Finite Group Databasebeta

← #27432359

Subgroups — #27432359

Order 13568.

842 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810951C211yesmaximal, characteristic
2C2×#16810862C211yesmaximal, characteristic
3#20279200C211yesmaximal, characteristic
4#20276196C211yesmaximal, characteristic, complemented
5#20276133C211yesmaximal, characteristic, complemented
6#20276093C211yesmaximal, characteristic, complemented
7#20276037C211yesmaximal, characteristic, complemented
8#20275483C211yesmaximal, characteristic, complemented
9#20275349C211yesmaximal, characteristic, complemented
10#20275056C211yesmaximal, characteristic, complemented
11#20275022C211yesmaximal, characteristic
12#20274895C211yesmaximal, characteristic
13#20274730C211yesmaximal, characteristic, complemented
14C2×#16810385C211yesmaximal, characteristic
15C53×#2360C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20#16810951—24no—
21#16810862—24no—
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133140C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34#16810563—21no—
35#16810563—21no—
36#16810529C2211yescharacteristic
37#16810529C2211yescharacteristic
38#16810489C2211yescharacteristic
39#16810489C2211yescharacteristic
40#16810484—21no—
41#16810484—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#389—21no—
55C53×#389C2211yescharacteristic
56C53×#387—21no—
57C53×#385—21no—
58C4×C4⋊C4×C53C2211yescharacteristic
59C4×C22⋊C4×C53—21no—
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C23×Dic53—21no—
63C2×#11719925—22no—
64C2×#11719925—22no—
65C2×#11719925—41no—
66C22×#88322—22no—
67C22×#88322—22no—
68C22×#88322—41no—
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73#13133196—22no—
74#13133196—22no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83#13133150—22no—
84#13133148—22no—
85#13133148—22no—
86C4⋊#88322—22no—
87C4⋊#88322—41no—
88#13133141—22no—
89#13133141—22no—
90#13133141—22no—
91#13133140—22no—
92#13133140—22no—
93#13133140—41no—
94C22⋊#88322—22no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×Q8×C53C2311yescharacteristic
104C23×C4×C53—21no—
105C23×C4×C53C2311yescharacteristic
106C22⋊Q8×C53—22no—
107C22⋊Q8×C53D414yes—
108C4×Q8×C53—22no—
109C4×Q8×C53—41no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53—41no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C42×C53—21no—
123C2×C42×C53—21no—
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53C2×D411yescharacteristic
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53C2×D412yes—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—42no—
148C22×C4×C53—42no—
149C22×C4×C53—81no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D412yes—
156C22×C4×C53#4112yes—
157C22×C4×C53C2411yescharacteristic
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—42no—
163C22×Dic53—42no—
164C22×Dic53—42no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—21no—
171C4⋊C4×C53—24no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53—42no—
178C4⋊C4×C53—81no—
179C4⋊C4×C53C2×D412yes—
180C4⋊C4×C53#4112yes—
181C22⋊C4×C53—24no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—22no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—42no—
189C42×C53—81no—
190C22⋊Dic53—42no—
191#11719925—44no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—42no—
199#11719919—42no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—44no—
212C2×#88322#4112yes—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53#12112yes—
218C23×C53#13512yes—
219C23×C53#13712yes—
220C23×C53C22×D411yescharacteristic
221Q8×C53—22no—
222Q8×C53—22no—
223Q8×C53—41no—
224Q8×C53—41no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
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—84no—
256C2×C4×C53#12112yes—
257C2×C4×C53C22×D411yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53H(5,𝔽2)11yescharacteristic
262C2×C4×C532−1+411yescharacteristic
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—84no—
276Dic106—41no—
277Dic106—41no—
278Dic106—41no—
279Dic106—41no—
280#88322—22no—
281#88322—41no—
282#88322—41no—
283#88322—41no—
284#25208—531nomaximal, Sylow
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53#46414yes—
292C22×C53C2×#12111yescharacteristic
293C22×C53#54611yescharacteristic
294C22×C53#54711yescharacteristic
295C22×C53C2×#13511yescharacteristic
296C22×C53C2×#13711yescharacteristic
297C22×C53#57711yescharacteristic
298C22×C53#57811yescharacteristic
299C4×C53—22no—
300C4×C53—41no—
301C4×C53—41no—
302C4×C53—42no—
303C4×C53—41no—
304C4×C53—42no—
305C4×C53—42no—
306C4×C53—41no—
307C4×C53—81no—
308C4×C53#52911yescharacteristic
309C4×C53#53911yescharacteristic
310C4×C53#54611yescharacteristic
311C4×C53#54711yescharacteristic
312Dic53—81no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316C2×#483—531no—
317C2×#467—531no—
318#2503—531no—
319#2360D5311yescharacteristic
320#2085—531no—
321#2074—531no—
322#2066—531no—
323#2056—531no—
324#1949—531no—
325#1927—531no—
326#1878—531no—
327#1872—531no—
328#1850—531no—
329#1805—531no—
330C2×#340—531no—
331C2×C53—24no—
332C2×C53#242511yescharacteristic
333C2×C53C2×#46411yescharacteristic
334C2×C53#304611yescharacteristic
335C2×C53#329611yescharacteristic
336C2×C53#331211yescharacteristic
337C2×C53#333611yescharacteristic
338C2×C53#334411yescharacteristic
339C22×Q16—531no—
340C23×C8—531no—
341C2×C22⋊Q8—531no—
342C2×C22⋊Q8C2×D5311yescharacteristic
343C2×C4×Q8C2×D5311yescharacteristic
344C22×C4⋊C4—531no—
345C22×C4⋊C4C2×D5311yescharacteristic
346#483—1064no—
347#467—1064no—
348C2×#107—531no—
349C2×#106—531no—
350C2×#106—531no—
351C2×C4⋊C8—531no—
352C2×C4⋊C8—531no—
353C2×#103—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×C22⋊C8—531no—
360#400—1061no—
361#400—1061no—
362#391—531no—
363#391—531no—
364#391C2×D5311yescharacteristic
365#389—21no—
366#389C2×D5311yescharacteristic
367#387—21no—
368#385—21no—
369#382—531no—
370#382—531no—
371#381—1061no—
372#381—1061no—
373C4×C4⋊C4C2×D5311yescharacteristic
374C4×C22⋊C4—21no—
375C4×C22⋊C4C2×D5311yescharacteristic
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#336—531no—
383#336—531no—
384C53#2520811yescharacteristic, complemented, Sylow
385C22×Q8—531no—
386C22×Q8C22×D5311yescharacteristic
387C23×C4—21no—
388C23×C4—1061no—
389C23×C4C22×D5311yescharacteristic
390C2×Q16—1062no—
391C2×Q16—1062no—
392C2×Q16—2121no—
393C22×C8—531no—
394C22×C8—531no—
395C22×C8—531no—
396C22×C8—531no—
397C22×C8—1062no—
398C22×C8—1062no—
399C22×C8—2121no—
400C22⋊Q8—22no—
401C22⋊Q8—1062no—
402C22⋊Q8—1062no—
403C22⋊Q8D21214yes—
404C4×Q8—22no—
405C4×Q8—41no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—41no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—1062no—
416C2×C4⋊C4—1062no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4C22⋊D5311yescharacteristic
425C2×C22⋊C4C22⋊D5311yescharacteristic
426C2×C42—21no—
427C2×C42—21no—
428C2×C42C22×D5311yescharacteristic
429C2×C42C22×D5311yescharacteristic
430#107—1062no—
431#106—1062no—
432#106—1062no—
433C4⋊C8—1062no—
434C4⋊C8—2121no—
435#103—1062no—
436#103—1062no—
437#103—1062no—
438#103—1062no—
439#103—1062no—
440#103—2121no—
441C22⋊C8—1062no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95C22⋊D5311yescharacteristic
455#95C22⋊D5311yescharacteristic
456C24C2×C22⋊D5311yescharacteristic
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—1061no—
463C2×Q8—1061no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8—1061no—
467C2×Q8—1061no—
468C2×Q8C2×D21212yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—81no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4—2122no—
487C22×C4C2×D21212yes—
488C22×C4#1171994512yes—
489C22×C4C2×C22⋊D5311yescharacteristic
490C22×C4C2×C22⋊D5311yescharacteristic
491C22×C4C2×C22⋊D5311yescharacteristic
492C22×C4C2×C22⋊D5311yescharacteristic
493C22×C4C2×C22⋊D5311yescharacteristic
494C22×C4C23×D5311yescharacteristic
495Q16—2124no—
496C2×C8—532no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—2124no—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4C2×D21212yes—
526C4⋊C4#1171994712yes—
527C22⋊C4—24no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—2122no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—81no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23#1313319112yes—
542C23#1313323712yes—
543C23#1313324812yes—
544C23C22×C22⋊D5311yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—41no—
548Q8—41no—
549Q8—2121no—
550Q8—2121no—
551Q8—2121no—
552Q8—2121no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—84no—
584C2×C4—4244no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4#1313319112yes—
598C2×C4C22×D21211yescharacteristic
599C2×C4#41×D5311yescharacteristic
600C2×C4C2×#1171994711yescharacteristic
601C2×C4C2×#1171994511yescharacteristic
602C2×C4#1313326211yescharacteristic
603C2×C4#1313326411yescharacteristic
604C8—1062no—
605C8—2121no—
606C8—2121no—
607C8—2121no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22#1681084514yes—
615C22C2×#1313319111yescharacteristic
616C22#1681126711yescharacteristic
617C22#1681127211yescharacteristic
618C22C2×#1313323711yescharacteristic
619C22C2×#1313324811yescharacteristic
620C22#1681143411yescharacteristic
621C22#1681143811yescharacteristic
622C4—22no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—42no—
628C4—41no—
629C4—42no—
630C4—81no—
631C4—4241no—
632C4—4241no—
633C4—4241no—
634C4—4241no—
635C4#1681115311yescharacteristic
636C4#1681121511yescharacteristic
637C4#1681126811yescharacteristic
638C4#1681127711yescharacteristic
639C2—24no—
640C2#2027826011yescharacteristic
641C2C2×#1681084511yescharacteristic
642C2#2028426511yescharacteristic
643C2#2028655011yescharacteristic
644C2#2028676611yescharacteristic
645C2#2028707711yescharacteristic
646C2#2028716411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2360C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2360D53
hypocenterC53#25208
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
4. #20276196C2semidirect
5. #20276133C2semidirect
6. #20276093C2semidirect
7. #20276037C2semidirect
8. #20275483C2semidirect
9. #20275349C2semidirect
10. #20275056C2semidirect
13. #20274730C2semidirect
384. C53#25208semidirect

Maximal subgroups

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

subgroupindex
C2×#168109512
C2×#168108622
#202792002
#202761962
#202761332
#202760932
#202760372
#202754832
#202753492
#202750562
#202750222
#202748952
#202747302
C2×#168103852
C53×#23602
#2520853