Finite Group Databasebeta

← #27537407

Subgroups — #27537407

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811060C211yesmaximal, characteristic
2C4×#13133240C211yesmaximal, characteristic, complemented
3C4×D8×D53C211yesmaximal, characteristic, complemented
4#20276592C211yesmaximal, characteristic, complemented
5#20276589C211yesmaximal, characteristic, complemented
6#20276577C211yesmaximal, characteristic, complemented
7SD32×Dic53C211yesmaximal, characteristic, complemented
8#20276564C211yesmaximal, characteristic, complemented
9#20276561C211yesmaximal, characteristic, complemented
10#20276559C211yesmaximal, characteristic, complemented
11#20276500C211yesmaximal, characteristic, complemented
12#20276429C211yesmaximal, characteristic, complemented
13#20276426C211yesmaximal, characteristic, complemented
14#20276308C211yesmaximal, characteristic, complemented
15C53×#2175C211yesmaximal, characteristic, complemented
16C2×#13133240C2211yescharacteristic
17C2×D8×D53C2211yescharacteristic
18C4×#11719949—21no—
19C4×D4×D53—21no—
20#16811060C418yescomplemented
21C2×#13133160C2211yescharacteristic
22C2×D848C2211yescharacteristic
23C2×#13133158C2211yescharacteristic
24C2×#13133156C2211yescharacteristic
25C2×#13133156C2211yescharacteristic
26Q16×Dic53C2211yescharacteristic, complemented
27C4×#11719924—21no—
28#16810708—21no—
29C4×D424C2211yescharacteristic
30C4×D424C2211yescharacteristic, complemented
31D8×Dic53C2211yescharacteristic
32#16810705C2211yescharacteristic
33#16810705C2211yescharacteristic, complemented
34C4×#11719922—21no—
35#16810703—21no—
36#16810660C2211yescharacteristic
37#107×D53C2211yescharacteristic
38#16810618—21no—
39#102×D53—21no—
40C4×C8×D53C2211yescharacteristic
41#16810451C2211yescharacteristic, complemented
42#16810450C2211yescharacteristic, complemented
43#16810448C2211yescharacteristic, complemented
44#16810427C2211yescharacteristic, complemented
45#16810425C2211yescharacteristic
46#16810425C2211yescharacteristic, complemented
47#16810424C2211yescharacteristic
48#16810423C2211yescharacteristic
49#16810423C2211yescharacteristic, complemented
50#16810399C2211yescharacteristic, complemented
51C16×Dic53C2211yescharacteristic, complemented
52C16×Dic53C2211yescharacteristic, complemented
53C2×SD32×C53C2211yescharacteristic
54C4×Q16×C53C2211yescharacteristic, complemented
55C4×D8×C53C2211yescharacteristic
56C53×#366C2211yescharacteristic, complemented
57C53×#358C2211yescharacteristic, complemented
58C53×#357C2211yescharacteristic
59C53×#346C2211yescharacteristic, complemented
60C2×#11719949—21no—
61C2×D4×D53—21no—
62C22×C4×D53—41no—
63#13133240—22no—
64#13133240C2×C414yescomplemented
65C2×#11719924—21no—
66C2×D424C2311yescharacteristic
67C2×D424C2311yescharacteristic
68C2×#11719922—21no—
69D8×D53—22no—
70D8×D53C2×C414yescomplemented
71C2×C8×D53D411yescharacteristic
72C2×C8×D53C2311yescharacteristic
73Q8×Dic53—21no—
74C4×D212—21no—
75C4×D212—21no—
76C4×D212—41no—
77D4×Dic53—21no—
78#13133179—21no—
79#13133179—21no—
80#13133179—41no—
81C4×C22⋊D53—41no—
82#13133177—41no—
83#13133175—21no—
84#13133175—41no—
85C4⋊C4×D53—21no—
86C22⋊C4×D53—41no—
87C42×D53—41no—
88C42×D53D411yescharacteristic
89#13133160C2×C414yescomplemented
90D848C2×C414yescomplemented
91#13133158C2×C414yescomplemented
92#13133156—22no—
93#13133156C2×C414yescomplemented
94C2×#11719908C2311yescharacteristic
95C2×#11719908C2311yescharacteristic
96#13133151C2311yescharacteristic
97#13133151C2311yescharacteristic
98#13133150—21no—
99#13133148—21no—
100#13133141—21no—
101#13133139—21no—
102#13133139—21no—
103#13133138—21no—
104#13133137—21no—
105#13133137—21no—
106C4×#88322D411yescharacteristic
107C8×Dic53D411yescharacteristic
108C8×Dic53C2311yescharacteristic
109C2×Q16×C53C2311yescharacteristic
110C2×D8×C53C2311yescharacteristic
111C4×Q8×C53—21no—
112C4×D4×C53—21no—
113SD32×C53C2×C414yescomplemented
114C2×C16×C53C2311yescharacteristic
115C2×C16×C53C2311yescharacteristic
116#107×C53C2311yescharacteristic
117#107×C53C2311yescharacteristic
118#103×C53—21no—
119#102×C53—21no—
120C4×C8×C53C2311yescharacteristic
121C23×D53—41no—
122C2×Q8×C53—21no—
123C2×D4×C53—21no—
124C22×C4×C53—41no—
125C2×C22⋊D53—41no—
126C22×Dic53—41no—
127#11719949—24no—
128#11719949—42no—
129D4×D53—24no—
130D4×D53—42no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—41no—
134C2×C4×D53—41no—
135C2×C4×D53—41no—
136C2×C4×D53—41no—
137C2×C4×D53—41no—
138C2×C4×D53—41no—
139C2×C4×D53—41no—
140C2×C4×D53—84no—
141C2×C4×D53D811yescharacteristic
142C2×C4×D53D811yescharacteristic
143C2×C4×D53C2×D411yescharacteristic
144Q16×C53—21no—
145Q16×C53C22×C412yescomplemented
146D8×C53—21no—
147D8×C53C22×C412yes—
148C2×C8×C53C2×D411yescharacteristic
149C2×C8×C53C2411yescharacteristic
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C42×C53—41no—
155C42×C53C2×D411yescharacteristic
156C22⋊Dic53—41no—
157#11719924—24no—
158#11719922—24no—
159#11719921—41no—
160#11719921—41no—
161#11719921—41no—
162#11719921—41no—
163#11719921—41no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—41no—
167#11719919—41no—
168C4×Dic53—41no—
169C4×Dic53—41no—
170C4×Dic53D811yescharacteristic
171C4×Dic53D811yescharacteristic
172C4×Dic53C2×D411yescharacteristic
173C2×#88322C2×D411yescharacteristic
174C2×#88322C2×D411yescharacteristic
175D424—21no—
176D424—21no—
177D424C22×C412yes—
178D424C22×C412yescomplemented
179C8×D53—22no—
180C8×D53C22×C412yes—
181C8×D53#4112yes—
182C16×C53—21no—
183C16×C53C22×C412yescomplemented
184#11719908—21no—
185#11719908C22×C412yescomplemented
186C23×C53—41no—
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—44no—
190C22×D53—82no—
191C22×D53—82no—
192C22×D53C2×D811yescharacteristic
193Q8×C53—22no—
194Q8×C53—41no—
195D4×C53—22no—
196D4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—82no—
201C2×C4×C53C2×D811yescharacteristic
202C2×C4×C53C2×D811yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C22⋊D53—44no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—82no—
208C2×Dic53C2×D811yescharacteristic
209C2×Dic53C2×D811yescharacteristic
210C2×Dic53C2×D811yescharacteristic
211D212—22no—
212D212—22no—
213D212—44no—
214D212—41no—
215D212—41no—
216C4×D53—22no—
217C4×D53—41no—
218C4×D53—44no—
219C4×D53—82no—
220C4×D53—82no—
221C4×D53—82no—
222C4×D53—82no—
223C4×D53—82no—
224C4×D53C4×D412yes—
225C4×D53#13512yes—
226C8×C53—21no—
227C8×C53C23×C411yescharacteristic
228C8×C53C2×#4111yescharacteristic
229#88322—21no—
230#88322C4×D412yes—
231C4×#509—531nomaximal, Sylow
232C22×C53—42no—
233C22×C53—81no—
234C22×C53—81no—
235C22×C53C22×D811yescharacteristic
236C2×D53—21no—
237C2×D53—21no—
238C2×D53—42no—
239C2×D53—42no—
240C2×D53—88no—
241C2×D53—81no—
242C2×D53—81no—
243C2×D53—81no—
244C2×D53—81no—
245C2×D53C4×D812yes—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—42no—
249C4×C53—81no—
250C4×C53—81no—
251C4×C53C2×C4×D411yescharacteristic
252C4×C53C2×#13511yescharacteristic
253Dic53—21no—
254Dic53—81no—
255Dic53—81no—
256Dic53C4×D812yes—
257Dic53#50912yescomplemented
258C2×#509—531no—
259C4×#135—531no—
260C2×C4×D8—531no—
261#2177—531no—
262#2177—531no—
263#2175—531no—
264#2175D5311yescharacteristic, complemented
265C4×SD32—531no—
266C4×SD32—531no—
267C4×D16—531no—
268C4×D16—531no—
269#2159—531no—
270#2140—531no—
271#2139—531no—
272C4×M32—531no—
273C2×C53—82no—
274C2×C53C2×C4×D811yescharacteristic
275C2×C53C2×#50911yescharacteristic
276C2×C53#341411yescharacteristic
277D53—22no—
278D53—82no—
279D53—82no—
280C2×#135—531no—
281C22×D8—531no—
282C4×#41—1061no—
283C2×C4×D4—1061no—
284#509—538no—
285C2×SD32—531no—
286C2×SD32C2×D5311yescharacteristic
287C2×D16—531no—
288C2×D16—531no—
289C2×M32—531no—
290C2×M32—531no—
291C4×Q16—531no—
292C4×Q16C2×D5311yescharacteristic, complemented
293C4×SD16—1061no—
294C4×SD16—1061no—
295C4×D8—531no—
296C4×D8—531no—
297C4×D8—531no—
298C4×D8—531no—
299C4×D8—531no—
300C4×D8—1061no—
301C4×D8—1061no—
302C4×D8C2×D5311yescharacteristic
303#427—531no—
304C2×#107—531no—
305#416—1061no—
306C2×#102—1061no—
307C2×C4×C8—531no—
308#367—531no—
309#367—531no—
310#366—531no—
311#366C2×D5311yescharacteristic, complemented
312#358—531no—
313#358C2×D5311yescharacteristic, complemented
314#357—531no—
315#357—531no—
316#357—531no—
317#357—531no—
318#357—531no—
319#357C2×D5311yescharacteristic
320#346—531no—
321#346C2×D5311yescharacteristic, complemented
322C4×C16—531no—
323C4×C16—531no—
324C53C4×#50911yescharacteristic, complemented, Sylow
325C2×#41—1061no—
326C22×D4—1061no—
327C23×C4—2121no—
328#135—534no—
329#135—1062no—
330C2×Q16C22×D5311yescharacteristic
331C2×SD16—1061no—
332C2×D8—534no—
333C2×D8—531no—
334C2×D8—531no—
335C2×D8—1062no—
336C2×D8—1061no—
337C2×D8C22×D5311yescharacteristic
338C22×C8—531no—
339C22×C8—531no—
340C4×Q8—21no—
341C4×Q8—1061no—
342C4×D4—21no—
343C4×D4—1061no—
344C4×D4—1061no—
345C4×D4—1061no—
346C4×D4—1061no—
347C4×D4—1061no—
348C4×D4—2121no—
349C4×D4—2121no—
350C4×D4—2121no—
351C4×D4—2121no—
352#117—1061no—
353#117—2121no—
354C2×C4⋊C4—1061no—
355C2×C22⋊C4—2121no—
356C2×C42—531no—
357C2×C42—2121no—
358SD32—534no—
359SD32C4×D5314yescomplemented
360D16—534no—
361D16—534no—
362M32—534no—
363M32—1062no—
364C2×C16—531no—
365C2×C16—531no—
366C2×C16C22×D5311yescharacteristic
367C2×C16C22×D5311yescharacteristic
368#107—531no—
369#107—531no—
370#107—1061no—
371#107C22×D5311yescharacteristic
372#107C22×D5311yescharacteristic
373#106—1061no—
374#103—21no—
375#103—1061no—
376#102—21no—
377#102—1061no—
378#102—1061no—
379#102—1061no—
380#102—1061no—
381#102—1061no—
382C4×C8—531no—
383C4×C8—531no—
384C4×C8—531no—
385C4×C8C22×D5311yescharacteristic
386C24—2121no—
387#41—1064no—
388#41—2122no—
389C2×Q8—21no—
390C2×D4—21no—
391C2×D4—1064no—
392C2×D4—1061no—
393C2×D4—1061no—
394C2×D4—2121no—
395C2×D4—2122no—
396C2×D4—2121no—
397C22×C4—41no—
398C22×C4—531no—
399C22×C4—531no—
400C22×C4—531no—
401C22×C4—4244no—
402C22×C4—2121no—
403C22×C4—2121no—
404C22×C4—2121no—
405C22×C4—2121no—
406C22×C4—2121no—
407C22×C4—2121no—
408C22×C4—2121no—
409Q16—21no—
410Q16C2×C4×D5312yescomplemented
411SD16—1064no—
412D8—21no—
413D8—532no—
414D8—532no—
415D8—1064no—
416D8—1061no—
417D8—1061no—
418D8C2×C4×D5312yes—
419C2×C8—531no—
420C2×C8—532no—
421C2×C8—532no—
422C2×C8—531no—
423C2×C8—1062no—
424C2×C8D4×D5311yescharacteristic
425C2×C8C23×D5311yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—1061no—
430C4⋊C4—1061no—
431C4⋊C4—2121no—
432C4⋊C4—2121no—
433C22⋊C4—41no—
434C22⋊C4—2121no—
435C22⋊C4—2121no—
436C22⋊C4—2121no—
437C22⋊C4—2121no—
438C22⋊C4—2121no—
439C22⋊C4—2121no—
440C42—41no—
441C42—531no—
442C42—531no—
443C42—531no—
444C42—2121no—
445C42—2121no—
446C42D4×D5311yescharacteristic
447C16—21no—
448C16—532no—
449C16—1061no—
450C16C2×C4×D5312yescomplemented
451C23—41no—
452C23—531no—
453C23—4242no—
454C23—4242no—
455C23—2124no—
456C23—2121no—
457C23—2121no—
458Q8—22no—
459Q8—41no—
460D4—22no—
461D4—41no—
462D4—1062no—
463D4—1062no—
464D4—2124no—
465D4—2124no—
466D4—2121no—
467D4—2121no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—531no—
473C2×C4—531no—
474C2×C4—532no—
475C2×C4—532no—
476C2×C4—531no—
477C2×C4—1062no—
478C2×C4—4242no—
479C2×C4—4242no—
480C2×C4—4242no—
481C2×C4—4242no—
482C2×C4—4242no—
483C2×C4—4242no—
484C2×C4—2121no—
485C2×C4—2124no—
486C2×C4—2121no—
487C2×C4—2121no—
488C2×C4D8×D5311yescharacteristic
489C2×C4D8×D5311yescharacteristic
490C2×C4C2×D4×D5311yescharacteristic
491C8—21no—
492C8—532no—
493C8—1061no—
494C8C22×C4×D5311yescharacteristic
495C8#41×D5311yescharacteristic
496C22—42no—
497C22—81no—
498C22—81no—
499C22—532no—
500C22—1061no—
501C22—1061no—
502C22—4248no—
503C22—4241no—
504C22—4241no—
505C22—4241no—
506C22—4241no—
507C22—2122no—
508C22—2122no—
509C22C2×D8×D5311yescharacteristic
510C4—21no—
511C4—21no—
512C4—42no—
513C4—81no—
514C4—81no—
515C4—532no—
516C4—532no—
517C4—1061no—
518C4—4241no—
519C4—4241no—
520C4C4×D4×D5311yescharacteristic
521C4#135×D5311yescharacteristic
522C2—82no—
523C2—1062no—
524C2—4242no—
525C2—4242no—
526C2C4×D8×D5311yescharacteristic
527C2#509×D5311yescharacteristic
528C2#2028785011yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2175C2
socleC22×C53C22×D8
prime radicalC2×#16811060C2
cosocleC2×C8×C53C24
hypercenter#2175D53
hypocenterC53C4×#509
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
2. C4×#13133240C2semidirect
3. C4×D8×D53C2semidirect
4. #20276592C2semidirect
5. #20276589C2semidirect
6. #20276577C2semidirect
7. SD32×Dic53C2semidirect
8. #20276564C2semidirect
9. #20276561C2semidirect
10. #20276559C2semidirect
11. #20276500C2semidirect
12. #20276429C2semidirect
13. #20276426C2semidirect
14. #20276308C2semidirect
15. C53×#2175C2semidirect
20. #16811060C4semidirect
26. Q16×Dic53C22semidirect
30. C4×D424C22semidirect
33. #16810705C22semidirect
41. #16810451C22semidirect
42. #16810450C22semidirect
43. #16810448C22semidirect
44. #16810427C22semidirect
46. #16810425C22semidirect
49. #16810423C22semidirect
50. #16810399C22semidirect
51. C16×Dic53C22semidirect
52. C16×Dic53C22semidirect
54. C4×Q16×C53C22semidirect
56. C53×#366C22semidirect
57. C53×#358C22semidirect
59. C53×#346C22semidirect
64. #13133240C2×C4semidirect
70. D8×D53C2×C4semidirect
89. #13133160C2×C4semidirect
90. D848C2×C4semidirect
91. #13133158C2×C4semidirect
93. #13133156C2×C4semidirect
113. SD32×C53C2×C4semidirect
145. Q16×C53C22×C4semidirect
178. D424C22×C4semidirect
183. C16×C53C22×C4semidirect
185. #11719908C22×C4semidirect
257. Dic53#509semidirect
264. #2175D53semidirect
292. C4×Q16C2×D53semidirect
311. #366C2×D53semidirect
313. #358C2×D53semidirect
321. #346C2×D53semidirect
324. C53C4×#509semidirect
359. SD32C4×D53semidirect
410. Q16C2×C4×D53semidirect
450. C16C2×C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168110602
C4×#131332402
C4×D8×D532
#202765922
#202765892
#202765772
SD32×Dic532
#202765642
#202765612
#202765592
#202765002
#202764292
#202764262
#202763082
C53×#21752
C4×#50953