Finite Group Databasebeta

← #27370126

Subgroups — #27370126

Order 13568.

608 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284719C211yesmaximal, characteristic
2#20284508C211yesmaximal, characteristic
3#20283749C211yesmaximal, characteristic
4#20274287C211yesmaximal, characteristic, complemented
5#20274259C211yesmaximal, characteristic, complemented
6#20274235C211yesmaximal, characteristic, complemented
7#20274211C211yesmaximal, characteristic, complemented
8#20274163C211yesmaximal, characteristic, complemented
9#20274135C211yesmaximal, characteristic, complemented
10#20274111C211yesmaximal, characteristic, complemented
11#20274087C211yesmaximal, characteristic, complemented
12#20273504C211yesmaximal, characteristic, complemented
13#20273445C211yesmaximal, characteristic, complemented
14#20273425C211yesmaximal, characteristic, complemented
15C53×#3231C211yesmaximal, characteristic, complemented
16C2×C4×D212C2211yescharacteristic
17#16810951—21no—
18#16810920—21no—
19#16810920—21no—
20#16810910C2211yescharacteristic
21#16810910C2211yescharacteristic
22#16810910C2211yescharacteristic
23#16810910C2211yescharacteristic
24#16810900C2211yescharacteristic
25#16810900C2211yescharacteristic
26#16810825C2211yescharacteristic
27#16810825C2211yescharacteristic
28#16810773—21no—
29C4×#11719924—21no—
30C4×#11719924—21no—
31#16810702C2211yescharacteristic
32#16810702C2211yescharacteristic
33#16810693—21no—
34#16810693—21no—
35C8×D212C2211yescharacteristic
36C8×D212C2211yescharacteristic
37#16810680C2211yescharacteristic
38#16810676C2211yescharacteristic
39#16810666—21no—
40#16810647C2211yescharacteristic
41#16810627—21no—
42#16810585—21no—
43C22⋊C8×D53—21no—
44#16810376C2211yescharacteristic, complemented
45#16810376C2211yescharacteristic, complemented
46#16810374C2211yescharacteristic, complemented
47#16810374C2211yescharacteristic, complemented
48#16810372C2211yescharacteristic, complemented
49#16810372C2211yescharacteristic, complemented
50#16810372C2211yescharacteristic, complemented
51#16810372C2211yescharacteristic, complemented
52#16810369C2211yescharacteristic, complemented
53#16810369C2211yescharacteristic, complemented
54#16810369C2211yescharacteristic, complemented
55#16810369C2211yescharacteristic, complemented
56C53×#533C2211yescharacteristic
57C53×#533C2211yescharacteristic
58C53×#500C2211yescharacteristic, complemented
59C53×#498C2211yescharacteristic, complemented
60C53×#487C2211yescharacteristic, complemented
61C53×#487C2211yescharacteristic, complemented
62C53×#485—21no—
63C53×#485—21no—
64C53×#479—21no—
65C53×#479—21no—
66C53×#431C2211yescharacteristic
67C22×D212D411yescharacteristic
68C22×C4×D53—21no—
69C2×#11719924—21no—
70C2×#11719924—21no—
71C2×#11719917—21no—
72C2×#11719912—21no—
73C2×#11719912—21no—
74C2×C8×D53—21no—
75C2×C8×D53—21no—
76C4×D212—21no—
77C4×D212—21no—
78C4×D212D411yescharacteristic
79C4×D212D411yescharacteristic
80C4×D212C2311yescharacteristic
81C4×D212C2311yescharacteristic
82C2×C4⋊Dic53D411yescharacteristic
83C2×#11719921—21no—
84#13133150—21no—
85#13133150—21no—
86#13133148—21no—
87#13133148—21no—
88#13133148—21no—
89#13133148—21no—
90C4⋊#88322—21no—
91C4⋊#88322C2311yescharacteristic
92C4⋊#88322C2311yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97#13133141—21no—
98#13133141—21no—
99#13133137—21no—
100#13133137—21no—
101#13133137—21no—
102#13133137—21no—
103#13133137—21no—
104#13133137—21no—
105C22⋊#88322—21no—
106C22⋊#88322—21no—
107#13133129—21no—
108#13133129—21no—
109#13133129—21no—
110#13133129—21no—
111#13133128—21no—
112C4×#88322—21no—
113C4⋊Q8×C53C2311yescharacteristic
114C4⋊Q8×C53C2311yescharacteristic
115C4⋊Q8×C53C2311yescharacteristic
116C4⋊Q8×C53C2311yescharacteristic
117#125×C53—21no—
118#125×C53—21no—
119C22⋊Q8×C53—21no—
120C22⋊Q8×C53—21no—
121C22⋊Q8×C53—21no—
122C22⋊Q8×C53—21no—
123C4×Q8×C53—21no—
124C4×Q8×C53—21no—
125C4×Q8×C53—21no—
126C4×Q8×C53—21no—
127#117×C53—21no—
128#117×C53—21no—
129C2×C42×C53C2311yescharacteristic
130#107×C53—21no—
131#107×C53—21no—
132#106×C53—21no—
133#106×C53—21no—
134C4⋊C8×C53D411yescharacteristic
135C4⋊C8×C53D411yescharacteristic
136#103×C53—21no—
137#103×C53—21no—
138#103×C53—21no—
139#103×C53—21no—
140C22⋊C8×C53D411yescharacteristic
141C22⋊C8×C53D411yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C23×D53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×Q8×C53—21no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53#4111yescharacteristic
152C22×Dic53—21no—
153C2×D212—22no—
154C2×D212—21no—
155C2×D212—41no—
156C2×D212C2×D411yescharacteristic
157C2×D212C2×D411yescharacteristic
158C2×C4×D53—21no—
159C2×C4×D53—21no—
160C2×C4×D53—21no—
161C2×C4×D53—21no—
162C2×C4×D53—81no—
163C2×C4×D53—81no—
164C2×C8×C53C2×D411yescharacteristic
165C2×C8×C53C2×D411yescharacteristic
166C2×C8×C53C2×D411yescharacteristic
167C2×C8×C53C2×D411yescharacteristic
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—41no—
177C4⋊C4×C53—41no—
178C4⋊C4×C53—41no—
179C4⋊C4×C53—41no—
180C22⋊C4×C53—41no—
181C22⋊C4×C53—41no—
182C42×C53—41no—
183C42×C53—41no—
184C42×C53C2×D411yescharacteristic
185C42×C53C2×D411yescharacteristic
186C42×C53#4111yescharacteristic
187C42×C53C2411yescharacteristic
188#11719924—42no—
189#11719924—42no—
190#11719921—21no—
191#11719921—21no—
192#11719921—21no—
193#11719921—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53C2×D411yescharacteristic
196C4⋊Dic53C2×D411yescharacteristic
197#11719917—42no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204#11719912—81no—
205#11719912—81no—
206C8×D53—81no—
207C8×D53—81no—
208C23×C53#12111yescharacteristic
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—81no—
214Q8×C53—41no—
215Q8×C53—41no—
216Q8×C53—41no—
217Q8×C53—41no—
218Q8×C53—81no—
219Q8×C53—81no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53#12111yescharacteristic
231C2×C4×C53#12111yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—81no—
238D212—21no—
239D212—21no—
240D212—21no—
241D212—42no—
242D212#13712yes—
243C4×D53—41no—
244C4×D53—41no—
245C4×D53—41no—
246C4×D53—41no—
247C4×D53—81no—
248C4×D53—81no—
249C8×C53—21no—
250C8×C53—21no—
251C8×C53—21no—
252C8×C53—21no—
253#88322—41no—
254#88322—41no—
255#88322—41no—
256#88322—41no—
257#88322—41no—
258#88322—41no—
259#19466—531nomaximal, Sylow
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53C2×#12111yescharacteristic
264C2×D53—22no—
265C2×D53—41no—
266C2×D53—41no—
267C2×D53—41no—
268C2×D53—41no—
269C2×D53—41no—
270C2×D53—84no—
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—21no—
274C4×C53—21no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53C2×#13711yescharacteristic
280C4×C53#57511yescharacteristic
281C4×C53#57711yescharacteristic
282C4×C53#57711yescharacteristic
283Dic53—41no—
284Dic53—41no—
285#3231D5311yescharacteristic, complemented
286#3101—531no—
287#3077—531no—
288#2974—531no—
289#1703—531no—
290#1699—531no—
291#1695—531no—
292#1691—531no—
293#1683—531no—
294#1679—531no—
295#1675—531no—
296#1671—531no—
297#1574—531no—
298#1559—531no—
299#1554—531no—
300C2×C53—41no—
301C2×C53#304211yescharacteristic
302C2×C53#305611yescharacteristic
303C2×C53#306711yescharacteristic
304D53—42no—
305D53—81no—
306#533C2×D5311yescharacteristic
307#533C2×D5311yescharacteristic
308C2×C4×D4—531no—
309#500C2×D5311yescharacteristic, complemented
310#498C2×D5311yescharacteristic, complemented
311#487C2×D5311yescharacteristic, complemented
312#487C2×D5311yescharacteristic, complemented
313#485—21no—
314#485—21no—
315#483—1061no—
316#479—21no—
317#479—21no—
318#478—1061no—
319#478—1061no—
320#476—531no—
321#476—531no—
322#476—531no—
323#476—531no—
324#474—531no—
325#474—531no—
326#461—531no—
327#461—531no—
328#450—1061no—
329C4×SD16—1061no—
330C4×SD16—1061no—
331#436—531no—
332#436—531no—
333#435—1061no—
334#435—1061no—
335C8×D4—531no—
336C8×D4—531no—
337#432—531no—
338#431—531no—
339#431C2×D5311yescharacteristic
340#428—1061no—
341#423—531no—
342#418—1061no—
343#407—1061no—
344C2×C22⋊C8—1061no—
345#335—531no—
346#335—531no—
347#334—531no—
348#334—531no—
349#333—531no—
350#333—531no—
351#333—531no—
352#333—531no—
353#332—531no—
354#332—531no—
355#332—531no—
356#332—531no—
357C53#1946611yescharacteristic, complemented, Sylow
358C22×D4—531no—
359C23×C4—1061no—
360C2×SD16—1061no—
361C2×SD16—1061no—
362C2×M16—1061no—
363C2×M16—1061no—
364C2×M16—1061no—
365C22×C8—1061no—
366C22×C8—1061no—
367C4⋊Q8C22×D5311yescharacteristic
368C4⋊Q8C22×D5311yescharacteristic
369C4⋊Q8C22×D5311yescharacteristic
370C4⋊Q8C22×D5311yescharacteristic
371#125—21no—
372#125—21no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×D4—531no—
382C4×D4—531no—
383C4×D4—531no—
384C4×D4—531no—
385C4×D4—1061no—
386C4×D4—1061no—
387#117—21no—
388#117—21no—
389C2×C4⋊C4—531no—
390C2×C22⋊C4—1061no—
391C2×C42C22×D5311yescharacteristic
392#107—21no—
393#107—21no—
394#107—1061no—
395#107—1061no—
396#106—21no—
397#106—21no—
398#106—1061no—
399#106—1061no—
400#106—1061no—
401#106—1061no—
402C4⋊C8—531no—
403C4⋊C8—531no—
404C4⋊C8—531no—
405C4⋊C8—531no—
406C4⋊C8—531no—
407C4⋊C8—531no—
408C4⋊C8—1061no—
409C4⋊C8C22⋊D5311yescharacteristic
410C4⋊C8C22⋊D5311yescharacteristic
411#103—21no—
412#103—21no—
413#103—21no—
414#103—21no—
415#103—1061no—
416#103—1061no—
417#102—1061no—
418#102—1061no—
419#102—1061no—
420#102—1061no—
421#102—1061no—
422#102—1061no—
423C22⋊C8—1061no—
424C22⋊C8—1061no—
425C22⋊C8—1061no—
426C22⋊C8—1061no—
427C22⋊C8—1061no—
428C22⋊C8—1061no—
429C22⋊C8C22⋊D5311yescharacteristic
430C22⋊C8C22⋊D5311yescharacteristic
431#97—1061no—
432C4×C8—1061no—
433C4×C8C22×D5311yescharacteristic
434C4×C8C22×D5311yescharacteristic
435C24—1061no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×D4—531no—
441C2×D4—531no—
442C2×D4—1062no—
443C2×D4—1061no—
444C2×D4—2121no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—4241no—
451C22×C4—4241no—
452C22×C4D4×D5311yescharacteristic
453C22×C4#1171994711yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455SD16—2122no—
456SD16—2122no—
457M16—4241no—
458M16—4241no—
459M16—2122no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—4241no—
467C2×C8—4241no—
468C2×C8C2×C22⋊D5311yescharacteristic
469C2×C8C2×C22⋊D5311yescharacteristic
470C2×C8C2×C22⋊D5311yescharacteristic
471C2×C8C2×C22⋊D5311yescharacteristic
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—531no—
485C4⋊C4—531no—
486C4⋊C4—1061no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—1061no—
490C22⋊C4—1061no—
491C22⋊C4—1061no—
492C22⋊C4—1061no—
493C42—41no—
494C42—41no—
495C42D4×D5311yescharacteristic
496C42#1171994711yescharacteristic
497C42C2×C22⋊D5311yescharacteristic
498C42C23×D5311yescharacteristic
499C23—1061no—
500C23—1061no—
501C23—4241no—
502C23—2122no—
503C23—2122no—
504C23#1313318811yescharacteristic
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—81no—
510Q8—81no—
511D4—532no—
512D4—1061no—
513D4—1061no—
514D4—1061no—
515D4—2122no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—4241no—
528C2×C4—4241no—
529C2×C4—4241no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4#1313318811yescharacteristic
535C2×C4#1313318811yescharacteristic
536C2×C4#1313318811yescharacteristic
537C2×C4C2×D4×D5311yescharacteristic
538C2×C4C22×C22⋊D5311yescharacteristic
539C2×C4C2×#1171994711yescharacteristic
540C8—21no—
541C8—21no—
542C8—21no—
543C8—21no—
544C8—2121no—
545C8—2121no—
546C8—2121no—
547C8—2121no—
548C8—2121no—
549C8—2121no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—1062no—
554C22—4244no—
555C22—2121no—
556C22—2121no—
557C22—2121no—
558C22—2121no—
559C22—2121no—
560C22C2×#1313318811yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—2121no—
570C4—2121no—
571C4#137×D5311yescharacteristic
572C4#1681142211yescharacteristic
573C4#1681143111yescharacteristic
574C4#1681143111yescharacteristic
575C2—41no—
576C2—4241no—
577C2—2122no—
578C2#2028422311yescharacteristic
579C2#2028433911yescharacteristic
580C2#2028443111yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3231C2
socleC22×C53C2×#121
prime radicalC22×D212D4
hypercenter#3231D53
hypocenterC53#19466
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3231

Complements

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

NH ≅ G/Ntype
4. #20274287C2semidirect
5. #20274259C2semidirect
6. #20274235C2semidirect
7. #20274211C2semidirect
8. #20274163C2semidirect
9. #20274135C2semidirect
10. #20274111C2semidirect
11. #20274087C2semidirect
12. #20273504C2semidirect
13. #20273445C2semidirect
14. #20273425C2semidirect
15. C53×#3231C2semidirect
44. #16810376C22semidirect
45. #16810376C22semidirect
46. #16810374C22semidirect
47. #16810374C22semidirect
48. #16810372C22semidirect
49. #16810372C22semidirect
50. #16810372C22semidirect
51. #16810372C22semidirect
52. #16810369C22semidirect
53. #16810369C22semidirect
54. #16810369C22semidirect
55. #16810369C22semidirect
58. C53×#500C22semidirect
59. C53×#498C22semidirect
60. C53×#487C22semidirect
61. C53×#487C22semidirect
285. #3231D53semidirect
309. #500C2×D53semidirect
310. #498C2×D53semidirect
311. #487C2×D53semidirect
312. #487C2×D53semidirect
357. C53#19466semidirect

Maximal subgroups

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

subgroupindex
#202847192
#202845082
#202837492
#202742872
#202742592
#202742352
#202742112
#202741632
#202741352
#202741112
#202740872
#202735042
#202734452
#202734252
C53×#32312
#1946653