Finite Group Databasebeta

← #27383889

Subgroups — #27383889

Order 13568.

1774 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D53C211yesmaximal, characteristic
2C2×#16810586C211yesmaximal, characteristic
3#20277094C211yesmaximal, characteristic
4#20274804C214yesmaximal, complemented
5#20274800C214yesmaximal, complemented
6#20274436C211yesmaximal, characteristic, complemented
7#20274435C211yesmaximal, characteristic, complemented
8#20274434C211yesmaximal, characteristic, complemented
9C53×#1736C211yesmaximal, characteristic, complemented
10C23×C4×D53—21no—
11C23×C4×D53C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919—21no—
16C2×C4⋊C4×D53—21no—
17C2×C4⋊C4×D53—21no—
18C2×C4⋊C4×D53—24no—
19C2×C4⋊C4×D53C418yescomplemented
20C2×C4⋊C4×D53C2212yes—
21C2×C4⋊C4×D53C2212yes—
22#16810586—24no—
23C2×#13133129C2212yes—
24C2×#13133129C2212yes—
25#16810484—21no—
26#16810480C2212yes—
27#16810480C2212yes—
28#16810478—21no—
29C4×#11719921—21no—
30C4×#11719921—21no—
31#16810386C2214yescomplemented
32#16810385C2214yescomplemented
33#16810385C2214yescomplemented
34C22×M16×C53C2211yescharacteristic
35C2×#117×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C53×#340C2214yescomplemented
38C24×D53D411yescharacteristic
39C23×Dic53—21no—
40C23×Dic53D411yescharacteristic
41C22×C4×D53—22no—
42C22×C4×D53—22no—
43C22×C4×D53—22no—
44C22×C4×D53—22no—
45C22×C4×D53—28no—
46C22×C4×D53—22no—
47C22×C4×D53—22no—
48C22×C4×D53—48no—
49C22×C4×D53C2×C414yes—
50C22×C4×D53D411yescharacteristic
51C22×C4×D53D411yescharacteristic
52C22×C4×D53D414yes—
53C22×C4×D53C2311yescharacteristic
54C22×C4×D53C2311yescharacteristic
55C2×#11719917—22no—
56C2×#11719917—22no—
57C2×#11719917—22no—
58C22×#88322C2312yes—
59#13133176—22no—
60#13133176—22no—
61C2×C4⋊Dic53—22no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53C2×C414yes—
64C2×C4⋊Dic53C2312yes—
65C2×C4⋊Dic53C2312yes—
66C2×#11719919—28no—
67C2×#11719919—22no—
68C2×#11719919—22no—
69C4⋊C4×D53—28no—
70C4⋊C4×D53—28no—
71C4⋊C4×D53—24no—
72C4⋊C4×D53—24no—
73C4⋊C4×D53C2×C418yes—
74C4⋊C4×D53C2×C418yes—
75C2×C22⋊Dic53—21no—
76C2×#11719921—22no—
77C2×#11719921—22no—
78C2×C4×Dic53—21no—
79#13133129—24no—
80#13133129—24no—
81#13133124—22no—
82#13133124—22no—
83C23×C4×C53—21no—
84C23×C4×C53C2311yescharacteristic
85C2×M16×C53—22no—
86C2×M16×C53—22no—
87C2×M16×C53—22no—
88C22×C8×C53C2312yes—
89#117×C53—22no—
90#117×C53—22no—
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53C2×C414yes—
94C2×C4⋊C4×C53C2312yes—
95C2×C4⋊C4×C53C2312yes—
96C2×C22⋊C4×C53—21no—
97C2×C42×C53—21no—
98C24×C53C2×D411yescharacteristic
99C23×D53—22no—
100C23×D53—22no—
101C23×D53—24no—
102C23×D53—24no—
103C23×D53C22⋊C414yes—
104C23×D53C2×D411yescharacteristic
105C23×D53C2×D411yescharacteristic
106C22×C4×C53—24no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—44no—
110C22×C4×C53C22×C412yes—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D412yes—
113C22×C4×C53#4112yes—
114C22×C4×C53#4112yes—
115C22×C4×C53#4112yes—
116C22×C4×C53#4112yes—
117C22×C4×C53C2411yescharacteristic
118C22×Dic53—22no—
119C22×Dic53—22no—
120C22×Dic53—24no—
121C22×Dic53—22no—
122C22×Dic53—22no—
123C22×Dic53—44no—
124C22×Dic53C22⋊C414yes—
125C22×Dic53C2×D411yescharacteristic
126C22×Dic53C2×D411yescharacteristic
127C2×C4×D53—28no—
128C2×C4×D53—28no—
129C2×C4×D53—28no—
130C2×C4×D53—28no—
131C2×C4×D53—22no—
132C2×C4×D53—22no—
133C2×C4×D53—22no—
134C2×C4×D53—22no—
135C2×C4×D53—21no—
136C2×C4×D53—21no—
137C2×C4×D53—21no—
138C2×C4×D53—21no—
139C2×C4×D53—21no—
140C2×C4×D53—21no—
141C2×C4×D53—21no—
142C2×C4×D53—21no—
143C2×C4×D53—28no—
144C2×C4×D53—28no—
145C2×C4×D53—48no—
146C2×C4×D53—432no—
147C2×C4×D53—48no—
148C2×C4×D53C22⋊C414yes—
149C2×C4×D53C22⋊C414yes—
150C2×C4×D53C22×C412yes—
151C2×C4×D53C22×C412yes—
152C2×C4×D53C2×D412yes—
153C2×C4×D53C2×D412yes—
154M16×C53—44no—
155C2×C8×C53—22no—
156C2×C8×C53—22no—
157C2×C8×C53—22no—
158C4⋊C4×C53—24no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53C22×C414yes—
163C22⋊C4×C53—42no—
164C42×C53—42no—
165C22⋊Dic53—42no—
166#11719921—44no—
167#11719921—44no—
168C4⋊Dic53—24no—
169C4⋊Dic53—22no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53C22×C414yes—
173#11719919—28no—
174#11719919—28no—
175C4×Dic53—42no—
176#11719917—44no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53C2×C22⋊C412yes—
183C23×C53#12312yes—
184C23×C53#12312yes—
185C23×C53C22×D411yescharacteristic
186C22×D53—216no—
187C22×D53—216no—
188C22×D53—21no—
189C22×D53—21no—
190C22×D53—21no—
191C22×D53—21no—
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—28no—
195C22×D53—28no—
196C22×D53—28no—
197C22×D53C2×C22⋊C412yes—
198C22×D53C2×C22⋊C412yes—
199C22×D53AGL(1,ℤ/8)14yes—
200C22×D53#13714yes—
201C2×C4×C53—24no—
202C2×C4×C53—48no—
203C2×C4×C53—44no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53C2×C22⋊C412yes—
208C2×C4×C53#11714yes—
209C2×C4×C53#11714yes—
210C2×C4×C53#12314yes—
211C2×C4×C53#12312yes—
212C2×C4×C53#12314yes—
213C2×C4×C53#12312yes—
214C2×C4×C53C23×C411yescharacteristic
215C2×C4×C53C22×D411yescharacteristic
216C2×C4×C53C2×#4111yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C53C2×#4111yescharacteristic
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—28no—
225C2×Dic53—24no—
226C2×Dic53—48no—
227C2×Dic53—44no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53C2×C22⋊C412yes—
232C2×Dic53C2×C22⋊C412yes—
233C4×D53—22no—
234C4×D53—22no—
235C4×D53—22no—
236C4×D53—22no—
237C4×D53—22no—
238C4×D53—22no—
239C4×D53—22no—
240C4×D53—22no—
241C4×D53—28no—
242C4×D53—28no—
243C4×D53—416no—
244C4×D53—416no—
245C8×C53—42no—
246#88322—42no—
247#20700—531nomaximal, Sylow
248C22×C53—24no—
249C22×C53—24no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53#38114yes—
253C22×C53C22×C22⋊C411yescharacteristic
254C22×C53C2×#12311yescharacteristic
255C22×C53C2×#12311yescharacteristic
256C22×C53C2×AGL(1,ℤ/8)12yes—
257C22×C53C2×#13712yes—
258C2×D53—232no—
259C2×D53—24no—
260C2×D53—24no—
261C2×D53—24no—
262C2×D53—24no—
263C2×D53—22no—
264C2×D53—22no—
265C2×D53—22no—
266C2×D53—22no—
267C2×D53#41814yes—
268C2×D53#41914yes—
269C4×C53—44no—
270C4×C53—82no—
271C4×C53#38114yes—
272C4×C53C2×#11711yescharacteristic
273C4×C53C2×#11711yescharacteristic
274C4×C53C2×#12311yescharacteristic
275C4×C53C2×#12311yescharacteristic
276Dic53—22no—
277Dic53—22no—
278Dic53—44no—
279Dic53—82no—
280C23×C4⋊C4—531no—
281C2×#407—531no—
282#2280—531no—
283#1824—534no—
284#1823—534no—
285#1736—531no—
286#1736—531no—
287#1736—531no—
288#1736D5311yescharacteristic, complemented
289C2×C53—24no—
290C2×C53C2×#38111yescharacteristic
291C2×C53C2×#41811yescharacteristic
292C2×C53C2×#41911yescharacteristic
293C2×C53#309212yes—
294C2×C53#309212yes—
295D53—24no—
296D53—24no—
297C24×C4—531no—
298C24×C4—1061no—
299C22×M16—531no—
300C22×M16C2×D5311yescharacteristic
301C2×#117—531no—
302C2×#117C2×D5311yescharacteristic
303C22×C4⋊C4—531no—
304C22×C4⋊C4—538no—
305C22×C4⋊C4—532no—
306C22×C4⋊C4—532no—
307C22×C4⋊C4—1061no—
308C22×C4⋊C4—1061no—
309C22×C4⋊C4—1064no—
310C22×C4⋊C4—1061no—
311C22×C4⋊C4C2×D5311yescharacteristic
312#407—1064no—
313C2×C22⋊C8—532no—
314C2×C22⋊C8—532no—
315#381—1061no—
316#380—532no—
317#380—532no—
318#379—1061no—
319C4×C22⋊C4—1061no—
320C4×C22⋊C4—1061no—
321#340—534no—
322#340—534no—
323#340—534no—
324#340C2×D5314yescomplemented
325C53#2070011yescharacteristic, complemented, Sylow
326C25—531no—
327C23×C4—21no—
328C23×C4—531no—
329C23×C4—534no—
330C23×C4—531no—
331C23×C4—531no—
332C23×C4—534no—
333C23×C4—531no—
334C23×C4—531no—
335C23×C4—1068no—
336C23×C4—1062no—
337C23×C4—1062no—
338C23×C4—1062no—
339C23×C4—1062no—
340C23×C4—1062no—
341C23×C4—1062no—
342C23×C4—1061no—
343C23×C4—2128no—
344C23×C4C22×D5311yescharacteristic
345C2×M16—22no—
346C2×M16—22no—
347C2×M16—22no—
348C2×M16—1062no—
349C2×M16—1062no—
350C2×M16—1062no—
351C22×C8—532no—
352C22×C8C22×D5312yes—
353#117—22no—
354#117—22no—
355#117—1062no—
356#117—1062no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—534no—
360C2×C4⋊C4—532no—
361C2×C4⋊C4—538no—
362C2×C4⋊C4—538no—
363C2×C4⋊C4—532no—
364C2×C4⋊C4—1068no—
365C2×C4⋊C4—1068no—
366C2×C4⋊C4—1064no—
367C2×C4⋊C4—1064no—
368C2×C4⋊C4—1068no—
369C2×C4⋊C4—1062no—
370C2×C4⋊C4—1062no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4C4×D5314yes—
374C2×C4⋊C4C22×D5312yes—
375C2×C4⋊C4C22×D5312yes—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—1062no—
378C2×C22⋊C4—1062no—
379C2×C22⋊C4—1061no—
380C2×C42—21no—
381C2×C42—1061no—
382C22⋊C8—1064no—
383C22⋊C8—1064no—
384#95—1062no—
385#95—1062no—
386C24—531no—
387C24—531no—
388C24—534no—
389C24—1064no—
390C24—1064no—
391C24—1062no—
392C24—1062no—
393C24D4×D5311yescharacteristic
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—44no—
398C22×C4—534no—
399C22×C4—531no—
400C22×C4—531no—
401C22×C4—534no—
402C22×C4—534no—
403C22×C4—532no—
404C22×C4—532no—
405C22×C4—532no—
406C22×C4—532no—
407C22×C4—1068no—
408C22×C4—1068no—
409C22×C4—1062no—
410C22×C4—1062no—
411C22×C4—1068no—
412C22×C4—1068no—
413C22×C4—1062no—
414C22×C4—1062no—
415C22×C4—1068no—
416C22×C4—1062no—
417C22×C4—1062no—
418C22×C4—1068no—
419C22×C4—1061no—
420C22×C4—1061no—
421C22×C4—1061no—
422C22×C4—1061no—
423C22×C4—1061no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1064no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—2128no—
431C22×C4—21232no—
432C22×C4—2128no—
433C22×C4—2124no—
434C22×C4C2×C4×D5312yes—
435C22×C4#1171994512yes—
436C22×C4#1171994512yes—
437C22×C4D4×D5311yescharacteristic
438C22×C4D4×D5312yes—
439C22×C4#1171994712yes—
440C22×C4#1171994912yes—
441C22×C4C23×D5311yescharacteristic
442M16—44no—
443M16—2124no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—1062no—
448C2×C8—1062no—
449C2×C8—1062no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—534no—
455C4⋊C4—1068no—
456C4⋊C4—1068no—
457C4⋊C4—1064no—
458C4⋊C4—1062no—
459C4⋊C4—1062no—
460C4⋊C4—1062no—
461C4⋊C4C2×C4×D5314yes—
462C22⋊C4—42no—
463C22⋊C4—2124no—
464C22⋊C4—2124no—
465C22⋊C4—2122no—
466C42—42no—
467C42—2122no—
468C23—22no—
469C23—22no—
470C23—532no—
471C23—532no—
472C23—534no—
473C23—534no—
474C23—1068no—
475C23—1068no—
476C23—10616no—
477C23—10616no—
478C23—1061no—
479C23—1061no—
480C23—1061no—
481C23—1061no—
482C23—1062no—
483C23—1062no—
484C23—1068no—
485C23C22⋊C4×D5312yes—
486C23#1313319912yes—
487C23#1313320212yes—
488C23C2×D4×D5311yescharacteristic
489C2×C4—24no—
490C2×C4—48no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—532no—
496C2×C4—532no—
497C2×C4—1061no—
498C2×C4—1061no—
499C2×C4—1061no—
500C2×C4—1061no—
501C2×C4—1068no—
502C2×C4—1062no—
503C2×C4—1062no—
504C2×C4—1062no—
505C2×C4—1062no—
506C2×C4—1062no—
507C2×C4—1062no—
508C2×C4—1062no—
509C2×C4—1062no—
510C2×C4—1068no—
511C2×C4—1068no—
512C2×C4—1064no—
513C2×C4—21216no—
514C2×C4—21216no—
515C2×C4—2122no—
516C2×C4—2122no—
517C2×C4—2122no—
518C2×C4—2128no—
519C2×C4—2124no—
520C2×C4C22⋊C4×D5312yes—
521C2×C4#1313317414yes—
522C2×C4#1313317514yes—
523C2×C4#1313319914yes—
524C2×C4#1313319912yes—
525C2×C4#1313320214yes—
526C2×C4#1313320212yes—
527C2×C4C22×C4×D5311yescharacteristic
528C2×C4C2×D4×D5311yescharacteristic
529C2×C4C2×#1171994711yescharacteristic
530C2×C4C2×#1171994511yescharacteristic
531C2×C4C2×#1171994511yescharacteristic
532C2×C4C2×#1171994911yescharacteristic
533C8—42no—
534C8—2122no—
535C22—22no—
536C22—22no—
537C22—24no—
538C22—24no—
539C22—534no—
540C22—534no—
541C22—10632no—
542C22—1062no—
543C22—1062no—
544C22—1062no—
545C22—1062no—
546C22—1064no—
547C22—1064no—
548C22—1064no—
549C22—1064no—
550C22#1681048514yes—
551C22C2×C22⋊C4×D5311yescharacteristic
552C22C2×#1313319911yescharacteristic
553C22C2×#1313320211yescharacteristic
554C22AGL(1,ℤ/8)×D5312yes—
555C22#137×D5312yes—
556C4—44no—
557C4—82no—
558C4—1062no—
559C4—1062no—
560C4—4242no—
561C4—2124no—
562C4#1681048514yes—
563C4C2×#1313317411yescharacteristic
564C4C2×#1313317511yescharacteristic
565C4C2×#1313319911yescharacteristic
566C4C2×#1313320211yescharacteristic
567C2—24no—
568C2—1064no—
569C2—1064no—
570C2C2×#1681048511yescharacteristic
571C2#418×D5311yescharacteristic
572C2#419×D5311yescharacteristic
573C2#2028463612yes—
574C2#2028464112yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1736C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenter#1736D53
hypocenterC53#20700
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #20274804C2semidirect
5. #20274800C2semidirect
6. #20274436C2semidirect
7. #20274435C2semidirect
8. #20274434C2semidirect
9. C53×#1736C2semidirect
19. C2×C4⋊C4×D53C4semidirect
31. #16810386C22semidirect
32. #16810385C22semidirect
33. #16810385C22semidirect
37. C53×#340C22semidirect
288. #1736D53semidirect
324. #340C2×D53semidirect
325. C53#20700semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D532
C2×#168105862
#202770942
#202748042
#202748002
#202744362
#202744352
#202744342
C53×#17362
#2070053