Finite Group Databasebeta

← #27394365

Subgroups — #27394365

Order 13568.

1194 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C8×D53C211yesmaximal, characteristic
2#20277097C211yesmaximal, characteristic
3C2×#16810480C211yesmaximal, characteristic
4#20274917C212yesmaximal, complemented
5#20274911C212yesmaximal, complemented
6#20274908C211yesmaximal, characteristic, complemented
7#20274907C211yesmaximal, characteristic, complemented
8#20274905C211yesmaximal, characteristic, complemented
9#20274741C212yesmaximal, complemented
10#20274739C212yesmaximal, complemented
11C53×#1852C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×C8×D53C2212yes—
14C2×#13133175C2211yescharacteristic
15C2×#13133175C2211yescharacteristic
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719921—21no—
18C2×#13133151C2212yescomplemented
19C2×#13133148C2212yescomplemented
20C2×#13133148C2212yescomplemented
21C2×C22⋊#88322C2211yescharacteristic
22C2×#13133129C2212yes—
23C22⋊C8×D53C418yescomplemented
24#16810485—21no—
25#16810483C2211yescharacteristic
26#16810480—21no—
27#16810480—21no—
28#16810480C2211yescharacteristic
29#16810480C2212yes—
30#16810480C2211yescharacteristic
31#16810480C2212yes—
32#16810478—21no—
33C22⋊C4×Dic53—21no—
34#16810470—21no—
35C2×#13133124—21no—
36#16810386C2212yescomplemented
37#16810385C2212yescomplemented
38#16810385C2212yescomplemented
39C22×C4⋊C4×C53C2211yescharacteristic
40C2×#107×C53C2212yescomplemented
41C2×C22⋊C8×C53C2211yescharacteristic
42C53×#380C2211yescharacteristic
43C53×#340C2212yescomplemented
44C24×D53D411yescharacteristic
45C23×Dic53—21no—
46C23×Dic53D411yescharacteristic
47C22×C4×D53—42no—
48C22×C4×D53—42no—
49C22×C4×D53C2×C414yes—
50C22×C4×D53D411yescharacteristic
51C22×C4×D53D411yescharacteristic
52C22×C4×D53Q814yes—
53C22×C4×D53C2311yescharacteristic
54C22×C4×D53C2311yescharacteristic
55C22×#88322C2312yes—
56C2×C8×D53—42no—
57C2×C8×D53—42no—
58C2×C8×D53C2×C418yescomplemented
59#13133175—42no—
60#13133175—42no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—42no—
63C2×C4⋊Dic53C2312yes—
64C2×C4⋊Dic53C2311yescharacteristic
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C22⋊Dic53—21no—
67C2×#11719921—22no—
68C2×#11719921—22no—
69C2×#11719921—21no—
70C2×#11719921—21no—
71C2×#11719921—21no—
72C2×#11719921—21no—
73C2×#11719921—44no—
74C2×C4×Dic53—21no—
75C2×C4×Dic53—21no—
76#13133151—42no—
77#13133148—42no—
78#13133148—42no—
79C22⋊#88322C2×C414yes—
80#13133129C2×C418yescomplemented
81#13133124—21no—
82#13133124—22no—
83#13133124—22no—
84#13133124—21no—
85#13133124—21no—
86C23×C4×C53—21no—
87C23×C4×C53C2311yescharacteristic
88C22×C8×C53C2312yes—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—42no—
91C2×C4⋊C4×C53C2312yes—
92C2×C4⋊C4×C53C2311yescharacteristic
93C2×C4⋊C4×C53C2311yescharacteristic
94C2×C22⋊C4×C53—21no—
95#107×C53—42no—
96C22⋊C8×C53C2×C414yes—
97#95×C53—21no—
98C24×C53C2×D411yescharacteristic
99C23×D53—22no—
100C23×D53—22no—
101C23×D53—22no—
102C23×D53—22no—
103C23×D53—41no—
104C23×D53—41no—
105C23×D53C4⋊C414yes—
106C23×D53C2×D411yescharacteristic
107C23×D53C2×D411yescharacteristic
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—22no—
113C22×C4×C53—42no—
114C22×C4×C53—42no—
115C22×C4×C53—42no—
116C22×C4×C53C22×C412yes—
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×Q812yes—
119C22×C4×C53C2411yescharacteristic
120C22×Dic53—21no—
121C22×Dic53—21no—
122C22×Dic53—22no—
123C22×Dic53—41no—
124C22×Dic53—41no—
125C22×Dic53—42no—
126C22×Dic53—42no—
127C22×Dic53—42no—
128C22×Dic53C4⋊C414yes—
129C22×Dic53C2×D411yescharacteristic
130C22×Dic53C2×D411yescharacteristic
131C2×C4×D53—22no—
132C2×C4×D53—22no—
133C2×C4×D53—22no—
134C2×C4×D53—22no—
135C2×C4×D53—48no—
136C2×C4×D53—48no—
137C2×C4×D53—41no—
138C2×C4×D53—41no—
139C2×C4×D53—41no—
140C2×C4×D53—41no—
141C2×C4×D53C4⋊C414yes—
142C2×C4×D53C4⋊C414yes—
143C2×C4×D53C22×C412yes—
144C2×C4×D53C22×C412yes—
145C2×C4×D53C2×Q812yes—
146C2×C4×D53C2×Q812yes—
147C2×C8×C53—22no—
148C2×C8×C53—22no—
149C2×C8×C53C22×C414yes—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—42no—
155C22⋊C4×C53—42no—
156C22⋊Dic53—42no—
157#11719921—44no—
158#11719921—42no—
159#11719921—42no—
160#11719921—42no—
161#11719921—44no—
162#11719921—42no—
163C4⋊Dic53—22no—
164C4⋊Dic53—22no—
165C4⋊Dic53—22no—
166C4⋊Dic53—22no—
167C4⋊Dic53—42no—
168C4×Dic53—42no—
169C4×Dic53—42no—
170C2×#88322—22no—
171C2×#88322—22no—
172C2×#88322C22×C414yes—
173C8×D53—48no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53C2×C4⋊C412yes—
179C23×C53#13512yes—
180C23×C53#13512yes—
181C23×C53C22×D411yescharacteristic
182C22×D53—21no—
183C22×D53—21no—
184C22×D53—21no—
185C22×D53—21no—
186C22×D53—21no—
187C22×D53—21no—
188C22×D53—21no—
189C22×D53—21no—
190C22×D53—21no—
191C22×D53—21no—
192C22×D53—28no—
193C22×D53—28no—
194C22×D53—44no—
195C22×D53—44no—
196C22×D53—44no—
197C22×D53—44no—
198C22×D53—44no—
199C22×D53C2×C4⋊C412yes—
200C22×D53C2×C4⋊C412yes—
201C22×D53AGL(1,ℤ/8)12yes—
202C22×D53#13712yes—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—42no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—42no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—42no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—84no—
217C2×C4×C53C2×C4⋊C412yes—
218C2×C4×C53C23×C411yescharacteristic
219C2×C4×C53C22×Q811yescharacteristic
220C2×C4×C53H(5,𝔽2)12yes—
221C2×C4×C53H(5,𝔽2)12yes—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—44no—
227C2×Dic53—42no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—42no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—42no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—84no—
237C2×Dic53C2×C4⋊C412yes—
238C2×Dic53C2×C4⋊C412yes—
239C4×D53—24no—
240C4×D53—24no—
241C4×D53—24no—
242C4×D53—24no—
243C4×D53—44no—
244C4×D53—44no—
245C8×C53—24no—
246#88322—24no—
247#21941—531nomaximal, Sylow
248C22×C53—22no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53#42714yes—
255C22×C53C22×C4⋊C411yescharacteristic
256C22×C53#53411yescharacteristic
257C22×C53#53411yescharacteristic
258C22×C53C2×#13511yescharacteristic
259C22×C53C2×#13511yescharacteristic
260C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C53C2×#13711yescharacteristic
262C2×D53—22no—
263C2×D53—22no—
264C2×D53—22no—
265C2×D53—22no—
266C2×D53—22no—
267C2×D53—22no—
268C2×D53—22no—
269C2×D53—22no—
270C2×D53—22no—
271C2×D53—416no—
272C2×D53—41no—
273C2×D53—41no—
274C2×D53—41no—
275C2×D53—41no—
276C2×D53#42814yes—
277C4×C53—22no—
278C4×C53—81no—
279C4×C53—82no—
280C4×C53—81no—
281C4×C53#51812yes—
282C4×C53#54312yes—
283Dic53—22no—
284Dic53—22no—
285Dic53—81no—
286Dic53—82no—
287Dic53—81no—
288C22×C22⋊C8—531no—
289#2280—531no—
290C2×#380—531no—
291#1854—532no—
292#1854—532no—
293#1853—531no—
294#1853—531no—
295#1852—531no—
296#1852D5311yescharacteristic, complemented
297#1808—532no—
298#1807—532no—
299C2×C53—24no—
300C2×C53#231511yescharacteristic
301C2×C53C2×#42711yescharacteristic
302C2×C53C2×#42811yescharacteristic
303C2×C53#318811yescharacteristic
304C2×C53#318811yescharacteristic
305C2×C53#319011yescharacteristic
306C2×C53#319011yescharacteristic
307D53—24no—
308D53—42no—
309C24×C4—531no—
310C23×C8—532no—
311C2×#117—531no—
312C2×#117—531no—
313C22×C4⋊C4—531no—
314C22×C4⋊C4C2×D5311yescharacteristic
315C22×C22⋊C4—1061no—
316C2×#107—532no—
317C2×#107C2×D5312yescomplemented
318C2×#106—532no—
319C2×#106—532no—
320C2×C22⋊C8—531no—
321C2×C22⋊C8—532no—
322C2×C22⋊C8—538no—
323C2×C22⋊C8C2×D5311yescharacteristic
324#381—1061no—
325#380—531no—
326#380—531no—
327#380—532no—
328#380—531no—
329#380—532no—
330#380—1061no—
331#380—1061no—
332#380C2×D5311yescharacteristic
333#379—1061no—
334C4×C22⋊C4—1061no—
335C4×C22⋊C4—1061no—
336C2×#95—1061no—
337#340—532no—
338#340—532no—
339#340—532no—
340#340C2×D5312yescomplemented
341C53#2194111yescharacteristic, complemented, Sylow
342C25—531no—
343C23×C4—21no—
344C23×C4—531no—
345C23×C4—531no—
346C23×C4—531no—
347C23×C4—534no—
348C23×C4—531no—
349C23×C4—531no—
350C23×C4—534no—
351C23×C4—1061no—
352C23×C4—2122no—
353C23×C4—2122no—
354C23×C4C22×D5311yescharacteristic
355C22×C8—532no—
356C22×C8—538no—
357C22×C8—2122no—
358C22×C8—2122no—
359C22×C8C22×D5312yes—
360#117—2122no—
361#117—2122no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4—532no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—1061no—
368C2×C4⋊C4—2122no—
369C2×C4⋊C4C22×D5312yes—
370C2×C4⋊C4C22×D5311yescharacteristic
371C2×C4⋊C4C22×D5311yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—1062no—
374C2×C22⋊C4—1062no—
375C2×C22⋊C4—1061no—
376C2×C22⋊C4—1061no—
377C2×C22⋊C4—1061no—
378C2×C22⋊C4—1061no—
379C2×C22⋊C4—1061no—
380C2×C22⋊C4—2124no—
381C2×C42—1061no—
382C2×C42—1061no—
383#107—42no—
384#107—2122no—
385#106—2122no—
386#106—2122no—
387C22⋊C8—534no—
388C22⋊C8—538no—
389C22⋊C8C4×D5314yes—
390#95—21no—
391#95—1062no—
392#95—1062no—
393#95—1061no—
394#95—1061no—
395#95—1061no—
396C24—531no—
397C24—531no—
398C24—534no—
399C24—1062no—
400C24—1062no—
401C24—1062no—
402C24—1062no—
403C24—2121no—
404C24—2121no—
405C24D4×D5311yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—531no—
415C22×C4—531no—
416C22×C4—534no—
417C22×C4—532no—
418C22×C4—532no—
419C22×C4—532no—
420C22×C4—532no—
421C22×C4—534no—
422C22×C4—534no—
423C22×C4—1062no—
424C22×C4—1062no—
425C22×C4—1062no—
426C22×C4—1062no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1062no—
430C22×C4—2128no—
431C22×C4—2128no—
432C22×C4—2121no—
433C22×C4—2121no—
434C22×C4—2121no—
435C22×C4—2121no—
436C22×C4—2121no—
437C22×C4—2121no—
438C22×C4—2122no—
439C22×C4—2122no—
440C22×C4—2122no—
441C22×C4C2×C4×D5312yes—
442C22×C4D4×D5311yescharacteristic
443C22×C4Q8×D5312yes—
444C22×C4C23×D5311yescharacteristic
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—534no—
448C2×C8—1062no—
449C2×C8—1062no—
450C2×C8—2128no—
451C2×C8C2×C4×D5314yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—1062no—
458C4⋊C4—1062no—
459C4⋊C4—1062no—
460C4⋊C4—1062no—
461C4⋊C4—2122no—
462C22⋊C4—42no—
463C22⋊C4—2124no—
464C22⋊C4—2122no—
465C22⋊C4—2122no—
466C22⋊C4—2124no—
467C22⋊C4—2122no—
468C22⋊C4—2122no—
469C22⋊C4—2122no—
470C42—2122no—
471C42—2122no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—532no—
477C23—532no—
478C23—532no—
479C23—532no—
480C23—1068no—
481C23—1061no—
482C23—1061no—
483C23—1061no—
484C23—1061no—
485C23—1061no—
486C23—1061no—
487C23—1061no—
488C23—1061no—
489C23—1068no—
490C23—1061no—
491C23—1061no—
492C23—2124no—
493C23—2124no—
494C23—2124no—
495C23—2124no—
496C23—2124no—
497C23C4⋊C4×D5312yes—
498C23#1313323612yes—
499C23#1313324012yes—
500C23C2×D4×D5311yescharacteristic
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—84no—
515C2×C4—532no—
516C2×C4—532no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1064no—
522C2×C4—1064no—
523C2×C4—1064no—
524C2×C4—1064no—
525C2×C4—4244no—
526C2×C4—2124no—
527C2×C4—2124no—
528C2×C4—2124no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2122no—
538C2×C4C4⋊C4×D5312yes—
539C2×C4C22×C4×D5311yescharacteristic
540C2×C4C2×Q8×D5311yescharacteristic
541C2×C4#1313326212yes—
542C2×C4#1313326312yes—
543C8—24no—
544C8—1064no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—534no—
552C22—1062no—
553C22—1062no—
554C22—1062no—
555C22—1062no—
556C22—1062no—
557C22—1062no—
558C22—1062no—
559C22—1062no—
560C22—1062no—
561C22—21216no—
562C22—2121no—
563C22—2121no—
564C22—2121no—
565C22—2121no—
566C22#1681066014yes—
567C22C2×C4⋊C4×D5311yescharacteristic
568C22#1681118311yescharacteristic
569C22#1681118411yescharacteristic
570C22C2×#1313323611yescharacteristic
571C22C2×#1313324011yescharacteristic
572C22AGL(1,ℤ/8)×D5311yescharacteristic
573C22#137×D5311yescharacteristic
574C4—22no—
575C4—82no—
576C4—81no—
577C4—81no—
578C4—1062no—
579C4—1062no—
580C4—4242no—
581C4—4241no—
582C4—4241no—
583C4#1681109512yes—
584C4#1681124812yes—
585C2—24no—
586C2—1064no—
587C2—2122no—
588C2#2027732611yescharacteristic
589C2C2×#1681066011yescharacteristic
590C2#428×D5311yescharacteristic
591C2#2028547711yescharacteristic
592C2#2028548511yescharacteristic
593C2#2028549511yescharacteristic
594C2#2028549911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1852C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenter#1852D53
hypocenterC53#21941
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
4. #20274917C2semidirect
5. #20274911C2semidirect
6. #20274908C2semidirect
7. #20274907C2semidirect
8. #20274905C2semidirect
9. #20274741C2semidirect
10. #20274739C2semidirect
11. C53×#1852C2semidirect
18. C2×#13133151C22semidirect
19. C2×#13133148C22semidirect
20. C2×#13133148C22semidirect
23. C22⋊C8×D53C4semidirect
36. #16810386C22semidirect
37. #16810385C22semidirect
38. #16810385C22semidirect
40. C2×#107×C53C22semidirect
43. C53×#340C22semidirect
58. C2×C8×D53C2×C4semidirect
80. #13133129C2×C4semidirect
296. #1852D53semidirect
317. C2×#107C2×D53semidirect
340. #340C2×D53semidirect
341. C53#21941semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C8×D532
#202770972
C2×#168104802
#202749172
#202749112
#202749082
#202749072
#202749052
#202747412
#202747392
C53×#18522
#2194153