Finite Group Databasebeta

← #27487197

Subgroups — #27487197

Order 13568.

724 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283079C211yesmaximal, characteristic
2#20277193C211yesmaximal, characteristic, complemented
3#20276215C211yesmaximal, characteristic, complemented
4#20276092C211yesmaximal, characteristic, complemented
5#20276085C211yesmaximal, characteristic
6#20276028C211yesmaximal, characteristic
7#20275571C211yesmaximal, characteristic
8#20275442C211yesmaximal, characteristic, complemented
9#20275400C211yesmaximal, characteristic
10#20275388C211yesmaximal, characteristic, complemented
11#20275339C211yesmaximal, characteristic
12#20275294C211yesmaximal, characteristic, complemented
13#20274983C211yesmaximal, characteristic, complemented
14#20274563C211yesmaximal, characteristic, complemented
15C53×#1880C211yesmaximal, characteristic
16C2×#13133210—21no—
17C2×#13133149C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133146C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×C8×Dic53C2211yescharacteristic
27C2×C8×Dic53C2211yescharacteristic
28#16810530C2211yescharacteristic
29#16810530C2211yescharacteristic
30#16810526C2211yescharacteristic
31#16810526C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810522C2211yescharacteristic
34#16810522C2211yescharacteristic
35#16810521C2211yescharacteristic
36#16810519C2211yescharacteristic
37#16810517—21no—
38#16810516—21no—
39#16810511—21no—
40#16810509—21no—
41#16810509—21no—
42C4×C4⋊Dic53—21no—
43C4×C4⋊Dic53C2211yescharacteristic
44#16810469—21no—
45C42×Dic53C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52C22×SD16×C53C2211yescharacteristic
53C2×C4×Q8×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C53×#390C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61#13133210—44no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×#11719919—21no—
70C2×#11719919—21no—
71C2×C22⋊Dic53—21no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53D411yescharacteristic
77C2×C4×Dic53D411yescharacteristic
78C2×C4×Dic53D411yescharacteristic
79C2×C4×Dic53D411yescharacteristic
80C2×C4×Dic53C2311yescharacteristic
81C2×C4×Dic53C2311yescharacteristic
82#13133149—22no—
83#13133148—22no—
84#13133148—41no—
85#13133146—22no—
86#13133146—22no—
87#13133141—22no—
88#13133141—22no—
89#13133138—22no—
90#13133138—41no—
91C8×Dic53—41no—
92C8×Dic53—41no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×Q8×C53C2311yescharacteristic
104C22×D4×C53C2311yescharacteristic
105C2×SD16×C53—22no—
106C2×SD16×C53—22no—
107C2×SD16×C53—41no—
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C4×Q8×C53—22no—
111C4×Q8×C53—22no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C42×C53—21no—
118C2×C42×C53C2311yescharacteristic
119#103×C53—41no—
120#102×C53—22no—
121C24×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53#4112yes—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148SD16×C53—44no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53#4112yes—
155C4⋊C4×C53—24no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53#4112yes—
160C22⋊C4×C53—42no—
161C22⋊C4×C53—42no—
162C42×C53—24no—
163C42×C53—21no—
164C42×C53—21no—
165C22⋊Dic53—42no—
166C22⋊Dic53—42no—
167C22⋊Dic53—42no—
168C22⋊Dic53—42no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—42no—
178C4⋊Dic53—42no—
179C4⋊Dic53—42no—
180#11719919—42no—
181#11719919—42no—
182C4×Dic53—21no—
183C4×Dic53—21no—
184C4×Dic53—21no—
185C4×Dic53—21no—
186C4×Dic53—22no—
187C4×Dic53—41no—
188C4×Dic53—41no—
189C4×Dic53—42no—
190C4×Dic53SD1614yescomplemented
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322#4112yes—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53C22×D411yescharacteristic
205Q8×C53—22no—
206Q8×C53—22no—
207Q8×C53—41no—
208Q8×C53—41no—
209D4×C53—41no—
210D4×C53—41no—
211D4×C53—41no—
212D4×C53—41no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53C2×SD1612yes—
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53H(5,𝔽2)11yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53C2×SD1612yes—
256C2×Dic53C2×SD1612yes—
257C2×Dic53#13512yes—
258C8×C53—22no—
259C8×C53—41no—
260#88322—22no—
261#88322—41no—
262#29434—531nomaximal, Sylow
263C22×C53—81no—
264C22×C53—81no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53#52511yescharacteristic
271C22×C53#53211yescharacteristic
272C22×C53#53711yescharacteristic
273C22×C53C22×SD1611yescharacteristic
274C22×C53C2×#13511yescharacteristic
275C22×C53#57511yescharacteristic
276C22×C53#57511yescharacteristic
277C4×C53—22no—
278C4×C53—41no—
279C4×C53—42no—
280C4×C53—42no—
281C4×C53—81no—
282C4×C53#55111yescharacteristic
283C4×C53#55211yescharacteristic
284C4×C53#55311yescharacteristic
285C4×C53#55611yescharacteristic
286Dic53—22no—
287Dic53—22no—
288Dic53—22no—
289Dic53—41no—
290Dic53—81no—
291Dic53—81no—
292Dic53—81no—
293Dic53—81no—
294#2860—531no—
295#2295—531no—
296#2088—531no—
297#2066—531no—
298#2065—531no—
299#2055—531no—
300#1968—531no—
301#1941—531no—
302#1934—531no—
303#1933—531no—
304#1926—531no—
305#1919—531no—
306#1880D5311yescharacteristic
307#1865—531no—
308C4×C4⋊C8—531no—
309C2×C53—161no—
310C2×C53#260111yescharacteristic
311C2×C53#310211yescharacteristic
312C2×C53#310311yescharacteristic
313C2×C53#316711yescharacteristic
314C2×C53#318311yescharacteristic
315C2×C53#324611yescharacteristic
316C2×C53#325011yescharacteristic
317C22×SD16C2×D5311yescharacteristic
318C2×#126—1061no—
319C2×C4×Q8C2×D5311yescharacteristic
320C2×#106—531no—
321C2×#106—531no—
322C2×#106—531no—
323C2×C4⋊C8—531no—
324C2×C4⋊C8—531no—
325C2×#103—531no—
326C2×#103—531no—
327C2×#103C2×D5311yescharacteristic
328C2×#102—531no—
329C2×#102—531no—
330C2×#102C2×D5311yescharacteristic
331C2×C4×C8—531no—
332C2×C4×C8—531no—
333#391—531no—
334#391—531no—
335#390—531no—
336#390—531no—
337#390C2×D5311yescharacteristic
338#389—531no—
339#389—531no—
340#389—531no—
341#389—531no—
342#389—531no—
343#388—1061no—
344#388—1061no—
345#388—1061no—
346#387—1061no—
347#387—1061no—
348C4×C4⋊C4—531no—
349C4×C4⋊C4—1061no—
350#376—1061no—
351C43—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340C2×D5311yescharacteristic
358#336—531no—
359#336C2×D5311yescharacteristic
360C53#2943411yescharacteristic, complemented, Sylow
361C22×Q8C22×D5311yescharacteristic
362C22×D4C22×D5311yescharacteristic
363C2×SD16—22no—
364C2×SD16—22no—
365C2×SD16—41no—
366C22×C8—531no—
367C22×C8—531no—
368C22×C8C22×D5311yescharacteristic
369C22×C8C22×D5311yescharacteristic
370#126—2124no—
371C4×Q8—22no—
372C4×Q8—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C42—21no—
392C2×C42—531no—
393C2×C42—531no—
394C2×C42—531no—
395C2×C42—531no—
396C2×C42—531no—
397C2×C42—531no—
398C2×C42—1061no—
399C2×C42C22×D5311yescharacteristic
400#106—1062no—
401#106—1062no—
402#106—2121no—
403C4⋊C8—1062no—
404C4⋊C8—1062no—
405#103—41no—
406#103—1062no—
407#103—1062no—
408#102—22no—
409#102—1062no—
410#102—2121no—
411C4×C8—2121no—
412C4×C8—2121no—
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—
422#95—1061no—
423C24—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8#1171994712yes—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4D4×D5311yescharacteristic
448C22×C4D4×D5311yescharacteristic
449C22×C4C23×D5311yescharacteristic
450SD16—44no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—532no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8#1171994512yes—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—2122no—
476C4⋊C4—2122no—
477C4⋊C4—2122no—
478C4⋊C4—2122no—
479C4⋊C4—2122no—
480C4⋊C4#1171994712yes—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—2122no—
484C22⋊C4—2122no—
485C22⋊C4—2122no—
486C22⋊C4—2122no—
487C42—24no—
488C42—21no—
489C42—21no—
490C42—534no—
491C42—1062no—
492C42—1061no—
493C42—1061no—
494C42—1061no—
495C42—1061no—
496C42—2121no—
497C42—2121no—
498C42—2122no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C2×D4×D5311yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—532no—
530C2×C4—532no—
531C2×C4—532no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
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×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4SD16×D5312yes—
554C2×C4#41×D5311yescharacteristic
555C2×C4C2×#1171994711yescharacteristic
556C2×C4C2×#1171994711yescharacteristic
557C2×C4C2×#1171994511yescharacteristic
558C2×C4#1313326211yescharacteristic
559C2×C4#1313326411yescharacteristic
560C8—22no—
561C8—41no—
562C8—1062no—
563C8—2121no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22#1681112911yescharacteristic
572C22#1681117011yescharacteristic
573C22#1681120211yescharacteristic
574C22C2×SD16×D5311yescharacteristic
575C22#135×D5311yescharacteristic
576C22#1681141911yescharacteristic
577C22#1681142211yescharacteristic
578C4—22no—
579C4—41no—
580C4—42no—
581C4—42no—
582C4—81no—
583C4—1062no—
584C4—1062no—
585C4—1062no—
586C4—4241no—
587C4—4241no—
588C4—4241no—
589C4—4241no—
590C4—2121no—
591C4#1681129911yescharacteristic
592C4#1681131011yescharacteristic
593C4#1681132011yescharacteristic
594C4#1681134311yescharacteristic
595C2—161no—
596C2#2028030811yescharacteristic
597C2#2028472611yescharacteristic
598C2#2028474111yescharacteristic
599C2#2028527111yescharacteristic
600C2#2028544011yescharacteristic
601C2#2028602911yescharacteristic
602C2#2028606511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1880C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#1880D53
hypocenterC53#29434
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
2. #20277193C2semidirect
3. #20276215C2semidirect
4. #20276092C2semidirect
8. #20275442C2semidirect
10. #20275388C2semidirect
12. #20275294C2semidirect
13. #20274983C2semidirect
14. #20274563C2semidirect
190. C4×Dic53SD16semidirect
360. C53#29434semidirect

Maximal subgroups

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

subgroupindex
#202830792
#202771932
#202762152
#202760922
#202760852
#202760282
#202755712
#202754422
#202754002
#202753882
#202753392
#202752942
#202749832
#202745632
C53×#18802
#2943453