Finite Group Databasebeta

← #27428646

Subgroups — #27428646

Order 13568.

776 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279026C211yesmaximal, characteristic
2#20277646C211yesmaximal, characteristic
3#20276285C211yesmaximal, characteristic, complemented
4#20276201C211yesmaximal, characteristic, complemented
5#20276150C211yesmaximal, characteristic, complemented
6#20276142C211yesmaximal, characteristic, complemented
7#20275292C211yesmaximal, characteristic, complemented
8#20275284C211yesmaximal, characteristic, complemented
9#20275005C211yesmaximal, characteristic
10#20274915C211yesmaximal, characteristic, complemented
11#20274910C211yesmaximal, characteristic, complemented
12#20274808C211yesmaximal, characteristic, complemented
13#20274742C211yesmaximal, characteristic, complemented
14C2×#16810385C211yesmaximal, characteristic, complemented
15C53×#1824C211yesmaximal, characteristic, complemented
16C23×#88322C2211yescharacteristic
17C2×C4×C22⋊D53C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133129C2211yescharacteristic
25C2×#13133129C2211yescharacteristic
26#16810561—21no—
27#16810558—21no—
28#16810519C2211yescharacteristic
29#16810519C2211yescharacteristic
30#16810515—21no—
31#16810510—21no—
32#16810502—21no—
33#16810500—21no—
34#16810496—21no—
35#16810488C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810480C2211yescharacteristic
38#16810480C2211yescharacteristic
39#16810480C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C22⋊C4×Dic53—21no—
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C53×#380C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×C22⋊D53D411yescharacteristic
60C23×Dic53—21no—
61C22×C4×D53C2311yescharacteristic
62C22×#88322—22no—
63C22×#88322—22no—
64C22×#88322—41no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C4×C22⋊D53—22no—
68C4×C22⋊D53—22no—
69#13133175—22no—
70#13133175—41no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53—41no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919D411yescharacteristic
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53D411yescharacteristic
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921D411yescharacteristic
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91#13133150—22no—
92#13133150—41no—
93#13133149—22no—
94#13133149—41no—
95#13133129—22no—
96#13133129—41no—
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—
106C23×C4×C53—21no—
107C23×C4×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—22no—
112C2×C4⋊C4×C53—22no—
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C22⋊C8×C53—22no—
119#95×C53—21no—
120C24×C53C2×D411yescharacteristic
121C23×D53C2×D411yescharacteristic
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—42no—
131C22×C4×C53—81no—
132C22×C4×C53—81no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53#4112yes—
135C22×C4×C53#4112yes—
136C22×C4×C53C2411yescharacteristic
137C2×C22⋊D53—22no—
138C2×C22⋊D53—22no—
139C2×C22⋊D53—22no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—42no—
145C22×Dic53—42no—
146C22×Dic53—81no—
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×C4×D53—21no—
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—42no—
171C22⋊C4×C53—42no—
172C22⋊Dic53—22no—
173C22⋊Dic53—42no—
174#11719921—22no—
175#11719921—42no—
176#11719921—81no—
177#11719921—81no—
178#11719921—81no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—42no—
187C4⋊Dic53#4112yes—
188#11719919—22no—
189#11719919—81no—
190#11719919—81no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—42no—
194C4×Dic53—81no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—44no—
201C2×#88322#4112yes—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53#12312yes—
207C23×C53#13512yes—
208C23×C53#13712yes—
209C23×C53C22×D411yescharacteristic
210C22×D53—21no—
211C22×D53—21no—
212C22×D53—21no—
213C22×D53—21no—
214C22×D53—21no—
215C22×D53—21no—
216C22×D53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—84no—
236C2×C4×C53#12312yes—
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C532−1+411yescharacteristic
243C22⋊D53—44no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—22no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
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—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—84no—
266C4×D53—41no—
267C4×D53—41no—
268C4×D53—41no—
269C4×D53—41no—
270C8×C53—41no—
271C8×C53—41no—
272#88322—22no—
273#88322—41no—
274#24947—531nomaximal, Sylow
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53#48514yes—
282C22×C53C2×#12311yescharacteristic
283C22×C53#54611yescharacteristic
284C22×C53#54711yescharacteristic
285C22×C53C2×#13511yescharacteristic
286C22×C53C2×#13711yescharacteristic
287C22×C53#57611yescharacteristic
288C22×C53#57711yescharacteristic
289C2×D53—41no—
290C2×D53—41no—
291C2×D53—41no—
292C2×D53—41no—
293C2×D53—41no—
294C2×D53—41no—
295C2×D53—41no—
296C4×C53—22no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53—81no—
300C4×C53—81no—
301C4×C53#52911yescharacteristic
302C4×C53#53911yescharacteristic
303C4×C53#55111yescharacteristic
304C4×C53#55211yescharacteristic
305Dic53—41no—
306Dic53—41no—
307Dic53—41no—
308Dic53—42no—
309Dic53—81no—
310Dic53—81no—
311Dic53—81no—
312#2487—531no—
313#2352—531no—
314#2101—531no—
315#2085—531no—
316#2077—531no—
317#2075—531no—
318#1919—531no—
319#1918—531no—
320#1869—531no—
321#1854—531no—
322#1853—531no—
323#1825—531no—
324#1824D5311yescharacteristic, complemented
325#1808—531no—
326C2×#340—531no—
327C2×C53—24no—
328C2×C53#241911yescharacteristic
329C2×C53C2×#48511yescharacteristic
330C2×C53#309611yescharacteristic
331C2×C53#330111yescharacteristic
332C2×C53#331411yescharacteristic
333C2×C53#333911yescharacteristic
334C2×C53#334411yescharacteristic
335D53—81no—
336C23×C8—531no—
337C2×C4×D4—531no—
338C2×#117—531no—
339C22×C4⋊C4—531no—
340C22×C4⋊C4C2×D5311yescharacteristic
341C2×#107—531no—
342C2×#107—531no—
343C2×#106—531no—
344C2×#106—531no—
345C2×C22⋊C8—531no—
346C2×C22⋊C8—531no—
347C2×C22⋊C8C2×D5311yescharacteristic
348#400—1061no—
349#399—1061no—
350#389—531no—
351#389—531no—
352#388—1061no—
353#387—1061no—
354#386—1061no—
355#386—1061no—
356#385—1061no—
357#382—531no—
358#380—531no—
359#380—531no—
360#380—531no—
361#380—531no—
362#380C2×D5311yescharacteristic
363C4×C4⋊C4—531no—
364C4×C22⋊C4—1061no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340C2×D5311yescharacteristic
374#340C2×D5311yescharacteristic
375#340C2×D5311yescharacteristic
376#340C2×D5311yescharacteristic
377#336—531no—
378#336—531no—
379C53#2494711yescharacteristic, complemented, Sylow
380C22×D4—531no—
381C23×C4—21no—
382C23×C4—531no—
383C23×C4—1061no—
384C23×C4C22×D5311yescharacteristic
385C22×C8—531no—
386C22×C8—531no—
387C22×C8—1062no—
388C22×C8—1062no—
389C22×C8—2121no—
390C22×C8C22×D5311yescharacteristic
391C22×C8C22×D5311yescharacteristic
392C4×D4—1062no—
393C4×D4—1062no—
394#117—1062no—
395#117—2121no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—1062no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—2121no—
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—531no—
415C2×C22⋊C4—531no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C42—531no—
422C2×C42—1061no—
423C2×C42—1061no—
424#107—1062no—
425#107—2121no—
426#106—1062no—
427#106—2121no—
428C22⋊C8—22no—
429C22⋊C8—1062no—
430C22⋊C8—2121no—
431#95—21no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441C24—531no—
442C24D4×D5311yescharacteristic
443C2×D4—1062no—
444C2×D4—1062no—
445C2×D4—1062no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—42no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—4241no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4#1171994512yes—
474C22×C4D4×D5311yescharacteristic
475C22×C4#1171994712yes—
476C22×C4C23×D5311yescharacteristic
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—532no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—2124no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—42no—
499C4⋊C4—532no—
500C4⋊C4—1062no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—4241no—
509C4⋊C4—4241no—
510C4⋊C4—2122no—
511C22⋊C4—42no—
512C22⋊C4—1062no—
513C22⋊C4—1062no—
514C22⋊C4—4241no—
515C22⋊C4—4241no—
516C22⋊C4—4241no—
517C22⋊C4—2122no—
518C22⋊C4—2122no—
519C42—1061no—
520C42—1061no—
521C42—4241no—
522C42—2122no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—1061no—
528C23—1061no—
529C23—1061no—
530C23—1061no—
531C23—1061no—
532C23—1061no—
533C23—1061no—
534C23#1313319912yes—
535C23#1313323812yes—
536C23#1313324912yes—
537C23C2×D4×D5311yescharacteristic
538D4—2124no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—84no—
558C2×C4—1061no—
559C2×C4—1062no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—4244no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
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—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4#1313319912yes—
585C2×C4#41×D5311yescharacteristic
586C2×C4#41×D5311yescharacteristic
587C2×C4C2×#1171994711yescharacteristic
588C2×C4C2×#1171994511yescharacteristic
589C2×C4#1313326311yescharacteristic
590C2×C4#1313326411yescharacteristic
591C8—41no—
592C8—41no—
593C8—1062no—
594C8—2121no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—2121no—
602C22—2121no—
603C22—2121no—
604C22—2121no—
605C22—2121no—
606C22—2121no—
607C22—2121no—
608C22#1681096214yes—
609C22C2×#1313319911yescharacteristic
610C22#1681126511yescharacteristic
611C22#1681127111yescharacteristic
612C22C2×#1313323811yescharacteristic
613C22C2×#1313324911yescharacteristic
614C22#1681142711yescharacteristic
615C22#1681143311yescharacteristic
616C4—22no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—4241no—
622C4—4241no—
623C4—4241no—
624C4—2122no—
625C4—2121no—
626C4—2121no—
627C4—2121no—
628C4#1681115011yescharacteristic
629C4#1681121711yescharacteristic
630C4#1681129711yescharacteristic
631C4#1681130611yescharacteristic
632C2—24no—
633C2—4241no—
634C2#2027819611yescharacteristic
635C2C2×#1681096211yescharacteristic
636C2#2028467111yescharacteristic
637C2#2028661511yescharacteristic
638C2#2028679411yescharacteristic
639C2#2028711511yescharacteristic
640C2#2028716811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1824C2
socleC23×C53C22×D4
prime radicalC22×C22⋊D53D4
hypercenter#1824D53
hypocenterC53#24947
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
3. #20276285C2semidirect
4. #20276201C2semidirect
5. #20276150C2semidirect
6. #20276142C2semidirect
7. #20275292C2semidirect
8. #20275284C2semidirect
10. #20274915C2semidirect
11. #20274910C2semidirect
12. #20274808C2semidirect
13. #20274742C2semidirect
14. C2×#16810385C2semidirect
15. C53×#1824C2semidirect
324. #1824D53semidirect
379. C53#24947semidirect

Maximal subgroups

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

subgroupindex
#202790262
#202776462
#202762852
#202762012
#202761502
#202761422
#202752922
#202752842
#202750052
#202749152
#202749102
#202748082
#202747422
C2×#168103852
C53×#18242
#2494753