Finite Group Databasebeta

← #27367570

Subgroups — #27367570

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286022C211yesmaximal, characteristic, complemented
2#20285633C211yesmaximal, characteristic, complemented
3#20283461C211yesmaximal, characteristic, complemented
4#20274191C212yesmaximal, complemented
5#20274177C212yesmaximal, complemented
6#20274067C212yesmaximal, complemented
7#20274053C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273364C211yesmaximal, characteristic, complemented
10C2×#16810367C211yesmaximal, characteristic, complemented
11C53×#3151C211yesmaximal, characteristic, complemented
12C2×#13133225—21no—
13#16811173C2211yescharacteristic
14#16810999—21no—
15#16810969—21no—
16#16810903—21no—
17#16810900C2212yescomplemented
18#16810857—22no—
19#16810818—21no—
20#16810815C2212yes—
21#16810815C2212yes—
22#16810815C2212yescomplemented
23#16810757—22no—
24#16810702C2212yescomplemented
25#16810696—22no—
26C8×D212C2212yescomplemented
27#16810676C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×C4⋊#88322C2211yescharacteristic
30#16810591—21no—
31#16810376C2212yescomplemented
32#16810374C2212yescomplemented
33#16810369C2211yescharacteristic, complemented
34#16810369C2211yescharacteristic, complemented
35#16810367C2212yes—
36#16810367C2212yes—
37#16810367C2211yescharacteristic, complemented
38#16810367C2211yescharacteristic, complemented
39C2×AGL(1,ℤ/8)×C53—21no—
40C2×AGL(1,ℤ/8)×C53—21no—
41C53×#532C2211yescharacteristic
42C2×#127×C53C2211yescharacteristic
43C53×#496—22no—
44C53×#493C2212yes—
45C53×#492C2212yes—
46C4×M16×C53C2211yescharacteristic
47C2×#11719945—21no—
48C2×#11719922—22no—
49C2×#11719922—22no—
50C2×#11719922—22no—
51#13133225—44no—
52C2×#11719912—22no—
53C22×#88322—21no—
54C22×#88322—21no—
55C2×C8×D53—22no—
56#13133217D411yescharacteristic
57#13133214D411yescharacteristic
58#13133205—21no—
59#13133191—21no—
60#13133191D412yes—
61C4×D212—21no—
62C4×D212C2312yes—
63#13133150—22no—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322C2311yescharacteristic
67C4⋊#88322C2311yescharacteristic
68C4⋊#88322C2312yes—
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2311yescharacteristic
71#13133138—22no—
72#13133138—22no—
73#13133138—22no—
74#13133137—22no—
75C22⋊#88322—21no—
76#13133129—22no—
77#13133129—22no—
78C4×#88322—21no—
79C2×#41×C53—21no—
80C22×D4×C53—21no—
81C22×D4×C53—21no—
82C22×D4×C53—21no—
83AGL(1,ℤ/8)×C53—44no—
84AGL(1,ℤ/8)×C53—44no—
85C2×SD16×C53—22no—
86C2×SD16×C53—22no—
87C2×D8×C53—22no—
88C2×D8×C53—22no—
89C2×M16×C53D411yescharacteristic
90C2×M16×C53D411yescharacteristic
91C4⋊Q8×C53C2311yescharacteristic
92#127×C53—21no—
93#127×C53—21no—
94#127×C53—21no—
95#127×C53C2312yes—
96#127×C53C2311yescharacteristic
97#124×C53—21no—
98#121×C53—21no—
99#121×C53—21no—
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102C2×C42×C53C2311yescharacteristic
103#97×C53D412yes—
104C4×C8×C53C2312yes—
105C24×C53—41no—
106#41×C53—42no—
107#41×C53—42no—
108C2×Q8×C53—21no—
109C2×D4×C53—21no—
110C2×D4×C53—21no—
111C2×D4×C53—22no—
112C2×D4×C53—22no—
113C2×D4×C53—21no—
114C2×D4×C53—22no—
115C2×D4×C53—41no—
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×C53—41no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53#4111yescharacteristic
128C2×C22⋊D53—22no—
129#11719945—42no—
130#11719945—42no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212C2×D412yes—
134C2×C4×D53—22no—
135C2×Dic106—21no—
136SD16×C53—44no—
137SD16×C53—44no—
138D8×C53—44no—
139D8×C53—44no—
140M16×C53—22no—
141M16×C53D814yescomplemented
142C2×C8×C53C2×D412yes—
143C2×C8×C53C2×D412yes—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C22⋊C4×C53—41no—
147C22⋊C4×C53—41no—
148C42×C53C2×D411yescharacteristic
149C42×C53C2×D411yescharacteristic
150C42×C53#4111yescharacteristic
151C42×C53C2411yescharacteristic
152#11719922—82no—
153#11719922—82no—
154#11719922—82no—
155#11719921—22no—
156#11719921—22no—
157C4⋊Dic53—21no—
158C4⋊Dic53C2×D412yes—
159C2×#88322—21no—
160C2×#88322—21no—
161C2×#88322—21no—
162C2×#88322—21no—
163C2×#88322—22no—
164C2×#88322—42no—
165C2×#88322—81no—
166#11719912—44no—
167C8×D53—44no—
168C23×C53—41no—
169C23×C53—42no—
170C23×C53—82no—
171C23×C53—82no—
172C23×C53—82no—
173C23×C53#12111yescharacteristic
174C22×D53—22no—
175Q8×C53—41no—
176Q8×C53—41no—
177D4×C53—42no—
178D4×C53—41no—
179D4×C53—42no—
180D4×C53—41no—
181D4×C53—42no—
182D4×C53—41no—
183D4×C53—41no—
184D4×C53—42no—
185D4×C53—81no—
186D4×C53—82no—
187D4×C53—82no—
188D4×C53—82no—
189D4×C53—82no—
190D4×C53—82no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—82no—
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53C2×D812yes—
200C2×C4×C53AGL(1,ℤ/8)12yes—
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C2×C4×C53C2×#4111yescharacteristic
204C22⋊D53—44no—
205C2×Dic53—22no—
206D212—22no—
207D212—22no—
208D212—42no—
209D212—81no—
210C4×D53—42no—
211C4×D53—42no—
212Dic106—42no—
213C8×C53—22no—
214C8×C53C2×D814yes—
215#88322—41no—
216#88322—41no—
217#88322—42no—
218#88322—41no—
219#88322—41no—
220#19230—531nomaximal, Sylow
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—81no—
224C22×C53—82no—
225C22×C53—82no—
226C22×C53—81no—
227C22×C53—82no—
228C22×C53—81no—
229C22×C53—164no—
230C22×C53#45912yes—
231C22×C53C2×#12111yescharacteristic
232C2×D53—42no—
233C2×D53—42no—
234C2×D53—42no—
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—81no—
239C4×C53#45912yes—
240C4×C53C22×D811yescharacteristic
241C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
242C4×C53#57611yescharacteristic
243C4×C53#57711yescharacteristic
244Dic53—42no—
245#3245—531no—
246#3205—531no—
247#3151D5311yescharacteristic, complemented
248#2935—531no—
249#1687—532no—
250#1685—532no—
251#1667—532no—
252#1665—532no—
253#1565—531no—
254#1540—531no—
255C2×#331—531no—
256C2×C53—22no—
257C2×C53—161no—
258C2×C53—162no—
259C2×C53C2×#45911yescharacteristic
260C2×C53#306211yescharacteristic
261C2×C53#306811yescharacteristic
262D53—82no—
263C2×AGL(1,ℤ/8)—21no—
264C2×AGL(1,ℤ/8)—21no—
265C2×#131—1061no—
266#532—531no—
267#532C2×D5311yescharacteristic
268C2×#127C2×D5311yescharacteristic
269#496—22no—
270#493—1061no—
271#493C2×D5312yes—
272#492C2×D5312yes—
273#486—1061no—
274#474—532no—
275#474—1061no—
276#466—1062no—
277#459—532no—
278#459—532no—
279#459—532no—
280#459—1061no—
281#447—1062no—
282#436—532no—
283#435—1062no—
284C8×D4—532no—
285#431—531no—
286C2×C4⋊C8—531no—
287C2×C4⋊C8—531no—
288#408—1061no—
289C4×M16C2×D5311yescharacteristic
290#335—532no—
291#334—532no—
292#332—531no—
293#332—531no—
294#331—532no—
295#331—532no—
296#331—531no—
297#331—531no—
298C53#1923011yescharacteristic, complemented, Sylow
299C2×#41—21no—
300C2×#41—1061no—
301C22×D4—21no—
302C22×D4—21no—
303C22×D4—21no—
304AGL(1,ℤ/8)—44no—
305AGL(1,ℤ/8)—44no—
306C2×SD16—22no—
307C2×SD16—22no—
308C2×D8—22no—
309C2×D8—22no—
310C2×D8—1062no—
311C2×D8—1062no—
312C2×D8—1062no—
313#131—2124no—
314C2×M16—1062no—
315C2×M16C22⋊D5311yescharacteristic
316C2×M16C22⋊D5311yescharacteristic
317C22×C8—1061no—
318C22×C8—1061no—
319C22×C8—1062no—
320C4⋊Q8—531no—
321C4⋊Q8C22×D5311yescharacteristic
322#127—21no—
323#127—21no—
324#127—21no—
325#127—531no—
326#127C22×D5312yes—
327#127C22×D5311yescharacteristic
328#124—21no—
329#124—1061no—
330#121—21no—
331#121—21no—
332#121—532no—
333#121—1061no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—532no—
337C4×D4—1061no—
338C2×C42C22×D5311yescharacteristic
339#107—1062no—
340C4⋊C8—531no—
341C4⋊C8—531no—
342C4⋊C8—532no—
343C4⋊C8—531no—
344C4⋊C8—531no—
345C4⋊C8—1061no—
346C4⋊C8—1061no—
347#102—1062no—
348#102—1062no—
349#102—1062no—
350#102—1062no—
351C22⋊C8—1061no—
352C22⋊C8—1062no—
353C22⋊C8—1062no—
354#97C22⋊D5312yes—
355C4×C8—1061no—
356C4×C8C22×D5312yes—
357C24—41no—
358#41—42no—
359#41—42no—
360#41—2122no—
361#41—2122no—
362C2×Q8—21no—
363C2×Q8—1061no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4—82no—
379C2×D4—532no—
380C2×D4—1062no—
381C2×D4—1061no—
382C2×D4—1061no—
383C22×C4—41no—
384C22×C4—1062no—
385C22×C4D4×D5311yescharacteristic
386C22×C4#1171994711yescharacteristic
387C22×C4C2×C22⋊D5311yescharacteristic
388SD16—44no—
389SD16—44no—
390D8—44no—
391D8—44no—
392D8—4242no—
393D8—4242no—
394D8—4242no—
395M16—22no—
396M16—2124no—
397M16#1171992214yescomplemented
398C2×C8—1061no—
399C2×C8—1061no—
400C2×C8—1062no—
401C2×C8—1061no—
402C2×C8—1061no—
403C2×C8—4241no—
404C2×C8—2122no—
405C2×C8—2124no—
406C2×C8C2×C22⋊D5312yes—
407C2×C8C2×C22⋊D5312yes—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—532no—
411C4⋊C4—1061no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—1062no—
415C22⋊C4—1062no—
416C42D4×D5311yescharacteristic
417C42#1171994711yescharacteristic
418C42C2×C22⋊D5311yescharacteristic
419C42C23×D5311yescharacteristic
420C23—41no—
421C23—42no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—1062no—
426C23#1313318811yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—2122no—
430D4—41no—
431D4—42no—
432D4—41no—
433D4—42no—
434D4—41no—
435D4—42no—
436D4—42no—
437D4—41no—
438D4—81no—
439D4—82no—
440D4—82no—
441D4—82no—
442D4—82no—
443D4—82no—
444D4—1062no—
445D4—1062no—
446D4—4241no—
447D4—2124no—
448D4—2122no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—82no—
454C2×C4—1062no—
455C2×C4—2122no—
456C2×C4—2122no—
457C2×C4#1313318811yescharacteristic
458C2×C4#1313318811yescharacteristic
459C2×C4#1313318811yescharacteristic
460C2×C4C2×#1171992212yes—
461C2×C4#1313324312yes—
462C2×C4C2×D4×D5311yescharacteristic
463C2×C4C22×C22⋊D5311yescharacteristic
464C2×C4C2×#1171994711yescharacteristic
465C8—22no—
466C8—2121no—
467C8—2121no—
468C8—2122no—
469C8—2121no—
470C8—2121no—
471C8C2×#1171992214yes—
472C22—21no—
473C22—21no—
474C22—82no—
475C22—81no—
476C22—81no—
477C22—82no—
478C22—82no—
479C22—81no—
480C22—164no—
481C22—2122no—
482C22—2122no—
483C22—2122no—
484C22#1681081512yes—
485C22C2×#1313318811yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—81no—
490C4—2122no—
491C4#1681081512yes—
492C4C22×#1171992211yescharacteristic
493C4C2×#1313324311yescharacteristic
494C4#1681142711yescharacteristic
495C4#1681143311yescharacteristic
496C2—22no—
497C2—162no—
498C2—161no—
499C2—4242no—
500C2C2×#1681081511yescharacteristic
501C2#2028438911yescharacteristic
502C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3151C2
socleC22×C53C2×#121
hypercenter#3151D53
hypocenterC53#19230
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. #20286022C2semidirect
2. #20285633C2semidirect
3. #20283461C2semidirect
4. #20274191C2semidirect
5. #20274177C2semidirect
6. #20274067C2semidirect
7. #20274053C2semidirect
8. #20273467C2semidirect
9. #20273364C2semidirect
10. C2×#16810367C2semidirect
11. C53×#3151C2semidirect
17. #16810900C22semidirect
22. #16810815C22semidirect
24. #16810702C22semidirect
26. C8×D212C22semidirect
31. #16810376C22semidirect
32. #16810374C22semidirect
33. #16810369C22semidirect
34. #16810369C22semidirect
37. #16810367C22semidirect
38. #16810367C22semidirect
141. M16×C53D8semidirect
247. #3151D53semidirect
298. C53#19230semidirect
397. M16#11719922semidirect

Maximal subgroups

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

subgroupindex
#202860222
#202856332
#202834612
#202741912
#202741772
#202740672
#202740532
#202734672
#202733642
C2×#168103672
C53×#31512
#1923053