Finite Group Databasebeta

← #27489367

Subgroups — #27489367

Order 13568.

644 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277359C211yesmaximal, characteristic
2#20276282C211yesmaximal, characteristic, complemented
3#20276239C211yesmaximal, characteristic
4#20276192C211yesmaximal, characteristic, complemented
5#20276119C211yesmaximal, characteristic
6#20275455C211yesmaximal, characteristic, complemented
7#20275440C211yesmaximal, characteristic, complemented
8#20275411C211yesmaximal, characteristic
9#20275397C211yesmaximal, characteristic
10#20275290C211yesmaximal, characteristic, complemented
11#20275283C211yesmaximal, characteristic, complemented
12#20274975C211yesmaximal, characteristic
13#20274975C211yesmaximal, characteristic
14#20274467C211yesmaximal, characteristic, complemented
15C53×#2647C211yesmaximal, characteristic, complemented
16C2×C4×D212C2211yescharacteristic
17C2×#13133150C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133137C2211yescharacteristic
22C2×#13133137C2211yescharacteristic
23C2×#13133137C2211yescharacteristic
24C2×#13133137C2211yescharacteristic
25C2×#13133128C2211yescharacteristic
26C2×C4×#88322C2211yescharacteristic
27#16810525C2211yescharacteristic
28#16810525C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810512—21no—
32#16810512—21no—
33#16810485—21no—
34#16810482—21no—
35C4×C4⋊Dic53C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C53×#400—21no—
49C53×#400—21no—
50C53×#398—21no—
51C53×#398—21no—
52C53×#389C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#384C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C53×#376C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C4×D53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C4×D212—22no—
66C4×D212—22no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×#11719921—21no—
73C2×#11719921—21no—
74C2×#11719921—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×C4×Dic53—21no—
78#13133150—22no—
79#13133148—22no—
80#13133148—22no—
81#13133148—22no—
82#13133137—22no—
83#13133137—22no—
84#13133137—22no—
85#13133137—22no—
86#13133128—22no—
87C4×#88322—22no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C42×C53C2311yescharacteristic
102C2×C42×C53C2311yescharacteristic
103C2×C42×C53C2311yescharacteristic
104#95×C53—21no—
105#95×C53—21no—
106#95×C53—21no—
107#95×C53—21no—
108#95×C53—21no—
109#95×C53—21no—
110#95×C53D411yescharacteristic
111#95×C53D411yescharacteristic
112#95×C53D411yescharacteristic
113#95×C53D411yescharacteristic
114C23×D53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2411yescharacteristic
126C22×Dic53—21no—
127C22×Dic53—21no—
128C22×Dic53—21no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212#4112yes—
135C2×C4×D53—24no—
136C2×C4×D53—81no—
137C2×C4×D53—81no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—81no—
147C4⋊C4×C53—81no—
148C4⋊C4×C53—81no—
149C4⋊C4×C53—81no—
150C4⋊C4×C53—81no—
151C4⋊C4×C53—81no—
152C42×C53—21no—
153C42×C53—21no—
154C42×C53—21no—
155C42×C53—21no—
156C42×C53—21no—
157C42×C53—21no—
158#11719921—24no—
159#11719921—81no—
160#11719921—81no—
161#11719921—81no—
162#11719921—81no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53#4112yes—
169#11719919—81no—
170#11719919—81no—
171C4×Dic53—81no—
172C2×#88322—21no—
173C2×#88322—21no—
174C2×#88322—21no—
175C2×#88322—21no—
176C2×#88322—21no—
177C2×#88322—21no—
178C2×#88322—21no—
179C2×#88322—21no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322#4112yes—
183C2×#88322#4112yes—
184C23×C53C22×D411yescharacteristic
185C22×D53—22no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—41no—
190C22×D53—41no—
191C22×D53—41no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
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—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—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C532−1+411yescharacteristic
227C2×C4×C532−1+411yescharacteristic
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—
237D212—22no—
238D212—22no—
239D212—41no—
240D212—41no—
241C4×D53—84no—
242#88322—22no—
243#88322—22no—
244#88322—41no—
245#88322—41no—
246#29590—531nomaximal, Sylow
247C22×C53#52511yescharacteristic
248C22×C53#53211yescharacteristic
249C22×C53#53611yescharacteristic
250C22×C53#57611yescharacteristic
251C22×C53#57711yescharacteristic
252C22×C53#57711yescharacteristic
253C22×C53#57711yescharacteristic
254C2×D53—42no—
255C2×D53—42no—
256C2×D53—42no—
257C2×D53—81no—
258C2×D53—81no—
259C2×D53—81no—
260C2×D53—81no—
261C4×C53—41no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53—81no—
268C4×C53—81no—
269C4×C53—81no—
270C4×C53—81no—
271C4×C53#55211yescharacteristic
272C4×C53#55211yescharacteristic
273C4×C53#55211yescharacteristic
274C4×C53#55211yescharacteristic
275Dic53—42no—
276Dic53—81no—
277Dic53—81no—
278#2647D5311yescharacteristic, complemented
279#2319—531no—
280#2101—531no—
281#2092—531no—
282#2084—531no—
283#2071—531no—
284#1944—531no—
285#1941—531no—
286#1936—531no—
287#1934—531no—
288#1919—531no—
289#1918—531no—
290#1864—531no—
291#1864—531no—
292#1746—531no—
293C2×C53#261111yescharacteristic
294C2×C53#311911yescharacteristic
295C2×C53#312311yescharacteristic
296C2×C53#317311yescharacteristic
297C2×C53#317911yescharacteristic
298C2×C53#324111yescharacteristic
299C2×C53#324411yescharacteristic
300D53—82no—
301C2×C4×D4—531no—
302C2×#107—531no—
303C2×#106—531no—
304C2×#106—531no—
305C2×#106—531no—
306C2×#102—531no—
307C2×#102—531no—
308C2×#102—531no—
309C2×#102—531no—
310C2×#97—531no—
311C2×C4×C8—531no—
312#400—21no—
313#400—21no—
314#398—21no—
315#398—21no—
316#390—531no—
317#390—531no—
318#389—531no—
319#389—531no—
320#389C2×D5311yescharacteristic
321#389C2×D5311yescharacteristic
322#389C2×D5311yescharacteristic
323#389C2×D5311yescharacteristic
324#388—1061no—
325#388—1061no—
326#384C2×D5311yescharacteristic
327#382C2×D5311yescharacteristic
328#381—1061no—
329#380—1061no—
330C4×C4⋊C4—531no—
331#376C2×D5311yescharacteristic
332#340—531no—
333#340—531no—
334#340—531no—
335#340—531no—
336#340—531no—
337#340—531no—
338#340—531no—
339#340—531no—
340#336—531no—
341#336—531no—
342#336—531no—
343#336—531no—
344C53#2959011yescharacteristic, complemented, Sylow
345C22×D4—531no—
346C23×C4—1061no—
347C22×C8—531no—
348C22×C8—531no—
349C22×C8—531no—
350C22×C8—531no—
351C4×D4—1062no—
352C4×D4—1062no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—531no—
360C2×C4⋊C4—531no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4C22×D5311yescharacteristic
365C2×C4⋊C4C22×D5311yescharacteristic
366C2×C4⋊C4C22×D5311yescharacteristic
367C2×C4⋊C4C22×D5311yescharacteristic
368C2×C22⋊C4—1061no—
369C2×C22⋊C4—1061no—
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—1061no—
372C2×C22⋊C4—1061no—
373C2×C42—1061no—
374C2×C42C22×D5311yescharacteristic
375C2×C42C22×D5311yescharacteristic
376C2×C42C22×D5311yescharacteristic
377#107—1062no—
378#106—1062no—
379#106—1062no—
380#106—1062no—
381#102—1062no—
382#102—1062no—
383#102—1062no—
384#102—1062no—
385#97—1062no—
386C4×C8—1062no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—1061no—
394#95—1061no—
395#95—1061no—
396#95C22⋊D5311yescharacteristic
397#95C22⋊D5311yescharacteristic
398#95C22⋊D5311yescharacteristic
399#95C22⋊D5311yescharacteristic
400C24—1061no—
401C2×D4—532no—
402C2×D4—1061no—
403C2×D4—1061no—
404C2×D4—1061no—
405C2×D4—1061no—
406C2×D4—1061no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—1064no—
412C22×C4—1061no—
413C22×C4—1061no—
414C22×C4—1061no—
415C22×C4—4241no—
416C22×C4—4241no—
417C22×C4C2×C22⋊D5311yescharacteristic
418C22×C4C2×C22⋊D5311yescharacteristic
419C22×C4C2×C22⋊D5311yescharacteristic
420C22×C4C2×C22⋊D5311yescharacteristic
421C22×C4C2×C22⋊D5311yescharacteristic
422C22×C4C2×C22⋊D5311yescharacteristic
423C22×C4C23×D5311yescharacteristic
424C2×C8—532no—
425C2×C8—532no—
426C2×C8—1061no—
427C2×C8—1061no—
428C2×C8—1061no—
429C2×C8—1061no—
430C2×C8—1061no—
431C2×C8—1061no—
432C2×C8—1061no—
433C2×C8—1061no—
434C2×C8—1061no—
435C2×C8—1061no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—532no—
451C4⋊C4—1061no—
452C4⋊C4—1061no—
453C4⋊C4—1061no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4—4241no—
457C4⋊C4—4241no—
458C22⋊C4—1064no—
459C22⋊C4—4241no—
460C22⋊C4—4241no—
461C22⋊C4—4241no—
462C22⋊C4—4241no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—4241no—
470C23—1062no—
471C23—2121no—
472C23—2121no—
473C23—2121no—
474C23—2121no—
475C23—2121no—
476C23—2121no—
477C23C22×C22⋊D5311yescharacteristic
478D4—1062no—
479D4—1062no—
480D4—2121no—
481D4—2121no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—1062no—
513C2×C4—4244no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4#41×D5311yescharacteristic
523C2×C4#41×D5311yescharacteristic
524C2×C4#41×D5311yescharacteristic
525C2×C4#41×D5311yescharacteristic
526C2×C4#1313326511yescharacteristic
527C2×C4#1313326511yescharacteristic
528C8—1062no—
529C8—1062no—
530C8—2121no—
531C8—2121no—
532C22—4241no—
533C22—4241no—
534C22—4241no—
535C22—4241no—
536C22—2122no—
537C22—2122no—
538C22—2122no—
539C22#1681113011yescharacteristic
540C22#1681116911yescharacteristic
541C22#1681119711yescharacteristic
542C22#1681142611yescharacteristic
543C22#1681143111yescharacteristic
544C22#1681143111yescharacteristic
545C22#1681143411yescharacteristic
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4—4241no—
557C4—4241no—
558C4—2122no—
559C4#1681130511yescharacteristic
560C4#1681130511yescharacteristic
561C4#1681130511yescharacteristic
562C4#1681130511yescharacteristic
563C2—4242no—
564C2#2028041511yescharacteristic
565C2#2028490811yescharacteristic
566C2#2028496211yescharacteristic
567C2#2028532611yescharacteristic
568C2#2028539211yescharacteristic
569C2#2028596811yescharacteristic
570C2#2028601311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2647C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2647D53
hypocenterC53#29590
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ #376 ◁ #2647

Complements

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

NH ≅ G/Ntype
2. #20276282C2semidirect
4. #20276192C2semidirect
6. #20275455C2semidirect
7. #20275440C2semidirect
10. #20275290C2semidirect
11. #20275283C2semidirect
14. #20274467C2semidirect
15. C53×#2647C2semidirect
278. #2647D53semidirect
344. C53#29590semidirect

Maximal subgroups

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

subgroupindex
#202773592
#202762822
#202762392
#202761922
#202761192
#202754552
#202754402
#202754112
#202753972
#202752902
#202752832
#202749752
#202744672
C53×#26472
#2959053