Finite Group Databasebeta

← #27351419

Subgroups — #27351419

Order 13568.

1046 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811159C211yesmaximal, characteristic
2C2×#16810673C211yesmaximal, characteristic, complemented
3#20277110C211yesmaximal, characteristic, complemented
4#20273291C214yesmaximal, complemented
5#20273287C211yesmaximal, characteristic, complemented
6#20273284C211yesmaximal, characteristic
7#20273282C211yesmaximal, characteristic
8#20273200C214yesmaximal, complemented
9C53×#1520C211yesmaximal, characteristic, complemented
10#16811159—21no—
11#16811159—21no—
12#16811159—21no—
13#16811159—21no—
14#16811159C412yescomplemented
15#16811159C412yescomplemented
16#16811159C2212yes—
17#16811159C2212yes—
18C2×#13133212—21no—
19C2×#13133208C411yescharacteristic, complemented
20C2×#13133206C411yescharacteristic, complemented
21C2×#13133201—21no—
22C2×#13133191C2211yescharacteristic
23C2×Q8×Dic53C411yescharacteristic, complemented
24C2×#13133179C411yescharacteristic, complemented
25C2×#13133177—21no—
26C2×#13133175—21no—
27C22×C4×Dic53C2211yescharacteristic
28#16810673—21no—
29#16810673—21no—
30#16810673C2212yescomplemented
31#16810673C2212yescomplemented
32C2×#13133146—21no—
33C2×C22⋊#88322C2211yescharacteristic
34C2×C8×Dic53—21no—
35#16810490C2212yescomplemented
36#16810488—21no—
37#16810484—21no—
38#16810483C2211yescharacteristic
39C4⋊C4×Dic53—21no—
40C22⋊C4×Dic53—21no—
41#16810468C2212yescomplemented
42#16810360C2214yescomplemented
43#16810358C2214yes—
44#16810356C2214yes—
45C2×C22⋊Q8×C53C2211yescharacteristic
46C2×C22⋊C8×C53C2211yescharacteristic
47C53×#380C2211yescharacteristic
48C53×#328C2214yescomplemented
49C22×D212C2×C411yescharacteristic
50C22×C22⋊D53—21no—
51C23×Dic53D411yescharacteristic
52C23×Dic53D411yescharacteristic
53C22×C4×D53—21no—
54C22×#88322—21no—
55#13133212—22no—
56#13133212—22no—
57#13133212—22no—
58#13133212—22no—
59#13133208—21no—
60#13133208—21no—
61#13133208C2×C412yes—
62#13133208C2×C412yes—
63#13133206—21no—
64#13133206—21no—
65#13133206C2×C412yes—
66#13133206C2×C412yes—
67#13133201—22no—
68#13133201—22no—
69#13133201—22no—
70#13133201—22no—
71#13133191—21no—
72#13133191—21no—
73#13133191C2×C412yes—
74#13133191C2312yes—
75Q8×Dic53—21no—
76Q8×Dic53—21no—
77Q8×Dic53C2×C412yes—
78Q8×Dic53C2×C412yes—
79#13133179—21no—
80#13133179—21no—
81#13133179C2×C412yes—
82#13133179C2×C412yes—
83#13133177—22no—
84#13133177—22no—
85#13133177—22no—
86#13133177—22no—
87#13133175—22no—
88#13133175—22no—
89#13133175—22no—
90#13133175—22no—
91C2×C4⋊Dic53—21no—
92C2×C4⋊Dic53—21no—
93C2×C4⋊Dic53—21no—
94C2×C4⋊Dic53C2×C411yescharacteristic
95C2×#11719919—21no—
96C2×C22⋊Dic53—21no—
97C2×#11719921—21no—
98C2×#11719921—21no—
99C2×#11719921—21no—
100C2×#11719921—21no—
101C2×C4×Dic53—21no—
102C2×C4×Dic53—22no—
103C2×C4×Dic53C2×C411yescharacteristic
104C2×C4×Dic53C2×C411yescharacteristic
105C2×C4×Dic53C2×C411yescharacteristic
106C2×C4×Dic53C2×C411yescharacteristic
107C2×C4×Dic53D411yescharacteristic
108C2×C4×Dic53D411yescharacteristic
109C2×C4×Dic53D411yescharacteristic
110C2×C4×Dic53D411yescharacteristic
111C2×C4×Dic53D411yescharacteristic
112C2×C4×Dic53D411yescharacteristic
113C2×C4×Dic53C2311yescharacteristic
114C2×C4×Dic53C2311yescharacteristic
115#13133146—22no—
116#13133146—22no—
117C22⋊#88322—21no—
118C22⋊#88322C2312yes—
119C8×Dic53—22no—
120C8×Dic53—22no—
121#13133124—22no—
122#13133124—22no—
123#13133124—22no—
124#13133124C2312yes—
125C22×Q8×C53C2×C411yescharacteristic
126C23×C4×C53C2311yescharacteristic
127C22×C8×C53—21no—
128C22⋊Q8×C53—21no—
129C22⋊Q8×C53—21no—
130C22⋊Q8×C53C2×C412yes—
131C22⋊Q8×C53C2312yes—
132C2×C4⋊C4×C53—21no—
133C2×C4⋊C4×C53—21no—
134C2×C4⋊C4×C53C2×C411yescharacteristic
135C2×C22⋊C4×C53—21no—
136C2×C22⋊C4×C53—21no—
137C22⋊C8×C53—21no—
138C22⋊C8×C53C2312yes—
139#95×C53C2312yes—
140C24×C53C2×D411yescharacteristic
141C23×D53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—42no—
144C2×Q8×C53C22×C412yes—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—22no—
148C22×C4×C53—42no—
149C22×C4×C53C22×C411yescharacteristic
150C22×C4×C53C22×C411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2411yescharacteristic
155C2×C22⋊D53—22no—
156C2×C22⋊D53—22no—
157C2×C22⋊D53—42no—
158C2×C22⋊D53—42no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—21no—
162C22×Dic53—22no—
163C22×Dic53—42no—
164C22×Dic53C22⋊C411yescharacteristic
165C22×Dic53C22⋊C411yescharacteristic
166C22×Dic53C22⋊C411yescharacteristic
167C22×Dic53C22⋊C411yescharacteristic
168C22×Dic53C22⋊C411yescharacteristic
169C22×Dic53C22⋊C411yescharacteristic
170C22×Dic53C22⋊C411yescharacteristic
171C22×Dic53C22⋊C411yescharacteristic
172C22×Dic53C2×D411yescharacteristic
173C22×Dic53C2×D411yescharacteristic
174C22×Dic53C2×D411yescharacteristic
175C22×Dic53C2×D411yescharacteristic
176C2×D212—21no—
177C2×D212—42no—
178C2×D212C22×C412yes—
179C2×C4×D53—22no—
180C2×C4×D53—22no—
181C2×C4×D53—44no—
182C2×C8×C53—22no—
183C2×C8×C53—42no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—22no—
187C4⋊C4×C53—42no—
188C4⋊C4×C53C22×C412yes—
189C22⋊C4×C53—22no—
190C22⋊C4×C53—22no—
191C22⋊C4×C53—42no—
192C22⋊Dic53—42no—
193#11719921—22no—
194#11719921—22no—
195#11719921—22no—
196#11719921—22no—
197#11719921—22no—
198#11719921—22no—
199#11719921—22no—
200#11719921—22no—
201C4⋊Dic53—21no—
202C4⋊Dic53—22no—
203C4⋊Dic53—22no—
204C4⋊Dic53—22no—
205C4⋊Dic53—22no—
206C4⋊Dic53—42no—
207C4⋊Dic53C22×C412yes—
208#11719919—22no—
209#11719919—22no—
210C4×Dic53—22no—
211C4×Dic53—22no—
212C4×Dic53—44no—
213C4×Dic53C22⋊C411yescharacteristic
214C4×Dic53C22⋊C411yescharacteristic
215C4×Dic53C22⋊C411yescharacteristic
216C4×Dic53C22⋊C411yescharacteristic
217C4×Dic53C22⋊C411yescharacteristic
218C4×Dic53C22⋊C411yescharacteristic
219C4×Dic53C22⋊C411yescharacteristic
220C4×Dic53C22⋊C411yescharacteristic
221C4×Dic53C22×C411yescharacteristic
222C4×Dic53C22×C411yescharacteristic
223C4×Dic53C22×C411yescharacteristic
224C4×Dic53C22×C411yescharacteristic
225C4×Dic53C2×D411yescharacteristic
226C4×Dic53C2×D411yescharacteristic
227C4×Dic53C2×D411yescharacteristic
228C4×Dic53C2×D411yescharacteristic
229C2×#88322—22no—
230C2×#88322—42no—
231C23×C53—41no—
232C23×C53—41no—
233C23×C53C2×C22⋊C411yescharacteristic
234C23×C53C2×C22⋊C411yescharacteristic
235C23×C53#13512yes—
236C23×C53#13512yes—
237C23×C53C22×D411yescharacteristic
238C22×D53—22no—
239C22×D53—42no—
240C22×D53—42no—
241C22×D53—42no—
242Q8×C53—42no—
243Q8×C53—42no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—42no—
257C2×C4×C53C2×C22⋊C411yescharacteristic
258C2×C4×C53C2×C22⋊C411yescharacteristic
259C2×C4×C53C23×C411yescharacteristic
260C2×C4×C53C22×D411yescharacteristic
261C22⋊D53—42no—
262C22⋊D53—42no—
263C22⋊D53—44no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—22no—
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—22no—
273C2×Dic53—22no—
274C2×Dic53—22no—
275C2×Dic53—22no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53C2×C22⋊C411yescharacteristic
284C2×Dic53C2×C22⋊C411yescharacteristic
285C2×Dic53C2×C22⋊C411yescharacteristic
286C2×Dic53C2×C22⋊C411yescharacteristic
287C2×Dic53C2×C22⋊C411yescharacteristic
288C2×Dic53C2×C22⋊C411yescharacteristic
289C2×Dic53C2×C22⋊C411yescharacteristic
290C2×Dic53C2×C22⋊C411yescharacteristic
291D212—42no—
292D212—42no—
293C4×D53—44no—
294C4×D53—44no—
295C8×C53—42no—
296#88322—42no—
297#17567—531nomaximal, Sylow
298C22×C53—21no—
299C22×C53—21no—
300C22×C53—21no—
301C22×C53—21no—
302C22×C53—21no—
303C22×C53—21no—
304C22×C53—42no—
305C22×C53—42no—
306C22×C53#41612yes—
307C22×C53#41612yes—
308C22×C53C22×C22⋊C411yescharacteristic
309C22×C53C2×#13511yescharacteristic
310C22×C53C2×#13511yescharacteristic
311C2×D53—42no—
312C2×D53—42no—
313C2×D53—42no—
314C2×D53—84no—
315C4×C53—21no—
316C4×C53—21no—
317C4×C53—21no—
318C4×C53—42no—
319C4×C53—42no—
320C4×C53—82no—
321C4×C53#41012yes—
322Dic53—21no—
323Dic53—21no—
324Dic53—21no—
325Dic53—21no—
326Dic53—21no—
327Dic53—21no—
328Dic53—21no—
329Dic53—42no—
330Dic53—82no—
331Dic53#42112yes—
332C2×#529—531no—
333C2×#431—531no—
334#2283—531no—
335#1521—534no—
336#1520—531no—
337#1520D5311yescharacteristic, complemented
338#1519—531no—
339#1519—531no—
340#1500—534no—
341C2×C53—21no—
342C2×C53—21no—
343C2×C53—22no—
344C2×C53—22no—
345C2×C53C2×#41011yescharacteristic
346C2×C53C2×#41611yescharacteristic
347C2×C53C2×#42111yescharacteristic
348D53—82no—
349#529—532no—
350#529—532no—
351#529—532no—
352#529—532no—
353#529—1061no—
354#529—1061no—
355#529—1061no—
356#529—1061no—
357C2×#126—1061no—
358C2×#125—531no—
359C2×#124—531no—
360C2×#123—1061no—
361C2×C22⋊Q8C2×D5311yescharacteristic
362C2×#121—531no—
363C2×C4×Q8—531no—
364C2×C4×D4—531no—
365C2×C4×D4—1061no—
366C2×#117—1061no—
367C22×C42—531no—
368#431—532no—
369#431—532no—
370#431—1061no—
371#431—1061no—
372C2×C4⋊C8—1061no—
373C2×C22⋊C8—531no—
374C2×C22⋊C8C2×D5311yescharacteristic
375C2×C4×C8—1061no—
376#383—532no—
377#382—1061no—
378#381—1061no—
379#380—531no—
380#380C2×D5311yescharacteristic
381C4×C4⋊C4—1061no—
382C4×C22⋊C4—1061no—
383#376—532no—
384#328—534no—
385#328C2×D5314yescomplemented
386#327—534no—
387#327—534no—
388C53#1756711yescharacteristic, complemented, Sylow
389C22×Q8C4×D5311yescharacteristic
390C22×D4—531no—
391C22×D4—1061no—
392C23×C4—531no—
393C23×C4—531no—
394C23×C4—1061no—
395C23×C4C22×D5311yescharacteristic
396C22×C8—21no—
397C22×C8—1061no—
398#126—1062no—
399#126—1062no—
400#126—1062no—
401#126—1062no—
402#125—532no—
403#125—532no—
404#125—1061no—
405#125—1061no—
406#124—532no—
407#124—532no—
408#124—1061no—
409#124—1061no—
410#123—1062no—
411#123—1062no—
412#123—1062no—
413#123—1062no—
414C22⋊Q8—21no—
415C22⋊Q8—21no—
416C22⋊Q8C4×D5312yes—
417C22⋊Q8C22×D5312yes—
418#121—532no—
419#121—532no—
420#121—1061no—
421#121—1061no—
422C4×Q8—532no—
423C4×Q8—532no—
424C4×Q8—1061no—
425C4×Q8—1061no—
426C4×D4—532no—
427C4×D4—532no—
428C4×D4—1061no—
429C4×D4—1061no—
430C4×D4—1062no—
431C4×D4—1062no—
432C4×D4—1062no—
433C4×D4—1062no—
434#117—1062no—
435#117—1062no—
436#117—1062no—
437#117—1062no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—531no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4C4×D5311yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—1061no—
449C2×C22⋊C4—1061no—
450C2×C22⋊C4—1061no—
451C2×C22⋊C4—1061no—
452C2×C22⋊C4—1061no—
453C2×C42—531no—
454C2×C42—531no—
455C2×C42—531no—
456C2×C42—531no—
457C2×C42—531no—
458C2×C42—531no—
459C2×C42—531no—
460C2×C42—531no—
461C2×C42—531no—
462C2×C42—531no—
463C2×C42—531no—
464C2×C42—531no—
465C2×C42—1061no—
466C2×C42—1062no—
467C4⋊C8—1062no—
468C4⋊C8—1062no—
469C22⋊C8—21no—
470C22⋊C8—532no—
471C22⋊C8—1061no—
472C22⋊C8C22×D5312yes—
473C4×C8—1062no—
474C4×C8—1062no—
475#95—532no—
476#95—1062no—
477#95—1062no—
478#95—1062no—
479#95C22×D5312yes—
480C24—1061no—
481C24D4×D5311yescharacteristic
482C2×Q8—21no—
483C2×Q8—42no—
484C2×Q8C2×C4×D5312yes—
485C2×D4—532no—
486C2×D4—1062no—
487C2×D4—1062no—
488C2×D4—1061no—
489C2×D4—2122no—
490C2×D4—2122no—
491C2×D4—2122no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—42no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—531no—
499C22×C4—531no—
500C22×C4—531no—
501C22×C4—531no—
502C22×C4—531no—
503C22×C4—531no—
504C22×C4—531no—
505C22×C4—531no—
506C22×C4—531no—
507C22×C4—531no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—1061no—
513C22×C4—1062no—
514C22×C4—2124no—
515C22×C4—2122no—
516C22×C4C2×C4×D5311yescharacteristic
517C22×C4C2×C4×D5311yescharacteristic
518C22×C4D4×D5311yescharacteristic
519C22×C4D4×D5311yescharacteristic
520C22×C4D4×D5311yescharacteristic
521C22×C4C23×D5311yescharacteristic
522C2×C8—22no—
523C2×C8—42no—
524C2×C8—1062no—
525C2×C8—2122no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—42no—
530C4⋊C4—532no—
531C4⋊C4—1062no—
532C4⋊C4—1062no—
533C4⋊C4—1062no—
534C4⋊C4—1062no—
535C4⋊C4—1062no—
536C4⋊C4—1062no—
537C4⋊C4—1061no—
538C4⋊C4—2122no—
539C4⋊C4C2×C4×D5312yes—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—42no—
543C22⋊C4—1062no—
544C22⋊C4—1062no—
545C22⋊C4—1062no—
546C22⋊C4—1062no—
547C22⋊C4—1062no—
548C22⋊C4—1062no—
549C22⋊C4—1062no—
550C22⋊C4—1062no—
551C22⋊C4—2122no—
552C42—531no—
553C42—531no—
554C42—531no—
555C42—531no—
556C42—531no—
557C42—531no—
558C42—531no—
559C42—531no—
560C42—531no—
561C42—531no—
562C42—531no—
563C42—531no—
564C42—531no—
565C42—531no—
566C42—531no—
567C42—531no—
568C42—1062no—
569C42—1062no—
570C42—2124no—
571C23—41no—
572C23—41no—
573C23—1062no—
574C23—2122no—
575C23—2122no—
576C23—2122no—
577C23C22⋊C4×D5311yescharacteristic
578C23C22⋊C4×D5311yescharacteristic
579C23#1313323812yes—
580C23#1313324012yes—
581C23C2×D4×D5311yescharacteristic
582Q8—42no—
583Q8—42no—
584D4—2122no—
585D4—2124no—
586D4—2122no—
587D4—2122no—
588D4—2122no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—531no—
603C2×C4—531no—
604C2×C4—531no—
605C2×C4—531no—
606C2×C4—531no—
607C2×C4—531no—
608C2×C4—531no—
609C2×C4—531no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—1062no—
615C2×C4—1062no—
616C2×C4—1062no—
617C2×C4—1061no—
618C2×C4—1061no—
619C2×C4—1061no—
620C2×C4—1061no—
621C2×C4—1062no—
622C2×C4—2124no—
623C2×C4—2124no—
624C2×C4—2121no—
625C2×C4—2121no—
626C2×C4—2121no—
627C2×C4—2122no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4C22⋊C4×D5311yescharacteristic
632C2×C4C22⋊C4×D5311yescharacteristic
633C2×C4C22×C4×D5311yescharacteristic
634C2×C4C2×D4×D5311yescharacteristic
635C8—42no—
636C8—2122no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—42no—
644C22—42no—
645C22—4244no—
646C22—2122no—
647C22—2122no—
648C22—2122no—
649C22#1681061812yes—
650C22#1681061812yes—
651C22C2×C22⋊C4×D5311yescharacteristic
652C22C2×#1313323811yescharacteristic
653C22C2×#1313324011yescharacteristic
654C4—21no—
655C4—21no—
656C4—21no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—532no—
661C4—1061no—
662C4—1061no—
663C4—1061no—
664C4—1061no—
665C4—1061no—
666C4—1061no—
667C4—1061no—
668C4—4242no—
669C4—2122no—
670C4#1681059512yes—
671C2—21no—
672C2—21no—
673C2—22no—
674C2—22no—
675C2—4242no—
676C2C2×#1681059511yescharacteristic
677C2C2×#1681061811yescharacteristic
678C2#421×D5311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1520C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#13133191C22
cosocleC22×C4×C53C24
hypercenter#1520D53
hypocenterC53#17567
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
2. C2×#16810673C2semidirect
3. #20277110C2semidirect
4. #20273291C2semidirect
5. #20273287C2semidirect
8. #20273200C2semidirect
9. C53×#1520C2semidirect
14. #16811159C4semidirect
15. #16811159C4semidirect
19. C2×#13133208C4semidirect
20. C2×#13133206C4semidirect
23. C2×Q8×Dic53C4semidirect
24. C2×#13133179C4semidirect
30. #16810673C22semidirect
31. #16810673C22semidirect
35. #16810490C22semidirect
41. #16810468C22semidirect
42. #16810360C22semidirect
48. C53×#328C22semidirect
337. #1520D53semidirect
385. #328C2×D53semidirect
388. C53#17567semidirect

Maximal subgroups

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

subgroupindex
C2×#168111592
C2×#168106732
#202771102
#202732912
#202732872
#202732842
#202732822
#202732002
C53×#15202
#1756753