Finite Group Databasebeta

← #27369499

Subgroups — #27369499

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#20285656C211yesmaximal, characteristic, complemented
2#20284503C211yesmaximal, characteristic, complemented
3#20283826C211yesmaximal, characteristic, complemented
4#20274129C212yesmaximal, complemented
5#20274122C212yesmaximal, complemented
6#20274061C212yesmaximal, complemented
7#20274054C212yesmaximal, complemented
8#20273487C211yesmaximal, characteristic, complemented
9#20273406C211yesmaximal, characteristic, complemented
10C2×#16810367C211yesmaximal, characteristic, complemented
11C53×#3038C211yesmaximal, characteristic
12#16811180C2211yescharacteristic
13#16810995C2212yes—
14#16810969C2212yes—
15#16810930—21no—
16#16810923—21no—
17#16810918C2211yescharacteristic, complemented
18#16810913C2211yescharacteristic, complemented
19#16810908C2211yescharacteristic, complemented
20#16810903C2211yescharacteristic, complemented
21#16810820—21no—
22#16810820—21no—
23#16810774—22no—
24Q8×#88322C2212yescomplemented
25D4×#88322—22no—
26D4×#88322C2212yescomplemented
27#16810676—21no—
28#16810676C2211yescharacteristic
29#16810659—21no—
30C2×C4⋊#88322—21no—
31C2×C4⋊#88322C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33#16810619—21no—
34#16810373C2212yescomplemented
35#16810373C2212yescomplemented
36#16810371C2212yescomplemented
37#16810368C2212yescomplemented
38#16810367C2212yes—
39#16810367C2212yes—
40C2×AGL(1,ℤ/8)×C53—21no—
41C2×#127×C53C2211yescharacteristic
42C4×#41×C53C2211yescharacteristic
43C53×#460—21no—
44C53×#460C2212yes—
45C53×#459—21no—
46C53×#459C2212yes—
47C53×#423C2211yescharacteristic
48C53×#418—21no—
49C2×#11719923—22no—
50C2×#11719923—22no—
51C22×#88322—22no—
52C22×#88322—21no—
53C22×#88322—21no—
54#13133217C2311yescharacteristic
55#13133217C2311yescharacteristic
56#13133209—21no—
57#13133196—21no—
58#13133196—21no—
59C4×Dic106—21no—
60C4×Dic106—21no—
61#13133176—21no—
62#13133150—21no—
63#13133150—21no—
64#13133148—21no—
65#13133148—21no—
66C4⋊#88322—21no—
67C4⋊#88322—21no—
68C4⋊#88322—22no—
69C4⋊#88322D411yescharacteristic
70C4⋊#88322D411yescharacteristic
71C4⋊#88322C2312yes—
72C4⋊#88322C2311yescharacteristic
73C4⋊#88322C2311yescharacteristic
74#13133141—21no—
75#13133141—21no—
76#13133138—22no—
77#13133138—22no—
78#13133138—21no—
79#13133138—21no—
80C22⋊#88322—22no—
81C22⋊#88322D412yes—
82C4×#88322—22no—
83C4×#88322C2312yes—
84C2×#41×C53D411yescharacteristic
85C22×D4×C53—21no—
86C22×D4×C53—21no—
87C22×D4×C53—21no—
88AGL(1,ℤ/8)×C53—44no—
89C2×SD16×C53—22no—
90C2×D8×C53—22no—
91C2×M16×C53—21no—
92#127×C53—21no—
93#127×C53—21no—
94#127×C53—21no—
95#127×C53C2312yes—
96C4×Q8×C53D411yescharacteristic
97C4×Q8×C53C2311yescharacteristic
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53D411yescharacteristic
101C4×D4×C53C2311yescharacteristic
102#117×C53—21no—
103#117×C53D411yescharacteristic
104C2×C42×C53—21no—
105C2×C42×C53C2311yescharacteristic
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#102×C53—22no—
109#102×C53—22no—
110C24×C53—41no—
111#41×C53—22no—
112#41×C53SD1614yescomplemented
113C2×Q8×C53C2×D411yescharacteristic
114C2×D4×C53—21no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—41no—
119C2×D4×C53—41no—
120C2×D4×C53—41no—
121C2×D4×C53—42no—
122C2×D4×C53—42no—
123C2×D4×C53—81no—
124C2×D4×C53—81no—
125C2×D4×C53—82no—
126C2×D4×C53C2×D411yescharacteristic
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53#4111yescharacteristic
133C2×Dic106—21no—
134C2×Dic106—21no—
135SD16×C53—44no—
136D8×C53—44no—
137M16×C53—42no—
138C2×C8×C53—22no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53C2×D411yescharacteristic
142C4⋊C4×C53C2×D411yescharacteristic
143C22⋊C4×C53—21no—
144C22⋊C4×C53—21no—
145C22⋊C4×C53—21no—
146C42×C53—21no—
147C42×C53—21no—
148C42×C53—21no—
149C42×C53C2×D411yescharacteristic
150C42×C53C2×D411yescharacteristic
151C42×C53#4111yescharacteristic
152C42×C53C2411yescharacteristic
153C22⋊Dic53—41no—
154#11719923—82no—
155#11719923—82no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160#11719919—41no—
161#11719919—41no—
162C4×Dic53—41no—
163C2×#88322—22no—
164C2×#88322—21no—
165C2×#88322—21no—
166C2×#88322—42no—
167C2×#88322—82no—
168C2×#88322—81no—
169C2×#88322C2×D412yes—
170C2×#88322C2×D412yes—
171C23×C53—21no—
172C23×C53—42no—
173C23×C53—82no—
174C23×C53—82no—
175C23×C53—82no—
176C23×C53#12111yescharacteristic
177Q8×C53—21no—
178Q8×C53C2×SD1612yes—
179D4×C53—24no—
180D4×C53—21no—
181D4×C53—42no—
182D4×C53—42no—
183D4×C53—42no—
184D4×C53—42no—
185D4×C53—82no—
186D4×C53—82no—
187D4×C53—82no—
188D4×C53—82no—
189D4×C53C2×SD1612yes—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—24no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—81no—
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53#12111yescharacteristic
202C2×C4×C53#12111yescharacteristic
203C2×C4×C53C2×SD1612yes—
204C2×C4×C53#13512yes—
205C2×C4×C53C22×D411yescharacteristic
206C2×C4×C53C22×D411yescharacteristic
207C2×C4×C53C2×#4111yescharacteristic
208C2×Dic53—41no—
209C2×Dic53—41no—
210Dic106—41no—
211Dic106—41no—
212Dic106—81no—
213C8×C53—42no—
214#88322—22no—
215#88322—22no—
216#88322—41no—
217#88322—41no—
218#88322—42no—
219#19409—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—22no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53—82no—
226C22×C53—82no—
227C22×C53—82no—
228C22×C53—164no—
229C22×C53#46512yes—
230C22×C53C2×#12111yescharacteristic
231C4×C53—21no—
232C4×C53—22no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—41no—
236C4×C53—41no—
237C4×C53#46512yes—
238C4×C53C22×SD1611yescharacteristic
239C4×C53C2×#13511yescharacteristic
240C4×C53#57611yescharacteristic
241C4×C53#57711yescharacteristic
242Dic53—81no—
243Dic53—81no—
244#3208—531no—
245#3076—531no—
246#3038D5311yescharacteristic
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—42no—
257C2×C53—162no—
258C2×C53#304511yescharacteristic
259C2×C53C2×#46511yescharacteristic
260C2×C53#306811yescharacteristic
261C2×AGL(1,ℤ/8)—21no—
262#533—531no—
263C2×#127C2×D5311yescharacteristic
264C4×#41C2×D5311yescharacteristic
265#492—532no—
266#486—532no—
267#479—1061no—
268#478—1061no—
269#477—531no—
270#476—531no—
271#475—531no—
272#474—531no—
273#460—21no—
274#460—1061no—
275#460—1061no—
276#460C2×D5312yes—
277#459—21no—
278#459C2×D5312yes—
279#450—1062no—
280C8×Q8—532no—
281C8×D4—532no—
282C8×D4—1062no—
283#431—531no—
284#431—1061no—
285#427—1061no—
286#423C2×D5311yescharacteristic
287C2×C4⋊C8—531no—
288C2×C4⋊C8—531no—
289C2×C4⋊C8—1061no—
290#418—21no—
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×#41C22⋊D5311yescharacteristic
300C22×D4—21no—
301C22×D4—21no—
302C22×D4—21no—
303AGL(1,ℤ/8)—44no—
304C2×SD16—22no—
305C2×SD16—1062no—
306C2×SD16—1062no—
307C2×D8—22no—
308C2×M16—21no—
309C22×C8—1062no—
310C22×C8—1061no—
311C22×C8—1061no—
312C4⋊Q8—531no—
313C4⋊Q8—531no—
314#127—21no—
315#127—21no—
316#127—21no—
317#127C22×D5312yes—
318#125—1061no—
319C22⋊Q8—1061no—
320C22⋊Q8—1061no—
321C4×Q8—1061no—
322C4×Q8—1061no—
323C4×Q8C22⋊D5311yescharacteristic
324C4×Q8C22×D5311yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22⋊D5311yescharacteristic
328C4×D4C22×D5311yescharacteristic
329#117—21no—
330#117—1061no—
331#117C22⋊D5311yescharacteristic
332C2×C42—21no—
333C2×C42C22×D5311yescharacteristic
334#107—1061no—
335#107—1061no—
336#106—1061no—
337#106—1061no—
338C4⋊C8—21no—
339C4⋊C8—531no—
340C4⋊C8—531no—
341C4⋊C8—532no—
342C4⋊C8—531no—
343C4⋊C8—531no—
344C4⋊C8—1062no—
345C4⋊C8—1061no—
346C4⋊C8—1061no—
347C4⋊C8C22×D5312yes—
348#103—1061no—
349#103—1061no—
350#102—22no—
351#102—22no—
352#102—1062no—
353#102—1062no—
354#102—1061no—
355#102—1061no—
356C22⋊C8—532no—
357C22⋊C8—1062no—
358C4×C8—532no—
359C4×C8—1062no—
360C24—41no—
361#41—22no—
362#41#1171992314yescomplemented
363C2×Q8—1061no—
364C2×Q8—1061no—
365C2×Q8C2×C22⋊D5311yescharacteristic
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—81no—
376C2×D4—81no—
377C2×D4—82no—
378C2×D4C2×C22⋊D5311yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4D4×D5311yescharacteristic
383C22×C4#1171994711yescharacteristic
384C22×C4C2×C22⋊D5311yescharacteristic
385SD16—44no—
386SD16—4242no—
387SD16—4242no—
388D8—44no—
389M16—42no—
390C2×C8—22no—
391C2×C8—532no—
392C2×C8—532no—
393C2×C8—1062no—
394C2×C8—1061no—
395C2×C8—1061no—
396C2×C8—4242no—
397C2×C8—4241no—
398C2×C8—2122no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—1061no—
402C4⋊C4—1061no—
403C4⋊C4—1061no—
404C4⋊C4—1061no—
405C4⋊C4—2121no—
406C4⋊C4—2121no—
407C4⋊C4C2×C22⋊D5311yescharacteristic
408C4⋊C4C2×C22⋊D5311yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—2121no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—2121no—
417C42D4×D5311yescharacteristic
418C42#1171994711yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C42C23×D5311yescharacteristic
421C23—21no—
422C23—42no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23#1313318911yescharacteristic
427Q8—21no—
428Q8—4241no—
429Q8—2121no—
430Q8—2121no—
431Q8C2×#1171992312yes—
432D4—24no—
433D4—21no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—82no—
439D4—82no—
440D4—82no—
441D4—82no—
442D4C2×#1171992312yes—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—2121no—
454C2×C4—2121no—
455C2×C4#1313318911yescharacteristic
456C2×C4#1313318911yescharacteristic
457C2×C4#1313318911yescharacteristic
458C2×C4C2×#1171992312yes—
459C2×C4#1313323712yes—
460C2×C4C2×D4×D5311yescharacteristic
461C2×C4C22×C22⋊D5311yescharacteristic
462C2×C4C2×#1171994711yescharacteristic
463C8—42no—
464C8—1062no—
465C8—1062no—
466C8—2121no—
467C8—2121no—
468C8—2122no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—164no—
478C22#1681085012yes—
479C22C2×#1313318911yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—4241no—
487C4—4241no—
488C4#1681085012yes—
489C4C22×#1171992311yescharacteristic
490C4C2×#1313323711yescharacteristic
491C4#1681142711yescharacteristic
492C4#1681143311yescharacteristic
493C2—22no—
494C2—42no—
495C2—162no—
496C2#2028425811yescharacteristic
497C2C2×#1681085011yescharacteristic
498C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3038C2
socleC22×C53C2×#121
hypercenter#3038D53
hypocenterC53#19409
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #20285656C2semidirect
2. #20284503C2semidirect
3. #20283826C2semidirect
4. #20274129C2semidirect
5. #20274122C2semidirect
6. #20274061C2semidirect
7. #20274054C2semidirect
8. #20273487C2semidirect
9. #20273406C2semidirect
10. C2×#16810367C2semidirect
17. #16810918C22semidirect
18. #16810913C22semidirect
19. #16810908C22semidirect
20. #16810903C22semidirect
24. Q8×#88322C22semidirect
26. D4×#88322C22semidirect
34. #16810373C22semidirect
35. #16810373C22semidirect
36. #16810371C22semidirect
37. #16810368C22semidirect
112. #41×C53SD16semidirect
298. C53#19409semidirect
362. #41#11719923semidirect

Maximal subgroups

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

subgroupindex
#202856562
#202845032
#202838262
#202741292
#202741222
#202740612
#202740542
#202734872
#202734062
C2×#168103672
C53×#30382
#1940953