Finite Group Databasebeta

← #27367913

Subgroups — #27367913

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285297C211yesmaximal, characteristic, complemented
2#20284814C211yesmaximal, characteristic
3C2×D4×#88322C211yesmaximal, characteristic
4#20274140C212yesmaximal, complemented
5#20274136C212yesmaximal, complemented
6#20274102C212yesmaximal, complemented
7#20274078C212yesmaximal, complemented
8#20273476C211yesmaximal, characteristic, complemented
9#20273416C211yesmaximal, characteristic, complemented
10#20273415C211yesmaximal, characteristic, complemented
11C53×#3110C211yesmaximal, characteristic
12C23×#88322—21no—
13#16811180C2211yescharacteristic
14#16811026C2211yescharacteristic, complemented
15#16811002C2211yescharacteristic, complemented
16#16810975C2211yescharacteristic, complemented
17#16810973C2211yescharacteristic, complemented
18#16810958—21no—
19#16810903C2212yes—
20#16810862—21no—
21#16810862—21no—
22#16810828C2212yes—
23#16810774—21no—
24#16810723—22no—
25C4×#11719925—21no—
26C4×#11719925—21no—
27D4×#88322—21no—
28D4×#88322—21no—
29D4×#88322C2212yes—
30D4×#88322C2212yes—
31#16810680C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33C2×C22⋊#88322—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810375C2212yescomplemented
36#16810373C2212yescomplemented
37#16810370C2212yescomplemented
38#16810369C2212yescomplemented
39#16810369C2212yescomplemented
40#16810368C2212yescomplemented
41C53×#533C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#484—21no—
44C53×#474C2212yes—
45C53×#461C2212yes—
46C53×#450—21no—
47C53×#440—22no—
48C53×#432C2211yescharacteristic
49C2×#11719925—21no—
50C2×#11719925—21no—
51C2×#11719923—22no—
52C22×#88322—22no—
53C22×#88322—22no—
54C22×#88322—21no—
55C22×#88322—44no—
56C22×#88322D411yescharacteristic
57C22×#88322D411yescharacteristic
58#13133217C2311yescharacteristic
59#13133217C2311yescharacteristic
60#13133209—21no—
61#13133196—21no—
62#13133196—21no—
63C4×Dic106—21no—
64C4×Dic106—21no—
65#13133176—21no—
66#13133150—22no—
67#13133150—21no—
68#13133150—21no—
69C4⋊#88322—21no—
70C4⋊#88322C2312yes—
71C4⋊#88322C2312yes—
72#13133141—21no—
73#13133141—21no—
74#13133140—22no—
75#13133140—21no—
76#13133140—21no—
77#13133138—22no—
78C22⋊#88322—22no—
79C22⋊#88322—22no—
80C22⋊#88322—21no—
81#13133128—21no—
82C4×#88322D411yescharacteristic
83C4×#88322D411yescharacteristic
84C4×#88322C2311yescharacteristic
85C4×#88322C2311yescharacteristic
86C22×D4×C53D411yescharacteristic
87C23×C4×C53—21no—
88C2×SD16×C53—22no—
89C4⋊Q8×C53C2311yescharacteristic
90C4⋊Q8×C53C2311yescharacteristic
91#125×C53—21no—
92C22⋊Q8×C53—21no—
93C22⋊Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53D412yes—
99C4×D4×C53C2312yes—
100#117×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#107×C53—22no—
105C4⋊C8×C53C2312yes—
106#103×C53—22no—
107#102×C53—22no—
108C22⋊C8×C53—21no—
109#97×C53—21no—
110C24×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×D4×C53—24no—
114C2×D4×C53—21no—
115C2×D4×C53C2×D412yes—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C2×Dic106—21no—
125C2×Dic106—21no—
126SD16×C53—44no—
127C2×C8×C53—22no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—41no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53C2×D412yes—
136C22⋊C4×C53—22no—
137C22⋊C4×C53—22no—
138C22⋊C4×C53—41no—
139C42×C53—41no—
140C42×C53C2×D411yescharacteristic
141C42×C53C2×D411yescharacteristic
142C42×C53#4111yescharacteristic
143C42×C53C2411yescharacteristic
144C22⋊Dic53—41no—
145#11719925—42no—
146#11719925—42no—
147#11719923—44no—
148C4⋊Dic53—21no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152#11719919—41no—
153#11719919—41no—
154C4×Dic53—41no—
155C2×#88322—22no—
156C2×#88322—22no—
157C2×#88322—22no—
158C2×#88322—41no—
159C2×#88322—44no—
160C2×#88322—42no—
161C2×#88322—44no—
162C2×#88322C2×D411yescharacteristic
163C2×#88322C2×D411yescharacteristic
164C2×#88322C2×D411yescharacteristic
165C2×#88322C2×D411yescharacteristic
166C23×C53—24no—
167C23×C53—22no—
168C23×C53—42no—
169C23×C53—42no—
170C23×C53#12111yescharacteristic
171Q8×C53—42no—
172Q8×C53—41no—
173Q8×C53—41no—
174D4×C53—24no—
175D4×C53—22no—
176D4×C53#13514yes—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—22no—
182C2×C4×C53—44no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53C22×D411yescharacteristic
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C2×#4111yescharacteristic
194C2×Dic53—41no—
195C2×Dic53—41no—
196Dic106—42no—
197Dic106—41no—
198Dic106—41no—
199C8×C53—42no—
200#88322—21no—
201#88322—21no—
202#88322—21no—
203#88322—42no—
204#88322—42no—
205#88322AGL(1,ℤ/8)12yes—
206#19259—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—24no—
210C22×C53—48no—
211C22×C53—42no—
212C22×C53—42no—
213C22×C53#47312yes—
214C22×C53C2×#12111yescharacteristic
215C4×C53—21no—
216C4×C53—21no—
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—42no—
220C4×C53—81no—
221C4×C53—81no—
222C4×C53C2×#13512yes—
223C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
224C4×C53#57511yescharacteristic
225Dic53—81no—
226Dic53—81no—
227#3170—531no—
228#3110—531no—
229#3110D5311yescharacteristic
230C2×C8×D4—531no—
231#1679—532no—
232#1679—532no—
233#1673—532no—
234#1669—532no—
235#1568—531no—
236#1552—531no—
237#1552—531no—
238C2×C53—22no—
239C2×C53—44no—
240C2×C53#303611yescharacteristic
241C2×C53#305011yescharacteristic
242C2×C53C2×#47311yescharacteristic
243C23×C8—1061no—
244#533—531no—
245#533C2×D5311yescharacteristic
246C2×C4×D4C2×D5311yescharacteristic
247#500—531no—
248#494—531no—
249#487—531no—
250#487—531no—
251#484—21no—
252#484—1061no—
253#474—532no—
254#474C2×D5312yes—
255#467—1061no—
256#467—1061no—
257#461—532no—
258#461C2×D5312yes—
259#450—21no—
260#450—1061no—
261#440—22no—
262#440—1062no—
263C4×Q16—1061no—
264C4×Q16—1061no—
265C8×D4—532no—
266C8×D4—532no—
267C8×D4—1061no—
268C8×D4—1061no—
269#432—531no—
270#432C2×D5311yescharacteristic
271C2×C4⋊C8—531no—
272C2×C22⋊C8—1061no—
273C2×C4×C8—531no—
274#335—532no—
275#334—532no—
276#332—532no—
277#332—532no—
278#332—532no—
279#332—532no—
280C53#1925911yescharacteristic, complemented, Sylow
281C22×D4C22⋊D5311yescharacteristic
282C23×C4—21no—
283C2×Q16—1061no—
284C2×Q16—1061no—
285C2×SD16—22no—
286C2×SD16—1062no—
287C22×C8—531no—
288C22×C8—531no—
289C22×C8—1062no—
290C22×C8—1062no—
291C22×C8—1061no—
292C22×C8—2124no—
293C4⋊Q8—531no—
294C4⋊Q8—531no—
295C4⋊Q8C22×D5311yescharacteristic
296C4⋊Q8C22×D5311yescharacteristic
297#125—21no—
298#125—1061no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C22⋊Q8—1061no—
302C22⋊Q8—1061no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8—1061no—
306C4×Q8—1061no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4C22⋊D5312yes—
310C4×D4C22×D5312yes—
311#117—21no—
312#117—1061no—
313C2×C4⋊C4C22⋊D5311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C42C22×D5311yescharacteristic
316#107—22no—
317#107—1061no—
318#107—1061no—
319#107—1062no—
320C4⋊C8—532no—
321C4⋊C8—532no—
322C4⋊C8—1061no—
323C4⋊C8C22×D5312yes—
324#103—22no—
325#103—1061no—
326#103—1061no—
327#103—1061no—
328#103—1061no—
329#103—1062no—
330#102—22no—
331#102—1062no—
332C22⋊C8—21no—
333C22⋊C8—1062no—
334C22⋊C8—1062no—
335C22⋊C8—1061no—
336#97—21no—
337#97—1061no—
338C4×C8—531no—
339C4×C8—531no—
340C4×C8—531no—
341C4×C8—531no—
342C24—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—1061no—
346C2×Q8—1061no—
347C2×D4—24no—
348C2×D4—21no—
349C2×D4C2×C22⋊D5312yes—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4D4×D5311yescharacteristic
356C22×C4#1171994711yescharacteristic
357C22×C4C2×C22⋊D5311yescharacteristic
358Q16—2122no—
359Q16—2122no—
360SD16—44no—
361SD16—2124no—
362C2×C8—22no—
363C2×C8—531no—
364C2×C8—531no—
365C2×C8—531no—
366C2×C8—531no—
367C2×C8—1062no—
368C2×C8—1062no—
369C2×C8—1062no—
370C2×C8—2124no—
371C2×C8—2121no—
372C2×C8—2124no—
373C2×C8—2122no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—1061no—
382C4⋊C4—1061no—
383C4⋊C4—1061no—
384C4⋊C4—1061no—
385C4⋊C4—2121no—
386C4⋊C4—2121no—
387C4⋊C4C2×C22⋊D5312yes—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—41no—
391C22⋊C4—2121no—
392C42—41no—
393C42—2121no—
394C42D4×D5311yescharacteristic
395C42#1171994711yescharacteristic
396C42C2×C22⋊D5311yescharacteristic
397C42C23×D5311yescharacteristic
398C23—24no—
399C23—22no—
400C23—42no—
401C23—42no—
402C23#1313318911yescharacteristic
403Q8—42no—
404Q8—41no—
405Q8—41no—
406Q8—2122no—
407Q8—2121no—
408Q8—2121no—
409D4—24no—
410D4—22no—
411D4#1313323714yes—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—2121no—
424C2×C4—2121no—
425C2×C4#1313318911yescharacteristic
426C2×C4#1313318911yescharacteristic
427C2×C4#1313318911yescharacteristic
428C2×C4C2×D4×D5311yescharacteristic
429C2×C4C22×C22⋊D5311yescharacteristic
430C2×C4C2×#1171994711yescharacteristic
431C8—42no—
432C8—532no—
433C8—1061no—
434C8—1061no—
435C8—1061no—
436C8—2122no—
437C8—2122no—
438C22—21no—
439C22—21no—
440C22—24no—
441C22—48no—
442C22—42no—
443C22—42no—
444C22#1681089612yes—
445C22C2×#1313318911yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—42no—
451C4—81no—
452C4—81no—
453C4—4241no—
454C4—4241no—
455C4C2×#1313323712yes—
456C4AGL(1,ℤ/8)×D5311yescharacteristic
457C4#1681141911yescharacteristic
458C2—22no—
459C2—44no—
460C2#2028417611yescharacteristic
461C2#2028429311yescharacteristic
462C2C2×#1681089611yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3110C2
socleC22×C53C2×#121
prime radicalC22×D4×C53D4
hypercenter#3110D53
hypocenterC53#19259
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #20285297C2semidirect
4. #20274140C2semidirect
5. #20274136C2semidirect
6. #20274102C2semidirect
7. #20274078C2semidirect
8. #20273476C2semidirect
9. #20273416C2semidirect
10. #20273415C2semidirect
14. #16811026C22semidirect
15. #16811002C22semidirect
16. #16810975C22semidirect
17. #16810973C22semidirect
35. #16810375C22semidirect
36. #16810373C22semidirect
37. #16810370C22semidirect
38. #16810369C22semidirect
39. #16810369C22semidirect
40. #16810368C22semidirect
280. C53#19259semidirect

Maximal subgroups

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

subgroupindex
#202852972
#202848142
C2×D4×#883222
#202741402
#202741362
#202741022
#202740782
#202734762
#202734162
#202734152
C53×#31102
#1925953