Finite Group Databasebeta

← #27411535

Subgroups — #27411535

Order 13568.

1164 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288120C211yesmaximal, characteristic, complemented
2C2×#16810596C211yesmaximal, characteristic
3C2×#16810596C211yesmaximal, characteristic
4C2×#16810596C211yesmaximal, characteristic
5#20275893C218yesmaximal, complemented
6#20274448C211yesmaximal, characteristic, complemented
7#20274447C211yesmaximal, characteristic, complemented
8C53×#1740C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10#16811254—22no—
11#16811251—22no—
12#16811251—22no—
13#16811249C2214yes—
14#16811180—22no—
15#16811176—22no—
16#16811176—22no—
17C2×#13133176C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20#16810596—22no—
21#16810596—22no—
22#16810596—22no—
23#16810596—22no—
24#16810596—22no—
25#16810596—22no—
26C2×#13133131C2214yes—
27C2×#13133131C2214yes—
28C2×#13133131C2214yes—
29#16810390C2214yes—
30#16810389C2214yes—
31C2×#117×C53C2211yescharacteristic
32C2×#117×C53C2211yescharacteristic
33C2×#117×C53C2211yescharacteristic
34C53×#342C2214yes—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945—21no—
44C2×#11719945—21no—
45C2×#11719945—21no—
46C2×#11719945—21no—
47C22×Dic106Q811yescharacteristic, complemented
48C22×D212Q811yescharacteristic, complemented
49C22×C22⋊D53C2312yes—
50C22×C4×D53Q812yescomplemented
51#13133217—24no—
52#13133216—24no—
53#13133216—24no—
54#13133209—24no—
55#13133208—24no—
56#13133208—24no—
57#13133196—28no—
58#13133194—28no—
59#13133194—28no—
60C4×Dic106—28no—
61#13133176—21no—
62#13133176—21no—
63#13133176—21no—
64#13133176—21no—
65#13133176—21no—
66#13133176—21no—
67#13133176—21no—
68#13133176—21no—
69#13133176D412yes—
70#13133176D412yes—
71#13133176D412yes—
72#13133176D412yes—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C22⋊Dic53C2312yes—
77C2×C22⋊Dic53C2312yes—
78C2×C22⋊Dic53C2312yes—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82#13133131—28no—
83#13133131—28no—
84#13133131—28no—
85C23×C4×C53C2311yescharacteristic
86#117×C53—21no—
87#117×C53—21no—
88#117×C53—21no—
89#117×C53—21no—
90#117×C53—21no—
91#117×C53—21no—
92#117×C53—21no—
93#117×C53—21no—
94#117×C53—21no—
95#117×C53—21no—
96#117×C53D412yes—
97#117×C53D412yes—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C22⋊C4×C53C2312yes—
102C2×C22⋊C4×C53C2312yes—
103C2×C22⋊C4×C53C2312yes—
104C2×C42×C53—21no—
105C2×C42×C53—21no—
106C2×C42×C53—21no—
107C24×C53C2411yescharacteristic
108C23×D53C2×Q812yes—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×Q811yescharacteristic
119C22×C4×C53C2×Q811yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C22×C4×C53#4111yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C22×C4×C53#4111yescharacteristic
125C22×C4×C53#4111yescharacteristic
126C2×C22⋊D53—22no—
127C2×C22⋊D53—22no—
128C2×C22⋊D53—22no—
129C2×C22⋊D53—22no—
130C2×C22⋊D53—22no—
131C2×C22⋊D53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53C2×Q812yes—
136#11719945—41no—
137#11719945—41no—
138#11719945—41no—
139#11719945—41no—
140#11719945—41no—
141#11719945—41no—
142#11719945—41no—
143#11719945—41no—
144#11719945—41no—
145#11719945—41no—
146#11719945—41no—
147#11719945—41no—
148#11719945—41no—
149#11719945—41no—
150#11719945—41no—
151#11719945—41no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×C4×D53—22no—
163C2×C4×D53—22no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—28no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—41no—
174C4⋊C4×C53—41no—
175C4⋊C4×C53—41no—
176C4⋊C4×C53—41no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—22no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53C2×D414yes—
183C42×C53—22no—
184C42×C53—22no—
185C42×C53—41no—
186C42×C53—41no—
187C42×C53—41no—
188C42×C53—41no—
189C22⋊Dic53—22no—
190C22⋊Dic53—22no—
191C22⋊Dic53—22no—
192C22⋊Dic53—22no—
193C22⋊Dic53C2×D414yes—
194C22⋊Dic53C2×D414yes—
195C4⋊Dic53—28no—
196C4⋊Dic53—22no—
197C4⋊Dic53—22no—
198C4⋊Dic53—22no—
199C4⋊Dic53—22no—
200C4⋊Dic53—41no—
201C4⋊Dic53—41no—
202#11719919—28no—
203#11719919—28no—
204C4×Dic53—22no—
205C4×Dic53—22no—
206C4×Dic53—22no—
207C4×Dic53—22no—
208C4×Dic53—41no—
209C4×Dic53—41no—
210C23×C53—81no—
211C23×C53C22×D411yescharacteristic
212C23×C53C22×D411yescharacteristic
213C23×C53C22×D411yescharacteristic
214C23×C53C22×Q811yescharacteristic
215C23×C53C2×#4111yescharacteristic
216C23×C53C2×#4111yescharacteristic
217C23×C53C2×#4111yescharacteristic
218C22×D53—22no—
219C22×D53—22no—
220C22×D53—22no—
221C22×D53—82no—
222C2×C4×C53—24no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—82no—
226C2×C4×C53—82no—
227C2×C4×C53—82no—
228C2×C4×C53#12011yescharacteristic
229C2×C4×C53#12011yescharacteristic
230C2×C4×C53#12011yescharacteristic
231C2×C4×C53#12011yescharacteristic
232C2×C4×C53C22⋊Q811yescharacteristic
233C2×C4×C53C22⋊Q811yescharacteristic
234C2×C4×C53C22⋊Q811yescharacteristic
235C2×C4×C53C22⋊Q811yescharacteristic
236C2×C4×C53C22⋊Q811yescharacteristic
237C2×C4×C53C22⋊Q811yescharacteristic
238C2×C4×C53C22⋊Q811yescharacteristic
239C2×C4×C53C22⋊Q811yescharacteristic
240C2×C4×C53C22⋊Q811yescharacteristic
241C2×C4×C53C22⋊Q811yescharacteristic
242C2×C4×C53C22⋊Q811yescharacteristic
243C2×C4×C53C22⋊Q811yescharacteristic
244C2×C4×C53#12411yescharacteristic
245C2×C4×C53#12411yescharacteristic
246C2×C4×C53#12411yescharacteristic
247C2×C4×C53#12411yescharacteristic
248C2×C4×C53#12411yescharacteristic
249C2×C4×C53#12411yescharacteristic
250C2×C4×C53#12411yescharacteristic
251C2×C4×C53#12411yescharacteristic
252C2×C4×C53#12411yescharacteristic
253C2×C4×C53#12411yescharacteristic
254C2×C4×C53#12411yescharacteristic
255C2×C4×C53#12411yescharacteristic
256C22⋊D53—42no—
257C22⋊D53—42no—
258C22⋊D53—42no—
259C22⋊D53—42no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—42no—
266C2×Dic53—82no—
267C2×Dic53—82no—
268C2×Dic53—82no—
269D212—41no—
270D212—41no—
271D212—41no—
272D212—41no—
273C4×D53—42no—
274C4×D53—42no—
275C4×D53—42no—
276C4×D53—42no—
277Dic106—41no—
278Dic106—41no—
279Dic106—41no—
280Dic106—41no—
281Dic106—88no—
282#23474—531nomaximal, Sylow
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53C2×#12011yescharacteristic
291C22×C53C2×C22⋊Q811yescharacteristic
292C22×C53C2×C22⋊Q811yescharacteristic
293C22×C53C2×C22⋊Q811yescharacteristic
294C22×C53C2×#12411yescharacteristic
295C22×C53C2×#12411yescharacteristic
296C22×C53C2×#12411yescharacteristic
297C2×D53—42no—
298C2×D53—82no—
299C2×D53—82no—
300C2×D53—82no—
301C4×C53—44no—
302C4×C53—82no—
303C4×C53—82no—
304C4×C53#39311yescharacteristic
305C4×C53#39311yescharacteristic
306C4×C53#39311yescharacteristic
307C4×C53#39311yescharacteristic
308C4×C53#39311yescharacteristic
309C4×C53#39311yescharacteristic
310C4×C53#39311yescharacteristic
311C4×C53#39311yescharacteristic
312Dic53—44no—
313Dic53—44no—
314Dic53—42no—
315Dic53—82no—
316#3458—531no—
317C2×#410—531no—
318C2×#410—531no—
319C2×#410—531no—
320#2032—538no—
321#1740—531no—
322#1740—531no—
323#1740D5311yescharacteristic, complemented
324C2×C53—21no—
325C2×C53—21no—
326C2×C53—21no—
327C2×C53—21no—
328C2×C53—21no—
329C2×C53—21no—
330C2×C53—21no—
331C2×C53C2×#39311yescharacteristic
332D53—82no—
333C22×#41—531no—
334#544—1062no—
335#544—1062no—
336#544—1062no—
337#543—534no—
338#533—1062no—
339#533—1062no—
340#533—1062no—
341C2×#117—531no—
342C2×#117—531no—
343C2×#117—531no—
344C2×#117C2×D5311yescharacteristic
345C2×#117C2×D5311yescharacteristic
346C2×#117C2×D5311yescharacteristic
347#410—1062no—
348#410—1062no—
349#410—1062no—
350#410—1062no—
351#410—1062no—
352#410—1062no—
353C2×#99—534no—
354C2×#99—534no—
355C2×#99—534no—
356#342—534no—
357#342—534no—
358#342C2×D5314yes—
359C53#2347411yescharacteristic, complemented, Sylow
360C2×#41—1061no—
361C2×#41—1061no—
362C2×#41—1061no—
363C2×#41—1061no—
364C2×#41—1061no—
365C2×#41—1061no—
366C2×#41—1061no—
367C2×#41—1061no—
368C2×#41—1061no—
369C2×#41—1061no—
370C2×#41—1061no—
371C2×#41—1061no—
372C22×Q8—531no—
373C22×D4—532no—
374C22×D4—531no—
375C23×C4—532no—
376C23×C4C22×D5311yescharacteristic
377C4⋊Q8—1064no—
378C4⋊Q8—1064no—
379C4⋊Q8—1064no—
380#125—1064no—
381#125—1064no—
382#125—1064no—
383C22⋊Q8—1068no—
384C22⋊Q8—1068no—
385C22⋊Q8—1068no—
386C4×Q8—1068no—
387#117—21no—
388#117—21no—
389#117—21no—
390#117—21no—
391#117—21no—
392#117—21no—
393#117—21no—
394#117—21no—
395#117—21no—
396#117—21no—
397#117—532no—
398#117—532no—
399#117—532no—
400#117—532no—
401#117—1061no—
402#117—1061no—
403#117—1061no—
404#117—1061no—
405#117—1061no—
406#117—1061no—
407#117—1061no—
408#117—1061no—
409#117D21212yes—
410#117D21212yes—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C22⋊C4—532no—
418C2×C22⋊C4—532no—
419C2×C22⋊C4—532no—
420C2×C22⋊C4C22×D5312yes—
421C2×C22⋊C4C22×D5312yes—
422C2×C22⋊C4C22×D5312yes—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42—1061no—
427C2×C42—1061no—
428C2×C42—1061no—
429#99—1068no—
430#99—1068no—
431#99—1068no—
432C24—532no—
433C24C23×D5311yescharacteristic
434#41—2121no—
435#41—2121no—
436#41—2121no—
437#41—2121no—
438#41—2121no—
439#41—2121no—
440#41—2121no—
441#41—2121no—
442#41—2121no—
443#41—2121no—
444#41—2121no—
445#41—2121no—
446#41—2121no—
447#41—2121no—
448#41—2121no—
449#41—2121no—
450C2×Q8—1068no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C2×Q8—1061no—
454C2×Q8—1061no—
455C2×Q8—1061no—
456C2×Q8—1061no—
457C2×D4—1062no—
458C2×D4—1062no—
459C2×D4—1062no—
460C2×D4—1062no—
461C2×D4—1062no—
462C2×D4—1062no—
463C2×D4—1061no—
464C2×D4—1061no—
465C2×D4—1061no—
466C2×D4—1061no—
467C2×D4—1061no—
468C2×D4—1061no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—532no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4C2×D21211yescharacteristic
483C22×C4C2×D21211yescharacteristic
484C22×C4#1171994511yescharacteristic
485C22×C4#1171994511yescharacteristic
486C22×C4#1171994511yescharacteristic
487C22×C4#1171994511yescharacteristic
488C22×C4D4×D5311yescharacteristic
489C22×C4D4×D5311yescharacteristic
490C22×C4D4×D5311yescharacteristic
491C22×C4D4×D5311yescharacteristic
492C22×C4#1171994711yescharacteristic
493C22×C4#1171994711yescharacteristic
494C22×C4Q8×D5311yescharacteristic
495C22×C4Q8×D5311yescharacteristic
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—1068no—
503C4⋊C4—1068no—
504C4⋊C4—1068no—
505C4⋊C4—1062no—
506C4⋊C4—1062no—
507C4⋊C4—1062no—
508C4⋊C4—1062no—
509C4⋊C4—2121no—
510C4⋊C4—2121no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—534no—
517C22⋊C4—534no—
518C22⋊C4—1062no—
519C22⋊C4—1062no—
520C22⋊C4—1062no—
521C22⋊C4—1062no—
522C22⋊C4C2×D21214yes—
523C42—22no—
524C42—22no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—1062no—
530C42—1062no—
531C42—1062no—
532C42—1062no—
533C42—2121no—
534C42—2121no—
535C23—81no—
536C23—1062no—
537C23—1062no—
538C23—1062no—
539C23—4242no—
540C23C2×D4×D5311yescharacteristic
541C23C2×D4×D5311yescharacteristic
542C23C22×D21211yescharacteristic
543C23C2×Q8×D5311yescharacteristic
544C23C2×#1171994711yescharacteristic
545C23C2×#1171994511yescharacteristic
546C23C2×#1171994511yescharacteristic
547Q8—4248no—
548Q8—2121no—
549Q8—2121no—
550Q8—2121no—
551Q8—2121no—
552D4—2122no—
553D4—2122no—
554D4—2122no—
555D4—2122no—
556D4—2121no—
557D4—2121no—
558D4—2121no—
559D4—2121no—
560C2×C4—24no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—1064no—
567C2×C4—1064no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—4242no—
572C2×C4—4242no—
573C2×C4—4242no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4#1313318711yescharacteristic
580C2×C4#1313318711yescharacteristic
581C2×C4#1313318711yescharacteristic
582C2×C4#1313318711yescharacteristic
583C2×C4#1313319511yescharacteristic
584C2×C4#1313319511yescharacteristic
585C2×C4#1313319511yescharacteristic
586C2×C4#1313319511yescharacteristic
587C2×C4#1313319511yescharacteristic
588C2×C4#1313319511yescharacteristic
589C2×C4#1313319511yescharacteristic
590C2×C4#1313319511yescharacteristic
591C2×C4#1313319711yescharacteristic
592C2×C4#1313319711yescharacteristic
593C2×C4#1313319711yescharacteristic
594C2×C4#1313319711yescharacteristic
595C2×C4#1313320311yescharacteristic
596C2×C4#1313320311yescharacteristic
597C2×C4#1313320311yescharacteristic
598C2×C4#1313320311yescharacteristic
599C2×C4#1313320311yescharacteristic
600C2×C4#1313320311yescharacteristic
601C2×C4#1313320311yescharacteristic
602C2×C4#1313320311yescharacteristic
603C2×C4#1313320511yescharacteristic
604C2×C4#1313320511yescharacteristic
605C2×C4#1313320511yescharacteristic
606C2×C4#1313320511yescharacteristic
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—4242no—
615C22—4242no—
616C22—4242no—
617C22—2122no—
618C22C2×#1313318711yescharacteristic
619C22C2×#1313319511yescharacteristic
620C22C2×#1313319511yescharacteristic
621C22C2×#1313319711yescharacteristic
622C22C2×#1313320311yescharacteristic
623C22C2×#1313320311yescharacteristic
624C22C2×#1313320511yescharacteristic
625C4—44no—
626C4—82no—
627C4—82no—
628C4—4242no—
629C4—2124no—
630C4—2124no—
631C4—2122no—
632C4#1681053611yescharacteristic
633C4#1681053611yescharacteristic
634C4#1681053611yescharacteristic
635C4#1681053611yescharacteristic
636C4#1681053611yescharacteristic
637C4#1681053611yescharacteristic
638C4#1681053611yescharacteristic
639C4#1681053611yescharacteristic
640C2—21no—
641C2—21no—
642C2—21no—
643C2—21no—
644C2—21no—
645C2—21no—
646C2—21no—
647C2—4242no—
648C2C2×#1681053611yescharacteristic
subgroupG/H
centerC4#16810536
commutator, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC53×#1740C2
socleC2×C53C2×#393
prime radicalC22×#11719945C22
hypercenter#1740D53
hypocenterC53#23474
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
1. #20288120C2semidirect
5. #20275893C2semidirect
6. #20274448C2semidirect
7. #20274447C2semidirect
8. C53×#1740C2semidirect
47. C22×Dic106Q8semidirect
48. C22×D212Q8semidirect
50. C22×C4×D53Q8semidirect
323. #1740D53semidirect
359. C53#23474semidirect

Maximal subgroups

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

subgroupindex
#202881202
C2×#168105962
#202758932
#202744482
#202744472
C53×#17402
#2347453