Finite Group Databasebeta

← #27419216

Subgroups — #27419216

Order 13568.

1558 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133205C211yesmaximal, characteristic
2C2×#16810845C212yesmaximal, complemented
3C2×#16810778C211yesmaximal, characteristic, complemented
4C2×#16810773C211yesmaximal, characteristic
5#20275956C212yesmaximal, complemented
6#20275808C212yesmaximal, complemented
7#20275647C212yesmaximal, complemented
8#20274695C211yesmaximal, characteristic, complemented
9#20274649C211yesmaximal, characteristic, complemented
10#20274645C211yesmaximal, characteristic
11C4×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
12C23×Dic106C2211yescharacteristic
13C23×D212C2211yescharacteristic
14C22×#11719925C2212yescomplemented
15C22×#11719924C2212yes—
16C2×#13133205—28no—
17C2×#13133205—21no—
18C2×#13133205—21no—
19C2×#13133205C2212yes—
20C2×#13133205C2212yes—
21C22×#11719921—21no—
22C22×#11719921—21no—
23#16810845—28no—
24#16810778—24no—
25#16810773—24no—
26C2×C4⋊#88322C2212yescomplemented
27C2×#13133140C2212yescomplemented
28C2×#13133140C2212yescomplemented
29C2×#13133140C2212yescomplemented
30C2×#13133137C2212yes—
31C2×#13133137C2212yes—
32C2×#13133137C2212yes—
33C2×C22⋊#88322C2211yescharacteristic
34C2×C22⋊#88322C2211yescharacteristic
35#16810378C2212yescomplemented
36C2×C22⋊Q8×C53C2211yescharacteristic
37C2×C4×Q8×C53C2211yescharacteristic, complemented
38C22×C42×C53C2211yescharacteristic
39C53×#391C2211yescharacteristic, complemented
40C53×#389C2211yescharacteristic, complemented
41C53×#387—21no—
42C53×#385—21no—
43C53×#380C2211yescharacteristic
44C4×C4⋊C4×C53—21no—
45C4×C4⋊C4×C53C2211yescharacteristic, complemented
46C4×C22⋊C4×C53—21no—
47C24×D53—21no—
48C22×Dic106—21no—
49C22×Dic106—24no—
50C22×Dic106—22no—
51C22×Dic106—22no—
52C22×Dic106—42no—
53C22×Dic106C2312yes—
54C22×D212—21no—
55C22×D212—24no—
56C22×D212—22no—
57C22×D212—22no—
58C22×D212—42no—
59C22×D212C2312yes—
60C23×Dic53—21no—
61C2×#11719925—24no—
62C2×#11719925—24no—
63C2×#11719925—42no—
64C2×#11719924—24no—
65C2×#11719924—24no—
66C2×#11719924—42no—
67C22×#88322C2312yes—
68C22×#88322C2312yes—
69#13133205—28no—
70#13133205—28no—
71#13133205—24no—
72#13133205—24no—
73#13133205—24no—
74#13133205D418yes—
75C2×#11719921—28no—
76C2×#11719921—28no—
77C2×#11719921—22no—
78C2×#11719921—22no—
79C2×#11719921—22no—
80C2×#11719921—22no—
81C4⋊#88322—24no—
82#13133140—24no—
83#13133140—24no—
84#13133140—42no—
85#13133137—24no—
86#13133137—24no—
87#13133137—42no—
88C22⋊#88322—22no—
89C22⋊#88322—22no—
90C22×Q8×C53C2311yescharacteristic
91C23×C4×C53D411yescharacteristic
92C23×C4×C53D411yescharacteristic
93C23×C4×C53C2311yescharacteristic
94C22⋊Q8×C53—22no—
95C22⋊Q8×C53D414yes—
96C4×Q8×C53—24no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C42×C53—24no—
106C2×C42×C53—21no—
107C2×C42×C53—21no—
108C2×C42×C53D411yescharacteristic
109C2×C42×C53D411yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112#95×C53—21no—
113#95×C53—21no—
114#95×C53—21no—
115#95×C53—21no—
116#95×C53—21no—
117C24×C53C2×D411yescharacteristic
118C23×D53—24no—
119C23×D53—22no—
120C23×D53—42no—
121C23×D53—42no—
122C23×D53—42no—
123C23×D53—42no—
124C23×D53—42no—
125C23×D53—42no—
126C2×Q8×C53—21no—
127C2×Q8×C53—22no—
128C2×Q8×C53—22no—
129C2×Q8×C53C2×D412yes—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—41no—
141C22×C4×C53—41no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D412yes—
148C22×C4×C53#4112yes—
149C22×C4×C53C2411yescharacteristic
150C22×Dic53—24no—
151C22×Dic53—22no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C2×D212—22no—
155C2×D212—22no—
156C2×D212—22no—
157C2×D212—22no—
158C2×D212—24no—
159C2×D212—22no—
160C2×D212—48no—
161C2×D212—48no—
162C2×D212—42no—
163C2×D212—42no—
164C2×D212C2×D414yes—
165C2×Dic106—22no—
166C2×Dic106—22no—
167C2×Dic106—22no—
168C2×Dic106—22no—
169C2×Dic106—24no—
170C2×Dic106—22no—
171C2×Dic106—48no—
172C2×Dic106—48no—
173C2×Dic106—42no—
174C2×Dic106—42no—
175C2×Dic106C2×D414yes—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—22no—
179C4⋊C4×C53—22no—
180C4⋊C4×C53—42no—
181C4⋊C4×C53—42no—
182C4⋊C4×C53C2×D412yes—
183C22⋊C4×C53—24no—
184C22⋊C4×C53—42no—
185C42×C53—22no—
186C42×C53—22no—
187C42×C53—22no—
188C42×C53—42no—
189C42×C53—42no—
190C42×C53C2×D414yes—
191#11719925—48no—
192#11719924—48no—
193#11719921—28no—
194#11719921—28no—
195#11719921—28no—
196#11719921—28no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53#12112yes—
208C23×C53#13512yes—
209C23×C53#13712yes—
210C23×C53C22×D411yescharacteristic
211C22×D53—24no—
212C22×D53—48no—
213C22×D53—48no—
214C22×D53—48no—
215C22×D53—42no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—84no—
222C22×D53—84no—
223C22×D53—84no—
224C22×D53—84no—
225Q8×C53—24no—
226Q8×C53—42no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—44no—
244C2×C4×C53—44no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53#12012yes—
256C2×C4×C53#12012yes—
257C2×C4×C53#12112yes—
258C2×C4×C53C22×D412yes—
259C2×C4×C53C22×D411yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53H(5,𝔽2)12yes—
262C2×Dic53—24no—
263C2×Dic53—48no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266D212—24no—
267D212—24no—
268D212—48no—
269D212—42no—
270D212—42no—
271Dic106—24no—
272Dic106—24no—
273Dic106—48no—
274Dic106—42no—
275Dic106—42no—
276#88322—42no—
277#88322—42no—
278#24217—531nomaximal, Sylow
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53#46414yescomplemented
286C22×C53C2×#12011yescharacteristic
287C22×C53C2×#12111yescharacteristic
288C22×C53#53511yescharacteristic
289C22×C53C2×#13511yescharacteristic
290C22×C53C2×#13711yescharacteristic
291C22×C53#57511yescharacteristic
292C22×C53#57511yescharacteristic
293C2×D53—44no—
294C2×D53—44no—
295C2×D53—44no—
296C2×D53—816no—
297C2×D53—82no—
298C2×D53—82no—
299C2×D53—82no—
300C2×D53—82no—
301C4×C53—22no—
302C4×C53—22no—
303C4×C53—22no—
304C4×C53—42no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53—42no—
308C4×C53—81no—
309C4×C53—81no—
310C4×C53D4212yes—
311C4×C53#54612yes—
312Dic53—44no—
313C22×#124—531no—
314C2×#464—532no—
315C2×#451—531no—
316C2×#450—531no—
317C4×C22⋊Q8D5311yescharacteristic, complemented
318#2042—532no—
319#2015—532no—
320#1983—532no—
321#1797—531no—
322#1787—531no—
323#1786—531no—
324C2×C53—24no—
325C2×C53#240811yescharacteristic
326C2×C53#300211yescharacteristic
327C2×C53#300711yescharacteristic
328C2×C53C2×#46411yescharacteristic
329C2×C53#304011yescharacteristic
330C2×C53#321111yescharacteristic
331C2×C53#321511yescharacteristic
332D53—84no—
333C23×Q8—531no—
334C23×D4—531no—
335C22×Q16—532no—
336C22×SD16—532no—
337C2×#124—532no—
338C2×#124—532no—
339C2×#124—1068no—
340C2×#124—1061no—
341C2×#124—1061no—
342C2×C22⋊Q8C2×D5311yescharacteristic
343C2×C4×Q8C2×D5311yescharacteristic, complemented
344C22×C22⋊C4—1061no—
345C22×C22⋊C4—1061no—
346C22×C42C2×D5311yescharacteristic
347#464—1068no—
348#451—1064no—
349#450—1064no—
350C2×C4⋊C8—532no—
351C2×#103—532no—
352C2×#103—532no—
353C2×#103—532no—
354C2×#102—532no—
355C2×#102—532no—
356C2×#102—532no—
357C2×C22⋊C8—531no—
358C2×C22⋊C8—531no—
359#391C2×D5311yescharacteristic, complemented
360#389C2×D5311yescharacteristic, complemented
361#387—21no—
362#385—21no—
363#380C2×D5311yescharacteristic
364C4×C4⋊C4—21no—
365C4×C4⋊C4C2×D5311yescharacteristic, complemented
366C4×C22⋊C4—21no—
367#336—532no—
368C53#2421711yescharacteristic, complemented, Sylow
369C25—1061no—
370C22×Q8—532no—
371C22×Q8—1064no—
372C22×Q8—1061no—
373C22×Q8—1062no—
374C22×Q8—1062no—
375C22×Q8—2122no—
376C22×Q8C22×D5311yescharacteristic
377C22×D4—532no—
378C22×D4—1064no—
379C22×D4—1062no—
380C22×D4—1062no—
381C22×D4—1061no—
382C22×D4—2122no—
383C23×C4—1061no—
384C23×C4C22⋊D5311yescharacteristic
385C23×C4C22⋊D5311yescharacteristic
386C23×C4C22×D5311yescharacteristic
387C2×Q16—1064no—
388C2×Q16—1064no—
389C2×Q16—2122no—
390C2×SD16—1064no—
391C2×SD16—1064no—
392C2×SD16—2122no—
393C22×C8—532no—
394C22×C8—532no—
395#124—538no—
396#124—1064no—
397#124—1064no—
398#124—1068no—
399#124—1068no—
400#124—1064no—
401C22⋊Q8—22no—
402C22⋊Q8D21214yes—
403C4×Q8—24no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1068no—
413C2×C22⋊C4—1068no—
414C2×C22⋊C4—1062no—
415C2×C22⋊C4—1062no—
416C2×C22⋊C4—1062no—
417C2×C22⋊C4—1062no—
418C2×C42—24no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42C22⋊D5311yescharacteristic
422C2×C42C22⋊D5311yescharacteristic
423C2×C42C22×D5311yescharacteristic
424C2×C42C22×D5311yescharacteristic
425C4⋊C8—1064no—
426#103—1064no—
427#103—1064no—
428#103—2122no—
429#102—1064no—
430#102—1064no—
431#102—2122no—
432C22⋊C8—1062no—
433C22⋊C8—1062no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439C24—1064no—
440C24—1062no—
441C24—2122no—
442C24—2122no—
443C24—2122no—
444C24—2122no—
445C24—2122no—
446C24—2122no—
447C24C2×C22⋊D5311yescharacteristic
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—534no—
452C2×Q8—1064no—
453C2×Q8—1062no—
454C2×Q8—1062no—
455C2×Q8—1062no—
456C2×Q8—1062no—
457C2×Q8—1062no—
458C2×Q8—2128no—
459C2×Q8—2128no—
460C2×Q8—2122no—
461C2×Q8—2122no—
462C2×Q8C2×D21212yes—
463C2×D4—534no—
464C2×D4—1064no—
465C2×D4—1062no—
466C2×D4—1062no—
467C2×D4—1062no—
468C2×D4—1062no—
469C2×D4—1062no—
470C2×D4—2128no—
471C2×D4—2128no—
472C2×D4—2122no—
473C2×D4—2122no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—1064no—
487C22×C4—1062no—
488C22×C4—2122no—
489C22×C4—2122no—
490C22×C4C2×D21212yes—
491C22×C4#1171994512yes—
492C22×C4C2×C22⋊D5311yescharacteristic
493C22×C4C2×C22⋊D5311yescharacteristic
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C22×C4C2×C22⋊D5311yescharacteristic
497C22×C4C23×D5311yescharacteristic
498Q16—2128no—
499SD16—2128no—
500C2×C8—1062no—
501C2×C8—1062no—
502C2×C8—1062no—
503C2×C8—1062no—
504C2×C8—1062no—
505C2×C8—1062no—
506C4⋊C4—24no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4C2×D21212yes—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—1068no—
516C22⋊C4—1068no—
517C22⋊C4—1068no—
518C22⋊C4—1068no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—42no—
523C42—42no—
524C42D4×D5314yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—1064no—
530C23—4244no—
531C23—4244no—
532C23—4244no—
533C23—4244no—
534C23—2128no—
535C23—2128no—
536C23—2128no—
537C23—2122no—
538C23—2122no—
539C23—2122no—
540C23—2122no—
541C23—2122no—
542C23—2122no—
543C23#1313319112yes—
544C23#1313323712yes—
545C23#1313324812yes—
546C23C22×C22⋊D5311yescharacteristic
547Q8—24no—
548Q8—42no—
549Q8—1064no—
550Q8—1064no—
551Q8—2128no—
552Q8—2122no—
553Q8—2122no—
554D4—1064no—
555D4—1064no—
556D4—2128no—
557D4—2122no—
558D4—2122no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—1064no—
588C2×C4—2128no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4#1313318512yes—
592C2×C4#1313318512yes—
593C2×C4#1313319112yes—
594C2×C4C2×D4×D5312yes—
595C2×C4C22×D21211yescharacteristic
596C2×C4C2×#1171994511yescharacteristic
597C2×C4#1313326312yes—
598C8—2122no—
599C8—2122no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—42416no—
607C22—4242no—
608C22—4242no—
609C22—4242no—
610C22—4242no—
611C22—2124no—
612C22—2124no—
613C22—2124no—
614C22#1681084514yescomplemented
615C22C2×#1313318511yescharacteristic
616C22C2×#1313319111yescharacteristic
617C22#1681118711yescharacteristic
618C22C2×#1313323711yescharacteristic
619C22C2×#1313324811yescharacteristic
620C22#1681142111yescharacteristic
621C22#1681142111yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—42no—
627C4—42no—
628C4—41no—
629C4—81no—
630C4—81no—
631C4—2124no—
632C4D4×D21212yes—
633C4#1681126912yes—
634C2—24no—
635C2—4244no—
636C2#2027807111yescharacteristic
637C2#2028393111yescharacteristic
638C2#2028397311yescharacteristic
639C2C2×#1681084511yescharacteristic
640C2#2028420911yescharacteristic
641C2#2028567611yescharacteristic
642C2#2028571411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×C22⋊Q8×C53C2
socleC23×C53C22×D4
prime radicalC23×D212C22
hypercenterC4×C22⋊Q8D53
hypocenterC53#24217
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16810845C2semidirect
3. C2×#16810778C2semidirect
5. #20275956C2semidirect
6. #20275808C2semidirect
7. #20275647C2semidirect
8. #20274695C2semidirect
9. #20274649C2semidirect
11. C4×C22⋊Q8×C53C2semidirect
14. C22×#11719925C22semidirect
26. C2×C4⋊#88322C22semidirect
27. C2×#13133140C22semidirect
28. C2×#13133140C22semidirect
29. C2×#13133140C22semidirect
35. #16810378C22semidirect
37. C2×C4×Q8×C53C22semidirect
39. C53×#391C22semidirect
40. C53×#389C22semidirect
45. C4×C4⋊C4×C53C22semidirect
285. C22×C53#464semidirect
317. C4×C22⋊Q8D53semidirect
343. C2×C4×Q8C2×D53semidirect
359. #391C2×D53semidirect
360. #389C2×D53semidirect
365. C4×C4⋊C4C2×D53semidirect
368. C53#24217semidirect
614. C22#16810845semidirect

Maximal subgroups

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

subgroupindex
C22×#131332052
C2×#168108452
C2×#168107782
C2×#168107732
#202759562
#202758082
#202756472
#202746952
#202746492
#202746452
C4×C22⋊Q8×C532
#2421753