Finite Group Databasebeta

← #27541260

Subgroups — #27541260

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287138C211yesmaximal, characteristic, complemented
2#20286402C211yesmaximal, characteristic, complemented
3#20276952C211yesmaximal, characteristic, complemented
4#20276924C211yesmaximal, characteristic, complemented
5#20276907C211yesmaximal, characteristic, complemented
6#20276853C211yesmaximal, characteristic, complemented
7#20276828C211yesmaximal, characteristic, complemented
8#20276801C211yesmaximal, characteristic, complemented
9#20276771C211yesmaximal, characteristic, complemented
10#20276640C211yesmaximal, characteristic, complemented
11#20276629C211yesmaximal, characteristic, complemented
12#20276592C211yesmaximal, characteristic, complemented
13#20276432C211yesmaximal, characteristic, complemented
14#20276423C211yesmaximal, characteristic, complemented
15C53×#2168C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17C22×D424C2211yescharacteristic
18#16811338—21no—
19#16811269—21no—
20C2×#13133162C2211yescharacteristic, complemented
21C2×D848C2211yescharacteristic
22C2×D848C2211yescharacteristic
23#16811028C2211yescharacteristic, complemented
24#16811007C2211yescharacteristic
25#16810948—21no—
26#16810948—21no—
27#16810914—21no—
28#16810864C2211yescharacteristic
29#16810859C2211yescharacteristic
30#16810859C2211yescharacteristic
31#16810859C2211yescharacteristic, complemented
32#16810839—21no—
33#16810776—21no—
34#16810770—21no—
35C4×D424C2211yescharacteristic
36C4×D424C2211yescharacteristic, complemented
37#16810661C2211yescharacteristic
38#16810661C2211yescharacteristic, complemented
39#16810451C2211yescharacteristic, complemented
40#16810449C2211yescharacteristic, complemented
41#16810449C2211yescharacteristic, complemented
42#16810428C2211yescharacteristic, complemented
43#16810428C2211yescharacteristic, complemented
44#16810428C2211yescharacteristic, complemented
45#16810425C2211yescharacteristic
46#16810425C2211yescharacteristic
47#16810425C2211yescharacteristic
48#16810425C2211yescharacteristic
49#16810425C2211yescharacteristic
50#16810425C2211yescharacteristic
51#16810425C2211yescharacteristic, complemented
52C2×M32×C53C2211yescharacteristic
53C22×C16×C53C2211yescharacteristic, complemented
54C8×D4×C53C2211yescharacteristic
55C53×C4⋊C16C2211yescharacteristic, complemented
56C53×C22⋊C16C2211yescharacteristic
57C53×C22⋊C16C2211yescharacteristic, complemented
58C53×#346C2211yescharacteristic, complemented
59C2×#11719945—21no—
60C22×D212—21no—
61#13133239—22no—
62#13133239—41no—
63C2×Dic212C2311yescharacteristic
64C2×#11719913—21no—
65C2×#11719913—21no—
66C2×D424—21no—
67C2×D424—22no—
68C2×D424—41no—
69C2×D424C2311yescharacteristic
70C2×D424C2311yescharacteristic
71C2×D424C2311yescharacteristic
72#13133217—21no—
73#13133205—21no—
74#13133203—41no—
75#13133203—41no—
76#13133202—41no—
77#13133196—21no—
78#13133191—21no—
79#13133191—21no—
80#13133191—21no—
81#13133190—41no—
82#13133188—41no—
83#13133185—41no—
84C4×D212—21no—
85C4×D212—21no—
86#13133176—21no—
87#13133176—21no—
88#13133175—41no—
89C22⋊C4×D53—41no—
90#13133162—41no—
91D848—22no—
92D848—41no—
93#13133151—21no—
94#13133151C2311yescharacteristic
95#13133151C2311yescharacteristic
96#13133151C2311yescharacteristic
97#13133151C2311yescharacteristic
98#13133149—21no—
99#13133149—21no—
100#13133142—21no—
101#13133139—21no—
102#13133139—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—
120C2×D4×C53D811yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C2×C22⋊D53—41no—
124C2×C22⋊D53—41no—
125C2×C22⋊D53—41no—
126C2×C22⋊D53—41no—
127#11719945—42no—
128#11719945—81no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—21no—
132C2×D212—42no—
133C2×D212—41no—
134C2×D212—81no—
135C2×C4×D53—41no—
136C2×C4×D53—41no—
137C2×C4×D53—41no—
138C2×Dic106—21no—
139C2×Dic106—41no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53C2×D411yescharacteristic
145C2×C8×C53C2×D411yescharacteristic
146C2×C8×C53C2×D411yescharacteristic
147C2×C8×C53C2411yescharacteristic
148C4⋊C4×C53D811yescharacteristic
149C22⋊C4×C53D811yescharacteristic
150C22⋊C4×C53D811yescharacteristic
151C42×C53C2×D411yescharacteristic
152C22⋊Dic53—41no—
153C22⋊Dic53—41no—
154C22⋊Dic53—41no—
155#11719921—41no—
156#11719921—41no—
157#11719921—41no—
158#11719921—41no—
159#11719921—41no—
160#11719921—41no—
161#11719921—41no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—41no—
167#11719919—41no—
168C4×Dic53—41no—
169C4×Dic53—41no—
170Dic212—21no—
171Dic212—21no—
172D424—21no—
173D424—21no—
174D424—21no—
175D424—21no—
176D424—21no—
177D424—42no—
178D424#4112yes—
179#11719913—42no—
180#11719913—81no—
181C16×C53—21no—
182C16×C53—21no—
183C16×C53—21no—
184C16×C53—21no—
185C23×C53C2×D811yescharacteristic
186C23×C53C2×D811yescharacteristic
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—41no—
190C22×D53—82no—
191C22×D53—82no—
192C22×D53—161no—
193D4×C53—41no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53C2×D811yescharacteristic
199C2×C4×C53C2×D811yescharacteristic
200C2×C4×C53C2×D811yescharacteristic
201C2×C4×C53C2×D811yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C22⋊D53—82no—
204C22⋊D53—161no—
205C22⋊D53—161no—
206C22⋊D53—161no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53—41no—
212D212—22no—
213D212—41no—
214D212—41no—
215D212—41no—
216D212—41no—
217D212—41no—
218D212—82no—
219C4×D53—82no—
220C4×D53—82no—
221C4×D53—161no—
222Dic106—41no—
223Dic106—41no—
224Dic106—161no—
225C8×C53—21no—
226C8×C53—21no—
227C8×C53—21no—
228C8×C53C2×#4111yescharacteristic
229C8×C53H(5,𝔽2)11yescharacteristic
230#33886—531nomaximal, Sylow
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53C22×D811yescharacteristic
238C2×D53—42no—
239C2×D53—81no—
240C2×D53—81no—
241C2×D53—81no—
242C2×D53—81no—
243C2×D53—81no—
244C2×D53—81no—
245C2×D53—81no—
246C2×D53—81no—
247C2×D53—164no—
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53#54611yescharacteristic
253C4×C53#57711yescharacteristic
254Dic53—81no—
255Dic53—81no—
256Dic53—81no—
257Dic53—81no—
258Dic53—81no—
259#3341—531no—
260#3280—531no—
261#2243—531no—
262#2237—531no—
263#2234—531no—
264#2224—531no—
265#2220—531no—
266#2215—531no—
267#2210—531no—
268#2186—531no—
269#2184—531no—
270#2177—531no—
271#2168D5311yescharacteristic, complemented
272#2140—531no—
273#2138—531no—
274C2×C53—41no—
275C2×C53—41no—
276C2×C53#329411yescharacteristic
277C2×C53#341411yescharacteristic
278C2×C53#341511yescharacteristic
279D53—82no—
280D53—161no—
281D53—161no—
282C2×#135—531no—
283C22×D8—531no—
284#555—1061no—
285#546—1061no—
286C2×SD32—531no—
287C2×D16—531no—
288C2×D16—531no—
289C2×M32C2×D5311yescharacteristic
290C22×C16C2×D5311yescharacteristic, complemented
291#500—531no—
292#495—531no—
293#482—1061no—
294#482—1061no—
295#476—1061no—
296#467—531no—
297#466—531no—
298#466—531no—
299#466—531no—
300#463—1061no—
301#450—1061no—
302#449—1061no—
303C4×D8—531no—
304C4×D8—531no—
305C8×D4C2×D5311yescharacteristic
306#427—531no—
307#427—531no—
308#367—531no—
309#366—531no—
310#366—531no—
311C4⋊C16C2×D5311yescharacteristic, complemented
312#358—531no—
313#358—531no—
314#358—531no—
315#357—531no—
316#357—531no—
317#357—531no—
318#357—531no—
319#357—531no—
320#357—531no—
321#357—531no—
322C22⋊C16C2×D5311yescharacteristic
323C22⋊C16C2×D5311yescharacteristic, complemented
324#346C2×D5311yescharacteristic, complemented
325C53#3388611yescharacteristic, complemented, Sylow
326C2×#41—1061no—
327C22×D4—1061no—
328#135—1062no—
329#135—2121no—
330C2×Q16—531no—
331C2×SD16—1061no—
332C2×SD16—1061no—
333C2×D8—531no—
334C2×D8—531no—
335C2×D8—531no—
336C2×D8—1062no—
337C2×D8—1061no—
338C2×D8—2121no—
339C22×C8C22×D5311yescharacteristic
340C22×C8C22×D5311yescharacteristic
341C4⋊Q8—1061no—
342#124—1061no—
343#124—2121no—
344#124—2121no—
345#123—2121no—
346C22⋊Q8—1061no—
347#121—1061no—
348#121—1061no—
349#121—1061no—
350#121—2121no—
351#121—2121no—
352#120—2121no—
353C4×D4—1061no—
354C4×D4—1061no—
355C4×D4D21211yescharacteristic
356#117—1061no—
357#117—1061no—
358#117—2121no—
359C2×C22⋊C4—2121no—
360SD32—2121no—
361D16—1062no—
362D16—2121no—
363M32—41no—
364C2×C16—41no—
365C2×C16C22×D5311yescharacteristic
366C2×C16C22×D5311yescharacteristic
367C2×C16C22×D5311yescharacteristic
368C2×C16C22×D5311yescharacteristic
369#107—531no—
370#107—531no—
371#107—531no—
372#107—531no—
373#107—1061no—
374#106—1061no—
375#106—1061no—
376C4⋊C8D21211yescharacteristic
377#103—1061no—
378#102—1061no—
379#102—1061no—
380#102—1061no—
381#102—1061no—
382#102—1061no—
383C22⋊C8D21211yescharacteristic
384C22⋊C8D21211yescharacteristic
385C4×C8C22×D5311yescharacteristic
386C24—2121no—
387#41—4241no—
388#41—2122no—
389C2×Q8—1061no—
390C2×Q8—2121no—
391C2×D4—1061no—
392C2×D4—1061no—
393C2×D4—1061no—
394C2×D4—4241no—
395C2×D4—2122no—
396C2×D4—2121no—
397C2×D4—2121no—
398C2×D4—2121no—
399C2×D4—2121no—
400C2×D4—2121no—
401C2×D4D42411yescharacteristic
402C22×C4—2121no—
403C22×C4—2121no—
404C22×C4—2121no—
405C22×C4C2×D21211yescharacteristic
406C22×C4C2×D21211yescharacteristic
407Q16—1061no—
408Q16—1061no—
409SD16—4241no—
410SD16—2122no—
411D8—532no—
412D8—1061no—
413D8—1061no—
414D8—1061no—
415D8—1061no—
416D8—1061no—
417D8—2122no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8C2×D21211yescharacteristic
423C2×C8C2×D21211yescharacteristic
424C2×C8C2×D21211yescharacteristic
425C2×C8C23×D5311yescharacteristic
426C4⋊C4—1061no—
427C4⋊C4—1061no—
428C4⋊C4—1061no—
429C4⋊C4—1061no—
430C4⋊C4—2121no—
431C4⋊C4—2121no—
432C4⋊C4D42411yescharacteristic
433C22⋊C4—2121no—
434C22⋊C4—2121no—
435C22⋊C4—2121no—
436C22⋊C4—2121no—
437C22⋊C4—2121no—
438C22⋊C4—2121no—
439C22⋊C4—2121no—
440C22⋊C4—2121no—
441C22⋊C4—2121no—
442C22⋊C4—2121no—
443C22⋊C4D42411yescharacteristic
444C22⋊C4D42411yescharacteristic
445C42—2121no—
446C42—2121no—
447C42C2×D21211yescharacteristic
448C16—21no—
449C16—21no—
450C16—21no—
451C16—21no—
452C23—8481no—
453C23—4242no—
454C23—4242no—
455C23—2121no—
456C23—2121no—
457C23—2121no—
458C23C2×D42411yescharacteristic
459C23C2×D42411yescharacteristic
460Q8—8481no—
461Q8—2121no—
462Q8—2121no—
463D4—41no—
464D4—8481no—
465D4—8481no—
466D4—8481no—
467D4—1062no—
468D4—4242no—
469D4—4242no—
470D4—2121no—
471D4—2121no—
472D4—2121no—
473D4—2121no—
474D4—2121no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—8481no—
480C2×C4—4242no—
481C2×C4—4242no—
482C2×C4—2121no—
483C2×C4—2121no—
484C2×C4—2121no—
485C2×C4—2121no—
486C2×C4—2121no—
487C2×C4C2×D42411yescharacteristic
488C2×C4C2×D42411yescharacteristic
489C2×C4C2×D42411yescharacteristic
490C2×C4C2×D42411yescharacteristic
491C2×C4C22×D21211yescharacteristic
492C8—21no—
493C8—21no—
494C8—21no—
495C8#41×D5311yescharacteristic
496C8#1313326211yescharacteristic
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—8484no—
504C22—4241no—
505C22—4241no—
506C22—4241no—
507C22—4241no—
508C22—4241no—
509C22—4241no—
510C22—4241no—
511C22—4241no—
512C22—2122no—
513C22C22×D42411yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—4241no—
519C4—4241no—
520C4—4241no—
521C4—4241no—
522C4—4241no—
523C4#1681126811yescharacteristic
524C4#1681143511yescharacteristic
525C2—41no—
526C2—41no—
527C2—8481no—
528C2—8481no—
529C2—4242no—
530C2#2028652911yescharacteristic
531C2#2028785111yescharacteristic
532C2#2028785611yescharacteristic
subgroupG/H
centerC22C22×D424
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2168C2
socleC22×C53C22×D8
hypercenter#2168D53
hypocenterC53#33886
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. #20287138C2semidirect
2. #20286402C2semidirect
3. #20276952C2semidirect
4. #20276924C2semidirect
5. #20276907C2semidirect
6. #20276853C2semidirect
7. #20276828C2semidirect
8. #20276801C2semidirect
9. #20276771C2semidirect
10. #20276640C2semidirect
11. #20276629C2semidirect
12. #20276592C2semidirect
13. #20276432C2semidirect
14. #20276423C2semidirect
15. C53×#2168C2semidirect
20. C2×#13133162C22semidirect
23. #16811028C22semidirect
31. #16810859C22semidirect
36. C4×D424C22semidirect
38. #16810661C22semidirect
39. #16810451C22semidirect
40. #16810449C22semidirect
41. #16810449C22semidirect
42. #16810428C22semidirect
43. #16810428C22semidirect
44. #16810428C22semidirect
51. #16810425C22semidirect
53. C22×C16×C53C22semidirect
55. C53×C4⋊C16C22semidirect
57. C53×C22⋊C16C22semidirect
58. C53×#346C22semidirect
271. #2168D53semidirect
290. C22×C16C2×D53semidirect
311. C4⋊C16C2×D53semidirect
323. C22⋊C16C2×D53semidirect
324. #346C2×D53semidirect
325. C53#33886semidirect

Maximal subgroups

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

subgroupindex
#202871382
#202864022
#202769522
#202769242
#202769072
#202768532
#202768282
#202768012
#202767712
#202766402
#202766292
#202765922
#202764322
#202764232
C53×#21682
#3388653