Finite Group Databasebeta

← #27370088

Subgroups — #27370088

Order 13568.

680 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285845C211yesmaximal, characteristic, complemented
2#20284806C211yesmaximal, characteristic
3C2×#16810828C211yesmaximal, characteristic, complemented
4#20274276C212yesmaximal, complemented
5#20274214C212yesmaximal, complemented
6#20274152C212yesmaximal, complemented
7#20274090C212yesmaximal, complemented
8#20273500C211yesmaximal, characteristic, complemented
9#20273431C211yesmaximal, characteristic, complemented
10#20273411C211yesmaximal, characteristic, complemented
11C53×#2974C211yesmaximal, characteristic, complemented
12C22×#11719923—21no—
13#16811173C2211yescharacteristic
14C2×#13133217C2211yescharacteristic
15#16811021C2212yescomplemented
16#16810970C2212yescomplemented
17#16810942—21no—
18#16810942—21no—
19#16810941—21no—
20#16810914—22no—
21#16810913C2211yescharacteristic
22#16810913C2211yescharacteristic
23#16810828—21no—
24#16810828—21no—
25#16810828C2212yes—
26#16810828C2212yes—
27#16810818C2211yescharacteristic
28#16810818C2211yescharacteristic
29#16810757—21no—
30#16810733—21no—
31#16810733—21no—
32#16810680C2211yescharacteristic
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133140—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810371C2212yescomplemented
38#16810368C2212yescomplemented
39#16810368C2212yescomplemented
40#16810366C2212yescomplemented
41C2×C4×D4×C53C2211yescharacteristic
42C53×#436C2212yes—
43C53×#435—22no—
44C8×D4×C53C2212yes—
45C53×#432C2211yescharacteristic
46C53×#431C2211yescharacteristic
47C53×#407—21no—
48C2×C22⋊C8×C53—21no—
49C2×#11719945—21no—
50C22×Dic106—21no—
51C2×#11719923—22no—
52C2×#11719923—22no—
53C2×#11719923—44no—
54C2×#11719922—21no—
55C2×#11719922—21no—
56C22×#88322—21no—
57#13133217—21no—
58#13133217—21no—
59#13133217—21no—
60#13133217C2311yescharacteristic
61#13133217C2312yes—
62#13133214C2311yescharacteristic
63#13133205—21no—
64#13133191—21no—
65#13133191—21no—
66C4×D212—21no—
67C4×D212—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70#13133149—22no—
71#13133149—22no—
72#13133148—21no—
73#13133148—21no—
74C4⋊#88322—21no—
75C4⋊#88322C2311yescharacteristic
76C4⋊#88322C2311yescharacteristic
77C4⋊#88322C2312yes—
78#13133140—22no—
79#13133140—22no—
80#13133139—22no—
81#13133139—22no—
82#13133138—21no—
83#13133138—21no—
84#13133137—21no—
85#13133137—21no—
86C22⋊#88322—21no—
87#13133128—21no—
88C22×D4×C53D411yescharacteristic
89C23×C4×C53—21no—
90C2×M16×C53—22no—
91C22×C8×C53—22no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53D411yescharacteristic
95C4×D4×C53D411yescharacteristic
96C4×D4×C53C2311yescharacteristic
97C4×D4×C53C2311yescharacteristic
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C22⋊C4×C53—21no—
100C2×C42×C53C2311yescharacteristic
101C4⋊C8×C53D412yes—
102C4⋊C8×C53C2312yes—
103C22⋊C8×C53—22no—
104C22⋊C8×C53—22no—
105C22⋊C8×C53—21no—
106C22⋊C8×C53D411yescharacteristic
107C22⋊C8×C53D411yescharacteristic
108#97×C53—21no—
109C4×C8×C53C2312yes—
110C24×C53—21no—
111C2×D4×C53—22no—
112C2×D4×C53—21no—
113C2×D4×C53—41no—
114C2×D4×C53C2×D411yescharacteristic
115C2×D4×C53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—82no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53#4111yescharacteristic
125C2×C22⋊D53—41no—
126C22×Dic53—41no—
127#11719945—82no—
128C2×D212—21no—
129C2×D212—21no—
130C2×D212—21no—
131C2×C4×D53—41no—
132C2×Dic106—22no—
133C2×Dic106—21no—
134C2×Dic106—42no—
135C2×Dic106—42no—
136C2×Dic106—41no—
137M16×C53—82no—
138C2×C8×C53—22no—
139C2×C8×C53—82no—
140C2×C8×C53C2×D412yes—
141C2×C8×C53C2×D412yes—
142C4⋊C4×C53—21no—
143C4⋊C4×C53C2×D411yescharacteristic
144C4⋊C4×C53C2×D411yescharacteristic
145C22⋊C4×C53—21no—
146C22⋊C4×C53—21no—
147C22⋊C4×C53—21no—
148C22⋊C4×C53—21no—
149C42×C53C2×D411yescharacteristic
150C42×C53C2×D411yescharacteristic
151C42×C53#4111yescharacteristic
152C42×C53C2411yescharacteristic
153#11719923—44no—
154#11719923—44no—
155#11719922—42no—
156#11719922—42no—
157#11719921—41no—
158#11719921—41no—
159C4⋊Dic53—21no—
160C4⋊Dic53—22no—
161C4⋊Dic53—21no—
162C4⋊Dic53—22no—
163C4⋊Dic53—41no—
164C4⋊Dic53—41no—
165C2×#88322—21no—
166C2×#88322—21no—
167C2×#88322—22no—
168C2×#88322—42no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—42no—
172C23×C53—42no—
173C23×C53—81no—
174C23×C53#12111yescharacteristic
175C22×D53—41no—
176D4×C53—21no—
177D4×C53—21no—
178D4×C53—21no—
179D4×C53—42no—
180D4×C53AGL(1,ℤ/8)12yes—
181C2×C4×C53—22no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—22no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—81no—
190C2×C4×C53—81no—
191C2×C4×C53—81no—
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53#12111yescharacteristic
195C2×C4×C53C22×D411yescharacteristic
196C2×C4×C53C22×D411yescharacteristic
197C2×C4×C53C2×#4111yescharacteristic
198C22⋊D53—82no—
199C2×Dic53—42no—
200C2×Dic53—41no—
201C2×Dic53—82no—
202D212—41no—
203D212—41no—
204D212—42no—
205D212—81no—
206C4×D53—82no—
207Dic106—42no—
208Dic106—42no—
209Dic106—82no—
210Dic106—81no—
211C8×C53—22no—
212C8×C53—22no—
213C8×C53—42no—
214#88322—42no—
215#88322—41no—
216#88322—41no—
217#19462—531nomaximal, Sylow
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—22no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53—41no—
226C22×C53—84no—
227C22×C53#47112yes—
228C22×C53C2×#12111yescharacteristic
229C2×D53—81no—
230C2×D53—81no—
231C2×D53—81no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—21no—
236C4×C53—41no—
237C4×C53—41no—
238C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
239C4×C53#57511yescharacteristic
240C4×C53#57712yes—
241Dic53—82no—
242Dic53—81no—
243#3227—531no—
244#3109—531no—
245C2×#461—531no—
246#2974D5311yescharacteristic, complemented
247#1701—532no—
248#1691—532no—
249#1681—532no—
250#1671—532no—
251#1573—531no—
252#1556—531no—
253#1551—531no—
254C2×C53—22no—
255C2×C53—42no—
256C2×C53—81no—
257C2×C53#304211yescharacteristic
258C2×C53#305511yescharacteristic
259C2×C53C2×#47111yescharacteristic
260D53—161no—
261C22×SD16—1061no—
262#532—531no—
263C2×C4⋊Q8—531no—
264C2×C4×D4C2×D5311yescharacteristic
265#498—532no—
266#486—532no—
267#481—1061no—
268#481—1061no—
269#481—1061no—
270#476—531no—
271#476—531no—
272#476—1062no—
273#461—532no—
274#461—532no—
275#461—1061no—
276#461—1061no—
277#459—531no—
278#459—531no—
279#447—1061no—
280#442—1061no—
281#442—1061no—
282#436C2×D5312yes—
283#435—22no—
284C8×D4C2×D5312yes—
285#432—531no—
286#432C2×D5311yescharacteristic
287#431C2×D5311yescharacteristic
288C2×C4⋊C8—531no—
289C2×#103—1061no—
290#407—21no—
291C2×C22⋊C8—21no—
292#335—532no—
293#334—532no—
294#333—532no—
295#332—532no—
296#332—532no—
297#331—532no—
298C53#1946211yescharacteristic, complemented, Sylow
299C2×#41—1061no—
300C22×Q8—1061no—
301C22×D4D21211yescharacteristic
302C23×C4—21no—
303C2×SD16—1062no—
304C2×SD16—1062no—
305C2×SD16—2124no—
306C2×D8—1061no—
307C2×D8—1061no—
308C2×M16—22no—
309C22×C8—22no—
310C22×C8—1061no—
311C4⋊Q8—532no—
312C4⋊Q8—531no—
313C4⋊Q8—1061no—
314C4⋊Q8—1061no—
315C4⋊Q8—1061no—
316#127—531no—
317#124—1061no—
318#121—1061no—
319#121—1061no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—1061no—
323C4×D4—1061no—
324C4×D4D21211yescharacteristic
325C4×D4D21211yescharacteristic
326C4×D4C22×D5311yescharacteristic
327C4×D4C22×D5311yescharacteristic
328C2×C4⋊C4—1061no—
329C2×C4⋊C4—1061no—
330C2×C4⋊C4D21211yescharacteristic
331C2×C22⋊C4—21no—
332C2×C42C22×D5311yescharacteristic
333#106—1062no—
334#106—1062no—
335#106—1061no—
336#106—1061no—
337C4⋊C8—532no—
338C4⋊C8—531no—
339C4⋊C8—531no—
340C4⋊C8—1061no—
341C4⋊C8C22⋊D5312yes—
342C4⋊C8C22×D5312yes—
343#103—1062no—
344#103—1062no—
345#102—1061no—
346#102—1061no—
347#102—1062no—
348#102—1062no—
349#102—1061no—
350#102—1061no—
351C22⋊C8—21no—
352C22⋊C8—22no—
353C22⋊C8—22no—
354C22⋊C8—1061no—
355C22⋊C8C22⋊D5311yescharacteristic
356C22⋊C8C22⋊D5311yescharacteristic
357#97—21no—
358#97—1061no—
359C4×C8C22×D5312yes—
360C24—21no—
361#41—4242no—
362C2×Q8—1062no—
363C2×Q8—1061no—
364C2×Q8—2122no—
365C2×Q8—2122no—
366C2×Q8—2121no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—1061no—
371C2×D4—1061no—
372C2×D4—1061no—
373C2×D4—2121no—
374C2×D4C2×D21211yescharacteristic
375C2×D4C2×D21211yescharacteristic
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—82no—
382C22×C4—2121no—
383C22×C4—2121no—
384C22×C4C2×D21211yescharacteristic
385C22×C4#1171994511yescharacteristic
386C22×C4C2×C22⋊D5311yescharacteristic
387SD16—2124no—
388SD16—2124no—
389D8—2122no—
390D8—2122no—
391M16—82no—
392C2×C8—22no—
393C2×C8—82no—
394C2×C8—1061no—
395C2×C8—1061no—
396C2×C8—1062no—
397C2×C8—2122no—
398C2×C8C2×C22⋊D5312yes—
399C2×C8C2×C22⋊D5312yes—
400C4⋊C4—21no—
401C4⋊C4—1062no—
402C4⋊C4—1061no—
403C4⋊C4—1061no—
404C4⋊C4—1062no—
405C4⋊C4—2121no—
406C4⋊C4—2121no—
407C4⋊C4C2×D21211yescharacteristic
408C4⋊C4C2×D21211yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—2121no—
414C22⋊C4—2121no—
415C42C2×D21211yescharacteristic
416C42#1171994511yescharacteristic
417C42C2×C22⋊D5311yescharacteristic
418C42C23×D5311yescharacteristic
419C23—21no—
420C23—21no—
421C23—42no—
422C23—42no—
423C23—81no—
424C23—2121no—
425C23#1313319111yescharacteristic
426Q8—4241no—
427Q8—4242no—
428Q8—2122no—
429Q8—2122no—
430D4—21no—
431D4—21no—
432D4—21no—
433D4—42no—
434D4—4241no—
435D4—4242no—
436D4—2121no—
437D4—2121no—
438D4—2122no—
439D4#1313324512yes—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—4242no—
452C2×C4—4242no—
453C2×C4—2121no—
454C2×C4—2122no—
455C2×C4#1313319111yescharacteristic
456C2×C4#1313319111yescharacteristic
457C2×C4#1313319111yescharacteristic
458C2×C4C22×C22⋊D5311yescharacteristic
459C2×C4C22×D21211yescharacteristic
460C2×C4C2×#1171994511yescharacteristic
461C8—22no—
462C8—22no—
463C8—42no—
464C8—2122no—
465C8—2121no—
466C8—2121no—
467C22—21no—
468C22—21no—
469C22—22no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—84no—
476C22—4241no—
477C22—4241no—
478C22—4241no—
479C22#1681088612yes—
480C22C2×#1313319111yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—41no—
486C4—41no—
487C4—4242no—
488C4—4241no—
489C4C2×#1313324511yescharacteristic
490C4#1681142311yescharacteristic
491C4#1681143412yes—
492C2—22no—
493C2—42no—
494C2—81no—
495C2—8481no—
496C2#2028422711yescharacteristic
497C2#2028433411yescharacteristic
498C2C2×#1681088611yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2974C2
socleC22×C53C2×#121
prime radical#20284806C2
hypercenter#2974D53
hypocenterC53#19462
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2974

Complements

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

NH ≅ G/Ntype
1. #20285845C2semidirect
3. C2×#16810828C2semidirect
4. #20274276C2semidirect
5. #20274214C2semidirect
6. #20274152C2semidirect
7. #20274090C2semidirect
8. #20273500C2semidirect
9. #20273431C2semidirect
10. #20273411C2semidirect
11. C53×#2974C2semidirect
15. #16811021C22semidirect
16. #16810970C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810371C22semidirect
38. #16810368C22semidirect
39. #16810368C22semidirect
40. #16810366C22semidirect
246. #2974D53semidirect
298. C53#19462semidirect

Maximal subgroups

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

subgroupindex
#202858452
#202848062
C2×#168108282
#202742762
#202742142
#202741522
#202740902
#202735002
#202734312
#202734112
C53×#29742
#1946253