Finite Group Databasebeta

← #27452734

Subgroups — #27452734

Order 13568.

732 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277784C211yesmaximal, characteristic
2#20276119C211yesmaximal, characteristic
3#20276086C211yesmaximal, characteristic
4#20275956C211yesmaximal, characteristic
5#20275956C211yesmaximal, characteristic
6#20275390C211yesmaximal, characteristic, complemented
7#20275349C211yesmaximal, characteristic, complemented
8#20275304C211yesmaximal, characteristic, complemented
9#20275297C211yesmaximal, characteristic, complemented
10#20275274C211yesmaximal, characteristic, complemented
11#20274968C211yesmaximal, characteristic, complemented
12#20274961C211yesmaximal, characteristic
13#20274717C211yesmaximal, characteristic, complemented
14C4×#13133137C211yesmaximal, characteristic
15C53×#2506C211yesmaximal, characteristic, complemented
16C2×#13133205C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133140C2211yescharacteristic
23C2×#13133140C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31#16810529C2211yescharacteristic
32#16810525C2211yescharacteristic
33#16810513—21no—
34#16810512—21no—
35#16810494—21no—
36C4×C4⋊Dic53C2211yescharacteristic
37#16810470—21no—
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#125×C53—21no—
49C53×#400—21no—
50C53×#398—21no—
51C53×#389C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#387—21no—
56C53×#384C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133205—22no—
67#13133205—22no—
68C4×Dic106—22no—
69C4×Dic106—41no—
70C4×D212—22no—
71C4×D212—41no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82#13133150—22no—
83#13133148—22no—
84C4⋊#88322—41no—
85#13133140—22no—
86#13133140—22no—
87#13133140—22no—
88#13133140—41no—
89#13133137—22no—
90#13133137—22no—
91#13133137—22no—
92#13133137—41no—
93C4×#88322—41no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#125×C53—44no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C42×C53—21no—
111C2×C42×C53C2311yescharacteristic
112C2×C42×C53C2311yescharacteristic
113C2×C42×C53C2311yescharacteristic
114#95×C53—21no—
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119#95×C53D411yescharacteristic
120#95×C53D411yescharacteristic
121C23×D53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212#4112yes—
142C2×C4×D53—42no—
143C2×C4×D53—42no—
144C2×C4×D53—81no—
145C2×Dic106—21no—
146C2×Dic106—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106#4112yes—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—42no—
161C4⋊C4×C53—42no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53—42no—
164C4⋊C4×C53—42no—
165C4⋊C4×C53—42no—
166C4⋊C4×C53#4112yes—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—42no—
174#11719921—24no—
175#11719921—24no—
176#11719921—42no—
177#11719921—42no—
178#11719921—81no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53#4112yes—
183#11719919—42no—
184C4×Dic53—42no—
185C4×Dic53—81no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C23×C53C22×D411yescharacteristic
199C22×D53—22no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
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—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—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53#13512yes—
235C2×C4×C53AGL(1,ℤ/8)12yes—
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53H(5,𝔽2)11yescharacteristic
241C2×C4×C532−1+411yescharacteristic
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251D212—22no—
252D212—22no—
253D212—41no—
254D212—41no—
255C4×D53—84no—
256Dic106—22no—
257Dic106—22no—
258Dic106—41no—
259Dic106—41no—
260#88322—41no—
261#88322—41no—
262#88322—41no—
263#88322—41no—
264#26761—531nomaximal, Sylow
265C22×C53#52511yescharacteristic
266C22×C53#54611yescharacteristic
267C22×C53#54811yescharacteristic
268C22×C53C2×#13511yescharacteristic
269C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C53#57711yescharacteristic
271C22×C53#57811yescharacteristic
272C2×D53—42no—
273C2×D53—42no—
274C2×D53—42no—
275C2×D53—81no—
276C2×D53—81no—
277C2×D53—81no—
278C2×D53—81no—
279C4×C53—22no—
280C4×C53—22no—
281C4×C53—41no—
282C4×C53—42no—
283C4×C53—41no—
284C4×C53—41no—
285C4×C53—41no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53—81no—
289C4×C53#52911yescharacteristic
290C4×C53#53911yescharacteristic
291C4×C53#55211yescharacteristic
292C4×C53#55311yescharacteristic
293Dic53—42no—
294Dic53—42no—
295Dic53—81no—
296#2506D5311yescharacteristic, complemented
297#2367—531no—
298#2071—531no—
299#2065—531no—
300#2042—531no—
301#2042—531no—
302#1933—531no—
303#1927—531no—
304#1921—531no—
305#1920—531no—
306#1916—531no—
307#1863—531no—
308#1862—531no—
309#1803—531no—
310C4×#102—531no—
311C2×C53#246011yescharacteristic
312C2×C53#310611yescharacteristic
313C2×C53#312011yescharacteristic
314C2×C53#330011yescharacteristic
315C2×C53#330611yescharacteristic
316C2×C53#336511yescharacteristic
317C2×C53#336711yescharacteristic
318D53—82no—
319C2×#125—21no—
320C2×#124—531no—
321C2×C4×Q8—531no—
322C2×C4×D4—531no—
323C2×#107—531no—
324C2×#106—531no—
325C2×C4⋊C8—531no—
326C2×#103—531no—
327C2×#103—531no—
328C2×#103—531no—
329C2×#103—531no—
330C2×#102—531no—
331C2×#102—531no—
332C2×#102—531no—
333C2×#102—531no—
334C2×C4×C8—531no—
335#400—21no—
336#398—21no—
337#391—531no—
338#390—531no—
339#389C2×D5311yescharacteristic
340#389C2×D5311yescharacteristic
341#389C2×D5311yescharacteristic
342#389C2×D5311yescharacteristic
343#388—1061no—
344#388—1061no—
345#387—21no—
346#385—1061no—
347#384C2×D5311yescharacteristic
348C4×C4⋊C4—531no—
349C4×C4⋊C4C2×D5311yescharacteristic
350C4×C4⋊C4C2×D5311yescharacteristic
351C4×C22⋊C4—1061no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#336—531no—
359#336—531no—
360#336—531no—
361#336—531no—
362C53#2676111yescharacteristic, complemented, Sylow
363C22×Q8—531no—
364C22×D4—531no—
365C23×C4—1061no—
366C22×C8—531no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8—531no—
370#125—44no—
371#124—1062no—
372#124—1062no—
373C4×Q8—1062no—
374C4×Q8—2121no—
375C4×D4—1062no—
376C4×D4—2121no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4C22⋊D5311yescharacteristic
387C2×C4⋊C4C22⋊D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C22⋊C4—1061no—
397C2×C42—21no—
398C2×C42—1061no—
399C2×C42—1061no—
400C2×C42C22×D5311yescharacteristic
401C2×C42C22×D5311yescharacteristic
402C2×C42C22×D5311yescharacteristic
403#107—1062no—
404#106—1062no—
405C4⋊C8—2121no—
406#103—1062no—
407#103—1062no—
408#103—1062no—
409#103—2121no—
410#102—1062no—
411#102—1062no—
412#102—1062no—
413#102—2121no—
414C4×C8—2121no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423#95C22⋊D5311yescharacteristic
424#95C22⋊D5311yescharacteristic
425C24—1061no—
426C2×Q8—532no—
427C2×Q8—1061no—
428C2×Q8—1061no—
429C2×Q8—1061no—
430C2×Q8—1061no—
431C2×Q8—1061no—
432C2×D4—532no—
433C2×D4—1061no—
434C2×D4—1061no—
435C2×D4—1061no—
436C2×D4—1061no—
437C2×D4—1061no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—4241no—
446C22×C4—2122no—
447C22×C4—2122no—
448C22×C4C2×C22⋊D5311yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450C22×C4C2×C22⋊D5311yescharacteristic
451C22×C4C2×C22⋊D5311yescharacteristic
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C23×D5311yescharacteristic
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—532no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—2122no—
487C4⋊C4#1171994912yes—
488C22⋊C4—1064no—
489C22⋊C4—1064no—
490C22⋊C4—4241no—
491C22⋊C4—2122no—
492C22⋊C4—2122no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—42no—
500C42—4241no—
501C42—2122no—
502C23—1062no—
503C23—2121no—
504C23—2121no—
505C23—2121no—
506C23—2121no—
507C23—2121no—
508C23—2121no—
509C23C22×C22⋊D5311yescharacteristic
510Q8—1062no—
511Q8—1062no—
512Q8—2121no—
513Q8—2121no—
514D4—1062no—
515D4—1062no—
516D4—2121no—
517D4—2121no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
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—41no—
545C2×C4—41no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—4244no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4#1313323712yes—
557C2×C4#1313324312yes—
558C2×C4#41×D5311yescharacteristic
559C2×C4#41×D5311yescharacteristic
560C2×C4#41×D5311yescharacteristic
561C2×C4C2×#1171994911yescharacteristic
562C2×C4#1313326311yescharacteristic
563C2×C4#1313326511yescharacteristic
564C8—2121no—
565C8—2121no—
566C8—2121no—
567C8—2121no—
568C22—4241no—
569C22—4241no—
570C22—4241no—
571C22—4241no—
572C22—2122no—
573C22—2122no—
574C22—2122no—
575C22#1681113011yescharacteristic
576C22#1681126111yescharacteristic
577C22#1681128411yescharacteristic
578C22C2×#1313323711yescharacteristic
579C22C2×#1313324311yescharacteristic
580C22#1681143111yescharacteristic
581C22#1681143811yescharacteristic
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—41no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—4241no—
593C4—2122no—
594C4—2122no—
595C4#1681115811yescharacteristic
596C4#1681122111yescharacteristic
597C4#1681130511yescharacteristic
598C4#1681132611yescharacteristic
599C2—4242no—
600C2#2027866311yescharacteristic
601C2#2028476911yescharacteristic
602C2#2028492311yescharacteristic
603C2#2028659611yescharacteristic
604C2#2028667011yescharacteristic
605C2#2028741111yescharacteristic
606C2#2028743311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2506C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2506D53
hypocenterC53#26761
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2506

Complements

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

NH ≅ G/Ntype
6. #20275390C2semidirect
7. #20275349C2semidirect
8. #20275304C2semidirect
9. #20275297C2semidirect
10. #20275274C2semidirect
11. #20274968C2semidirect
13. #20274717C2semidirect
15. C53×#2506C2semidirect
296. #2506D53semidirect
362. C53#26761semidirect

Maximal subgroups

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

subgroupindex
#202777842
#202761192
#202760862
#202759562
#202753902
#202753492
#202753042
#202752972
#202752742
#202749682
#202749612
#202747172
C4×#131331372
C53×#25062
#2676153