Finite Group Databasebeta

← #27541125

Subgroups — #27541125

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287084C211yesmaximal, characteristic, complemented
2D4×D424C211yesmaximal, characteristic
3#20276957C211yesmaximal, characteristic, complemented
4#20276907C211yesmaximal, characteristic, complemented
5#20276896C211yesmaximal, characteristic, complemented
6#20276873C211yesmaximal, characteristic, complemented
7#20276828C211yesmaximal, characteristic, complemented
8#20276811C211yesmaximal, characteristic, complemented
9#20276756C211yesmaximal, characteristic, complemented
10#20276646C211yesmaximal, characteristic, complemented
11#20276646C211yesmaximal, characteristic, complemented
12#20276581C211yesmaximal, characteristic, complemented
13#20276432C211yesmaximal, characteristic, complemented
14C2×#16810425C211yesmaximal, characteristic, complemented
15C53×#2168C211yesmaximal, characteristic, complemented
16C22×D424C2211yescharacteristic
17C22×D424C2211yescharacteristic
18#16811313—21no—
19D4×D212—21no—
20C2×#13133162C2211yescharacteristic
21C2×#13133162C2211yescharacteristic, complemented
22C2×#13133162C2211yescharacteristic, complemented
23#16811025C2211yescharacteristic, complemented
24#16811000C2211yescharacteristic
25#16810965—21no—
26#16810954—21no—
27#16810931—21no—
28#16810864C2211yescharacteristic, complemented
29#16810864C2211yescharacteristic, complemented
30#16810859C2211yescharacteristic
31#16810859C2211yescharacteristic
32#16810819—21no—
33#16810759—21no—
34#16810759—21no—
35C4×Dic212C2211yescharacteristic, complemented
36C4×D424C2211yescharacteristic
37#16810661C2211yescharacteristic, complemented
38C2×#13133151C2211yescharacteristic, complemented
39#16810451C2211yescharacteristic, complemented
40#16810449C2211yescharacteristic, complemented
41#16810449C2211yescharacteristic, complemented
42#16810428C2211yescharacteristic, complemented
43#16810425C2211yescharacteristic
44#16810425C2211yescharacteristic
45#16810425C2211yescharacteristic
46#16810425C2211yescharacteristic, complemented
47#16810425C2211yescharacteristic, complemented
48#16810425C2211yescharacteristic, complemented
49#16810425C2211yescharacteristic, complemented
50#16810425C2211yescharacteristic, complemented
51#16810425C2211yescharacteristic, complemented
52C2×M32×C53C2211yescharacteristic, complemented
53C22×C16×C53C2211yescharacteristic, complemented
54C8×D4×C53C2211yescharacteristic
55C53×C4⋊C16C2211yescharacteristic, complemented
56C53×C22⋊C16C2211yescharacteristic, complemented
57C53×C22⋊C16C2211yescharacteristic, complemented
58C53×#346C2211yescharacteristic, complemented
59C22×D212—21no—
60C22×D212—21no—
61C2×D4×D53—41no—
62C2×Dic212C2311yescharacteristic
63C2×D424—21no—
64C2×D424—21no—
65C2×D424—21no—
66C2×D424—22no—
67C2×D424—22no—
68C2×D424—41no—
69C2×D424—41no—
70C2×D424C2311yescharacteristic
71C2×D424C2311yescharacteristic
72C2×D424C2311yescharacteristic
73#13133214—21no—
74#13133210—41no—
75#13133209—21no—
76#13133207—41no—
77#13133198—41no—
78#13133196—21no—
79#13133196—21no—
80#13133192—41no—
81#13133191—21no—
82#13133191—21no—
83#13133187—41no—
84#13133187—41no—
85C4×Dic106—21no—
86C4×D212—21no—
87#13133176—21no—
88#13133173—41no—
89C2×C4⋊Dic53—21no—
90#13133162—22no—
91#13133162—41no—
92#13133162—41no—
93#13133151—21no—
94#13133151C2311yescharacteristic
95#13133151C2311yescharacteristic
96#13133151C2311yescharacteristic
97#13133151C2311yescharacteristic
98#13133149—21no—
99#13133149—21no—
100#13133142—21no—
101#13133142—21no—
102#13133142—21no—
103#13133139—21no—
104#13133139—21no—
105#13133139—21no—
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C4×D4×C53D411yescharacteristic
109M32×C53—41no—
110C2×C16×C53—41no—
111C2×C16×C53C2311yescharacteristic
112C2×C16×C53C2311yescharacteristic
113C2×C16×C53C2311yescharacteristic
114C2×C16×C53C2311yescharacteristic
115C4⋊C8×C53D411yescharacteristic
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C23×D53—41no—
120C23×D53—41no—
121C2×D4×C53D811yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C2×C22⋊D53—41no—
125C2×C22⋊D53—41no—
126C22×Dic53—41no—
127D4×D53—84no—
128C2×D212—21no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—42no—
132C2×D212—42no—
133C2×D212—41no—
134C2×D212—41no—
135C2×D212—41no—
136C2×D212—41no—
137C2×D212—81no—
138C2×D212—81no—
139C2×C4×D53—41no—
140C2×Dic106—21no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53C2×D411yescharacteristic
146C2×C8×C53C2×D411yescharacteristic
147C2×C8×C53C2×D411yescharacteristic
148C2×C8×C53C2411yescharacteristic
149C4⋊C4×C53D811yescharacteristic
150C22⋊C4×C53D811yescharacteristic
151C22⋊C4×C53D811yescharacteristic
152C42×C53C2×D411yescharacteristic
153C22⋊Dic53—41no—
154C22⋊Dic53—41no—
155C22⋊Dic53—41no—
156#11719921—41no—
157#11719921—41no—
158#11719921—41no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—41no—
164#11719919—41no—
165#11719919—41no—
166#11719919—41no—
167#11719919—41no—
168#11719919—41no—
169C4×Dic53—41no—
170C4×Dic53—41no—
171Dic212—21no—
172Dic212—21no—
173D424—21no—
174D424—21no—
175D424—21no—
176D424—21no—
177D424—21no—
178D424—42no—
179D424—42no—
180D424—81no—
181D424#4112yes—
182C16×C53—21no—
183C16×C53—21no—
184C16×C53—21no—
185C16×C53—21no—
186C23×C53C2×D811yescharacteristic
187C23×C53C2×D811yescharacteristic
188C22×D53—41no—
189C22×D53—41no—
190C22×D53—41no—
191C22×D53—82no—
192C22×D53—82no—
193C22×D53—82no—
194C22×D53—82no—
195C22×D53—161no—
196C22×D53—161no—
197D4×C53—22no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53C2×D811yescharacteristic
203C2×C4×C53C2×D811yescharacteristic
204C2×C4×C53C2×D811yescharacteristic
205C2×C4×C53C2×D811yescharacteristic
206C2×C4×C53C22×D411yescharacteristic
207C22⋊D53—82no—
208C22⋊D53—82no—
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53—41no—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—82no—
215D212—22no—
216D212—41no—
217D212—41no—
218D212—41no—
219D212—41no—
220D212—41no—
221D212—82no—
222D212—82no—
223D212—82no—
224D212—161no—
225C4×D53—82no—
226Dic106—41no—
227Dic106—41no—
228C8×C53—21no—
229C8×C53—21no—
230C8×C53—21no—
231C8×C53C2×#4111yescharacteristic
232C8×C53H(5,𝔽2)11yescharacteristic
233#33877—531nomaximal, Sylow
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53#50912yes—
240C22×C53C22×D811yescharacteristic
241C2×D53—42no—
242C2×D53—81no—
243C2×D53—81no—
244C2×D53—81no—
245C2×D53—81no—
246C2×D53—81no—
247C2×D53—81no—
248C2×D53—81no—
249C2×D53—81no—
250C2×D53—164no—
251C2×D53—164no—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—21no—
256C4×C53#54611yescharacteristic
257C4×C53#57711yescharacteristic
258Dic53—81no—
259Dic53—81no—
260Dic53—81no—
261Dic53—81no—
262Dic53—81no—
263#3336—531no—
264D4×D8—531no—
265#2244—531no—
266#2234—531no—
267#2232—531no—
268#2228—531no—
269#2220—531no—
270#2217—531no—
271#2207—531no—
272#2187—531no—
273#2187—531no—
274#2175—531no—
275#2168D5311yescharacteristic, complemented
276#2140—531no—
277C2×#357—531no—
278C2×C53—22no—
279C2×C53—41no—
280C2×C53#329411yescharacteristic
281C2×C53C2×#50911yescharacteristic
282C2×C53#341611yescharacteristic
283D53—82no—
284D53—161no—
285D53—161no—
286C22×D8—531no—
287C22×D8—531no—
288#552—1061no—
289D42—1061no—
290C2×SD32—531no—
291C2×SD32—531no—
292C2×SD32—531no—
293C2×M32C2×D5311yescharacteristic, complemented
294C22×C16C2×D5311yescharacteristic, complemented
295#499—531no—
296#493—531no—
297#485—1061no—
298#483—1061no—
299#479—1061no—
300#467—531no—
301#467—531no—
302#466—531no—
303#466—531no—
304#459—1061no—
305#447—1061no—
306#447—1061no—
307C4×Q16—531no—
308C4×D8—531no—
309C8×D4C2×D5311yescharacteristic
310#427—531no—
311C2×#107—531no—
312#367—531no—
313#366—531no—
314#366—531no—
315C4⋊C16C2×D5311yescharacteristic, complemented
316#358—531no—
317#357—531no—
318#357—531no—
319#357—531no—
320#357—531no—
321#357—531no—
322#357—531no—
323#357—531no—
324#357—531no—
325#357—531no—
326C22⋊C16C2×D5311yescharacteristic, complemented
327C22⋊C16C2×D5311yescharacteristic, complemented
328#346C2×D5311yescharacteristic, complemented
329C53#3387711yescharacteristic, complemented, Sylow
330C22×D4—1061no—
331C22×D4—1061no—
332C22×D4—2121no—
333C2×Q16—531no—
334C2×D8—531no—
335C2×D8—531no—
336C2×D8—531no—
337C2×D8—1062no—
338C2×D8—1062no—
339C2×D8—1061no—
340C2×D8—1061no—
341C2×D8—1061no—
342C2×D8—2121no—
343C2×D8—2121no—
344C22×C8C22×D5311yescharacteristic
345C22×C8C22×D5311yescharacteristic
346#127—1061no—
347#126—2121no—
348#125—1061no—
349#125—2121no—
350#123—2121no—
351C22⋊Q8—1061no—
352C22⋊Q8—1061no—
353#121—1061no—
354#121—1061no—
355#121—2121no—
356#120—2121no—
357#120—2121no—
358C4×Q8—1061no—
359C4×D4—1061no—
360C4×D4D21211yescharacteristic
361#117—1061no—
362#117—2121no—
363C2×C4⋊C4—1061no—
364SD32—1062no—
365SD32—2121no—
366SD32—2121no—
367M32—41no—
368C2×C16—41no—
369C2×C16C22×D5311yescharacteristic
370C2×C16C22×D5311yescharacteristic
371C2×C16C22×D5311yescharacteristic
372C2×C16C22×D5311yescharacteristic
373#107—531no—
374#107—531no—
375#107—531no—
376#107—531no—
377#107—1061no—
378#106—1061no—
379#106—1061no—
380C4⋊C8D21211yescharacteristic
381#103—1061no—
382#103—1061no—
383#103—1061no—
384#102—1061no—
385#102—1061no—
386#102—1061no—
387C22⋊C8D21211yescharacteristic
388C22⋊C8D21211yescharacteristic
389C4×C8C22×D5311yescharacteristic
390C24—2121no—
391C24—2121no—
392C2×Q8—1061no—
393C2×D4—1061no—
394C2×D4—1061no—
395C2×D4—1061no—
396C2×D4—4241no—
397C2×D4—4241no—
398C2×D4—4244no—
399C2×D4—2122no—
400C2×D4—2122no—
401C2×D4—2121no—
402C2×D4—2121no—
403C2×D4—2121no—
404C2×D4—2121no—
405C2×D4—2121no—
406C2×D4—2121no—
407C2×D4D42411yescharacteristic
408C22×C4—2121no—
409C22×C4—2121no—
410C22×C4C2×D21211yescharacteristic
411C22×C4C2×D21211yescharacteristic
412Q16—1061no—
413Q16—1061no—
414D8—532no—
415D8—1061no—
416D8—1061no—
417D8—1061no—
418D8—1061no—
419D8—1061no—
420D8—4241no—
421D8—2122no—
422D8—2122no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8C2×D21211yescharacteristic
428C2×C8C2×D21211yescharacteristic
429C2×C8C2×D21211yescharacteristic
430C2×C8C23×D5311yescharacteristic
431C4⋊C4—1061no—
432C4⋊C4—1061no—
433C4⋊C4—1061no—
434C4⋊C4—1061no—
435C4⋊C4—2121no—
436C4⋊C4—2121no—
437C4⋊C4—2121no—
438C4⋊C4—2121no—
439C4⋊C4—2121no—
440C4⋊C4—2121no—
441C4⋊C4D42411yescharacteristic
442C22⋊C4—2121no—
443C22⋊C4—2121no—
444C22⋊C4—2121no—
445C22⋊C4—2121no—
446C22⋊C4—2121no—
447C22⋊C4—2121no—
448C22⋊C4D42411yescharacteristic
449C22⋊C4D42411yescharacteristic
450C42—2121no—
451C42—2121no—
452C42C2×D21211yescharacteristic
453C16—21no—
454C16—21no—
455C16—21no—
456C16—21no—
457C23—8481no—
458C23—8481no—
459C23—4242no—
460C23—4242no—
461C23—4242no—
462C23—4242no—
463C23—2121no—
464C23—2121no—
465C23—2121no—
466C23C2×D42411yescharacteristic
467C23C2×D42411yescharacteristic
468Q8—2121no—
469Q8—2121no—
470D4—22no—
471D4—8481no—
472D4—1062no—
473D4—4242no—
474D4—4242no—
475D4—4242no—
476D4—4242no—
477D4—4242no—
478D4—2121no—
479D4—2121no—
480D4—2121no—
481D4—2121no—
482D4—2121no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—4242no—
488C2×C4—4242no—
489C2×C4—2121no—
490C2×C4—2121no—
491C2×C4—2121no—
492C2×C4—2121no—
493C2×C4—2121no—
494C2×C4C2×D42411yescharacteristic
495C2×C4C2×D42411yescharacteristic
496C2×C4C2×D42411yescharacteristic
497C2×C4C2×D42411yescharacteristic
498C2×C4C22×D21211yescharacteristic
499C8—21no—
500C8—21no—
501C8—21no—
502C8#41×D5311yescharacteristic
503C8#1313326211yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—8484no—
510C22—8484no—
511C22—4241no—
512C22—4241no—
513C22—4241no—
514C22—4241no—
515C22—4241no—
516C22—4241no—
517C22—4241no—
518C22—4241no—
519C22—2122no—
520C22#1681106112yes—
521C22C22×D42411yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—4241no—
527C4—4241no—
528C4—4241no—
529C4—4241no—
530C4—4241no—
531C4#1681126811yescharacteristic
532C4#1681143511yescharacteristic
533C2—22no—
534C2—41no—
535C2—8481no—
536C2—8481no—
537C2—4242no—
538C2#2028652911yescharacteristic
539C2C2×#1681106111yescharacteristic
540C2#2028786311yescharacteristic
subgroupG/H
centerC22C22×D424
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2168C2
socleC22×C53C22×D8
prime radicalD4×D424C2
hypercenter#2168D53
hypocenterC53#33877
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #2168

Complements

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

NH ≅ G/Ntype
1. #20287084C2semidirect
3. #20276957C2semidirect
4. #20276907C2semidirect
5. #20276896C2semidirect
6. #20276873C2semidirect
7. #20276828C2semidirect
8. #20276811C2semidirect
9. #20276756C2semidirect
10. #20276646C2semidirect
11. #20276646C2semidirect
12. #20276581C2semidirect
13. #20276432C2semidirect
14. C2×#16810425C2semidirect
15. C53×#2168C2semidirect
21. C2×#13133162C22semidirect
22. C2×#13133162C22semidirect
23. #16811025C22semidirect
28. #16810864C22semidirect
29. #16810864C22semidirect
35. C4×Dic212C22semidirect
37. #16810661C22semidirect
38. C2×#13133151C22semidirect
39. #16810451C22semidirect
40. #16810449C22semidirect
41. #16810449C22semidirect
42. #16810428C22semidirect
46. #16810425C22semidirect
47. #16810425C22semidirect
48. #16810425C22semidirect
49. #16810425C22semidirect
50. #16810425C22semidirect
51. #16810425C22semidirect
52. C2×M32×C53C22semidirect
53. C22×C16×C53C22semidirect
55. C53×C4⋊C16C22semidirect
56. C53×C22⋊C16C22semidirect
57. C53×C22⋊C16C22semidirect
58. C53×#346C22semidirect
275. #2168D53semidirect
293. C2×M32C2×D53semidirect
294. C22×C16C2×D53semidirect
315. C4⋊C16C2×D53semidirect
326. C22⋊C16C2×D53semidirect
327. C22⋊C16C2×D53semidirect
328. #346C2×D53semidirect
329. C53#33877semidirect

Maximal subgroups

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

subgroupindex
#202870842
D4×D4242
#202769572
#202769072
#202768962
#202768732
#202768282
#202768112
#202767562
#202766462
#202765812
#202764322
C2×#168104252
C53×#21682
#3387753