C2×#12411yescharacteristic
273C22×C53#53511yescharacteristic
274C22×C53#57611yescharacteristic
275C22×C53#57711yescharacteristic
276C22×C53#57711yescharacteristic
277C22×C53#57811yescharacteristic
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—42no—
281C2×D53—81no—
282C2×D53—81no—
283C2×D53—81no—
284C2×D53—81no—
285C4×C53—22no—
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—22no—
289C4×C53—41no—
290C4×C53—41no—
291C4×C53—81no—
292C4×C53—81no—
293C4×C53—81no—
294C4×C53—81no—
295C4×C53#55411yescharacteristic
296C4×C53#55411yescharacteristic
297C4×C53#55511yescharacteristic
298C4×C53#55511yescharacteristic
299Dic53—42no—
300Dic53—81no—
301Dic53—81no—
302#2660D5311yescharacteristic, complemented
303#2326—531no—
304#2026—531no—
305#2025—531no—
306#2024—531no—
307#2023—531no—
308#1959—531no—
309#1958—531no—
310#1836—531no—
311#1834—531no—
312#1805—531no—
313#1804—531no—
314#1803—531no—
315#1802—531no—
316#1768—531no—
317C2×C53#256311yescharacteristic
318C2×C53#308311yescharacteristic
319C2×C53#308411yescharacteristic
320C2×C53#313611yescharacteristic
321C2×C53#313711yescharacteristic
322C2×C53#322411yescharacteristic
323C2×C53#322511yescharacteristic
324D53—82no—
325C2×C4⋊Q8C2×D5311yescharacteristic
326C2×#125C2×D5311yescharacteristic
327C2×#124—531no—
328C2×C4×Q8—531no—
329C2×C4×D4—531no—
330C2×C4⋊C8—531no—
331C2×C4⋊C8—531no—
332C2×#103—531no—
333C2×#103—531no—
334C2×#103—531no—
335C2×#103—531no—
336C2×#102—531no—
337C2×#102—531no—
338C2×#102—531no—
339C2×#102—531no—
340C2×#97—531no—
341C2×C4×C8—531no—
342#398—21no—
343#398—21no—
344#395—21no—
345#395—21no—
346#391—531no—
347#390—531no—
348#388—1061no—
349#388—1061no—
350#385—1061no—
351#385—1061no—
352#384—531no—
353#384C2×D5311yescharacteristic
354#384C2×D5311yescharacteristic
355#382C2×D5311yescharacteristic
356#382C2×D5311yescharacteristic
357#376C2×D5311yescharacteristic
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#336—531no—
367#336—531no—
368C53#2915111yescharacteristic, complemented, Sylow
369C22×Q8—21no—
370C22×Q8—531no—
371C22×D4—531no—
372C23×C4—1061no—
373C22×C8—531no—
374C22×C8—531no—
375C22×C8—531no—
376C22×C8—531no—
377C4⋊Q8—41no—
378C4⋊Q8—41no—
379#125—41no—
380#125—41no—
381#124—534no—
382#124—1062no—
383C4×Q8—1062no—
384C4×Q8—1062no—
385C4×D4—1062no—
386C4×D4—1062no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C42—1061no—
409C2×C42C22×D5311yescharacteristic
410C2×C42C22×D5311yescharacteristic
411C2×C42C22×D5311yescharacteristic
412C4⋊C8—1062no—
413C4⋊C8—1062no—
414#103—2121no—
415#103—2121no—
416#103—2121no—
417#103—2121no—
418#102—2121no—
419#102—2121no—
420#102—2121no—
421#102—2121no—
422#97—1062no—
423C4×C8—1062no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1061no—
429#95—1061no—
430#95C22⋊D5311yescharacteristic
431#95C22⋊D5311yescharacteristic
432#95C22⋊D5311yescharacteristic
433#95C22⋊D5311yescharacteristic
434C24—1061no—
435C2×Q8—42no—
436C2×Q8—81no—
437C2×Q8—81no—
438C2×Q8—532no—
439C2×Q8—1061no—
440C2×Q8—1061no—
441C2×Q8—1061no—
442C2×Q8—1061no—
443C2×Q8—1061no—
444C2×D4—532no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—1061no—
448C2×D4—1061no—
449C2×D4—1061no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—4241no—
458C22×C4—4241no—
459C22×C4—2122no—
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C2×C22⋊D5311yescharacteristic
464C22×C4C2×C22⋊D5311yescharacteristic
465C22×C4C2×C22⋊D5311yescharacteristic
466C22×C4C23×D5311yescharacteristic
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—
478C2×C8—1061no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—4241no—
499C4⋊C4—4241no—
500C4⋊C4—2122no—
501C22⋊C4—1064no—
502C22⋊C4—1064no—
503C22⋊C4—4241no—
504C22⋊C4—4241no—
505C22⋊C4—2122no—
506C42—21no—
507C42—21no—
508C42—2122no—
509C42Q8×D5312yes—
510C42#1171994912yes—
511C42#1171994912yes—
512C42#1171994912yes—
513C23—1062no—
514C23—2121no—
515C23—2121no—
516C23—2121no—
517C23—2121no—
518C23—2121no—
519C23—2121no—
520C23C22×C22⋊D5311yescharacteristic
521Q8—84no—
522Q8—1062no—
523Q8—1062no—
524Q8—2121no—
525Q8—2121no—
526D4—1062no—
527D4—1062no—
528D4—2121no—
529D4—2121no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
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—41no—
555C2×C4—41no—
556C2×C4—1062no—
557C2×C4—4244no—
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#1313319312yes—
567C2×C4#1313319312yes—
568C2×C4#1313320612yes—
569C2×C4#1313320612yes—
570C2×C4C2×Q8×D5311yescharacteristic
571C2×C4C2×#1171994911yescharacteristic
572C2×C4C2×#1171994911yescharacteristic
573C2×C4C2×#1171994911yescharacteristic
574C2×C4#1313326211yescharacteristic
575C2×C4#1313326211yescharacteristic
576C8—2121no—
577C8—2121no—
578C8—2121no—
579C8—2121no—
580C22—4241no—
581C22—4241no—
582C22—4241no—
583C22—4241no—
584C22—2122no—
585C22—2122no—
586C22—2122no—
587C22C2×#1313319311yescharacteristic
588C22C2×#1313320611yescharacteristic
589C22#1681119011yescharacteristic
590C22#1681142611yescharacteristic
591C22#1681143111yescharacteristic
592C22#1681143411yescharacteristic
593C22#1681143811yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4—4241no—
605C4—4241no—
606C4—2122no—
607C4#1681133411yescharacteristic
608C4#1681133411yescharacteristic
609C4#1681133911yescharacteristic
610C4#1681133911yescharacteristic
611C2—4242no—
612C2#2027988911yescharacteristic
613C2#2028456211yescharacteristic
614C2#2028457311yescharacteristic
615C2#2028507411yescharacteristic
616C2#2028508511yescharacteristic
617C2#2028580911yescharacteristic
618C2#2028582411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2660C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2660D53
hypocenterC53#29151
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ #376 ◁ #2660

Complements

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

NH ≅ G/Ntype
2. #20275863C2semidirect
4. #20275853C2semidirect
8. #20274848C2semidirect
9. #20274842C2semidirect
10. #20274727C2semidirect
12. #20274717C2semidirect
14. #20274572C2semidirect
15. C53×#2660C2semidirect
67. #13133205Q8semidirect
302. #2660D53semidirect
368. C53#29151semidirect

Maximal subgroups

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

subgroupindex
#202774262
#202758632
#202758582
#202758532
#202758482
#202755292
#202755242
#202748482
#202748422
#202747272
#202747222
#202747172
#202747122
#202745722
C53×#26602
#2915153