Finite Group Databasebeta

← #27465637

Subgroups — #27465637

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277397C211yesmaximal, characteristic
2#20277324C211yesmaximal, characteristic
3#20276148C211yesmaximal, characteristic
4#20276143C211yesmaximal, characteristic
5#20275912C211yesmaximal, characteristic
6#20275912C211yesmaximal, characteristic
7#20275485C211yesmaximal, characteristic, complemented
8#20275479C211yesmaximal, characteristic, complemented
9#20275357C211yesmaximal, characteristic, complemented
10#20275350C211yesmaximal, characteristic, complemented
11#20275301C211yesmaximal, characteristic, complemented
12#20275294C211yesmaximal, characteristic, complemented
13#20274731C211yesmaximal, characteristic, complemented
14#20274721C211yesmaximal, characteristic, complemented
15C53×#1798C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×#13133151C2211yescharacteristic
19C2×#13133149C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133142C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26#16810529C2211yescharacteristic
27#16810521C2211yescharacteristic
28#16810518—21no—
29#16810517—21no—
30#16810516—21no—
31#16810516—21no—
32#16810511—21no—
33#16810509—21no—
34#16810509—21no—
35#16810504—21no—
36#16810493C2211yescharacteristic
37#16810489C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810483C2211yescharacteristic
40#16810483C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42C4×C4⋊Dic53C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×C4⋊C8×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60#13133196—22no—
61#13133196—22no—
62C4×Dic106—22no—
63C4×Dic106—22no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
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×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84#13133151—41no—
85#13133149—22no—
86#13133149—22no—
87#13133149—41no—
88#13133142—22no—
89#13133142—22no—
90#13133142—41no—
91#13133142—41no—
92#13133124—21no—
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—
103C23×C4×C53C2311yescharacteristic
104C22×C8×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108#117×C53—22no—
109#117×C53—22no—
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53D411yescharacteristic
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C42×C53C2311yescharacteristic
115C2×C42×C53C2311yescharacteristic
116C4⋊C8×C53—22no—
117C4⋊C8×C53—22no—
118C22⋊C8×C53—41no—
119C22⋊C8×C53—41no—
120C24×C53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
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×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C2×Dic106—21no—
142C2×Dic106—21no—
143C2×Dic106—21no—
144C2×Dic106—21no—
145C2×Dic106—21no—
146C2×Dic106#4112yes—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—22no—
161C22⋊C4×C53—22no—
162C22⋊C4×C53—22no—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53#4112yes—
166C42×C53#4112yes—
167C22⋊Dic53—24no—
168C22⋊Dic53—81no—
169C22⋊Dic53—81no—
170C22⋊Dic53—81no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—81no—
185C4⋊Dic53—81no—
186C4⋊Dic53#4112yes—
187#11719919—24no—
188#11719919—42no—
189#11719919—81no—
190C4×Dic53—42no—
191C4×Dic53—81no—
192C4×Dic53—81no—
193C4×Dic53—81no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53C22×D411yescharacteristic
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
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—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53#12312yes—
220C2×C4×C53#12312yes—
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53H(5,𝔽2)11yescharacteristic
226C2×C4×C532−1+411yescharacteristic
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
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—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251Dic106—22no—
252Dic106—22no—
253Dic106—41no—
254Dic106—41no—
255C8×C53—41no—
256C8×C53—41no—
257C8×C53—41no—
258C8×C53—41no—
259#27638—531nomaximal, Sylow
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53C2×#12311yescharacteristic
268C22×C53#52511yescharacteristic
269C22×C53#53711yescharacteristic
270C22×C53#57711yescharacteristic
271C22×C53#57711yescharacteristic
272C22×C53#57711yescharacteristic
273C22×C53#57811yescharacteristic
274C4×C53—22no—
275C4×C53—22no—
276C4×C53—41no—
277C4×C53—41no—
278C4×C53—41no—
279C4×C53#55211yescharacteristic
280C4×C53#55311yescharacteristic
281C4×C53#55511yescharacteristic
282C4×C53#55611yescharacteristic
283Dic53—42no—
284Dic53—42no—
285Dic53—81no—
286Dic53—81no—
287Dic53—81no—
288Dic53—81no—
289Dic53—81no—
290Dic53—81no—
291#2323—531no—
292#2314—531no—
293#2077—531no—
294#2076—531no—
295#2035—531no—
296#2035—531no—
297#1949—531no—
298#1948—531no—
299#1928—531no—
300#1927—531no—
301#1920—531no—
302#1919—531no—
303#1805—531no—
304#1803—531no—
305#1798D5311yescharacteristic
306C2×C53—81no—
307C2×C53#248511yescharacteristic
308C2×C53#309711yescharacteristic
309C2×C53#309911yescharacteristic
310C2×C53#312011yescharacteristic
311C2×C53#312611yescharacteristic
312C2×C53#325811yescharacteristic
313C2×C53#326311yescharacteristic
314C2×C22⋊Q8—531no—
315C2×C4×Q8—531no—
316C2×#117C2×D5311yescharacteristic
317C2×#107—531no—
318C2×#106—531no—
319C2×#106—531no—
320C2×#106—531no—
321C2×C4⋊C8C2×D5311yescharacteristic
322C2×C4⋊C8C2×D5311yescharacteristic
323C2×#103—531no—
324C2×#103—531no—
325C2×#103—531no—
326C2×#103—531no—
327C2×C22⋊C8C2×D5311yescharacteristic
328C2×C22⋊C8C2×D5311yescharacteristic
329#391—531no—
330#389—531no—
331#389—1061no—
332#388—1061no—
333#388—1061no—
334#388—1061no—
335#388—1061no—
336#387—1061no—
337#387—1061no—
338#387—1061no—
339#384—531no—
340#382—531no—
341#380—531no—
342#380—531no—
343#380—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#336C2×D5311yescharacteristic
356#336C2×D5311yescharacteristic
357C53#2763811yescharacteristic, complemented, Sylow
358C22×Q8—531no—
359C23×C4C22×D5311yescharacteristic
360C22×C8C22×D5311yescharacteristic
361C22×C8C22×D5311yescharacteristic
362C22×C8C22×D5311yescharacteristic
363C22×C8C22×D5311yescharacteristic
364C22⋊Q8—1062no—
365C22⋊Q8—1062no—
366C4×Q8—1062no—
367C4×Q8—1062no—
368#117—22no—
369#117—22no—
370C2×C4⋊C4—531no—
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⋊C4D21211yescharacteristic
383C2×C4⋊C4D21211yescharacteristic
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4D21211yescharacteristic
389C2×C22⋊C4D21211yescharacteristic
390C2×C42—1061no—
391C2×C42—1061no—
392C2×C42—1061no—
393C2×C42—1061no—
394C2×C42C22×D5311yescharacteristic
395C2×C42C22×D5311yescharacteristic
396#107—2121no—
397#106—1062no—
398#106—1062no—
399#106—2121no—
400C4⋊C8—22no—
401C4⋊C8—22no—
402#103—1062no—
403#103—1062no—
404#103—2121no—
405#103—2121no—
406C22⋊C8—41no—
407C22⋊C8—41no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419C24C2×D21211yescharacteristic
420C2×Q8—532no—
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4C2×D21211yescharacteristic
441C22×C4C2×D21211yescharacteristic
442C22×C4C2×D21211yescharacteristic
443C22×C4C2×D21211yescharacteristic
444C22×C4C2×D21211yescharacteristic
445C22×C4C23×D5311yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—532no—
461C4⋊C4—1064no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
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—4241no—
476C4⋊C4—4241no—
477C4⋊C4—4241no—
478C4⋊C4—2122no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—1064no—
482C22⋊C4—4241no—
483C22⋊C4—4241no—
484C22⋊C4—4241no—
485C42—21no—
486C42—21no—
487C42—4241no—
488C42—4241no—
489C42—4241no—
490C42—2122no—
491C42#1171994712yes—
492C42#1171994712yes—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C22×D21211yescharacteristic
501Q8—1062no—
502Q8—1062no—
503Q8—2121no—
504Q8—2121no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—1062no—
523C2×C4—1062no—
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—2121no—
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×C4—2121no—
545C2×C4—2121no—
546C2×C4#1313320112yes—
547C2×C4#1313320112yes—
548C2×C4#41×D5311yescharacteristic
549C2×C4#41×D5311yescharacteristic
550C2×C4C2×#1171994711yescharacteristic
551C2×C4C2×#1171994711yescharacteristic
552C2×C4#1313326211yescharacteristic
553C2×C4#1313326511yescharacteristic
554C8—41no—
555C8—41no—
556C8—41no—
557C8—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22C2×#1313320111yescharacteristic
566C22#1681113511yescharacteristic
567C22#1681120411yescharacteristic
568C22#1681143511yescharacteristic
569C22#1681143511yescharacteristic
570C22#1681143511yescharacteristic
571C22#1681144011yescharacteristic
572C4—22no—
573C4—22no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—4241no—
578C4—4241no—
579C4—4241no—
580C4—4241no—
581C4—4241no—
582C4—4241no—
583C4—2122no—
584C4—2122no—
585C4#1681130711yescharacteristic
586C4#1681132011yescharacteristic
587C4#1681133711yescharacteristic
588C4#1681134411yescharacteristic
589C2—81no—
590C2#2027901311yescharacteristic
591C2#2028468511yescharacteristic
592C2#2028470611yescharacteristic
593C2#2028493111yescharacteristic
594C2#2028500711yescharacteristic
595C2#2028614311yescharacteristic
596C2#2028618611yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1798C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1798D53
hypocenterC53#27638
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1798

Complements

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

NH ≅ G/Ntype
7. #20275485C2semidirect
8. #20275479C2semidirect
9. #20275357C2semidirect
10. #20275350C2semidirect
11. #20275301C2semidirect
12. #20275294C2semidirect
13. #20274731C2semidirect
14. #20274721C2semidirect
357. C53#27638semidirect

Maximal subgroups

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

subgroupindex
#202773972
#202773242
#202761482
#202761432
#202759122
#202754852
#202754792
#202753572
#202753502
#202753012
#202752942
#202747312
#202747212
C53×#17982
#2763853