Finite Group Databasebeta

← #27383902

Subgroups — #27383902

Order 13568.

1542 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D53C211yesmaximal, characteristic
2C2×#16810585C211yesmaximal, characteristic
3#380×D53C211yesmaximal, characteristic
4#20274811C211yesmaximal, characteristic, complemented
5#20274810C211yesmaximal, characteristic, complemented
6#20274807C211yesmaximal, characteristic, complemented
7#20274804C212yesmaximal, complemented
8#20274800C212yesmaximal, complemented
9#20274433C212yesmaximal, complemented
10#20274433C212yesmaximal, complemented
11C53×#1825C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×#11719912C2212yes—
14C2×#13133176C2211yescharacteristic
15C2×#13133175C2211yescharacteristic
16C2×#13133175C2211yescharacteristic
17C2×#13133174—21no—
18C2×#13133173—21no—
19#117×D53—24no—
20#117×D53C418yescomplemented
21C2×C4⋊C4×D53C2212yes—
22C2×C4⋊C4×D53C2211yescharacteristic
23C2×C4⋊C4×D53C2211yescharacteristic
24C2×C22⋊C4×D53—21no—
25C2×C22⋊C4×D53—21no—
26C2×C42×D53—21no—
27#16810585—24no—
28C2×C22⋊#88322C2211yescharacteristic
29C2×#13133129C2212yes—
30#16810483C2211yescharacteristic
31#16810482—21no—
32#16810481—21no—
33#16810480C2212yes—
34#95×D53—21no—
35#16810386C2212yescomplemented
36#16810386C2212yescomplemented
37#16810385C2212yescomplemented
38#16810385C2212yescomplemented
39#16810385C2212yescomplemented
40#16810385C2212yescomplemented
41C2×#117×C53C2211yescharacteristic
42C2×C22⋊C8×C53C2211yescharacteristic
43C53×#380C2211yescharacteristic
44C53×#340C2212yescomplemented
45C53×#340C2212yescomplemented
46C24×D53D411yescharacteristic
47C23×Dic53D411yescharacteristic
48C22×C4×D53—28no—
49C22×C4×D53—21no—
50C22×C4×D53—21no—
51C22×C4×D53—22no—
52C22×C4×D53C2×C414yes—
53C22×C4×D53D411yescharacteristic
54C22×C4×D53D411yescharacteristic
55C22×C4×D53D414yes—
56C22×C4×D53C2311yescharacteristic
57C22×C4×D53C2311yescharacteristic
58C2×#11719912—24no—
59C2×#11719912—28no—
60C22×#88322C2312yes—
61#13133176—22no—
62#13133176C2×C414yescomplemented
63#13133175—22no—
64#13133175—22no—
65#13133175C2×C414yes—
66#13133175C2×C414yes—
67#13133174—28no—
68#13133173—28no—
69C2×C4⋊Dic53C2312yes—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—22no—
75C4⋊C4×D53—22no—
76C4⋊C4×D53—22no—
77C4⋊C4×D53—28no—
78C4⋊C4×D53C2×C414yescomplemented
79C4⋊C4×D53C2×C414yescomplemented
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—22no—
85C22⋊C4×D53—28no—
86C22⋊C4×D53—44no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C42×D53—28no—
91C22⋊#88322—22no—
92#13133129—24no—
93#13133124—21no—
94#13133124—22no—
95C23×C4×C53C2311yescharacteristic
96C22×C8×C53C2312yes—
97#117×C53—22no—
98#117×C53C2×C414yescomplemented
99C2×C4⋊C4×C53C2312yes—
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C42×C53—21no—
105C22⋊C8×C53—22no—
106#95×C53—21no—
107C24×C53C2×D411yescharacteristic
108C23×D53—24no—
109C23×D53—22no—
110C23×D53—22no—
111C23×D53—42no—
112C23×D53C22⋊C414yes—
113C23×D53C2×D411yescharacteristic
114C23×D53C2×D411yescharacteristic
115C22×C4×C53—24no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53C22×C412yes—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D412yes—
122C22×C4×C53C2411yescharacteristic
123C22×Dic53—24no—
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53—22no—
127C22×Dic53C22⋊C414yes—
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130C2×C4×D53—24no—
131C2×C4×D53—24no—
132C2×C4×D53—24no—
133C2×C4×D53—216no—
134C2×C4×D53—216no—
135C2×C4×D53—24no—
136C2×C4×D53—24no—
137C2×C4×D53—44no—
138C2×C4×D53—42no—
139C2×C4×D53—42no—
140C2×C4×D53—44no—
141C2×C4×D53—44no—
142C2×C4×D53—44no—
143C2×C4×D53C22⋊C414yes—
144C2×C4×D53C22⋊C414yes—
145C2×C4×D53C22×C412yes—
146C2×C4×D53C22×C412yes—
147C2×C4×D53C2×D412yes—
148C2×C4×D53C2×D412yes—
149C2×C4×D53#4118yes—
150C2×C8×C53—22no—
151C2×C8×C53—24no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53C22×C412yes—
156C4⋊C4×C53C22×C412yes—
157C22⋊C4×C53—24no—
158C22⋊C4×C53—42no—
159C42×C53—24no—
160C22⋊Dic53—24no—
161C22⋊Dic53—42no—
162#11719921—24no—
163#11719921—24no—
164#11719921—44no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—24no—
168C4⋊Dic53C22×C412yes—
169C4⋊Dic53C22×C412yes—
170#11719919—24no—
171#11719919—24no—
172#11719919—44no—
173C4×Dic53—24no—
174C4×Dic53—24no—
175C4×Dic53—24no—
176C2×#88322—22no—
177C2×#88322—24no—
178#11719912—48no—
179C23×C53—22no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—22no—
183C23×C53C2×C22⋊C412yes—
184C23×C53C22×D411yescharacteristic
185C22×D53—22no—
186C22×D53—21no—
187C22×D53—21no—
188C22×D53—22no—
189C22×D53—22no—
190C22×D53—28no—
191C22×D53—28no—
192C22×D53—44no—
193C22×D53—48no—
194C22×D53—48no—
195C22×D53C2×C22⋊C412yes—
196C22×D53C2×C22⋊C412yes—
197C22×D53#12314yes—
198C22×D53AGL(1,ℤ/8)12yes—
199C22×D53#13712yes—
200C2×C4×C53—22no—
201C2×C4×C53—24no—
202C2×C4×C53—24no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53C2×C22⋊C412yes—
212C2×C4×C53C23×C411yescharacteristic
213C2×C4×C53C22×D411yescharacteristic
214C2×C4×C53C2×#4114yes—
215C2×Dic53—22no—
216C2×Dic53—28no—
217C2×Dic53—22no—
218C2×Dic53—22no—
219C2×Dic53—41no—
220C2×Dic53—41no—
221C2×Dic53—42no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53C2×C22⋊C412yes—
226C2×Dic53C2×C22⋊C412yes—
227C2×Dic53#12314yes—
228C4×D53—28no—
229C4×D53—24no—
230C4×D53—24no—
231C4×D53—28no—
232C4×D53—48no—
233C4×D53—48no—
234C4×D53—88no—
235C4×D53#11718yes—
236C4×D53#12318yes—
237C8×C53—42no—
238#88322—42no—
239#20702—531nomaximal, Sylow
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53C22×C22⋊C411yescharacteristic
246C22×C53C2×#12312yes—
247C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C53C2×#13711yescharacteristic
249C22×C53#57512yes—
250C2×D53—24no—
251C2×D53—24no—
252C2×D53—24no—
253C2×D53—24no—
254C2×D53—22no—
255C2×D53—22no—
256C2×D53—22no—
257C2×D53—22no—
258C2×D53—416no—
259C2×D53#38114yes—
260C2×D53#41714yes—
261C4×C53—22no—
262C4×C53—42no—
263C4×C53—42no—
264C4×C53—82no—
265C4×C53C2×#11712yes—
266C4×C53C2×#12312yes—
267Dic53—22no—
268Dic53—42no—
269Dic53—42no—
270Dic53—82no—
271Dic53#38114yes—
272C22×#117—531no—
273C2×#407—531no—
274C2×#380—531no—
275#1825—531no—
276#1825—531no—
277#1825—531no—
278#1825D5311yescharacteristic, complemented
279#1824—532no—
280#1823—532no—
281#1736—532no—
282#1736—532no—
283C2×C53—42no—
284C2×C53C2×#38111yescharacteristic
285C2×C53C2×#41711yescharacteristic
286C2×C53#289611yescharacteristic
287C2×C53#309412yes—
288C2×C53#309512yes—
289D53—24no—
290D53—24no—
291C24×C4—531no—
292C22×M16—532no—
293C2×#117—531no—
294C2×#117—538no—
295C2×#117—531no—
296C2×#117—531no—
297C2×#117—1061no—
298C2×#117—1061no—
299C2×#117—1064no—
300C2×#117C2×D5311yescharacteristic
301C22×C4⋊C4—531no—
302C22×C4⋊C4—531no—
303C22×C4⋊C4—532no—
304C22×C22⋊C4—1061no—
305C22×C22⋊C4—1061no—
306C22×C42—1061no—
307#407—1064no—
308C2×C22⋊C8—531no—
309C2×C22⋊C8—532no—
310C2×C22⋊C8C2×D5311yescharacteristic
311#380—531no—
312#380—532no—
313#380—1061no—
314#380—1061no—
315#380C2×D5311yescharacteristic
316C2×#95—1061no—
317#340—532no—
318#340—532no—
319#340—532no—
320#340—532no—
321#340—532no—
322#340—532no—
323#340C2×D5312yescomplemented
324#340C2×D5312yescomplemented
325C53#2070211yescharacteristic, complemented, Sylow
326C25—531no—
327C23×C4—531no—
328C23×C4—531no—
329C23×C4—534no—
330C23×C4—531no—
331C23×C4—534no—
332C23×C4—531no—
333C23×C4—531no—
334C23×C4—1068no—
335C23×C4—1061no—
336C23×C4—1061no—
337C23×C4—1062no—
338C23×C4C22×D5311yescharacteristic
339C2×M16—1064no—
340C2×M16—1068no—
341C22×C8—532no—
342C22×C8C22×D5312yes—
343#117—22no—
344#117—534no—
345#117—534no—
346#117—534no—
347#117—1068no—
348#117—1068no—
349#117—1062no—
350#117—1062no—
351#117—1062no—
352#117C4×D5314yescomplemented
353C2×C4⋊C4—531no—
354C2×C4⋊C4—531no—
355C2×C4⋊C4—534no—
356C2×C4⋊C4—534no—
357C2×C4⋊C4—532no—
358C2×C4⋊C4—1062no—
359C2×C4⋊C4—1062no—
360C2×C4⋊C4—1068no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—1062no—
364C2×C4⋊C4C22×D5312yes—
365C2×C4⋊C4C22×D5311yescharacteristic
366C2×C4⋊C4C22×D5311yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—1068no—
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—1061no—
372C2×C22⋊C4—1061no—
373C2×C22⋊C4—1062no—
374C2×C22⋊C4—1061no—
375C2×C22⋊C4—2124no—
376C2×C42—21no—
377C2×C42—1068no—
378C2×C42—1061no—
379C2×C42—1061no—
380C2×C42—1061no—
381C22⋊C8—22no—
382C22⋊C8—1064no—
383C22⋊C8—1062no—
384#95—21no—
385#95—1062no—
386#95—1061no—
387C24—531no—
388C24—531no—
389C24—534no—
390C24—1062no—
391C24—1064no—
392C24—1062no—
393C24—2122no—
394C24D4×D5311yescharacteristic
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—531no—
400C22×C4—532no—
401C22×C4—532no—
402C22×C4—538no—
403C22×C4—534no—
404C22×C4—531no—
405C22×C4—534no—
406C22×C4—534no—
407C22×C4—532no—
408C22×C4—532no—
409C22×C4—1064no—
410C22×C4—10616no—
411C22×C4—10616no—
412C22×C4—1064no—
413C22×C4—1064no—
414C22×C4—1064no—
415C22×C4—1064no—
416C22×C4—1064no—
417C22×C4—1061no—
418C22×C4—1061no—
419C22×C4—1062no—
420C22×C4—2124no—
421C22×C4—2124no—
422C22×C4—2124no—
423C22×C4—2122no—
424C22×C4—2122no—
425C22×C4—2124no—
426C22×C4C2×C4×D5312yes—
427C22×C4D4×D5311yescharacteristic
428C22×C4D4×D5312yes—
429C22×C4C23×D5311yescharacteristic
430M16—2128no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—1062no—
434C2×C8—1064no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—24no—
438C4⋊C4—532no—
439C4⋊C4—532no—
440C4⋊C4—1064no—
441C4⋊C4—1064no—
442C4⋊C4—1061no—
443C4⋊C4—1061no—
444C4⋊C4—1064no—
445C4⋊C4—2124no—
446C4⋊C4C2×C4×D5312yes—
447C4⋊C4C2×C4×D5312yes—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4—1064no—
451C22⋊C4—1064no—
452C22⋊C4—1064no—
453C22⋊C4—2124no—
454C22⋊C4—2122no—
455C42—24no—
456C42—1064no—
457C42—1064no—
458C42—1064no—
459C23—22no—
460C23—21no—
461C23—21no—
462C23—22no—
463C23—532no—
464C23—532no—
465C23—534no—
466C23—532no—
467C23—532no—
468C23—1068no—
469C23—1068no—
470C23—1062no—
471C23—1061no—
472C23—1061no—
473C23—1062no—
474C23—1062no—
475C23—2124no—
476C23—2128no—
477C23—2128no—
478C23C22⋊C4×D5312yes—
479C23C2×D4×D5311yescharacteristic
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—532no—
492C2×C4—534no—
493C2×C4—538no—
494C2×C4—538no—
495C2×C4—532no—
496C2×C4—1068no—
497C2×C4—1068no—
498C2×C4—1068no—
499C2×C4—1062no—
500C2×C4—1064no—
501C2×C4—1064no—
502C2×C4—1062no—
503C2×C4—1062no—
504C2×C4—4248no—
505C2×C4—2128no—
506C2×C4—2128no—
507C2×C4—2122no—
508C2×C4—2122no—
509C2×C4—2122no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4—2122no—
513C2×C4C22⋊C4×D5312yes—
514C2×C4C22×C4×D5311yescharacteristic
515C2×C4C2×D4×D5311yescharacteristic
516C2×C4#41×D5314yes—
517C8—42no—
518C8—2122no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—42no—
523C22—42no—
524C22—534no—
525C22—534no—
526C22—1062no—
527C22—1062no—
528C22—1062no—
529C22—1062no—
530C22—1064no—
531C22—1064no—
532C22—1064no—
533C22—1064no—
534C22—21216no—
535C22C2×C22⋊C4×D5311yescharacteristic
536C22#123×D5312yes—
537C22AGL(1,ℤ/8)×D5311yescharacteristic
538C22#137×D5311yescharacteristic
539C22#1681142012yes—
540C4—22no—
541C4—42no—
542C4—42no—
543C4—82no—
544C4—534no—
545C4—1062no—
546C4—4242no—
547C4—2122no—
548C4—2122no—
549C4#117×D5312yes—
550C4#123×D5312yes—
551C2—42no—
552C2—1064no—
553C2—1064no—
554C2#381×D5311yescharacteristic
555C2#417×D5311yescharacteristic
556C2#2028323011yescharacteristic
557C2#2028465412yes—
558C2#2028466212yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1825C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenter#1825D53
hypocenterC53#20702
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
4. #20274811C2semidirect
5. #20274810C2semidirect
6. #20274807C2semidirect
7. #20274804C2semidirect
8. #20274800C2semidirect
9. #20274433C2semidirect
10. #20274433C2semidirect
11. C53×#1825C2semidirect
20. #117×D53C4semidirect
35. #16810386C22semidirect
36. #16810386C22semidirect
37. #16810385C22semidirect
38. #16810385C22semidirect
39. #16810385C22semidirect
40. #16810385C22semidirect
44. C53×#340C22semidirect
45. C53×#340C22semidirect
62. #13133176C2×C4semidirect
78. C4⋊C4×D53C2×C4semidirect
79. C4⋊C4×D53C2×C4semidirect
98. #117×C53C2×C4semidirect
278. #1825D53semidirect
323. #340C2×D53semidirect
324. #340C2×D53semidirect
325. C53#20702semidirect
352. #117C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D532
C2×#168105852
#380×D532
#202748112
#202748102
#202748072
#202748042
#202748002
#202744332
C53×#18252
#2070253