Finite Group Databasebeta

← #28328404

Subgroups — #28328404

Order 13568.

652 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D53⋊#444C211yesmaximal, characteristic, complemented
2#20291469C211yesmaximal, characteristic, complemented
3#20291126C212yesmaximal, complemented
4#20291124C212yesmaximal, complemented
5#20291122C212yesmaximal, complemented
6#20291120C212yesmaximal, complemented
7#20291062C211yesmaximal, characteristic, complemented
8#20291042C211yesmaximal, characteristic, complemented
9#20291035C211yesmaximal, characteristic, complemented
10C2×#16811513C211yesmaximal, characteristic, complemented
11#20287848C211yesmaximal, characteristic
12D53⋊#131—21no—
13#41×#6157—21no—
14C2×#13133292C2211yescharacteristic
15#16811670C2212yes—
16#16811669C2212yes—
17#16811668—22no—
18#16811664C2212yescomplemented
19#16811663C2212yescomplemented
20#16811662—22no—
21D53⋊#108C2211yescharacteristic
22#16811633C2211yescharacteristic
23D53⋊#104—21no—
24#16811613—21no—
25C2×#13133271C2211yescharacteristic
26#16811548C2212yescomplemented
27#16811547C2212yescomplemented
28#16811542C2212yescomplemented
29#16811541C2212yescomplemented
30#16811540C2212yes—
31#16811539C2212yes—
32#16811513C2212yescomplemented
33#16811513C2212yescomplemented
34#135×D53C2211yescharacteristic
35#135×D53C2211yescharacteristic
36#16811065C411yescharacteristic, complemented
37#16811065C2211yescharacteristic
38#16811062C411yescharacteristic, complemented
39#16811062C2211yescharacteristic
40#16811060C411yescharacteristic, complemented
41#16811060C2211yescharacteristic
42#16811057C411yescharacteristic, complemented
43#16811057C2211yescharacteristic
44#16811055C411yescharacteristic, complemented
45#16811053C411yescharacteristic, complemented
46#16811053C411yescharacteristic, complemented
47C2×#13133156C2211yescharacteristic
48C53×#508C411yescharacteristic, complemented
49#13133305—22no—
50#13133304—22no—
51#13133304—42no—
52D53⋊M16—21no—
53D53⋊M16—41no—
54C2×D53⋊C8—41no—
55C2×D53⋊C8D411yescharacteristic
56Q8×#6157—22no—
57D4×#6157—22no—
58D4×#6157—42no—
59#13133296—21no—
60#13133296—41no—
61C2×C4×#6157—41no—
62C2×C4×#6157D411yescharacteristic
63#13133292—21no—
64#13133292C2312yes—
65C2×#11719910—21no—
66#13133288—21no—
67#13133287C2312yes—
68#13133286—21no—
69#13133285C2312yes—
70#13133282—22no—
71#13133281—22no—
72#13133280—22no—
73#13133279—22no—
74#13133271D412yes—
75#13133271C2312yes—
76#41×D53—21no—
77#41×D53—21no—
78#13133240C2×C411yescharacteristic
79#13133240C2311yescharacteristic
80#13133239C2×C411yescharacteristic
81#13133239C2×C411yescharacteristic, complemented
82#13133238—21no—
83#13133238—21no—
84#13133237—21no—
85#13133237—21no—
86#13133236C2×C411yescharacteristic
87#13133236C2311yescharacteristic
88Q16×D53C2×C411yescharacteristic, complemented
89Q16×D53C2311yescharacteristic
90SD16×D53—21no—
91SD16×D53—21no—
92D8×D53C2×C411yescharacteristic, complemented
93D8×D53C2311yescharacteristic
94C2×C8×D53C2311yescharacteristic
95Dic424C2×C411yescharacteristic, complemented
96#13133163C2×C411yescharacteristic, complemented
97#13133163C2×C411yescharacteristic, complemented
98#13133162C2×C411yescharacteristic, complemented
99#13133162C2×C411yescharacteristic, complemented
100#13133161C2×C411yescharacteristic, complemented
101#13133161C2×C411yescharacteristic, complemented
102#13133160C2×C411yescharacteristic, complemented
103#13133160C2×C411yescharacteristic, complemented
104D848C2×C411yescharacteristic, complemented
105#13133158C2×C411yescharacteristic, complemented
106#13133158C2×C411yescharacteristic, complemented
107#13133156C2×C411yescharacteristic, complemented
108#13133156C2×C411yescharacteristic, complemented
109#13133156C2311yescharacteristic
110#13133156C2311yescharacteristic
111C2×#11719908C2×C411yescharacteristic, complemented
112#135×C53C2×C411yescharacteristic
113#135×C53C2×C411yescharacteristic, complemented
114Q32×C53C2×C411yescharacteristic, complemented
115SD32×C53C2×C411yescharacteristic, complemented
116SD32×C53C2×C411yescharacteristic, complemented
117D16×C53C2×C411yescharacteristic, complemented
118C2×C16×C53C2×C411yescharacteristic, complemented
119C22×#6157—21no—
120C22×#6157—42no—
121#41×C53—21no—
122#41×C53—21no—
123#11719949—21no—
124#11719949—21no—
125Q8×D53—21no—
126Q8×D53—21no—
127#11719947—21no—
128#11719947—21no—
129#11719947—41no—
130#11719947—41no—
131D4×D53—21no—
132D4×D53—21no—
133D4×D53—41no—
134D4×D53—41no—
135#11719945—21no—
136#11719945—21no—
137#11719945—41no—
138#11719945—41no—
139C2×C4×D53—41no—
140C2×C4×D53—41no—
141C2×C4×D53C2×D411yescharacteristic
142C22⋊#6157—42no—
143C22⋊#6157—41no—
144#11719940—42no—
145#11719940—41no—
146C2×#88324—21no—
147C2×#88324—42no—
148C4⋊#6157—22no—
149C4⋊#6157—42no—
150C4×#6157—21no—
151C4×#6157—42no—
152C4×#6157—41no—
153C4×#6157C2×D412yes—
154#11719936—22no—
155#11719936—42no—
156D53⋊C8—21no—
157D53⋊C8—42no—
158D53⋊C8—41no—
159D53⋊C8C2×D412yes—
160Q16×C53C22×C411yescharacteristic
161Q16×C53C22×C411yescharacteristic
162SD16×C53—21no—
163SD16×C53—21no—
164D8×C53C22×C411yescharacteristic
165D8×C53C22×C411yescharacteristic
166C2×C8×C53C22×C411yescharacteristic
167#11719925—21no—
168#11719925—21no—
169#11719924—21no—
170#11719924—21no—
171#11719923—21no—
172#11719923—21no—
173#11719922—21no—
174#11719922—21no—
175C2×#88322#4111yescharacteristic
176Dic212C22×C411yescharacteristic
177Dic212C22×C411yescharacteristic
178D424C22×C411yescharacteristic
179D424C22×C411yescharacteristic
180#11719913—21no—
181#11719913—21no—
182C8×D53C22×C411yescharacteristic
183C8×D53C2×D411yescharacteristic
184C8×D53#4111yescharacteristic
185C8×D53C2411yescharacteristic
186#11719910—22no—
187C16×C53C22×C411yescharacteristic
188C16×C53C22×C411yescharacteristic
189#11719908C22×C411yescharacteristic
190#11719908C22×C411yescharacteristic
191C22×D53—41no—
192C22×D53—41no—
193C22×D53AGL(1,ℤ/8)11yescharacteristic
194C2×#6157—22no—
195C2×#6157—42no—
196C2×#6157—81no—
197C2×#6157—84no—
198Q8×C53—21no—
199Q8×C53—21no—
200D4×C53—21no—
201D4×C53—21no—
202D4×C53—41no—
203D4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53C4×D411yescharacteristic
207C22⋊D53—41no—
208C22⋊D53—41no—
209C22⋊D53—41no—
210C22⋊D53—41no—
211C22⋊D53—41no—
212C22⋊D53—41no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53AGL(1,ℤ/8)11yescharacteristic
216D212—21no—
217D212—21no—
218D212—41no—
219D212—41no—
220C4×D53—41no—
221C4×D53—41no—
222C4×D53—41no—
223C4×D53—41no—
224C4×D53—41no—
225C4×D53—41no—
226C4×D53C4×D411yescharacteristic
227C4×D53AGL(1,ℤ/8)11yescharacteristic
228C4×D53AGL(1,ℤ/8)11yescharacteristic
229C4×D53C22×D411yescharacteristic
230Dic106—21no—
231Dic106—21no—
232Dic106—41no—
233Dic106—41no—
234#88324—22no—
235#88324—42no—
236C8×C53C4×D411yescharacteristic
237C8×C53C23×C411yescharacteristic
238#88322C4×D411yescharacteristic
239#88322C2×#4111yescharacteristic
240#33468—531nomaximal, Sylow
241C22×C53—41no—
242C22×C53—41no—
243C22×C53#44211yescharacteristic
244C2×D53—41no—
245C2×D53—41no—
246C2×D53—41no—
247C2×D53—82no—
248C2×D53—82no—
249C2×D53#44211yescharacteristic
250C2×D53C2×AGL(1,ℤ/8)11yescharacteristic
251#6157—42no—
252#6157—82no—
253C4×C53—41no—
254C4×C53—41no—
255C4×C53#44211yescharacteristic
256C4×C53C2×C4×D411yescharacteristic
257Dic53—41no—
258Dic53—41no—
259Dic53#44211yescharacteristic
260Dic53C2×AGL(1,ℤ/8)11yescharacteristic
261#3414—531no—
262C2×#444—531no—
263#2943—531no—
264AGL(1,ℤ/16)—532no—
265AGL(1,ℤ/16)—532no—
266#2180—532no—
267#2180—532no—
268#2155—531no—
269#2148—531no—
270#2145—531no—
271C2×#347—531no—
272C2×C53—21no—
273C2×C53—81no—
274C2×C53—81no—
275C2×C53C2×#44211yescharacteristic
276D53—21no—
277D53—21no—
278D53—81no—
279D53—81no—
280C2×#135—531no—
281C2×#135—531no—
282C2×#131—1061no—
283C4×#41—1061no—
284#510—531no—
285#510—531no—
286#510—531no—
287#510—531no—
288#509—531no—
289#509—531no—
290#509—531no—
291#509—531no—
292#508—531no—
293#508—531no—
294#508—531no—
295#508#615711yescharacteristic, complemented
296C2×M32—531no—
297C2×M32—531no—
298#444—532no—
299#444—532no—
300#444—1062no—
301#442—532no—
302#441—532no—
303#440—1062no—
304C2×#108—531no—
305#427—531no—
306C2×#104—1061no—
307#416—1061no—
308C2×#97—531no—
309#365—532no—
310#365—532no—
311#362—532no—
312#361—532no—
313#360—532no—
314#360—532no—
315#347—532no—
316#347—532no—
317C53#3346811yescharacteristic, complemented, Sylow
318C2×#41—1061no—
319C2×#41—1061no—
320#135—531no—
321#135—531no—
322#135—531no—
323#135—531no—
324#135—531no—
325#135—531no—
326#135—1061no—
327#135—1061no—
328#135—1061no—
329#135—1061no—
330#135C2×#615711yescharacteristic
331#135C2×#615711yescharacteristic, complemented
332C2×Q16—531no—
333C2×Q16—531no—
334C2×SD16—1061no—
335C2×SD16—1061no—
336C2×D8—531no—
337C2×D8—531no—
338#131—1062no—
339#131—1062no—
340#131—2122no—
341C2×M16—1061no—
342C2×M16—2121no—
343C22×C8—531no—
344C22×C8—531no—
345C22×C8—2121no—
346C4×Q8—1062no—
347C4×D4—1062no—
348C4×D4—2122no—
349#117—1061no—
350#117—2121no—
351C2×C42—531no—
352C2×C42—2121no—
353Q32—531no—
354Q32—531no—
355Q32—531no—
356Q32C2×#615711yescharacteristic, complemented
357SD32—531no—
358SD32—531no—
359SD32—531no—
360SD32—531no—
361SD32—531no—
362SD32—531no—
363SD32C2×#615711yescharacteristic, complemented
364SD32C2×#615711yescharacteristic, complemented
365D16—531no—
366D16—531no—
367D16—531no—
368D16C2×#615711yescharacteristic, complemented
369M32—531no—
370M32—531no—
371M32—531no—
372M32—531no—
373M32—532no—
374M32—1061no—
375C2×C16—531no—
376C2×C16—1061no—
377C2×C16C2×#615711yescharacteristic, complemented
378#108—532no—
379#108—1061no—
380#107—1061no—
381#106—532no—
382#104—1062no—
383#104—1062no—
384#103—1062no—
385#102—1062no—
386#97—532no—
387#97—532no—
388#41—21no—
389#41—21no—
390#41—1061no—
391#41—1061no—
392#41—1061no—
393#41—1061no—
394#41—1061no—
395#41—1061no—
396#41—2121no—
397#41—2121no—
398#41—2121no—
399#41—2121no—
400C2×Q8—1061no—
401C2×Q8—1061no—
402C2×D4—1061no—
403C2×D4—1061no—
404C2×D4—2121no—
405C2×D4—2121no—
406C22×C4—531no—
407C22×C4—1061no—
408C22×C4—2121no—
409C22×C4—2121no—
410C22×C4—2122no—
411Q16—531no—
412Q16—531no—
413Q16—1061no—
414Q16—1061no—
415Q16C22×#615711yescharacteristic
416Q16C22×#615711yescharacteristic
417SD16—21no—
418SD16—21no—
419SD16—1061no—
420SD16—1061no—
421SD16—1061no—
422SD16—1061no—
423SD16—1061no—
424SD16—1061no—
425D8—531no—
426D8—531no—
427D8—1061no—
428D8—1061no—
429D8C22×#615711yescharacteristic
430D8C22×#615711yescharacteristic
431M16—1062no—
432M16—2122no—
433M16—2122no—
434M16—2121no—
435C2×C8—531no—
436C2×C8—531no—
437C2×C8—531no—
438C2×C8—531no—
439C2×C8—531no—
440C2×C8—532no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—2122no—
444C2×C8—2122no—
445C2×C8—2121no—
446C2×C8C22×#615711yescharacteristic
447C4⋊C4—1062no—
448C4⋊C4—2122no—
449C22⋊C4—2122no—
450C22⋊C4—2121no—
451C42—532no—
452C42—1061no—
453C42—2121no—
454C42—2122no—
455C16—531no—
456C16—531no—
457C16—1062no—
458C16C22×#615711yescharacteristic
459C16C22×#615711yescharacteristic
460C23—531no—
461C23—2121no—
462C23—2121no—
463Q8—21no—
464Q8—21no—
465Q8—1061no—
466Q8—1061no—
467Q8—2121no—
468Q8—2121no—
469D4—21no—
470D4—21no—
471D4—41no—
472D4—41no—
473D4—1061no—
474D4—1061no—
475D4—2121no—
476D4—2121no—
477D4—2121no—
478D4—2121no—
479D4—2121no—
480D4—2121no—
481D4—2121no—
482D4—2121no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—531no—
486C2×C4—531no—
487C2×C4—531no—
488C2×C4—531no—
489C2×C4—531no—
490C2×C4—1062no—
491C2×C4—4244no—
492C2×C4—4241no—
493C2×C4—2121no—
494C2×C4—2121no—
495C2×C4—2121no—
496C2×C4—2121no—
497C2×C4—2121no—
498C2×C4—2121no—
499C2×C4—2122no—
500C2×C4—2121no—
501C2×C4—2121no—
502C2×C4D4×#615711yescharacteristic
503C8—531no—
504C8—531no—
505C8—1062no—
506C8—2122no—
507C8D4×#615711yescharacteristic
508C8C23×#615711yescharacteristic
509C22—41no—
510C22—41no—
511C22—531no—
512C22—531no—
513C22—4242no—
514C22—4242no—
515C22—2121no—
516C22—2121no—
517C22—2121no—
518C22#1681166411yescharacteristic
519C4—41no—
520C4—41no—
521C4—531no—
522C4—531no—
523C4—4242no—
524C4—2122no—
525C4—2121no—
526C4—2121no—
527C4#1681166411yescharacteristic
528C4C2×D4×#615711yescharacteristic
529C2—21no—
530C2—81no—
531C2—81no—
532C2—1061no—
533C2—1061no—
534C2—4241no—
535C2—4241no—
536C2C2×#1681166411yescharacteristic
subgroupG/H
centerC4#16811664
commutatorC8×C53C23×C4
FrattiniC8C23×#6157
FittingC53×#508C4
socleC2×C53C2×#442
prime radical#20287848C2
cosocleC8×D53C24
hypercenter#508#6157
hypocenterC53#33468
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. D53⋊#444C2semidirect
2. #20291469C2semidirect
3. #20291126C2semidirect
4. #20291124C2semidirect
5. #20291122C2semidirect
6. #20291120C2semidirect
7. #20291062C2semidirect
8. #20291042C2semidirect
9. #20291035C2semidirect
10. C2×#16811513C2semidirect
18. #16811664C22semidirect
19. #16811663C22semidirect
26. #16811548C22semidirect
27. #16811547C22semidirect
28. #16811542C22semidirect
29. #16811541C22semidirect
32. #16811513C22semidirect
33. #16811513C22semidirect
36. #16811065C4semidirect
38. #16811062C4semidirect
40. #16811060C4semidirect
42. #16811057C4semidirect
44. #16811055C4semidirect
45. #16811053C4semidirect
46. #16811053C4semidirect
48. C53×#508C4semidirect
81. #13133239C2×C4semidirect
88. Q16×D53C2×C4semidirect
92. D8×D53C2×C4semidirect
95. Dic424C2×C4semidirect
96. #13133163C2×C4semidirect
97. #13133163C2×C4semidirect
98. #13133162C2×C4semidirect
99. #13133162C2×C4semidirect
100. #13133161C2×C4semidirect
101. #13133161C2×C4semidirect
102. #13133160C2×C4semidirect
103. #13133160C2×C4semidirect
104. D848C2×C4semidirect
105. #13133158C2×C4semidirect
106. #13133158C2×C4semidirect
107. #13133156C2×C4semidirect
108. #13133156C2×C4semidirect
111. C2×#11719908C2×C4semidirect
113. #135×C53C2×C4semidirect
114. Q32×C53C2×C4semidirect
115. SD32×C53C2×C4semidirect
116. SD32×C53C2×C4semidirect
117. D16×C53C2×C4semidirect
118. C2×C16×C53C2×C4semidirect
295. #508#6157semidirect
317. C53#33468semidirect
331. #135C2×#6157semidirect
356. Q32C2×#6157semidirect
363. SD32C2×#6157semidirect
364. SD32C2×#6157semidirect
368. D16C2×#6157semidirect
377. C2×C16C2×#6157semidirect

Maximal subgroups

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

subgroupindex
D53⋊#4442
#202914692
#202911262
#202911242
#202911222
#202911202
#202910622
#202910422
#202910352
C2×#168115132
#202878482
#3346853