Finite Group Databasebeta

← #27367578

Subgroups — #27367578

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#20285630C211yesmaximal, characteristic, complemented
2#20285168C211yesmaximal, characteristic, complemented
3#20283459C211yesmaximal, characteristic, complemented
4#20274192C212yesmaximal, complemented
5#20274182C212yesmaximal, complemented
6#20274068C212yesmaximal, complemented
7#20274058C212yesmaximal, complemented
8#20273465C211yesmaximal, characteristic, complemented
9#20273364C211yesmaximal, characteristic, complemented
10C2×#16810366C211yesmaximal, characteristic, complemented
11C53×#3245C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13C2×#13133242—21no—
14C2×#13133224—21no—
15C2×#13133214C2211yescharacteristic
16#16811013—22no—
17#16810999—21no—
18#16810999C2212yes—
19#16810997C2212yes—
20#16810971—21no—
21#16810818C2212yes—
22#16810818C2212yes—
23#16810756—22no—
24#16810701C2212yescomplemented
25#16810695—22no—
26D4×#88322C2212yescomplemented
27#16810676C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29#16810590—21no—
30C4×#11719917C2211yescharacteristic
31#16810375C2212yescomplemented
32#16810373C2212yescomplemented
33#16810369C2211yescharacteristic, complemented
34#16810369C2211yescharacteristic, complemented
35#16810367C2211yescharacteristic, complemented
36#16810367C2211yescharacteristic, complemented
37#16810366C2212yes—
38#16810366C2212yes—
39C53×#532C2211yescharacteristic
40C53×#532C2211yescharacteristic
41C53×#474—21no—
42C53×#474C2212yescomplemented
43C53×#466—22no—
44C53×#459—21no—
45C53×#459C2212yescomplemented
46C2×C4⋊C8×C53C2211yescharacteristic
47C22×D212—21no—
48C22×D212—21no—
49C22×D212—21no—
50#13133242—44no—
51#13133242—44no—
52C2×#11719924—22no—
53C2×#11719924—22no—
54C2×#11719922—22no—
55C2×#11719922—22no—
56C2×#11719922—22no—
57C2×#11719922—22no—
58#13133224—44no—
59C2×#11719917—22no—
60C2×#11719917D411yescharacteristic
61C2×#11719917D411yescharacteristic
62C22×#88322—22no—
63C22×#88322—21no—
64#13133214—21no—
65#13133214—21no—
66#13133214—21no—
67#13133214C2312yes—
68C4⋊#88322—21no—
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2311yescharacteristic
71C4⋊#88322C2311yescharacteristic
72C4⋊#88322C2311yescharacteristic
73#13133137—22no—
74#13133137—22no—
75C22⋊#88322—22no—
76C22⋊#88322—22no—
77C22⋊#88322—21no—
78#13133128D412yes—
79C4×#88322—21no—
80C4×#88322C2312yes—
81C2×#41×C53—21no—
82C2×#41×C53—21no—
83C2×D8×C53—22no—
84C22×C8×C53—21no—
85C4⋊Q8×C53D411yescharacteristic
86C4⋊Q8×C53C2311yescharacteristic
87#127×C53D411yescharacteristic
88#127×C53C2311yescharacteristic
89#124×C53—21no—
90#124×C53—21no—
91#121×C53—21no—
92#121×C53—21no—
93#121×C53—21no—
94#121×C53D412yes—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53C2312yes—
99C2×C42×C53C2311yescharacteristic
100#107×C53—22no—
101C4⋊C8×C53—21no—
102C4⋊C8×C53C2312yes—
103#102×C53—22no—
104#102×C53—22no—
105C23×D53—41no—
106#41×C53—42no—
107#41×C53—42no—
108#41×C53—42no—
109#41×C53—42no—
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×D4×C53—22no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—41no—
119C2×D4×C53C2×D412yes—
120C22×C4×C53—22no—
121C22×C4×C53—41no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53#4111yescharacteristic
125C2×D212—22no—
126C2×D212—22no—
127C2×D212—22no—
128C2×D212—42no—
129C2×D212—42no—
130C2×D212—41no—
131C2×D212—41no—
132C2×D212—41no—
133C2×D212—81no—
134C2×D212—81no—
135C2×D212—82no—
136D8×C53—82no—
137C2×C8×C53—22no—
138C2×C8×C53—81no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53C2×D412yes—
143C22⋊C4×C53—22no—
144C22⋊C4×C53—22no—
145C22⋊C4×C53—41no—
146C22⋊C4×C53—41no—
147C42×C53C2×D411yescharacteristic
148C42×C53C2×D411yescharacteristic
149C42×C53#4111yescharacteristic
150C42×C53C2411yescharacteristic
151#11719924—44no—
152#11719924—44no—
153#11719922—44no—
154#11719922—44no—
155#11719922—82no—
156#11719922—82no—
157#11719917—22no—
158#11719917—44no—
159#11719917D814yescomplemented
160C2×#88322—21no—
161C2×#88322—21no—
162C2×#88322—21no—
163C2×#88322—21no—
164C2×#88322—42no—
165C2×#88322—44no—
166C2×#88322C2×D412yes—
167C2×#88322C2×D412yes—
168C23×C53—22no—
169C23×C53—41no—
170C23×C53#12111yescharacteristic
171C22×D53—42no—
172C22×D53—82no—
173C22×D53—82no—
174C22×D53—82no—
175Q8×C53—42no—
176Q8×C53—41no—
177Q8×C53—41no—
178D4×C53—22no—
179D4×C53—22no—
180D4×C53—44no—
181D4×C53—41no—
182D4×C53—41no—
183D4×C53—41no—
184D4×C53—41no—
185D4×C53—42no—
186D4×C53—81no—
187D4×C53—81no—
188D4×C53—82no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—22no—
192C2×C4×C53—41no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53—41no—
196C2×C4×C53—82no—
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53C2×D812yes—
201C2×C4×C53AGL(1,ℤ/8)12yes—
202C2×C4×C53C22×D411yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C2×C4×C53C2×#4111yescharacteristic
205D212—42no—
206D212—42no—
207D212—42no—
208D212—42no—
209D212—82no—
210D212—82no—
211D212—82no—
212D212—82no—
213C8×C53—42no—
214#88322—22no—
215#88322—41no—
216#88322—41no—
217#88322—41no—
218#88322—41no—
219#88322C2×D814yes—
220#19230—531nomaximal, Sylow
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—42no—
224C22×C53—42no—
225C22×C53—42no—
226C22×C53—81no—
227C22×C53—81no—
228C22×C53—81no—
229C22×C53#45912yes—
230C22×C53C2×#12111yescharacteristic
231C2×D53—82no—
232C2×D53—82no—
233C2×D53—82no—
234C2×D53—164no—
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—42no—
239C4×C53—81no—
240C4×C53#45912yes—
241C4×C53C22×D811yescharacteristic
242C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
243C4×C53#57611yescharacteristic
244C4×C53#57711yescharacteristic
245#3245D5311yescharacteristic, complemented
246#3205—531no—
247#3151—531no—
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—82no—
258C2×C53—161no—
259C2×C53C2×#45911yescharacteristic
260C2×C53#306211yescharacteristic
261C2×C53#306811yescharacteristic
262D53—162no—
263C2×AGL(1,ℤ/8)—1061no—
264C2×AGL(1,ℤ/8)—1061no—
265C2×#131—1061no—
266#532C2×D5311yescharacteristic
267#532C2×D5311yescharacteristic
268C2×#127—531no—
269#496—1062no—
270#493—532no—
271#493—1061no—
272#492—532no—
273#486—1061no—
274#474—21no—
275#474C2×D5312yescomplemented
276#466—22no—
277#459—21no—
278#459—532no—
279#459—532no—
280#459C2×D5312yescomplemented
281#447—1062no—
282#436—532no—
283#435—1062no—
284C8×D4—532no—
285#431—531no—
286C2×C4⋊C8—531no—
287C2×C4⋊C8C2×D5311yescharacteristic
288#408—1061no—
289C4×M16—531no—
290#335—532no—
291#334—532no—
292#332—531no—
293#332—531no—
294#331—531no—
295#331—531no—
296#331—532no—
297#331—532no—
298C53#1923011yescharacteristic, complemented, Sylow
299C2×#41—21no—
300C2×#41—21no—
301C22×D4—1061no—
302C22×D4—1061no—
303C22×D4—1061no—
304AGL(1,ℤ/8)—2124no—
305AGL(1,ℤ/8)—2124no—
306C2×SD16—1062no—
307C2×SD16—1062no—
308C2×D8—22no—
309C2×D8—1062no—
310C2×D8—1062no—
311C2×D8—1062no—
312C2×D8—1062no—
313#131—2124no—
314C2×M16—531no—
315C2×M16—531no—
316C2×M16—1062no—
317C22×C8—21no—
318C22×C8—1062no—
319C22×C8—1061no—
320C4⋊Q8C22⋊D5311yescharacteristic
321C4⋊Q8C22×D5311yescharacteristic
322#127—532no—
323#127—1061no—
324#127—1061no—
325#127—1061no—
326#127C22⋊D5311yescharacteristic
327#127C22×D5311yescharacteristic
328#124—21no—
329#124—21no—
330#121—21no—
331#121—21no—
332#121—21no—
333#121C22⋊D5312yes—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4C22×D5312yes—
338C2×C42C22×D5311yescharacteristic
339#107—22no—
340C4⋊C8—21no—
341C4⋊C8—531no—
342C4⋊C8—531no—
343C4⋊C8—531no—
344C4⋊C8—531no—
345C4⋊C8—1061no—
346C4⋊C8C22×D5312yes—
347#102—22no—
348#102—22no—
349#102—1062no—
350#102—1062no—
351C22⋊C8—1062no—
352C22⋊C8—1062no—
353C22⋊C8—1061no—
354#97—532no—
355C4×C8—532no—
356C4×C8—1061no—
357C24—2121no—
358#41—42no—
359#41—42no—
360#41—42no—
361#41—42no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—41no—
371C2×D4—1062no—
372C2×D4—1062no—
373C2×D4—1062no—
374C2×D4—4241no—
375C2×D4—4241no—
376C2×D4—4242no—
377C2×D4—2121no—
378C2×D4—2121no—
379C2×D4—2121no—
380C2×D4—2122no—
381C2×D4—2122no—
382C2×D4C2×C22⋊D5312yes—
383C22×C4—22no—
384C22×C4—41no—
385C22×C4D4×D5311yescharacteristic
386C22×C4#1171994711yescharacteristic
387C22×C4C2×C22⋊D5311yescharacteristic
388SD16—2124no—
389SD16—2124no—
390D8—82no—
391D8—4242no—
392D8—4242no—
393D8—2124no—
394D8—2124no—
395M16—534no—
396M16—1062no—
397M16—2124no—
398C2×C8—22no—
399C2×C8—81no—
400C2×C8—532no—
401C2×C8—532no—
402C2×C8—1061no—
403C2×C8—1061no—
404C2×C8—1061no—
405C2×C8—1061no—
406C2×C8—2124no—
407C2×C8—2122no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4C2×C22⋊D5312yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C42D4×D5311yescharacteristic
417C42#1171994711yescharacteristic
418C42C2×C22⋊D5311yescharacteristic
419C42C23×D5311yescharacteristic
420C23—22no—
421C23—41no—
422C23—4242no—
423C23—4242no—
424C23—4242no—
425C23—2122no—
426C23#1313318911yescharacteristic
427Q8—42no—
428Q8—41no—
429Q8—41no—
430D4—22no—
431D4—22no—
432D4—44no—
433D4—42no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—81no—
439D4—81no—
440D4—82no—
441D4—4242no—
442D4—4242no—
443D4—4242no—
444D4—4242no—
445D4—2122no—
446D4—2122no—
447D4—2122no—
448D4—2122no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—82no—
457C2×C4#1313318911yescharacteristic
458C2×C4#1313318911yescharacteristic
459C2×C4#1313318911yescharacteristic
460C2×C4D8×D5312yes—
461C2×C4#1313324112yes—
462C2×C4C2×D4×D5311yescharacteristic
463C2×C4C22×C22⋊D5311yescharacteristic
464C2×C4C2×#1171994711yescharacteristic
465C8—42no—
466C8—534no—
467C8—1062no—
468C8—2121no—
469C8—2121no—
470C8—2121no—
471C8—2121no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—8484no—
481C22—4242no—
482C22—4242no—
483C22—4242no—
484C22#1681081612yes—
485C22C2×#1313318911yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—42no—
490C4—81no—
491C4#1681081612yes—
492C4C2×D8×D5311yescharacteristic
493C4C2×#1313324111yescharacteristic
494C4#1681142611yescharacteristic
495C4#1681143111yescharacteristic
496C2—22no—
497C2—82no—
498C2—161no—
499C2—8482no—
500C2C2×#1681081611yescharacteristic
501C2#2028439711yescharacteristic
502C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3245C2
socleC22×C53C2×#121
hypercenter#3245D53
hypocenterC53#19230
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #20285630C2semidirect
2. #20285168C2semidirect
3. #20283459C2semidirect
4. #20274192C2semidirect
5. #20274182C2semidirect
6. #20274068C2semidirect
7. #20274058C2semidirect
8. #20273465C2semidirect
9. #20273364C2semidirect
10. C2×#16810366C2semidirect
11. C53×#3245C2semidirect
24. #16810701C22semidirect
26. D4×#88322C22semidirect
31. #16810375C22semidirect
32. #16810373C22semidirect
33. #16810369C22semidirect
34. #16810369C22semidirect
35. #16810367C22semidirect
36. #16810367C22semidirect
42. C53×#474C22semidirect
45. C53×#459C22semidirect
159. #11719917D8semidirect
245. #3245D53semidirect
275. #474C2×D53semidirect
280. #459C2×D53semidirect
298. C53#19230semidirect

Maximal subgroups

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

subgroupindex
#202856302
#202851682
#202834592
#202741922
#202741822
#202740682
#202740582
#202734652
#202733642
C2×#168103662
C53×#32452
#1923053