Finite Group Databasebeta

← #27429388

Subgroups — #27429388

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280071C211yesmaximal, characteristic
2#20277627C211yesmaximal, characteristic, complemented
3#20276277C211yesmaximal, characteristic, complemented
4#20276276C211yesmaximal, characteristic, complemented
5#20276121C211yesmaximal, characteristic
6#20275653C211yesmaximal, characteristic, complemented
7#20275486C211yesmaximal, characteristic, complemented
8#20275339C211yesmaximal, characteristic
9#20275337C211yesmaximal, characteristic
10#20275308C211yesmaximal, characteristic, complemented
11#20275303C211yesmaximal, characteristic, complemented
12#20274847C211yesmaximal, characteristic, complemented
13#20274713C211yesmaximal, characteristic
14#20274542C211yesmaximal, characteristic
15C53×#2071C211yesmaximal, characteristic
16C2×C4×Dic106C2211yescharacteristic
17C2×#13133182—21no—
18C2×D4×Dic53C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133127C2211yescharacteristic
25#16810559—21no—
26#16810535—21no—
27#16810526C2211yescharacteristic
28#16810522C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810519—21no—
31#16810518—21no—
32#16810518—21no—
33#16810498—21no—
34#16810495—21no—
35#16810492C2211yescharacteristic
36#16810492C2211yescharacteristic
37#16810488C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#102×C53C2211yescharacteristic
53C2×#102×C53C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#384C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
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#13133148—22no—
91#13133146—22no—
92#13133138—22no—
93#13133138—22no—
94#13133127—22no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102C22×D4×C53C2311yescharacteristic
103C22×C8×C53C2311yescharacteristic
104C22×C8×C53C2311yescharacteristic
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C42×C53C2311yescharacteristic
113#102×C53—22no—
114#102×C53—22no—
115C24×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53#4112yes—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2411yescharacteristic
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—42no—
133C22×Dic53—81no—
134C22×Dic53—81no—
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C2×Dic106—22no—
140C2×Dic106—22no—
141C2×Dic106—22no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C4⋊C4×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—81no—
156C22⋊C4×C53—42no—
157C22⋊C4×C53—81no—
158C42×C53—21no—
159C42×C53—21no—
160C22⋊Dic53—42no—
161C22⋊Dic53—42no—
162C22⋊Dic53—81no—
163C4⋊Dic53—21no—
164C4⋊Dic53—22no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—81no—
172C4⋊Dic53#4112yes—
173#11719919—22no—
174#11719919—22no—
175#11719919—42no—
176#11719919—42no—
177#11719919—81no—
178#11719919—81no—
179#11719919—81no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—42no—
183C4×Dic53—42no—
184C4×Dic53—42no—
185C4×Dic53#4112yes—
186C4×Dic53#4112yes—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C23×C53—22no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53C22×D411yescharacteristic
201D4×C53—22no—
202D4×C53—22no—
203D4×C53—41no—
204D4×C53—41no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
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#13712yes—
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C532−1+411yescharacteristic
225C2×C4×C532−1+411yescharacteristic
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—22no—
234C2×Dic53—21no—
235C2×Dic53—21no—
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—84no—
248C2×Dic53#12312yes—
249C2×Dic53#12312yes—
250C2×Dic53AGL(1,ℤ/8)12yes—
251Dic106—28no—
252C8×C53—41no—
253C8×C53—41no—
254#88322—41no—
255#88322—41no—
256#25002—531nomaximal, Sylow
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53C2×#12311yescharacteristic
265C22×C53#54811yescharacteristic
266C22×C53#54811yescharacteristic
267C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C53C2×#13711yescharacteristic
269C22×C53#57611yescharacteristic
270C22×C53#57711yescharacteristic
271C4×C53—22no—
272C4×C53—41no—
273C4×C53—81no—
274C4×C53—81no—
275C4×C53—81no—
276C4×C53#52911yescharacteristic
277C4×C53#54111yescharacteristic
278C4×C53#55211yescharacteristic
279C4×C53#55611yescharacteristic
280Dic53—22no—
281Dic53—22no—
282Dic53—22no—
283Dic53—41no—
284Dic53—42no—
285Dic53—81no—
286Dic53—81no—
287Dic53—81no—
288#2580—531no—
289Q8×C4⋊C4—531no—
290#2100—531no—
291#2099—531no—
292#2071—531no—
293#2071D5311yescharacteristic
294#1985—531no—
295#1949—531no—
296#1926—531no—
297#1925—531no—
298#1921—531no—
299#1920—531no—
300#1836—531no—
301#1802—531no—
302#1762—531no—
303C2×C53—82no—
304C2×C53#242111yescharacteristic
305C2×C53#309211yescharacteristic
306C2×C53#309611yescharacteristic
307C2×C53#336611yescharacteristic
308C2×C53#337211yescharacteristic
309C2×C53#337311yescharacteristic
310C2×C53#337511yescharacteristic
311C2×C4×Q8—531no—
312C2×C4×Q8—1061no—
313C2×C4×D4—531no—
314C2×#107—531no—
315C2×#106—531no—
316C2×C4⋊C8—531no—
317C2×#102—531no—
318C2×#102—531no—
319C2×#102C2×D5311yescharacteristic
320C2×#102C2×D5311yescharacteristic
321C2×#97—531no—
322#399—1061no—
323#393—1061no—
324#390—531no—
325#390C2×D5311yescharacteristic
326#389—531no—
327#389—531no—
328#389—1061no—
329#389—1061no—
330#389—1061no—
331#385—1061no—
332#385—1061no—
333#384—531no—
334#384—531no—
335#384C2×D5311yescharacteristic
336#382—531no—
337C4×C4⋊C4—531no—
338C4×C4⋊C4—531no—
339C4×C4⋊C4—531no—
340#340—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340C2×D5311yescharacteristic
349#340C2×D5311yescharacteristic
350#336—531no—
351#336—531no—
352#336—531no—
353#336C2×D5311yescharacteristic
354C53#2500211yescharacteristic, complemented, Sylow
355C22×Q8—531no—
356C22×D4C22×D5311yescharacteristic
357C23×C4—1061no—
358C22×C8—531no—
359C22×C8—531no—
360C22×C8C22×D5311yescharacteristic
361C22×C8C22×D5311yescharacteristic
362C4×Q8—1062no—
363C4×Q8—1062no—
364C4×Q8—2124no—
365C4×D4—1062no—
366C4×D4—1062no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—531no—
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—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4C22×D5311yescharacteristic
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×C42—531no—
391C2×C42—531no—
392C2×C42—1061no—
393C2×C42—1061no—
394C2×C42—1061no—
395C2×C42C22×D5311yescharacteristic
396#107—1062no—
397#106—1062no—
398C4⋊C8—1062no—
399#102—22no—
400#102—22no—
401#102—1062no—
402#102—1062no—
403#97—1062no—
404#95—1061no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411C24—21no—
412C2×Q8—1062no—
413C2×Q8—1062no—
414C2×Q8—1062no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4#1171994712yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—531no—
427C22×C4—531no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—4241no—
433C22×C4—4241no—
434C22×C4—2122no—
435C22×C4D4×D5311yescharacteristic
436C22×C4D4×D5311yescharacteristic
437C22×C4C23×D5311yescharacteristic
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—81no—
458C4⋊C4—532no—
459C4⋊C4—1062no—
460C4⋊C4—1062no—
461C4⋊C4—1062no—
462C4⋊C4—1061no—
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—
475C22⋊C4—42no—
476C22⋊C4—81no—
477C22⋊C4—4241no—
478C22⋊C4—2122no—
479C22⋊C4—2122no—
480C42—21no—
481C42—21no—
482C42—532no—
483C42—532no—
484C42—1061no—
485C42—1061no—
486C42—2122no—
487C42—2122no—
488C42—2122no—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23C2×D4×D5311yescharacteristic
497Q8—1068no—
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—532no—
517C2×C4—532no—
518C2×C4—532no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1062no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—4244no—
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#1313324612yes—
542C2×C4#41×D5311yescharacteristic
543C2×C4#41×D5311yescharacteristic
544C2×C4#41×D5311yescharacteristic
545C2×C4C2×#1171994711yescharacteristic
546C2×C4#1313326411yescharacteristic
547C2×C4#1313326411yescharacteristic
548C8—41no—
549C8—41no—
550C8—2121no—
551C8—2121no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22#123×D5311yescharacteristic
560C22#1681127911yescharacteristic
561C22#1681128111yescharacteristic
562C22AGL(1,ℤ/8)×D5311yescharacteristic
563C22C2×#1313324611yescharacteristic
564C22#1681142411yescharacteristic
565C22#1681143211yescharacteristic
566C4—22no—
567C4—41no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—1062no—
572C4—1062no—
573C4—1062no—
574C4—4241no—
575C4—4241no—
576C4—4241no—
577C4—2122no—
578C4—2121no—
579C4#1681115011yescharacteristic
580C4#1681123011yescharacteristic
581C4#1681131011yescharacteristic
582C4#1681134011yescharacteristic
583C2—82no—
584C2#2027821111yescharacteristic
585C2#2028463711yescharacteristic
586C2#2028466611yescharacteristic
587C2#2028741711yescharacteristic
588C2#2028750011yescharacteristic
589C2#2028750411yescharacteristic
590C2#2028753411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2071C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2071D53
hypocenterC53#25002
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
2. #20277627C2semidirect
3. #20276277C2semidirect
4. #20276276C2semidirect
6. #20275653C2semidirect
7. #20275486C2semidirect
10. #20275308C2semidirect
11. #20275303C2semidirect
12. #20274847C2semidirect
354. C53#25002semidirect

Maximal subgroups

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

subgroupindex
#202800712
#202776272
#202762772
#202762762
#202761212
#202756532
#202754862
#202753392
#202753372
#202753082
#202753032
#202748472
#202747132
#202745422
C53×#20712
#2500253