Finite Group Databasebeta

← #27422815

Subgroups — #27422815

Order 13568.

1188 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810773C211yesmaximal, characteristic
2C2×#16810680C211yesmaximal, characteristic, complemented
3C2×#16810525C211yesmaximal, characteristic
4#20276100C212yesmaximal, complemented
5#20276030C212yesmaximal, complemented
6#20275599C212yesmaximal, complemented
7#20275411C212yesmaximal, complemented
8#20275023C211yesmaximal, characteristic, complemented
9#20274902C211yesmaximal, characteristic, complemented
10#20274645C211yesmaximal, characteristic
11C53×#2855C211yesmaximal, characteristic, complemented
12C23×D212C2211yescharacteristic
13C22×#11719924C2212yes—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719921—21no—
16C22×#11719921—21no—
17#16810773—24no—
18#16810680—24no—
19C2×#13133150C2212yescomplemented
20C2×C4⋊#88322C2212yescomplemented
21C2×#13133141C2212yescomplemented
22C2×#13133137C2212yes—
23C2×#13133137C2212yes—
24C2×#13133137C2212yes—
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×#13133128C2212yescomplemented
28#16810525—21no—
29#16810525—21no—
30#16810525C2212yes—
31#16810525C2212yes—
32#16810385C2212yescomplemented
33#16810385C2212yescomplemented
34C2×C22⋊Q8×C53C2211yescharacteristic
35C22×C42×C53C2211yescharacteristic
36C53×#391C2211yescharacteristic, complemented
37C53×#391C2211yescharacteristic, complemented
38C53×#389C2211yescharacteristic, complemented
39C53×#389C2211yescharacteristic, complemented
40C53×#387—21no—
41C53×#387—21no—
42C53×#385—21no—
43C53×#381—21no—
44C53×#380C2211yescharacteristic
45C24×D53—21no—
46C22×D212—21no—
47C22×D212—24no—
48C22×D212—22no—
49C22×D212—22no—
50C22×D212—42no—
51C22×D212C2312yes—
52C23×Dic53—21no—
53C2×#11719924—24no—
54C2×#11719924—24no—
55C2×#11719924—42no—
56C22×#88322C2312yes—
57C22×#88322C2312yes—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53—42no—
60C2×C4⋊Dic53C2312yes—
61C2×#11719921—22no—
62C2×#11719921—22no—
63C2×#11719921—22no—
64C2×#11719921—22no—
65C2×#11719921—44no—
66C2×#11719921—44no—
67#13133150—42no—
68C4⋊#88322—24no—
69#13133141—42no—
70#13133137—24no—
71#13133137—24no—
72#13133137—42no—
73C22⋊#88322—22no—
74C22⋊#88322D414yes—
75#13133128—24no—
76C22×Q8×C53C2311yescharacteristic
77C23×C4×C53D411yescharacteristic
78C23×C4×C53D411yescharacteristic
79C23×C4×C53C2311yescharacteristic
80C22⋊Q8×C53—22no—
81C22⋊Q8×C53—22no—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C4⋊C4×C53C2311yescharacteristic
88C2×C22⋊C4×C53—21no—
89C2×C22⋊C4×C53—21no—
90C2×C42×C53—22no—
91C2×C42×C53—22no—
92C2×C42×C53D411yescharacteristic
93C2×C42×C53D411yescharacteristic
94C2×C42×C53C2311yescharacteristic
95C2×C42×C53C2311yescharacteristic
96#95×C53—21no—
97#95×C53—21no—
98#95×C53—21no—
99#95×C53—21no—
100#95×C53—21no—
101#95×C53—21no—
102#95×C53—21no—
103C24×C53C2×D411yescharacteristic
104C23×D53—24no—
105C23×D53—22no—
106C23×D53—42no—
107C23×D53—42no—
108C23×D53—42no—
109C23×D53—42no—
110C23×D53—42no—
111C23×D53—42no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—22no—
115C2×Q8×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—41no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53#4112yes—
135C22×C4×C53C2411yescharacteristic
136C22×Dic53—22no—
137C22×Dic53—42no—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C2×D212—22no—
141C2×D212—22no—
142C2×D212—22no—
143C2×D212—22no—
144C2×D212—24no—
145C2×D212—22no—
146C2×D212—48no—
147C2×D212—48no—
148C2×D212—42no—
149C2×D212—42no—
150C2×D212#4114yes—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—42no—
156C4⋊C4×C53—42no—
157C4⋊C4×C53—42no—
158C22⋊C4×C53—42no—
159C22⋊C4×C53—42no—
160C42×C53—22no—
161C42×C53—22no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53—21no—
166#11719924—48no—
167#11719921—44no—
168#11719921—44no—
169#11719921—44no—
170#11719921—44no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—42no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322C2×D414yes—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53#12112yes—
185C23×C53AGL(1,ℤ/8)12yes—
186C23×C53#13712yes—
187C23×C53C22×D411yescharacteristic
188C22×D53—24no—
189C22×D53—48no—
190C22×D53—48no—
191C22×D53—48no—
192C22×D53—42no—
193C22×D53—42no—
194C22×D53—42no—
195C22×D53—42no—
196C22×D53—42no—
197C22×D53—42no—
198C22×D53—84no—
199C22×D53—84no—
200C22×D53—84no—
201C22×D53—84no—
202Q8×C53—42no—
203Q8×C53—42no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—24no—
209C2×C4×C53—24no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—44no—
220C2×C4×C53—44no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53#12112yes—
234C2×C4×C53AGL(1,ℤ/8)12yes—
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C2×#4112yes—
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53H(5,𝔽2)12yes—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—84no—
243D212—24no—
244D212—24no—
245D212—48no—
246D212—42no—
247D212—42no—
248#88322—24no—
249#88322—42no—
250#24507—531nomaximal, Sylow
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53#46814yes—
258C22×C53C2×#12111yescharacteristic
259C22×C53#52511yescharacteristic
260C22×C53#53511yescharacteristic
261C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C53C2×#13711yescharacteristic
264C22×C53#57511yescharacteristic
265C2×D53—44no—
266C2×D53—44no—
267C2×D53—44no—
268C2×D53—816no—
269C2×D53—82no—
270C2×D53—82no—
271C2×D53—82no—
272C2×D53—82no—
273C4×C53—22no—
274C4×C53—22no—
275C4×C53—41no—
276C4×C53—41no—
277C4×C53—41no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53#54612yes—
283C4×C53#55112yes—
284Dic53—82no—
285C2×#450—531no—
286C2×#432—531no—
287#2855D5311yescharacteristic, complemented
288C2×#390—531no—
289#2067—532no—
290#2055—532no—
291#1973—532no—
292#1936—532no—
293#1872—531no—
294#1852—531no—
295#1786—531no—
296C2×C53—24no—
297C2×C53#241111yescharacteristic
298C2×C53C2×#46811yescharacteristic
299C2×C53#305511yescharacteristic
300C2×C53#311011yescharacteristic
301C2×C53#311411yescharacteristic
302C2×C53#321511yescharacteristic
303C2×C53#321711yescharacteristic
304D53—84no—
305C23×D4—531no—
306C22×SD16—532no—
307C2×C22⋊Q8C2×D5311yescharacteristic
308C22×C4⋊C4—531no—
309C22×C22⋊C4—1061no—
310C22×C22⋊C4—1061no—
311C22×C42C2×D5311yescharacteristic
312#450—1064no—
313#432—1064no—
314C2×#107—532no—
315C2×C4⋊C8—532no—
316C2×#103—532no—
317C2×#102—532no—
318C2×#102—532no—
319C2×#102—532no—
320C2×C22⋊C8—531no—
321C2×C22⋊C8—531no—
322C2×#97—532no—
323#391C2×D5311yescharacteristic, complemented
324#391C2×D5311yescharacteristic, complemented
325#390—532no—
326#390—532no—
327#390—1061no—
328#390—1061no—
329#389C2×D5311yescharacteristic, complemented
330#389C2×D5311yescharacteristic, complemented
331#387—21no—
332#387—21no—
333#385—21no—
334#381—21no—
335#380C2×D5311yescharacteristic
336#340—532no—
337#340—532no—
338C53#2450711yescharacteristic, complemented, Sylow
339C25—1061no—
340C22×Q8C22×D5311yescharacteristic
341C22×D4—532no—
342C22×D4—1064no—
343C22×D4—1062no—
344C22×D4—1062no—
345C22×D4—1061no—
346C22×D4—2122no—
347C23×C4—1061no—
348C23×C4C22⋊D5311yescharacteristic
349C23×C4C22⋊D5311yescharacteristic
350C23×C4C22×D5311yescharacteristic
351C2×SD16—1064no—
352C2×SD16—1064no—
353C2×SD16—2122no—
354C22×C8—532no—
355C22×C8—532no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—532no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—2122no—
364C2×C4⋊C4C22×D5311yescharacteristic
365C2×C4⋊C4C22×D5311yescharacteristic
366C2×C4⋊C4C22×D5311yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—1062no—
370C2×C22⋊C4—1062no—
371C2×C22⋊C4—1062no—
372C2×C22⋊C4—1062no—
373C2×C22⋊C4—2124no—
374C2×C22⋊C4—2124no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42C22⋊D5311yescharacteristic
378C2×C42C22⋊D5311yescharacteristic
379C2×C42C22×D5311yescharacteristic
380C2×C42C22×D5311yescharacteristic
381#107—2122no—
382C4⋊C8—1064no—
383#103—2122no—
384#102—1064no—
385#102—1064no—
386#102—2122no—
387C22⋊C8—534no—
388C22⋊C8—1062no—
389#97—1064no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397C24—1064no—
398C24—1062no—
399C24—2122no—
400C24—2122no—
401C24—2122no—
402C24—2122no—
403C24—2122no—
404C24—2122no—
405C24C2×C22⋊D5311yescharacteristic
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×D4—534no—
411C2×D4—1064no—
412C2×D4—1062no—
413C2×D4—1062no—
414C2×D4—1062no—
415C2×D4—1062no—
416C2×D4—1062no—
417C2×D4—2128no—
418C2×D4—2128no—
419C2×D4—2122no—
420C2×D4—2122no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—41no—
433C22×C4—1062no—
434C22×C4—2122no—
435C22×C4—2122no—
436C22×C4—2122no—
437C22×C4D4×D5312yes—
438C22×C4#1171994712yes—
439C22×C4C2×C22⋊D5311yescharacteristic
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C2×C22⋊D5311yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C23×D5311yescharacteristic
445SD16—2128no—
446C2×C8—534no—
447C2×C8—1062no—
448C2×C8—1062no—
449C2×C8—1062no—
450C2×C8—1062no—
451C2×C8—1062no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—1062no—
460C4⋊C4—1062no—
461C4⋊C4—2122no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—2124no—
465C22⋊C4—2124no—
466C22⋊C4—2124no—
467C22⋊C4—2124no—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—1064no—
479C23—4244no—
480C23—4244no—
481C23—4244no—
482C23—4244no—
483C23—2128no—
484C23—2128no—
485C23—2128no—
486C23—2122no—
487C23—2122no—
488C23—2122no—
489C23—2122no—
490C23—2122no—
491C23—2122no—
492C23#1313318912yes—
493C23#1313324212yes—
494C23#1313324812yes—
495C23C22×C22⋊D5311yescharacteristic
496Q8—42no—
497Q8—42no—
498D4—1064no—
499D4—1064no—
500D4—2128no—
501D4—2122no—
502D4—2122no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—4244no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4#1313318912yes—
537C2×C4#1313324212yes—
538C2×C4C2×D4×D5311yescharacteristic
539C2×C4#41×D5312yes—
540C2×C4C2×#1171994711yescharacteristic
541C2×C4#1313326312yes—
542C8—1064no—
543C8—2122no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—42416no—
551C22—4242no—
552C22—4242no—
553C22—4242no—
554C22—4242no—
555C22—2124no—
556C22—2124no—
557C22—2124no—
558C22#1681087114yes—
559C22C2×#1313318911yescharacteristic
560C22#1681113011yescharacteristic
561C22#1681118711yescharacteristic
562C22C2×#1313324211yescharacteristic
563C22C2×#1313324211yescharacteristic
564C22C2×#1313324811yescharacteristic
565C22#1681142111yescharacteristic
566C4—22no—
567C4—22no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—4242no—
576C4#1681126512yes—
577C4#1681129612yes—
578C2—24no—
579C2—4244no—
580C2#2027810111yescharacteristic
581C2C2×#1681087111yescharacteristic
582C2#2028433611yescharacteristic
583C2#2028481011yescharacteristic
584C2#2028484911yescharacteristic
585C2#2028571411yescharacteristic
586C2#2028573011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2855C2
socleC23×C53C22×D4
prime radicalC23×D212C22
hypercenter#2855D53
hypocenterC53#24507
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
2. C2×#16810680C2semidirect
4. #20276100C2semidirect
5. #20276030C2semidirect
6. #20275599C2semidirect
7. #20275411C2semidirect
8. #20275023C2semidirect
9. #20274902C2semidirect
11. C53×#2855C2semidirect
19. C2×#13133150C22semidirect
20. C2×C4⋊#88322C22semidirect
21. C2×#13133141C22semidirect
27. C2×#13133128C22semidirect
32. #16810385C22semidirect
33. #16810385C22semidirect
36. C53×#391C22semidirect
37. C53×#391C22semidirect
38. C53×#389C22semidirect
39. C53×#389C22semidirect
287. #2855D53semidirect
323. #391C2×D53semidirect
324. #391C2×D53semidirect
329. #389C2×D53semidirect
330. #389C2×D53semidirect
338. C53#24507semidirect

Maximal subgroups

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

subgroupindex
C2×#168107732
C2×#168106802
C2×#168105252
#202761002
#202760302
#202755992
#202754112
#202750232
#202749022
#202746452
C53×#28552
#2450753