Finite Group Databasebeta

← #27465232

Subgroups — #27465232

Order 13568.

1052 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277098C211yesmaximal, characteristic
2C2×#16810483C211yesmaximal, characteristic
3#20276148C212yesmaximal, complemented
4#20276143C212yesmaximal, complemented
5#20274916C212yesmaximal, complemented
6#20274910C211yesmaximal, characteristic, complemented
7#20274905C211yesmaximal, characteristic, complemented
8#20274811C212yesmaximal, complemented
9#20274802C211yesmaximal, characteristic, complemented
10#20274802C211yesmaximal, characteristic, complemented
11C53×#1779C211yesmaximal, characteristic
12C2×#13133176C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C22⋊Dic53—21no—
17C2×#13133151C2212yescomplemented
18C2×#13133149C2212yescomplemented
19#16810486—21no—
20#16810483—21no—
21#16810483—21no—
22#16810483C2212yes—
23#16810483C2212yes—
24#16810483C2211yescharacteristic
25#16810483C2211yescharacteristic
26#16810483C2211yescharacteristic
27#16810483C2211yescharacteristic
28#16810479—21no—
29C4×C22⋊Dic53—21no—
30C4×C22⋊Dic53—21no—
31C2×#13133124—21no—
32#16810386C2212yescomplemented
33#16810386C2212yescomplemented
34#16810386C2212yescomplemented
35#16810386C2212yescomplemented
36#16810386C2212yescomplemented
37#16810386C2212yescomplemented
38C24×C4×C53C2211yescharacteristic
39C2×C22⋊C8×C53C2211yescharacteristic
40C2×C22⋊C8×C53C2211yescharacteristic
41C2×C22⋊C8×C53C2212yes—
42C2×C22⋊C8×C53C2212yes—
43C23×Dic53—21no—
44C23×Dic53—21no—
45#13133176—24no—
46#13133176—42no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—22no—
51C2×C4⋊Dic53—42no—
52C2×C4⋊Dic53C2312yes—
53C2×C4⋊Dic53C2312yes—
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2311yescharacteristic
56C2×C4⋊Dic53C2311yescharacteristic
57C2×C4⋊Dic53C2311yescharacteristic
58C2×C22⋊Dic53—22no—
59C2×C22⋊Dic53—22no—
60C2×C22⋊Dic53—21no—
61C2×C22⋊Dic53—21no—
62C2×C22⋊Dic53—21no—
63C2×C22⋊Dic53—21no—
64C2×C22⋊Dic53—21no—
65C2×C22⋊Dic53—21no—
66C2×C22⋊Dic53—44no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53—21no—
69#13133151—42no—
70#13133149—42no—
71#13133124—22no—
72#13133124—22no—
73#13133124—21no—
74#13133124—21no—
75#13133124—21no—
76#13133124—21no—
77C25×C53D411yescharacteristic
78C23×C4×C53—22no—
79C23×C4×C53—22no—
80C23×C4×C53—24no—
81C23×C4×C53—42no—
82C23×C4×C53D411yescharacteristic
83C23×C4×C53D411yescharacteristic
84C23×C4×C53D411yescharacteristic
85C23×C4×C53C2311yescharacteristic
86C23×C4×C53C2311yescharacteristic
87C23×C4×C53C2311yescharacteristic
88C22×C8×C53C2312yes—
89C22×C8×C53C2312yes—
90C22⋊C8×C53—22no—
91C22⋊C8×C53—22no—
92C22⋊C8×C53—24no—
93C22⋊C8×C53—42no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—41no—
100C24×C53—41no—
101C24×C53C2×D411yescharacteristic
102C24×C53C2×D411yescharacteristic
103C24×C53C2×D411yescharacteristic
104C22×C4×C53—22no—
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—41no—
120C22×C4×C53—41no—
121C22×C4×C53—41no—
122C22×C4×C53—48no—
123C22×C4×C53—48no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53#4112yes—
128C22×C4×C53#4112yes—
129C22×C4×C53#4114yes—
130C22×C4×C53C2411yescharacteristic
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—42no—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C22×Dic53—42no—
141C22×Dic53—42no—
142C22×Dic53—42no—
143C2×C8×C53—22no—
144C2×C8×C53—22no—
145C2×C8×C53—22no—
146C2×C8×C53—22no—
147C2×C8×C53—22no—
148C2×C8×C53—22no—
149C22⋊Dic53—42no—
150C22⋊Dic53—44no—
151C22⋊Dic53—44no—
152C22⋊Dic53—42no—
153C22⋊Dic53—42no—
154C22⋊Dic53—42no—
155C22⋊Dic53—42no—
156C22⋊Dic53—42no—
157C4⋊Dic53—22no—
158C4⋊Dic53—22no—
159C4⋊Dic53—22no—
160C4⋊Dic53—22no—
161C4⋊Dic53—22no—
162C4⋊Dic53—22no—
163C4⋊Dic53—22no—
164C4⋊Dic53—22no—
165C4⋊Dic53—42no—
166C4⋊Dic53—42no—
167C4×Dic53—42no—
168C4×Dic53—42no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—28no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—44no—
187C23×C53—44no—
188C23×C53—44no—
189C23×C53—44no—
190C23×C53—44no—
191C23×C53—44no—
192C23×C53#12312yes—
193C23×C53#12312yes—
194C23×C53#13512yes—
195C23×C53AGL(1,ℤ/8)12yes—
196C23×C53#13712yes—
197C23×C53C22×D411yescharacteristic
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—24no—
203C2×C4×C53—24no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—44no—
210C2×C4×C53—44no—
211C2×C4×C53—44no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53#12312yes—
215C2×C4×C53#12312yes—
216C2×C4×C53C2×#4111yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C53C2×#4112yes—
219C2×C4×C53H(5,𝔽2)12yes—
220C2×Dic53—42no—
221C2×Dic53—41no—
222C2×Dic53—41no—
223C2×Dic53—42no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—42no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—42no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—84no—
239C2×Dic53—84no—
240C8×C53—42no—
241C8×C53—42no—
242#27602—531nomaximal, Sylow
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53—22no—
248C22×C53—22no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—416no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53#48214yes—
265C22×C53C2×#12311yescharacteristic
266C22×C53C2×#12311yescharacteristic
267C22×C53#53411yescharacteristic
268C22×C53C2×#13511yescharacteristic
269C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C53C2×#13711yescharacteristic
271C22×C53#57511yescharacteristic
272C4×C53—22no—
273C4×C53—22no—
274C4×C53—41no—
275C4×C53#55112yes—
276C4×C53#55112yes—
277Dic53—82no—
278Dic53—81no—
279Dic53—81no—
280Dic53—81no—
281Dic53—81no—
282Dic53—82no—
283#2280—531no—
284C2×#380—531no—
285#2077—532no—
286#2076—532no—
287#1854—532no—
288#1853—531no—
289#1852—531no—
290#1825—532no—
291#1823—531no—
292#1823—531no—
293#1779D5311yescharacteristic
294C2×C53—24no—
295C2×C53—42no—
296C2×C53—42no—
297C2×C53#248311yescharacteristic
298C2×C53C2×#48211yescharacteristic
299C2×C53#309111yescharacteristic
300C2×C53#309211yescharacteristic
301C2×C53#309511yescharacteristic
302C2×C53#319011yescharacteristic
303C2×C53#319411yescharacteristic
304C24×C4C2×D5311yescharacteristic
305C2×#117—531no—
306C2×#117—531no—
307C22×C4⋊C4—531no—
308C22×C4⋊C4—531no—
309C22×C22⋊C4—1061no—
310C2×#107—532no—
311C2×#106—532no—
312C2×C22⋊C8C2×D5311yescharacteristic
313C2×C22⋊C8C2×D5311yescharacteristic
314C2×C22⋊C8C2×D5312yes—
315C2×C22⋊C8C2×D5312yes—
316#381—1061no—
317#380—531no—
318#380—532no—
319#380—531no—
320#380—531no—
321#380—531no—
322#380—532no—
323#380—1061no—
324#380—1061no—
325#379—1061no—
326C4×C22⋊C4—1061no—
327C4×C22⋊C4—1061no—
328C2×#95—1061no—
329#340—532no—
330#340—532no—
331#340—532no—
332#340—532no—
333#340—532no—
334#340—532no—
335C53#2760211yescharacteristic, complemented, Sylow
336C25D21211yescharacteristic
337C23×C4—22no—
338C23×C4—22no—
339C23×C4—24no—
340C23×C4—42no—
341C23×C4—1061no—
342C23×C4—1061no—
343C23×C4D21211yescharacteristic
344C23×C4D21211yescharacteristic
345C23×C4D21211yescharacteristic
346C23×C4C22×D5311yescharacteristic
347C23×C4C22×D5311yescharacteristic
348C23×C4C22×D5311yescharacteristic
349C22×C8C22×D5312yes—
350C22×C8C22×D5312yes—
351#117—1064no—
352#117—2122no—
353C2×C4⋊C4—532no—
354C2×C4⋊C4—532no—
355C2×C4⋊C4—531no—
356C2×C4⋊C4—531no—
357C2×C4⋊C4—531no—
358C2×C4⋊C4—531no—
359C2×C4⋊C4—1062no—
360C2×C4⋊C4—1062no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—2122no—
364C2×C22⋊C4—1062no—
365C2×C22⋊C4—1062no—
366C2×C22⋊C4—1061no—
367C2×C22⋊C4—1061no—
368C2×C22⋊C4—1061no—
369C2×C22⋊C4—1061no—
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—1061no—
372C2×C22⋊C4—2124no—
373C2×C42—1061no—
374C2×C42—1061no—
375#107—2122no—
376#106—2122no—
377C22⋊C8—22no—
378C22⋊C8—22no—
379C22⋊C8—24no—
380C22⋊C8—42no—
381#95—1062no—
382#95—1062no—
383#95—1061no—
384#95—1061no—
385#95—1061no—
386#95—1061no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—41no—
393C24—41no—
394C24C2×D21211yescharacteristic
395C24C2×D21211yescharacteristic
396C24C2×D21211yescharacteristic
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—1061no—
418C22×C4—1061no—
419C22×C4—1061no—
420C22×C4—1061no—
421C22×C4—1062no—
422C22×C4—1062no—
423C22×C4—2122no—
424C22×C4—2122no—
425C22×C4—2122no—
426C22×C4—2122no—
427C22×C4—2122no—
428C22×C4—2122no—
429C22×C4C2×D21211yescharacteristic
430C22×C4C2×D21211yescharacteristic
431C22×C4C2×D21211yescharacteristic
432C22×C4#1171994712yes—
433C22×C4#1171994712yes—
434C22×C4#1171994714yes—
435C22×C4C23×D5311yescharacteristic
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C4⋊C4—1062no—
443C4⋊C4—1062no—
444C4⋊C4—1062no—
445C4⋊C4—1062no—
446C4⋊C4—1062no—
447C4⋊C4—1062no—
448C4⋊C4—1062no—
449C4⋊C4—1062no—
450C4⋊C4—2122no—
451C4⋊C4—2122no—
452C22⋊C4—2124no—
453C22⋊C4—2124no—
454C22⋊C4—2122no—
455C22⋊C4—2122no—
456C22⋊C4—2122no—
457C22⋊C4—2122no—
458C22⋊C4—2122no—
459C22⋊C4—2122no—
460C42—2122no—
461C42—2122no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—28no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23#1313320112yes—
486C23#1313320112yes—
487C23#1313323912yes—
488C23#1313324512yes—
489C23#1313325012yes—
490C23C22×D21211yescharacteristic
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—4244no—
508C2×C4—4244no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4—2122no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4—2122no—
515C2×C4—2122no—
516C2×C4—2122no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2122no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4—2122no—
527C2×C4#1313320112yes—
528C2×C4#1313320112yes—
529C2×C4C2×#1171994711yescharacteristic
530C2×C4C2×#1171994711yescharacteristic
531C2×C4C2×#1171994712yes—
532C2×C4#1313326212yes—
533C8—42no—
534C8—42no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—416no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22#1681094814yes—
557C22C2×#1313320111yescharacteristic
558C22C2×#1313320111yescharacteristic
559C22#1681118511yescharacteristic
560C22C2×#1313323911yescharacteristic
561C22C2×#1313324511yescharacteristic
562C22C2×#1313325011yescharacteristic
563C22#1681142311yescharacteristic
564C4—22no—
565C4—22no—
566C4—41no—
567C4—4241no—
568C4—4242no—
569C4—4241no—
570C4—4242no—
571C4—4241no—
572C4—4241no—
573C4#1681130412yes—
574C4#1681130412yes—
575C2—24no—
576C2—42no—
577C2—42no—
578C2#2027899611yescharacteristic
579C2C2×#1681094811yescharacteristic
580C2#2028463011yescharacteristic
581C2#2028464011yescharacteristic
582C2#2028466411yescharacteristic
583C2#2028549811yescharacteristic
584C2#2028553111yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1779C2
socleC23×C53C22×D4
prime radicalC25×C53D4
hypercenter#1779D53
hypocenterC53#27602
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
3. #20276148C2semidirect
4. #20276143C2semidirect
5. #20274916C2semidirect
6. #20274910C2semidirect
7. #20274905C2semidirect
8. #20274811C2semidirect
9. #20274802C2semidirect
10. #20274802C2semidirect
17. C2×#13133151C22semidirect
18. C2×#13133149C22semidirect
32. #16810386C22semidirect
33. #16810386C22semidirect
34. #16810386C22semidirect
35. #16810386C22semidirect
36. #16810386C22semidirect
37. #16810386C22semidirect
335. C53#27602semidirect

Maximal subgroups

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

subgroupindex
#202770982
C2×#168104832
#202761482
#202761432
#202749162
#202749102
#202749052
#202748112
#202748022
C53×#17792
#2760253