Finite Group Databasebeta

← #27530984

Subgroups — #27530984

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#20276294C211yesmaximal, characteristic, complemented
3#20276293C211yesmaximal, characteristic, complemented
4#20276170C211yesmaximal, characteristic, complemented
5#20276165C211yesmaximal, characteristic, complemented
6#20276001C211yesmaximal, characteristic
7#20275427C211yesmaximal, characteristic, complemented
8#20275423C211yesmaximal, characteristic, complemented
9#20275077C211yesmaximal, characteristic
10#20275074C211yesmaximal, characteristic
11#20274954C211yesmaximal, characteristic, complemented
12#20274935C211yesmaximal, characteristic, complemented
13#20274933C211yesmaximal, characteristic, complemented
14#20274855C211yesmaximal, characteristic, complemented
15C53×#1861C211yesmaximal, characteristic, complemented
16#16811421C2211yescharacteristic
17#16811389C2211yescharacteristic
18#16810671C2211yescharacteristic
19#16810671C2211yescharacteristic
20#16810651C2211yescharacteristic
21#16810650C2211yescharacteristic
22#16810647C2211yescharacteristic
23#16810639C2211yescharacteristic
24C2×#13133145C2211yescharacteristic
25#16810619C2211yescharacteristic
26#16810617C2211yescharacteristic
27#16810607C2211yescharacteristic
28#16810605C2211yescharacteristic
29#16810591C2211yescharacteristic
30#16810590C2211yescharacteristic
31C2×#13133128C2211yescharacteristic
32#16810394C2211yescharacteristic
33#16810394C2211yescharacteristic
34#16810393C2211yescharacteristic
35#16810387C2211yescharacteristic
36#16810387C2211yescharacteristic
37#16810387C2211yescharacteristic
38#16810387C2211yescharacteristic
39#16810387C2211yescharacteristic
40#16810387C2211yescharacteristic
41#16810383C2211yescharacteristic
42#16810381C2211yescharacteristic
43#16810379C2211yescharacteristic
44C53×#568C2211yescharacteristic
45C53×#430C2211yescharacteristic
46C53×#423C2211yescharacteristic
47C53×#421C2211yescharacteristic
48C2×#104×C53C2211yescharacteristic
49C53×#344C2211yescharacteristic
50C53×#339C2211yescharacteristic
51C2×#11719945C2311yescharacteristic
52#13133248—21no—
53#13133247—21no—
54#13133243—21no—
55#13133242—21no—
56#13133237—41no—
57#13133225—41no—
58#13133223—41no—
59C2×#11719917C2311yescharacteristic
60C2×#11719917C2311yescharacteristic
61C2×#11719917C2311yescharacteristic
62C2×#11719917C2311yescharacteristic
63M16×D53—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133176C2311yescharacteristic
67#13133152—41no—
68#13133152—41no—
69C4⋊#88322—41no—
70C4⋊#88322—41no—
71#13133146—41no—
72#13133145—22no—
73#13133143—41no—
74#13133141—21no—
75#13133140—21no—
76#13133138—21no—
77#13133137—21no—
78#13133136—21no—
79#13133135—21no—
80#13133134—21no—
81#13133133—21no—
82C22⋊#88322—41no—
83#13133129—41no—
84#13133128—22no—
85C2×#41×C53C2311yescharacteristic
86#131×C53—41no—
87#131×C53—41no—
88C2×M16×C53—41no—
89C2×M16×C53C2311yescharacteristic
90C2×M16×C53C2311yescharacteristic
91C2×M16×C53C2311yescharacteristic
92C2×M16×C53C2311yescharacteristic
93#117×C53C2311yescharacteristic
94C2×C42×C53C2311yescharacteristic
95#108×C53—41no—
96C4⋊C8×C53—41no—
97#104×C53—22no—
98#104×C53—41no—
99#41×C53—21no—
100#41×C53—21no—
101#41×C53—21no—
102#41×C53—21no—
103C2×Q8×C53#4111yescharacteristic
104C2×D4×C53—21no—
105C2×D4×C53#4111yescharacteristic
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53C2411yescharacteristic
109C2×C22⋊D53—21no—
110#11719945—21no—
111#11719945—21no—
112#11719945—21no—
113#11719945—21no—
114C2×D212#4111yescharacteristic
115C2×C4×D53—21no—
116C2×Dic106#4111yescharacteristic
117M16×C53—21no—
118M16×C53—21no—
119M16×C53—21no—
120M16×C53—21no—
121M16×C53—21no—
122M16×C53—21no—
123M16×C53—21no—
124M16×C53—21no—
125M16×C53—42no—
126C2×C8×C53—21no—
127C2×C8×C53—21no—
128C2×C8×C53—21no—
129C2×C8×C53—21no—
130C2×C8×C53—42no—
131C4⋊C4×C53—21no—
132C22⋊C4×C53—21no—
133C42×C53—21no—
134C42×C53—21no—
135C42×C53#4112yes—
136C22⋊Dic53—21no—
137#11719925—41no—
138#11719924—41no—
139#11719923—41no—
140#11719922—41no—
141C4⋊Dic53—21no—
142C4×Dic53—21no—
143#11719917—21no—
144#11719917—21no—
145#11719917—21no—
146#11719917—21no—
147#11719917—21no—
148#11719917—21no—
149#11719917—21no—
150#11719917—21no—
151C2×#88322—21no—
152C2×#88322—21no—
153C2×#88322—21no—
154C2×#88322—21no—
155C2×#88322—21no—
156C2×#88322—21no—
157C2×#88322—21no—
158C2×#88322—21no—
159C2×#88322—21no—
160C2×#88322—21no—
161#11719912—42no—
162C8×D53—42no—
163C23×C53—21no—
164C23×C532−1+411yescharacteristic
165C22×D53—21no—
166Q8×C53—21no—
167Q8×C53—21no—
168D4×C53—21no—
169D4×C53—21no—
170D4×C53—41no—
171D4×C53—41no—
172C2×C4×C53—21no—
173C2×C4×C53—22no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—81no—
178C2×C4×C53C2×#4111yescharacteristic
179C2×C4×C53C2×#4111yescharacteristic
180C2×C4×C53C2×#4111yescharacteristic
181C2×C4×C53H(5,𝔽2)11yescharacteristic
182C2×C4×C53H(5,𝔽2)11yescharacteristic
183C2×C4×C53H(5,𝔽2)11yescharacteristic
184C22⋊D53—41no—
185C22⋊D53—41no—
186C2×Dic53—21no—
187C2×Dic53—41no—
188D212—21no—
189D212—21no—
190C4×D53—41no—
191C4×D53—41no—
192Dic106—21no—
193Dic106—21no—
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—81no—
208C22×C53#53911yescharacteristic
209C22×C53#53911yescharacteristic
210C22×C53#53911yescharacteristic
211C2×D53—41no—
212C2×D53—81no—
213C4×C53—42no—
214C4×C53—41no—
215C4×C53—81no—
216C4×C53#55111yescharacteristic
217C4×C53#55311yescharacteristic
218C4×C53#55311yescharacteristic
219C4×C53H(3,𝔽4)11yescharacteristic
220Dic53—41no—
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#1881—531no—
232#1861—531no—
233#1861D5311yescharacteristic, complemented
234#1858—531no—
235#1858—531no—
236#1838—531no—
237C2×C53—21no—
238C2×C53—21no—
239C2×C53—21no—
240C2×C53—81no—
241C2×C53#278211yescharacteristic
242D53—81no—
243#575—531no—
244#568—531no—
245#568C2×D5311yescharacteristic
246#430—531no—
247#430—531no—
248#430C2×D5311yescharacteristic
249#424—531no—
250#424—531no—
251#423—531no—
252#423C2×D5311yescharacteristic
253#421—531no—
254#421C2×D5311yescharacteristic
255C2×#104—531no—
256C2×#104C2×D5311yescharacteristic
257#416—531no—
258#416—531no—
259#413—531no—
260#413—531no—
261#408—531no—
262#408—531no—
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#341—531no—
274#339—531no—
275#339C2×D5311yescharacteristic
276#338—531no—
277H(3,ℤ/4)—531no—
278C53#3270311yescharacteristic, complemented, Sylow
279C2×#41—531no—
280C2×#41C22×D5311yescharacteristic
281#137—1061no—
282#137—1061no—
283AGL(1,ℤ/8)—1061no—
284AGL(1,ℤ/8)—1061no—
285#135—2121no—
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—2121no—
296C2×M16C22×D5311yescharacteristic
297C2×M16C22×D5311yescharacteristic
298C2×M16C22×D5311yescharacteristic
299C2×M16C22×D5311yescharacteristic
300C22×C8—531no—
301C22×C8—531no—
302#117—531no—
303#117C22×D5311yescharacteristic
304C2×C42C22×D5311yescharacteristic
305#108—41no—
306#108—2121no—
307#108—2121no—
308C4⋊C8—41no—
309C4⋊C8—2121no—
310C4⋊C8—2121no—
311C4⋊C8—2121no—
312#104—22no—
313#104—41no—
314#104—1062no—
315#104—2121no—
316#103—1061no—
317#103—1061no—
318#102—1061no—
319#102—1061no—
320#101—1061no—
321#101—1061no—
322#100—1061no—
323#100—1061no—
324C22⋊C8—2121no—
325C22⋊C8—2121no—
326#97—1062no—
327#41—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331#41—1061no—
332#41—1061no—
333#41—1061no—
334#41—1061no—
335C2×Q8—531no—
336C2×Q8#1171994911yescharacteristic
337C2×D4—21no—
338C2×D4—531no—
339C2×D4—1061no—
340C2×D4#1171994911yescharacteristic
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—1061no—
344C22×C4C23×D5311yescharacteristic
345Q16—2121no—
346SD16—2121no—
347SD16—2121no—
348D8—2121no—
349M16—21no—
350M16—21no—
351M16—21no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—21no—
356M16—21no—
357M16—42no—
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—42no—
372C2×C8—1061no—
373C2×C8—1061no—
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—21no—
389C42—1061no—
390C42#1171994512yes—
391C23—21no—
392C23—1061no—
393C23#1313326411yescharacteristic
394Q8—21no—
395Q8—21no—
396Q8—1061no—
397Q8—1061no—
398D4—21no—
399D4—21no—
400D4—41no—
401D4—41no—
402D4—1061no—
403D4—1061no—
404D4—2121no—
405D4—2121no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—81no—
412C2×C4—1061no—
413C2×C4—2121no—
414C2×C4—2121no—
415C2×C4—2121no—
416C2×C4#41×D5311yescharacteristic
417C2×C4C2×#1171994511yescharacteristic
418C2×C4C2×#1171994911yescharacteristic
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—81no—
435C22—4241no—
436C22—2121no—
437C22#1681121511yescharacteristic
438C22#1681121611yescharacteristic
439C22#1681122211yescharacteristic
440C4—42no—
441C4—41no—
442C4—81no—
443C4—4241no—
444C4—2121no—
445C4#1681130011yescharacteristic
446C4#1681132111yescharacteristic
447C4#1681132311yescharacteristic
448C4#1681136611yescharacteristic
449C2—21no—
450C2—21no—
451C2—21no—
452C2—81no—
453C2—4241no—
454C2#2028238711yescharacteristic
subgroupG/H
centerC4#16811366
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1861C2
socleC2×C53#2782
prime radical#16811421C22
hypercenter#1861D53
hypocenterC53#32703
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1861

Complements

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

NH ≅ G/Ntype
1. #20276298C2semidirect
2. #20276294C2semidirect
3. #20276293C2semidirect
4. #20276170C2semidirect
5. #20276165C2semidirect
7. #20275427C2semidirect
8. #20275423C2semidirect
11. #20274954C2semidirect
12. #20274935C2semidirect
13. #20274933C2semidirect
14. #20274855C2semidirect
15. C53×#1861C2semidirect
233. #1861D53semidirect
278. C53#32703semidirect

Maximal subgroups

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

subgroupindex
#202762982
#202762942
#202762932
#202761702
#202761652
#202760012
#202754272
#202754232
#202750772
#202750742
#202749542
#202749352
#202749332
#202748552
C53×#18612
#3270353