Finite Group Databasebeta

← #27395257

Subgroups — #27395257

Order 13568.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810774C211yesmaximal, characteristic
2#20277243C211yesmaximal, characteristic, complemented
3C2×#16810526C211yesmaximal, characteristic
4#20275570C212yesmaximal, complemented
5#20275395C212yesmaximal, complemented
6#20275345C212yesmaximal, complemented
7#20275067C212yesmaximal, complemented
8#20275020C211yesmaximal, characteristic, complemented
9#20274907C211yesmaximal, characteristic, complemented
10C22⋊C8×Dic53C211yesmaximal, characteristic, complemented
11C53×#1786C211yesmaximal, characteristic
12C22×#11719923C2212yes—
13C2×#13133196C2211yescharacteristic
14C2×#13133182C2211yescharacteristic, complemented
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C22⋊Dic53—21no—
17C22×C22⋊Dic53—21no—
18C22×C4×Dic53C2211yescharacteristic
19#16810774C418yescomplemented
20C2×#13133148C2212yescomplemented
21C2×#13133142C2212yescomplemented
22C2×#13133138C2212yes—
23C2×C22⋊#88322C2211yescharacteristic
24C2×C8×Dic53C2212yescomplemented
25#16810528C2211yescharacteristic, complemented
26#16810526—21no—
27#16810526—21no—
28#16810526C2212yes—
29#16810526C2212yes—
30#16810519C2211yescharacteristic, complemented
31#16810505—21no—
32#16810495—21no—
33C4⋊C4×Dic53C2211yescharacteristic, complemented
34#16810474—21no—
35C22⋊C4×Dic53—21no—
36#16810386C2212yescomplemented
37#16810385C2212yescomplemented
38#16810377C2212yescomplemented
39C23×D4×C53C2211yescharacteristic
40C2×#102×C53C2212yes—
41C2×#102×C53C2212yes—
42C2×C22⋊C8×C53C2211yescharacteristic
43C53×#380C2211yescharacteristic
44C22×Dic106C2311yescharacteristic
45C23×Dic53—21no—
46C23×Dic53D411yescharacteristic
47C23×Dic53D411yescharacteristic
48C2×#11719923—42no—
49C2×#11719923—42no—
50C2×#11719923C2×C418yescomplemented
51C22×#88322C2312yes—
52#13133196—22no—
53#13133196C2×C414yes—
54#13133182—42no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—42no—
57C2×C4⋊Dic53C2312yes—
58C2×C4⋊Dic53C2311yescharacteristic
59C2×#11719919—21no—
60C2×#11719919—21no—
61C2×#11719919—21no—
62C2×C22⋊Dic53—21no—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53—22no—
66C2×C22⋊Dic53—22no—
67C2×C22⋊Dic53—44no—
68C2×C22⋊Dic53—44no—
69C2×C4×Dic53—21no—
70C2×C4×Dic53—21no—
71C2×C4×Dic53—42no—
72C2×C4×Dic53D411yescharacteristic
73C2×C4×Dic53D411yescharacteristic
74C2×C4×Dic53C2311yescharacteristic
75C2×C4×Dic53C2311yescharacteristic
76#13133148—42no—
77#13133142—42no—
78#13133138C2×C418yescomplemented
79C22⋊#88322C2×C414yes—
80C8×Dic53—42no—
81#13133124—21no—
82#13133124—21no—
83#13133124—21no—
84#13133124—21no—
85C25×C53—21no—
86C22×D4×C53—21no—
87C22×D4×C53—22no—
88C22×D4×C53—42no—
89C22×D4×C53—42no—
90C22×D4×C53C2×C414yes—
91C22×D4×C53C2312yes—
92C23×C4×C53C2311yescharacteristic
93C22×C8×C53C2312yes—
94C2×C4⋊C4×C53C2311yescharacteristic
95C2×C4⋊C4×C53C2311yescharacteristic
96C2×C22⋊C4×C53—21no—
97#102×C53—42no—
98#102×C53—42no—
99C22⋊C8×C53—22no—
100#95×C53—21no—
101C24×C53—24no—
102C24×C53—22no—
103C24×C53—42no—
104C24×C53—42no—
105C24×C53—42no—
106C24×C53—42no—
107C24×C53—42no—
108C24×C53—42no—
109C24×C53C2×D411yescharacteristic
110C2×D4×C53—24no—
111C2×D4×C53—22no—
112C2×D4×C53—22no—
113C2×D4×C53—22no—
114C2×D4×C53—22no—
115C2×D4×C53—22no—
116C2×D4×C53—48no—
117C2×D4×C53—42no—
118C2×D4×C53—42no—
119C2×D4×C53—48no—
120C2×D4×C53C22×C414yes—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53C22×C412yes—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53#4112yes—
128C22×C4×C53C2411yescharacteristic
129C22×Dic53—22no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—41no—
135C22×Dic53—41no—
136C22×Dic53—41no—
137C22×Dic53—42no—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C22×Dic53SD1614yescomplemented
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C2×Dic106—22no—
146C2×Dic106—22no—
147C2×Dic106—21no—
148C2×Dic106C22×C412yes—
149C2×C8×C53—22no—
150C2×C8×C53—22no—
151C2×C8×C53—22no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C22⋊C4×C53—42no—
155C22⋊Dic53—24no—
156C22⋊Dic53—44no—
157C22⋊Dic53—44no—
158C22⋊Dic53—44no—
159C22⋊Dic53—44no—
160#11719923—48no—
161C4⋊Dic53—21no—
162C4⋊Dic53—22no—
163C4⋊Dic53—22no—
164C4⋊Dic53—42no—
165C4⋊Dic53C22×C412yes—
166#11719919—24no—
167#11719919—42no—
168#11719919—42no—
169C4×Dic53—22no—
170C4×Dic53—22no—
171C4×Dic53—41no—
172C4×Dic53—41no—
173C4×Dic53—42no—
174C4×Dic53—42no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322C22×C414yes—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—24no—
183C23×C53—48no—
184C23×C53—48no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53—48no—
192C23×C53—84no—
193C23×C53—84no—
194C23×C53—84no—
195C23×C53—84no—
196C23×C53C4×D412yes—
197C23×C53C2×SD1612yes—
198C23×C53#13512yes—
199C23×C53C22×D411yescharacteristic
200D4×C53—24no—
201D4×C53—24no—
202D4×C53—48no—
203D4×C53—42no—
204D4×C53—42no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—24no—
208C2×C4×C53—24no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53C4×D412yes—
216C2×C4×C53C23×C411yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C53H(5,𝔽2)12yes—
219C2×C4×C53H(5,𝔽2)12yes—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—22no—
227C2×Dic53—22no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—44no—
232C2×Dic53—44no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—84no—
241C2×Dic53C2×SD1612yes—
242C2×Dic53C2×SD1612yes—
243C2×Dic53AGL(1,ℤ/8)12yes—
244Dic106—24no—
245Dic106—42no—
246C8×C53—42no—
247#88322—24no—
248#22043—531nomaximal, Sylow
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—44no—
256C22×C53—44no—
257C22×C53—44no—
258C22×C53—816no—
259C22×C53—82no—
260C22×C53—82no—
261C22×C53—82no—
262C22×C53—82no—
263C22×C53C4×SD1614yes—
264C22×C53C2×C4×D411yescharacteristic
265C22×C53#53411yescharacteristic
266C22×C53#53511yescharacteristic
267C22×C53C22×SD1611yescharacteristic
268C22×C53C2×#13511yescharacteristic
269C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C53#57511yescharacteristic
271C4×C53—22no—
272C4×C53—81no—
273C4×C53—81no—
274C4×C53#51812yes—
275C4×C53#53812yes—
276Dic53—22no—
277Dic53—22no—
278Dic53—22no—
279Dic53—42no—
280Dic53—41no—
281Dic53—42no—
282Dic53—82no—
283C2×#450—531no—
284C4×C22⋊Q8—531no—
285C2×#390—531no—
286#1968—532no—
287#1934—532no—
288#1926—532no—
289#1880—532no—
290#1872—531no—
291#1853—531no—
292#1786D5311yescharacteristic
293C4×C22⋊C8—531no—
294C2×C53—24no—
295C2×C53—84no—
296C2×C53#232111yescharacteristic
297C2×C53C2×C4×SD1611yescharacteristic
298C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
299C2×C53#318711yescharacteristic
300C2×C53#318811yescharacteristic
301C2×C53#320811yescharacteristic
302C2×C53#321111yescharacteristic
303C23×D4C2×D5311yescharacteristic
304C22×SD16—532no—
305C2×C22⋊Q8—531no—
306C2×C4×Q8—531no—
307C22×C4⋊C4—531no—
308C22×C22⋊C4—1061no—
309C22×C22⋊C4—1061no—
310C22×C42—531no—
311#450—538no—
312C2×#106—532no—
313C2×#103—532no—
314C2×#102—532no—
315C2×#102C2×D5312yes—
316C2×#102C2×D5312yes—
317C2×C22⋊C8—531no—
318C2×C22⋊C8C2×D5311yescharacteristic
319C2×C4×C8—532no—
320#391—531no—
321#390—532no—
322#390—532no—
323#390—1061no—
324#390—1061no—
325#389—531no—
326#387—1061no—
327#385—1061no—
328#380C2×D5311yescharacteristic
329C4×C4⋊C4—531no—
330C4×C4⋊C4—1061no—
331C4×C22⋊C4—1061no—
332#340—532no—
333#340—532no—
334#336—532no—
335C53#2204311yescharacteristic, complemented, Sylow
336C25—21no—
337C22×Q8—531no—
338C22×D4—21no—
339C22×D4—22no—
340C22×D4—42no—
341C22×D4—42no—
342C22×D4C4×D5314yes—
343C22×D4C22×D5312yes—
344C23×C4—531no—
345C23×C4—531no—
346C23×C4—1061no—
347C23×C4C22×D5311yescharacteristic
348C2×SD16—538no—
349C2×SD16—2122no—
350C2×SD16—2122no—
351C22×C8—532no—
352C22×C8C22×D5312yes—
353C22⋊Q8—534no—
354C22⋊Q8—1062no—
355C4×Q8—2122no—
356C2×C4⋊C4—531no—
357C2×C4⋊C4—532no—
358C2×C4⋊C4—1061no—
359C2×C4⋊C4—1061no—
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4—2122no—
363C2×C4⋊C4C22×D5311yescharacteristic
364C2×C4⋊C4C22×D5311yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—1061no—
367C2×C22⋊C4—1062no—
368C2×C22⋊C4—1062no—
369C2×C22⋊C4—1062no—
370C2×C22⋊C4—1062no—
371C2×C22⋊C4—2124no—
372C2×C22⋊C4—2124no—
373C2×C42—531no—
374C2×C42—531no—
375C2×C42—531no—
376C2×C42—531no—
377C2×C42—1061no—
378C2×C42—1061no—
379C2×C42—2122no—
380#106—2122no—
381#103—2122no—
382#102—42no—
383#102—42no—
384#102—538no—
385C22⋊C8—22no—
386C22⋊C8—534no—
387C4×C8—2122no—
388#95—21no—
389#95—1061no—
390#95—1061no—
391#95—1061no—
392#95—1061no—
393C24—24no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24D4×D5311yescharacteristic
402C2×Q8—532no—
403C2×Q8—1061no—
404C2×Q8—1062no—
405C2×Q8—1062no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4C2×C4×D5314yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—531no—
422C22×C4—531no—
423C22×C4—531no—
424C22×C4—534no—
425C22×C4—531no—
426C22×C4—1062no—
427C22×C4—1062no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1062no—
431C22×C4—2121no—
432C22×C4—2121no—
433C22×C4—2121no—
434C22×C4—2122no—
435C22×C4—2122no—
436C22×C4—2122no—
437C22×C4C2×C4×D5312yes—
438C22×C4D4×D5311yescharacteristic
439C22×C4#1171994712yes—
440C22×C4C23×D5311yescharacteristic
441SD16—2128no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—534no—
446C2×C8—1062no—
447C2×C8—1062no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—532no—
451C4⋊C4—1064no—
452C4⋊C4—1061no—
453C4⋊C4—1062no—
454C4⋊C4—1062no—
455C4⋊C4—2122no—
456C4⋊C4—2122no—
457C4⋊C4—2122no—
458C22⋊C4—42no—
459C22⋊C4—1064no—
460C22⋊C4—2124no—
461C22⋊C4—2124no—
462C22⋊C4—2124no—
463C22⋊C4—2124no—
464C42—1062no—
465C42—1062no—
466C42—2121no—
467C42—2121no—
468C42—2122no—
469C42—2122no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—24no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—48no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23#1313317712yes—
489C23SD16×D5312yes—
490C23#1313323612yes—
491C23C2×D4×D5311yescharacteristic
492Q8—1064no—
493Q8—2122no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—532no—
510C2×C4—532no—
511C2×C4—532no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—4244no—
524C2×C4—2124no—
525C2×C4—2124no—
526C2×C4—2122no—
527C2×C4—2122no—
528C2×C4—2122no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4#1313317712yes—
534C2×C4C22×C4×D5311yescharacteristic
535C2×C4C2×#1171994711yescharacteristic
536C2×C4#1313326212yes—
537C2×C4#1313326212yes—
538C8—42no—
539C8—1064no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22#1681071414yes—
555C22C2×#1313317711yescharacteristic
556C22#1681118311yescharacteristic
557C22#1681118811yescharacteristic
558C22C2×SD16×D5311yescharacteristic
559C22C2×#1313323611yescharacteristic
560C22AGL(1,ℤ/8)×D5311yescharacteristic
561C22#1681141911yescharacteristic
562C4—22no—
563C4—81no—
564C4—81no—
565C4—1062no—
566C4—1062no—
567C4—1062no—
568C4—4242no—
569C4—2121no—
570C4—2122no—
571C4—2122no—
572C4#1681109612yes—
573C4#1681120812yes—
574C2—24no—
575C2—84no—
576C2#2027737011yescharacteristic
577C2C2×#1681071411yescharacteristic
578C2#2028351911yescharacteristic
579C2#2028546911yescharacteristic
580C2#2028547711yescharacteristic
581C2#2028565411yescharacteristic
582C2#2028567711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1786C2
socleC23×C53C22×D4
prime radicalC23×D4×C53C22
cosocleC22×C4×C53C24
hypercenter#1786D53
hypocenterC53#22043
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. #20277243C2semidirect
4. #20275570C2semidirect
5. #20275395C2semidirect
6. #20275345C2semidirect
7. #20275067C2semidirect
8. #20275020C2semidirect
9. #20274907C2semidirect
10. C22⋊C8×Dic53C2semidirect
14. C2×#13133182C22semidirect
19. #16810774C4semidirect
20. C2×#13133148C22semidirect
21. C2×#13133142C22semidirect
24. C2×C8×Dic53C22semidirect
25. #16810528C22semidirect
30. #16810519C22semidirect
33. C4⋊C4×Dic53C22semidirect
36. #16810386C22semidirect
37. #16810385C22semidirect
38. #16810377C22semidirect
50. C2×#11719923C2×C4semidirect
78. #13133138C2×C4semidirect
140. C22×Dic53SD16semidirect
335. C53#22043semidirect

Maximal subgroups

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

subgroupindex
C2×#168107742
#202772432
C2×#168105262
#202755702
#202753952
#202753452
#202750672
#202750202
#202749072
C22⋊C8×Dic532
C53×#17862
#2204353