Finite Group Databasebeta

← #27350371

Subgroups — #27350371

Order 13568.

922 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811306C211yesmaximal, characteristic
2#20283802C212yesmaximal, complemented
3#20273083C212yesmaximal, complemented
4#20273082C212yesmaximal, complemented
5#20273069C212yesmaximal, complemented
6#20273054C211yesmaximal, characteristic
7#20273049C211yesmaximal, characteristic, complemented
8#20273046C211yesmaximal, characteristic, complemented
9#20273034C211yesmaximal, characteristic
10#20273026C211yesmaximal, characteristic, complemented
11C53×#1458C211yesmaximal, characteristic, complemented
12#16811306—22no—
13#16811306—22no—
14#16811306—22no—
15#16811306—22no—
16C2×#13133212C2211yescharacteristic
17C2×#13133211C2211yescharacteristic
18C2×#13133209C411yescharacteristic
19C2×#13133207C411yescharacteristic
20C2×#13133207C2211yescharacteristic
21C2×#13133201C411yescharacteristic
22C2×#13133200C411yescharacteristic
23C2×#13133195C411yescharacteristic
24C2×#13133194C411yescharacteristic
25C2×C4×Dic106C2211yescharacteristic
26C2×#13133177C2211yescharacteristic
27C2×#13133174C411yescharacteristic
28C2×#13133173C2211yescharacteristic
29C22×#11719919C411yescharacteristic
30#16810753—22no—
31#16810752—22no—
32#16810752—22no—
33C8×Dic106—22no—
34#16810680C2212yes—
35#16810677C2212yes—
36#16810677C2212yes—
37#16810673C2212yes—
38#16810673C2212yes—
39#16810349C2212yes—
40#16810348C2212yes—
41#16810348C2212yes—
42#16810347C2211yescharacteristic
43#16810347C2211yescharacteristic
44#16810346C2212yes—
45#16810346C2212yes—
46C2×#117×C53C2211yescharacteristic
47C53×#431C2212yes—
48C53×#324C2212yes—
49C53×#323C2211yescharacteristic
50C53×#323C2211yescharacteristic
51C22×Dic106C2×C411yescharacteristic
52C22×C22⋊D53C2×C411yescharacteristic
53C23×Dic53C2×C411yescharacteristic
54C23×Dic53C2×C411yescharacteristic
55C22×C4×D53C2×C411yescharacteristic
56#13133212—22no—
57#13133212—22no—
58#13133211—22no—
59#13133211—22no—
60#13133209—22no—
61#13133209—22no—
62#13133207—22no—
63#13133207—22no—
64#13133207—22no—
65#13133207—22no—
66#13133201—22no—
67#13133201—22no—
68#13133200—22no—
69#13133200—22no—
70#13133195—22no—
71#13133195—22no—
72#13133194—22no—
73#13133194—22no—
74C4×Dic106D412yes—
75C4×Dic106D412yes—
76C4×Dic106D412yes—
77C4×Dic106D412yes—
78#13133177—22no—
79#13133177—22no—
80#13133174—21no—
81#13133174—21no—
82#13133174—21no—
83#13133174—21no—
84#13133173—22no—
85#13133173—22no—
86C2×C4⋊Dic53C2×C411yescharacteristic
87C2×C4⋊Dic53C2×C411yescharacteristic
88C2×C4⋊Dic53C2×C411yescharacteristic
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×#11719919C2×C411yescharacteristic
94C2×#11719919C2×C411yescharacteristic
95C2×#11719919C2×C411yescharacteristic
96C2×#11719919C2×C411yescharacteristic
97C2×#11719919C2×C411yescharacteristic
98C2×#11719919C2×C411yescharacteristic
99C2×#11719919C2×C411yescharacteristic
100C2×#11719919C2×C411yescharacteristic
101C2×#11719919C2×C411yescharacteristic
102C2×#11719919C2×C411yescharacteristic
103C2×#11719919C2311yescharacteristic
104C2×C22⋊Dic53C2×C411yescharacteristic
105C2×#11719921C2×C411yescharacteristic
106C2×#11719921C2×C411yescharacteristic
107C2×#11719921C2×C411yescharacteristic
108C2×#11719921C2×C411yescharacteristic
109C2×#11719921C2311yescharacteristic
110C2×C4×Dic53C2×C411yescharacteristic
111C2×C4×Dic53C2311yescharacteristic
112C2×C4×Dic53C2311yescharacteristic
113C4⋊#88322—22no—
114#13133146—22no—
115#13133146—22no—
116#13133146—22no—
117#13133146—22no—
118C22⋊#88322C2312yes—
119C22⋊#88322C2312yes—
120#13133128—22no—
121#13133127—22no—
122#13133127—22no—
123C8×Dic53—22no—
124C8×Dic53—22no—
125C23×C4×C53C2×C411yescharacteristic
126#117×C53—22no—
127#117×C53—22no—
128C2×C4⋊C4×C53C2×C411yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C22⋊C4×C53C2×C411yescharacteristic
131C2×C22⋊C4×C53C2311yescharacteristic
132C2×C42×C53C2×C411yescharacteristic
133C2×C42×C53C2311yescharacteristic
134C4⋊C8×C53—22no—
135C22⋊C8×C53C2312yes—
136C22⋊C8×C53C2312yes—
137C4×C8×C53—22no—
138C24×C53C22×C411yescharacteristic
139C23×D53C22×C411yescharacteristic
140C22×C4×C53—22no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—41no—
144C22×C4×C53C22×C411yescharacteristic
145C22×C4×C53C22×C411yescharacteristic
146C22×C4×C53C22×C411yescharacteristic
147C22×C4×C53C22×C411yescharacteristic
148C22×C4×C53C22×C411yescharacteristic
149C22×C4×C53C2411yescharacteristic
150C2×C22⋊D53—22no—
151C2×C22⋊D53—22no—
152C2×C22⋊D53—41no—
153C22×Dic53—22no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—41no—
159C22×Dic53—42no—
160C22×Dic53C22×C411yescharacteristic
161C22×Dic53C22×C411yescharacteristic
162C22×Dic53C22×C411yescharacteristic
163C22×Dic53C22×C411yescharacteristic
164C22×Dic53C22×C411yescharacteristic
165C22×Dic53C22×C411yescharacteristic
166C22×Dic53C22×C411yescharacteristic
167C2×C4×D53—21no—
168C2×C4×D53—21no—
169C2×C4×D53—42no—
170C2×Dic106—42no—
171C2×Dic106C22⋊C412yes—
172C2×Dic106C22⋊C412yes—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—22no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53C2×D412yes—
182C42×C53C2×D412yes—
183C22⋊Dic53—22no—
184#11719921—21no—
185#11719921—21no—
186#11719921—22no—
187#11719921—22no—
188#11719921—21no—
189#11719921—21no—
190#11719921—22no—
191C4⋊Dic53—22no—
192C4⋊Dic53—22no—
193C4⋊Dic53C22⋊C412yes—
194C4⋊Dic53C22⋊C412yes—
195#11719919—21no—
196#11719919—21no—
197#11719919—22no—
198#11719919—22no—
199#11719919—21no—
200#11719919—21no—
201#11719919—21no—
202#11719919—21no—
203#11719919—22no—
204#11719919—21no—
205#11719919—21no—
206#11719919—22no—
207#11719919—22no—
208#11719919C22⋊C411yescharacteristic
209#11719919C22⋊C411yescharacteristic
210#11719919C22⋊C411yescharacteristic
211#11719919C22⋊C411yescharacteristic
212#11719919C22⋊C412yes—
213#11719919C22⋊C412yes—
214C4×Dic53—21no—
215C4×Dic53—21no—
216C4×Dic53C2×D412yes—
217C4×Dic53C2×D412yes—
218C4×Dic53C2×D412yes—
219C4×Dic53C2×D412yes—
220C2×#88322—22no—
221C2×#88322—22no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53#13112yes—
227C23×C53C23×C411yescharacteristic
228C22×D53—21no—
229C22×D53—21no—
230C22×D53—21no—
231C22×D53—41no—
232C22×D53—41no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53C2×C22⋊C412yes—
247C2×C4×C53C2×C22⋊C411yescharacteristic
248C2×C4×C53C2×C22⋊C411yescharacteristic
249C2×C4×C53C22×D412yes—
250C22⋊D53—44no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—22no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—41no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53C2×C22⋊C411yescharacteristic
269C2×Dic53C2×C22⋊C412yes—
270C2×Dic53C2×C22⋊C411yescharacteristic
271C2×Dic53C2×C22⋊C411yescharacteristic
272C2×Dic53C2×C22⋊C412yes—
273C2×Dic53C2×C22⋊C411yescharacteristic
274C4×D53—42no—
275C4×D53—42no—
276Dic106—24no—
277Dic106—42no—
278C8×C53—42no—
279C8×C53—42no—
280#88322—42no—
281#88322—42no—
282#17423—531nomaximal, Sylow
283C22×C53—21no—
284C22×C53—21no—
285C22×C53—22no—
286C22×C53—42no—
287C22×C53—42no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53C22×C22⋊C411yescharacteristic
291C22×C53C2×#13111yescharacteristic
292C22×C53#56811yescharacteristic
293C2×D53—41no—
294C2×D53—41no—
295C2×D53—41no—
296C2×D53—81no—
297C2×D53—81no—
298C4×C53—21no—
299C4×C53—21no—
300C4×C53—22no—
301C4×C53—22no—
302C4×C53—41no—
303C4×C53—41no—
304C4×C53—41no—
305Dic53—21no—
306Dic53—21no—
307Dic53—22no—
308Dic53—22no—
309Dic53—21no—
310Dic53—41no—
311Dic53—41no—
312Dic53—41no—
313Dic53#41312yes—
314C2×#552—531no—
315#2984—532no—
316#1473—532no—
317#1473—532no—
318#1470—532no—
319#1466—531no—
320#1464—531no—
321#1464—531no—
322#1460—531no—
323#1458—531no—
324#1458D5311yescharacteristic, complemented
325C2×C53—21no—
326C2×C53—21no—
327C2×C53—42no—
328C2×C53#288011yescharacteristic
329C2×C53#288411yescharacteristic
330C2×C53C2×#41311yescharacteristic
331D53—81no—
332#552—1062no—
333#552—1062no—
334#552—1062no—
335#552—1062no—
336C2×#126—531no—
337C2×#126—531no—
338C2×#125—531no—
339C2×#125—531no—
340C2×#125—531no—
341C2×#123—531no—
342C2×#123—531no—
343C2×C22⋊Q8—531no—
344C2×C22⋊Q8—531no—
345C2×C4×Q8—531no—
346C2×C4×D4—531no—
347C2×#117—531no—
348C2×#117—531no—
349C2×#117C2×D5311yescharacteristic
350C22×C4⋊C4—531no—
351#446—1062no—
352#446—1062no—
353#446—1062no—
354C8×Q8—1062no—
355#432—532no—
356#432—532no—
357#432—532no—
358#431—532no—
359#431—532no—
360#431C2×D5312yes—
361#324—532no—
362#324—532no—
363#324—532no—
364#324C2×D5312yes—
365#323—531no—
366#323—531no—
367#323—532no—
368#323—532no—
369#323C2×D5311yescharacteristic
370#323C2×D5311yescharacteristic
371C53#1742311yescharacteristic, complemented, Sylow
372C22×Q8—531no—
373C22×D4—531no—
374C23×C4—531no—
375C23×C4—531no—
376C23×C4—531no—
377C23×C4C4×D5311yescharacteristic
378#126—1062no—
379#126—1062no—
380#126—1062no—
381#126—1062no—
382#125—1062no—
383#125—1062no—
384#125—1062no—
385#125—1062no—
386#125—1062no—
387#125—1062no—
388#123—1062no—
389#123—1062no—
390#123—1062no—
391#123—1062no—
392C22⋊Q8—1062no—
393C22⋊Q8—1062no—
394C22⋊Q8—1062no—
395C22⋊Q8—1062no—
396C4×Q8—532no—
397C4×Q8—532no—
398C4×Q8—532no—
399C4×Q8—532no—
400C4×D4—1062no—
401C4×D4—1062no—
402#117—22no—
403#117—22no—
404#117—1061no—
405#117—1061no—
406#117—1061no—
407#117—1061no—
408#117—1062no—
409#117—1062no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4C4×D5311yescharacteristic
429C2×C4⋊C4C22×D5311yescharacteristic
430C2×C22⋊C4—531no—
431C2×C22⋊C4—531no—
432C2×C22⋊C4—531no—
433C2×C22⋊C4—531no—
434C2×C22⋊C4—531no—
435C2×C22⋊C4—531no—
436C2×C22⋊C4C4×D5311yescharacteristic
437C2×C22⋊C4C22×D5311yescharacteristic
438C2×C42—531no—
439C2×C42—531no—
440C2×C42—531no—
441C2×C42C4×D5311yescharacteristic
442C2×C42C22×D5311yescharacteristic
443C4⋊C8—22no—
444C4⋊C8—1062no—
445C4⋊C8—1062no—
446C4⋊C8—1062no—
447C4⋊C8—1062no—
448C4⋊C8—1062no—
449C22⋊C8—532no—
450C22⋊C8—532no—
451C22⋊C8C22×D5312yes—
452C22⋊C8C22×D5312yes—
453#97—1062no—
454#97—1062no—
455#97—1062no—
456C4×C8—22no—
457C4×C8—1062no—
458C4×C8—1062no—
459C24—531no—
460C24C2×C4×D5311yescharacteristic
461C2×Q8—532no—
462C2×Q8—532no—
463C2×Q8—2122no—
464C2×D4—1062no—
465C2×D4—1062no—
466C2×D4—2121no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—41no—
471C22×C4—531no—
472C22×C4—531no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1062no—
485C22×C4—2122no—
486C22×C4—2121no—
487C22×C4—2122no—
488C22×C4C2×C4×D5311yescharacteristic
489C22×C4C2×C4×D5311yescharacteristic
490C22×C4C2×C4×D5311yescharacteristic
491C22×C4C2×C4×D5311yescharacteristic
492C22×C4C2×C4×D5311yescharacteristic
493C22×C4C23×D5311yescharacteristic
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—1062no—
497C2×C8—1062no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—531no—
501C4⋊C4—531no—
502C4⋊C4—531no—
503C4⋊C4—531no—
504C4⋊C4—532no—
505C4⋊C4—532no—
506C4⋊C4—532no—
507C4⋊C4—532no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1062no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1062no—
516C4⋊C4—1062no—
517C4⋊C4—1062no—
518C4⋊C4—1062no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1062no—
522C4⋊C4—1062no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—1061no—
526C22⋊C4—1061no—
527C22⋊C4—1062no—
528C22⋊C4—1062no—
529C22⋊C4—1061no—
530C22⋊C4—1061no—
531C22⋊C4—1062no—
532C22⋊C4—1062no—
533C42—21no—
534C42—21no—
535C42—532no—
536C42—532no—
537C42—532no—
538C42—532no—
539C42—1061no—
540C42—1061no—
541C42D4×D5312yes—
542C42D4×D5312yes—
543C23—21no—
544C23—21no—
545C23—41no—
546C23—41no—
547C23—1061no—
548C23—1061no—
549C23—1061no—
550C23—2121no—
551C23—2121no—
552C23#1313322312yes—
553C23C22×C4×D5311yescharacteristic
554Q8—1064no—
555Q8—2122no—
556D4—2124no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—531no—
571C2×C4—531no—
572C2×C4—531no—
573C2×C4—532no—
574C2×C4—532no—
575C2×C4—531no—
576C2×C4—1062no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2121no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4C22⋊C4×D5311yescharacteristic
596C2×C4C22⋊C4×D5311yescharacteristic
597C2×C4C22⋊C4×D5312yes—
598C2×C4C2×D4×D5312yes—
599C8—42no—
600C8—42no—
601C8—2122no—
602C8—2122no—
603C22—21no—
604C22—21no—
605C22—22no—
606C22—42no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—4241no—
611C22—4241no—
612C22—2121no—
613C22—2121no—
614C22—2121no—
615C22C2×C22⋊C4×D5311yescharacteristic
616C22C2×#1313322311yescharacteristic
617C22#1681138911yescharacteristic
618C4—21no—
619C4—21no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—532no—
626C4—1061no—
627C4—1061no—
628C4—1062no—
629C4—1061no—
630C4—1062no—
631C4—2121no—
632C4—2121no—
633C4—2121no—
634C2—21no—
635C2—21no—
636C2—42no—
637C2—4241no—
638C2#2028316111yescharacteristic
639C2#2028317711yescharacteristic
640C2#413×D5311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1458C2
socleC22×C53C22×C22⋊C4
prime radicalC22×C22⋊D53C2×C4
cosocleC22×C4×C53C24
hypercenter#1458D53
hypocenterC53#17423
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #20283802C2semidirect
3. #20273083C2semidirect
4. #20273082C2semidirect
5. #20273069C2semidirect
7. #20273049C2semidirect
8. #20273046C2semidirect
10. #20273026C2semidirect
11. C53×#1458C2semidirect
324. #1458D53semidirect
371. C53#17423semidirect

Maximal subgroups

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

subgroupindex
C2×#168113062
#202838022
#202730832
#202730822
#202730692
#202730542
#202730492
#202730462
#202730342
#202730262
C53×#14582
#1742353