Finite Group Databasebeta

← #27465483

Subgroups — #27465483

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×Dic53C211yesmaximal, characteristic
2#20277117C211yesmaximal, characteristic, complemented
3#20276151C211yesmaximal, characteristic, complemented
4#20276148C211yesmaximal, characteristic, complemented
5#20275894C211yesmaximal, characteristic
6#20275477C211yesmaximal, characteristic, complemented
7#20275470C211yesmaximal, characteristic, complemented
8#20275345C211yesmaximal, characteristic, complemented
9#20275344C211yesmaximal, characteristic, complemented
10#20275310C211yesmaximal, characteristic, complemented
11#20275309C211yesmaximal, characteristic, complemented
12#20274713C211yesmaximal, characteristic, complemented
13#20274713C211yesmaximal, characteristic, complemented
14#20274693C211yesmaximal, characteristic, complemented
15C53×#2033C211yesmaximal, characteristic
16C2×D4×Dic53—21no—
17C2×D4×Dic53C2211yescharacteristic
18C2×#13133173—21no—
19C22×C4×Dic53C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28#16810526C2211yescharacteristic
29#16810522C2211yescharacteristic
30#16810516—21no—
31#16810503—21no—
32#16810492C2211yescharacteristic
33#16810492C2211yescharacteristic
34#16810487—21no—
35#16810483C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810481—21no—
38C4⋊C4×Dic53C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41#16810474—21no—
42C22⋊C4×Dic53—21no—
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#121×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C23×Dic53—21no—
60C23×Dic53D411yescharacteristic
61C23×Dic53D411yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64D4×Dic53—22no—
65D4×Dic53—22no—
66D4×Dic53—44no—
67#13133173—44no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53—22no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133150—22no—
92#13133149—41no—
93#13133148—41no—
94#13133146—22no—
95#13133146—22no—
96#13133138—22no—
97#13133138—41no—
98C22⋊#88322—41no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C22×D4×C53—21no—
106C22×D4×C53C2311yescharacteristic
107C23×C4×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110#121×C53—22no—
111#121×C53—22no—
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117#107×C53—22no—
118#102×C53—22no—
119#102×C53—41no—
120C22⋊C8×C53—41no—
121#95×C53—21no—
122C24×C53—21no—
123C24×C53C2×D411yescharacteristic
124C2×D4×C53—24no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—42no—
131C2×D4×C53—42no—
132C2×D4×C53#4112yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53—42no—
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160C22×Dic53C2×D411yescharacteristic
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53#4112yes—
173C22⋊C4×C53—24no—
174C22⋊C4×C53—42no—
175C22⋊Dic53—42no—
176C22⋊Dic53—42no—
177C22⋊Dic53—42no—
178C22⋊Dic53—42no—
179C22⋊Dic53—42no—
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—42no—
189#11719919—42no—
190#11719919—42no—
191#11719919—42no—
192#11719919—42no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—22no—
196C4×Dic53—22no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C4×Dic53—42no—
200C4×Dic53#4112yes—
201C4×Dic53#4112yes—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—22no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53AGL(1,ℤ/8)12yes—
222C23×C53C22×D411yescharacteristic
223D4×C53—22no—
224D4×C53—22no—
225D4×C53—48no—
226D4×C53—41no—
227D4×C53—41no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—21no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53H(5,𝔽2)11yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C2×Dic53—28no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—44no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
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—84no—
272C2×Dic53#12312yes—
273C2×Dic53#12312yes—
274C2×Dic53#13512yes—
275C2×Dic53AGL(1,ℤ/8)12yes—
276C2×Dic53#13712yes—
277C8×C53—41no—
278C8×C53—41no—
279#88322—41no—
280#88322—41no—
281#27627—531nomaximal, Sylow
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—42no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—81no—
293C22×C53—81no—
294C22×C53—81no—
295C22×C53—81no—
296C22×C53C2×#12311yescharacteristic
297C22×C53#52511yescharacteristic
298C22×C53#53711yescharacteristic
299C22×C53C2×#13511yescharacteristic
300C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C53C2×#13711yescharacteristic
303C4×C53—42no—
304C4×C53—41no—
305C4×C53—81no—
306C4×C53—81no—
307C4×C53#55211yescharacteristic
308C4×C53#55311yescharacteristic
309C4×C53#55511yescharacteristic
310C4×C53#55611yescharacteristic
311Dic53—22no—
312Dic53—22no—
313Dic53—41no—
314Dic53—81no—
315Dic53—81no—
316Dic53—81no—
317Dic53—81no—
318Dic53#48214yes—
319C4×#121—531no—
320#2285—531no—
321#2077—531no—
322#2077—531no—
323#2033—531no—
324#2033D5311yescharacteristic
325#1948—531no—
326#1947—531no—
327#1926—531no—
328#1926—531no—
329#1921—531no—
330#1921—531no—
331#1802—531no—
332#1802—531no—
333#1797—531no—
334C2×C53—42no—
335C2×C53—82no—
336C2×C53#248511yescharacteristic
337C2×C53C2×#48211yescharacteristic
338C2×C53#309211yescharacteristic
339C2×C53#310711yescharacteristic
340C2×C53#310911yescharacteristic
341C2×C53#324711yescharacteristic
342C2×C53#325411yescharacteristic
343C2×#121C2×D5311yescharacteristic
344C2×C4×D4—531no—
345C2×C4×D4—1061no—
346C2×#117—1061no—
347C22×C42—531no—
348C2×#107—531no—
349C2×#107C2×D5311yescharacteristic
350C2×#106—531no—
351C2×#106—531no—
352C2×C4⋊C8—531no—
353C2×C4⋊C8—531no—
354C2×#102—531no—
355C2×#102—531no—
356C2×#102C2×D5311yescharacteristic
357C2×#102C2×D5311yescharacteristic
358C2×C22⋊C8—531no—
359C2×C22⋊C8C2×D5311yescharacteristic
360#390—531no—
361#389—531no—
362#388—1061no—
363#386—1061no—
364#384—531no—
365#384—531no—
366#382—1061no—
367#380—531no—
368#380—531no—
369#380—1061no—
370#380C2×D5311yescharacteristic
371C4×C4⋊C4—531no—
372C4×C4⋊C4—531no—
373C4×C4⋊C4—531no—
374C4×C4⋊C4—1061no—
375C4×C22⋊C4—1061no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340C2×D5311yescharacteristic
384#336—531no—
385#336—531no—
386C53#2762711yescharacteristic, complemented, Sylow
387C22×D4—21no—
388C22×D4C22×D5311yescharacteristic
389C23×C4—531no—
390C23×C4—531no—
391C23×C4—1061no—
392C23×C4C22×D5311yescharacteristic
393C22×C8—531no—
394C22×C8—531no—
395C22×C8C22×D5311yescharacteristic
396C22×C8C22×D5311yescharacteristic
397#121—22no—
398#121—22no—
399C4×D4—1062no—
400C4×D4—1062no—
401C4×D4—2124no—
402#117—2124no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C42—531no—
423C2×C42—531no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—1062no—
427C2×C42—1062no—
428C2×C42—1061no—
429C2×C42—1061no—
430C2×C42—1061no—
431#107—22no—
432#107—1062no—
433#106—2121no—
434#106—2121no—
435C4⋊C8—1062no—
436C4⋊C8—1062no—
437#102—22no—
438#102—41no—
439#102—1062no—
440#102—2121no—
441C22⋊C8—41no—
442C22⋊C8—2121no—
443#95—21no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450C24—21no—
451C24D4×D5311yescharacteristic
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4#1171994712yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—531no—
471C22×C4—531no—
472C22×C4—531no—
473C22×C4—531no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4—2122no—
487C22×C4D4×D5311yescharacteristic
488C22×C4C23×D5311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—2122no—
515C4⋊C4—2122no—
516C4⋊C4—2122no—
517C4⋊C4—2122no—
518C4⋊C4—2122no—
519C4⋊C4#1171994712yes—
520C22⋊C4—24no—
521C22⋊C4—42no—
522C22⋊C4—2122no—
523C22⋊C4—2122no—
524C22⋊C4—2122no—
525C22⋊C4—2122no—
526C22⋊C4—2122no—
527C42—532no—
528C42—532no—
529C42—1062no—
530C42—1062no—
531C42—1061no—
532C42—1061no—
533C42—2122no—
534C42—2122no—
535C42—2122no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23#1313324112yes—
550C23C2×D4×D5311yescharacteristic
551D4—22no—
552D4—22no—
553D4—48no—
554D4—41no—
555D4—41no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—532no—
573C2×C4—532no—
574C2×C4—532no—
575C2×C4—532no—
576C2×C4—532no—
577C2×C4—1068no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—4244no—
586C2×C4—2124no—
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—2121no—
598C2×C4—2121no—
599C2×C4#41×D5311yescharacteristic
600C2×C4#41×D5311yescharacteristic
601C2×C4C2×#1171994711yescharacteristic
602C2×C4C2×#1171994711yescharacteristic
603C2×C4#1313326211yescharacteristic
604C2×C4#1313326411yescharacteristic
605C8—41no—
606C8—41no—
607C8—2121no—
608C8—2121no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22#123×D5311yescharacteristic
624C22#1681112911yescharacteristic
625C22#1681120011yescharacteristic
626C22#135×D5311yescharacteristic
627C22AGL(1,ℤ/8)×D5311yescharacteristic
628C22C2×#1313324111yescharacteristic
629C22#137×D5311yescharacteristic
630C4—42no—
631C4—41no—
632C4—81no—
633C4—81no—
634C4—534no—
635C4—1062no—
636C4—1062no—
637C4—4241no—
638C4—4241no—
639C4—4241no—
640C4—4241no—
641C4—2121no—
642C4#1681131011yescharacteristic
643C4#1681131611yescharacteristic
644C4#1681133711yescharacteristic
645C4#1681134211yescharacteristic
646C2—42no—
647C2—82no—
648C2#2027900511yescharacteristic
649C2#482×D5311yescharacteristic
650C2#2028463411yescharacteristic
651C2#2028477911yescharacteristic
652C2#2028480111yescharacteristic
653C2#2028603411yescharacteristic
654C2#2028609711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2033C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2033D53
hypocenterC53#27627
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
2. #20277117C2semidirect
3. #20276151C2semidirect
4. #20276148C2semidirect
6. #20275477C2semidirect
7. #20275470C2semidirect
8. #20275345C2semidirect
9. #20275344C2semidirect
10. #20275310C2semidirect
11. #20275309C2semidirect
12. #20274713C2semidirect
13. #20274713C2semidirect
14. #20274693C2semidirect
386. C53#27627semidirect

Maximal subgroups

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

subgroupindex
#121×Dic532
#202771172
#202761512
#202761482
#202758942
#202754772
#202754702
#202753452
#202753442
#202753102
#202753092
#202747132
#202746932
C53×#20332
#2762753