Finite Group Databasebeta

← #27430266

Subgroups — #27430266

Order 13568.

732 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#20280071C211yesmaximal, characteristic
2#20277627C211yesmaximal, characteristic, complemented
3#20276241C211yesmaximal, characteristic
4#20276194C211yesmaximal, characteristic, complemented
5#20276187C211yesmaximal, characteristic, complemented
6#20275653C211yesmaximal, characteristic, complemented
7#20275484C211yesmaximal, characteristic, complemented
8#20275339C211yesmaximal, characteristic
9#20275338C211yesmaximal, characteristic
10#20275308C211yesmaximal, characteristic, complemented
11#20275302C211yesmaximal, characteristic, complemented
12#20274880C211yesmaximal, characteristic, complemented
13#20274713C211yesmaximal, characteristic
14#20274542C211yesmaximal, characteristic
15C53×#2094C211yesmaximal, characteristic
16C2×C4×Dic106C2211yescharacteristic
17C2×#13133182—21no—
18C2×D4×Dic53C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133127C2211yescharacteristic
27#16810559—21no—
28#16810535—21no—
29#16810526C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810519—21no—
33#16810518—21no—
34#16810518—21no—
35#16810498—21no—
36#16810495—21no—
37#16810492C2211yescharacteristic
38#16810492C2211yescharacteristic
39#16810488C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×C4⋊Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#106×C53C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C53×#390C2211yescharacteristic
57C53×#384C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106D411yescharacteristic
60C23×Dic53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C4×Dic106—22no—
64C4×Dic106—22no—
65#13133182—44no—
66D4×Dic53—22no—
67D4×Dic53—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53D411yescharacteristic
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
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—41no—
91#13133150—41no—
92#13133148—41no—
93#13133146—22no—
94#13133138—41no—
95#13133138Q814yescomplemented
96#13133127—22no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C22×D4×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C2×C4⋊C4×C53—21no—
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—22no—
116#106×C53—41no—
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—21no—
135C22×Dic53—21no—
136C22×Dic53—42no—
137C22×Dic53—81no—
138C22×Dic53—81no—
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C2×Dic106—22no—
144C2×Dic106—22no—
145C2×Dic106—22no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53#4112yes—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—42no—
159C4⋊C4×C53—81no—
160C22⋊C4×C53—42no—
161C22⋊C4×C53—81no—
162C42×C53—21no—
163C42×C53—21no—
164C22⋊Dic53—42no—
165C22⋊Dic53—42no—
166C22⋊Dic53—81no—
167C4⋊Dic53—21no—
168C4⋊Dic53—22no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—81no—
176C4⋊Dic53C2×Q812yes—
177#11719919—22no—
178#11719919—22no—
179#11719919—42no—
180#11719919—42no—
181#11719919—81no—
182#11719919—81no—
183#11719919—81no—
184C4×Dic53—21no—
185C4×Dic53—21no—
186C4×Dic53—42no—
187C4×Dic53—42no—
188C4×Dic53—42no—
189C4×Dic53#4112yes—
190C4×Dic53#4112yes—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322C2×Q812yes—
197C23×C53—22no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53C22×D411yescharacteristic
205D4×C53—22no—
206D4×C53—22no—
207D4×C53—41no—
208D4×C53—41no—
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×C53—41no—
223C2×C4×C53#13712yes—
224C2×C4×C53C22×Q811yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53H(5,𝔽2)11yescharacteristic
229C2×C4×C532−1+411yescharacteristic
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—22no—
238C2×Dic53—21no—
239C2×Dic53—21no—
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—
251C2×Dic53—84no—
252C2×Dic53#12312yes—
253C2×Dic53#12312yes—
254C2×Dic53AGL(1,ℤ/8)12yes—
255Dic106—28no—
256C8×C53—22no—
257C8×C53—41no—
258#88322—22no—
259#88322—41no—
260#25068—531nomaximal, Sylow
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—81no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53C2×#12311yescharacteristic
269C22×C53D4×Q811yescharacteristic
270C22×C53#55011yescharacteristic
271C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C53C2×#13711yescharacteristic
273C22×C53#57611yescharacteristic
274C22×C53#57711yescharacteristic
275C4×C53—22no—
276C4×C53—41no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53#53311yescharacteristic
281C4×C53#54211yescharacteristic
282C4×C53#55211yescharacteristic
283C4×C53#55411yescharacteristic
284Dic53—22no—
285Dic53—22no—
286Dic53—22no—
287Dic53—41no—
288Dic53—42no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292#2580—531no—
293Q8×C4⋊C4—531no—
294#2094D5311yescharacteristic
295#2093—531no—
296#2084—531no—
297#2083—531no—
298#1985—531no—
299#1949—531no—
300#1926—531no—
301#1925—531no—
302#1921—531no—
303#1920—531no—
304#1847—531no—
305#1802—531no—
306#1762—531no—
307C2×C53—82no—
308C2×C53#242311yescharacteristic
309C2×C53#309211yescharacteristic
310C2×C53#309611yescharacteristic
311C2×C53#338411yescharacteristic
312C2×C53#338511yescharacteristic
313C2×C53#339911yescharacteristic
314C2×C53#340011yescharacteristic
315C2×C4×Q8—531no—
316C2×C4×Q8—1061no—
317C2×C4×D4—531no—
318C2×#107—531no—
319C2×#107—531no—
320C2×#107—531no—
321C2×#106—531no—
322C2×#106C2×D5311yescharacteristic
323C2×#106C2×D5311yescharacteristic
324C2×C4⋊C8—531no—
325C2×#102—531no—
326C2×#102—531no—
327C2×#102C2×D5311yescharacteristic
328C2×#102C2×D5311yescharacteristic
329C2×#97—531no—
330#399—1061no—
331#393—1061no—
332#390—531no—
333#390C2×D5311yescharacteristic
334#389—531no—
335#389—531no—
336#389—1061no—
337#389—1061no—
338#389—1061no—
339#385—1061no—
340#385—1061no—
341#384—531no—
342#384—531no—
343#384C2×D5311yescharacteristic
344#382—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#340—531no—
354#336—531no—
355#336—531no—
356#336—531no—
357#336C2×D5311yescharacteristic
358C53#2506811yescharacteristic, complemented, Sylow
359C22×Q8—531no—
360C22×D4C22×D5311yescharacteristic
361C23×C4—1061no—
362C22×C8—531no—
363C22×C8—531no—
364C22×C8C22×D5311yescharacteristic
365C22×C8C22×D5311yescharacteristic
366C4×Q8—1062no—
367C4×Q8—1062no—
368C4×Q8—2124no—
369C4×D4—1062no—
370C4×D4—1062no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C42—531no—
395C2×C42—531no—
396C2×C42—1061no—
397C2×C42—1061no—
398C2×C42—1061no—
399C2×C42C22×D5311yescharacteristic
400#107—1062no—
401#107—2121no—
402#107—2121no—
403#106—22no—
404#106—41no—
405#106—2121no—
406C4⋊C8—1062no—
407#102—22no—
408#102—41no—
409#102—534no—
410#102—2121no—
411#97—1062no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419C24—21no—
420C2×Q8—1062no—
421C2×Q8—1062no—
422C2×Q8—1062no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4Q8×D5312yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—531no—
433C22×C4—531no—
434C22×C4—531no—
435C22×C4—531no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—4241no—
441C22×C4—4241no—
442C22×C4—2122no—
443C22×C4D4×D5311yescharacteristic
444C22×C4D4×D5311yescharacteristic
445C22×C4C23×D5311yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—532no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8#1171994712yes—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—42no—
465C4⋊C4—81no—
466C4⋊C4—532no—
467C4⋊C4—1062no—
468C4⋊C4—1062no—
469C4⋊C4—1062no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—4241no—
478C4⋊C4—4241no—
479C4⋊C4—4241no—
480C4⋊C4—4241no—
481C4⋊C4—2122no—
482C4⋊C4—2122no—
483C22⋊C4—42no—
484C22⋊C4—81no—
485C22⋊C4—4241no—
486C22⋊C4—2122no—
487C22⋊C4—2122no—
488C42—21no—
489C42—21no—
490C42—532no—
491C42—532no—
492C42—1061no—
493C42—1061no—
494C42—2122no—
495C42—2122no—
496C42—2122no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23C2×D4×D5311yescharacteristic
505Q8—1068no—
506D4—22no—
507D4—22no—
508D4—41no—
509D4—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
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—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
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#1313324612yes—
550C2×C4C2×Q8×D5311yescharacteristic
551C2×C4#41×D5311yescharacteristic
552C2×C4#41×D5311yescharacteristic
553C2×C4C2×#1171994711yescharacteristic
554C2×C4#1313326211yescharacteristic
555C2×C4#1313326411yescharacteristic
556C8—22no—
557C8—41no—
558C8—1062no—
559C8—2121no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22#123×D5311yescharacteristic
568C22#1681128611yescharacteristic
569C22#1681129311yescharacteristic
570C22AGL(1,ℤ/8)×D5311yescharacteristic
571C22C2×#1313324611yescharacteristic
572C22#1681142711yescharacteristic
573C22#1681142911yescharacteristic
574C4—22no—
575C4—41no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—1062no—
580C4—1062no—
581C4—1062no—
582C4—4241no—
583C4—4241no—
584C4—4241no—
585C4—2121no—
586C4—2122no—
587C4#1681117511yescharacteristic
588C4#1681124111yescharacteristic
589C4#1681131011yescharacteristic
590C4#1681132711yescharacteristic
591C2—82no—
592C2#2027823311yescharacteristic
593C2#2028463711yescharacteristic
594C2#2028467211yescharacteristic
595C2#2028760911yescharacteristic
596C2#2028762911yescharacteristic
597C2#2028775411yescharacteristic
598C2#2028777311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2094C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2094D53
hypocenterC53#25068
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2094

Complements

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

NH ≅ G/Ntype
2. #20277627C2semidirect
4. #20276194C2semidirect
5. #20276187C2semidirect
6. #20275653C2semidirect
7. #20275484C2semidirect
10. #20275308C2semidirect
11. #20275302C2semidirect
12. #20274880C2semidirect
95. #13133138Q8semidirect
358. C53#25068semidirect

Maximal subgroups

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

subgroupindex
#202800712
#202776272
#202762412
#202761942
#202761872
#202756532
#202754842
#202753392
#202753382
#202753082
#202753022
#202748802
#202747132
#202745422
C53×#20942
#2506853