Finite Group Databasebeta

← #27447924

Subgroups — #27447924

Order 13568.

840 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277751C211yesmaximal, characteristic
2#20276272C211yesmaximal, characteristic, complemented
3#20276216C211yesmaximal, characteristic, complemented
4#20276086C211yesmaximal, characteristic, complemented
5#20275848C211yesmaximal, characteristic, complemented
6#20275664C211yesmaximal, characteristic, complemented
7#20275304C211yesmaximal, characteristic, complemented
8#20275151C211yesmaximal, characteristic
9#20275032C211yesmaximal, characteristic
10#20274961C211yesmaximal, characteristic, complemented
11#20274912C211yesmaximal, characteristic, complemented
12#20274900C211yesmaximal, characteristic, complemented
13#20274712C211yesmaximal, characteristic, complemented
14#20274434C211yesmaximal, characteristic, complemented
15C53×#2440C211yesmaximal, characteristic, complemented
16C22×#11719917C2211yescharacteristic
17C2×#13133191C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133137C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29#16810525C2211yescharacteristic
30#16810523—21no—
31#16810521C2211yescharacteristic
32#16810512—21no—
33#16810499—21no—
34C4×C4⋊Dic53C2211yescharacteristic
35#16810470—21no—
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C4⋊Q8×C53C2211yescharacteristic
49C2×C22⋊Q8×C53—21no—
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#398—21no—
52C53×#395—21no—
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#385—21no—
56C53×#384C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C22×C4×D53—21no—
62C2×#11719917—22no—
63C2×#11719917—41no—
64C2×#11719917—41no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133191—22no—
70#13133191Q814yescomplemented
71C4×D212—22no—
72C4×D212—41no—
73#13133176—22no—
74#13133176—41no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86#13133150—22no—
87#13133148—41no—
88C4⋊#88322—41no—
89C4⋊#88322—41no—
90#13133137—22no—
91#13133137—41no—
92#13133137—41no—
93#13133137—41no—
94C22⋊#88322—22no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53—21no—
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100C4⋊Q8×C53—22no—
101C4⋊Q8×C53—41no—
102C22⋊Q8×C53—44no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—22no—
110C2×C4⋊C4×C53—41no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53D411yescharacteristic
124#95×C53D411yescharacteristic
125C24×C53C2×D411yescharacteristic
126C23×D53—21no—
127C2×Q8×C53—42no—
128C2×Q8×C53—42no—
129C2×Q8×C53—81no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—42no—
139C22×C4×C53—42no—
140C22×C4×C53—81no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×Q812yes—
147C22×C4×C53#4112yes—
148C22×C4×C53C2411yescharacteristic
149C2×C22⋊D53—24no—
150C2×C22⋊D53—81no—
151C2×C22⋊D53—81no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212C2×Q812yes—
161C2×C4×D53—42no—
162C2×C4×D53—42no—
163C2×C4×D53—81no—
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—21no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53—42no—
175C4⋊C4×C53—42no—
176C4⋊C4×C53—81no—
177C4⋊C4×C53#4112yes—
178C22⋊C4×C53—22no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—42no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C22⋊Dic53—42no—
186#11719921—24no—
187#11719921—42no—
188#11719921—42no—
189#11719921—81no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53C2×Q812yes—
195C4⋊Dic53#4112yes—
196#11719919—81no—
197C4×Dic53—42no—
198C4×Dic53—81no—
199#11719917—44no—
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—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53C22⋊Q812yes—
219C23×C53C22×D411yescharacteristic
220C22×D53—22no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227Q8×C53—84no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
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—41no—
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—41no—
256C2×C4×C53—84no—
257C2×C4×C53C22⋊Q812yes—
258C2×C4×C53AGL(1,ℤ/8)12yes—
259C2×C4×C53#13712yes—
260C2×C4×C53C22×Q811yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53H(5,𝔽2)11yescharacteristic
265C2×C4×C53H(5,𝔽2)11yescharacteristic
266C22⋊D53—84no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276D212—22no—
277D212—22no—
278D212—41no—
279D212—41no—
280C4×D53—84no—
281#88322—41no—
282#88322—41no—
283#88322—41no—
284#88322—41no—
285#26427—531nomaximal, Sylow
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53C2×C22⋊Q811yescharacteristic
294C22×C53#54611yescharacteristic
295C22×C53#54611yescharacteristic
296C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C53C2×#13711yescharacteristic
298C22×C53#57611yescharacteristic
299C22×C53#57711yescharacteristic
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C4×C53—22no—
308C4×C53—22no—
309C4×C53—22no—
310C4×C53—41no—
311C4×C53—41no—
312C4×C53—42no—
313C4×C53—81no—
314C4×C53—81no—
315C4×C53—81no—
316C4×C53#52911yescharacteristic
317C4×C53#53811yescharacteristic
318C4×C53#55411yescharacteristic
319C4×C53#55411yescharacteristic
320Dic53—42no—
321Dic53—42no—
322Dic53—81no—
323#2440D5311yescharacteristic, complemented
324#2363—531no—
325#2099—531no—
326#2088—531no—
327#2065—531no—
328#2023—531no—
329#1987—531no—
330#1921—531no—
331#1895—531no—
332#1874—531no—
333#1862—531no—
334#1854—531no—
335#1851—531no—
336#1802—531no—
337#1736—531no—
338C2×C53—42no—
339C2×C53#245011yescharacteristic
340C2×C53#307911yescharacteristic
341C2×C53#308111yescharacteristic
342C2×C53#330611yescharacteristic
343C2×C53#332011yescharacteristic
344C2×C53#332511yescharacteristic
345C2×C53#333311yescharacteristic
346D53—82no—
347C22×M16—531no—
348C2×C4⋊Q8C2×D5311yescharacteristic
349C2×C22⋊Q8—21no—
350C2×#121—531no—
351C2×C4×D4—531no—
352C2×#117—531no—
353C22×C4⋊C4C2×D5311yescharacteristic
354C2×#107—531no—
355C2×#106—531no—
356C2×C4⋊C8—531no—
357C2×C4⋊C8—531no—
358C2×#102—531no—
359C2×#102—531no—
360C2×#102—531no—
361C2×#102—531no—
362C2×C22⋊C8—531no—
363#398—21no—
364#395—21no—
365#390—531no—
366#390—1061no—
367#389—531no—
368#389C2×D5311yescharacteristic
369#389C2×D5311yescharacteristic
370#388—1061no—
371#386—1061no—
372#385—21no—
373#384C2×D5311yescharacteristic
374#380C2×D5311yescharacteristic
375C4×C4⋊C4—531no—
376C4×C22⋊C4—1061no—
377C4×C22⋊C4C2×D5311yescharacteristic
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#336—531no—
389#336—531no—
390C53#2642711yescharacteristic, complemented, Sylow
391C22×Q8—21no—
392C22×D4—531no—
393C22×D4—1061no—
394C23×C4—21no—
395C23×C4—1061no—
396C23×C4C22×D5311yescharacteristic
397C2×M16—1062no—
398C2×M16—2121no—
399C2×M16—2121no—
400C22×C8—531no—
401C22×C8—531no—
402C22×C8—531no—
403C22×C8—531no—
404C4⋊Q8—22no—
405C4⋊Q8—41no—
406C22⋊Q8—44no—
407#121—534no—
408#121—1062no—
409C4×D4—1062no—
410C4×D4—2121no—
411#117—1062no—
412#117—2121no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—41no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C4⋊C4C22×D5311yescharacteristic
427C2×C4⋊C4C22×D5311yescharacteristic
428C2×C4⋊C4C22×D5311yescharacteristic
429C2×C22⋊C4—21no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4C22⋊D5311yescharacteristic
436C2×C22⋊C4C22⋊D5311yescharacteristic
437C2×C42—1061no—
438C2×C42—1061no—
439C2×C42C22×D5311yescharacteristic
440C2×C42C22×D5311yescharacteristic
441#107—1062no—
442#106—2121no—
443C4⋊C8—2121no—
444C4⋊C8—2121no—
445#102—1062no—
446#102—2121no—
447#102—2121no—
448#102—2121no—
449C22⋊C8—1062no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—1061no—
454#95—1061no—
455#95C22⋊D5311yescharacteristic
456#95C22⋊D5311yescharacteristic
457C24—1061no—
458C24C2×C22⋊D5311yescharacteristic
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—81no—
462C2×D4—532no—
463C2×D4—1064no—
464C2×D4—1061no—
465C2×D4—1061no—
466C2×D4—1061no—
467C2×D4—1061no—
468C2×D4—1061no—
469C2×D4—4241no—
470C2×D4—4241no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—81no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—4241no—
486C22×C4—2122no—
487C22×C4—2122no—
488C22×C4Q8×D5312yes—
489C22×C4#1171994912yes—
490C22×C4C2×C22⋊D5311yescharacteristic
491C22×C4C2×C22⋊D5311yescharacteristic
492C22×C4C2×C22⋊D5311yescharacteristic
493C22×C4C2×C22⋊D5311yescharacteristic
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C23×D5311yescharacteristic
496M16—2124no—
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—1061no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—81no—
522C4⋊C4—532no—
523C4⋊C4—532no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—4241no—
529C4⋊C4#1171994712yes—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—42no—
533C22⋊C4—1064no—
534C22⋊C4—4241no—
535C22⋊C4—2122no—
536C22⋊C4—2122no—
537C22⋊C4—2122no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—4241no—
543C42—2122no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—1062no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23—2121no—
556C23—2121no—
557C23#1313319312yes—
558C23C22×C22⋊D5311yescharacteristic
559Q8—84no—
560D4—1062no—
561D4—1062no—
562D4—4244no—
563D4—2121no—
564D4—2121no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—84no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—4244no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4#1313319312yes—
605C2×C4#1313324312yes—
606C2×C4#1313324812yes—
607C2×C4C2×Q8×D5311yescharacteristic
608C2×C4#41×D5311yescharacteristic
609C2×C4C2×#1171994711yescharacteristic
610C2×C4C2×#1171994911yescharacteristic
611C2×C4#1313326211yescharacteristic
612C2×C4#1313326311yescharacteristic
613C8—2121no—
614C8—2121no—
615C8—2121no—
616C8—2121no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—4241no—
625C22—4241no—
626C22—4241no—
627C22—4241no—
628C22—2122no—
629C22—2122no—
630C22—2122no—
631C22C2×#1313319311yescharacteristic
632C22#1681126111yescharacteristic
633C22#1681126711yescharacteristic
634C22C2×#1313324311yescharacteristic
635C22C2×#1313324811yescharacteristic
636C22#1681142611yescharacteristic
637C22#1681143111yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—42no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—4241no—
648C4—2122no—
649C4—2122no—
650C4#1681115911yescharacteristic
651C4#1681121311yescharacteristic
652C4#1681132711yescharacteristic
653C4#1681133111yescharacteristic
654C2—42no—
655C2—4242no—
656C2#2027853511yescharacteristic
657C2#2028452411yescharacteristic
658C2#2028454011yescharacteristic
659C2#2028667011yescharacteristic
660C2#2028688611yescharacteristic
661C2#2028695711yescharacteristic
662C2#2028704711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2440C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2440D53
hypocenterC53#26427
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
2. #20276272C2semidirect
3. #20276216C2semidirect
4. #20276086C2semidirect
5. #20275848C2semidirect
6. #20275664C2semidirect
7. #20275304C2semidirect
10. #20274961C2semidirect
11. #20274912C2semidirect
12. #20274900C2semidirect
13. #20274712C2semidirect
14. #20274434C2semidirect
15. C53×#2440C2semidirect
70. #13133191Q8semidirect
323. #2440D53semidirect
390. C53#26427semidirect

Maximal subgroups

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

subgroupindex
#202777512
#202762722
#202762162
#202760862
#202758482
#202756642
#202753042
#202751512
#202750322
#202749612
#202749122
#202749002
#202747122
#202744342
C53×#24402
#2642753