Finite Group Databasebeta

← #27387109

Subgroups — #27387109

Order 13568.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810724C211yesmaximal, characteristic
2#20277190C211yesmaximal, characteristic, complemented
3C4×D4×Dic53C211yesmaximal, characteristic
4#20275359C211yesmaximal, characteristic, complemented
5#20275346C211yesmaximal, characteristic, complemented
6#20275306C211yesmaximal, characteristic, complemented
7#20275287C211yesmaximal, characteristic, complemented
8#20275051C211yesmaximal, characteristic
9#20274974C211yesmaximal, characteristic
10#20274962C211yesmaximal, characteristic
11#20274611C211yesmaximal, characteristic, complemented
12#20274547C211yesmaximal, characteristic, complemented
13#20274542C211yesmaximal, characteristic
14#20274462C211yesmaximal, characteristic, complemented
15C53×#1877C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C22×C4×Dic53—21no—
20#16810724C418yescomplemented
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133127C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30#16810530C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810508—21no—
34#16810506—21no—
35C4⋊C4×Dic53C2211yescharacteristic
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4×C4⋊Dic53C2211yescharacteristic
39#16810474—21no—
40C22⋊C4×Dic53—21no—
41C22⋊C4×Dic53—21no—
42C4×C22⋊Dic53—21no—
43#16810468—21no—
44C42×Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×SD16×C53C2211yescharacteristic
54C2×C4×D4×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#391C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C23×Dic53—21no—
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64Q8×Dic53—22no—
65Q8×Dic53C2×C414yescomplemented
66D4×Dic53—22no—
67D4×Dic53—22no—
68D4×Dic53—22no—
69D4×Dic53C2×C414yes—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—44no—
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133151C2×C414yescomplemented
92#13133150—22no—
93#13133148—22no—
94#13133141—22no—
95#13133141C2×C414yescomplemented
96#13133138—22no—
97#13133138C2×C414yes—
98#13133127—22no—
99#13133127C2×C414yescomplemented
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C22×Q8×C53C2311yescharacteristic
109C22×D4×C53C2311yescharacteristic
110C23×C4×C53—21no—
111C2×SD16×C53—22no—
112C2×SD16×C53—22no—
113C2×SD16×C53C2×C414yes—
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C4×D4×C53—22no—
117C4×D4×C53—22no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121C2×C42×C53C2311yescharacteristic
122#103×C53—22no—
123#102×C53—22no—
124#95×C53—21no—
125#95×C53—21no—
126C24×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53C22×C412yes—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53C22×C412yes—
137C2×D4×C53#4112yes—
138C2×D4×C53#4112yes—
139C22×C4×C53—24no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—24no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—24no—
153C22×Dic53—21no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53C2×D411yescharacteristic
162SD16×C53—28no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53C22×C412yes—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53#4112yes—
173C22⋊C4×C53—24no—
174C42×C53—21no—
175C42×C53—21no—
176C22⋊Dic53—24no—
177C22⋊Dic53—24no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—24no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53C22×C412yes—
185C4⋊Dic53C22×C412yes—
186C4⋊Dic53#4112yes—
187#11719919—42no—
188#11719919—42no—
189C4×Dic53—22no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—24no—
193C4×Dic53—22no—
194C4×Dic53—41no—
195C4×Dic53—41no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C4×Dic53—42no—
200C4×Dic53C22×C412yes—
201C4×Dic53#4112yes—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322C22×C412yes—
208C23×C53—22no—
209C23×C53—22no—
210C23×C53—22no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53C22×D411yescharacteristic
216Q8×C53—22no—
217Q8×C53—22no—
218Q8×C53—41no—
219Q8×C53—41no—
220D4×C53—22no—
221D4×C53—22no—
222D4×C53—22no—
223D4×C53#11714yes—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—48no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53AGL(1,ℤ/8)12yes—
240C2×C4×C53C23×C411yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—48no—
259C2×Dic53—48no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53C4×D412yes—
270C2×Dic53C4×D412yes—
271C2×Dic53#13712yes—
272C8×C53—22no—
273C8×C53—41no—
274#88322—22no—
275#88322—41no—
276#21121—531nomaximal, Sylow
277C22×C53—24no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53C2×C4×D411yescharacteristic
285C22×C53#52511yescharacteristic
286C22×C53#52511yescharacteristic
287C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C53C2×#13711yescharacteristic
289C22×C53#57511yescharacteristic
290C22×C53#57511yescharacteristic
291C4×C53—22no—
292C4×C53—42no—
293C4×C53—41no—
294C4×C53—42no—
295C4×C53—81no—
296C4×C53C2×#11711yescharacteristic
297C4×C53C4×#4111yescharacteristic
298C4×C53#52911yescharacteristic
299C4×C53#52911yescharacteristic
300Dic53—22no—
301Dic53—22no—
302Dic53—22no—
303Dic53—42no—
304Dic53—41no—
305Dic53—42no—
306Dic53—42no—
307Dic53—81no—
308C2×#440—531no—
309#2295—531no—
310C42×D4—531no—
311#1928—531no—
312#1927—531no—
313#1921—531no—
314#1918—531no—
315#1877—531no—
316#1877D5311yescharacteristic
317#1864—531no—
318#1862—531no—
319#1776—531no—
320#1763—531no—
321#1762—531no—
322#1745—531no—
323C2×C53—44no—
324C2×C53#229011yescharacteristic
325C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
326C2×C53C4×#13711yescharacteristic
327C2×C53#311011yescharacteristic
328C2×C53#311111yescharacteristic
329C2×C53#311411yescharacteristic
330C2×C53#311611yescharacteristic
331C22×SD16C2×D5311yescharacteristic
332C2×C4×Q8—531no—
333C2×C4×D4—531no—
334C2×C4×D4—531no—
335C2×C4×D4C2×D5311yescharacteristic
336C22×C42—1061no—
337#440—538no—
338C2×#107—531no—
339C2×#107—531no—
340C2×#106—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103C2×D5311yescharacteristic
344C2×#102—531no—
345C2×#102—531no—
346C2×#102C2×D5311yescharacteristic
347C2×#97—531no—
348C2×#97—531no—
349#391—531no—
350#391C2×D5311yescharacteristic
351#389—531no—
352#389—531no—
353#387—1061no—
354#387—1061no—
355C4×C4⋊C4—531no—
356C4×C4⋊C4—531no—
357C4×C4⋊C4—531no—
358C4×C4⋊C4—531no—
359C4×C4⋊C4—1061no—
360C4×C22⋊C4—1061no—
361C4×C22⋊C4—1061no—
362C4×C22⋊C4—1061no—
363#376—1061no—
364C43—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340C2×D5311yescharacteristic
371#336—531no—
372#336—531no—
373#336—531no—
374#336C2×D5311yescharacteristic
375C53#2112111yescharacteristic, complemented, Sylow
376C22×Q8C22×D5311yescharacteristic
377C22×D4C22×D5311yescharacteristic
378C23×C4—21no—
379C23×C4—1061no—
380C23×C4—1061no—
381C2×SD16—22no—
382C2×SD16—22no—
383C2×SD16C4×D5314yes—
384C22×C8—531no—
385C22×C8—531no—
386C22×C8C22×D5311yescharacteristic
387C22×C8C22×D5311yescharacteristic
388C4×Q8—534no—
389C4×Q8—1062no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—534no—
393C4×D4—1062no—
394C4×D4—1062no—
395C4×D4—1062no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C42—531no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42—1061no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—2124no—
420C2×C42C22×D5311yescharacteristic
421#107—534no—
422#107—1062no—
423#106—1062no—
424#103—22no—
425#103—534no—
426#103—1062no—
427#102—22no—
428#102—534no—
429#102—1062no—
430#97—534no—
431#97—1062no—
432#95—21no—
433#95—21no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442C24—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8C2×C4×D5312yes—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D5312yes—
453C2×D4#1171994712yes—
454C2×D4#1171994712yes—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—1064no—
466C22×C4—1064no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4D4×D5311yescharacteristic
476C22×C4D4×D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478SD16—28no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—532no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8C2×C4×D5312yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—532no—
495C4⋊C4—532no—
496C4⋊C4—532no—
497C4⋊C4—1064no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4#1171994712yes—
506C22⋊C4—24no—
507C22⋊C4—1064no—
508C22⋊C4—1064no—
509C42—21no—
510C42—21no—
511C42—532no—
512C42—532no—
513C42—1062no—
514C42—1064no—
515C42—1062no—
516C42—1061no—
517C42—1061no—
518C42—2121no—
519C42—2121no—
520C42—2122no—
521C42—2122no—
522C42—2122no—
523C42—2122no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C2×D4×D5311yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1313317314yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—532no—
556C2×C4—532no—
557C2×C4—532no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1061no—
567C2×C4—1062no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—2128no—
571C2×C4—2128no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4#1313324412yes—
582C2×C4C22×C4×D5311yescharacteristic
583C2×C4#41×D5311yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4C2×#1171994711yescharacteristic
586C2×C4C2×#1171994711yescharacteristic
587C2×C4C2×#1171994711yescharacteristic
588C8—22no—
589C8—41no—
590C8—1062no—
591C8—2121no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C4×D4×D5311yescharacteristic
600C22#1681112911yescharacteristic
601C22#1681113311yescharacteristic
602C22C2×#1313324411yescharacteristic
603C22#137×D5311yescharacteristic
604C22#1681141911yescharacteristic
605C22#1681142011yescharacteristic
606C4—22no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—1062no—
612C4—1062no—
613C4—1062no—
614C4—4241no—
615C4—2122no—
616C4—2122no—
617C4—2122no—
618C4—2121no—
619C4C2×#1313317311yescharacteristic
620C4C4×#1171994711yescharacteristic
621C4#1681115011yescharacteristic
622C4#1681115411yescharacteristic
623C2—44no—
624C2#2027715511yescharacteristic
625C2C4×#1313324411yescharacteristic
626C2#2028353711yescharacteristic
627C2#2028481311yescharacteristic
628C2#2028482111yescharacteristic
629C2#2028485411yescharacteristic
630C2#2028487011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1877C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1877D53
hypocenterC53#21121
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
2. #20277190C2semidirect
4. #20275359C2semidirect
5. #20275346C2semidirect
6. #20275306C2semidirect
7. #20275287C2semidirect
11. #20274611C2semidirect
12. #20274547C2semidirect
14. #20274462C2semidirect
20. #16810724C4semidirect
65. Q8×Dic53C2×C4semidirect
91. #13133151C2×C4semidirect
95. #13133141C2×C4semidirect
99. #13133127C2×C4semidirect
375. C53#21121semidirect

Maximal subgroups

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

subgroupindex
C2×#168107242
#202771902
C4×D4×Dic532
#202753592
#202753462
#202753062
#202752872
#202750512
#202749742
#202749622
#202746112
#202745472
#202745422
#202744622
C53×#18772
#2112153