Finite Group Databasebeta

← #27357964

Subgroups — #27357964

Order 13568.

972 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285487C211yesmaximal, characteristic, complemented
2#20285472C211yesmaximal, characteristic
3#20280528C211yesmaximal, characteristic, complemented
4#20273746C214yesmaximal, complemented
5#20273678C214yesmaximal, complemented
6#20273287C211yesmaximal, characteristic, complemented
7#20273283C211yesmaximal, characteristic
8C2×#16810349C211yesmaximal, characteristic, complemented
9C2×C53×#539C211yesmaximal, characteristic
10C2×#13133196C2211yescharacteristic
11#16810958—21no—
12#16810956C2212yescomplemented
13#16810953—21no—
14#16810939C2212yes—
15#16810862—22no—
16#16810850—22no—
17#16810779—21no—
18#16810778C2212yescomplemented
19#16810774—21no—
20#16810761C2212yes—
21C2×C22⋊#88322C2211yescharacteristic
22C2×C22⋊#88322C2211yescharacteristic
23#16810562C2212yescomplemented
24#16810556—22no—
25#16810543C2212yescomplemented
26#16810496—21no—
27#16810486—21no—
28#16810483C2211yescharacteristic
29#16810360C2214yescomplemented
30#16810357C2214yes—
31#16810349C2214yescomplemented
32C53×#539—22no—
33C53×#539—22no—
34C53×#539—22no—
35C53×#539C2214yes—
36C2×#126×C53—21no—
37C2×#125×C53—21no—
38C2×#123×C53—21no—
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53—21no—
41C2×C22⋊Q8×C53C2211yescharacteristic
42C2×#121×C53C2211yescharacteristic
43C2×C4×D4×C53—21no—
44C22×C4⋊C4×C53C2211yescharacteristic
45C22×Dic106C2311yescharacteristic
46C2×#11719925—22no—
47C2×#11719923—22no—
48C22×#88322—21no—
49C22×#88322—21no—
50#13133196—21no—
51#13133196—21no—
52#13133196—21no—
53#13133196—21no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53C2311yescharacteristic
56C2×#11719919—21no—
57C2×#11719919—22no—
58C2×C22⋊Dic53—21no—
59C2×C22⋊Dic53—21no—
60#13133150—22no—
61#13133148—22no—
62#13133141—22no—
63#13133140—22no—
64#13133140—22no—
65#13133138—22no—
66C22⋊#88322—21no—
67C22⋊#88322—21no—
68C22⋊#88322C2312yes—
69C22⋊#88322C2312yes—
70#13133124—22no—
71#13133124—22no—
72#13133124—22no—
73#13133124C2312yes—
74C22×Q8×C53D411yescharacteristic
75C22×D4×C53—21no—
76C22×D4×C53D411yescharacteristic
77C23×C4×C53—21no—
78C23×C4×C53D411yescharacteristic
79C23×C4×C53D411yescharacteristic
80C23×C4×C53C2311yescharacteristic
81#126×C53—24no—
82#126×C53—24no—
83#125×C53—24no—
84#125×C53—24no—
85#123×C53—24no—
86#123×C53—24no—
87#123×C53—24no—
88#123×C53—24no—
89C22⋊Q8×C53—24no—
90C22⋊Q8×C53—24no—
91C22⋊Q8×C53—21no—
92C22⋊Q8×C53—21no—
93C22⋊Q8×C53D412yes—
94C22⋊Q8×C53C2312yes—
95#121×C53—21no—
96#121×C53—21no—
97#121×C53D412yes—
98#121×C53C2312yes—
99C4×D4×C53—24no—
100C4×D4×C53—24no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53D412yes—
112C2×C4⋊C4×C53C2312yes—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C42×C53—21no—
119C24×C53—21no—
120C24×C53C2×D411yescharacteristic
121C2×Q8×C53—21no—
122C2×Q8×C53—81no—
123C2×Q8×C53C2×D412yes—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—21no—
127C2×D4×C53—42no—
128C2×D4×C53—42no—
129C2×D4×C53—81no—
130C2×D4×C53C2×D412yes—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—24no—
140C22×C4×C53—21no—
141C22×C4×C53—81no—
142C22×C4×C53—81no—
143C22×C4×C53—82no—
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—22no—
154C22×Dic53—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—81no—
158C4⋊C4×C53—24no—
159C4⋊C4×C53—24no—
160C4⋊C4×C53—24no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53C2×D412yes—
170C4⋊C4×C53C2×D412yes—
171C22⋊C4×C53—24no—
172C22⋊C4×C53—24no—
173C22⋊C4×C53—24no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C42×C53—24no—
179C22⋊Dic53—41no—
180C22⋊Dic53—41no—
181C22⋊Dic53—42no—
182#11719925—82no—
183#11719923—82no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—81no—
187#11719919—41no—
188#11719919—41no—
189#11719919—82no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—81no—
193C2×#88322—81no—
194C23×C53—22no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53#12011yescharacteristic
201C23×C53#12011yescharacteristic
202C23×C53#13512yes—
203C23×C53#13712yes—
204C23×C53C22×D411yescharacteristic
205Q8×C53—42no—
206Q8×C53—81no—
207D4×C53—42no—
208D4×C53—42no—
209D4×C53—42no—
210D4×C53—44no—
211D4×C53—81no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—81no—
233C2×C4×C53—84no—
234C2×C4×C53#12011yescharacteristic
235C2×C4×C53#12011yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—42no—
245Dic106—42no—
246Dic106—42no—
247#88322—42no—
248#88322—42no—
249#18289—531nomaximal, Sylow
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—21no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—84no—
262C22×C53#45212yes—
263C22×C53#45212yes—
264C22×C53C2×#12011yescharacteristic
265C22×C53C2×#13511yescharacteristic
266C22×C53C2×#13711yescharacteristic
267C4×C53—21no—
268C4×C53—21no—
269C4×C53—21no—
270C4×C53—21no—
271C4×C53—42no—
272C4×C53—42no—
273C4×C53—42no—
274C4×C53—42no—
275C4×C53—42no—
276Dic53—82no—
277Dic53—81no—
278Dic53—81no—
279C2×#539D5311yescharacteristic
280#3189—531no—
281#3187—531no—
282#2627—531no—
283#1615—534no—
284#1603—534no—
285#1520—531no—
286#1519—531no—
287C2×#324—531no—
288C2×C53—21no—
289C2×C53—22no—
290C2×C53—21no—
291C2×C53—22no—
292C2×C53—82no—
293C2×C53C2×#45211yescharacteristic
294C2×C53#302011yescharacteristic
295C2×C53#302611yescharacteristic
296#539—22no—
297#539—22no—
298#539—22no—
299#539C2×D5314yes—
300C2×#126—21no—
301C2×#125—21no—
302C2×#123—21no—
303C2×#123—21no—
304C2×C22⋊Q8—21no—
305C2×C22⋊Q8—531no—
306C2×C22⋊Q8C2×D5311yescharacteristic
307C2×#121C2×D5311yescharacteristic
308C2×C4×D4—21no—
309C22×C4⋊C4C2×D5311yescharacteristic
310#484—532no—
311#484—1061no—
312#483—1061no—
313#481—532no—
314#467—1062no—
315#465—1062no—
316#451—532no—
317#451—1061no—
318#450—1061no—
319#448—532no—
320C2×C22⋊C8—531no—
321C2×C22⋊C8—531no—
322#400—532no—
323#398—1062no—
324#395—532no—
325#385—1061no—
326#381—1061no—
327#380—531no—
328#328—534no—
329#327—534no—
330#324—534no—
331C53#1828911yescharacteristic, complemented, Sylow
332C22×Q8—531no—
333C22×Q8C22⋊D5311yescharacteristic
334C22×D4—21no—
335C22×D4C22⋊D5311yescharacteristic
336C23×C4—21no—
337C23×C4C22⋊D5311yescharacteristic
338C23×C4C22⋊D5311yescharacteristic
339C23×C4C22×D5311yescharacteristic
340C2×Q16—1062no—
341C2×SD16—1062no—
342C22×C8—1061no—
343C22×C8—1061no—
344#126—24no—
345#126—24no—
346#125—24no—
347#125—24no—
348#123—24no—
349#123—24no—
350#123—24no—
351#123—24no—
352C22⋊Q8—24no—
353C22⋊Q8—24no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—1061no—
357C22⋊Q8—1061no—
358C22⋊Q8—1061no—
359C22⋊Q8—1061no—
360C22⋊Q8C22⋊D5312yes—
361C22⋊Q8C22×D5312yes—
362#121—21no—
363#121—21no—
364#121C22⋊D5312yes—
365#121C22×D5312yes—
366C4×D4—24no—
367C4×D4—24no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4C22⋊D5312yes—
381C2×C4⋊C4C22⋊D5311yescharacteristic
382C2×C4⋊C4C22⋊D5311yescharacteristic
383C2×C4⋊C4C22×D5312yes—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C42—21no—
392#107—1062no—
393#106—1062no—
394#103—1062no—
395#103—1062no—
396#103—1062no—
397#102—1062no—
398C22⋊C8—532no—
399C22⋊C8—532no—
400C22⋊C8—1061no—
401C22⋊C8—1061no—
402#95—532no—
403#95—1062no—
404#95—1062no—
405#95—1062no—
406C24—21no—
407C24C2×C22⋊D5311yescharacteristic
408C2×Q8—21no—
409C2×Q8—81no—
410C2×Q8—1061no—
411C2×Q8—1061no—
412C2×Q8—4241no—
413C2×Q8C2×C22⋊D5312yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4C2×C22⋊D5312yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—82no—
434C22×C4—1062no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4C2×C22⋊D5312yes—
438C22×C4C2×C22⋊D5312yes—
439C22×C4C2×C22⋊D5311yescharacteristic
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C2×C22⋊D5311yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C23×D5311yescharacteristic
445Q16—4242no—
446SD16—4242no—
447C2×C8—1062no—
448C2×C8—1062no—
449C2×C8—4241no—
450C2×C8—4241no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—4242no—
465C4⋊C4—4241no—
466C4⋊C4—2121no—
467C4⋊C4—2121no—
468C4⋊C4C2×C22⋊D5312yes—
469C4⋊C4C2×C22⋊D5312yes—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—2121no—
478C22⋊C4—2121no—
479C22⋊C4—2122no—
480C42—24no—
481C23—22no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23#1313318611yescharacteristic
488C23#1313318611yescharacteristic
489C23#1313323712yes—
490C23#1313324712yes—
491C23C22×C22⋊D5311yescharacteristic
492Q8—42no—
493Q8—81no—
494Q8—2122no—
495Q8—2122no—
496D4—42no—
497D4—44no—
498D4—42no—
499D4—42no—
500D4—81no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—81no—
522C2×C4—84no—
523C2×C4—2122no—
524C2×C4—2122no—
525C2×C4—2122no—
526C2×C4—2122no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2122no—
530C2×C4#1313318611yescharacteristic
531C2×C4#1313318611yescharacteristic
532C2×C4C22×C22⋊D5311yescharacteristic
533C2×C4C22×C22⋊D5311yescharacteristic
534C8—2122no—
535C8—2122no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—84no—
548C22#1681078512yes—
549C22#1681078512yes—
550C22C2×#1313318611yescharacteristic
551C22C2×#1313323711yescharacteristic
552C22C2×#1313324711yescharacteristic
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—4242no—
563C4—4241no—
564C4—4241no—
565C2—21no—
566C2—22no—
567C2—22no—
568C2—21no—
569C2—82no—
570C2C2×#1681078511yescharacteristic
571C2#2028407911yescharacteristic
572C2#2028411211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#539C2
socleC22×C53C2×#120
prime radicalC2×#121×C53C22
hypercenterC2×#539D53
hypocenterC53#18289
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#539

Complements

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

NH ≅ G/Ntype
1. #20285487C2semidirect
3. #20280528C2semidirect
4. #20273746C2semidirect
5. #20273678C2semidirect
6. #20273287C2semidirect
8. C2×#16810349C2semidirect
12. #16810956C22semidirect
18. #16810778C22semidirect
23. #16810562C22semidirect
25. #16810543C22semidirect
29. #16810360C22semidirect
31. #16810349C22semidirect
331. C53#18289semidirect

Maximal subgroups

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

subgroupindex
#202854872
#202854722
#202805282
#202737462
#202736782
#202732872
#202732832
C2×#168103492
C2×C53×#5392
#1828953