Finite Group Databasebeta

← #27411534

Subgroups — #27411534

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×#16810597C211yesmaximal, characteristic
3C2×#16810597C211yesmaximal, characteristic
4#20275892C218yesmaximal, complemented
5#20274448C211yesmaximal, characteristic, complemented
6#20274447C211yesmaximal, characteristic, complemented
7#20274447C211yesmaximal, characteristic, complemented
8C2×C53×#410C211yesmaximal, characteristic
9#16811254—22no—
10#16811251—22no—
11#16811251—22no—
12#16811249C2214yes—
13#16811180—22no—
14#16811176—22no—
15#16811176—22no—
16C2×#13133176C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20#16810597—22no—
21#16810597—22no—
22#16810597—22no—
23#16810597—22no—
24C2×#13133132C2214yes—
25C2×#13133132C2214yes—
26#16810390C2214yes—
27#16810389C2214yes—
28#16810389C2214yes—
29C22×#41×C53C2211yescharacteristic
30C2×#117×C53C2211yescharacteristic
31C2×#117×C53C2211yescharacteristic
32C53×#410—22no—
33C53×#410—22no—
34C2×#99×C53C2214yes—
35#13133217—24no—
36#13133216—24no—
37#13133216—24no—
38#13133209—24no—
39#13133208—24no—
40#13133208—24no—
41#13133196—28no—
42#13133194—28no—
43#13133194—28no—
44C4×Dic106—28no—
45#13133176—21no—
46#13133176—21no—
47#13133176—21no—
48#13133176—21no—
49#13133176—21no—
50#13133176—21no—
51#13133176—21no—
52#13133176—21no—
53#13133176—21no—
54#13133176—21no—
55#13133176—21no—
56#13133176—21no—
57#13133176D412yes—
58#13133176D412yes—
59#13133176D412yes—
60#13133176D412yes—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C22⋊Dic53C2312yes—
66C2×C22⋊Dic53C2312yes—
67C2×C22⋊Dic53C2312yes—
68C2×C22⋊Dic53C2312yes—
69C2×C4×Dic53—21no—
70C2×C4×Dic53—21no—
71C2×C4×Dic53—21no—
72C2×C4×Dic53—21no—
73#13133132—28no—
74#13133132—28no—
75C2×#41×C53—21no—
76C2×#41×C53—21no—
77C2×#41×C53—21no—
78C2×#41×C53—21no—
79C2×#41×C53—21no—
80C2×#41×C53—21no—
81C2×#41×C53—21no—
82C2×#41×C53—21no—
83C2×#41×C53—21no—
84C2×#41×C53—21no—
85C2×#41×C53—21no—
86C2×#41×C53—21no—
87C22×Q8×C53Q811yescharacteristic, complemented
88C22×D4×C53Q811yescharacteristic, complemented
89C22×D4×C53C2312yes—
90C23×C4×C53Q812yescomplemented
91C23×C4×C53C2311yescharacteristic
92#117×C53—21no—
93#117×C53—21no—
94#117×C53—21no—
95#117×C53—21no—
96#117×C53—21no—
97#117×C53—21no—
98#117×C53D412yes—
99#117×C53D412yes—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C22⋊C4×C53C2312yes—
103C2×C22⋊C4×C53C2312yes—
104C2×C42×C53—21no—
105C2×C42×C53—21no—
106#99×C53—28no—
107C24×C53C2×Q812yes—
108C24×C53C2411yescharacteristic
109#41×C53—41no—
110#41×C53—41no—
111#41×C53—41no—
112#41×C53—41no—
113#41×C53—41no—
114#41×C53—41no—
115#41×C53—41no—
116#41×C53—41no—
117#41×C53—41no—
118#41×C53—41no—
119#41×C53—41no—
120#41×C53—41no—
121#41×C53—41no—
122#41×C53—41no—
123#41×C53—41no—
124#41×C53—41no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×D4×C53—22no—
132C2×D4×C53—22no—
133C2×D4×C53—22no—
134C2×D4×C53—22no—
135C2×D4×C53—22no—
136C2×D4×C53—22no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—22no—
149C22×C4×C53—22no—
150C22×C4×C53—22no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×Q812yes—
158C22×C4×C53C2×Q811yescharacteristic
159C22×C4×C53C2×Q811yescharacteristic
160C22×C4×C53#4111yescharacteristic
161C22×C4×C53#4111yescharacteristic
162C22×C4×C53#4111yescharacteristic
163C22×C4×C53#4111yescharacteristic
164C22×C4×C53#4111yescharacteristic
165C22×C4×C53#4111yescharacteristic
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—22no—
170C2×Dic106—28no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—41no—
174C4⋊C4×C53—41no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53C2×D414yes—
179C42×C53—22no—
180C42×C53—22no—
181C42×C53—41no—
182C42×C53—41no—
183C22⋊Dic53—22no—
184C22⋊Dic53—22no—
185C22⋊Dic53—22no—
186C22⋊Dic53—22no—
187C22⋊Dic53—22no—
188C22⋊Dic53—22no—
189C22⋊Dic53C2×D414yes—
190C22⋊Dic53C2×D414yes—
191C4⋊Dic53—28no—
192C4⋊Dic53—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53—22no—
195C4⋊Dic53—22no—
196C4⋊Dic53—41no—
197C4⋊Dic53—41no—
198C4⋊Dic53—41no—
199C4⋊Dic53—41no—
200#11719919—28no—
201#11719919—28no—
202C4×Dic53—22no—
203C4×Dic53—22no—
204C4×Dic53—22no—
205C4×Dic53—22no—
206C4×Dic53—41no—
207C4×Dic53—41no—
208C4×Dic53—41no—
209C4×Dic53—41no—
210C23×C53—22no—
211C23×C53—22no—
212C23×C53—22no—
213C23×C53—81no—
214C23×C53—82no—
215C23×C53C22×D411yescharacteristic
216C23×C53C22×D411yescharacteristic
217C23×C53C22×D411yescharacteristic
218C23×C53C22×Q811yescharacteristic
219C23×C53C2×#4111yescharacteristic
220C23×C53C2×#4111yescharacteristic
221C23×C53C2×#4111yescharacteristic
222Q8×C53—41no—
223Q8×C53—41no—
224Q8×C53—41no—
225Q8×C53—41no—
226D4×C53—42no—
227D4×C53—42no—
228D4×C53—42no—
229D4×C53—42no—
230D4×C53—41no—
231D4×C53—41no—
232D4×C53—41no—
233D4×C53—41no—
234C2×C4×C53—24no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—82no—
244C2×C4×C53—82no—
245C2×C4×C53#12011yescharacteristic
246C2×C4×C53#12011yescharacteristic
247C2×C4×C53#12011yescharacteristic
248C2×C4×C53#12011yescharacteristic
249C2×C4×C53C22⋊Q811yescharacteristic
250C2×C4×C53C22⋊Q811yescharacteristic
251C2×C4×C53C22⋊Q811yescharacteristic
252C2×C4×C53C22⋊Q811yescharacteristic
253C2×C4×C53C22⋊Q811yescharacteristic
254C2×C4×C53C22⋊Q811yescharacteristic
255C2×C4×C53C22⋊Q811yescharacteristic
256C2×C4×C53C22⋊Q811yescharacteristic
257C2×C4×C53C22⋊Q811yescharacteristic
258C2×C4×C53C22⋊Q811yescharacteristic
259C2×C4×C53C22⋊Q811yescharacteristic
260C2×C4×C53C22⋊Q811yescharacteristic
261C2×C4×C53#12411yescharacteristic
262C2×C4×C53#12411yescharacteristic
263C2×C4×C53#12411yescharacteristic
264C2×C4×C53#12411yescharacteristic
265C2×C4×C53#12411yescharacteristic
266C2×C4×C53#12411yescharacteristic
267C2×C4×C53#12411yescharacteristic
268C2×C4×C53#12411yescharacteristic
269C2×C4×C53#12411yescharacteristic
270C2×C4×C53#12411yescharacteristic
271C2×C4×C53#12411yescharacteristic
272C2×C4×C53#12411yescharacteristic
273C2×Dic53—24no—
274C2×Dic53—24no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—82no—
278C2×Dic53—82no—
279C2×Dic53—82no—
280C2×Dic53—82no—
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×C53—42no—
291C22×C53—82no—
292C22×C53—82no—
293C22×C53—82no—
294C22×C53C2×#12011yescharacteristic
295C22×C53C2×C22⋊Q811yescharacteristic
296C22×C53C2×C22⋊Q811yescharacteristic
297C22×C53C2×C22⋊Q811yescharacteristic
298C22×C53C2×#12411yescharacteristic
299C22×C53C2×#12411yescharacteristic
300C22×C53C2×#12411yescharacteristic
301C4×C53—44no—
302C4×C53—42no—
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—82no—
315Dic53—82no—
316#3458—531no—
317C2×#410—531no—
318C2×#410—531no—
319C2×#410D5311yescharacteristic
320#2032—538no—
321#1740—531no—
322#1740—531no—
323#1740—531no—
324C2×C53—21no—
325C2×C53—21no—
326C2×C53—21no—
327C2×C53—21no—
328C2×C53—21no—
329C2×C53—21no—
330C2×C53—21no—
331C2×C53—82no—
332C2×C53C2×#39311yescharacteristic
333C22×#41C2×D5311yescharacteristic
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×#117—531no—
345C2×#117C2×D5311yescharacteristic
346C2×#117C2×D5311yescharacteristic
347#410—22no—
348#410—22no—
349#410—1062no—
350#410—1062no—
351#410—1062no—
352#410—1062no—
353C2×#99—534no—
354C2×#99—534no—
355C2×#99C2×D5314yes—
356#342—534no—
357#342—534no—
358#342—534no—
359C53#2347411yescharacteristic, complemented, Sylow
360C2×#41—21no—
361C2×#41—21no—
362C2×#41—21no—
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41—21no—
367C2×#41—21no—
368C2×#41—21no—
369C2×#41—21no—
370C2×#41—21no—
371C2×#41—21no—
372C22×Q8Dic10611yescharacteristic, complemented
373C22×D4Dic10611yescharacteristic, complemented
374C22×D4C22×D5312yes—
375C23×C4Dic10612yescomplemented
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—532no—
394#117—532no—
395#117—532no—
396#117—532no—
397#117—1061no—
398#117—1061no—
399#117—1061no—
400#117—1061no—
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#117C22⋊D5312yes—
410#117C22⋊D5312yes—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—1061no—
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⋊C4—532no—
421C2×C22⋊C4C22×D5312yes—
422C2×C22⋊C4C22×D5312yes—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—1061no—
426C2×C42—1061no—
427C2×C42—1061no—
428C2×C42—1061no—
429#99—28no—
430#99—1068no—
431#99—1068no—
432C24C2×Dic10612yes—
433C24C23×D5311yescharacteristic
434#41—41no—
435#41—41no—
436#41—41no—
437#41—41no—
438#41—41no—
439#41—41no—
440#41—41no—
441#41—41no—
442#41—41no—
443#41—41no—
444#41—41no—
445#41—41no—
446#41—41no—
447#41—41no—
448#41—41no—
449#41—41no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—1068no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4C2×Dic10612yes—
482C22×C4C2×Dic10611yescharacteristic
483C22×C4C2×Dic10611yescharacteristic
484C22×C4#1171994511yescharacteristic
485C22×C4#1171994511yescharacteristic
486C22×C4D4×D5311yescharacteristic
487C22×C4D4×D5311yescharacteristic
488C22×C4D4×D5311yescharacteristic
489C22×C4D4×D5311yescharacteristic
490C22×C4#1171994711yescharacteristic
491C22×C4#1171994711yescharacteristic
492C22×C4#1171994711yescharacteristic
493C22×C4#1171994711yescharacteristic
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—1068no—
501C4⋊C4—1068no—
502C4⋊C4—1068no—
503C4⋊C4—1062no—
504C4⋊C4—1062no—
505C4⋊C4—1062no—
506C4⋊C4—1062no—
507C4⋊C4—2121no—
508C4⋊C4—2121no—
509C4⋊C4—2121no—
510C4⋊C4—2121no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—534no—
515C22⋊C4—534no—
516C22⋊C4—1062no—
517C22⋊C4—1062no—
518C22⋊C4—1062no—
519C22⋊C4—1062no—
520C22⋊C4—1062no—
521C22⋊C4—1062no—
522C22⋊C4C2×C22⋊D5314yes—
523C42—22no—
524C42—22no—
525C42—41no—
526C42—41no—
527C42—1062no—
528C42—1062no—
529C42—1062no—
530C42—1062no—
531C42—2121no—
532C42—2121no—
533C42—2121no—
534C42—2121no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—81no—
539C23—82no—
540C23C2×D4×D5311yescharacteristic
541C23C2×D4×D5311yescharacteristic
542C23C22×C22⋊D5311yescharacteristic
543C23C22×Dic10611yescharacteristic
544C23C2×#1171994711yescharacteristic
545C23C2×#1171994711yescharacteristic
546C23C2×#1171994511yescharacteristic
547Q8—41no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—4248no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—41no—
557D4—41no—
558D4—41no—
559D4—41no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—1064no—
572C2×C4—1064no—
573C2×C4—4242no—
574C2×C4—4242no—
575C2×C4—4242no—
576C2×C4—4242no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4#1313318511yescharacteristic
580C2×C4#1313318511yescharacteristic
581C2×C4#1313318511yescharacteristic
582C2×C4#1313318511yescharacteristic
583C2×C4#1313319411yescharacteristic
584C2×C4#1313319411yescharacteristic
585C2×C4#1313319411yescharacteristic
586C2×C4#1313319411yescharacteristic
587C2×C4#1313319411yescharacteristic
588C2×C4#1313319411yescharacteristic
589C2×C4#1313319411yescharacteristic
590C2×C4#1313319411yescharacteristic
591C2×C4#1313319611yescharacteristic
592C2×C4#1313319611yescharacteristic
593C2×C4#1313319611yescharacteristic
594C2×C4#1313319611yescharacteristic
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#1313320411yescharacteristic
604C2×C4#1313320411yescharacteristic
605C2×C4#1313320411yescharacteristic
606C2×C4#1313320411yescharacteristic
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22C2×#1313318511yescharacteristic
619C22C2×#1313319411yescharacteristic
620C22C2×#1313319411yescharacteristic
621C22C2×#1313319611yescharacteristic
622C22C2×#1313320311yescharacteristic
623C22C2×#1313320311yescharacteristic
624C22C2×#1313320411yescharacteristic
625C4—44no—
626C4—42no—
627C4—82no—
628C4—4242no—
629C4—4242no—
630C4—2124no—
631C4—2124no—
632C4#1681053511yescharacteristic
633C4#1681053511yescharacteristic
634C4#1681053511yescharacteristic
635C4#1681053511yescharacteristic
636C4#1681053511yescharacteristic
637C4#1681053511yescharacteristic
638C4#1681053511yescharacteristic
639C4#1681053511yescharacteristic
640C2—21no—
641C2—21no—
642C2—21no—
643C2—21no—
644C2—21no—
645C2—21no—
646C2—21no—
647C2—82no—
648C2C2×#1681053511yescharacteristic
subgroupG/H
centerC4#16810535
commutator, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC2×C53×#410C2
socleC2×C53C2×#393
prime radicalC22×#41×C53C22
hypercenterC2×#410D53
hypocenterC53#23474
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
1. #20288120C2semidirect
4. #20275892C2semidirect
5. #20274448C2semidirect
6. #20274447C2semidirect
7. #20274447C2semidirect
87. C22×Q8×C53Q8semidirect
88. C22×D4×C53Q8semidirect
90. C23×C4×C53Q8semidirect
359. C53#23474semidirect
372. C22×Q8Dic106semidirect
373. C22×D4Dic106semidirect
375. C23×C4Dic106semidirect

Maximal subgroups

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

subgroupindex
#202881202
C2×#168105972
#202758922
#202744482
#202744472
C2×C53×#4102
#2347453