Finite Group Databasebeta

← #27351817

Subgroups — #27351817

Order 13568.

1038 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288554C211yesmaximal, characteristic
2#20277655C211yesmaximal, characteristic, complemented
3#20273293C212yesmaximal, complemented
4#20273288C211yesmaximal, characteristic, complemented
5#20273284C211yesmaximal, characteristic, complemented
6#20273272C212yesmaximal, complemented
7#20273207C212yesmaximal, complemented
8#20273202C212yesmaximal, complemented
9C2×#16810361C211yesmaximal, characteristic, complemented
10C2×#16810358C211yesmaximal, characteristic, complemented
11C53×#2972C211yesmaximal, characteristic, complemented
12#16811322—22no—
13#16811319C412yescomplemented
14#16811313C412yescomplemented
15#16811302—22no—
16#16811282C412yescomplemented
17#16811269C412yescomplemented
18#16811220—21no—
19#16811211—21no—
20#16811182—21no—
21#16811157C2212yes—
22#16811134C2212yes—
23C2×#13133196C2211yescharacteristic
24C2×#13133191C2211yescharacteristic
25#16811096—21no—
26#16810516—21no—
27#16810509—21no—
28#16810503—21no—
29#16810491C2211yescharacteristic, complemented
30#16810486C2211yescharacteristic, complemented
31#16810483C2211yescharacteristic
32C22⋊C4×Dic53—21no—
33C4×C22⋊Dic53C2211yescharacteristic, complemented
34#16810469C2211yescharacteristic, complemented
35C2×#13133124C2211yescharacteristic
36#16810361C2212yescomplemented
37#16810361C2212yescomplemented
38#16810361C2212yescomplemented
39#16810361C2212yescomplemented
40#16810358C2212yes—
41#16810358C2212yes—
42#16810358C2212yes—
43#16810358C2212yes—
44C2×C4×D4×C53C2211yescharacteristic
45C8×D4×C53—22no—
46C8×D4×C53—22no—
47C53×#431C2212yescomplemented
48C2×C22⋊C8×C53C2211yescharacteristic
49C2×C22⋊C8×C53C2211yescharacteristic
50C2×C22⋊C8×C53C2212yescomplemented
51C2×#11719945C2×C412yescomplemented
52C22×Dic106C2×C411yescharacteristic
53C22×D212C2×C411yescharacteristic
54C22×C22⋊D53—21no—
55C23×Dic53—21no—
56#13133212—22no—
57#13133211—22no—
58#13133210—22no—
59#13133209C2×C412yescomplemented
60#13133207—22no—
61#13133205C2×C412yescomplemented
62#13133203—22no—
63#13133202—22no—
64#13133200—22no—
65#13133200—22no—
66#13133199—22no—
67#13133198—22no—
68#13133196—21no—
69#13133196—21no—
70#13133196C2×C412yes—
71#13133196C2×C412yescomplemented
72#13133196C2312yes—
73#13133195—22no—
74#13133191—21no—
75#13133191—21no—
76#13133191C2×C412yes—
77#13133191C2×C412yescomplemented
78#13133191C2312yes—
79#13133190—22no—
80#13133189—22no—
81#13133189—22no—
82#13133186—22no—
83#13133185—22no—
84C4×Dic106C2×C412yescomplemented
85C4×D212C2×C412yescomplemented
86C4×C22⋊D53—22no—
87C4×C22⋊D53—22no—
88C4×C22⋊D53—22no—
89#13133177—22no—
90#13133177—22no—
91#13133176C2×C412yescomplemented
92#13133174—22no—
93#13133173—22no—
94C2×C4⋊Dic53—21no—
95C2×C4⋊Dic53C2×C411yescharacteristic
96C2×C4⋊Dic53C2×C411yescharacteristic
97C2×#11719919—21no—
98C2×C22⋊Dic53—21no—
99C2×C22⋊Dic53—21no—
100C2×C22⋊Dic53—21no—
101C2×C22⋊Dic53—21no—
102C2×#11719921—21no—
103C22⋊C4×D53—22no—
104C2×C4×Dic53—21no—
105C2×C4×Dic53—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124C2311yescharacteristic
112#13133124C2311yescharacteristic
113#13133124C2311yescharacteristic
114#13133124C2311yescharacteristic
115C22×D4×C53D411yescharacteristic
116C23×C4×C53C2311yescharacteristic
117C23×C4×C53C2311yescharacteristic
118C22×C8×C53—21no—
119C22×C8×C53—21no—
120C22×C8×C53—22no—
121C4×D4×C53C2×C412yes—
122C4×D4×C53C2×C412yes—
123C4×D4×C53D412yes—
124C4×D4×C53D412yes—
125C2×C4⋊C4×C53D411yescharacteristic
126C2×C22⋊C4×C53D411yescharacteristic
127C2×C22⋊C4×C53D411yescharacteristic
128C2×C42×C53C2311yescharacteristic
129C4⋊C8×C53—22no—
130C22⋊C8×C53—21no—
131C22⋊C8×C53—21no—
132C22⋊C8×C53—22no—
133C22⋊C8×C53C2312yes—
134C22⋊C8×C53C2312yes—
135C4×C8×C53—22no—
136C24×C53C2×D411yescharacteristic
137C24×C53C2×D411yescharacteristic
138C23×D53—21no—
139C2×D4×C53—22no—
140C2×D4×C53—22no—
141C2×D4×C53C22⋊C412yes—
142C2×D4×C53C22⋊C412yes—
143C22×C4×C53—81no—
144C22×C4×C53—81no—
145C22×C4×C53C22×C411yescharacteristic
146C22×C4×C53C22×C411yescharacteristic
147C22×C4×C53C22×C412yes—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D412yes—
155C22×C4×C53C2411yescharacteristic
156C2×C22⋊D53—22no—
157C2×C22⋊D53—22no—
158C2×C22⋊D53—22no—
159C2×C22⋊D53—22no—
160C2×C22⋊D53—42no—
161C2×C22⋊D53—42no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—21no—
166C22×Dic53—21no—
167C22×Dic53—21no—
168C22×Dic53—21no—
169C22×Dic53—42no—
170C22×Dic53—42no—
171C22×Dic53—81no—
172#11719945—82no—
173C2×D212—21no—
174C2×D212—81no—
175C2×D212C22×C412yes—
176C2×C4×D53—22no—
177C2×C4×D53—22no—
178C2×Dic106—21no—
179C2×Dic106—81no—
180C2×Dic106C22×C412yes—
181C2×C8×C53—22no—
182C2×C8×C53—22no—
183C2×C8×C53—81no—
184C2×C8×C53—81no—
185C2×C8×C53—82no—
186C4⋊C4×C53C22⋊C412yes—
187C4⋊C4×C53C22⋊C412yes—
188C22⋊C4×C53C22⋊C412yes—
189C22⋊C4×C53C22⋊C412yes—
190C22⋊C4×C53C22⋊C412yes—
191C22⋊C4×C53C22⋊C412yes—
192C42×C53C22×C412yes—
193C42×C53C2×D412yes—
194C22⋊Dic53—22no—
195C22⋊Dic53—22no—
196C22⋊Dic53—22no—
197C22⋊Dic53—22no—
198C22⋊Dic53—42no—
199C22⋊Dic53—42no—
200C22⋊Dic53—81no—
201#11719921—22no—
202#11719921—22no—
203#11719921—22no—
204#11719921—22no—
205#11719921—22no—
206#11719921—22no—
207C4⋊Dic53—21no—
208C4⋊Dic53—21no—
209C4⋊Dic53—81no—
210C4⋊Dic53C22×C412yes—
211C4⋊Dic53C22×C412yes—
212#11719919—22no—
213#11719919—22no—
214#11719919—22no—
215#11719919—22no—
216#11719919—22no—
217#11719919—22no—
218C4×Dic53—22no—
219C4×Dic53—22no—
220C4×Dic53—42no—
221C4×Dic53—81no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53C2×C22⋊C412yes—
229C23×C53C2×C22⋊C411yescharacteristic
230C23×C53C2×C22⋊C411yescharacteristic
231C23×C53#13512yes—
232C23×C53#13512yes—
233C23×C53C22×D411yescharacteristic
234C22×D53—22no—
235C22×D53—42no—
236C22×D53—81no—
237C22×D53—81no—
238D4×C53—24no—
239D4×C53—24no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—42no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—42no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53C2×C22⋊C411yescharacteristic
253C2×C4×C53C2×C22⋊C412yes—
254C2×C4×C53C2×C22⋊C412yes—
255C2×C4×C53C2×C22⋊C411yescharacteristic
256C2×C4×C53C2×C22⋊C412yes—
257C2×C4×C53C23×C411yescharacteristic
258C2×C4×C53C22×D411yescharacteristic
259C22⋊D53—42no—
260C22⋊D53—42no—
261C22⋊D53—44no—
262C22⋊D53—82no—
263C22⋊D53—82no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—42no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—81no—
281C2×Dic53—81no—
282C2×Dic53—84no—
283D212—42no—
284D212—81no—
285C4×D53—82no—
286C4×D53—82no—
287Dic106—42no—
288Dic106—81no—
289C8×C53—42no—
290C8×C53—42no—
291#17615—531nomaximal, Sylow
292C22×C53—21no—
293C22×C53—21no—
294C22×C53—21no—
295C22×C53—21no—
296C22×C53—21no—
297C22×C53—21no—
298C22×C53—22no—
299C22×C53—22no—
300C22×C53—22no—
301C22×C53—42no—
302C22×C53—42no—
303C22×C53—81no—
304C22×C53—81no—
305C22×C53#41612yes—
306C22×C53#41612yes—
307C22×C53C22×C22⋊C411yescharacteristic
308C22×C53C2×#13511yescharacteristic
309C22×C53C2×#13511yescharacteristic
310C2×D53—42no—
311C2×D53—42no—
312C2×D53—42no—
313C2×D53—162no—
314C4×C53—21no—
315C4×C53—21no—
316C4×C53—21no—
317C4×C53—22no—
318C4×C53—21no—
319C4×C53—22no—
320C4×C53—22no—
321Dic53—42no—
322Dic53—42no—
323Dic53—42no—
324Dic53—42no—
325Dic53—81no—
326Dic53—81no—
327Dic53—81no—
328#3505—531no—
329#2972D5311yescharacteristic, complemented
330#2353—531no—
331#1521—532no—
332#1520—531no—
333#1519—531no—
334#1516—532no—
335#1501—532no—
336#1500—532no—
337C2×#328—531no—
338C2×#327—531no—
339C2×C53—21no—
340C2×C53—21no—
341C2×C53—22no—
342C2×C53—22no—
343C2×C53—42no—
344C2×C53#288311yescharacteristic
345C2×C53C2×#41611yescharacteristic
346C2×C53#290111yescharacteristic
347D53—82no—
348#553—532no—
349#553—1062no—
350#552—532no—
351#551—1062no—
352#548—532no—
353#546—532no—
354#539—1061no—
355#538—1061no—
356#534—1061no—
357#529—532no—
358#525—532no—
359C2×C22⋊Q8—531no—
360C2×#121—531no—
361#518—1061no—
362C2×C4×D4C2×D5311yescharacteristic
363C8×D4—22no—
364C8×D4—22no—
365#431C2×D5312yescomplemented
366C2×C22⋊C8C2×D5311yescharacteristic
367C2×C22⋊C8C2×D5311yescharacteristic
368C2×C22⋊C8C2×D5312yescomplemented
369#388—1061no—
370#387—1061no—
371#386—1061no—
372#383—531no—
373#381—531no—
374#380—531no—
375C4×C22⋊C4—531no—
376C4×C22⋊C4—1061no—
377#376—531no—
378C2×#95—531no—
379#328—532no—
380#328—532no—
381#328—532no—
382#328—532no—
383#327—532no—
384#327—532no—
385#327—532no—
386#327—532no—
387C53#1761511yescharacteristic, complemented, Sylow
388C2×#41—532no—
389C22×Q8—531no—
390C22×D4—531no—
391C22×D4—1061no—
392C22×D4D21211yescharacteristic
393C23×C4—1061no—
394C23×C4C22×D5311yescharacteristic
395C23×C4C22×D5311yescharacteristic
396C22×C8—21no—
397C22×C8—21no—
398C22×C8—22no—
399#126—1062no—
400#126—1062no—
401#126—1062no—
402#125—532no—
403#125—1062no—
404#124—532no—
405#124—1062no—
406#123—1062no—
407#123—1062no—
408#123—1062no—
409#123—1062no—
410#123—1062no—
411C22⋊Q8—532no—
412C22⋊Q8—532no—
413C22⋊Q8—532no—
414C22⋊Q8—1061no—
415C22⋊Q8—1061no—
416C22⋊Q8—1062no—
417#121—532no—
418#121—532no—
419#121—532no—
420#121—1062no—
421#121—1062no—
422#121—1061no—
423#121—1061no—
424#121—1062no—
425#120—1062no—
426#120—1062no—
427C4×Q8—532no—
428C4×D4—532no—
429C4×D4—1062no—
430C4×D4—1062no—
431C4×D4—1062no—
432C4×D4—1062no—
433C4×D4—1062no—
434C4×D4C4×D5312yes—
435C4×D4C4×D5312yes—
436C4×D4C22⋊D5312yes—
437C4×D4C22⋊D5312yes—
438#117—532no—
439#117—1062no—
440#117—1062no—
441C2×C4⋊C4—531no—
442C2×C4⋊C4—531no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4D21211yescharacteristic
446C2×C22⋊C4—1061no—
447C2×C22⋊C4—1061no—
448C2×C22⋊C4—1062no—
449C2×C22⋊C4—1061no—
450C2×C22⋊C4—1061no—
451C2×C22⋊C4—1061no—
452C2×C22⋊C4D21211yescharacteristic
453C2×C22⋊C4D21211yescharacteristic
454C2×C42—1061no—
455C2×C42—1061no—
456C2×C42C22×D5311yescharacteristic
457C4⋊C8—22no—
458C22⋊C8—21no—
459C22⋊C8—21no—
460C22⋊C8—22no—
461C22⋊C8C22×D5312yes—
462C22⋊C8C22×D5312yes—
463C4×C8—22no—
464#95—531no—
465#95—531no—
466#95—531no—
467#95—531no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473C24—1061no—
474C24C2×D21211yescharacteristic
475C24C2×D21211yescharacteristic
476#41—4242no—
477C2×Q8—532no—
478C2×Q8—1061no—
479C2×Q8—4241no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—532no—
483C2×D4—1062no—
484C2×D4—1062no—
485C2×D4—1062no—
486C2×D4—1062no—
487C2×D4—1061no—
488C2×D4—4241no—
489C2×D4—2122no—
490C2×D4—2122no—
491C2×D4#1171992112yes—
492C2×D4#1171992112yes—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—1061no—
504C22×C4—4241no—
505C22×C4—2122no—
506C22×C4—2122no—
507C22×C4C2×C4×D5311yescharacteristic
508C22×C4C2×C4×D5311yescharacteristic
509C22×C4C2×C4×D5312yes—
510C22×C4C2×D21211yescharacteristic
511C22×C4C2×D21211yescharacteristic
512C22×C4C2×D21211yescharacteristic
513C22×C4C2×D21211yescharacteristic
514C22×C4C2×C22⋊D5311yescharacteristic
515C22×C4C2×C22⋊D5311yescharacteristic
516C22×C4C2×C22⋊D5312yes—
517C22×C4C23×D5311yescharacteristic
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—81no—
521C2×C8—81no—
522C2×C8—82no—
523C4⋊C4—532no—
524C4⋊C4—532no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1062no—
528C4⋊C4—1062no—
529C4⋊C4—1062no—
530C4⋊C4—1062no—
531C4⋊C4—1062no—
532C4⋊C4—1062no—
533C4⋊C4—4241no—
534C4⋊C4#1171992112yes—
535C4⋊C4#1171992112yes—
536C22⋊C4—1062no—
537C22⋊C4—1062no—
538C22⋊C4—1062no—
539C22⋊C4—1062no—
540C22⋊C4—1062no—
541C22⋊C4—1062no—
542C22⋊C4—1062no—
543C22⋊C4—1062no—
544C22⋊C4—1062no—
545C22⋊C4—1062no—
546C22⋊C4—4241no—
547C22⋊C4—2122no—
548C22⋊C4—2122no—
549C22⋊C4#1171992112yes—
550C22⋊C4#1171992112yes—
551C22⋊C4#1171992112yes—
552C22⋊C4#1171992112yes—
553C42—1062no—
554C42—1062no—
555C42—4241no—
556C42—2122no—
557C42C2×C4×D5312yes—
558C42C2×C22⋊D5312yes—
559C23—21no—
560C23—21no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—1062no—
566C23—4241no—
567C23—4241no—
568C23—2122no—
569C23C2×#1171992112yes—
570C23C2×#1171992111yescharacteristic
571C23C2×#1171992111yescharacteristic
572C23#1313323912yes—
573C23#1313323912yes—
574C23C22×D21211yescharacteristic
575Q8—4241no—
576Q8—2122no—
577D4—24no—
578D4—24no—
579D4—4241no—
580D4—4242no—
581D4—4242no—
582D4—2122no—
583D4—2124no—
584D4—2122no—
585D4—2122no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1062no—
602C2×C4—4242no—
603C2×C4—4242no—
604C2×C4—4244no—
605C2×C4—4241no—
606C2×C4—4241no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2122no—
618C2×C4—2121no—
619C2×C4C2×#1171992112yes—
620C2×C4C2×#1171992112yes—
621C2×C4C2×#1171992112yes—
622C2×C4C2×#1171992111yescharacteristic
623C2×C4C2×#1171992111yescharacteristic
624C2×C4C22×C4×D5311yescharacteristic
625C2×C4C22×C22⋊D5311yescharacteristic
626C8—42no—
627C8—42no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—42no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—8482no—
642C22—2122no—
643C22—2122no—
644C22—2122no—
645C22#1681062012yes—
646C22#1681062012yes—
647C22C22×#1171992111yescharacteristic
648C22C2×#1313323911yescharacteristic
649C22C2×#1313323911yescharacteristic
650C4—21no—
651C4—21no—
652C4—22no—
653C4—21no—
654C4—21no—
655C4—22no—
656C4—22no—
657C4—4241no—
658C4—4241no—
659C4—4241no—
660C4—2122no—
661C4—2122no—
662C4—2122no—
663C4—2122no—
664C2—21no—
665C2—21no—
666C2—22no—
667C2—22no—
668C2—42no—
669C2—4242no—
670C2#2028317511yescharacteristic
671C2C2×#1681062011yescharacteristic
672C2#2028326011yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2972C2
socleC22×C53C22×C22⋊C4
prime radical#20288554C2
cosocleC22×C4×C53C24
hypercenter#2972D53
hypocenterC53#17615
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #20277655C2semidirect
3. #20273293C2semidirect
4. #20273288C2semidirect
5. #20273284C2semidirect
6. #20273272C2semidirect
7. #20273207C2semidirect
8. #20273202C2semidirect
9. C2×#16810361C2semidirect
10. C2×#16810358C2semidirect
11. C53×#2972C2semidirect
13. #16811319C4semidirect
14. #16811313C4semidirect
16. #16811282C4semidirect
17. #16811269C4semidirect
29. #16810491C22semidirect
30. #16810486C22semidirect
33. C4×C22⋊Dic53C22semidirect
34. #16810469C22semidirect
36. #16810361C22semidirect
37. #16810361C22semidirect
38. #16810361C22semidirect
39. #16810361C22semidirect
47. C53×#431C22semidirect
50. C2×C22⋊C8×C53C22semidirect
51. C2×#11719945C2×C4semidirect
59. #13133209C2×C4semidirect
61. #13133205C2×C4semidirect
71. #13133196C2×C4semidirect
77. #13133191C2×C4semidirect
84. C4×Dic106C2×C4semidirect
85. C4×D212C2×C4semidirect
91. #13133176C2×C4semidirect
329. #2972D53semidirect
365. #431C2×D53semidirect
368. C2×C22⋊C8C2×D53semidirect
387. C53#17615semidirect

Maximal subgroups

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

subgroupindex
#202885542
#202776552
#202732932
#202732882
#202732842
#202732722
#202732072
#202732022
C2×#168103612
C2×#168103582
C53×#29722
#1761553