Finite Group Databasebeta

← #27530976

Subgroups — #27530976

Order 13568.

480 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276298C211yesmaximal, characteristic, complemented
2#20276297C211yesmaximal, characteristic, complemented
3#20276292C211yesmaximal, characteristic, complemented
4#20276169C211yesmaximal, characteristic, complemented
5#20276165C211yesmaximal, characteristic, complemented
6#20275999C211yesmaximal, characteristic
7#20275422C211yesmaximal, characteristic, complemented
8#20275421C211yesmaximal, characteristic, complemented
9#20275079C211yesmaximal, characteristic
10#20274955C211yesmaximal, characteristic, complemented
11#20274953C211yesmaximal, characteristic, complemented
12#20274935C211yesmaximal, characteristic, complemented
13#20274934C211yesmaximal, characteristic, complemented
14#20274851C211yesmaximal, characteristic, complemented
15C53×#1881C211yesmaximal, characteristic
16#16811390C2211yescharacteristic
17#16810671C2211yescharacteristic
18#16810671C2211yescharacteristic
19#16810671C2211yescharacteristic
20#16810650C2211yescharacteristic
21#16810649C2211yescharacteristic
22#16810646C2211yescharacteristic
23#16810646C2211yescharacteristic
24#16810640C2211yescharacteristic
25C2×#13133144C2211yescharacteristic
26C2×#13133144C2211yescharacteristic
27#16810619C2211yescharacteristic
28#16810607C2211yescharacteristic
29#16810607C2211yescharacteristic
30#16810590C2211yescharacteristic
31C2×C8×Dic53C2211yescharacteristic
32#16810395C2211yescharacteristic
33#16810394C2211yescharacteristic
34#16810393C2211yescharacteristic
35#16810388C2211yescharacteristic
36#16810388C2211yescharacteristic
37#16810387C2211yescharacteristic
38#16810387C2211yescharacteristic
39#16810387C2211yescharacteristic
40#16810384C2211yescharacteristic
41#16810382C2211yescharacteristic
42#16810381C2211yescharacteristic
43#16810379C2211yescharacteristic
44C53×#575C2211yescharacteristic
45C53×#568C2211yescharacteristic
46C53×#421C2211yescharacteristic
47C53×#416C2211yescharacteristic
48C53×#408C2211yescharacteristic
49C53×#344C2211yescharacteristic
50C53×#341C2211yescharacteristic
51#13133224—41no—
52#13133224—41no—
53C2×#11719917—41no—
54C2×#11719917C2311yescharacteristic
55C2×#11719917C2311yescharacteristic
56C2×#11719917C2311yescharacteristic
57C2×#11719917C2311yescharacteristic
58C2×#11719917C2311yescharacteristic
59C22×#88322C2311yescharacteristic
60#13133176C2311yescharacteristic
61C2×C4×Dic53C2311yescharacteristic
62#13133152—41no—
63#13133152—41no—
64#13133152—41no—
65C4⋊#88322—41no—
66#13133146—41no—
67#13133146—41no—
68#13133146—41no—
69#13133144—21no—
70#13133144—21no—
71#13133144—21no—
72#13133144—21no—
73#13133144—41no—
74#13133141—21no—
75#13133138—21no—
76#13133136—21no—
77#13133136—21no—
78#13133134—21no—
79#13133134—21no—
80C22⋊#88322—41no—
81C8×Dic53—21no—
82C8×Dic53—21no—
83C2×#41×C53C2311yescharacteristic
84C2×#41×C53C2311yescharacteristic
85#137×C53—21no—
86#137×C53—21no—
87AGL(1,ℤ/8)×C53—21no—
88AGL(1,ℤ/8)×C53—21no—
89#135×C53—41no—
90#131×C53—41no—
91#131×C53—41no—
92C2×M16×C53—41no—
93C2×M16×C53C2311yescharacteristic
94C2×M16×C53C2311yescharacteristic
95C2×M16×C53C2311yescharacteristic
96C22×C8×C53C2311yescharacteristic
97#117×C53C2311yescharacteristic
98#104×C53—41no—
99#103×C53—21no—
100#102×C53—21no—
101C22⋊C8×C53—41no—
102#41×C53—21no—
103#41×C53—21no—
104#41×C53—21no—
105#41×C53—21no—
106#41×C53—21no—
107#41×C53—21no—
108#41×C53—21no—
109#41×C53—21no—
110C2×Q8×C53#4111yescharacteristic
111C2×Q8×C53#4111yescharacteristic
112C2×D4×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53#4111yescharacteristic
115C2×D4×C53#4111yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53C2411yescharacteristic
119C22×Dic53—21no—
120Q16×C53—41no—
121SD16×C53—41no—
122SD16×C53—41no—
123D8×C53—41no—
124M16×C53—21no—
125M16×C53—21no—
126M16×C53—21no—
127M16×C53—21no—
128M16×C53—21no—
129M16×C53—21no—
130M16×C53—42no—
131C2×C8×C53—21no—
132C2×C8×C53—21no—
133C2×C8×C53—21no—
134C2×C8×C53—21no—
135C2×C8×C53—21no—
136C2×C8×C53—21no—
137C2×C8×C53—42no—
138C4⋊C4×C53—21no—
139C22⋊C4×C53—21no—
140C42×C53—21no—
141C22⋊Dic53—21no—
142C4⋊Dic53—21no—
143C4×Dic53—21no—
144C4×Dic53—21no—
145C4×Dic53#4111yescharacteristic
146C4×Dic53#4111yescharacteristic
147#11719917—21no—
148#11719917—21no—
149#11719917—21no—
150#11719917—21no—
151#11719917—21no—
152#11719917—21no—
153#11719917—21no—
154#11719917—21no—
155#11719917—21no—
156#11719917—21no—
157#11719917—42no—
158C2×#88322—21no—
159C2×#88322—21no—
160C2×#88322—21no—
161C2×#88322—21no—
162C2×#88322—21no—
163C2×#88322—21no—
164C2×#88322—21no—
165C2×#88322—21no—
166C2×#88322—42no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C532−1+411yescharacteristic
170Q8×C53—21no—
171Q8×C53—21no—
172Q8×C53—21no—
173Q8×C53—21no—
174D4×C53—21no—
175D4×C53—21no—
176D4×C53—21no—
177D4×C53—21no—
178D4×C53—41no—
179D4×C53—41no—
180D4×C53—41no—
181D4×C53—41no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53C2×#4111yescharacteristic
190C2×C4×C53C2×#4111yescharacteristic
191C2×C4×C53C2×#4111yescharacteristic
192C2×C4×C53H(5,𝔽2)11yescharacteristic
193C2×C4×C53H(5,𝔽2)11yescharacteristic
194C2×C4×C53H(5,𝔽2)11yescharacteristic
195C2×Dic53—21no—
196C2×Dic53—21no—
197C2×Dic53—42no—
198C2×Dic53—41no—
199C8×C53—41no—
200C8×C53—41no—
201C8×C53—41no—
202C8×C53—41no—
203#88322—41no—
204#88322—41no—
205#88322—41no—
206#88322—41no—
207#88322—41no—
208#88322—41no—
209#32702—531nomaximal, Sylow
210C22×C53—41no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—81no—
214C22×C53—81no—
215C22×C53#53911yescharacteristic
216C22×C53#53911yescharacteristic
217C22×C53#53911yescharacteristic
218C4×C53—41no—
219C4×C53—41no—
220C4×C53—81no—
221C4×C53#55111yescharacteristic
222C4×C53#55311yescharacteristic
223C4×C53#55311yescharacteristic
224C4×C53H(3,𝔽4)11yescharacteristic
225Dic53—22no—
226Dic53—41no—
227Dic53—81no—
228#2103—531no—
229#2103—531no—
230#2102—531no—
231#2080—531no—
232#2080—531no—
233#2050—531no—
234#1937—531no—
235#1937—531no—
236#1881—531no—
237#1881D5311yescharacteristic
238#1861—531no—
239#1861—531no—
240#1858—531no—
241#1858—531no—
242#1837—531no—
243C2×C53—21no—
244C2×C53—21no—
245C2×C53—21no—
246C2×C53—81no—
247C2×C53—81no—
248C2×C53#278211yescharacteristic
249#575C2×D5311yescharacteristic
250#568—531no—
251#568C2×D5311yescharacteristic
252#430—531no—
253#430—531no—
254#430—531no—
255#424—531no—
256#424—531no—
257#423—531no—
258#423—531no—
259#421—531no—
260#421C2×D5311yescharacteristic
261C2×#104—531no—
262C2×#104—531no—
263#416—531no—
264#416C2×D5311yescharacteristic
265#413—531no—
266#413—531no—
267#408—531no—
268#408C2×D5311yescharacteristic
269C2×C4×C8—531no—
270#344—531no—
271#344—531no—
272#344—531no—
273#344C2×D5311yescharacteristic
274#341—531no—
275#341—531no—
276#341—531no—
277#341—531no—
278#341—531no—
279#341C2×D5311yescharacteristic
280#339—531no—
281#339—531no—
282#338—531no—
283H(3,ℤ/4)—531no—
284C53#3270211yescharacteristic, complemented, Sylow
285C2×#41C22×D5311yescharacteristic
286C2×#41C22×D5311yescharacteristic
287#137—21no—
288#137—21no—
289AGL(1,ℤ/8)—21no—
290AGL(1,ℤ/8)—21no—
291#135—41no—
292#131—41no—
293#131—41no—
294#131—2121no—
295#131—2121no—
296C2×M16—41no—
297C2×M16—531no—
298C2×M16—531no—
299C2×M16—531no—
300C2×M16—531no—
301C2×M16—531no—
302C2×M16—2121no—
303C2×M16C22×D5311yescharacteristic
304C2×M16C22×D5311yescharacteristic
305C2×M16C22×D5311yescharacteristic
306C22×C8—531no—
307C22×C8C22×D5311yescharacteristic
308#117—531no—
309#117C22×D5311yescharacteristic
310C2×C42—531no—
311#108—2121no—
312#108—2121no—
313#108—2121no—
314C4⋊C8—2121no—
315C4⋊C8—2121no—
316C4⋊C8—2121no—
317C4⋊C8—2121no—
318#104—41no—
319#104—1061no—
320#104—1061no—
321#104—1061no—
322#104—1061no—
323#104—2121no—
324#103—21no—
325#103—1061no—
326#102—21no—
327#102—1061no—
328#101—1061no—
329#101—1061no—
330#100—1061no—
331#100—1061no—
332C22⋊C8—41no—
333C22⋊C8—2121no—
334C4×C8—1061no—
335C4×C8—1061no—
336#41—21no—
337#41—21no—
338#41—21no—
339#41—21no—
340#41—21no—
341#41—21no—
342#41—21no—
343#41—21no—
344C2×Q8#1171994511yescharacteristic
345C2×Q8#1171994711yescharacteristic
346C2×D4—21no—
347C2×D4—21no—
348C2×D4#1171994511yescharacteristic
349C2×D4#1171994711yescharacteristic
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—1061no—
353C22×C4C23×D5311yescharacteristic
354Q16—41no—
355SD16—41no—
356SD16—41no—
357D8—41no—
358M16—21no—
359M16—21no—
360M16—21no—
361M16—21no—
362M16—21no—
363M16—21no—
364M16—42no—
365M16—1061no—
366M16—1061no—
367M16—1061no—
368M16—1061no—
369M16—1061no—
370M16—1061no—
371M16—1061no—
372M16—1061no—
373M16—1061no—
374M16—1061no—
375M16—2122no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—42no—
383C2×C8—1061no—
384C2×C8—1061no—
385C2×C8—1061no—
386C2×C8—1061no—
387C2×C8—1061no—
388C2×C8—1061no—
389C2×C8—1061no—
390C2×C8—1061no—
391C2×C8—2122no—
392C4⋊C4—21no—
393C4⋊C4—1061no—
394C22⋊C4—21no—
395C22⋊C4—1061no—
396C42—21no—
397C42—531no—
398C42—531no—
399C42—1061no—
400C42—1061no—
401C23—21no—
402C23—21no—
403C23#1313326411yescharacteristic
404Q8—21no—
405Q8—21no—
406Q8—21no—
407Q8—21no—
408D4—21no—
409D4—21no—
410D4—21no—
411D4—21no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—41no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—1061no—
424C2×C4—1061no—
425C2×C4—2122no—
426C2×C4—2121no—
427C2×C4#41×D5311yescharacteristic
428C2×C4C2×#1171994711yescharacteristic
429C2×C4C2×#1171994511yescharacteristic
430C2×C4#1313326211yescharacteristic
431C2×C4#1313326211yescharacteristic
432C2×C4#1313326211yescharacteristic
433C8—41no—
434C8—41no—
435C8—41no—
436C8—41no—
437C8—2121no—
438C8—2121no—
439C8—2121no—
440C8—2121no—
441C8—2121no—
442C8—2121no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—81no—
447C22—81no—
448C22#1681121511yescharacteristic
449C22#1681121611yescharacteristic
450C22#1681121811yescharacteristic
451C4—41no—
452C4—41no—
453C4—81no—
454C4—1062no—
455C4—4241no—
456C4—2121no—
457C4#1681129911yescharacteristic
458C4#1681131611yescharacteristic
459C4#1681132211yescharacteristic
460C4#1681136611yescharacteristic
461C2—21no—
462C2—21no—
463C2—21no—
464C2—81no—
465C2—81no—
466C2#2028238811yescharacteristic
subgroupG/H
centerC4#16811366
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1881C2
socleC2×C53#2782
prime radicalC53×#575C22
hypercenter#1881D53
hypocenterC53#32702
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1881

Complements

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

NH ≅ G/Ntype
1. #20276298C2semidirect
2. #20276297C2semidirect
3. #20276292C2semidirect
4. #20276169C2semidirect
5. #20276165C2semidirect
7. #20275422C2semidirect
8. #20275421C2semidirect
10. #20274955C2semidirect
11. #20274953C2semidirect
12. #20274935C2semidirect
13. #20274934C2semidirect
14. #20274851C2semidirect
284. C53#32702semidirect

Maximal subgroups

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

subgroupindex
#202762982
#202762972
#202762922
#202761692
#202761652
#202759992
#202754222
#202754212
#202750792
#202749552
#202749532
#202749352
#202749342
#202748512
C53×#18812
#3270253