Finite Group Databasebeta

← #27489219

Subgroups — #27489219

Order 13568.

732 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283074C211yesmaximal, characteristic, complemented
2#20277180C211yesmaximal, characteristic
3#20276281C211yesmaximal, characteristic, complemented
4#20276246C211yesmaximal, characteristic
5#20276189C211yesmaximal, characteristic, complemented
6#20276121C211yesmaximal, characteristic
7#20275454C211yesmaximal, characteristic, complemented
8#20275452C211yesmaximal, characteristic, complemented
9#20275414C211yesmaximal, characteristic
10#20275409C211yesmaximal, characteristic
11#20275294C211yesmaximal, characteristic, complemented
12#20275290C211yesmaximal, characteristic, complemented
13#20274962C211yesmaximal, characteristic
14#20274462C211yesmaximal, characteristic, complemented
15C53×#1862C211yesmaximal, characteristic
16C2×#13133207—21no—
17C2×#13133176—21no—
18C2×#13133150C2211yescharacteristic
19C2×#13133149C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133127C2211yescharacteristic
24C2×#13133127C2211yescharacteristic
25#16810526C2211yescharacteristic
26#16810526C2211yescharacteristic
27#16810522C2211yescharacteristic
28#16810522C2211yescharacteristic
29#16810522C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810520—21no—
32#16810519C2211yescharacteristic
33#16810518—21no—
34#16810516—21no—
35#16810516—21no—
36#16810492C2211yescharacteristic
37#16810492C2211yescharacteristic
38#16810487—21no—
39C4×C4⋊Dic53C2211yescharacteristic
40C4×C22⋊Dic53—21no—
41C42×Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C4×D4×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61#13133207—44no—
62#13133176—44no—
63C2×C4⋊Dic53—21no—
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×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53D411yescharacteristic
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87#13133150—22no—
88#13133149—22no—
89#13133148—22no—
90#13133138—22no—
91#13133138—22no—
92#13133127—22no—
93#13133127—22no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100C22×D4×C53C2311yescharacteristic
101C23×C4×C53—21no—
102C22×C8×C53C2311yescharacteristic
103C22×C8×C53C2311yescharacteristic
104C4×D4×C53—22no—
105C4×D4×C53—22no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111C2×C42×C53C2311yescharacteristic
112#107×C53—22no—
113#102×C53—22no—
114#102×C53—22no—
115#95×C53—21no—
116C24×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53#4112yes—
123C22×C4×C53—24no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2411yescharacteristic
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53#4112yes—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—42no—
152C4⋊C4×C53#4112yes—
153C22⋊C4×C53—24no—
154C42×C53—21no—
155C42×C53—21no—
156C22⋊Dic53—42no—
157C22⋊Dic53—42no—
158C22⋊Dic53—42no—
159C22⋊Dic53—42no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—42no—
169C4⋊Dic53—42no—
170C4⋊Dic53—42no—
171#11719919—42no—
172#11719919—42no—
173#11719919—42no—
174#11719919—42no—
175#11719919—42no—
176#11719919—42no—
177C4×Dic53—22no—
178C4×Dic53—21no—
179C4×Dic53—21no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—22no—
183C4×Dic53—22no—
184C4×Dic53—41no—
185C4×Dic53—42no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322#4112yes—
192C23×C53—22no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53C22×D411yescharacteristic
200D4×C53—22no—
201D4×C53—22no—
202D4×C53—41no—
203D4×C53—41no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—22no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—48no—
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×C53AGL(1,ℤ/8)12yes—
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C532−1+411yescharacteristic
225C2×C4×C532−1+411yescharacteristic
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53AGL(1,ℤ/8)12yes—
249C2×Dic53#13712yes—
250C8×C53—22no—
251C8×C53—41no—
252#88322—22no—
253#88322—41no—
254#29580—531nomaximal, Sylow
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—81no—
259C22×C53—81no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53#52511yescharacteristic
263C22×C53#53211yescharacteristic
264C22×C53#53611yescharacteristic
265C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C53C2×#13711yescharacteristic
268C22×C53#57511yescharacteristic
269C4×C53—22no—
270C4×C53—42no—
271C4×C53—41no—
272C4×C53—81no—
273C4×C53—81no—
274C4×C53#55211yescharacteristic
275C4×C53#55211yescharacteristic
276C4×C53#55211yescharacteristic
277C4×C53#55211yescharacteristic
278Dic53—22no—
279Dic53—22no—
280Dic53—41no—
281Dic53—41no—
282Dic53—81no—
283Dic53—81no—
284Dic53—81no—
285Dic53—81no—
286#2859—531no—
287#2293—531no—
288#2100—531no—
289#2094—531no—
290#2083—531no—
291#2071—531no—
292#1943—531no—
293#1943—531no—
294#1936—531no—
295#1936—531no—
296#1919—531no—
297#1919—531no—
298#1862—531no—
299#1862D5311yescharacteristic
300#1745—531no—
301C2×C53—82no—
302C2×C53#261111yescharacteristic
303C2×C53#310911yescharacteristic
304C2×C53#311611yescharacteristic
305C2×C53#317111yescharacteristic
306C2×C53#318211yescharacteristic
307C2×C53#323411yescharacteristic
308C2×C53#323911yescharacteristic
309C2×#125—1061no—
310C2×C4×D4C2×D5311yescharacteristic
311C2×#117—1061no—
312C2×#107—531no—
313C2×#107C2×D5311yescharacteristic
314C2×#106—531no—
315C2×#106—531no—
316C2×#102—531no—
317C2×#102—531no—
318C2×#102C2×D5311yescharacteristic
319C2×#102C2×D5311yescharacteristic
320C2×#97—531no—
321C2×#97—531no—
322#390—531no—
323#390—531no—
324#389—531no—
325#389—531no—
326#389—531no—
327#389—531no—
328#389—531no—
329#389—1061no—
330#389—1061no—
331#389C2×D5311yescharacteristic
332#388—1061no—
333#388—1061no—
334#384—531no—
335#384—531no—
336#382—1061no—
337C4×C4⋊C4—531no—
338C4×C22⋊C4—1061no—
339C43—531no—
340#340—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340C2×D5311yescharacteristic
348#336—531no—
349#336—531no—
350#336—531no—
351#336C2×D5311yescharacteristic
352C53#2958011yescharacteristic, complemented, Sylow
353C22×D4C22×D5311yescharacteristic
354C23×C4—21no—
355C22×C8—531no—
356C22×C8—531no—
357C22×C8C22×D5311yescharacteristic
358C22×C8C22×D5311yescharacteristic
359#125—2124no—
360C4×D4—22no—
361C4×D4—22no—
362#117—2124no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—531no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1061no—
382C2×C22⋊C4—1061no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4—1061no—
385C2×C42—531no—
386C2×C42—531no—
387C2×C42—531no—
388C2×C42—531no—
389C2×C42—531no—
390C2×C42—531no—
391C2×C42—1061no—
392C2×C42C22×D5311yescharacteristic
393#107—22no—
394#107—1062no—
395#106—1062no—
396#106—1062no—
397#102—22no—
398#102—22no—
399#102—1062no—
400#102—1062no—
401#97—1062no—
402#97—1062no—
403#95—21no—
404#95—1061no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410C24—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4#1171994712yes—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—531no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4D4×D5311yescharacteristic
432C22×C4D4×D5311yescharacteristic
433C22×C4C23×D5311yescharacteristic
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—532no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8#1171994512yes—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—42no—
452C4⋊C4—1061no—
453C4⋊C4—1061no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—2122no—
461C4⋊C4—2122no—
462C4⋊C4—2122no—
463C4⋊C4—2122no—
464C4⋊C4—2122no—
465C4⋊C4—2122no—
466C4⋊C4—2122no—
467C4⋊C4—2122no—
468C4⋊C4—2122no—
469C4⋊C4#1171994712yes—
470C22⋊C4—24no—
471C22⋊C4—2122no—
472C22⋊C4—2122no—
473C22⋊C4—2122no—
474C22⋊C4—2122no—
475C42—21no—
476C42—21no—
477C42—1062no—
478C42—1062no—
479C42—1062no—
480C42—1061no—
481C42—1061no—
482C42—1061no—
483C42—1061no—
484C42—2121no—
485C42—2122no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23C2×D4×D5311yescharacteristic
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—48no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—532no—
514C2×C4—532no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4#1313324112yes—
538C2×C4#41×D5311yescharacteristic
539C2×C4C2×#1171994711yescharacteristic
540C2×C4C2×#1171994711yescharacteristic
541C2×C4C2×#1171994511yescharacteristic
542C2×C4#1313326411yescharacteristic
543C2×C4#1313326511yescharacteristic
544C8—22no—
545C8—41no—
546C8—1062no—
547C8—2121no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22#1681112911yescharacteristic
556C22#1681117011yescharacteristic
557C22#1681119611yescharacteristic
558C22AGL(1,ℤ/8)×D5311yescharacteristic
559C22C2×#1313324111yescharacteristic
560C22#137×D5311yescharacteristic
561C22#1681142011yescharacteristic
562C4—22no—
563C4—42no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—1062no—
568C4—1062no—
569C4—4241no—
570C4—4241no—
571C4—4241no—
572C4—4241no—
573C4—2121no—
574C4—2121no—
575C4#1681130711yescharacteristic
576C4#1681130811yescharacteristic
577C4#1681131011yescharacteristic
578C4#1681131311yescharacteristic
579C2—82no—
580C2#2028040911yescharacteristic
581C2#2028480111yescharacteristic
582C2#2028487011yescharacteristic
583C2#2028530811yescharacteristic
584C2#2028543011yescharacteristic
585C2#2028590311yescharacteristic
586C2#2028594411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1862C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#1862D53
hypocenterC53#29580
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
1. #20283074C2semidirect
3. #20276281C2semidirect
5. #20276189C2semidirect
7. #20275454C2semidirect
8. #20275452C2semidirect
11. #20275294C2semidirect
12. #20275290C2semidirect
14. #20274462C2semidirect
352. C53#29580semidirect

Maximal subgroups

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

subgroupindex
#202830742
#202771802
#202762812
#202762462
#202761892
#202761212
#202754542
#202754522
#202754142
#202754092
#202752942
#202752902
#202749622
#202744622
C53×#18622
#2958053