Finite Group Databasebeta

← #27438410

Subgroups — #27438410

Order 13568.

1318 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811173C211yesmaximal, characteristic
2C2×#16811033C211yesmaximal, characteristic, complemented
3C2×#16810989C211yesmaximal, characteristic, complemented
4#20275904C212yesmaximal, complemented
5#20275897C212yesmaximal, complemented
6#20275687C211yesmaximal, characteristic, complemented
7#20275538C211yesmaximal, characteristic, complemented
8#20274975C212yesmaximal, complemented
9#20274961C212yesmaximal, complemented
10#20274477C211yesmaximal, characteristic, complemented
11C53×#2851C211yesmaximal, characteristic, complemented
12C22×#11719945—21no—
13#16811173—24no—
14#16811173—24no—
15C2×#13133217C2211yescharacteristic
16C2×#13133214C2211yescharacteristic
17C2×#13133205—21no—
18C2×#13133191—21no—
19C2×#13133191C2212yes—
20C2×C4×D212—21no—
21C2×C4×D212C2212yes—
22#16811033—24no—
23#16810989—24no—
24C2×#13133150C2212yes—
25C2×#13133148C2212yes—
26C2×#13133137C2212yes—
27C2×#13133137C2212yes—
28C2×#13133137C2212yes—
29C2×#13133137C2212yes—
30C2×C22⋊#88322C2212yes—
31C2×#13133128C2211yescharacteristic
32C2×#13133128C2211yescharacteristic
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C2×#125×C53C2211yescharacteristic
37C22×C42×C53C2211yescharacteristic
38C53×#389C2211yescharacteristic
39C53×#389C2211yescharacteristic
40C53×#387—21no—
41C53×#387—21no—
42C53×#385—21no—
43C53×#385—21no—
44C53×#382C2211yescharacteristic
45C53×#380C2211yescharacteristic
46C53×#380C2211yescharacteristic
47C2×#11719945—28no—
48C2×#11719945—44no—
49C2×#11719945—44no—
50C22×Dic106—21no—
51C22×D212—21no—
52C22×D212—21no—
53C22×D212C2312yes—
54C22×C22⋊D53—22no—
55C22×C4×D53—22no—
56C22×#88322C2312yes—
57C22×#88322C2312yes—
58#13133217—22no—
59#13133217D414yes—
60#13133214—22no—
61#13133214D414yes—
62#13133205—28no—
63#13133191—28no—
64#13133191—24no—
65#13133191—24no—
66C4×D212—28no—
67C4×D212—24no—
68C4×D212—24no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2312yes—
71C2×#11719921—22no—
72C2×#11719921—22no—
73#13133150—24no—
74#13133148—24no—
75#13133137—24no—
76#13133137—24no—
77#13133137—24no—
78#13133137—24no—
79C22⋊#88322—24no—
80#13133128—22no—
81#13133128—22no—
82C23×C4×C53D411yescharacteristic
83C23×C4×C53D411yescharacteristic
84C23×C4×C53C2311yescharacteristic
85#125×C53—22no—
86#125×C53—22no—
87C2×C4⋊C4×C53—21no—
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53C2311yescharacteristic
92C2×C4⋊C4×C53C2311yescharacteristic
93C2×C4⋊C4×C53C2311yescharacteristic
94C2×C4⋊C4×C53C2311yescharacteristic
95C2×C22⋊C4×C53—21no—
96C2×C22⋊C4×C53—21no—
97C2×C42×C53—22no—
98C2×C42×C53—22no—
99C2×C42×C53D411yescharacteristic
100C2×C42×C53D411yescharacteristic
101C2×C42×C53C2311yescharacteristic
102C2×C42×C53C2311yescharacteristic
103#95×C53—21no—
104#95×C53—21no—
105#95×C53—21no—
106#95×C53—21no—
107#95×C53—21no—
108#95×C53—21no—
109C24×C53C2×D411yescharacteristic
110C23×D53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2411yescharacteristic
131C2×C22⋊D53—28no—
132C2×C22⋊D53—44no—
133C2×C22⋊D53—44no—
134C22×Dic53—22no—
135#11719945—432no—
136C2×D212—22no—
137C2×D212—22no—
138C2×D212—22no—
139C2×D212—22no—
140C2×D212—24no—
141C2×D212—24no—
142C2×D212—22no—
143C2×D212—42no—
144C2×D212—42no—
145C2×D212—42no—
146C2×D212—42no—
147C2×D212C2×D414yes—
148C2×C4×D53—28no—
149C2×C4×D53—44no—
150C2×C4×D53—44no—
151C2×Dic106—24no—
152C2×Dic106—42no—
153C2×Dic106—42no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—42no—
161C4⋊C4×C53—42no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53—42no—
164C22⋊C4×C53—42no—
165C22⋊C4×C53—42no—
166C42×C53—22no—
167C42×C53—22no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53C2×D412yes—
171C42×C53C2×D412yes—
172#11719921—28no—
173#11719921—28no—
174C4⋊Dic53—24no—
175C4⋊Dic53—22no—
176C4⋊Dic53C2×D414yes—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322#4114yescomplemented
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53AGL(1,ℤ/8)12yes—
190C23×C53C22×D411yescharacteristic
191C22×D53—24no—
192C22×D53—42no—
193C22×D53—42no—
194C22×D53—42no—
195C22×D53—42no—
196C22×D53—42no—
197C22×D53—42no—
198C2×C4×C53—28no—
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—21no—
211C2×C4×C53—44no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
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—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53#12012yes—
227C2×C4×C53#12012yes—
228C2×C4×C53AGL(1,ℤ/8)12yes—
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C2×#4112yes—
232C2×C4×C532−1+412yes—
233C22⋊D53—416no—
234C2×Dic53—24no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237D212—24no—
238D212—24no—
239D212—48no—
240D212—48no—
241D212—42no—
242D212—42no—
243C4×D53—416no—
244Dic106—48no—
245#88322—24no—
246#88322—42no—
247#25676—531nomaximal, Sylow
248C22×C53—22no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53C2×#12011yescharacteristic
256C22×C53#53211yescharacteristic
257C22×C53#53611yescharacteristic
258C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C53#57511yescharacteristic
261C22×C53#57511yescharacteristic
262C2×D53—44no—
263C2×D53—44no—
264C2×D53—44no—
265C2×D53—82no—
266C2×D53—82no—
267C2×D53—82no—
268C2×D53—82no—
269C4×C53—22no—
270C4×C53—22no—
271C4×C53—22no—
272C4×C53—41no—
273C4×C53—41no—
274C4×C53—81no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53#54712yes—
279C4×C53#54812yes—
280Dic53—44no—
281C2×#532—531no—
282C2×#501—531no—
283C2×#490—531no—
284#2851D5311yescharacteristic, complemented
285#2034—532no—
286#2033—532no—
287#1993—531no—
288#1961—531no—
289#1864—532no—
290#1862—532no—
291#1749—531no—
292C2×C53—42no—
293C2×C53#243211yescharacteristic
294C2×C53#300311yescharacteristic
295C2×C53#300511yescharacteristic
296C2×C53#317111yescharacteristic
297C2×C53#318011yescharacteristic
298C2×C53#323411yescharacteristic
299C2×C53#323511yescharacteristic
300D53—84no—
301C22×#41—1061no—
302#532—1064no—
303#532—1064no—
304C2×C4⋊Q8—531no—
305C2×#127—531no—
306C2×#125C2×D5311yescharacteristic
307C2×#124—1061no—
308C2×#121—532no—
309C2×#121—1061no—
310C2×C4×D4—532no—
311C2×C4×D4—1061no—
312C22×C42C2×D5311yescharacteristic
313#501—1064no—
314#490—1064no—
315C2×#107—532no—
316C2×#106—532no—
317C2×#102—532no—
318C2×#102—532no—
319C2×#102—532no—
320C2×#102—532no—
321C2×C22⋊C8—532no—
322C2×#97—531no—
323C2×#97—531no—
324#389C2×D5311yescharacteristic
325#389C2×D5311yescharacteristic
326#387—21no—
327#387—21no—
328#385—21no—
329#385—21no—
330#382C2×D5311yescharacteristic
331#380C2×D5311yescharacteristic
332#380C2×D5311yescharacteristic
333#340—532no—
334#340—532no—
335#336—532no—
336C53#2567611yescharacteristic, complemented, Sylow
337C2×#41—1068no—
338C2×#41—2124no—
339C2×#41—2124no—
340C22×Q8—1061no—
341C22×D4—532no—
342C22×D4—1062no—
343C22×D4—1061no—
344C22×D4—1061no—
345C23×C4—1062no—
346C23×C4C22⋊D5311yescharacteristic
347C23×C4C22⋊D5311yescharacteristic
348C23×C4C22×D5311yescharacteristic
349C22×C8—532no—
350C22×C8—532no—
351C4⋊Q8—534no—
352C4⋊Q8—1062no—
353#127—534no—
354#127—1062no—
355#125—22no—
356#125—22no—
357#124—1068no—
358#121—1064no—
359#121—1064no—
360#121—1068no—
361C4×D4—1064no—
362C4×D4—1064no—
363C4×D4—1068no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—532no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4C22×D5311yescharacteristic
371C2×C4⋊C4C22×D5311yescharacteristic
372C2×C4⋊C4C22×D5311yescharacteristic
373C2×C4⋊C4C22×D5311yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—1062no—
377C2×C22⋊C4—1062no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42C22⋊D5311yescharacteristic
381C2×C42C22⋊D5311yescharacteristic
382C2×C42C22×D5311yescharacteristic
383C2×C42C22×D5311yescharacteristic
384#107—1064no—
385#106—1064no—
386#102—1064no—
387#102—1064no—
388#102—1064no—
389#102—1064no—
390C22⋊C8—1064no—
391#97—1062no—
392#97—1062no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399C24—1062no—
400C24C2×C22⋊D5311yescharacteristic
401#41—21232no—
402C2×Q8—1064no—
403C2×Q8—2122no—
404C2×Q8—2122no—
405C2×D4—534no—
406C2×D4—1068no—
407C2×D4—1064no—
408C2×D4—1064no—
409C2×D4—1062no—
410C2×D4—1062no—
411C2×D4—1062no—
412C2×D4—1062no—
413C2×D4—1062no—
414C2×D4—2122no—
415C2×D4—2122no—
416C2×D4—2122no—
417C2×D4—2122no—
418C2×D4—2124no—
419C2×D4—2124no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—1068no—
435C22×C4—1062no—
436C22×C4—2124no—
437C22×C4—2124no—
438C22×C4C2×C22⋊D5311yescharacteristic
439C22×C4C2×C22⋊D5311yescharacteristic
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C2×C22⋊D5311yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C23×D5311yescharacteristic
444C2×C8—534no—
445C2×C8—1062no—
446C2×C8—1062no—
447C2×C8—1062no—
448C2×C8—1062no—
449C2×C8—1062no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—534no—
461C4⋊C4—1064no—
462C4⋊C4—1062no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—1068no—
466C22⋊C4—1068no—
467C42—22no—
468C42—22no—
469C42—21no—
470C42—21no—
471C42D4×D5312yes—
472C42D4×D5312yes—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—1064no—
480C23—2122no—
481C23—2122no—
482C23—2122no—
483C23—2122no—
484C23—2122no—
485C23—2122no—
486C23#1313324212yes—
487C23C22×C22⋊D5311yescharacteristic
488Q8—2128no—
489D4—1064no—
490D4—1064no—
491D4—2128no—
492D4—2128no—
493D4—21216no—
494D4—2122no—
495D4—2122no—
496C2×C4—28no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
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—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—1064no—
525C2×C4—21216no—
526C2×C4—2122no—
527C2×C4—2122no—
528C2×C4#1313318512yes—
529C2×C4#1313318512yes—
530C2×C4#1313324212yes—
531C2×C4C2×D4×D5311yescharacteristic
532C2×C4C2×D4×D5311yescharacteristic
533C2×C4#41×D5312yes—
534C2×C4#1313326512yes—
535C8—1064no—
536C8—2122no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—4242no—
545C22—4242no—
546C22—4242no—
547C22—4242no—
548C22—2124no—
549C22—2124no—
550C22—2124no—
551C22C2×#1313318511yescharacteristic
552C22#1681116911yescharacteristic
553C22#1681119711yescharacteristic
554C22C2×#1313324211yescharacteristic
555C22C2×#1313324211yescharacteristic
556C22#1681142111yescharacteristic
557C22#1681142111yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—2124no—
568C4#1681127512yes—
569C4#1681128312yes—
570C2—42no—
571C2—4244no—
572C2#2027834411yescharacteristic
573C2#2028394111yescharacteristic
574C2#2028395511yescharacteristic
575C2#2028531111yescharacteristic
576C2#2028541411yescharacteristic
577C2#2028590511yescharacteristic
578C2#2028591311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2851C2
socleC23×C53C22×D4
prime radicalC2×#16811173C2
hypercenter#2851D53
hypocenterC53#25676
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
2. C2×#16811033C2semidirect
3. C2×#16810989C2semidirect
4. #20275904C2semidirect
5. #20275897C2semidirect
6. #20275687C2semidirect
7. #20275538C2semidirect
8. #20274975C2semidirect
9. #20274961C2semidirect
10. #20274477C2semidirect
11. C53×#2851C2semidirect
182. C2×#88322#41semidirect
284. #2851D53semidirect
336. C53#25676semidirect

Maximal subgroups

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

subgroupindex
C2×#168111732
C2×#168110332
C2×#168109892
#202759042
#202758972
#202756872
#202755382
#202749752
#202749612
#202744772
C53×#28512
#2567653