Finite Group Databasebeta

← #27428521

Subgroups — #27428521

Order 13568.

834 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279028C211yesmaximal, characteristic
2C4⋊C4×C22⋊D53C211yesmaximal, characteristic
3#20276280C211yesmaximal, characteristic, complemented
4#20276189C211yesmaximal, characteristic, complemented
5#20276146C211yesmaximal, characteristic, complemented
6#20276143C211yesmaximal, characteristic, complemented
7#20275309C211yesmaximal, characteristic, complemented
8#20275306C211yesmaximal, characteristic, complemented
9#20275004C211yesmaximal, characteristic
10#20274901C211yesmaximal, characteristic, complemented
11#20274805C211yesmaximal, characteristic, complemented
12#20274804C211yesmaximal, characteristic, complemented
13#20274741C211yesmaximal, characteristic, complemented
14#340×D53C211yesmaximal, characteristic, complemented
15C53×#1852C211yesmaximal, characteristic, complemented
16C22×C8×D53C2211yescharacteristic
17C2×C4×C22⋊D53—21no—
18C2×C4×C22⋊D53C2211yescharacteristic
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×C4⋊C4×D53C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×#13133129C2211yescharacteristic
25#16810564—21no—
26#16810559—21no—
27#16810522C2211yescharacteristic
28#16810519C2211yescharacteristic
29#16810518—21no—
30#16810499—21no—
31#16810497—21no—
32#16810496—21no—
33#16810494—21no—
34#16810492C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810480C2211yescharacteristic
38#16810480C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×C22⋊D53D411yescharacteristic
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C22×C4×D53C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C2×C8×D53—22no—
66C2×C8×D53—22no—
67C2×C8×D53—41no—
68C4×C22⋊D53—22no—
69C4×C22⋊D53—22no—
70C4×C22⋊D53—44no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919D411yescharacteristic
82C4⋊C4×D53—22no—
83C4⋊C4×D53—22no—
84C4⋊C4×D53—22no—
85C4⋊C4×D53—41no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53D411yescharacteristic
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921D411yescharacteristic
92C2×C4×Dic53—21no—
93C2×C4×Dic53C2311yescharacteristic
94#13133151—22no—
95#13133150—41no—
96C22⋊#88322—41no—
97#13133129—22no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C23×C4×C53—21no—
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—22no—
111C2×C4⋊C4×C53—41no—
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C22⋊C4×C53—21no—
117#107×C53—22no—
118#107×C53—41no—
119C22⋊C8×C53—22no—
120#95×C53—21no—
121C24×C53C2×D411yescharacteristic
122C23×D53C2×D411yescharacteristic
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—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53—81no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2411yescharacteristic
138C2×C22⋊D53—22no—
139C2×C22⋊D53—22no—
140C2×C22⋊D53Q1614yes—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C2×C4×D53—21no—
149C2×C4×D53—21no—
150C2×C4×D53—21no—
151C2×C4×D53—21no—
152C2×C4×D53—42no—
153C2×C4×D53—42no—
154C2×C4×D53—42no—
155C2×C4×D53—81no—
156C2×C4×D53—81no—
157C2×C4×D53—81no—
158C2×C4×D53#4112yes—
159C2×C4×D53#4112yes—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53#4112yes—
166C4⋊C4×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—21no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53#4112yes—
175C22⋊C4×C53—81no—
176C22⋊Dic53—22no—
177C22⋊Dic53—42no—
178C22⋊Dic53—81no—
179#11719921—22no—
180#11719921—42no—
181#11719921—42no—
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—81no—
191#11719919—22no—
192#11719919—42no—
193#11719919—42no—
194#11719919—42no—
195#11719919—81no—
196#11719919—81no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—42no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C8×D53—44no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53C2×Q1612yes—
214C23×C53C22×D411yescharacteristic
215C22×D53—21no—
216C22×D53—21no—
217C22×D53—21no—
218C22×D53—21no—
219C22×D53#12312yes—
220C22×D53C2×Q1612yes—
221C22×D53AGL(1,ℤ/8)12yes—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
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—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—84no—
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53H(5,𝔽2)11yescharacteristic
247C2×C4×C532−1+411yescharacteristic
248C22⋊D53—28no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
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—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53#12312yes—
269C2×Dic53C2×Q1612yes—
270C4×D53—22no—
271C4×D53—22no—
272C4×D53—22no—
273C4×D53—22no—
274C4×D53—84no—
275C4×D53—84no—
276C8×C53—22no—
277C8×C53—41no—
278#88322—41no—
279#88322—41no—
280#24939—531nomaximal, Sylow
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53C2×#12311yescharacteristic
289C22×C53#54611yescharacteristic
290C22×C53#54711yescharacteristic
291C22×C53C22×Q1611yescharacteristic
292C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C53#57611yescharacteristic
294C22×C53#57711yescharacteristic
295C2×D53—22no—
296C2×D53—22no—
297C2×D53—22no—
298C2×D53—22no—
299C2×D53—22no—
300C2×D53—22no—
301C2×D53#48414yes—
302C4×C53—41no—
303C4×C53—42no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53—81no—
307C4×C53#52911yescharacteristic
308C4×C53#53911yescharacteristic
309C4×C53#55111yescharacteristic
310C4×C53#55211yescharacteristic
311Dic53—22no—
312Dic53—22no—
313Dic53—41no—
314Dic53—81no—
315Dic53—81no—
316Dic53—81no—
317Dic53—81no—
318#2487—531no—
319D4×C4⋊C4—531no—
320#2100—531no—
321#2083—531no—
322#2076—531no—
323#2076—531no—
324#1921—531no—
325#1921—531no—
326#1869—531no—
327#1852—531no—
328#1852D5311yescharacteristic, complemented
329#1824—531no—
330#1824—531no—
331#1808—531no—
332C2×#340—531no—
333C2×C53—42no—
334C2×C53#241911yescharacteristic
335C2×C53C2×#48411yescharacteristic
336C2×C53#309611yescharacteristic
337C2×C53#329911yescharacteristic
338C2×C53#331811yescharacteristic
339C2×C53#333811yescharacteristic
340C2×C53#334511yescharacteristic
341D53—24no—
342C23×C8—531no—
343C2×C4×D4—531no—
344C2×C4×D4—1061no—
345C22×C4⋊C4—531no—
346C22×C4⋊C4—531no—
347C22×C4⋊C4C2×D5311yescharacteristic
348C2×#107—531no—
349C2×#107—531no—
350C2×#107C2×D5311yescharacteristic
351C2×#107C2×D5311yescharacteristic
352C2×C22⋊C8—531no—
353C2×C22⋊C8—531no—
354C2×C22⋊C8C2×D5311yescharacteristic
355#400—1061no—
356#399—1061no—
357#389—531no—
358#389—531no—
359#389—1061no—
360#386—1061no—
361#385—1061no—
362#385—1061no—
363#385—1061no—
364#384—531no—
365#380—531no—
366#380—531no—
367#380—531no—
368#380—531no—
369#380C2×D5311yescharacteristic
370C4×C4⋊C4—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340C2×D5311yescharacteristic
382#340C2×D5311yescharacteristic
383#336—531no—
384#336—531no—
385C53#2493911yescharacteristic, complemented, Sylow
386C22×D4—531no—
387C23×C4—21no—
388C23×C4—531no—
389C23×C4—1061no—
390C23×C4—1061no—
391C23×C4C22×D5311yescharacteristic
392C22×C8—531no—
393C22×C8—531no—
394C22×C8—1062no—
395C22×C8—1062no—
396C22×C8—2121no—
397C22×C8C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399C4×D4—1062no—
400C4×D4—1062no—
401C4×D4—2124no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—41no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1062no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—2121no—
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C22⋊C4—21no—
425C2×C22⋊C4—531no—
426C2×C22⋊C4—531no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C42—531no—
432C2×C42—1061no—
433#107—22no—
434#107—41no—
435#107—1062no—
436#107—2121no—
437C22⋊C8—22no—
438C22⋊C8—1062no—
439C22⋊C8—2121no—
440#95—21no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448C24—531no—
449C24D4×D5311yescharacteristic
450C2×D4—534no—
451C2×D4—1062no—
452C2×D4—1062no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—81no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—532no—
469C22×C4—532no—
470C22×C4—531no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—4241no—
480C22×C4—4241no—
481C22×C4—4241no—
482C22×C4—2122no—
483C22×C4—2122no—
484C22×C4—2122no—
485C22×C4D4×D5311yescharacteristic
486C22×C4C23×D5311yescharacteristic
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—2124no—
499C2×C8#1171994512yes—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—42no—
508C4⋊C4—1062no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—4241no—
518C4⋊C4—4241no—
519C4⋊C4—4241no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C4⋊C4#1171994712yes—
524C22⋊C4—81no—
525C22⋊C4—1062no—
526C22⋊C4—1062no—
527C22⋊C4—4241no—
528C22⋊C4—2122no—
529C22⋊C4—2122no—
530C22⋊C4—2122no—
531C42—1061no—
532C42—1061no—
533C42—2122no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—532no—
541C23—532no—
542C23—532no—
543C23—1061no—
544C23—1061no—
545C23—1061no—
546C23—1061no—
547C23Q16×D5312yes—
548C23C2×D4×D5311yescharacteristic
549D4—1068no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
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—84no—
570C2×C4—532no—
571C2×C4—532no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—4244no—
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#41×D5311yescharacteristic
598C2×C4#41×D5311yescharacteristic
599C2×C4C2×#1171994711yescharacteristic
600C2×C4C2×#1171994511yescharacteristic
601C2×C4#1313326211yescharacteristic
602C2×C4#1313326511yescharacteristic
603C8—22no—
604C8—41no—
605C8—2121no—
606C8—2121no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—534no—
615C22—1062no—
616C22—1062no—
617C22—1062no—
618C22—1062no—
619C22—1062no—
620C22—1062no—
621C22#123×D5311yescharacteristic
622C22#1681126011yescharacteristic
623C22#1681127411yescharacteristic
624C22C2×Q16×D5311yescharacteristic
625C22AGL(1,ℤ/8)×D5311yescharacteristic
626C22#1681142411yescharacteristic
627C22#1681143211yescharacteristic
628C4—42no—
629C4—41no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—1062no—
634C4—1062no—
635C4—4241no—
636C4—4241no—
637C4—4241no—
638C4—4241no—
639C4—2121no—
640C4#1681115511yescharacteristic
641C4#1681121911yescharacteristic
642C4#1681130211yescharacteristic
643C4#1681130711yescharacteristic
644C2—42no—
645C2—1064no—
646C2#2027819111yescharacteristic
647C2#484×D5311yescharacteristic
648C2#2028466611yescharacteristic
649C2#2028658411yescharacteristic
650C2#2028684911yescharacteristic
651C2#2028710511yescharacteristic
652C2#2028718611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1852C2
socleC23×C53C22×D4
prime radicalC22×C22⋊D53D4
hypercenter#1852D53
hypocenterC53#24939
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
3. #20276280C2semidirect
4. #20276189C2semidirect
5. #20276146C2semidirect
6. #20276143C2semidirect
7. #20275309C2semidirect
8. #20275306C2semidirect
10. #20274901C2semidirect
11. #20274805C2semidirect
12. #20274804C2semidirect
13. #20274741C2semidirect
14. #340×D53C2semidirect
15. C53×#1852C2semidirect
328. #1852D53semidirect
385. C53#24939semidirect

Maximal subgroups

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

subgroupindex
#202790282
C4⋊C4×C22⋊D532
#202762802
#202761892
#202761462
#202761432
#202753092
#202753062
#202750042
#202749012
#202748052
#202748042
#202747412
#340×D532
C53×#18522
#2493953