Finite Group Databasebeta

← #27369491

Subgroups — #27369491

Order 13568.

704 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284497C211yesmaximal, characteristic, complemented
2#20284191C211yesmaximal, characteristic
3#20283823C211yesmaximal, characteristic, complemented
4#20274128C212yesmaximal, complemented
5#20274121C212yesmaximal, complemented
6#20274060C212yesmaximal, complemented
7#20274053C212yesmaximal, complemented
8#20273486C211yesmaximal, characteristic, complemented
9#20273405C211yesmaximal, characteristic, complemented
10C2×#16810367C211yesmaximal, characteristic, complemented
11C53×#3208C211yesmaximal, characteristic, complemented
12C2×#13133243—21no—
13C4×#11719945C2211yescharacteristic
14#16810927—21no—
15#16810920—21no—
16#16810915C2211yescharacteristic, complemented
17#16810910C2211yescharacteristic, complemented
18#16810905C2211yescharacteristic, complemented
19#16810900C2211yescharacteristic, complemented
20#16810820—21no—
21#16810820C2212yes—
22#16810815—21no—
23#16810815C2212yes—
24C8×Dic106C2212yescomplemented
25C8×D212C2212yescomplemented
26#16810686—22no—
27#16810675—21no—
28#16810659—21no—
29#16810647C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31C2×C4⋊#88322C2211yescharacteristic
32C4⋊C8×D53—21no—
33#16810628—21no—
34#16810617—21no—
35#16810374C2212yescomplemented
36#16810374C2212yescomplemented
37#16810372C2212yescomplemented
38#16810369C2212yescomplemented
39#16810367C2212yes—
40#16810367C2212yes—
41C53×#533C2211yescharacteristic
42C2×#127×C53C2211yescharacteristic
43C53×#492C2212yes—
44C53×#486C2212yes—
45C53×#460—21no—
46C53×#460—21no—
47C53×#450—22no—
48C53×#431C2211yescharacteristic
49C2×#11719945D411yescharacteristic
50#13133243—44no—
51C2×#11719923—22no—
52C2×#11719922—22no—
53C2×#11719917—21no—
54C22×#88322—21no—
55C22×#88322—21no—
56C2×C8×D53—22no—
57C4×Dic106D411yescharacteristic
58C4×Dic106C2311yescharacteristic
59C4×D212D411yescharacteristic
60C4×D212C2311yescharacteristic
61C4×C22⋊D53—21no—
62C4×C22⋊D53—21no—
63#13133176D411yescharacteristic
64#13133174—21no—
65C42×D53—21no—
66#13133150—21no—
67#13133150—21no—
68#13133148—21no—
69#13133148—21no—
70C4⋊#88322—21no—
71C4⋊#88322—21no—
72C4⋊#88322—21no—
73C4⋊#88322C2312yes—
74C4⋊#88322C2312yes—
75C4⋊#88322C2311yescharacteristic
76C4⋊#88322C2311yescharacteristic
77#13133146—22no—
78#13133140—21no—
79#13133140—21no—
80#13133138—22no—
81#13133138—22no—
82#13133137—21no—
83#13133137—21no—
84#13133129—22no—
85C8×Dic53—22no—
86C22×D4×C53—21no—
87C22×D4×C53—21no—
88C22×D4×C53—21no—
89C2×SD16×C53—22no—
90C2×SD16×C53—22no—
91C4⋊Q8×C53C2311yescharacteristic
92C4⋊Q8×C53C2311yescharacteristic
93#127×C53—21no—
94#127×C53—21no—
95#127×C53—21no—
96#127×C53C2312yes—
97#125×C53—21no—
98C22⋊Q8×C53—21no—
99C22⋊Q8×C53—21no—
100C4×Q8×C53—21no—
101C4×Q8×C53—21no—
102#117×C53—21no—
103C2×C42×C53C2311yescharacteristic
104C4⋊C8×C53D411yescharacteristic
105C4⋊C8×C53D411yescharacteristic
106#102×C53—22no—
107#102×C53—22no—
108C22⋊C8×C53D412yes—
109C4×C8×C53C2312yes—
110C24×C53—41no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×D4×C53—22no—
114C2×D4×C53—22no—
115C2×D4×C53—22no—
116C2×D4×C53—41no—
117C2×D4×C53—41no—
118C2×D4×C53—41no—
119C2×D4×C53—42no—
120C2×D4×C53—42no—
121C2×D4×C53—81no—
122C2×D4×C53—81no—
123C2×D4×C53—82no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53#4111yescharacteristic
127C2×C22⋊D53—21no—
128#11719945—22no—
129#11719945SD1614yescomplemented
130C2×D212C2×D411yescharacteristic
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—21no—
134C2×Dic106C2×D411yescharacteristic
135SD16×C53—82no—
136SD16×C53—82no—
137C2×C8×C53C2×D412yes—
138C2×C8×C53C2×D412yes—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C42×C53—41no—
147C42×C53C2×D411yescharacteristic
148C42×C53C2×D411yescharacteristic
149C42×C53#4111yescharacteristic
150C42×C53C2411yescharacteristic
151C22⋊Dic53—21no—
152#11719923—44no—
153#11719922—44no—
154#11719921—21no—
155#11719921—21no—
156C4⋊Dic53C2×D411yescharacteristic
157C4⋊Dic53C2×D411yescharacteristic
158#11719919—21no—
159#11719919—21no—
160C4×Dic53—21no—
161C4×Dic53—21no—
162C4×Dic53—21no—
163#11719917—42no—
164C2×#88322—22no—
165C2×#88322—21no—
166C2×#88322—21no—
167C2×#88322—22no—
168C2×#88322—42no—
169C2×#88322—81no—
170C8×D53—82no—
171C23×C53—42no—
172C23×C53—82no—
173C23×C53—82no—
174C23×C53—82no—
175C23×C53#12111yescharacteristic
176C22×D53—21no—
177Q8×C53—41no—
178Q8×C53—41no—
179Q8×C53—81no—
180D4×C53—42no—
181D4×C53—42no—
182D4×C53—42no—
183D4×C53—42no—
184D4×C53—82no—
185D4×C53—82no—
186D4×C53—82no—
187D4×C53—82no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53#12111yescharacteristic
195C2×C4×C53#12111yescharacteristic
196C2×C4×C53C2×SD1612yes—
197C2×C4×C53#13512yes—
198C2×C4×C53C22×D411yescharacteristic
199C2×C4×C53C22×D411yescharacteristic
200C2×C4×C53C2×#4111yescharacteristic
201C22⋊D53—24no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205D212—21no—
206D212C2×SD1612yes—
207C4×D53—24no—
208C4×D53—41no—
209C4×D53—41no—
210C4×D53—81no—
211Dic106—21no—
212Dic106C2×SD1612yes—
213C8×C53—22no—
214C8×C53—22no—
215#88322—42no—
216#88322—41no—
217#88322—41no—
218#88322—42no—
219#19409—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—82no—
223C22×C53—82no—
224C22×C53—82no—
225C22×C53—164no—
226C22×C53#46512yes—
227C22×C53C2×#12111yescharacteristic
228C2×D53—22no—
229C2×D53—41no—
230C2×D53—41no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—81no—
235C4×C53—81no—
236C4×C53#46512yes—
237C4×C53C22×SD1611yescharacteristic
238C4×C53C2×#13511yescharacteristic
239C4×C53#57611yescharacteristic
240C4×C53#57711yescharacteristic
241Dic53—22no—
242Dic53—41no—
243Dic53—41no—
244#3208D5311yescharacteristic, complemented
245#3076—531no—
246#3038—531no—
247#2988—531no—
248#1678—532no—
249#1677—532no—
250#1666—532no—
251#1665—532no—
252#1570—531no—
253#1550—531no—
254C2×#331—531no—
255C2×C53—22no—
256C2×C53—162no—
257C2×C53#304511yescharacteristic
258C2×C53C2×#46511yescharacteristic
259C2×C53#306811yescharacteristic
260D53—42no—
261C2×AGL(1,ℤ/8)—1061no—
262#533C2×D5311yescharacteristic
263C2×#127C2×D5311yescharacteristic
264C4×#41—531no—
265#492C2×D5312yes—
266#486C2×D5312yes—
267#479—1061no—
268#478—1061no—
269#477—531no—
270#476—531no—
271#475—531no—
272#474—531no—
273#460—21no—
274#460—21no—
275#460—532no—
276#460—1061no—
277#459—532no—
278#459—1061no—
279#450—22no—
280C8×Q8—532no—
281C8×D4—532no—
282C8×D4—1062no—
283#431—1061no—
284#431C2×D5311yescharacteristic
285#427—1061no—
286#423—531no—
287C2×C4⋊C8—531no—
288C2×C4⋊C8—531no—
289C2×C4⋊C8—1061no—
290#418—1061no—
291#416—1061no—
292#334—532no—
293#334—532no—
294#333—532no—
295#332—532no—
296#331—532no—
297#331—532no—
298C53#1940911yescharacteristic, complemented, Sylow
299C2×#41—531no—
300C22×D4—21no—
301C22×D4—21no—
302C22×D4—21no—
303AGL(1,ℤ/8)—2124no—
304C2×SD16—22no—
305C2×SD16—22no—
306C2×SD16—1062no—
307C2×D8—1062no—
308C2×M16—1061no—
309C22×C8—1061no—
310C22×C8—1061no—
311C22×C8—1062no—
312C4⋊Q8C22×D5311yescharacteristic
313C4⋊Q8C22×D5311yescharacteristic
314#127—21no—
315#127—21no—
316#127—21no—
317#127C22×D5312yes—
318#125—21no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—531no—
324C4×Q8—531no—
325C4×D4—531no—
326C4×D4—531no—
327C4×D4—1061no—
328C4×D4—1061no—
329#117—21no—
330#117—531no—
331#117—1061no—
332C2×C42—1061no—
333C2×C42C22×D5311yescharacteristic
334#107—1061no—
335#107—1061no—
336#106—1061no—
337#106—1061no—
338C4⋊C8—532no—
339C4⋊C8—531no—
340C4⋊C8—531no—
341C4⋊C8—532no—
342C4⋊C8—1062no—
343C4⋊C8—1061no—
344C4⋊C8—1061no—
345C4⋊C8—1061no—
346C4⋊C8C22⋊D5311yescharacteristic
347C4⋊C8C22⋊D5311yescharacteristic
348#103—1061no—
349#103—1061no—
350#102—22no—
351#102—22no—
352#102—1061no—
353#102—1061no—
354#102—1062no—
355#102—1062no—
356C22⋊C8—1062no—
357C22⋊C8C22⋊D5312yes—
358C4×C8—1062no—
359C4×C8C22×D5312yes—
360C24—41no—
361#41—534no—
362#41—1062no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—531no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—81no—
375C2×D4—81no—
376C2×D4—82no—
377C2×D4—531no—
378C2×D4—1061no—
379C22×C4—1061no—
380C22×C4—1061no—
381C22×C4—1061no—
382C22×C4D4×D5311yescharacteristic
383C22×C4#1171994711yescharacteristic
384C22×C4C2×C22⋊D5311yescharacteristic
385SD16—82no—
386SD16—82no—
387SD16—2124no—
388D8—2124no—
389M16—2122no—
390C2×C8—1062no—
391C2×C8—1061no—
392C2×C8—1061no—
393C2×C8—1062no—
394C2×C8—4242no—
395C2×C8—4241no—
396C2×C8—2122no—
397C2×C8C2×C22⋊D5312yes—
398C2×C8C2×C22⋊D5312yes—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—531no—
406C4⋊C4—531no—
407C4⋊C4—1061no—
408C4⋊C4—1061no—
409C22⋊C4—41no—
410C22⋊C4—1061no—
411C22⋊C4—1061no—
412C22⋊C4—1061no—
413C42—41no—
414C42—1061no—
415C42—1061no—
416C42—1061no—
417C42D4×D5311yescharacteristic
418C42#1171994711yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C42C23×D5311yescharacteristic
421C23—42no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—1061no—
426C23#1313318811yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—81no—
430Q8—532no—
431Q8—1061no—
432D4—42no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—82no—
437D4—82no—
438D4—82no—
439D4—82no—
440D4—532no—
441D4—1064no—
442D4—1061no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—1061no—
449C2×C4—1064no—
450C2×C4—1061no—
451C2×C4—1061no—
452C2×C4—4241no—
453C2×C4—2121no—
454C2×C4—2121no—
455C2×C4#1313318811yescharacteristic
456C2×C4#1313318811yescharacteristic
457C2×C4#1313318811yescharacteristic
458C2×C4SD16×D5312yes—
459C2×C4#1313323812yes—
460C2×C4C2×D4×D5311yescharacteristic
461C2×C4C22×C22⋊D5311yescharacteristic
462C2×C4C2×#1171994711yescharacteristic
463C8—22no—
464C8—22no—
465C8—2122no—
466C8—2121no—
467C8—2121no—
468C8—2122no—
469C22—21no—
470C22—21no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—164no—
475C22—1062no—
476C22—2121no—
477C22—2121no—
478C22#1681084912yes—
479C22C2×#1313318811yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—81no—
484C4—81no—
485C4—1062no—
486C4—2121no—
487C4—2121no—
488C4#1681084912yes—
489C4C2×SD16×D5311yescharacteristic
490C4C2×#1313323811yescharacteristic
491C4#1681142611yescharacteristic
492C4#1681143411yescharacteristic
493C2—22no—
494C2—162no—
495C2—2122no—
496C2#2028425011yescharacteristic
497C2C2×#1681084911yescharacteristic
498C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3208C2
socleC22×C53C2×#121
prime radical#20284191C2
hypercenter#3208D53
hypocenterC53#19409
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #20284497C2semidirect
3. #20283823C2semidirect
4. #20274128C2semidirect
5. #20274121C2semidirect
6. #20274060C2semidirect
7. #20274053C2semidirect
8. #20273486C2semidirect
9. #20273405C2semidirect
10. C2×#16810367C2semidirect
11. C53×#3208C2semidirect
16. #16810915C22semidirect
17. #16810910C22semidirect
18. #16810905C22semidirect
19. #16810900C22semidirect
24. C8×Dic106C22semidirect
25. C8×D212C22semidirect
35. #16810374C22semidirect
36. #16810374C22semidirect
37. #16810372C22semidirect
38. #16810369C22semidirect
129. #11719945SD16semidirect
244. #3208D53semidirect
298. C53#19409semidirect

Maximal subgroups

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

subgroupindex
#202844972
#202841912
#202838232
#202741282
#202741212
#202740602
#202740532
#202734862
#202734052
C2×#168103672
C53×#32082
#1940953