Finite Group Databasebeta

← #27447025

Subgroups — #27447025

Order 13568.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810662C211yesmaximal, characteristic
2#20278435C211yesmaximal, characteristic
3#20277738C211yesmaximal, characteristic
4#20276261C211yesmaximal, characteristic, complemented
5#20276208C211yesmaximal, characteristic, complemented
6#20276106C211yesmaximal, characteristic, complemented
7#20276050C211yesmaximal, characteristic, complemented
8#20275480C211yesmaximal, characteristic, complemented
9#20275295C211yesmaximal, characteristic, complemented
10#20275145C211yesmaximal, characteristic
11#20275019C211yesmaximal, characteristic
12#20274983C211yesmaximal, characteristic, complemented
13#20274911C211yesmaximal, characteristic
14#20274896C211yesmaximal, characteristic
15C2×C53×#479C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C2×#13133193C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×C4⋊C4×D53C2211yescharacteristic
20#16810662—24no—
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133129C2211yescharacteristic
29#16810561—21no—
30#16810556—21no—
31#16810530C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810514—21no—
36#16810513—21no—
37#16810504—21no—
38#16810494—21no—
39#16810488C2211yescharacteristic
40#16810480—21no—
41#16810480C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4×#11719919—21no—
44C4×#11719921C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#125×C53C2211yescharacteristic
54C2×C4×Q8×C53C2211yescharacteristic
55C53×#479—24no—
56C2×#107×C53C2211yescharacteristic
57C2×#106×C53C2211yescharacteristic
58C2×C4⋊C8×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#103×C53C2211yescharacteristic
61C22×C4×D53—21no—
62C22×C4×D53C2311yescharacteristic
63C2×#11719912—22no—
64C2×#11719912—22no—
65C2×#11719912—22no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133193—22no—
69#13133193Q814yes—
70#13133175—22no—
71#13133175—41no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C4⋊C4×D53—22no—
82C4⋊C4×D53—41no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921D411yescharacteristic
87C2×#11719921D411yescharacteristic
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91#13133151—22no—
92#13133150—22no—
93#13133148—22no—
94#13133148—41no—
95#13133146—22no—
96#13133141—22no—
97#13133141—41no—
98#13133129—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124D411yescharacteristic
109#13133124D411yescharacteristic
110C22×Q8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113#125×C53—22no—
114#125×C53—41no—
115C4×Q8×C53—22no—
116C4×Q8×C53—41no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53—21no—
119C2×C4⋊C4×C53—21no—
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C42×C53—21no—
124C2×C42×C53C2311yescharacteristic
125#107×C53—22no—
126#106×C53D414yes—
127C4⋊C8×C53—22no—
128#103×C53—22no—
129#103×C53—22no—
130C23×D53C2×D411yescharacteristic
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53C2×Q812yes—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—42no—
156C2×C4×D53—42no—
157C2×C4×D53—81no—
158C2×C4×D53C2×Q812yes—
159C2×C4×D53#4112yes—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53C2×D412yes—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53C2×D412yes—
174C4⋊C4×C53C2×D412yes—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—42no—
178#11719921—24no—
179#11719921—41no—
180#11719921—41no—
181#11719921—42no—
182#11719921—42no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—81no—
191C4⋊Dic53C2×Q812yes—
192#11719919—24no—
193#11719919—42no—
194#11719919—42no—
195#11719919—81no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—42no—
199C4×Dic53—42no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322#4112yes—
206#11719912—28no—
207C23×C53C22×D411yescharacteristic
208C22×D53—21no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53—21no—
213C22×D53—21no—
214C22×D53C22⋊Q812yes—
215Q8×C53—22no—
216Q8×C53—22no—
217Q8×C53—41no—
218Q8×C53—41no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×Q811yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C532−1+411yescharacteristic
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—22no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53C22⋊Q812yes—
264C4×D53—22no—
265C4×D53—22no—
266C4×D53—22no—
267C4×D53—22no—
268C4×D53—84no—
269C8×C53—22no—
270C8×C53—41no—
271#88322—22no—
272#88322—41no—
273#26367—531nomaximal, Sylow
274C22×C53C2×C22⋊Q811yescharacteristic
275C22×C53D4211yescharacteristic
276C22×C53#54711yescharacteristic
277C22×C53#57511yescharacteristic
278C22×C53#57511yescharacteristic
279C22×C53#57711yescharacteristic
280C22×C53#57811yescharacteristic
281C2×D53—22no—
282C2×D53—22no—
283C2×D53—22no—
284C2×D53—41no—
285C2×D53—41no—
286C2×D53—41no—
287C2×D53—41no—
288C4×C53—42no—
289C4×C53—41no—
290C4×C53—42no—
291C4×C53—42no—
292C4×C53—41no—
293C4×C53—81no—
294C4×C53#52511yescharacteristic
295C4×C53#53911yescharacteristic
296C4×C53D4×Q811yescharacteristic
297C4×C53#55411yescharacteristic
298Dic53—22no—
299Dic53—42no—
300Dic53—41no—
301Dic53—41no—
302Dic53—81no—
303Dic53—81no—
304Dic53—81no—
305C2×#479D5311yescharacteristic, complemented
306C2×#428—531no—
307#2442—531no—
308#2362—531no—
309#2097—531no—
310#2087—531no—
311#2068—531no—
312#2059—531no—
313#1949—531no—
314#1919—531no—
315#1894—531no—
316#1872—531no—
317#1865—531no—
318#1854—531no—
319#1851—531no—
320C2×C53#244911yescharacteristic
321C2×C53#308011yescharacteristic
322C2×C53#308211yescharacteristic
323C2×C53#326911yescharacteristic
324C2×C53#327511yescharacteristic
325C2×C53#334911yescharacteristic
326C2×C53#335111yescharacteristic
327D53—42no—
328C22×M16—531no—
329C2×#125C2×D5311yescharacteristic
330C2×C22⋊Q8—531no—
331C2×C4×Q8C2×D5311yescharacteristic
332C2×#117—531no—
333C22×C4⋊C4—531no—
334#479—24no—
335#428—1064no—
336C2×#107—531no—
337C2×#107—531no—
338C2×#107C2×D5311yescharacteristic
339C2×#106—531no—
340C2×#106—531no—
341C2×#106C2×D5311yescharacteristic
342C2×C4⋊C8—531no—
343C2×C4⋊C8C2×D5311yescharacteristic
344C2×#103—531no—
345C2×#103—531no—
346C2×#103C2×D5311yescharacteristic
347C2×#103C2×D5311yescharacteristic
348C2×C22⋊C8—531no—
349#400—1061no—
350#398—1061no—
351#391—531no—
352#389—531no—
353#389—531no—
354#389—531no—
355#388—1061no—
356#388—1061no—
357#387—1061no—
358#385—1061no—
359#382—531no—
360#380—531no—
361#380—1061no—
362C4×C4⋊C4—531no—
363C4×C4⋊C4—1061no—
364C4×C22⋊C4—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#336—531no—
372#336—531no—
373C53#2636711yescharacteristic, complemented, Sylow
374C22×Q8C22×D5311yescharacteristic
375C23×C4—531no—
376C23×C4—1061no—
377C2×M16—1062no—
378C2×M16—1062no—
379C2×M16—1062no—
380C22×C8—531no—
381C22×C8—531no—
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384#125—22no—
385#125—41no—
386C22⋊Q8—534no—
387C22⋊Q8—1062no—
388C4×Q8—22no—
389C4×Q8—41no—
390#117—1062no—
391#117—2121no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—2121no—
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C22⋊C4—531no—
410C2×C22⋊C4—531no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C42—21no—
415C2×C42—531no—
416C2×C42—1061no—
417C2×C42—1061no—
418C2×C42C22×D5311yescharacteristic
419#107—22no—
420#107—1062no—
421#107—1062no—
422#106—1062no—
423#106—2121no—
424#106C22⋊D5314yes—
425C4⋊C8—22no—
426C4⋊C8—1062no—
427#103—22no—
428#103—22no—
429#103—1062no—
430#103—2121no—
431C22⋊C8—1062no—
432#95—531no—
433#95—531no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443C24—531no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8Q8×D5312yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—531no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—4241no—
468C22×C4—2122no—
469C22×C4—2122no—
470C22×C4D4×D5311yescharacteristic
471C22×C4D4×D5311yescharacteristic
472C22×C4C23×D5311yescharacteristic
473M16—1068no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—532no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8C2×C22⋊D5312yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—532no—
494C4⋊C4—1064no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—4241no—
503C4⋊C4—4241no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4C2×C22⋊D5312yes—
507C4⋊C4C2×C22⋊D5312yes—
508C22⋊C4—1064no—
509C22⋊C4—2121no—
510C22⋊C4—2121no—
511C22⋊C4—2122no—
512C22⋊C4—2122no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42—1061no—
517C42—1061no—
518C42—2122no—
519C42—2122no—
520C23—532no—
521C23—1061no—
522C23—1061no—
523C23—1061no—
524C23—1061no—
525C23—1061no—
526C23—1061no—
527C23C2×D4×D5311yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—532no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1062no—
554C2×C4—1062no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1062no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—4244no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4C22×C22⋊D5311yescharacteristic
577C2×C4C2×Q8×D5311yescharacteristic
578C2×C4#41×D5311yescharacteristic
579C2×C4#41×D5311yescharacteristic
580C2×C4#1313326211yescharacteristic
581C2×C4#1313326511yescharacteristic
582C8—22no—
583C8—41no—
584C8—1062no—
585C8—2121no—
586C22—1062no—
587C22—1062no—
588C22—1062no—
589C22—2121no—
590C22—2121no—
591C22—2121no—
592C22—2121no—
593C22C22⋊Q8×D5311yescharacteristic
594C22D4×C22⋊D5311yescharacteristic
595C22#1681127511yescharacteristic
596C22#1681142011yescharacteristic
597C22#1681142211yescharacteristic
598C22#1681143211yescharacteristic
599C22#1681143711yescharacteristic
600C4—41no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—42no—
605C4—81no—
606C4—1062no—
607C4—4241no—
608C4—4241no—
609C4—4241no—
610C4—2121no—
611C4—2121no—
612C4—2122no—
613C4#1681113011yescharacteristic
614C4#1681121911yescharacteristic
615C4Q8×C22⋊D5311yescharacteristic
616C4#1681132711yescharacteristic
617C2—2122no—
618C2#2027852111yescharacteristic
619C2#2028453411yescharacteristic
620C2#2028455611yescharacteristic
621C2#2028627011yescharacteristic
622C2#2028634011yescharacteristic
623C2#2028725211yescharacteristic
624C2#2028727211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#479C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenterC2×#479D53
hypocenterC53#26367
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
4. #20276261C2semidirect
5. #20276208C2semidirect
6. #20276106C2semidirect
7. #20276050C2semidirect
8. #20275480C2semidirect
9. #20275295C2semidirect
12. #20274983C2semidirect
15. C2×C53×#479C2semidirect
305. C2×#479D53semidirect
373. C53#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#168106622
#202784352
#202777382
#202762612
#202762082
#202761062
#202760502
#202754802
#202752952
#202751452
#202750192
#202749832
#202749112
#202748962
C2×C53×#4792
#2636753