Finite Group Databasebeta

← #27469653

Subgroups — #27469653

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280542C211yesmaximal, characteristic
2#20277336C211yesmaximal, characteristic
3#20276198C211yesmaximal, characteristic, complemented
4#20276196C211yesmaximal, characteristic, complemented
5#20276192C211yesmaximal, characteristic, complemented
6#20275686C211yesmaximal, characteristic, complemented
7#20275674C211yesmaximal, characteristic, complemented
8#20274917C211yesmaximal, characteristic
9#20274911C211yesmaximal, characteristic
10#20274909C211yesmaximal, characteristic
11#20274901C211yesmaximal, characteristic
12#20274882C211yesmaximal, characteristic, complemented
13#20274878C211yesmaximal, characteristic, complemented
14#20274480C211yesmaximal, characteristic
15C53×#2083C211yesmaximal, characteristic, complemented
16C2×#13133216C2211yescharacteristic
17C2×#13133175C2211yescharacteristic
18C2×#13133174C2211yescharacteristic
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133129C2211yescharacteristic
27C2×#13133129C2211yescharacteristic
28C2×#13133127C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30#16810564—21no—
31#16810558—21no—
32#16810543—21no—
33#16810536—21no—
34#16810521C2211yescharacteristic
35#16810518—21no—
36#16810518—21no—
37#16810494—21no—
38#16810494—21no—
39#16810492C2211yescharacteristic
40#16810489C2211yescharacteristic
41#16810488C2211yescharacteristic
42#16810488C2211yescharacteristic
43#16810480C2211yescharacteristic
44#16810480C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52C2×#107×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#384C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106—21no—
60C22×C4×D53—21no—
61C22×C4×D53C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133216—22no—
65#13133216—22no—
66#13133175—41no—
67#13133175—41no—
68#13133174—22no—
69#13133174—22no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919D411yescharacteristic
80C2×#11719919D411yescharacteristic
81C4⋊C4×D53—41no—
82C4⋊C4×D53—41no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921D411yescharacteristic
87C2×#11719921D411yescharacteristic
88C2×C4×Dic53—21no—
89C2×C4×Dic53C2311yescharacteristic
90#13133151—41no—
91#13133150—41no—
92#13133149—41no—
93#13133148—41no—
94#13133148—41no—
95#13133148—41no—
96#13133129Q814yescomplemented
97#13133129Q814yescomplemented
98#13133127—22no—
99C8×Dic53—22no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C42×C53C2311yescharacteristic
119#107×C53—41no—
120#107×C53—41no—
121#95×C53—21no—
122C23×D53C2×D411yescharacteristic
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2411yescharacteristic
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53C2×D411yescharacteristic
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C2×C4×D53—21no—
138C2×C4×D53—21no—
139C2×C4×D53—21no—
140C2×C4×D53—21no—
141C2×C4×D53—42no—
142C2×C4×D53—81no—
143C2×C4×D53—81no—
144C2×C4×D53C2×Q812yes—
145C2×C4×D53C2×Q812yes—
146C2×Dic106—42no—
147C2×Dic106—81no—
148C2×Dic106—81no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53C2×Q812yes—
154C2×C8×C53C2×Q812yes—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—81no—
164C4⋊C4×C53—81no—
165C4⋊C4×C53—81no—
166C42×C53—21no—
167C42×C53—21no—
168#11719921—22no—
169#11719921—22no—
170#11719921—42no—
171#11719921—81no—
172#11719921—81no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181#11719919—22no—
182#11719919—22no—
183#11719919—42no—
184#11719919—42no—
185#11719919—81no—
186#11719919—81no—
187#11719919—81no—
188C4×Dic53—42no—
189C4×Dic53#4112yes—
190C4×Dic53#4112yes—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322C2×Q812yes—
196C2×#88322C2×Q812yes—
197C23×C53C22×D411yescharacteristic
198C22×D53—21no—
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—21no—
202C22×D53—21no—
203C22×D53—21no—
204C22×D53C4⋊Q812yes—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
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×C53C22×Q811yescharacteristic
224C2×C4×C53C22×Q811yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53H(5,𝔽2)11yescharacteristic
228C2×C4×C53H(5,𝔽2)11yescharacteristic
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
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—41no—
246C2×Dic53—41no—
247C2×Dic53#12312yes—
248C2×Dic53#12312yes—
249C2×Dic53C4⋊Q812yes—
250C4×D53—22no—
251C4×D53—22no—
252C4×D53—22no—
253C4×D53—22no—
254C4×D53—84no—
255Dic106—84no—
256C8×C53—22no—
257C8×C53—22no—
258#88322—22no—
259#88322—22no—
260#27959—531nomaximal, Sylow
261C22×C53C2×#12311yescharacteristic
262C22×C53C2×C4⋊Q811yescharacteristic
263C22×C53#53511yescharacteristic
264C22×C53#57611yescharacteristic
265C22×C53#57711yescharacteristic
266C22×C53#57711yescharacteristic
267C22×C53#57811yescharacteristic
268C2×D53—22no—
269C2×D53—22no—
270C2×D53—22no—
271C2×D53—41no—
272C2×D53—41no—
273C2×D53—41no—
274C2×D53—41no—
275C4×C53—41no—
276C4×C53—41no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53#55411yescharacteristic
282C4×C53#55411yescharacteristic
283C4×C53#55411yescharacteristic
284C4×C53#55411yescharacteristic
285Dic53—22no—
286Dic53—22no—
287Dic53—22no—
288Dic53—81no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292#2628—531no—
293#2317—531no—
294#2085—531no—
295#2085—531no—
296#2084—531no—
297#2083D5311yescharacteristic, complemented
298#1993—531no—
299#1990—531no—
300#1854—531no—
301#1854—531no—
302#1853—531no—
303#1852—531no—
304#1847—531no—
305#1846—531no—
306#1750—531no—
307C2×C53#250211yescharacteristic
308C2×C53#309611yescharacteristic
309C2×C53#309711yescharacteristic
310C2×C53#316311yescharacteristic
311C2×C53#316411yescharacteristic
312C2×C53#322411yescharacteristic
313C2×C53#322511yescharacteristic
314D53—42no—
315C2×C4⋊Q8—531no—
316C2×#117—531no—
317C2×#117—531no—
318C22×C4⋊C4—531no—
319C2×#107—531no—
320C2×#107—531no—
321C2×#107C2×D5311yescharacteristic
322C2×#107C2×D5311yescharacteristic
323C2×#106—531no—
324C2×#106—531no—
325C2×#106—531no—
326C2×#106—531no—
327C2×C22⋊C8—531no—
328C2×C22⋊C8—531no—
329C2×#97—531no—
330C2×C4×C8—531no—
331#400—1061no—
332#399—1061no—
333#395—1061no—
334#393—1061no—
335#389—531no—
336#389—1061no—
337#389—1061no—
338#389C2×D5311yescharacteristic
339#385—1061no—
340#385—1061no—
341#384—531no—
342#384C2×D5311yescharacteristic
343#382—531no—
344#382—531no—
345#382—531no—
346#380—531no—
347#380—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340C2×D5311yescharacteristic
355#340C2×D5311yescharacteristic
356#336—531no—
357#336C2×D5311yescharacteristic
358C53#2795911yescharacteristic, complemented, Sylow
359C22×Q8—1061no—
360C23×C4—531no—
361C23×C4—1061no—
362C22×C8—531no—
363C22×C8—531no—
364C22×C8C22×D5311yescharacteristic
365C22×C8C22×D5311yescharacteristic
366C4⋊Q8—1062no—
367C4⋊Q8—1062no—
368#117—1062no—
369#117—1062no—
370#117—2121no—
371#117—2121no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—2121no—
387C2×C4⋊C4—2121no—
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C22⋊C4—531no—
393C2×C22⋊C4—531no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C22⋊C4—1061no—
397C2×C42—531no—
398C2×C42—1061no—
399C2×C42C22×D5311yescharacteristic
400#107—41no—
401#107—41no—
402#107—2121no—
403#107—2121no—
404#106—2121no—
405#106—2121no—
406#106—2121no—
407#106—2121no—
408C22⋊C8—534no—
409C22⋊C8—534no—
410#97—1062no—
411C4×C8—1062no—
412#95—21no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422C24—531no—
423C2×Q8—4241no—
424C2×Q8—4241no—
425C2×Q8—2122no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—531no—
433C22×C4—532no—
434C22×C4—532no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—4241no—
444C22×C4—4241no—
445C22×C4—2122no—
446C22×C4D4×D5311yescharacteristic
447C22×C4D4×D5311yescharacteristic
448C22×C4C23×D5311yescharacteristic
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—532no—
454C2×C8—532no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8Q8×D5312yes—
460C2×C8Q8×D5312yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—1062no—
473C4⋊C4—1062no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—4241no—
483C4⋊C4—4241no—
484C4⋊C4—4241no—
485C4⋊C4—2122no—
486C4⋊C4—2122no—
487C22⋊C4—1062no—
488C22⋊C4—1062no—
489C22⋊C4—4241no—
490C22⋊C4—4241no—
491C22⋊C4—2122no—
492C42—21no—
493C42—21no—
494C42—532no—
495C42—532no—
496C42—2122no—
497C23—532no—
498C23—1061no—
499C23—1061no—
500C23—1061no—
501C23—1061no—
502C23—1061no—
503C23—1061no—
504C23C2×D4×D5311yescharacteristic
505Q8—4244no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—532no—
525C2×C4—532no—
526C2×C4—532no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—1062no—
536C2×C4—1062no—
537C2×C4—4244no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4C2×Q8×D5311yescharacteristic
551C2×C4C2×Q8×D5311yescharacteristic
552C2×C4#41×D5311yescharacteristic
553C2×C4#41×D5311yescharacteristic
554C2×C4#1313326211yescharacteristic
555C2×C4#1313326311yescharacteristic
556C8—22no—
557C8—22no—
558C8—1062no—
559C8—1062no—
560C22—1062no—
561C22—1062no—
562C22—1062no—
563C22—2121no—
564C22—2121no—
565C22—2121no—
566C22—2121no—
567C22#123×D5311yescharacteristic
568C22C4⋊Q8×D5311yescharacteristic
569C22#1681118911yescharacteristic
570C22#1681142411yescharacteristic
571C22#1681143211yescharacteristic
572C22#1681143311yescharacteristic
573C22#1681143611yescharacteristic
574C4—41no—
575C4—41no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4—1062no—
581C4—1062no—
582C4—1062no—
583C4—4241no—
584C4—4241no—
585C4—4241no—
586C4—4241no—
587C4#1681132711yescharacteristic
588C4#1681132711yescharacteristic
589C4#1681132911yescharacteristic
590C4#1681132911yescharacteristic
591C2—2122no—
592C2#2027918811yescharacteristic
593C2#2028466611yescharacteristic
594C2#2028468311yescharacteristic
595C2#2028524211yescharacteristic
596C2#2028524811yescharacteristic
597C2#2028580611yescharacteristic
598C2#2028583111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2083C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2083D53
hypocenterC53#27959
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2083

Complements

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

NH ≅ G/Ntype
3. #20276198C2semidirect
4. #20276196C2semidirect
5. #20276192C2semidirect
6. #20275686C2semidirect
7. #20275674C2semidirect
12. #20274882C2semidirect
13. #20274878C2semidirect
15. C53×#2083C2semidirect
96. #13133129Q8semidirect
97. #13133129Q8semidirect
297. #2083D53semidirect
358. C53#27959semidirect

Maximal subgroups

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

subgroupindex
#202805422
#202773362
#202761982
#202761962
#202761922
#202756862
#202756742
#202749172
#202749112
#202749092
#202749012
#202748822
#202748782
#202744802
C53×#20832
#2795953