Finite Group Databasebeta

← #27469765

Subgroups — #27469765

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280540C211yesmaximal, characteristic
2#20277339C211yesmaximal, characteristic
3#20276287C211yesmaximal, characteristic, complemented
4#20276286C211yesmaximal, characteristic, complemented
5#20276281C211yesmaximal, characteristic, complemented
6#20276275C211yesmaximal, characteristic, complemented
7#20275686C211yesmaximal, characteristic, complemented
8#20275674C211yesmaximal, characteristic, complemented
9#20274847C211yesmaximal, characteristic, complemented
10#20274844C211yesmaximal, characteristic, complemented
11#20274811C211yesmaximal, characteristic
12#20274807C211yesmaximal, characteristic
13#20274801C211yesmaximal, characteristic
14#20274483C211yesmaximal, characteristic
15C53×#1824C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×C22⋊#88322C2211yescharacteristic
20C2×#13133127C2211yescharacteristic
21C2×C8×Dic53C2211yescharacteristic
22#16810565—21no—
23#16810559—21no—
24#16810544—21no—
25#16810535—21no—
26#16810522C2211yescharacteristic
27#16810519—21no—
28#16810519—21no—
29#16810519C2211yescharacteristic
30#16810496—21no—
31#16810496—21no—
32#16810492C2211yescharacteristic
33#16810492C2211yescharacteristic
34#16810488C2211yescharacteristic
35#16810488C2211yescharacteristic
36#16810488C2211yescharacteristic
37#16810483C2211yescharacteristic
38#16810386C2211yescharacteristic
39#16810386C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C53×#380C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133216—22no—
63#13133216—22no—
64#13133176—22no—
65#13133176—22no—
66#13133176—22no—
67#13133176—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53D411yescharacteristic
70C2×C4⋊Dic53D411yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53D411yescharacteristic
84C2×C22⋊Dic53D411yescharacteristic
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88C22⋊#88322—22no—
89#13133127—22no—
90C8×Dic53—22no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100C23×C4×C53—21no—
101C23×C4×C53C2311yescharacteristic
102C22×C8×C53C2311yescharacteristic
103C22×C8×C53C2311yescharacteristic
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—22no—
106C2×C4⋊C4×C53—22no—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C22⋊C8×C53—22no—
113#95×C53—21no—
114C24×C53C2×D411yescharacteristic
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—42no—
124C22×C4×C53—81no—
125C22×C4×C53—81no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53#4112yes—
128C22×C4×C53#4112yes—
129C22×C4×C53C2411yescharacteristic
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53C2×D411yescharacteristic
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C2×Dic106—42no—
139C2×Dic106—81no—
140C2×Dic106—81no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—42no—
156C22⋊C4×C53—81no—
157C22⋊Dic53—22no—
158C22⋊Dic53—22no—
159C22⋊Dic53—42no—
160C22⋊Dic53—81no—
161C4⋊Dic53—22no—
162C4⋊Dic53—22no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—81no—
172#11719919—42no—
173#11719919—81no—
174#11719919—81no—
175#11719919—81no—
176#11719919—81no—
177#11719919—81no—
178C4×Dic53—21no—
179C4×Dic53—21no—
180C4×Dic53—42no—
181C4×Dic53#4112yes—
182C4×Dic53#4112yes—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53#12412yes—
196C23×C53C22×D411yescharacteristic
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
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—84no—
216C2×C4×C53#12412yes—
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C53C2×#4111yescharacteristic
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C532−1+411yescharacteristic
222C2×C4×C532−1+411yescharacteristic
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
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—
244C2×Dic53—41no—
245C2×Dic53#12312yes—
246C2×Dic53#12312yes—
247Dic106—84no—
248C8×C53—41no—
249C8×C53—41no—
250#88322—41no—
251#88322—41no—
252#27971—531nomaximal, Sylow
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53C2×#12311yescharacteristic
261C22×C53C2×#12411yescharacteristic
262C22×C53#53611yescharacteristic
263C22×C53#57611yescharacteristic
264C22×C53#57711yescharacteristic
265C22×C53#57711yescharacteristic
266C22×C53#57811yescharacteristic
267C4×C53—22no—
268C4×C53—81no—
269C4×C53—81no—
270C4×C53—81no—
271C4×C53—81no—
272C4×C53#55211yescharacteristic
273C4×C53#55211yescharacteristic
274C4×C53#55211yescharacteristic
275C4×C53#55211yescharacteristic
276Dic53—22no—
277Dic53—22no—
278Dic53—41no—
279Dic53—41no—
280Dic53—81no—
281Dic53—81no—
282Dic53—81no—
283Dic53—81no—
284#2628—531no—
285#2317—531no—
286#2101—531no—
287#2101—531no—
288#2100—531no—
289#2099—531no—
290#1993—531no—
291#1990—531no—
292#1836—531no—
293#1835—531no—
294#1825—531no—
295#1825—531no—
296#1824D5311yescharacteristic
297#1823—531no—
298#1750—531no—
299C2×C53—42no—
300C2×C53#250311yescharacteristic
301C2×C53#309611yescharacteristic
302C2×C53#309711yescharacteristic
303C2×C53#313611yescharacteristic
304C2×C53#313711yescharacteristic
305C2×C53#324311yescharacteristic
306C2×C53#324411yescharacteristic
307C2×C4⋊Q8—531no—
308C2×#117—531no—
309C2×#117—531no—
310C22×C4⋊C4C2×D5311yescharacteristic
311C2×C22⋊C8—531no—
312C2×C22⋊C8C2×D5311yescharacteristic
313C2×#97—531no—
314C2×C4×C8—531no—
315#400—1061no—
316#399—1061no—
317#395—1061no—
318#393—1061no—
319#389—531no—
320#389—531no—
321#389—1061no—
322#389—1061no—
323#385—1061no—
324#385—1061no—
325#384—531no—
326#384—531no—
327#382—531no—
328#382—531no—
329#382—531no—
330#380—531no—
331#380C2×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#340C2×D5311yescharacteristic
345#340C2×D5311yescharacteristic
346#340C2×D5311yescharacteristic
347#340C2×D5311yescharacteristic
348#336—531no—
349#336—531no—
350C53#2797111yescharacteristic, complemented, Sylow
351C22×Q8—1061no—
352C23×C4—21no—
353C23×C4C22×D5311yescharacteristic
354C22×C8—531no—
355C22×C8—531no—
356C22×C8C22×D5311yescharacteristic
357C22×C8C22×D5311yescharacteristic
358C4⋊Q8—1062no—
359C4⋊Q8—1062no—
360#117—1062no—
361#117—1062no—
362#117—1062no—
363#117—1062no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—531no—
386C2×C22⋊C4—531no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4—1061no—
389C2×C42—531no—
390C2×C42—531no—
391C2×C42—1061no—
392C22⋊C8—22no—
393C22⋊C8—1062no—
394#97—1062no—
395C4×C8—1062no—
396#95—21no—
397#95—1061no—
398#95—1061no—
399#95—1061no—
400#95—1061no—
401#95—1061no—
402#95—1061no—
403#95—1061no—
404#95—1061no—
405#95—1061no—
406C24D4×D5311yescharacteristic
407C2×Q8—4241no—
408C2×Q8—4241no—
409C2×Q8—2122no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—42no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—531no—
422C22×C4—531no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4#1171994512yes—
430C22×C4D4×D5311yescharacteristic
431C22×C4#1171994712yes—
432C22×C4C23×D5311yescharacteristic
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—42no—
454C4⋊C4—1062no—
455C4⋊C4—1062no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—4241no—
465C4⋊C4—4241no—
466C4⋊C4—4241no—
467C4⋊C4—4241no—
468C4⋊C4—4241no—
469C4⋊C4—4241no—
470C4⋊C4—2122no—
471C22⋊C4—81no—
472C22⋊C4—1062no—
473C22⋊C4—1062no—
474C22⋊C4—4241no—
475C22⋊C4—2122no—
476C42—532no—
477C42—532no—
478C42—1061no—
479C42—1061no—
480C42—2122no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23#1313320312yes—
488C23C2×D4×D5311yescharacteristic
489Q8—4244no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
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—84no—
509C2×C4—532no—
510C2×C4—532no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
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#1313320312yes—
534C2×C4#41×D5311yescharacteristic
535C2×C4#41×D5311yescharacteristic
536C2×C4C2×#1171994711yescharacteristic
537C2×C4C2×#1171994511yescharacteristic
538C2×C4#1313326411yescharacteristic
539C2×C4#1313326411yescharacteristic
540C8—41no—
541C8—41no—
542C8—2121no—
543C8—2121no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22#123×D5311yescharacteristic
552C22C2×#1313320311yescharacteristic
553C22#1681119511yescharacteristic
554C22#1681142711yescharacteristic
555C22#1681142911yescharacteristic
556C22#1681143311yescharacteristic
557C22#1681143611yescharacteristic
558C4—22no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4—1062no—
564C4—1062no—
565C4—4241no—
566C4—4241no—
567C4—4241no—
568C4—4241no—
569C4—2121no—
570C4—2121no—
571C4#1681130611yescharacteristic
572C4#1681130611yescharacteristic
573C4#1681131011yescharacteristic
574C4#1681131011yescharacteristic
575C2—42no—
576C2#2027919811yescharacteristic
577C2#2028467211yescharacteristic
578C2#2028468311yescharacteristic
579C2#2028506511yescharacteristic
580C2#2028507811yescharacteristic
581C2#2028600211yescharacteristic
582C2#2028600911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1824C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1824D53
hypocenterC53#27971
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
3. #20276287C2semidirect
4. #20276286C2semidirect
5. #20276281C2semidirect
6. #20276275C2semidirect
7. #20275686C2semidirect
8. #20275674C2semidirect
9. #20274847C2semidirect
10. #20274844C2semidirect
350. C53#27971semidirect

Maximal subgroups

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

subgroupindex
#202805402
#202773392
#202762872
#202762862
#202762812
#202762752
#202756862
#202756742
#202748472
#202748442
#202748112
#202748072
#202748012
#202744832
C53×#18242
#2797153