Finite Group Databasebeta

← #27512604

Subgroups — #27512604

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279133C211yesmaximal, characteristic
2#20278400C211yesmaximal, characteristic
3#20276286C211yesmaximal, characteristic, complemented
4#20276275C211yesmaximal, characteristic, complemented
5#20276215C211yesmaximal, characteristic, complemented
6#20276148C211yesmaximal, characteristic
7#20276146C211yesmaximal, characteristic
8#20276041C211yesmaximal, characteristic, complemented
9#20275674C211yesmaximal, characteristic, complemented
10#20275484C211yesmaximal, characteristic, complemented
11#20275441C211yesmaximal, characteristic, complemented
12#20275324C211yesmaximal, characteristic
13#20275309C211yesmaximal, characteristic, complemented
14#20274810C211yesmaximal, characteristic
15C53×#1854C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133210—21no—
18C2×#13133150C2211yescharacteristic
19C2×#13133149C2211yescharacteristic
20C2×#13133146C2211yescharacteristic
21C2×C22⋊#88322C2211yescharacteristic
22C2×C8×Dic53C2211yescharacteristic
23#16810564—21no—
24#16810551—21no—
25#16810546—21no—
26#16810535—21no—
27#16810529—21no—
28#16810522C2211yescharacteristic
29#16810522C2211yescharacteristic
30#16810519C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810510—21no—
33#16810507—21no—
34#16810492C2211yescharacteristic
35#16810488C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810483C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×C22⋊Dic53C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133216—22no—
63#13133216—41no—
64#13133210—44no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53D411yescharacteristic
79C2×C22⋊Dic53D411yescharacteristic
80C2×C4×Dic53—21no—
81C2×C4×Dic53C2311yescharacteristic
82C2×C4×Dic53C2311yescharacteristic
83#13133150—41no—
84#13133149—22no—
85#13133146—22no—
86C22⋊#88322—41no—
87C8×Dic53—41no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124D411yescharacteristic
98#13133124D411yescharacteristic
99C23×C4×C53C2311yescharacteristic
100C22×C8×C53C2311yescharacteristic
101C22×C8×C53C2311yescharacteristic
102#117×C53—22no—
103#117×C53—41no—
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C42×C53—21no—
111#107×C53—22no—
112#106×C53—41no—
113C22⋊C8×C53—22no—
114#95×C53—21no—
115C24×C53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2411yescharacteristic
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C22×Dic53C2×D411yescharacteristic
135C22×Dic53C2×D411yescharacteristic
136C2×Dic106—42no—
137C2×Dic106—42no—
138C2×Dic106—81no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53#4112yes—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53#4112yes—
153C22⋊C4×C53—42no—
154C22⋊C4×C53—42no—
155C42×C53—42no—
156C22⋊Dic53—22no—
157C22⋊Dic53—22no—
158C22⋊Dic53—42no—
159C22⋊Dic53—81no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—42no—
169C4⋊Dic53—81no—
170#11719919—42no—
171#11719919—42no—
172#11719919—42no—
173#11719919—42no—
174#11719919—42no—
175C4×Dic53—21no—
176C4×Dic53—21no—
177C4×Dic53—21no—
178C4×Dic53—21no—
179C4×Dic53—42no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53C22×D411yescharacteristic
194C2×C4×C53—21no—
195C2×C4×C53—22no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53C2×#4111yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C2×C4×C53H(5,𝔽2)11yescharacteristic
216C2×C4×C532−1+411yescharacteristic
217C2×C4×C532−1+411yescharacteristic
218C2×C4×C532−1+411yescharacteristic
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
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#13512yes—
242C2×Dic53AGL(1,ℤ/8)12yes—
243Dic106—84no—
244C8×C53—22no—
245C8×C53—41no—
246#88322—41no—
247#88322—41no—
248#31423—531nomaximal, Sylow
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53#53611yescharacteristic
257C22×C53#54611yescharacteristic
258C22×C53#54711yescharacteristic
259C22×C53C2×#13511yescharacteristic
260C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C53#57711yescharacteristic
262C22×C53#57811yescharacteristic
263C4×C53—41no—
264C4×C53—42no—
265C4×C53—81no—
266C4×C53—81no—
267C4×C53—81no—
268C4×C53#53911yescharacteristic
269C4×C53#54111yescharacteristic
270C4×C53#55611yescharacteristic
271C4×C53#56311yescharacteristic
272Dic53—22no—
273Dic53—22no—
274Dic53—41no—
275Dic53—41no—
276Dic53—81no—
277Dic53—81no—
278Dic53—81no—
279Dic53—81no—
280#2496—531no—
281#2439—531no—
282#2101—531no—
283#2099—531no—
284#2088—531no—
285#2077—531no—
286#2076—531no—
287#2057—531no—
288#1990—531no—
289#1949—531no—
290#1941—531no—
291#1923—531no—
292#1921—531no—
293#1854D5311yescharacteristic
294#1825—531no—
295C2×C53—81no—
296C2×C53#268911yescharacteristic
297C2×C53#322811yescharacteristic
298C2×C53#324211yescharacteristic
299C2×C53#330011yescharacteristic
300C2×C53#330611yescharacteristic
301C2×C53#334011yescharacteristic
302C2×C53#335311yescharacteristic
303C2×C4⋊Q8—531no—
304C2×#126—1061no—
305C2×#117C2×D5311yescharacteristic
306C2×#107—531no—
307C2×#107C2×D5311yescharacteristic
308C2×#106—531no—
309C2×#106C2×D5311yescharacteristic
310C2×C4⋊C8—531no—
311C2×C22⋊C8—531no—
312C2×C22⋊C8C2×D5311yescharacteristic
313C2×C4×C8—531no—
314#400—1061no—
315#397—1061no—
316#396—1061no—
317#393—1061no—
318#391—1061no—
319#389—531no—
320#389—531no—
321#389—531no—
322#389—531no—
323#387—1061no—
324#387—1061no—
325#384—531no—
326#382—531no—
327#380—531no—
328#380—531no—
329#380C2×D5311yescharacteristic
330C4×C4⋊C4—531no—
331C4×C22⋊C4—531no—
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#340—531no—
341#340—531no—
342#340C2×D5311yescharacteristic
343#340C2×D5311yescharacteristic
344#336—531no—
345#336—531no—
346C53#3142311yescharacteristic, complemented, Sylow
347C22×Q8—1061no—
348C23×C4C22×D5311yescharacteristic
349C22×C8—531no—
350C22×C8—531no—
351C22×C8C22×D5311yescharacteristic
352C22×C8C22×D5311yescharacteristic
353C4⋊Q8—1062no—
354C4⋊Q8—2121no—
355#126—2124no—
356#117—22no—
357#117—41no—
358C2×C4⋊C4—531no—
359C2×C4⋊C4—531no—
360C2×C4⋊C4—531no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4—1061no—
365C2×C4⋊C4—1061no—
366C2×C4⋊C4—1061no—
367C2×C4⋊C4—1061no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4C22×D5311yescharacteristic
370C2×C4⋊C4C22×D5311yescharacteristic
371C2×C4⋊C4C22×D5311yescharacteristic
372C2×C4⋊C4C22×D5311yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—531no—
376C2×C22⋊C4—531no—
377C2×C22⋊C4—1061no—
378C2×C22⋊C4—1061no—
379C2×C42—21no—
380C2×C42—531no—
381C2×C42—531no—
382C2×C42—1061no—
383#107—22no—
384#107—2121no—
385#106—41no—
386#106—1062no—
387C4⋊C8—1062no—
388C22⋊C8—22no—
389C22⋊C8—2121no—
390C4×C8—2121no—
391#95—21no—
392#95—531no—
393#95—531no—
394#95—1061no—
395#95—1061no—
396#95—1061no—
397#95—1061no—
398#95—1061no—
399#95—1061no—
400#95—1061no—
401#95—1061no—
402#95—1061no—
403C24D4×D5311yescharacteristic
404C2×Q8—4241no—
405C2×Q8—2122no—
406C2×Q8—2122no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—531no—
418C22×C4—531no—
419C22×C4—531no—
420C22×C4—531no—
421C22×C4—1061no—
422C22×C4—1061no—
423C22×C4—1061no—
424C22×C4—1061no—
425C22×C4D4×D5311yescharacteristic
426C22×C4C23×D5311yescharacteristic
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—1061no—
433C2×C8—1061no—
434C2×C8—1061no—
435C2×C8—1061no—
436C2×C8—1061no—
437C2×C8—1061no—
438C2×C8#1171994512yes—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—1061no—
447C4⋊C4—1061no—
448C4⋊C4—1061no—
449C4⋊C4—1061no—
450C4⋊C4—1061no—
451C4⋊C4—1061no—
452C4⋊C4—1061no—
453C4⋊C4—1061no—
454C4⋊C4—4241no—
455C4⋊C4—2122no—
456C4⋊C4—2122no—
457C4⋊C4—2122no—
458C4⋊C4—2122no—
459C4⋊C4—2122no—
460C4⋊C4—2122no—
461C4⋊C4#1171994712yes—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—1062no—
465C22⋊C4—1062no—
466C22⋊C4—4241no—
467C22⋊C4—2122no—
468C42—42no—
469C42—1061no—
470C42—1061no—
471C42—1061no—
472C42—1061no—
473C42—2122no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23C2×D4×D5311yescharacteristic
482Q8—4244no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—532no—
503C2×C4—532no—
504C2×C4—1061no—
505C2×C4—1061no—
506C2×C4—1061no—
507C2×C4—1061no—
508C2×C4—1061no—
509C2×C4—1061no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1061no—
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—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4C2×#1171994711yescharacteristic
527C2×C4C2×#1171994511yescharacteristic
528C2×C4#1313326211yescharacteristic
529C2×C4#1313326411yescharacteristic
530C2×C4#1313326411yescharacteristic
531C2×C4#1313326511yescharacteristic
532C8—22no—
533C8—41no—
534C8—2121no—
535C8—2121no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22#1681119511yescharacteristic
544C22#1681126011yescharacteristic
545C22#1681127411yescharacteristic
546C22#135×D5311yescharacteristic
547C22AGL(1,ℤ/8)×D5311yescharacteristic
548C22#1681143011yescharacteristic
549C22#1681143911yescharacteristic
550C4—41no—
551C4—42no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—1062no—
556C4—1062no—
557C4—4241no—
558C4—4241no—
559C4—4241no—
560C4—4241no—
561C4—2121no—
562C4—2121no—
563C4#1681121511yescharacteristic
564C4#1681123311yescharacteristic
565C4#1681134311yescharacteristic
566C4#1681137411yescharacteristic
567C2—81no—
568C2#2028117011yescharacteristic
569C2#2028584911yescharacteristic
570C2#2028598911yescharacteristic
571C2#2028659511yescharacteristic
572C2#2028666911yescharacteristic
573C2#2028712411yescharacteristic
574C2#2028729511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1854C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1854D53
hypocenterC53#31423
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
3. #20276286C2semidirect
4. #20276275C2semidirect
5. #20276215C2semidirect
8. #20276041C2semidirect
9. #20275674C2semidirect
10. #20275484C2semidirect
11. #20275441C2semidirect
13. #20275309C2semidirect
346. C53#31423semidirect

Maximal subgroups

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

subgroupindex
#202791332
#202784002
#202762862
#202762752
#202762152
#202761482
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C53×#18542
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