Finite Group Databasebeta

← #27512596

Subgroups — #27512596

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279137C211yesmaximal, characteristic
2#20276282C211yesmaximal, characteristic, complemented
3#20276273C211yesmaximal, characteristic, complemented
4#20276216C211yesmaximal, characteristic, complemented
5#20276151C211yesmaximal, characteristic
6#20276146C211yesmaximal, characteristic
7#20276040C211yesmaximal, characteristic, complemented
8#20275677C211yesmaximal, characteristic, complemented
9#20275481C211yesmaximal, characteristic, complemented
10#20275440C211yesmaximal, characteristic, complemented
11#20275323C211yesmaximal, characteristic
12#20275304C211yesmaximal, characteristic, complemented
13#20274914C211yesmaximal, characteristic
14#20274810C211yesmaximal, characteristic
15C53×#2439C211yesmaximal, characteristic
16C2×#13133210—21no—
17C2×#13133150C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×C4×#88322C2211yescharacteristic
25#16810563—21no—
26#16810548—21no—
27#16810521C2211yescharacteristic
28#16810521C2211yescharacteristic
29#16810508—21no—
30#16810493C2211yescharacteristic
31#16810483C2211yescharacteristic
32#16810483C2211yescharacteristic
33C4×C4⋊Dic53C2211yescharacteristic
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
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×#117×C53C2211yescharacteristic
50C53×#396—21no—
51C53×#393—21no—
52C53×#391—21no—
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#387—21no—
56C53×#382C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133210—44no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—21no—
74C2×C4×Dic53—21no—
75#13133150—22no—
76#13133150—41no—
77#13133148—22no—
78#13133148—41no—
79C4⋊#88322—22no—
80C22⋊#88322—22no—
81C22⋊#88322—41no—
82C4×#88322—41no—
83#13133124—21no—
84#13133124—21no—
85#13133124—21no—
86#13133124—21no—
87#13133124—21no—
88C22×Q8×C53—21no—
89C23×C4×C53C2311yescharacteristic
90C4⋊Q8×C53—22no—
91C4⋊Q8×C53—41no—
92#117×C53—22no—
93#117×C53—41no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53D411yescharacteristic
104C2×C22⋊C4×C53D411yescharacteristic
105C2×C42×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C2×C42×C53C2311yescharacteristic
108#95×C53—21no—
109#95×C53—21no—
110#95×C53—21no—
111#95×C53—21no—
112#95×C53—21no—
113#95×C53D411yescharacteristic
114#95×C53D411yescharacteristic
115C24×C53C2×D411yescharacteristic
116C2×Q8×C53—42no—
117C2×Q8×C53—42no—
118C2×Q8×C53—81no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2411yescharacteristic
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—42no—
147C4⋊C4×C53—42no—
148C4⋊C4×C53—81no—
149C4⋊C4×C53#4112yes—
150C22⋊C4×C53—22no—
151C22⋊C4×C53—22no—
152C22⋊C4×C53—42no—
153C22⋊C4×C53—81no—
154C42×C53—21no—
155C42×C53—21no—
156C42×C53—21no—
157C42×C53—21no—
158C42×C53—42no—
159C22⋊Dic53—42no—
160C22⋊Dic53—42no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—42no—
170C4⋊Dic53—42no—
171#11719919—42no—
172#11719919—42no—
173C4×Dic53—42no—
174C2×#88322—21no—
175C2×#88322—21no—
176C2×#88322—21no—
177C2×#88322—21no—
178C2×#88322—21no—
179C2×#88322—21no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322#4112yes—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53C22×D411yescharacteristic
194Q8×C53—84no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—22no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53#13512yes—
225C2×C4×C53AGL(1,ℤ/8)12yes—
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53H(5,𝔽2)11yescharacteristic
229C2×C4×C532−1+411yescharacteristic
230C2×C4×C532−1+411yescharacteristic
231C2×C4×C532−1+411yescharacteristic
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244#88322—22no—
245#88322—41no—
246#88322—41no—
247#88322—41no—
248#31423—531nomaximal, Sylow
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53#53611yescharacteristic
257C22×C53#54611yescharacteristic
258C22×C53#54711yescharacteristic
259C22×C53C2×#13511yescharacteristic
260C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C53#57711yescharacteristic
262C22×C53#57811yescharacteristic
263C4×C53—22no—
264C4×C53—22no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53—42no—
269C4×C53—81no—
270C4×C53—81no—
271C4×C53—81no—
272C4×C53#53911yescharacteristic
273C4×C53#54111yescharacteristic
274C4×C53#55611yescharacteristic
275C4×C53#56311yescharacteristic
276Dic53—81no—
277Dic53—81no—
278Dic53—81no—
279Dic53—81no—
280#2496—531no—
281#2439D5311yescharacteristic
282#2101—531no—
283#2099—531no—
284#2088—531no—
285#2077—531no—
286#2076—531no—
287#2057—531no—
288#1990—531no—
289#1949—531no—
290#1941—531no—
291#1923—531no—
292#1921—531no—
293#1854—531no—
294#1825—531no—
295C2×C53—81no—
296C2×C53#268911yescharacteristic
297C2×C53#322811yescharacteristic
298C2×C53#324211yescharacteristic
299C2×C53#330011yescharacteristic
300C2×C53#330611yescharacteristic
301C2×C53#334011yescharacteristic
302C2×C53#335311yescharacteristic
303C2×C4⋊Q8C2×D5311yescharacteristic
304C2×#126—1061no—
305C2×#117C2×D5311yescharacteristic
306C2×#107—531no—
307C2×#107—531no—
308C2×#106—531no—
309C2×#106—531no—
310C2×C4⋊C8—531no—
311C2×C22⋊C8—531no—
312C2×C22⋊C8—531no—
313C2×C4×C8—531no—
314#400—1061no—
315#397—1061no—
316#396—21no—
317#393—21no—
318#391—21no—
319#389—531no—
320#389—531no—
321#389C2×D5311yescharacteristic
322#389C2×D5311yescharacteristic
323#387—21no—
324#387—1061no—
325#384—531no—
326#382C2×D5311yescharacteristic
327#380—531no—
328#380—531no—
329#380C2×D5311yescharacteristic
330C4×C4⋊C4—531no—
331C4×C22⋊C4C2×D5311yescharacteristic
332#340—531no—
333#340—531no—
334#340—531no—
335#340—531no—
336#340—531no—
337#340—531no—
338#340—531no—
339#340—531no—
340#340—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#336—531no—
345#336—531no—
346C53#3142311yescharacteristic, complemented, Sylow
347C22×Q8—21no—
348C23×C4C22×D5311yescharacteristic
349C22×C8—531no—
350C22×C8—531no—
351C22×C8—531no—
352C22×C8—531no—
353C4⋊Q8—22no—
354C4⋊Q8—41no—
355#126—2124no—
356#117—22no—
357#117—41no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4—531no—
363C2×C4⋊C4—531no—
364C2×C4⋊C4—531no—
365C2×C4⋊C4—1061no—
366C2×C4⋊C4—1061no—
367C2×C4⋊C4—1061no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4C22×D5311yescharacteristic
370C2×C4⋊C4C22×D5311yescharacteristic
371C2×C4⋊C4C22×D5311yescharacteristic
372C2×C4⋊C4C22×D5311yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—1061no—
376C2×C22⋊C4—1061no—
377C2×C22⋊C4C22⋊D5311yescharacteristic
378C2×C22⋊C4C22⋊D5311yescharacteristic
379C2×C42—21no—
380C2×C42—1061no—
381C2×C42C22×D5311yescharacteristic
382C2×C42C22×D5311yescharacteristic
383#107—1062no—
384#107—2121no—
385#106—1062no—
386#106—2121no—
387C4⋊C8—1062no—
388C22⋊C8—1062no—
389C22⋊C8—2121no—
390C4×C8—2121no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—1061no—
397#95—1061no—
398#95—1061no—
399#95—1061no—
400#95—1061no—
401#95C22⋊D5311yescharacteristic
402#95C22⋊D5311yescharacteristic
403C24C2×C22⋊D5311yescharacteristic
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—81no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—1061no—
418C22×C4—1061no—
419C22×C4—1061no—
420C22×C4—1061no—
421C22×C4C2×C22⋊D5311yescharacteristic
422C22×C4C2×C22⋊D5311yescharacteristic
423C22×C4C2×C22⋊D5311yescharacteristic
424C22×C4C2×C22⋊D5311yescharacteristic
425C22×C4C2×C22⋊D5311yescharacteristic
426C22×C4C23×D5311yescharacteristic
427C2×C8—532no—
428C2×C8—1061no—
429C2×C8—1061no—
430C2×C8—1061no—
431C2×C8—1061no—
432C2×C8—1061no—
433C2×C8—1061no—
434C2×C8—1061no—
435C2×C8—1061no—
436C2×C8—1061no—
437C2×C8—1061no—
438C2×C8—1061no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—81no—
449C4⋊C4—1061no—
450C4⋊C4—1061no—
451C4⋊C4—1061no—
452C4⋊C4—1061no—
453C4⋊C4—1061no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—2122no—
458C4⋊C4—2122no—
459C4⋊C4—2122no—
460C4⋊C4—2122no—
461C4⋊C4#1171994712yes—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—42no—
465C22⋊C4—81no—
466C22⋊C4—2122no—
467C22⋊C4—2122no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—42no—
473C42—2122no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23C22×C22⋊D5311yescharacteristic
482Q8—84no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4#1313323712yes—
525C2×C4#1313324212yes—
526C2×C4#41×D5311yescharacteristic
527C2×C4C2×#1171994711yescharacteristic
528C2×C4#1313326211yescharacteristic
529C2×C4#1313326411yescharacteristic
530C2×C4#1313326511yescharacteristic
531C2×C4#1313326511yescharacteristic
532C8—1062no—
533C8—2121no—
534C8—2121no—
535C8—2121no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22#1681119711yescharacteristic
544C22#1681126711yescharacteristic
545C22#1681127211yescharacteristic
546C22C2×#1313323711yescharacteristic
547C22C2×#1313324211yescharacteristic
548C22#1681143411yescharacteristic
549C22#1681143811yescharacteristic
550C4—22no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—4241no—
560C4—4241no—
561C4—4241no—
562C4—4241no—
563C4#1681121911yescharacteristic
564C4#1681123311yescharacteristic
565C4#1681134411yescharacteristic
566C4#1681137411yescharacteristic
567C2—81no—
568C2#2028116211yescharacteristic
569C2#2028585111yescharacteristic
570C2#2028598311yescharacteristic
571C2#2028660211yescharacteristic
572C2#2028667611yescharacteristic
573C2#2028712811yescharacteristic
574C2#2028729311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2439C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2439D53
hypocenterC53#31423
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2439

Complements

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

NH ≅ G/Ntype
2. #20276282C2semidirect
3. #20276273C2semidirect
4. #20276216C2semidirect
7. #20276040C2semidirect
8. #20275677C2semidirect
9. #20275481C2semidirect
10. #20275440C2semidirect
12. #20275304C2semidirect
346. C53#31423semidirect

Maximal subgroups

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

subgroupindex
#202791372
#202762822
#202762732
#202762162
#202761512
#202761462
#202760402
#202756772
#202754812
#202754402
#202753232
#202753042
#202749142
#202748102
C53×#24392
#3142353