Finite Group Databasebeta

← #27367894

Subgroups — #27367894

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285270C211yesmaximal, characteristic, complemented
2#20284814C211yesmaximal, characteristic, complemented
3C2×D4×#88322C211yesmaximal, characteristic, complemented
4#20274168C212yesmaximal, complemented
5#20274150C212yesmaximal, complemented
6#20274092C212yesmaximal, complemented
7#20274054C212yesmaximal, complemented
8#20273476C211yesmaximal, characteristic, complemented
9#20273416C211yesmaximal, characteristic, complemented
10#20273410C211yesmaximal, characteristic, complemented
11C53×#3109C211yesmaximal, characteristic
12C23×#88322—21no—
13#16811180C2211yescharacteristic
14#16811019C2211yescharacteristic, complemented
15#16810995C2211yescharacteristic, complemented
16#16810975C2211yescharacteristic, complemented
17#16810969C2211yescharacteristic, complemented
18#16810958—21no—
19#16810903C2212yes—
20#16810850—21no—
21#16810850—21no—
22#16810828C2212yes—
23#16810774—21no—
24#16810723—22no—
25C4×#11719923—21no—
26C4×#11719923—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#16810372C2212yescomplemented
38#16810370C2212yescomplemented
39#16810368C2212yescomplemented
40#16810367C2212yescomplemented
41C53×#532C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#481—21no—
44C53×#476C2212yes—
45C53×#459C2212yes—
46C53×#447—21no—
47C53×#442—22no—
48C53×#432C2211yescharacteristic
49C2×#11719923—21no—
50C2×#11719923—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#13133148—21no—
68#13133148—21no—
69C4⋊#88322—21no—
70C4⋊#88322C2312yes—
71C4⋊#88322C2312yes—
72#13133140—22no—
73#13133140—21no—
74#13133140—21no—
75#13133138—21no—
76#13133138—21no—
77#13133138—22no—
78C22⋊#88322—22no—
79C22⋊#88322—22no—
80C22⋊#88322—21no—
81#13133128—21no—
82C4×#88322D411yescharacteristic
83C4×#88322D411yescharacteristic
84C4×#88322C2311yescharacteristic
85C4×#88322C2311yescharacteristic
86C2×#41×C53—21no—
87C22×D4×C53D411yescharacteristic
88C23×C4×C53—21no—
89C2×D8×C53—22no—
90C4⋊Q8×C53C2311yescharacteristic
91#127×C53C2311yescharacteristic
92#124×C53—21no—
93#121×C53—21no—
94#121×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53D412yes—
100C4×D4×C53C2312yes—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#106×C53—22no—
105C4⋊C8×C53C2312yes—
106#102×C53—22no—
107#102×C53—22no—
108C22⋊C8×C53—21no—
109#97×C53—21no—
110C24×C53—21no—
111#41×C53—82no—
112C2×Q8×C53—21no—
113C2×D4×C53—24no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—41no—
119C2×D4×C53C2×D412yes—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—42no—
124C22×C4×C53—42no—
125C22×C4×C53—41no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53#4111yescharacteristic
129C2×Dic106—21no—
130C2×Dic106—21no—
131D8×C53—44no—
132C2×C8×C53—22no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53C2×D412yes—
137C22⋊C4×C53—22no—
138C22⋊C4×C53—22no—
139C22⋊C4×C53—41no—
140C22⋊C4×C53—41no—
141C42×C53C2×D411yescharacteristic
142C42×C53C2×D411yescharacteristic
143C42×C53#4111yescharacteristic
144C42×C53C2411yescharacteristic
145C22⋊Dic53—41no—
146#11719923—42no—
147#11719923—44no—
148#11719923—42no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153#11719919—41no—
154#11719919—41no—
155C4×Dic53—41no—
156C2×#88322—22no—
157C2×#88322—22no—
158C2×#88322—22no—
159C2×#88322—41no—
160C2×#88322—44no—
161C2×#88322—42no—
162C2×#88322—44no—
163C2×#88322C2×D411yescharacteristic
164C2×#88322C2×D411yescharacteristic
165C2×#88322C2×D411yescharacteristic
166C2×#88322C2×D411yescharacteristic
167C23×C53—24no—
168C23×C53—22no—
169C23×C53—42no—
170C23×C53—42no—
171C23×C53—41no—
172C23×C53#12111yescharacteristic
173Q8×C53—81no—
174D4×C53—24no—
175D4×C53—22no—
176D4×C53—42no—
177D4×C53—41no—
178D4×C53—41no—
179D4×C53—81no—
180D4×C53—82no—
181D4×C53#13514yes—
182C2×C4×C53—22no—
183C2×C4×C53—22no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—22no—
187C2×C4×C53—44no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—41no—
192C2×C4×C53—82no—
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53#12111yescharacteristic
195C2×C4×C53#12111yescharacteristic
196C2×C4×C53C22×D411yescharacteristic
197C2×C4×C53C22×D411yescharacteristic
198C2×C4×C53C2×#4111yescharacteristic
199C2×Dic53—41no—
200C2×Dic53—41no—
201Dic106—42no—
202Dic106—41no—
203Dic106—41no—
204C8×C53—42no—
205#88322—21no—
206#88322—21no—
207#88322—21no—
208#88322—42no—
209#88322—42no—
210#88322AGL(1,ℤ/8)12yes—
211#19257—531nomaximal, Sylow
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—24no—
215C22×C53—48no—
216C22×C53—42no—
217C22×C53—42no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53—81no—
221C22×C53#47112yes—
222C22×C53C2×#12111yescharacteristic
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—42no—
228C4×C53—81no—
229C4×C53C2×#13512yes—
230C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
231C4×C53#57511yescharacteristic
232Dic53—81no—
233Dic53—81no—
234#3167—531no—
235#3110—531no—
236#3109D5311yescharacteristic
237C2×C8×D4—531no—
238#1683—532no—
239#1681—532no—
240#1671—532no—
241#1665—532no—
242#1568—531no—
243#1552—531no—
244#1551—531no—
245C2×C53—22no—
246C2×C53—44no—
247C2×C53—161no—
248C2×C53#303611yescharacteristic
249C2×C53#305011yescharacteristic
250C2×C53C2×#47111yescharacteristic
251C23×C8—1061no—
252#533—531no—
253#532C2×D5311yescharacteristic
254C2×C4×D4C2×D5311yescharacteristic
255#498—531no—
256#492—531no—
257#487—531no—
258#486—531no—
259#484—1061no—
260#481—21no—
261#476C2×D5312yes—
262#474—532no—
263#465—1061no—
264#465—1061no—
265#461—532no—
266#459C2×D5312yes—
267#450—1061no—
268#447—21no—
269#442—22no—
270#440—1062no—
271C4×SD16—1061no—
272C4×SD16—1061no—
273C8×D4—532no—
274C8×D4—532no—
275C8×D4—1061no—
276C8×D4—1061no—
277#432—531no—
278#432C2×D5311yescharacteristic
279C2×C4⋊C8—531no—
280C2×C22⋊C8—1061no—
281C2×C4×C8—531no—
282#335—532no—
283#334—532no—
284#333—532no—
285#332—532no—
286#332—532no—
287#331—532no—
288C53#1925711yescharacteristic, complemented, Sylow
289C2×#41—21no—
290C22×D4C22⋊D5311yescharacteristic
291C23×C4—21no—
292C2×SD16—1061no—
293C2×SD16—1061no—
294C2×SD16—1062no—
295C2×D8—22no—
296C22×C8—531no—
297C22×C8—531no—
298C22×C8—1062no—
299C22×C8—1062no—
300C22×C8—1061no—
301C22×C8—2124no—
302C4⋊Q8—531no—
303C4⋊Q8—531no—
304C4⋊Q8C22×D5311yescharacteristic
305#127C22×D5311yescharacteristic
306#125—1061no—
307#124—21no—
308C22⋊Q8—1061no—
309C22⋊Q8—1061no—
310#121—21no—
311#121—21no—
312C4×Q8—1061no—
313C4×Q8—1061no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4C22⋊D5312yes—
319C4×D4C22×D5312yes—
320#117—1061no—
321C2×C4⋊C4C22⋊D5311yescharacteristic
322C2×C22⋊C4—21no—
323C2×C42C22×D5311yescharacteristic
324#107—1062no—
325#106—22no—
326#106—1061no—
327#106—1061no—
328C4⋊C8—532no—
329C4⋊C8—532no—
330C4⋊C8—1061no—
331C4⋊C8C22×D5312yes—
332#103—1061no—
333#103—1061no—
334#103—1062no—
335#102—22no—
336#102—22no—
337#102—1061no—
338#102—1061no—
339#102—1062no—
340C22⋊C8—21no—
341C22⋊C8—1062no—
342C22⋊C8—1062no—
343C22⋊C8—1061no—
344#97—21no—
345#97—1061no—
346C4×C8—531no—
347C4×C8—531no—
348C4×C8—531no—
349C4×C8—531no—
350C24—21no—
351#41—82no—
352C2×Q8—21no—
353C2×Q8—1061no—
354C2×Q8—1061no—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—41no—
361C2×D4C2×C22⋊D5312yes—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4D4×D5311yescharacteristic
369C22×C4#1171994711yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371SD16—2122no—
372SD16—2122no—
373SD16—2124no—
374D8—44no—
375C2×C8—22no—
376C2×C8—531no—
377C2×C8—531no—
378C2×C8—531no—
379C2×C8—531no—
380C2×C8—1062no—
381C2×C8—1062no—
382C2×C8—1062no—
383C2×C8—2124no—
384C2×C8—2121no—
385C2×C8—2124no—
386C2×C8—2122no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—1061no—
391C4⋊C4—1061no—
392C4⋊C4—1061no—
393C4⋊C4—1061no—
394C4⋊C4—2121no—
395C4⋊C4—2121no—
396C4⋊C4C2×C22⋊D5312yes—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—2121no—
402C42—2121no—
403C42D4×D5311yescharacteristic
404C42#1171994711yescharacteristic
405C42C2×C22⋊D5311yescharacteristic
406C42C23×D5311yescharacteristic
407C23—24no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23—41no—
412C23#1313318911yescharacteristic
413Q8—81no—
414Q8—2122no—
415Q8—2121no—
416Q8—2121no—
417D4—24no—
418D4—22no—
419D4—42no—
420D4—41no—
421D4—41no—
422D4—81no—
423D4—82no—
424D4#1313323714yes—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—82no—
436C2×C4—2121no—
437C2×C4—2121no—
438C2×C4#1313318911yescharacteristic
439C2×C4#1313318911yescharacteristic
440C2×C4#1313318911yescharacteristic
441C2×C4C2×D4×D5311yescharacteristic
442C2×C4C22×C22⋊D5311yescharacteristic
443C2×C4C2×#1171994711yescharacteristic
444C8—42no—
445C8—532no—
446C8—1061no—
447C8—1061no—
448C8—1061no—
449C8—2122no—
450C8—2122no—
451C22—21no—
452C22—21no—
453C22—24no—
454C22—48no—
455C22—42no—
456C22—42no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22#1681088412yes—
461C22C2×#1313318911yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—81no—
468C4—4241no—
469C4—4241no—
470C4C2×#1313323712yes—
471C4AGL(1,ℤ/8)×D5311yescharacteristic
472C4#1681141911yescharacteristic
473C2—22no—
474C2—44no—
475C2—161no—
476C2#2028417611yescharacteristic
477C2#2028429311yescharacteristic
478C2C2×#1681088411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3109C2
socleC22×C53C2×#121
hypercenter#3109D53
hypocenterC53#19257
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
1. #20285270C2semidirect
2. #20284814C2semidirect
3. C2×D4×#88322C2semidirect
4. #20274168C2semidirect
5. #20274150C2semidirect
6. #20274092C2semidirect
7. #20274054C2semidirect
8. #20273476C2semidirect
9. #20273416C2semidirect
10. #20273410C2semidirect
14. #16811019C22semidirect
15. #16810995C22semidirect
16. #16810975C22semidirect
17. #16810969C22semidirect
35. #16810375C22semidirect
36. #16810373C22semidirect
37. #16810372C22semidirect
38. #16810370C22semidirect
39. #16810368C22semidirect
40. #16810367C22semidirect
288. C53#19257semidirect

Maximal subgroups

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

subgroupindex
#202852702
#202848142
C2×D4×#883222
#202741682
#202741502
#202740922
#202740542
#202734762
#202734162
#202734102
C53×#31092
#1925753