Finite Group Databasebeta

← #27443092

Subgroups — #27443092

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279749C211yesmaximal, characteristic, complemented
2#20277663C211yesmaximal, characteristic
3#20276248C211yesmaximal, characteristic
4#20276246C211yesmaximal, characteristic
5#20276087C211yesmaximal, characteristic
6#20275479C211yesmaximal, characteristic, complemented
7#20275477C211yesmaximal, characteristic, complemented
8#20275475C211yesmaximal, characteristic, complemented
9#20275430C211yesmaximal, characteristic, complemented
10#20275298C211yesmaximal, characteristic, complemented
11#20275297C211yesmaximal, characteristic, complemented
12#20274870C211yesmaximal, characteristic, complemented
13#20274713C211yesmaximal, characteristic
14#102×Dic53C211yesmaximal, characteristic
15C53×#2065C211yesmaximal, characteristic
16C2×#13133207—21no—
17C2×D4×Dic53C2211yescharacteristic
18C2×#13133173—21no—
19C2×#13133149C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29#16810557—21no—
30#16810553—21no—
31#16810526C2211yescharacteristic
32#16810520—21no—
33#16810519C2211yescharacteristic
34#16810517—21no—
35#16810511—21no—
36#16810493C2211yescharacteristic
37#16810492C2211yescharacteristic
38#16810492C2211yescharacteristic
39#16810492C2211yescharacteristic
40#16810481—21no—
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4×#11719919C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#106×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C53×#390C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133207—44no—
63D4×Dic53—22no—
64D4×Dic53Q814yes—
65#13133173—44no—
66C2×C4⋊Dic53—21no—
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#13133149—41no—
90#13133149—41no—
91#13133148—22no—
92#13133148—22no—
93#13133148—41no—
94#13133146—22no—
95#13133146—22no—
96#13133138—22no—
97#13133138—41no—
98C8×Dic53—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124D411yescharacteristic
104#13133124D411yescharacteristic
105C22×D4×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C42×C53C2311yescharacteristic
115#106×C53—41no—
116C4⋊C8×C53—22no—
117#102×C53—22no—
118#102×C53—41no—
119#95×C53—21no—
120C24×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53C2×Q812yes—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—21no—
134C22×Dic53—24no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53#4112yes—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—42no—
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—21no—
171C4⋊Dic53—42no—
172C4⋊Dic53—81no—
173C4⋊Dic53C2×Q812yes—
174#11719919—22no—
175#11719919—42no—
176#11719919—42no—
177#11719919—42no—
178#11719919—42no—
179#11719919—81no—
180#11719919—81no—
181#11719919—81no—
182#11719919SD1614yes—
183C4×Dic53—21no—
184C4×Dic53—21no—
185C4×Dic53—42no—
186C4×Dic53—42no—
187C4×Dic53—42no—
188C4×Dic53C2×Q812yes—
189C4×Dic53#4112yes—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322#4112yes—
196C23×C53—22no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53C22×D411yescharacteristic
204D4×C53—22no—
205D4×C53—22no—
206D4×C53—41no—
207D4×C53—41no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
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×C53C2×SD1612yes—
223C2×C4×C53C22×Q811yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53H(5,𝔽2)11yescharacteristic
228C2×C4×C532−1+411yescharacteristic
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—22no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—48no—
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×Dic53C22⋊Q812yes—
250C2×Dic53C22⋊Q812yes—
251C2×Dic53C2×SD1612yes—
252C2×Dic53C2×SD1612yes—
253C2×Dic53AGL(1,ℤ/8)12yes—
254C8×C53—22no—
255C8×C53—41no—
256#88322—22no—
257#88322—41no—
258#26090—531nomaximal, Sylow
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53—81no—
266C22×C53C2×C22⋊Q811yescharacteristic
267C22×C53#54711yescharacteristic
268C22×C53#55011yescharacteristic
269C22×C53C22×SD1611yescharacteristic
270C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C53#57711yescharacteristic
272C22×C53#57711yescharacteristic
273C4×C53—22no—
274C4×C53—41no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53#53311yescharacteristic
279C4×C53#54411yescharacteristic
280C4×C53#55211yescharacteristic
281C4×C53#55311yescharacteristic
282Dic53—22no—
283Dic53—22no—
284Dic53—41no—
285Dic53—42no—
286Dic53—81no—
287Dic53—81no—
288Dic53—81no—
289Dic53#47614yes—
290#2552—531no—
291#2354—531no—
292#2094—531no—
293#2094—531no—
294#2065—531no—
295#2065D5311yescharacteristic
296#1948—531no—
297#1948—531no—
298#1948—531no—
299#1939—531no—
300#1920—531no—
301#1920—531no—
302#1844—531no—
303#1802—531no—
304C4×#102—531no—
305C2×C53—82no—
306C2×C53#244111yescharacteristic
307C2×C53C2×#47611yescharacteristic
308C2×C53#308411yescharacteristic
309C2×C53#333511yescharacteristic
310C2×C53#335411yescharacteristic
311C2×C53#339211yescharacteristic
312C2×C53#339711yescharacteristic
313C2×#125—1061no—
314C2×C4×D4—531no—
315C2×#117—1061no—
316C2×#106—531no—
317C2×#106—531no—
318C2×#106—531no—
319C2×#106—531no—
320C2×#106—531no—
321C2×#106C2×D5311yescharacteristic
322C2×C4⋊C8—531no—
323C2×C4⋊C8—531no—
324C2×C4⋊C8C2×D5311yescharacteristic
325C2×#102—531no—
326C2×#102—531no—
327C2×#102C2×D5311yescharacteristic
328C2×#102C2×D5311yescharacteristic
329C2×C4×C8—531no—
330#398—1061no—
331#398—1061no—
332#390—531no—
333#390C2×D5311yescharacteristic
334#389—531no—
335#389—1061no—
336#389C2×D5311yescharacteristic
337#388—1061no—
338#388—1061no—
339#384—531no—
340#384—531no—
341#384—531no—
342#384—531no—
343#380—1061no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340C2×D5311yescharacteristic
354#336—531no—
355#336—531no—
356C53#2609011yescharacteristic, complemented, Sylow
357C22×D4C22×D5311yescharacteristic
358C23×C4—1061no—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8C22×D5311yescharacteristic
362C22×C8C22×D5311yescharacteristic
363#125—2124no—
364C4×D4—534no—
365C4×D4—1062no—
366#117—2124no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
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⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C42—531no—
392C2×C42—531no—
393C2×C42—1061no—
394C2×C42—1061no—
395C2×C42—1061no—
396C2×C42C22×D5311yescharacteristic
397#106—41no—
398#106—1062no—
399#106—1062no—
400#106—2121no—
401#106—2121no—
402#106—2121no—
403C4⋊C8—22no—
404C4⋊C8—1062no—
405C4⋊C8—1062no—
406#102—22no—
407#102—41no—
408#102—1062no—
409#102—2121no—
410C4×C8—1062no—
411#95—21no—
412#95—531no—
413#95—531no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418C24—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4Q8×D5312yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—531no—
429C22×C4—531no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—1064no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—2122no—
438C22×C4—2122no—
439C22×C4D4×D5311yescharacteristic
440C22×C4D4×D5311yescharacteristic
441C22×C4C23×D5311yescharacteristic
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—532no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8#1171994712yes—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—534no—
462C4⋊C4—532no—
463C4⋊C4—1062no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—4241no—
472C4⋊C4—4241no—
473C4⋊C4—4241no—
474C4⋊C4—4241no—
475C4⋊C4—2122no—
476C4⋊C4—2122no—
477C4⋊C4—2122no—
478C4⋊C4—2122no—
479C4⋊C4—2122no—
480C22⋊C4—42no—
481C22⋊C4—81no—
482C22⋊C4—1064no—
483C22⋊C4—4241no—
484C22⋊C4—2122no—
485C42—21no—
486C42—21no—
487C42—532no—
488C42—532no—
489C42—1061no—
490C42—1061no—
491C42—2122no—
492C42—2122no—
493C42—2122no—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23C2×D4×D5311yescharacteristic
502D4—22no—
503D4—22no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
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—41no—
520C2×C4—532no—
521C2×C4—532no—
522C2×C4—532no—
523C2×C4—532no—
524C2×C4—532no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1062no—
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×C4—2121no—
545C2×C4SD16×D5312yes—
546C2×C4C2×Q8×D5311yescharacteristic
547C2×C4#41×D5311yescharacteristic
548C2×C4#41×D5311yescharacteristic
549C2×C4C2×#1171994711yescharacteristic
550C2×C4#1313326211yescharacteristic
551C2×C4#1313326411yescharacteristic
552C8—22no—
553C8—41no—
554C8—1062no—
555C8—2121no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22C22⋊Q8×D5311yescharacteristic
564C22#1681127311yescharacteristic
565C22#1681129111yescharacteristic
566C22C2×SD16×D5311yescharacteristic
567C22AGL(1,ℤ/8)×D5311yescharacteristic
568C22#1681143011yescharacteristic
569C22#1681143311yescharacteristic
570C4—22no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—534no—
576C4—1062no—
577C4—1062no—
578C4—4241no—
579C4—4241no—
580C4—4241no—
581C4—2122no—
582C4—2121no—
583C4#1681117511yescharacteristic
584C4#1681125211yescharacteristic
585C4#1681130911yescharacteristic
586C4#1681131611yescharacteristic
587C2—82no—
588C2#2027842311yescharacteristic
589C2#476×D5311yescharacteristic
590C2#2028457811yescharacteristic
591C2#2028706711yescharacteristic
592C2#2028730511yescharacteristic
593C2#2028769511yescharacteristic
594C2#2028773811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2065C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2065D53
hypocenterC53#26090
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2065

Complements

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

NH ≅ G/Ntype
1. #20279749C2semidirect
6. #20275479C2semidirect
7. #20275477C2semidirect
8. #20275475C2semidirect
9. #20275430C2semidirect
10. #20275298C2semidirect
11. #20275297C2semidirect
12. #20274870C2semidirect
356. C53#26090semidirect

Maximal subgroups

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

subgroupindex
#202797492
#202776632
#202762482
#202762462
#202760872
#202754792
#202754772
#202754752
#202754302
#202752982
#202752972
#202748702
#202747132
#102×Dic532
C53×#20652
#2609053