Finite Group Databasebeta

← #27425364

Subgroups — #27425364

Order 13568.

784 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810931C211yesmaximal, characteristic, complemented
2#20283025C211yesmaximal, characteristic
3C4×#13133196C211yesmaximal, characteristic
4#20276148C211yesmaximal, characteristic
5#20276143C211yesmaximal, characteristic
6#20275919C211yesmaximal, characteristic
7#20275912C211yesmaximal, characteristic
8#20275670C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20275458C211yesmaximal, characteristic, complemented
11#20275453C211yesmaximal, characteristic, complemented
12#20275405C211yesmaximal, characteristic, complemented
13#20275405C211yesmaximal, characteristic, complemented
14#20274731C211yesmaximal, characteristic, complemented
15C2×C53×#432C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19#16810931—24no—
20C2×#13133151C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28#16810529C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810507—21no—
33#16810507—21no—
34#16810507—21no—
35#16810496—21no—
36#16810496—21no—
37#16810496—21no—
38#16810489C2211yescharacteristic
39#16810483C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42C4×#11719919—21no—
43C4×C4⋊Dic53C2211yescharacteristic
44C4×C22⋊Dic53—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C22×C42×C53C2211yescharacteristic
54C53×#432—24no—
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C2×#97×C53C2211yescharacteristic
60C2×#97×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62#13133209—22no—
63#13133209—22no—
64#13133196—22no—
65#13133196—22no—
66C4×Dic106—22no—
67C4×Dic106—22no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88#13133151—22no—
89#13133151—22no—
90#13133149—22no—
91#13133149—22no—
92#13133142—22no—
93#13133142—22no—
94#13133142—22no—
95#13133142—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107C23×C4×C53D411yescharacteristic
108C23×C4×C53D411yescharacteristic
109C23×C4×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C2×C42×C53—22no—
115C2×C42×C53—22no—
116C2×C42×C53D411yescharacteristic
117C2×C42×C53D411yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C4⋊C8×C53—22no—
121C4⋊C8×C53D414yes—
122C22⋊C8×C53—22no—
123C22⋊C8×C53—22no—
124#97×C53—22no—
125#97×C53—22no—
126C24×C53C2×D411yescharacteristic
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—41no—
137C22×C4×C53—41no—
138C22×C4×C53—41no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106#4112yes—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53C2×D412yes—
170C2×C8×C53C2×D412yes—
171C42×C53—22no—
172C42×C53—22no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53C2×D412yes—
176C42×C53#4112yes—
177C22⋊Dic53—24no—
178C22⋊Dic53—42no—
179C22⋊Dic53—42no—
180C22⋊Dic53—42no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53#4112yes—
195#11719919—24no—
196#11719919—42no—
197#11719919—42no—
198#11719919—42no—
199#11719919—42no—
200#11719919—42no—
201#11719919—42no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53#13712yes—
211C23×C53C22×D411yescharacteristic
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—44no—
226C2×C4×C53—44no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53#12112yes—
232C2×C4×C53#12112yes—
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C532−1+411yescharacteristic
238C2×C4×C532−1+411yescharacteristic
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
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—
263Dic106—22no—
264Dic106—22no—
265Dic106—41no—
266Dic106—41no—
267C8×C53—22no—
268C8×C53—22no—
269C8×C53—41no—
270C8×C53—41no—
271#24707—531nomaximal, Sylow
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53C2×#12111yescharacteristic
280C22×C53#52511yescharacteristic
281C22×C53#53611yescharacteristic
282C22×C53C2×#13711yescharacteristic
283C22×C53#57511yescharacteristic
284C22×C53#57511yescharacteristic
285C22×C53#57511yescharacteristic
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53—41no—
291C4×C53#54711yescharacteristic
292C4×C53#54811yescharacteristic
293C4×C53#55211yescharacteristic
294C4×C53#55611yescharacteristic
295Dic53—42no—
296Dic53—42no—
297Dic53—81no—
298Dic53—81no—
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302Dic53—81no—
303C2×#479—531no—
304C2×#432D5311yescharacteristic
305#2851—531no—
306C4×C22⋊Q8—531no—
307#2077—531no—
308#2076—531no—
309#2036—531no—
310#2035—531no—
311#1989—531no—
312#1949—531no—
313#1944—531no—
314#1943—531no—
315#1935—531no—
316#1935—531no—
317#1805—531no—
318C2×C53—42no—
319C2×C53#241511yescharacteristic
320C2×C53#305411yescharacteristic
321C2×C53#305611yescharacteristic
322C2×C53#311111yescharacteristic
323C2×C53#311711yescharacteristic
324C2×C53#323611yescharacteristic
325C2×C53#324011yescharacteristic
326C2×#125—531no—
327C2×C22⋊Q8—531no—
328C2×C4×Q8—531no—
329C22×C42C2×D5311yescharacteristic
330#479—1064no—
331#432—24no—
332C2×#107—531no—
333C2×#107—531no—
334C2×#106—531no—
335C2×#106—531no—
336C2×C4⋊C8C2×D5311yescharacteristic
337C2×C4⋊C8C2×D5311yescharacteristic
338C2×#103—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×C22⋊C8C2×D5311yescharacteristic
343C2×C22⋊C8C2×D5311yescharacteristic
344C2×#97C2×D5311yescharacteristic
345C2×#97C2×D5311yescharacteristic
346#391—531no—
347#389—531no—
348#389—531no—
349#389—531no—
350#387—1061no—
351#387—1061no—
352#387—1061no—
353#385—1061no—
354#385—1061no—
355#385—1061no—
356#382—531no—
357#380—531no—
358#380—531no—
359#380—531no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4—1061no—
362C4×C22⋊C4—1061no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371C53#2470711yescharacteristic, complemented, Sylow
372C22×Q8—531no—
373C23×C4D21211yescharacteristic
374C23×C4D21211yescharacteristic
375C23×C4C22×D5311yescharacteristic
376C22×C8C22×D5311yescharacteristic
377C22×C8C22×D5311yescharacteristic
378C22×C8C22×D5311yescharacteristic
379C22×C8C22×D5311yescharacteristic
380#125—1062no—
381#125—1062no—
382C22⋊Q8—1062no—
383C22⋊Q8—1062no—
384C4×Q8—1062no—
385C4×Q8—1062no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—1061no—
407C2×C42—1061no—
408C2×C42D21211yescharacteristic
409C2×C42D21211yescharacteristic
410C2×C42C22×D5311yescharacteristic
411C2×C42C22×D5311yescharacteristic
412#107—1062no—
413#107—1062no—
414#106—1062no—
415#106—1062no—
416C4⋊C8—22no—
417C4⋊C8C22⋊D5314yes—
418#103—1062no—
419#103—1062no—
420#103—1062no—
421#103—1062no—
422C22⋊C8—22no—
423C22⋊C8—22no—
424#97—22no—
425#97—22no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437C24C2×D21211yescharacteristic
438C2×Q8—532no—
439C2×Q8—1061no—
440C2×Q8—1061no—
441C2×Q8—1061no—
442C2×Q8—1061no—
443C2×Q8—1061no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4C2×D21211yescharacteristic
465C22×C4C2×D21211yescharacteristic
466C22×C4C2×D21211yescharacteristic
467C22×C4C2×D21211yescharacteristic
468C22×C4C2×D21211yescharacteristic
469C22×C4C23×D5311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8C2×C22⋊D5312yes—
481C2×C8C2×C22⋊D5312yes—
482C4⋊C4—532no—
483C4⋊C4—1064no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C22⋊C4—1064no—
504C22⋊C4—2122no—
505C22⋊C4—2122no—
506C22⋊C4—2122no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—2122no—
512C42—2122no—
513C42#1171994512yes—
514C42C2×C22⋊D5312yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23#1313325012yes—
522C23C22×D21211yescharacteristic
523Q8—1062no—
524Q8—1062no—
525Q8—2121no—
526Q8—2121no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4#1313319112yes—
571C2×C4#1313319112yes—
572C2×C4C22×C22⋊D5311yescharacteristic
573C2×C4#41×D5311yescharacteristic
574C2×C4#41×D5311yescharacteristic
575C2×C4C2×#1171994511yescharacteristic
576C2×C4#1313326411yescharacteristic
577C2×C4#1313326411yescharacteristic
578C8—22no—
579C8—22no—
580C8—41no—
581C8—41no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#1313319111yescharacteristic
590C22#1681113511yescharacteristic
591C22#1681119911yescharacteristic
592C22C2×#1313325011yescharacteristic
593C22#1681142311yescharacteristic
594C22#1681142311yescharacteristic
595C22#1681142311yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—4241no—
602C4—4241no—
603C4—4241no—
604C4—4241no—
605C4—4241no—
606C4—4241no—
607C4—2122no—
608C4—2122no—
609C4#1681127211yescharacteristic
610C4#1681128011yescharacteristic
611C4#1681131211yescharacteristic
612C4#1681134111yescharacteristic
613C2—42no—
614C2#2027813811yescharacteristic
615C2#2028432411yescharacteristic
616C2#2028434511yescharacteristic
617C2#2028482611yescharacteristic
618C2#2028488811yescharacteristic
619C2#2028592311yescharacteristic
620C2#2028595711yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#432C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#432D53
hypocenterC53#24707
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
1. C2×#16810931C2semidirect
8. #20275670C2semidirect
9. #20275485C2semidirect
10. #20275458C2semidirect
11. #20275453C2semidirect
12. #20275405C2semidirect
13. #20275405C2semidirect
14. #20274731C2semidirect
371. C53#24707semidirect

Maximal subgroups

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

subgroupindex
C2×#168109312
#202830252
C4×#131331962
#202761482
#202761432
#202759192
#202759122
#202756702
#202754852
#202754582
#202754532
#202754052
#202747312
C2×C53×#4322
#2470753