Finite Group Databasebeta

← #27393829

Subgroups — #27393829

Order 13568.

1234 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811160C211yesmaximal, characteristic
2C2×#16810679C211yesmaximal, characteristic, complemented
3#384×D53C211yesmaximal, characteristic, complemented
4#20275682C212yesmaximal, complemented
5#20275528C212yesmaximal, complemented
6#20275031C212yesmaximal
7#20275019C212yesmaximal, complemented
8#20274720C211yesmaximal, characteristic, complemented
9#20274716C211yesmaximal, characteristic, complemented
10#20274712C211yesmaximal, characteristic, complemented
11C53×#1803C211yesmaximal, characteristic, complemented
12#16811160—24no—
13#16811160C418yescomplemented
14C2×#13133212—21no—
15C2×#13133208C2211yescharacteristic
16C2×#13133206C2211yescharacteristic
17C2×#13133202—21no—
18C2×#13133193C2211yescharacteristic
19C2×#13133192C2211yescharacteristic
20C2×C4×D212—21no—
21C2×C4×D212C2211yescharacteristic
22C2×#13133174—21no—
23C2×C4⋊C4×D53—21no—
24C2×C4⋊C4×D53—21no—
25C2×C4⋊C4×D53C2212yes—
26C2×C42×D53C2211yescharacteristic
27#16810679—24no—
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133141C2212yes—
30C2×#13133139C2212yes—
31C2×#13133137C2212yes—
32C2×#13133129C2212yes—
33C2×#13133127C2212yes—
34#16810493C2211yescharacteristic
35#16810492—21no—
36#16810492—21no—
37#16810492C2212yes—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C2×C4⋊C8×C53C2211yescharacteristic
43C2×#103×C53C2212yes—
44C53×#384C2211yescharacteristic
45C53×#340C2212yes—
46C22×D212—21no—
47C22×D212C2311yescharacteristic
48C22×C4×D53—21no—
49C22×C4×D53—22no—
50C22×C4×D53D411yescharacteristic
51C22×C4×D53D411yescharacteristic
52C22×C4×D53C2311yescharacteristic
53C22×#88322C2312yes—
54#13133212—28no—
55#13133208—22no—
56#13133208C2×C414yescomplemented
57#13133206—22no—
58#13133206C2×C414yescomplemented
59#13133202—28no—
60#13133193—22no—
61#13133193C2×C414yes—
62#13133192—22no—
63#13133192C2×C414yes—
64C4×D212—22no—
65C4×D212—28no—
66C4×D212C2×C414yescomplemented
67#13133174—28no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2312yes—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—22no—
76C2×#11719919—22no—
77C2×#11719919—22no—
78C4⋊C4×D53—42no—
79C4⋊C4×D53—42no—
80C4⋊C4×D53—44no—
81C4⋊C4×D53—44no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C42×D53C2×C414yes—
91C42×D53D414yes—
92C4⋊#88322—22no—
93#13133141—42no—
94#13133139—42no—
95#13133137—42no—
96#13133129—24no—
97#13133127—24no—
98C22×Q8×C53C2311yescharacteristic
99C22×C8×C53C2312yes—
100C4×Q8×C53—22no—
101C4×Q8×C53C2×C414yescomplemented
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53C2312yes—
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C42×C53—21no—
108C2×C42×C53C2311yescharacteristic
109C4⋊C8×C53—22no—
110#103×C53—42no—
111C23×D53—21no—
112C23×D53C2×D411yescharacteristic
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53—21no—
117C2×Q8×C53—21no—
118C2×Q8×C53C22×C412yes—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2411yescharacteristic
125C22×Dic53—21no—
126C22×Dic53—22no—
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130C2×D212—21no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212—24no—
135C2×D212—21no—
136C2×D212—42no—
137C2×D212—42no—
138C2×D212C22×C412yes—
139C2×C4×D53—22no—
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—21no—
144C2×C4×D53—21no—
145C2×C4×D53—21no—
146C2×C4×D53—21no—
147C2×C4×D53—24no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C4×D53—44no—
151C2×C4×D53—44no—
152C2×C4×D53—44no—
153C2×C4×D53C22⋊C414yes—
154C2×C4×D53C22⋊C414yes—
155C2×C4×D53C22×C412yes—
156C2×C4×D53C2×D412yes—
157C2×C8×C53—22no—
158C2×C8×C53—22no—
159C2×C8×C53—22no—
160C4⋊C4×C53—24no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—42no—
165C4⋊C4×C53—42no—
166C4⋊C4×C53C22×C412yes—
167C42×C53—24no—
168C42×C53C22×C412yes—
169C42×C53C2×D412yes—
170#11719921—24no—
171#11719921—24no—
172#11719921—24no—
173#11719921—24no—
174#11719921—24no—
175C4⋊Dic53—21no—
176C4⋊Dic53—24no—
177C4⋊Dic53—22no—
178C4⋊Dic53—22no—
179C4⋊Dic53—42no—
180C4⋊Dic53—42no—
181C4⋊Dic53C22×C412yes—
182#11719919—24no—
183#11719919—24no—
184#11719919—44no—
185#11719919—44no—
186#11719919—44no—
187C4×Dic53C22⋊C414yes—
188C4×Dic53C22⋊C414yes—
189C4×Dic53C22×C412yes—
190C4×Dic53C2×D412yes—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—22no—
194C23×C53C22×D411yescharacteristic
195C22×D53—21no—
196C22×D53—21no—
197C22×D53—21no—
198C22×D53—21no—
199C22×D53—22no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53C2×C22⋊C412yes—
207C22×D53AGL(1,ℤ/8)12yes—
208C22×D53#13712yes—
209Q8×C53—22no—
210Q8×C53—22no—
211Q8×C53—41no—
212Q8×C53—41no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53C2×C22⋊C412yes—
225C2×C4×C53C2×C22⋊C412yes—
226C2×C4×C53AGL(1,ℤ/8)12yes—
227C2×C4×C53#13712yes—
228C2×C4×C53C23×C411yescharacteristic
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53H(5,𝔽2)11yescharacteristic
231C2×C4×C53H(5,𝔽2)11yescharacteristic
232C2×C4×C532−1+411yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—22no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53C2×C22⋊C412yes—
247C2×Dic53C2×C22⋊C412yes—
248C2×Dic53C2×C22⋊C412yes—
249D212—22no—
250D212—22no—
251D212—48no—
252D212—41no—
253D212—41no—
254C4×D53—28no—
255C4×D53—22no—
256C4×D53—22no—
257C4×D53—22no—
258C4×D53—22no—
259C4×D53—28no—
260C4×D53—48no—
261C4×D53—88no—
262C8×C53—42no—
263#88322—42no—
264#21867—531nomaximal, Sylow
265C22×C53C22×C22⋊C411yescharacteristic
266C22×C53#53511yescharacteristic
267C22×C53#53611yescharacteristic
268C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C53C2×#13711yescharacteristic
271C22×C53C2×#13711yescharacteristic
272C2×D53—22no—
273C2×D53—22no—
274C2×D53—22no—
275C2×D53—22no—
276C2×D53—22no—
277C2×D53—22no—
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—42no—
281C2×D53—81no—
282C2×D53—81no—
283C2×D53—81no—
284C2×D53—81no—
285C2×D53#41714yes—
286C4×C53—22no—
287C4×C53—42no—
288C4×C53—42no—
289C4×C53—82no—
290C4×C53#41714yes—
291C4×C53#52011yescharacteristic
292C4×C53#52011yescharacteristic
293C4×C53#53711yescharacteristic
294C4×C53#53711yescharacteristic
295Dic53—22no—
296Dic53—22no—
297Dic53—22no—
298Dic53—42no—
299Dic53—82no—
300C2×#529—531no—
301C2×#432—531no—
302C2×#384—531no—
303#1992—532no—
304#1959—532no—
305#1874—532no—
306#1872—532no—
307#1803—531no—
308#1803D5311yescharacteristic, complemented
309#1802—531no—
310#1802—531no—
311C2×C53#231111yescharacteristic
312C2×C53C2×#41711yescharacteristic
313C2×C53C2×#41711yescharacteristic
314C2×C53#321711yescharacteristic
315C2×C53#321811yescharacteristic
316C2×C53#323811yescharacteristic
317C2×C53#323911yescharacteristic
318D53—24no—
319D53—82no—
320#529—538no—
321#529—1064no—
322C2×#126—1061no—
323C2×#125—531no—
324C2×#124—531no—
325C2×#123—1061no—
326C2×C22⋊Q8—531no—
327C2×#121—531no—
328C2×C4×Q8C2×D5311yescharacteristic
329C2×C4×D4—531no—
330C2×C4×D4—1061no—
331C2×#117—1061no—
332C22×C4⋊C4—532no—
333C22×C4⋊C4—1061no—
334C22×C4⋊C4—1061no—
335C22×C42—531no—
336#432—1064no—
337C2×C4⋊C8—531no—
338C2×C4⋊C8C2×D5311yescharacteristic
339C2×#103—532no—
340C2×#103C2×D5312yes—
341C2×#102—532no—
342C2×#102—532no—
343C2×C22⋊C8—532no—
344C2×#97—532no—
345#384—532no—
346#384—531no—
347#384—1061no—
348#384—1061no—
349#384C2×D5311yescharacteristic
350#340—532no—
351#340—532no—
352#340—532no—
353#340C2×D5312yes—
354C53#2186711yescharacteristic, complemented, Sylow
355C22×Q8C22×D5311yescharacteristic
356C22×D4—531no—
357C22×D4—1061no—
358C23×C4—531no—
359C23×C4—531no—
360C23×C4—531no—
361C23×C4—1062no—
362C23×C4—1061no—
363C22×C8—532no—
364C22×C8C22×D5312yes—
365#126—1068no—
366#125—534no—
367#125—1062no—
368#124—534no—
369#124—1062no—
370#123—1068no—
371C22⋊Q8—534no—
372C22⋊Q8—1062no—
373#121—534no—
374#121—1062no—
375C4×Q8—22no—
376C4×Q8C4×D5314yescomplemented
377C4×D4—534no—
378C4×D4—1068no—
379C4×D4—1062no—
380#117—1068no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—532no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1062no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—2122no—
395C2×C4⋊C4—2122no—
396C2×C4⋊C4—2124no—
397C2×C4⋊C4—2124no—
398C2×C4⋊C4C22×D5312yes—
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C22⋊C4—1061no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C42—21no—
406C2×C42—531no—
407C2×C42—531no—
408C2×C42—531no—
409C2×C42—534no—
410C2×C42—534no—
411C2×C42C22×D5311yescharacteristic
412C4⋊C8—22no—
413C4⋊C8—1062no—
414#103—42no—
415#103—2122no—
416#102—2122no—
417#102—2122no—
418C22⋊C8—1064no—
419#97—1064no—
420C24—531no—
421C24—1061no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8C2×C4×D5312yes—
428C2×D4—532no—
429C2×D4—1064no—
430C2×D4—1061no—
431C2×D4—1061no—
432C2×D4—1061no—
433C2×D4—1061no—
434C2×D4—1061no—
435C2×D4—2122no—
436C2×D4—2122no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—531no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—534no—
444C22×C4—534no—
445C22×C4—532no—
446C22×C4—532no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1064no—
457C22×C4—1062no—
458C22×C4—2124no—
459C22×C4—2124no—
460C22×C4—2124no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4D4×D5311yescharacteristic
464C22×C4D4×D5311yescharacteristic
465C22×C4C23×D5311yescharacteristic
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—1062no—
470C2×C8—1062no—
471C2×C8—1062no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—532no—
479C4⋊C4—1064no—
480C4⋊C4—1064no—
481C4⋊C4—1064no—
482C4⋊C4—1061no—
483C4⋊C4—1062no—
484C4⋊C4—1062no—
485C4⋊C4—2122no—
486C4⋊C4—2124no—
487C4⋊C4—2124no—
488C4⋊C4—2124no—
489C4⋊C4—2122no—
490C4⋊C4C2×C4×D5312yes—
491C22⋊C4—1064no—
492C22⋊C4—1064no—
493C22⋊C4—1064no—
494C22⋊C4—1064no—
495C22⋊C4—1064no—
496C42—24no—
497C42—534no—
498C42—534no—
499C42—532no—
500C42—532no—
501C42C2×C4×D5312yes—
502C42D4×D5312yes—
503C23—532no—
504C23—532no—
505C23—532no—
506C23—1061no—
507C23—1061no—
508C23—1061no—
509C23—1061no—
510C23—1062no—
511C23—2121no—
512C23—2121no—
513C23—2121no—
514C23—2121no—
515C23—2121no—
516C23—2121no—
517C23C2×D4×D5311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—1062no—
523D4—1062no—
524D4—2128no—
525D4—2121no—
526D4—2121no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—532no—
539C2×C4—532no—
540C2×C4—532no—
541C2×C4—1068no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—1068no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—4248no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2128no—
561C2×C4C22⋊C4×D5312yes—
562C2×C4C22⋊C4×D5312yes—
563C2×C4#1313324412yes—
564C2×C4#1313324912yes—
565C2×C4C22×C4×D5311yescharacteristic
566C2×C4C2×D4×D5311yescharacteristic
567C2×C4#1313326211yescharacteristic
568C2×C4#1313326311yescharacteristic
569C2×C4#1313326411yescharacteristic
570C2×C4#1313326511yescharacteristic
571C8—42no—
572C8—2122no—
573C22—534no—
574C22—1062no—
575C22—1062no—
576C22—1062no—
577C22—1062no—
578C22—1062no—
579C22—1062no—
580C22—4241no—
581C22—4241no—
582C22—4241no—
583C22—4241no—
584C22—2122no—
585C22—2122no—
586C22—2122no—
587C22C2×C22⋊C4×D5311yescharacteristic
588C22#1681118911yescharacteristic
589C22#1681119611yescharacteristic
590C22AGL(1,ℤ/8)×D5311yescharacteristic
591C22C2×#1313324411yescharacteristic
592C22#137×D5311yescharacteristic
593C22C2×#1313324911yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—82no—
598C4—1062no—
599C4—1062no—
600C4—1062no—
601C4—4242no—
602C4—2122no—
603C4#1681062314yes—
604C4#1681110311yescharacteristic
605C4#1681110411yescharacteristic
606C4#1681120211yescharacteristic
607C4#1681120311yescharacteristic
608C2—1064no—
609C2—4242no—
610C2#2027730411yescharacteristic
611C2#417×D5311yescharacteristic
612C2C2×#1681062311yescharacteristic
613C2#2028573711yescharacteristic
614C2#2028574611yescharacteristic
615C2#2028594011yescharacteristic
616C2#2028594911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1803C2
socleC23×C53C22×D4
prime radicalC2×#13133192C22
cosocleC22×C4×C53C24
hypercenter#1803D53
hypocenterC53#21867
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#16810679C2semidirect
3. #384×D53C2semidirect
4. #20275682C2semidirect
5. #20275528C2semidirect
7. #20275019C2semidirect
8. #20274720C2semidirect
9. #20274716C2semidirect
10. #20274712C2semidirect
11. C53×#1803C2semidirect
13. #16811160C4semidirect
56. #13133208C2×C4semidirect
58. #13133206C2×C4semidirect
66. C4×D212C2×C4semidirect
101. C4×Q8×C53C2×C4semidirect
308. #1803D53semidirect
354. C53#21867semidirect
376. C4×Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111602
C2×#168106792
#384×D532
#202756822
#202755282
#202750312
#202750192
#202747202
#202747162
#202747122
C53×#18032
#2186753