Finite Group Databasebeta

← #27442223

Subgroups — #27442223

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279747C211yesmaximal, characteristic, complemented
2#20277663C211yesmaximal, characteristic
3#20276103C211yesmaximal, characteristic
4#20276101C211yesmaximal, characteristic
5#20275859C211yesmaximal, characteristic
6#20275673C211yesmaximal, characteristic, complemented
7#20275669C211yesmaximal, characteristic, complemented
8#20275472C211yesmaximal, characteristic, complemented
9#20275432C211yesmaximal, characteristic, complemented
10#20275308C211yesmaximal, characteristic, complemented
11#20275305C211yesmaximal, characteristic, complemented
12#20274841C211yesmaximal, characteristic, complemented
13#20274713C211yesmaximal, characteristic
14#102×Dic53C211yesmaximal, characteristic
15C53×#2025C211yesmaximal, characteristic
16C2×#13133208C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×#13133173—21no—
19C2×#13133151C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×C8×Dic53C2211yescharacteristic
27#16810554—21no—
28#16810553—21no—
29#16810526C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810518—21no—
33#16810517—21no—
34#16810511—21no—
35#16810492—21no—
36#16810492C2211yescharacteristic
37#16810492C2211yescharacteristic
38#16810481—21no—
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4×#11719919C2211yescharacteristic
42C4×C4⋊Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#125×C53C2211yescharacteristic
53C2×#102×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C53×#390C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133208—22no—
63#13133208—41no—
64D4×Dic53—22no—
65D4×Dic53Q814yes—
66#13133173—44no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919D411yescharacteristic
80C2×#11719919D411yescharacteristic
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133151—22no—
90#13133150—22no—
91C4⋊#88322—41no—
92#13133146—22no—
93#13133146—41no—
94#13133138—22no—
95#13133138—41no—
96C8×Dic53—22no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124D411yescharacteristic
103#13133124D411yescharacteristic
104C22×D4×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107#125×C53—22no—
108#125×C53—41no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C42×C53C2311yescharacteristic
117#102×C53—22no—
118#102×C53—41no—
119C24×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53C2×Q812yes—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2411yescharacteristic
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—24no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—42no—
138C22×Dic53—42no—
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×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—42no—
155C4⋊C4×C53—42no—
156C4⋊C4×C53#4112yes—
157C22⋊C4×C53—42no—
158C22⋊C4×C53—81no—
159C42×C53—21no—
160C42×C53—21no—
161C22⋊Dic53—24no—
162C22⋊Dic53—42no—
163C22⋊Dic53—81no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—42no—
171C4⋊Dic53C2×Q812yes—
172C4⋊Dic53#4112yes—
173#11719919—22no—
174#11719919—22no—
175#11719919—42no—
176#11719919—42no—
177#11719919—42no—
178#11719919—81no—
179#11719919—81no—
180#11719919—81no—
181#11719919—81no—
182C4×Dic53—21no—
183C4×Dic53—21no—
184C4×Dic53—42no—
185C4×Dic53—42no—
186C4×Dic53—42no—
187C4×Dic53C2×Q812yes—
188C4×Dic53#4112yes—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C23×C53—22no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53C22×D411yescharacteristic
203D4×C53—22no—
204D4×C53—22no—
205D4×C53—41no—
206D4×C53—41no—
207C2×C4×C53—22no—
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—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×C53C22×Q811yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53H(5,𝔽2)11yescharacteristic
227C2×C4×C532−1+411yescharacteristic
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—22no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—22no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—48no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53C22⋊Q812yes—
250C2×Dic53C22⋊Q812yes—
251C2×Dic53#13512yes—
252C2×Dic53AGL(1,ℤ/8)12yes—
253C8×C53—41no—
254C8×C53—41no—
255#88322—41no—
256#88322—41no—
257#26025—531nomaximal, Sylow
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53C2×C22⋊Q811yescharacteristic
266C22×C53#54611yescharacteristic
267C22×C53#54811yescharacteristic
268C22×C53C2×#13511yescharacteristic
269C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C53#57611yescharacteristic
271C22×C53#57811yescharacteristic
272C4×C53—41no—
273C4×C53—42no—
274C4×C53—41no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53#53311yescharacteristic
278C4×C53#54411yescharacteristic
279C4×C53#55511yescharacteristic
280C4×C53#55611yescharacteristic
281Dic53—22no—
282Dic53—42no—
283Dic53—41no—
284Dic53—42no—
285Dic53—41no—
286Dic53—81no—
287Dic53—81no—
288Dic53#47814yes—
289#2552—531no—
290#2354—531no—
291#2067—531no—
292#2067—531no—
293#2025—531no—
294#2025D5311yescharacteristic
295#1989—531no—
296#1989—531no—
297#1947—531no—
298#1939—531no—
299#1921—531no—
300#1921—531no—
301#1834—531no—
302#1802—531no—
303C4×#102—531no—
304C2×C53—82no—
305C2×C53#243911yescharacteristic
306C2×C53C2×#47811yescharacteristic
307C2×C53#308311yescharacteristic
308C2×C53#330511yescharacteristic
309C2×C53#333011yescharacteristic
310C2×C53#336411yescharacteristic
311C2×C53#336611yescharacteristic
312C2×#125—531no—
313C2×#125C2×D5311yescharacteristic
314C2×C4×D4—531no—
315C2×#117—1061no—
316C2×#107—531no—
317C2×#107—531no—
318C2×C4⋊C8—531no—
319C2×C4⋊C8—531no—
320C2×C4⋊C8—531no—
321C2×#102—531no—
322C2×#102—531no—
323C2×#102C2×D5311yescharacteristic
324C2×#102C2×D5311yescharacteristic
325C2×C4×C8—531no—
326#398—1061no—
327#398—1061no—
328#390—531no—
329#390C2×D5311yescharacteristic
330#389—531no—
331#389—531no—
332#389—1061no—
333#388—1061no—
334#388—1061no—
335#384—531no—
336#384—531no—
337#384—1061no—
338#380—1061no—
339C4×C4⋊C4—531no—
340C4×C4⋊C4—531no—
341C4×C4⋊C4—531no—
342C4×C4⋊C4—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340C2×D5311yescharacteristic
352#340C2×D5311yescharacteristic
353#336—531no—
354#336C2×D5311yescharacteristic
355C53#2602511yescharacteristic, complemented, Sylow
356C22×D4C22×D5311yescharacteristic
357C23×C4—1061no—
358C22×C8—531no—
359C22×C8—531no—
360C22×C8C22×D5311yescharacteristic
361C22×C8C22×D5311yescharacteristic
362#125—22no—
363#125—41no—
364#125—1062no—
365#125—2121no—
366C4×D4—534no—
367C4×D4—1062no—
368#117—2124no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1061no—
393C2×C42—531no—
394C2×C42—531no—
395C2×C42—1061no—
396C2×C42—1061no—
397C2×C42—1061no—
398C2×C42C22×D5311yescharacteristic
399#107—1062no—
400#107—1062no—
401C4⋊C8—1062no—
402C4⋊C8—2121no—
403C4⋊C8—2121no—
404#102—22no—
405#102—41no—
406#102—1062no—
407#102—2121no—
408C4×C8—1062no—
409#95—531no—
410#95—531no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4Q8×D5312yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—531no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—531no—
430C22×C4—1064no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—2122no—
436C22×C4—2122no—
437C22×C4D4×D5311yescharacteristic
438C22×C4D4×D5311yescharacteristic
439C22×C4C23×D5311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—532no—
460C4⋊C4—532no—
461C4⋊C4—1062no—
462C4⋊C4—1062no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—4241no—
470C4⋊C4—4241no—
471C4⋊C4—4241no—
472C4⋊C4—4241no—
473C4⋊C4—2122no—
474C4⋊C4—2122no—
475C4⋊C4—2122no—
476C4⋊C4—2122no—
477C4⋊C4#1171994712yes—
478C22⋊C4—42no—
479C22⋊C4—81no—
480C22⋊C4—1064no—
481C22⋊C4—4241no—
482C22⋊C4—2122no—
483C42—21no—
484C42—21no—
485C42—532no—
486C42—532no—
487C42—1061no—
488C42—1061no—
489C42—2122no—
490C42—2122no—
491C42—2122no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23C2×D4×D5311yescharacteristic
500D4—22no—
501D4—22no—
502D4—41no—
503D4—41no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—532no—
520C2×C4—532no—
521C2×C4—532no—
522C2×C4—532no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1062no—
526C2×C4—1062no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—2128no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4C2×Q8×D5311yescharacteristic
545C2×C4#41×D5311yescharacteristic
546C2×C4#41×D5311yescharacteristic
547C2×C4C2×#1171994711yescharacteristic
548C2×C4#1313326211yescharacteristic
549C2×C4#1313326411yescharacteristic
550C8—41no—
551C8—41no—
552C8—2121no—
553C8—2121no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22C22⋊Q8×D5311yescharacteristic
562C22#1681126011yescharacteristic
563C22#1681128111yescharacteristic
564C22#135×D5311yescharacteristic
565C22AGL(1,ℤ/8)×D5311yescharacteristic
566C22#1681142711yescharacteristic
567C22#1681143711yescharacteristic
568C4—41no—
569C4—42no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—534no—
574C4—1062no—
575C4—4241no—
576C4—4241no—
577C4—2121no—
578C4—2122no—
579C4—2122no—
580C4—2121no—
581C4#1681117511yescharacteristic
582C4#1681125211yescharacteristic
583C4#1681133511yescharacteristic
584C4#1681134211yescharacteristic
585C2—82no—
586C2#2027839911yescharacteristic
587C2#478×D5311yescharacteristic
588C2#2028456711yescharacteristic
589C2#2028665811yescharacteristic
590C2#2028701311yescharacteristic
591C2#2028739711yescharacteristic
592C2#2028741911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2025C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2025D53
hypocenterC53#26025
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2025

Complements

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

NH ≅ G/Ntype
1. #20279747C2semidirect
6. #20275673C2semidirect
7. #20275669C2semidirect
8. #20275472C2semidirect
9. #20275432C2semidirect
10. #20275308C2semidirect
11. #20275305C2semidirect
12. #20274841C2semidirect
355. C53#26025semidirect

Maximal subgroups

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

subgroupindex
#202797472
#202776632
#202761032
#202761012
#202758592
#202756732
#202756692
#202754722
#202754322
#202753082
#202753052
#202748412
#202747132
#102×Dic532
C53×#20252
#2602553