Finite Group Databasebeta

← #27350411

Subgroups — #27350411

Order 13568.

916 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×#16811341C211yesmaximal, characteristic
2#20283802C212yesmaximal, complemented
3#20273082C212yesmaximal, complemented
4#20273081C212yesmaximal, complemented
5#20273065C212yesmaximal
6#20273057C211yesmaximal, characteristic, complemented
7#20273049C211yesmaximal, characteristic, complemented
8#20273048C211yesmaximal, characteristic, complemented
9#20273037C211yesmaximal, characteristic, complemented
10#20273028C211yesmaximal, characteristic, complemented
11C53×#1458C211yesmaximal, characteristic, complemented
12#16811341—22no—
13#16811341—22no—
14#16811341—22no—
15#16811341—22no—
16C2×#13133216C411yescharacteristic
17C2×#13133211C411yescharacteristic
18C2×#13133211C2211yescharacteristic
19C2×#13133210C411yescharacteristic
20C2×#13133210C2211yescharacteristic
21C2×#13133203C411yescharacteristic
22C2×#13133203C2211yescharacteristic
23C2×#13133197C411yescharacteristic
24C2×#13133196C411yescharacteristic
25C2×C4×Dic106C411yescharacteristic
26C2×C4×Dic106C2211yescharacteristic
27C2×#13133182C2211yescharacteristic
28C2×C4×C22⋊D53C411yescharacteristic
29C2×#13133175C2211yescharacteristic
30#16810753—22no—
31#16810752—22no—
32#16810752—22no—
33C8×Dic106—22no—
34#16810680C2212yes—
35#16810677C2212yes—
36#16810677C2212yes—
37#16810673C2212yes—
38#16810673C2212yes—
39#16810349C2211yescharacteristic
40#16810349C2211yescharacteristic
41#16810348C2212yes—
42#16810348C2212yes—
43#16810347C2212yes—
44#16810346C2212yes—
45#16810346C2212yes—
46C2×#117×C53C2211yescharacteristic
47C53×#431C2212yes—
48C53×#324C2211yescharacteristic
49C53×#324C2211yescharacteristic
50C53×#323C2212yes—
51C22×Dic106C2×C411yescharacteristic
52C22×Dic106C2×C411yescharacteristic
53C22×Dic106C2×C411yescharacteristic
54C22×C22⋊D53C2×C411yescharacteristic
55C22×C4×D53C2×C411yescharacteristic
56#13133216—22no—
57#13133216—22no—
58#13133211—22no—
59#13133211—22no—
60#13133211—22no—
61#13133211—22no—
62#13133210—22no—
63#13133210—22no—
64#13133210—22no—
65#13133210—22no—
66#13133203—22no—
67#13133203—22no—
68#13133203—22no—
69#13133203—22no—
70#13133197—22no—
71#13133197—22no—
72#13133196—22no—
73#13133196—22no—
74C4×Dic106—21no—
75C4×Dic106—21no—
76C4×Dic106—21no—
77C4×Dic106—21no—
78C4×Dic106D412yes—
79C4×Dic106D412yes—
80C4×Dic106D412yes—
81C4×Dic106D412yes—
82#13133182—22no—
83#13133182—22no—
84C4×C22⋊D53—21no—
85C4×C22⋊D53—21no—
86C4×C22⋊D53—21no—
87C4×C22⋊D53—21no—
88#13133175—22no—
89#13133175—22no—
90C2×C4⋊Dic53C2×C411yescharacteristic
91C2×C4⋊Dic53C2×C411yescharacteristic
92C2×C4⋊Dic53C2×C411yescharacteristic
93C2×#11719919C2×C411yescharacteristic
94C2×#11719919C2×C411yescharacteristic
95C2×#11719919C2×C411yescharacteristic
96C2×#11719919C2×C411yescharacteristic
97C2×#11719919C2×C411yescharacteristic
98C2×#11719919C2×C411yescharacteristic
99C2×#11719919C2311yescharacteristic
100C2×C22⋊Dic53C2×C411yescharacteristic
101C2×C22⋊Dic53C2×C411yescharacteristic
102C2×C22⋊Dic53C2×C411yescharacteristic
103C2×#11719921C2×C411yescharacteristic
104C2×#11719921C2×C411yescharacteristic
105C2×#11719921C2×C411yescharacteristic
106C2×#11719921C2×C411yescharacteristic
107C2×#11719921C2311yescharacteristic
108C2×C4×Dic53C2×C411yescharacteristic
109C2×C4×Dic53C2×C411yescharacteristic
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—21no—
141C22×C4×C53—21no—
142C22×C4×C53—22no—
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—21no—
151C2×C22⋊D53—21no—
152C2×C22⋊D53—42no—
153C22×Dic53C22×C411yescharacteristic
154C22×Dic53C22×C411yescharacteristic
155C22×Dic53C22×C411yescharacteristic
156C22×Dic53C22×C411yescharacteristic
157C22×Dic53C22×C411yescharacteristic
158C22×Dic53C22×C411yescharacteristic
159C22×Dic53C22×C411yescharacteristic
160C2×C4×D53—22no—
161C2×C4×D53—21no—
162C2×C4×D53—21no—
163C2×C4×D53—41no—
164C2×Dic106—22no—
165C2×Dic106—22no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—42no—
169C2×Dic106—41no—
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—21no—
184C22⋊Dic53—21no—
185C22⋊Dic53—22no—
186C22⋊Dic53—22no—
187#11719921—21no—
188#11719921—21no—
189#11719921—22no—
190#11719921—22no—
191#11719921—22no—
192#11719921—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53C22⋊C412yes—
197C4⋊Dic53C22⋊C412yes—
198#11719919—21no—
199#11719919—21no—
200#11719919—22no—
201#11719919—22no—
202#11719919—22no—
203#11719919—22no—
204#11719919C22⋊C411yescharacteristic
205#11719919C22⋊C411yescharacteristic
206#11719919C22⋊C411yescharacteristic
207#11719919C22⋊C411yescharacteristic
208#11719919C22⋊C412yes—
209#11719919C22⋊C412yes—
210C4×Dic53—21no—
211C4×Dic53—21no—
212C4×Dic53—21no—
213C4×Dic53—21no—
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—21no—
225C23×C53—41no—
226C23×C53—41no—
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—41no—
240C2×C4×C53—41no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53C2×C22⋊C412yes—
246C2×C4×C53C2×C22⋊C411yescharacteristic
247C2×C4×C53C2×C22⋊C411yescharacteristic
248C2×C4×C53#13112yes—
249C2×C4×C53C22×D412yes—
250C22⋊D53—42no—
251C22⋊D53—42no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—22no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53C2×C22⋊C411yescharacteristic
266C2×Dic53C2×C22⋊C412yes—
267C2×Dic53C2×C22⋊C412yes—
268C2×Dic53C2×C22⋊C411yescharacteristic
269C2×Dic53C2×C22⋊C411yescharacteristic
270C2×Dic53C2×C22⋊C411yescharacteristic
271C4×D53—42no—
272C4×D53—42no—
273Dic106—24no—
274Dic106—42no—
275Dic106—44no—
276Dic106—42no—
277Dic106—42no—
278C8×C53—42no—
279C8×C53—42no—
280#88322—42no—
281#88322—42no—
282#17428—531nomaximal, Sylow
283C22×C53—21no—
284C22×C53—21no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—81no—
289C22×C53—81no—
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—22no—
299C4×C53—21no—
300C4×C53—21no—
301C4×C53—22no—
302C4×C53—22no—
303C4×C53—41no—
304C4×C53—41no—
305Dic53—22no—
306Dic53—21no—
307Dic53—21no—
308Dic53—22no—
309Dic53—21no—
310Dic53—41no—
311Dic53—41no—
312Dic53—41no—
313Dic53#41312yes—
314C2×#556—531no—
315#2984—532no—
316#1473—532no—
317#1473—532no—
318#1469—532no—
319#1467—531no—
320#1464—531no—
321#1464—531no—
322#1461—531no—
323#1458—531no—
324#1458D5311yescharacteristic, complemented
325C2×C53—21no—
326C2×C53—21no—
327C2×C53—81no—
328C2×C53#288011yescharacteristic
329C2×C53#288411yescharacteristic
330C2×C53C2×#41311yescharacteristic
331D53—81no—
332#556—1062no—
333#556—1062no—
334#556—1062no—
335#556—1062no—
336C2×C4⋊Q8—531no—
337C2×#126—531no—
338C2×#126—531no—
339C2×#126—531no—
340C2×#126—531no—
341C2×#124—531no—
342C2×#124—531no—
343C2×C22⋊Q8—531no—
344C2×C22⋊Q8—531no—
345C2×C4×Q8—531no—
346C2×C4×Q8—531no—
347C2×C4×Q8—531no—
348C2×C4×D4—531no—
349C2×#117—531no—
350C2×#117C2×D5311yescharacteristic
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—531no—
362#324—531no—
363#324—532no—
364#324—532no—
365#324C2×D5311yescharacteristic
366#324C2×D5311yescharacteristic
367#323—532no—
368#323—532no—
369#323—532no—
370#323C2×D5312yes—
371C53#1742811yescharacteristic, complemented, Sylow
372C22×Q8—531no—
373C22×Q8—531no—
374C22×Q8—531no—
375C22×D4—531no—
376C23×C4—531no—
377C23×C4C4×D5311yescharacteristic
378C4⋊Q8—1062no—
379C4⋊Q8—1062no—
380#126—1062no—
381#126—1062no—
382#126—1062no—
383#126—1062no—
384#126—1062no—
385#126—1062no—
386#126—1062no—
387#126—1062no—
388#124—1062no—
389#124—1062no—
390#124—1062no—
391#124—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×Q8—1061no—
401C4×Q8—1061no—
402C4×Q8—1061no—
403C4×Q8—1061no—
404C4×Q8—1062no—
405C4×Q8—1062no—
406C4×D4—1061no—
407C4×D4—1061no—
408C4×D4—1061no—
409C4×D4—1061no—
410#117—22no—
411#117—22no—
412#117—1062no—
413#117—1062no—
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⋊C4C4×D5311yescharacteristic
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C22⋊C4—531no—
427C2×C22⋊C4—531no—
428C2×C22⋊C4—531no—
429C2×C22⋊C4—531no—
430C2×C22⋊C4—531no—
431C2×C22⋊C4—531no—
432C2×C22⋊C4—531no—
433C2×C22⋊C4—531no—
434C2×C22⋊C4C4×D5311yescharacteristic
435C2×C22⋊C4C22×D5311yescharacteristic
436C2×C42—531no—
437C2×C42—531no—
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—1062no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8—1062no—
467C2×Q8—2121no—
468C2×Q8—2122no—
469C2×Q8—2122no—
470C2×D4—1061no—
471C2×D4—1061no—
472C2×D4—2122no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—41no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1062no—
487C22×C4—2121no—
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—1062no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1062no—
512C4⋊C4—1062no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1062no—
516C4⋊C4—1062no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—1062no—
520C22⋊C4—1062no—
521C22⋊C4—1061no—
522C22⋊C4—1061no—
523C22⋊C4—1062no—
524C22⋊C4—1062no—
525C22⋊C4—1061no—
526C22⋊C4—1061no—
527C22⋊C4—1062no—
528C22⋊C4—1062no—
529C42—21no—
530C42—21no—
531C42—532no—
532C42—532no—
533C42—532no—
534C42—532no—
535C42—1061no—
536C42—1061no—
537C42—1061no—
538C42—1061no—
539C42—1061no—
540C42—1061no—
541C42D4×D5312yes—
542C42D4×D5312yes—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—41no—
547C23—41no—
548C23—1061no—
549C23—1061no—
550C23—1061no—
551C23—2121no—
552C23—2121no—
553C23C22×C4×D5311yescharacteristic
554Q8—1064no—
555Q8—2122no—
556Q8—2124no—
557Q8—2122no—
558Q8—2122no—
559D4—2122no—
560D4—2122no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—531no—
574C2×C4—531no—
575C2×C4—531no—
576C2×C4—531no—
577C2×C4—532no—
578C2×C4—532no—
579C2×C4—1062no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4C22⋊C4×D5311yescharacteristic
595C2×C4C22⋊C4×D5311yescharacteristic
596C2×C4C22⋊C4×D5312yes—
597C2×C4#1313322312yes—
598C2×C4C2×D4×D5312yes—
599C8—42no—
600C8—42no—
601C8—2122no—
602C8—2122no—
603C22—21no—
604C22—21no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—81no—
609C22—81no—
610C22—4241no—
611C22—4241no—
612C22—2121no—
613C22—2121no—
614C22—2121no—
615C22C2×C22⋊C4×D5311yescharacteristic
616C22C2×#1313322311yescharacteristic
617C22#1681138911yescharacteristic
618C4—22no—
619C4—21no—
620C4—21no—
621C4—22no—
622C4—22no—
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—81no—
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#17428
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. #20273082C2semidirect
4. #20273081C2semidirect
6. #20273057C2semidirect
7. #20273049C2semidirect
8. #20273048C2semidirect
9. #20273037C2semidirect
10. #20273028C2semidirect
11. C53×#1458C2semidirect
324. #1458D53semidirect
371. C53#17428semidirect

Maximal subgroups

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

subgroupindex
C2×#168113412
#202838022
#202730822
#202730812
#202730652
#202730572
#202730492
#202730482
#202730372
#202730282
C53×#14582
#1742853