Finite Group Databasebeta

← #27530985

Subgroups — #27530985

Order 13568.

476 conjugacy classes of subgroups.

The subgroup lattice has 456 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#20275429C211yesmaximal, characteristic, complemented
8#20275428C211yesmaximal, characteristic, complemented
9#20275079C211yesmaximal, characteristic
10#20274955C211yesmaximal, characteristic, complemented
11#20274953C211yesmaximal, characteristic, complemented
12#20274935C211yesmaximal, characteristic, complemented
13#20274934C211yesmaximal, characteristic, complemented
14#20274854C211yesmaximal, 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×#13133127C2211yescharacteristic
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—22no—
70#13133144—22no—
71#13133144—41no—
72#13133141—21no—
73#13133138—21no—
74#13133136—21no—
75#13133136—21no—
76#13133134—21no—
77#13133134—21no—
78C22⋊#88322—41no—
79#13133127—22no—
80C2×#41×C53C2311yescharacteristic
81C2×#41×C53C2311yescharacteristic
82#137×C53—21no—
83#137×C53—21no—
84AGL(1,ℤ/8)×C53—21no—
85AGL(1,ℤ/8)×C53—21no—
86#135×C53—41no—
87#131×C53—41no—
88#131×C53—41no—
89C2×M16×C53—41no—
90C2×M16×C53C2311yescharacteristic
91C2×M16×C53C2311yescharacteristic
92C2×M16×C53C2311yescharacteristic
93C22×C8×C53C2311yescharacteristic
94#117×C53C2311yescharacteristic
95#104×C53—41no—
96#103×C53—21no—
97#102×C53—21no—
98C22⋊C8×C53—41no—
99#41×C53—21no—
100#41×C53—21no—
101#41×C53—21no—
102#41×C53—21no—
103#41×C53—21no—
104#41×C53—21no—
105#41×C53—21no—
106#41×C53—21no—
107C2×Q8×C53#4111yescharacteristic
108C2×Q8×C53#4111yescharacteristic
109C2×D4×C53—21no—
110C2×D4×C53—21no—
111C2×D4×C53#4111yescharacteristic
112C2×D4×C53#4111yescharacteristic
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53C2411yescharacteristic
116C22×Dic53—21no—
117Q16×C53—41no—
118SD16×C53—41no—
119SD16×C53—41no—
120D8×C53—41no—
121M16×C53—21no—
122M16×C53—21no—
123M16×C53—21no—
124M16×C53—21no—
125M16×C53—21no—
126M16×C53—21no—
127M16×C53—42no—
128C2×C8×C53—21no—
129C2×C8×C53—21no—
130C2×C8×C53—21no—
131C2×C8×C53—21no—
132C2×C8×C53—21no—
133C2×C8×C53—21no—
134C2×C8×C53—42no—
135C4⋊C4×C53—21no—
136C22⋊C4×C53—21no—
137C42×C53—21no—
138C22⋊Dic53—21no—
139C4⋊Dic53—21no—
140C4×Dic53—21no—
141C4×Dic53—21no—
142C4×Dic53#4112yes—
143#11719917—21no—
144#11719917—21no—
145#11719917—21no—
146#11719917—21no—
147#11719917—21no—
148#11719917—21no—
149#11719917—21no—
150#11719917—21no—
151#11719917—21no—
152#11719917—21no—
153#11719917—42no—
154C2×#88322—21no—
155C2×#88322—21no—
156C2×#88322—21no—
157C2×#88322—21no—
158C2×#88322—21no—
159C2×#88322—21no—
160C2×#88322—21no—
161C2×#88322—21no—
162C2×#88322—42no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C532−1+411yescharacteristic
166Q8×C53—21no—
167Q8×C53—21no—
168Q8×C53—21no—
169Q8×C53—21no—
170D4×C53—21no—
171D4×C53—21no—
172D4×C53—21no—
173D4×C53—21no—
174D4×C53—41no—
175D4×C53—41no—
176D4×C53—41no—
177D4×C53—41no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53C2×#4111yescharacteristic
186C2×C4×C53C2×#4111yescharacteristic
187C2×C4×C53C2×#4111yescharacteristic
188C2×C4×C53H(5,𝔽2)11yescharacteristic
189C2×C4×C53H(5,𝔽2)11yescharacteristic
190C2×C4×C53H(5,𝔽2)11yescharacteristic
191C2×Dic53—22no—
192C2×Dic53—41no—
193C2×Dic53—81no—
194C8×C53—41no—
195C8×C53—41no—
196C8×C53—41no—
197C8×C53—41no—
198#88322—41no—
199#88322—41no—
200#88322—41no—
201#88322—41no—
202#88322—41no—
203#88322—41no—
204#32703—531nomaximal, Sylow
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—81no—
209C22×C53—81no—
210C22×C53#53911yescharacteristic
211C22×C53#53911yescharacteristic
212C22×C53#53911yescharacteristic
213C4×C53—41no—
214C4×C53—41no—
215C4×C53—81no—
216C4×C53#55111yescharacteristic
217C4×C53#55311yescharacteristic
218C4×C53#55311yescharacteristic
219C4×C53H(3,𝔽4)11yescharacteristic
220Dic53—42no—
221Dic53—81no—
222#2103—531no—
223#2103—531no—
224#2102—531no—
225#2080—531no—
226#2080—531no—
227#2050—531no—
228#1938—531no—
229#1938—531no—
230#1881—531no—
231#1881D5311yescharacteristic
232#1861—531no—
233#1861—531no—
234#1858—531no—
235#1858—531no—
236#1838—531no—
237C2×C53—21no—
238C2×C53—21no—
239C2×C53—21no—
240C2×C53—81no—
241C2×C53—81no—
242C2×C53#278211yescharacteristic
243#575C2×D5311yescharacteristic
244#568—531no—
245#568C2×D5311yescharacteristic
246#430—531no—
247#430—531no—
248#430—531no—
249#424—531no—
250#424—531no—
251#423—531no—
252#423—531no—
253#421—531no—
254#421C2×D5311yescharacteristic
255C2×#104—531no—
256C2×#104—531no—
257#416—531no—
258#416C2×D5311yescharacteristic
259#413—531no—
260#413—531no—
261#408—531no—
262#408C2×D5311yescharacteristic
263C2×#97—531no—
264#344—531no—
265#344—531no—
266#344—531no—
267#344C2×D5311yescharacteristic
268#341—531no—
269#341—531no—
270#341—531no—
271#341—531no—
272#341—531no—
273#341C2×D5311yescharacteristic
274#339—531no—
275#339—531no—
276#338—531no—
277H(3,ℤ/4)—531no—
278C53#3270311yescharacteristic, complemented, Sylow
279C2×#41C22×D5311yescharacteristic
280C2×#41C22×D5311yescharacteristic
281#137—21no—
282#137—21no—
283AGL(1,ℤ/8)—21no—
284AGL(1,ℤ/8)—21no—
285#135—41no—
286#131—41no—
287#131—41no—
288#131—2121no—
289#131—2121no—
290C2×M16—41no—
291C2×M16—531no—
292C2×M16—531no—
293C2×M16—531no—
294C2×M16—531no—
295C2×M16—531no—
296C2×M16—2121no—
297C2×M16C22×D5311yescharacteristic
298C2×M16C22×D5311yescharacteristic
299C2×M16C22×D5311yescharacteristic
300C22×C8—531no—
301C22×C8C22×D5311yescharacteristic
302#117—531no—
303#117C22×D5311yescharacteristic
304C2×C42—531no—
305#108—2121no—
306#108—2121no—
307#108—2121no—
308C4⋊C8—2121no—
309C4⋊C8—2121no—
310C4⋊C8—2121no—
311C4⋊C8—2121no—
312#104—41no—
313#104—1062no—
314#104—1062no—
315#104—2121no—
316#103—21no—
317#103—1061no—
318#102—21no—
319#102—1061no—
320#101—1061no—
321#101—1061no—
322#100—1061no—
323#100—1061no—
324C22⋊C8—41no—
325C22⋊C8—2121no—
326#97—1062no—
327#41—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331#41—21no—
332#41—21no—
333#41—21no—
334#41—21no—
335C2×Q8#1171994511yescharacteristic
336C2×Q8#1171994711yescharacteristic
337C2×D4—21no—
338C2×D4—21no—
339C2×D4#1171994511yescharacteristic
340C2×D4#1171994711yescharacteristic
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—1061no—
344C22×C4C23×D5311yescharacteristic
345Q16—41no—
346SD16—41no—
347SD16—41no—
348D8—41no—
349M16—21no—
350M16—21no—
351M16—21no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—42no—
356M16—1061no—
357M16—1061no—
358M16—1061no—
359M16—1061no—
360M16—1061no—
361M16—1061no—
362M16—1061no—
363M16—1061no—
364M16—1061no—
365M16—1061no—
366M16—2122no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—42no—
374C2×C8—1061no—
375C2×C8—1061no—
376C2×C8—1061no—
377C2×C8—1061no—
378C2×C8—1061no—
379C2×C8—1061no—
380C2×C8—1061no—
381C2×C8—1061no—
382C2×C8—2122no—
383C4⋊C4—21no—
384C4⋊C4—1061no—
385C22⋊C4—21no—
386C22⋊C4—1061no—
387C42—21no—
388C42—532no—
389C42—1061no—
390C42—1061no—
391C23—21no—
392C23—21no—
393C23#1313326411yescharacteristic
394Q8—21no—
395Q8—21no—
396Q8—21no—
397Q8—21no—
398D4—21no—
399D4—21no—
400D4—21no—
401D4—21no—
402D4—41no—
403D4—41no—
404D4—41no—
405D4—41no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—1062no—
414C2×C4—4241no—
415C2×C4—2121no—
416C2×C4#41×D5311yescharacteristic
417C2×C4C2×#1171994711yescharacteristic
418C2×C4C2×#1171994511yescharacteristic
419C2×C4#1313326211yescharacteristic
420C2×C4#1313326211yescharacteristic
421C2×C4#1313326211yescharacteristic
422C8—41no—
423C8—41no—
424C8—41no—
425C8—41no—
426C8—2121no—
427C8—2121no—
428C8—2121no—
429C8—2121no—
430C8—2121no—
431C8—2121no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22#1681121511yescharacteristic
438C22#1681121611yescharacteristic
439C22#1681121811yescharacteristic
440C4—41no—
441C4—41no—
442C4—81no—
443C4—4241no—
444C4—2122no—
445C4#1681129911yescharacteristic
446C4#1681131611yescharacteristic
447C4#1681132211yescharacteristic
448C4#1681136611yescharacteristic
449C2—21no—
450C2—21no—
451C2—21no—
452C2—81no—
453C2—81no—
454C2#2028238811yescharacteristic
subgroupG/H
centerC4#16811366
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1881C2
socleC2×C53#2782
prime radicalC53×#575C22
hypercenter#1881D53
hypocenterC53#32703
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. #20275429C2semidirect
8. #20275428C2semidirect
10. #20274955C2semidirect
11. #20274953C2semidirect
12. #20274935C2semidirect
13. #20274934C2semidirect
14. #20274854C2semidirect
278. C53#32703semidirect

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
#202754292
#202754282
#202750792
#202749552
#202749532
#202749352
#202749342
#202748542
C53×#18812
#3270353