Finite Group Databasebeta

← #27475277

Subgroups — #27475277

Order 13568.

918 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810991C211yesmaximal, characteristic, complemented
2C2×#16810930C211yesmaximal, characteristic, complemented
3C2×#16810825C211yesmaximal, characteristic
4C2×#16810721C211yesmaximal, characteristic
5#20279915C211yesmaximal, characteristic
6#20276263C211yesmaximal, characteristic, complemented
7#20276253C211yesmaximal, characteristic, complemented
8#20275798C211yesmaximal, characteristic
9#20275798C211yesmaximal, characteristic
10#20275599C211yesmaximal, characteristic
11#20275572C211yesmaximal, characteristic
12#20275536C211yesmaximal, characteristic, complemented
13#20275518C211yesmaximal, characteristic, complemented
14#20274590C211yesmaximal, characteristic, complemented
15C53×#2546C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C22×#11719924C2211yescharacteristic
18C22×#11719924C2211yescharacteristic
19C2×#13133209C2211yescharacteristic
20C2×C4×D212C2211yescharacteristic
21#16810991—24no—
22#16810930—24no—
23#16810825—24no—
24#16810721—24no—
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133141C2211yescharacteristic
32C2×#13133141C2211yescharacteristic
33C2×#13133141C2211yescharacteristic
34C2×#13133141C2211yescharacteristic
35C2×#13133141C2211yescharacteristic
36C2×#13133141C2211yescharacteristic
37C2×#13133141C2211yescharacteristic
38C2×#13133137C2211yescharacteristic
39C2×#13133128C2211yescharacteristic
40C2×#13133128C2211yescharacteristic
41C2×C4×#88322C2211yescharacteristic
42#16810545—22no—
43#16810525C2211yescharacteristic
44#16810525C2211yescharacteristic
45#16810497—22no—
46#16810493C2211yescharacteristic
47#16810489C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C4⋊Q8×C53C2211yescharacteristic
51C2×C4⋊Q8×C53C2211yescharacteristic
52C2×C4⋊Q8×C53C2211yescharacteristic
53C2×C4×Q8×C53C2211yescharacteristic
54C53×#398—22no—
55C53×#391C2211yescharacteristic
56C53×#391C2211yescharacteristic
57C53×#389—22no—
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C4×D53—21no—
61C2×#11719924—22no—
62C2×#11719924—22no—
63C2×#11719924—22no—
64C2×#11719924—22no—
65C2×#11719924—22no—
66C2×#11719924—22no—
67C2×#11719924—22no—
68C2×#11719924—41no—
69C2×#11719924—41no—
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73C22×#88322C2311yescharacteristic
74#13133209—22no—
75#13133209—22no—
76C4×D212—22no—
77C4×D212—22no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719921—22no—
85C2×#11719921—21no—
86C2×#11719921—22no—
87#13133150—22no—
88#13133150—22no—
89#13133148—22no—
90C4⋊#88322D414yes—
91#13133141—22no—
92#13133141—22no—
93#13133141—22no—
94#13133141—22no—
95#13133141—22no—
96#13133141—22no—
97#13133141—22no—
98#13133141—22no—
99#13133141—22no—
100#13133137—22no—
101#13133128—41no—
102#13133128D414yes—
103C4×#88322—41no—
104#13133124—22no—
105C22×Q8×C53C2311yescharacteristic
106C22×Q8×C53C2311yescharacteristic
107C22×Q8×C53C2311yescharacteristic
108C4⋊Q8×C53—22no—
109C4⋊Q8×C53—22no—
110C4⋊Q8×C53—22no—
111C4⋊Q8×C53—22no—
112C4⋊Q8×C53—41no—
113C4⋊Q8×C53—41no—
114C4×Q8×C53—22no—
115C4×Q8×C53—22no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53—22no—
119C2×C4⋊C4×C53—22no—
120C2×C4⋊C4×C53—21no—
121C2×C4⋊C4×C53D412yes—
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C42×C53—21no—
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126C2×C42×C53C2311yescharacteristic
127#95×C53—22no—
128#95×C53—22no—
129#95×C53D412yes—
130C23×D53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×Q8×C53#4112yes—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D412yes—
157C22×C4×C53C2411yescharacteristic
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C2×D212—21no—
162C2×D212—21no—
163C2×D212—21no—
164C2×D212—21no—
165C2×D212—21no—
166C2×D212#4112yes—
167C2×C4×D53—42no—
168C2×C4×D53—82no—
169C4⋊C4×C53—24no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—44no—
173C4⋊C4×C53—44no—
174C4⋊C4×C53—42no—
175C4⋊C4×C53—42no—
176C4⋊C4×C53—42no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—42no—
182C42×C53C2×D412yes—
183C42×C53C2×D412yes—
184#11719924—28no—
185#11719924—44no—
186#11719924—44no—
187#11719921—42no—
188#11719921—44no—
189#11719921—82no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—82no—
197#11719919—42no—
198#11719919—42no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322C2×D412yes—
208C2×#88322C2×D412yes—
209C2×#88322C2×D412yes—
210C2×#88322C2×D412yes—
211C23×C53C22×D411yescharacteristic
212C22×D53—22no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—42no—
216C22×D53—42no—
217Q8×C53—22no—
218Q8×C53—22no—
219Q8×C53—41no—
220Q8×C53—41no—
221Q8×C53—41no—
222Q8×C53—41no—
223Q8×C53—41no—
224Q8×C53—41no—
225Q8×C53—41no—
226Q8×C53—41no—
227Q8×C53—41no—
228Q8×C53—41no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
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—41no—
245C2×C4×C53—41no—
246C2×C4×C53—42no—
247C2×C4×C53#12712yes—
248C2×C4×C53#12712yes—
249C2×C4×C53#13714yes—
250C2×C4×C53C22×D411yescharacteristic
251C2×C4×C53C22×D411yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53H(5,𝔽2)11yescharacteristic
255C2×C4×C532−1+411yescharacteristic
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262D212—22no—
263D212—22no—
264D212—41no—
265D212—41no—
266C4×D53—84no—
267#88322—22no—
268#88322—22no—
269#88322—22no—
270#88322—22no—
271#28433—531nomaximal, Sylow
272C22×C53C2×#12711yescharacteristic
273C22×C53#54611yescharacteristic
274C22×C53#54811yescharacteristic
275C22×C53C2×#13712yes—
276C22×C53#57712yes—
277C2×D53—44no—
278C2×D53—42no—
279C2×D53—81no—
280C2×D53—81no—
281C2×D53—82no—
282C4×C53—22no—
283C4×C53—22no—
284C4×C53—24no—
285C4×C53—42no—
286C4×C53—42no—
287C4×C53—81no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53#52511yescharacteristic
291C4×C53#53711yescharacteristic
292C4×C53#54611yescharacteristic
293C4×C53#54811yescharacteristic
294Dic53—81no—
295Dic53—81no—
296Dic53—81no—
297C2×#491—531no—
298C2×#479—531no—
299C2×#461—531no—
300C2×#440—531no—
301#2566—531no—
302#2546D5311yescharacteristic, complemented
303#2097—531no—
304#2095—531no—
305#2013—531no—
306#2013—531no—
307#1973—531no—
308#1968—531no—
309#1960—531no—
310#1956—531no—
311#1771—531no—
312C2×C53#254011yescharacteristic
313C2×C53#315211yescharacteristic
314C2×C53#315411yescharacteristic
315C2×C53#332012yes—
316C2×C53#337212yes—
317D53—82no—
318C22×SD16—531no—
319C22×SD16—531no—
320C22×SD16—531no—
321C2×C4⋊Q8C2×D5311yescharacteristic
322C2×C4⋊Q8C2×D5311yescharacteristic
323C2×C4⋊Q8C2×D5311yescharacteristic
324C2×#125—531no—
325C2×C4×Q8C2×D5311yescharacteristic
326C2×C4×D4—531no—
327#491—1064no—
328#479—1064no—
329#461—1064no—
330#440—1064no—
331C2×#107—531no—
332C2×#107—531no—
333C2×#106—531no—
334C2×C4⋊C8—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#103—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#102—531no—
345C2×#97—531no—
346C2×#97—531no—
347C2×C4×C8—531no—
348#398—22no—
349#396—1062no—
350#391C2×D5311yescharacteristic
351#391C2×D5311yescharacteristic
352#390—531no—
353#390—531no—
354#389—22no—
355#385—1062no—
356#384—531no—
357#382—531no—
358C4×C4⋊C4C2×D5311yescharacteristic
359#336—531no—
360#336—531no—
361C53#2843311yescharacteristic, complemented, Sylow
362C22×Q8C22×D5311yescharacteristic
363C22×Q8C22×D5311yescharacteristic
364C22×Q8C22×D5311yescharacteristic
365C22×D4—531no—
366C23×C4—1061no—
367C2×SD16—1062no—
368C2×SD16—1062no—
369C2×SD16—1062no—
370C2×SD16—1062no—
371C2×SD16—1062no—
372C2×SD16—1062no—
373C2×SD16—1062no—
374C2×SD16—2121no—
375C2×SD16—2121no—
376C22×C8—531no—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8—531no—
380C4⋊Q8—22no—
381C4⋊Q8—22no—
382C4⋊Q8—22no—
383C4⋊Q8—22no—
384C4⋊Q8—41no—
385C4⋊Q8—41no—
386#125—1062no—
387#125—1062no—
388C4×Q8—22no—
389C4×Q8—22no—
390C4×D4—1062no—
391C4×D4—1062no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4C22⋊D5312yes—
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C22⋊C4—1062no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1062no—
408C2×C42—21no—
409C2×C42C22×D5311yescharacteristic
410C2×C42C22×D5311yescharacteristic
411C2×C42C22×D5311yescharacteristic
412#107—1062no—
413#107—1062no—
414#106—1062no—
415C4⋊C8—534no—
416#103—1062no—
417#103—1062no—
418#103—1062no—
419#103—1062no—
420#103—1062no—
421#103—1062no—
422#103—1062no—
423#103—1062no—
424#103—1062no—
425#102—1062no—
426#97—534no—
427#97—2121no—
428C4×C8—2121no—
429#95—22no—
430#95—22no—
431#95—1062no—
432#95C22⋊D5312yes—
433C24—1061no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8#1171994712yes—
452C2×D4—532no—
453C2×D4—1061no—
454C2×D4—1061no—
455C2×D4—1061no—
456C2×D4—1061no—
457C2×D4—1061no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—4242no—
466C22×C4—2122no—
467C22×C4C2×C22⋊D5312yes—
468C22×C4C2×C22⋊D5311yescharacteristic
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C2×C22⋊D5312yes—
471C22×C4C23×D5311yescharacteristic
472SD16—1068no—
473SD16—2124no—
474SD16—2124no—
475C2×C8—532no—
476C2×C8—532no—
477C2×C8—532no—
478C2×C8—532no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—4242no—
502C4⋊C4—2122no—
503C4⋊C4—2122no—
504C22⋊C4—4242no—
505C22⋊C4—2122no—
506C22⋊C4—2124no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42D4×D5312yes—
513C42D4×D5312yes—
514C23—1062no—
515C23—2121no—
516C23—2122no—
517C23—2121no—
518C23—2122no—
519C23C22×C22⋊D5311yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531Q8—41no—
532D4—1062no—
533D4—1062no—
534D4—2121no—
535D4—2121no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
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—42no—
554C2×C4—4244no—
555C2×C4—2122no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2122no—
560C2×C4—2122no—
561C2×C4#1313321312yes—
562C2×C4#1313321312yes—
563C2×C4#1313324814yes—
564C2×C4C2×D4×D5311yescharacteristic
565C2×C4C2×D4×D5311yescharacteristic
566C2×C4#41×D5311yescharacteristic
567C2×C4C2×#1171994711yescharacteristic
568C2×C4#1313326211yescharacteristic
569C2×C4#1313326411yescharacteristic
570C8—1062no—
571C8—1062no—
572C8—1062no—
573C8—1062no—
574C22—4241no—
575C22—4242no—
576C22—4241no—
577C22—2124no—
578C22—2122no—
579C22C2×#1313321311yescharacteristic
580C22#1681126711yescharacteristic
581C22#1681128011yescharacteristic
582C22C2×#1313324812yes—
583C22#1681143112yes—
584C4—22no—
585C4—24no—
586C4—22no—
587C4—42no—
588C4—42no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—4241no—
593C4—4241no—
594C4—4241no—
595C4#1681113311yescharacteristic
596C4#1681120211yescharacteristic
597C4#1681127011yescharacteristic
598C4#1681127911yescharacteristic
599C2—4242no—
600C2#2027965111yescharacteristic
601C2#2028517211yescharacteristic
602C2#2028519311yescharacteristic
603C2#2028688612yes—
604C2#2028749212yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2546C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2546D53
hypocenterC53#28433
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2546

Complements

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

NH ≅ G/Ntype
1. C2×#16810991C2semidirect
2. C2×#16810930C2semidirect
6. #20276263C2semidirect
7. #20276253C2semidirect
12. #20275536C2semidirect
13. #20275518C2semidirect
14. #20274590C2semidirect
15. C53×#2546C2semidirect
302. #2546D53semidirect
361. C53#28433semidirect

Maximal subgroups

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

subgroupindex
C2×#168109912
C2×#168109302
C2×#168108252
C2×#168107212
#202799152
#202762632
#202762532
#202757982
#202755992
#202755722
#202755362
#202755182
#202745902
C53×#25462
#2843353