Finite Group Databasebeta

← #27367658

Subgroups — #27367658

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286031C211yesmaximal, characteristic, complemented
2#20285656C211yesmaximal, characteristic, complemented
3#20283662C211yesmaximal, characteristic, complemented
4#20274196C212yesmaximal, complemented
5#20274184C212yesmaximal, complemented
6#20274072C212yesmaximal, complemented
7#20274060C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273364C211yesmaximal, characteristic, complemented
10C2×#16810367C211yesmaximal, characteristic, complemented
11C53×#3151C211yesmaximal, characteristic
12#16811180C2211yescharacteristic
13#16810995—21no—
14#16810969—21no—
15#16810913—21no—
16#16810905C2212yescomplemented
17#16810850—22no—
18#16810828—21no—
19#16810820C2212yes—
20#16810820C2212yes—
21#16810820C2212yescomplemented
22#16810774—22no—
23#16810753C2212yescomplemented
24#16810751—22no—
25C8×Dic106C2212yescomplemented
26#16810676C2211yescharacteristic
27#16810651—21no—
28C2×C4⋊#88322C2211yescharacteristic
29C2×C4⋊#88322C2211yescharacteristic
30#16810581—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×#11719923—22no—
48C2×#11719923—22no—
49C2×#11719923—22no—
50C22×#88322—21no—
51C22×#88322—21no—
52#13133217D411yescharacteristic
53#13133217D411yescharacteristic
54#13133209—21no—
55#13133196—21no—
56#13133196D412yes—
57C4×Dic106—21no—
58C4×Dic106C2312yes—
59#13133176—21no—
60#13133148—22no—
61C4⋊#88322—21no—
62C4⋊#88322—21no—
63C4⋊#88322C2311yescharacteristic
64C4⋊#88322C2311yescharacteristic
65C4⋊#88322C2311yescharacteristic
66C4⋊#88322C2312yes—
67C4⋊#88322C2311yescharacteristic
68#13133146—22no—
69#13133146—22no—
70#13133140—22no—
71#13133138—22no—
72#13133138—22no—
73#13133138—22no—
74C22⋊#88322—21no—
75#13133127—22no—
76C4×#88322—21no—
77C8×Dic53—22no—
78C2×#41×C53—21no—
79C22×D4×C53—21no—
80C22×D4×C53—21no—
81C22×D4×C53—21no—
82AGL(1,ℤ/8)×C53—44no—
83AGL(1,ℤ/8)×C53—44no—
84C2×SD16×C53—22no—
85C2×SD16×C53—22no—
86C2×D8×C53—22no—
87C2×D8×C53—22no—
88C2×M16×C53D411yescharacteristic
89C2×M16×C53D411yescharacteristic
90C4⋊Q8×C53C2311yescharacteristic
91#127×C53—21no—
92#127×C53—21no—
93#127×C53—21no—
94#127×C53C2312yes—
95#127×C53C2311yescharacteristic
96#124×C53—21no—
97#121×C53—21no—
98#121×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53—21no—
101C2×C42×C53C2311yescharacteristic
102#97×C53D412yes—
103C4×C8×C53C2312yes—
104C24×C53—41no—
105#41×C53—42no—
106#41×C53—42no—
107C2×Q8×C53—21no—
108C2×D4×C53—21no—
109C2×D4×C53—21no—
110C2×D4×C53—22no—
111C2×D4×C53—22no—
112C2×D4×C53—21no—
113C2×D4×C53—22no—
114C2×D4×C53—41no—
115C2×D4×C53—41no—
116C2×D4×C53—41no—
117C2×D4×C53—41no—
118C2×D4×C53—42no—
119C2×D4×C53—42no—
120C2×D4×C53—81no—
121C2×D4×C53—81no—
122C2×D4×C53—82no—
123C22×C4×C53—41no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53#4111yescharacteristic
127C2×Dic106—21no—
128C2×Dic106C2×D412yes—
129SD16×C53—44no—
130SD16×C53—44no—
131D8×C53—44no—
132D8×C53—44no—
133M16×C53—22no—
134M16×C53SD1614yescomplemented
135C2×C8×C53C2×D412yes—
136C2×C8×C53C2×D412yes—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C22⋊C4×C53—41no—
140C22⋊C4×C53—41no—
141C42×C53C2×D411yescharacteristic
142C42×C53C2×D411yescharacteristic
143C42×C53#4111yescharacteristic
144C42×C53C2411yescharacteristic
145C22⋊Dic53—22no—
146#11719923—82no—
147#11719923—82no—
148#11719923—82no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152C4⋊Dic53C2×D412yes—
153#11719919—22no—
154#11719919—22no—
155C4×Dic53—22no—
156C2×#88322—21no—
157C2×#88322—21no—
158C2×#88322—21no—
159C2×#88322—21no—
160C2×#88322—22no—
161C2×#88322—42no—
162C2×#88322—81no—
163C23×C53—41no—
164C23×C53—42no—
165C23×C53—82no—
166C23×C53—82no—
167C23×C53—82no—
168C23×C53#12111yescharacteristic
169Q8×C53—41no—
170Q8×C53—41no—
171D4×C53—42no—
172D4×C53—41no—
173D4×C53—42no—
174D4×C53—41no—
175D4×C53—42no—
176D4×C53—41no—
177D4×C53—41no—
178D4×C53—42no—
179D4×C53—81no—
180D4×C53—82no—
181D4×C53—82no—
182D4×C53—82no—
183D4×C53—82no—
184D4×C53—82no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—82no—
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53C2×SD1612yes—
194C2×C4×C53AGL(1,ℤ/8)12yes—
195C2×C4×C53C22×D411yescharacteristic
196C2×C4×C53C22×D411yescharacteristic
197C2×C4×C53C2×#4111yescharacteristic
198C2×Dic53—22no—
199C2×Dic53—22no—
200Dic106—22no—
201Dic106—22no—
202Dic106—81no—
203C8×C53—22no—
204C8×C53C2×SD1614yes—
205#88322—41no—
206#88322—41no—
207#88322—42no—
208#88322—41no—
209#88322—41no—
210#19238—531nomaximal, Sylow
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—81no—
214C22×C53—82no—
215C22×C53—82no—
216C22×C53—81no—
217C22×C53—82no—
218C22×C53—81no—
219C22×C53—164no—
220C22×C53#46012yes—
221C22×C53C2×#12111yescharacteristic
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—81no—
226C4×C53#46012yes—
227C4×C53C22×SD1611yescharacteristic
228C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
229C4×C53#57611yescharacteristic
230C4×C53#57711yescharacteristic
231Dic53—42no—
232Dic53—42no—
233#3246—531no—
234#3208—531no—
235#3151D5311yescharacteristic
236#2962—531no—
237#1688—532no—
238#1686—532no—
239#1668—532no—
240#1666—532no—
241#1565—531no—
242#1540—531no—
243C2×#331—531no—
244C2×C53—22no—
245C2×C53—161no—
246C2×C53—162no—
247C2×C53C2×#46011yescharacteristic
248C2×C53#306211yescharacteristic
249C2×C53#306811yescharacteristic
250C2×AGL(1,ℤ/8)—21no—
251C2×AGL(1,ℤ/8)—21no—
252#533—531no—
253#532C2×D5311yescharacteristic
254C2×#127C2×D5311yescharacteristic
255#496—22no—
256#493C2×D5312yes—
257#492—1061no—
258#492C2×D5312yes—
259#486—1061no—
260#476—1061no—
261#475—532no—
262#465—1062no—
263#461—1061no—
264#460—532no—
265#460—532no—
266#460—532no—
267#450—1062no—
268#446—532no—
269#446—1062no—
270C8×Q8—532no—
271#431—531no—
272#424—1061no—
273C2×C4⋊C8—531no—
274C2×C4⋊C8—531no—
275#405—1061no—
276C4×M16C2×D5311yescharacteristic
277#335—532no—
278#334—532no—
279#332—531no—
280#332—531no—
281#331—532no—
282#331—532no—
283#331—531no—
284#331—531no—
285C53#1923811yescharacteristic, complemented, Sylow
286C2×#41—21no—
287C22×D4—21no—
288C22×D4—21no—
289C22×D4—21no—
290AGL(1,ℤ/8)—44no—
291AGL(1,ℤ/8)—44no—
292C2×SD16—22no—
293C2×SD16—22no—
294C2×SD16—1062no—
295C2×SD16—1062no—
296C2×SD16—1062no—
297C2×D8—22no—
298C2×D8—22no—
299C2×M16C22⋊D5311yescharacteristic
300C2×M16C22⋊D5311yescharacteristic
301C22×C8—1061no—
302C22×C8—1061no—
303C4⋊Q8—531no—
304C4⋊Q8—531no—
305C4⋊Q8C22×D5311yescharacteristic
306#127—21no—
307#127—21no—
308#127—21no—
309#127C22×D5312yes—
310#127C22×D5311yescharacteristic
311#125—1061no—
312#124—21no—
313C22⋊Q8—532no—
314C22⋊Q8—1061no—
315#121—21no—
316#121—21no—
317C4×Q8—532no—
318C4×Q8—1061no—
319C4×D4—21no—
320C4×D4—21no—
321#117—1061no—
322C2×C42C22×D5311yescharacteristic
323#106—1062no—
324C4⋊C8—531no—
325C4⋊C8—531no—
326C4⋊C8—532no—
327C4⋊C8—531no—
328C4⋊C8—531no—
329C4⋊C8—1062no—
330C4⋊C8—1062no—
331C4⋊C8—1061no—
332C4⋊C8—1061no—
333#103—1062no—
334#102—1062no—
335#102—1062no—
336#102—1062no—
337C22⋊C8—1061no—
338#97—1062no—
339#97C22⋊D5312yes—
340C4×C8—1062no—
341C4×C8—1061no—
342C4×C8C22×D5312yes—
343C24—41no—
344#41—42no—
345#41—42no—
346C2×Q8—21no—
347C2×Q8—532no—
348C2×Q8—1061no—
349C2×D4—21no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—22no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—81no—
362C2×D4—81no—
363C2×D4—82no—
364C22×C4—41no—
365C22×C4D4×D5311yescharacteristic
366C22×C4#1171994711yescharacteristic
367C22×C4C2×C22⋊D5311yescharacteristic
368SD16—44no—
369SD16—44no—
370SD16—4242no—
371SD16—4242no—
372SD16—4242no—
373D8—44no—
374D8—44no—
375M16—22no—
376M16#1171992314yescomplemented
377C2×C8—1061no—
378C2×C8—1061no—
379C2×C8—1062no—
380C2×C8—1061no—
381C2×C8—1061no—
382C2×C8—4241no—
383C2×C8—2122no—
384C2×C8C2×C22⋊D5312yes—
385C2×C8C2×C22⋊D5312yes—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—532no—
389C4⋊C4—1061no—
390C4⋊C4—1061no—
391C4⋊C4—1062no—
392C4⋊C4—1061no—
393C4⋊C4—1062no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—1062no—
397C42—1062no—
398C42D4×D5311yescharacteristic
399C42#1171994711yescharacteristic
400C42C2×C22⋊D5311yescharacteristic
401C42C23×D5311yescharacteristic
402C23—41no—
403C23—42no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23#1313318811yescharacteristic
408Q8—41no—
409Q8—41no—
410Q8—1062no—
411Q8—1062no—
412Q8—4241no—
413D4—41no—
414D4—42no—
415D4—41no—
416D4—42no—
417D4—41no—
418D4—42no—
419D4—42no—
420D4—41no—
421D4—81no—
422D4—82no—
423D4—82no—
424D4—82no—
425D4—82no—
426D4—82no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—82no—
432C2×C4—1062no—
433C2×C4—1062no—
434C2×C4#1313318811yescharacteristic
435C2×C4#1313318811yescharacteristic
436C2×C4#1313318811yescharacteristic
437C2×C4C2×#1171992312yes—
438C2×C4#1313324312yes—
439C2×C4C2×D4×D5311yescharacteristic
440C2×C4C22×C22⋊D5311yescharacteristic
441C2×C4C2×#1171994711yescharacteristic
442C8—22no—
443C8—2121no—
444C8—2121no—
445C8—2122no—
446C8—2121no—
447C8—2121no—
448C8C2×#1171992314yes—
449C22—21no—
450C22—21no—
451C22—82no—
452C22—81no—
453C22—82no—
454C22—81no—
455C22—81no—
456C22—82no—
457C22—164no—
458C22#1681082012yes—
459C22C2×#1313318811yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—81no—
464C4—2122no—
465C4—2122no—
466C4#1681082012yes—
467C4C22×#1171992311yescharacteristic
468C4C2×#1313324311yescharacteristic
469C4#1681142711yescharacteristic
470C4#1681143311yescharacteristic
471C2—22no—
472C2—162no—
473C2—161no—
474C2C2×#1681082011yescharacteristic
475C2#2028438911yescharacteristic
476C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3151C2
socleC22×C53C2×#121
hypercenter#3151D53
hypocenterC53#19238
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. #20286031C2semidirect
2. #20285656C2semidirect
3. #20283662C2semidirect
4. #20274196C2semidirect
5. #20274184C2semidirect
6. #20274072C2semidirect
7. #20274060C2semidirect
8. #20273467C2semidirect
9. #20273364C2semidirect
10. C2×#16810367C2semidirect
16. #16810905C22semidirect
21. #16810820C22semidirect
23. #16810753C22semidirect
25. C8×Dic106C22semidirect
31. #16810376C22semidirect
32. #16810374C22semidirect
33. #16810369C22semidirect
34. #16810369C22semidirect
37. #16810367C22semidirect
38. #16810367C22semidirect
134. M16×C53SD16semidirect
285. C53#19238semidirect
376. M16#11719923semidirect

Maximal subgroups

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

subgroupindex
#202860312
#202856562
#202836622
#202741962
#202741842
#202740722
#202740602
#202734672
#202733642
C2×#168103672
C53×#31512
#1923853