Finite Group Databasebeta

← #27425453

Subgroups — #27425453

Order 13568.

746 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810919C211yesmaximal, characteristic, complemented
2#20280536C211yesmaximal, characteristic
3#20277397C211yesmaximal, characteristic
4#20276143C211yesmaximal, characteristic
5#20276138C211yesmaximal, characteristic
6#20275919C211yesmaximal, characteristic
7#20275912C211yesmaximal, characteristic
8#20275666C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20275458C211yesmaximal, characteristic, complemented
11#20275448C211yesmaximal, characteristic, complemented
12#20275405C211yesmaximal, characteristic, complemented
13#20275391C211yesmaximal, characteristic, complemented
14#20274731C211yesmaximal, characteristic, complemented
15C2×C53×#433C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19#16810919—24no—
20C2×#13133151C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28#16810562—21no—
29#16810559—21no—
30#16810543—21no—
31#16810535—21no—
32#16810529C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810521C2211yescharacteristic
36#16810518—21no—
37#16810517—21no—
38#16810511—21no—
39#16810504—21no—
40#16810489C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810483C2211yescharacteristic
43#16810483C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C53×#433—24no—
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C2×#97×C53C2211yescharacteristic
60C2×C4×C8×C53C2211yescharacteristic
61C22×Dic106—21no—
62C22×Dic106C2311yescharacteristic
63#13133217—22no—
64#13133217—22no—
65#13133196—22no—
66#13133196—22no—
67C4×Dic106—22no—
68C4×Dic106—22no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
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#13133151—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×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112#117×C53—22no—
113#117×C53—22no—
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53D411yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×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—
125C4×C8×C53—22no—
126C24×C53C2×D411yescharacteristic
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—42no—
153C2×Dic106—81no—
154C2×Dic106—81no—
155C2×Dic106#4112yes—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
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×C53C2×D412yes—
167C2×C8×C53C2×D412yes—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53C2×D412yes—
175C42×C53#4112yes—
176C22⋊Dic53—24no—
177C22⋊Dic53—81no—
178C22⋊Dic53—81no—
179C22⋊Dic53—81no—
180C4⋊Dic53—21no—
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—42no—
194C4⋊Dic53#4112yes—
195#11719919—24no—
196#11719919—42no—
197#11719919—81no—
198#11719919—81no—
199#11719919—81no—
200C4×Dic53—42no—
201C4×Dic53—81no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53C22×D411yescharacteristic
210C2×C4×C53—21no—
211C2×C4×C53—21no—
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—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53#12112yes—
228C2×C4×C53#12112yes—
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×C4×C532−1+411yescharacteristic
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
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—
259Dic106—22no—
260Dic106—22no—
261Dic106—41no—
262Dic106—41no—
263Dic106—84no—
264C8×C53—22no—
265C8×C53—22no—
266C8×C53—41no—
267C8×C53—41no—
268#24714—531nomaximal, Sylow
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53C2×#12111yescharacteristic
277C22×C53#52511yescharacteristic
278C22×C53#53611yescharacteristic
279C22×C53#57611yescharacteristic
280C22×C53#57711yescharacteristic
281C22×C53#57811yescharacteristic
282C22×C53#57811yescharacteristic
283C4×C53—22no—
284C4×C53—22no—
285C4×C53—41no—
286C4×C53—41no—
287C4×C53—41no—
288C4×C53#54711yescharacteristic
289C4×C53#54811yescharacteristic
290C4×C53#55211yescharacteristic
291C4×C53#55611yescharacteristic
292Dic53—42no—
293Dic53—42no—
294Dic53—81no—
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298Dic53—81no—
299Dic53—81no—
300C2×#477—531no—
301C2×#433D5311yescharacteristic
302#2628—531no—
303#2323—531no—
304#2076—531no—
305#2075—531no—
306#2036—531no—
307#2035—531no—
308#1988—531no—
309#1949—531no—
310#1944—531no—
311#1942—531no—
312#1935—531no—
313#1933—531no—
314#1805—531no—
315C2×C53—81no—
316C2×C53#241511yescharacteristic
317C2×C53#305911yescharacteristic
318C2×C53#306211yescharacteristic
319C2×C53#312111yescharacteristic
320C2×C53#312411yescharacteristic
321C2×C53#324211yescharacteristic
322C2×C53#324311yescharacteristic
323C2×C4⋊Q8—531no—
324C2×C22⋊Q8—531no—
325C2×C4×Q8—531no—
326C2×#117C2×D5311yescharacteristic
327#477—1064no—
328#433—24no—
329C2×#107—531no—
330C2×#107—531no—
331C2×#107—531no—
332C2×#106—531no—
333C2×C4⋊C8C2×D5311yescharacteristic
334C2×C4⋊C8C2×D5311yescharacteristic
335C2×#103—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#103—531no—
339C2×C22⋊C8C2×D5311yescharacteristic
340C2×C22⋊C8C2×D5311yescharacteristic
341C2×#97C2×D5311yescharacteristic
342C2×C4×C8C2×D5311yescharacteristic
343#400—1061no—
344#399—1061no—
345#395—1061no—
346#393—1061no—
347#391—531no—
348#389—531no—
349#389—531no—
350#389—531no—
351#389—1061no—
352#388—1061no—
353#388—1061no—
354#387—1061no—
355#382—531no—
356#380—531no—
357#380—531no—
358#380—531no—
359C4×C4⋊C4—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368C53#2471411yescharacteristic, complemented, Sylow
369C22×Q8—531no—
370C22×Q8—1061no—
371C23×C4C22×D5311yescharacteristic
372C22×C8C22×D5311yescharacteristic
373C22×C8C22×D5311yescharacteristic
374C22×C8C22×D5311yescharacteristic
375C22×C8C22×D5311yescharacteristic
376C4⋊Q8—1062no—
377C4⋊Q8—1062no—
378C22⋊Q8—1062no—
379C22⋊Q8—1062no—
380C4×Q8—1062no—
381C4×Q8—1062no—
382#117—22no—
383#117—22no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4D21211yescharacteristic
398C2×C4⋊C4D21211yescharacteristic
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4D21211yescharacteristic
404C2×C22⋊C4D21211yescharacteristic
405C2×C42—1061no—
406C2×C42—1061no—
407C2×C42C22×D5311yescharacteristic
408C2×C42C22×D5311yescharacteristic
409#107—1062no—
410#107—1062no—
411#107—1062no—
412#106—1062no—
413C4⋊C8—22no—
414C4⋊C8C22⋊D5314yes—
415#103—1062no—
416#103—1062no—
417#103—1062no—
418#103—1062no—
419C22⋊C8—22no—
420C22⋊C8—22no—
421#97—22no—
422C4×C8—22no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434C24C2×D21211yescharacteristic
435C2×Q8—532no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C2×Q8—1061no—
441C2×Q8—4241no—
442C2×Q8—4241no—
443C2×Q8—2122no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4C2×D21211yescharacteristic
459C22×C4C2×D21211yescharacteristic
460C22×C4C2×D21211yescharacteristic
461C22×C4C2×D21211yescharacteristic
462C22×C4C2×D21211yescharacteristic
463C22×C4C23×D5311yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8C2×C22⋊D5312yes—
475C2×C8C2×C22⋊D5312yes—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—532no—
479C4⋊C4—1064no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
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—4241no—
494C4⋊C4—4241no—
495C4⋊C4—4241no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—1064no—
501C22⋊C4—4241no—
502C22⋊C4—4241no—
503C22⋊C4—4241no—
504C42—21no—
505C42—21no—
506C42—4241no—
507C42—2122no—
508C42#1171994512yes—
509C42C2×C22⋊D5312yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23C22×D21211yescharacteristic
518Q8—1062no—
519Q8—1062no—
520Q8—4244no—
521Q8—2121no—
522Q8—2121no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
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—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—1062no—
541C2×C4—1062no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
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#1313319112yes—
565C2×C4#1313319112yes—
566C2×C4C22×C22⋊D5311yescharacteristic
567C2×C4#41×D5311yescharacteristic
568C2×C4#41×D5311yescharacteristic
569C2×C4C2×#1171994511yescharacteristic
570C2×C4#1313326411yescharacteristic
571C2×C4#1313326411yescharacteristic
572C8—22no—
573C8—22no—
574C8—41no—
575C8—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1313319111yescharacteristic
584C22#1681113511yescharacteristic
585C22#1681119911yescharacteristic
586C22#1681142811yescharacteristic
587C22#1681143511yescharacteristic
588C22#1681144011yescharacteristic
589C22#1681144011yescharacteristic
590C4—22no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—4241no—
596C4—4241no—
597C4—4241no—
598C4—4241no—
599C4—4241no—
600C4—4241no—
601C4—2122no—
602C4—2122no—
603C4#1681127211yescharacteristic
604C4#1681128011yescharacteristic
605C4#1681131211yescharacteristic
606C4#1681134111yescharacteristic
607C2—81no—
608C2#2027813811yescharacteristic
609C2#2028437011yescharacteristic
610C2#2028439511yescharacteristic
611C2#2028494611yescharacteristic
612C2#2028497911yescharacteristic
613C2#2028598511yescharacteristic
614C2#2028600011yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#433C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#433D53
hypocenterC53#24714
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
1. C2×#16810919C2semidirect
8. #20275666C2semidirect
9. #20275485C2semidirect
10. #20275458C2semidirect
11. #20275448C2semidirect
12. #20275405C2semidirect
13. #20275391C2semidirect
14. #20274731C2semidirect
368. C53#24714semidirect

Maximal subgroups

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

subgroupindex
C2×#168109192
#202805362
#202773972
#202761432
#202761382
#202759192
#202759122
#202756662
#202754852
#202754582
#202754482
#202754052
#202753912
#202747312
C2×C53×#4332
#2471453