Finite Group Databasebeta

← #27366633

Subgroups — #27366633

Order 13568.

662 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#20285225C211yesmaximal, characteristic, complemented
2#20284724C211yesmaximal, characteristic
3#20284719C211yesmaximal, characteristic
4C2×#16810700C211yesmaximal, characteristic
5#20274237C212yesmaximal, complemented
6#20274227C212yesmaximal, complemented
7#20274217C212yesmaximal, complemented
8#20274213C212yesmaximal, complemented
9#20273461C211yesmaximal, characteristic, complemented
10#20273373C211yesmaximal, characteristic, complemented
11C53×#1542C211yesmaximal, characteristic, complemented
12C22×#11719912—21no—
13#16811180C2211yescharacteristic
14C2×C4×D212C2211yescharacteristic
15#16811026C2211yescharacteristic, complemented
16#16811026C2211yescharacteristic, complemented
17#16811022—21no—
18#16811022—21no—
19#16811019C2211yescharacteristic, complemented
20#16811019C2211yescharacteristic, complemented
21#16810954—21no—
22#16810951—21no—
23#16810914C2212yes—
24#16810910C2212yes—
25#16810829C2212yes—
26#16810825C2212yes—
27#16810776—21no—
28#16810773—21no—
29C4×#11719913—22no—
30C4×#11719924—22no—
31#16810700—21no—
32#16810700—21no—
33#16810700C2212yes—
34#16810700C2212yes—
35#16810676C2211yescharacteristic
36#16810659—21no—
37#16810659—21no—
38C2×#13133129—21no—
39C2×C4×#88322C2211yescharacteristic
40#16810375C2212yescomplemented
41#16810373C2212yescomplemented
42#16810371C2212yescomplemented
43#16810368C2212yescomplemented
44C53×#533C2211yescharacteristic
45C53×#431C2211yescharacteristic
46C2×C4⋊C8×C53C2211yescharacteristic
47C53×#333C2212yescomplemented
48C53×#332C2212yescomplemented
49C22×D212D411yescharacteristic
50C22×C4×D53—21no—
51C2×#11719913—22no—
52C2×#11719924—22no—
53C2×#11719912—22no—
54C2×#11719912—22no—
55C2×#11719912—44no—
56C22×#88322D411yescharacteristic
57C22×#88322D411yescharacteristic
58#13133217C2311yescharacteristic
59#13133217C2311yescharacteristic
60#13133209—21no—
61#13133196—21no—
62#13133196—21no—
63C4×Dic106—21no—
64C4×Dic106—21no—
65C4×D212—21no—
66C4×D212—21no—
67C4×D212D412yes—
68C4×D212C2312yes—
69#13133176—21no—
70C2×C4⋊Dic53D411yescharacteristic
71C2×#11719921—21no—
72#13133150—21no—
73#13133150—21no—
74#13133150—21no—
75#13133150—21no—
76#13133149—22no—
77#13133148—22no—
78#13133148—21no—
79#13133148—21no—
80#13133148—21no—
81#13133148—21no—
82C4⋊#88322C2312yes—
83#13133142—22no—
84#13133141—22no—
85#13133139—22no—
86#13133137—22no—
87C22⋊#88322—21no—
88#13133129—22no—
89#13133129—22no—
90C4×#88322—21no—
91C4×#88322D411yescharacteristic
92C4×#88322D411yescharacteristic
93C4×#88322C2311yescharacteristic
94C4×#88322C2311yescharacteristic
95C22×C8×C53—21no—
96C4⋊Q8×C53C2311yescharacteristic
97C4⋊Q8×C53C2311yescharacteristic
98#125×C53—21no—
99C22⋊Q8×C53—21no—
100C22⋊Q8×C53—21no—
101C4×Q8×C53—21no—
102C4×Q8×C53—21no—
103#117×C53—21no—
104C2×C42×C53C2311yescharacteristic
105C4⋊C8×C53—21no—
106C4⋊C8×C53C2312yes—
107C4⋊C8×C53C2312yes—
108C22⋊C8×C53—21no—
109C4×C8×C53—21no—
110C23×D53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C22×Dic53—21no—
117C2×D212—21no—
118C2×D212—42no—
119C2×D212C2×D412yes—
120C2×C4×D53—22no—
121C2×C4×D53—22no—
122C2×C4×D53—42no—
123C2×C4×D53—42no—
124C2×Dic106—21no—
125C2×Dic106—21no—
126C2×C8×C53—22no—
127C2×C8×C53—22no—
128C2×C8×C53—42no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—41no—
134C4⋊C4×C53—41no—
135C22⋊C4×C53—41no—
136C42×C53—41no—
137C42×C53C2×D411yescharacteristic
138C42×C53C2×D411yescharacteristic
139C42×C53#4111yescharacteristic
140C42×C53C2411yescharacteristic
141C22⋊Dic53—41no—
142#11719924—44no—
143#11719921—22no—
144#11719921—22no—
145C4⋊Dic53—21no—
146C4⋊Dic53—21no—
147C4⋊Dic53—21no—
148C4⋊Dic53—21no—
149C4⋊Dic53—21no—
150C4⋊Dic53C2×D412yes—
151#11719919—41no—
152#11719919—41no—
153C4×Dic53—41no—
154C2×#88322—22no—
155C2×#88322—22no—
156C2×#88322SD1614yescomplemented
157C2×#88322C2×D411yescharacteristic
158C2×#88322C2×D411yescharacteristic
159C2×#88322C2×D411yescharacteristic
160C2×#88322C2×D411yescharacteristic
161#11719913—44no—
162#11719912—44no—
163#11719912—44no—
164C23×C53#12111yescharacteristic
165C22×D53—22no—
166C22×D53—42no—
167C22×D53—42no—
168C22×D53—42no—
169Q8×C53—42no—
170Q8×C53—81no—
171C2×C4×C53—22no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53C2×SD1612yes—
179C2×C4×C53#13712yes—
180C2×C4×C53C22×D411yescharacteristic
181C2×C4×C53C22×D411yescharacteristic
182C2×C4×C53C2×#4111yescharacteristic
183C2×Dic53—22no—
184C2×Dic53—42no—
185C2×Dic53—41no—
186C2×Dic53—41no—
187D212—22no—
188D212—22no—
189D212—42no—
190C4×D53—42no—
191C4×D53—42no—
192C4×D53—44no—
193Dic106—42no—
194Dic106—81no—
195C8×C53—42no—
196C8×C53—42no—
197#88322—21no—
198#88322—21no—
199#88322—42no—
200#88322C2×SD1612yes—
201#88322C2×SD1612yes—
202#19137—531nomaximal, Sylow
203C22×C53—21no—
204C22×C53—21no—
205C22×C53#46112yes—
206C22×C53C2×#12111yescharacteristic
207C2×D53—42no—
208C2×D53—42no—
209C2×D53—42no—
210C2×D53—84no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—81no—
215C4×C53—81no—
216C4×C53#46112yes—
217C4×C53C22×SD1611yescharacteristic
218C4×C53C2×#13711yescharacteristic
219C4×C53#57512yes—
220Dic53—42no—
221Dic53—81no—
222Dic53—81no—
223#3160—531no—
224#3101—531no—
225#3101—531no—
226C2×#436—531no—
227#1695—532no—
228#1693—532no—
229#1691—532no—
230#1691—532no—
231#1564—531no—
232#1542—531no—
233#1542D5311yescharacteristic, complemented
234C2×C53—22no—
235C2×C53C2×#46111yescharacteristic
236C2×C53#305411yescharacteristic
237C2×C53#306711yescharacteristic
238D53—82no—
239C22×M16—1061no—
240#533—531no—
241#533C2×D5311yescharacteristic
242C2×C4×D4—531no—
243#500—531no—
244#500—531no—
245#499—1061no—
246#499—1061no—
247#498—531no—
248#498—531no—
249#483—1061no—
250#483—1061no—
251#476—532no—
252#476—532no—
253#461—532no—
254#461—532no—
255#450—1061no—
256#450—1061no—
257C4×SD16—1062no—
258C4×SD16—1062no—
259#436—532no—
260#436—532no—
261#436—1061no—
262#436—1061no—
263#431—531no—
264#431C2×D5311yescharacteristic
265#427—1061no—
266#427—1061no—
267C2×C4⋊C8C2×D5311yescharacteristic
268C2×C22⋊C8—1061no—
269C2×C4×C8—531no—
270#335—532no—
271#334—532no—
272#333—532no—
273#333C2×D5312yescomplemented
274#332—532no—
275#332C2×D5312yescomplemented
276C53#1913711yescharacteristic, complemented, Sylow
277C22×D4—531no—
278C23×C4—1061no—
279C2×SD16—1062no—
280C2×SD16—1062no—
281C2×M16—1062no—
282C2×M16—1062no—
283C2×M16—2124no—
284C22×C8—21no—
285C22×C8—531no—
286C22×C8—531no—
287C4⋊Q8—531no—
288C4⋊Q8—531no—
289C4⋊Q8C22×D5311yescharacteristic
290C4⋊Q8C22×D5311yescharacteristic
291#125—21no—
292#125—1061no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—1061no—
296C22⋊Q8—1061no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—1061no—
300C4×Q8—1061no—
301C4×D4—532no—
302C4×D4—532no—
303C4×D4—1061no—
304C4×D4—1061no—
305#117—21no—
306#117—1061no—
307C2×C4⋊C4—531no—
308C2×C22⋊C4—1061no—
309C2×C42C22×D5311yescharacteristic
310#107—1061no—
311#107—1061no—
312#107—1061no—
313#107—1061no—
314#106—1062no—
315#106—1061no—
316#106—1061no—
317#106—1061no—
318#106—1061no—
319#106—1062no—
320C4⋊C8—21no—
321C4⋊C8—532no—
322C4⋊C8C22×D5312yes—
323C4⋊C8C22×D5312yes—
324#103—1062no—
325#103—1062no—
326#102—1062no—
327#102—1062no—
328C22⋊C8—21no—
329C22⋊C8—1062no—
330C22⋊C8—1062no—
331C22⋊C8—1061no—
332C4×C8—21no—
333C4×C8—531no—
334C4×C8—531no—
335C4×C8—531no—
336C4×C8—531no—
337C4×C8—1061no—
338C24—1061no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—1061no—
342C2×Q8—1061no—
343C2×D4—532no—
344C2×D4—1061no—
345C2×D4—2122no—
346C22×C4—1061no—
347C22×C4—1062no—
348C22×C4—1062no—
349C22×C4—2122no—
350C22×C4—2122no—
351C22×C4#1171994511yescharacteristic
352C22×C4D4×D5311yescharacteristic
353C22×C4D4×D5311yescharacteristic
354SD16—2124no—
355SD16—2124no—
356M16—2124no—
357M16—2124no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—42no—
361C2×C8—534no—
362C2×C8—531no—
363C2×C8—531no—
364C2×C8—531no—
365C2×C8—531no—
366C2×C8—1062no—
367C2×C8—1062no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—532no—
375C4⋊C4—1061no—
376C4⋊C4—1061no—
377C4⋊C4—1061no—
378C4⋊C4—1061no—
379C4⋊C4—1061no—
380C4⋊C4—2121no—
381C4⋊C4—2121no—
382C22⋊C4—41no—
383C22⋊C4—1062no—
384C22⋊C4—1062no—
385C22⋊C4—2121no—
386C42—41no—
387C42—2121no—
388C42#1171994511yescharacteristic
389C42D4×D5311yescharacteristic
390C42D4×D5311yescharacteristic
391C42C23×D5311yescharacteristic
392C23—1062no—
393C23—2122no—
394C23—2122no—
395C23—2122no—
396C23#1313319011yescharacteristic
397Q8—42no—
398Q8—81no—
399Q8—4241no—
400Q8—2122no—
401D4—1062no—
402D4—1062no—
403D4—2122no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—1062no—
409C2×C4—2121no—
410C2×C4—2121no—
411C2×C4—2122no—
412C2×C4—2124no—
413C2×C4—2122no—
414C2×C4—2122no—
415C2×C4#1313319011yescharacteristic
416C2×C4#1313319011yescharacteristic
417C2×C4#1313319011yescharacteristic
418C2×C4SD16×D5312yes—
419C2×C4#1313324912yes—
420C2×C4C2×D4×D5311yescharacteristic
421C2×C4C2×D4×D5311yescharacteristic
422C2×C4C2×#1171994511yescharacteristic
423C8—42no—
424C8—42no—
425C8—532no—
426C8—532no—
427C8—1061no—
428C8—1061no—
429C8—2122no—
430C22—21no—
431C22—21no—
432C22—4244no—
433C22—2122no—
434C22—2122no—
435C22—2122no—
436C22#1681082712yes—
437C22C2×#1313319011yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—81no—
442C4—81no—
443C4—4241no—
444C4—4241no—
445C4—2122no—
446C4#1681082712yes—
447C4C2×SD16×D5311yescharacteristic
448C4C2×#1313324911yescharacteristic
449C4#1681142012yes—
450C2—22no—
451C2—4242no—
452C2C2×#1681082711yescharacteristic
453C2#2028432311yescharacteristic
454C2#2028443411yescharacteristic
subgroupG/H
centerC2×C4#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1542C2
socleC22×C53C2×#121
prime radicalC22×D212D4
hypercenter#1542D53
hypocenterC53#19137
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
1. #20285225C2semidirect
5. #20274237C2semidirect
6. #20274227C2semidirect
7. #20274217C2semidirect
8. #20274213C2semidirect
9. #20273461C2semidirect
10. #20273373C2semidirect
11. C53×#1542C2semidirect
15. #16811026C22semidirect
16. #16811026C22semidirect
19. #16811019C22semidirect
20. #16811019C22semidirect
40. #16810375C22semidirect
41. #16810373C22semidirect
42. #16810371C22semidirect
43. #16810368C22semidirect
47. C53×#333C22semidirect
48. C53×#332C22semidirect
156. C2×#88322SD16semidirect
233. #1542D53semidirect
273. #333C2×D53semidirect
275. #332C2×D53semidirect
276. C53#19137semidirect

Maximal subgroups

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

subgroupindex
#202852252
#202847242
#202847192
C2×#168107002
#202742372
#202742272
#202742172
#202742132
#202734612
#202733732
C53×#15422
#1913753