Finite Group Databasebeta

← #27370084

Subgroups — #27370084

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#20285843C211yesmaximal, characteristic, complemented
2#20284806C211yesmaximal, characteristic
3#20283750C211yesmaximal, characteristic, complemented
4#20274278C212yesmaximal, complemented
5#20274212C212yesmaximal, complemented
6#20274154C212yesmaximal, complemented
7#20274088C212yesmaximal, complemented
8#20273498C211yesmaximal, characteristic, complemented
9#20273432C211yesmaximal, characteristic, complemented
10#20273408C211yesmaximal, characteristic, complemented
11C2×C53×#461C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16811019C2212yescomplemented
14#16810971C2212yescomplemented
15#16810941—21no—
16#16810941—21no—
17#16810941—21no—
18#16810913C2211yescharacteristic
19#16810913C2211yescharacteristic
20#16810910—22no—
21#16810818C2211yescharacteristic
22#16810818C2211yescharacteristic
23#16810757—21no—
24#16810733—21no—
25#16810733—21no—
26#16810701C2212yes—
27#16810695—22no—
28D4×#88322C2212yes—
29#16810680C2211yescharacteristic
30#16810680C2211yescharacteristic
31#16810676C2211yescharacteristic
32#16810587—21no—
33C2×C22⋊#88322—21no—
34#16810375C2212yescomplemented
35#16810373C2212yescomplemented
36#16810371C2212yescomplemented
37#16810369C2212yescomplemented
38#16810369C2212yescomplemented
39#16810366C2212yescomplemented
40C22×SD16×C53—21no—
41C2×C4⋊Q8×C53C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#461—21no—
44C53×#461—21no—
45C53×#461C2212yes—
46C53×#461C2212yes—
47C2×C4⋊C8×C53C2211yescharacteristic
48C2×#103×C53—21no—
49C2×#11719945—21no—
50C2×#11719922—21no—
51C2×#11719922—21no—
52C2×#11719917—22no—
53C22×#88322—22no—
54#13133217C2311yescharacteristic
55#13133214C2311yescharacteristic
56#13133205—21no—
57#13133191—21no—
58#13133191—21no—
59C4×D212—21no—
60C4×D212—21no—
61#13133148—21no—
62#13133148—21no—
63#13133148—22no—
64#13133148—22no—
65C4⋊#88322D412yes—
66C4⋊#88322C2311yescharacteristic
67C4⋊#88322C2311yescharacteristic
68C4⋊#88322C2312yes—
69#13133138—21no—
70#13133138—21no—
71#13133137—21no—
72#13133137—21no—
73#13133137—22no—
74#13133137—22no—
75C22⋊#88322—22no—
76C22⋊#88322—22no—
77C22⋊#88322—21no—
78C22⋊#88322—21no—
79C22⋊#88322D411yescharacteristic
80C22⋊#88322D411yescharacteristic
81#13133128—21no—
82#13133128—21no—
83C4×#88322C2312yes—
84C22×Q8×C53—21no—
85C22×D4×C53D411yescharacteristic
86C23×C4×C53—21no—
87C2×SD16×C53—22no—
88C2×SD16×C53—22no—
89C2×SD16×C53—44no—
90C22×C8×C53—21no—
91C4⋊Q8×C53—21no—
92C4⋊Q8×C53—21no—
93C4⋊Q8×C53—21no—
94C4⋊Q8×C53C2312yes—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53D411yescharacteristic
98C4×D4×C53D411yescharacteristic
99C4×D4×C53C2311yescharacteristic
100C4×D4×C53C2311yescharacteristic
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C22⋊C4×C53—21no—
105C2×C42×C53C2311yescharacteristic
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#103×C53—22no—
109#103×C53—22no—
110C24×C53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53—42no—
113C2×Q8×C53—42no—
114C2×Q8×C53—41no—
115C2×D4×C53—22no—
116C2×D4×C53—21no—
117C2×D4×C53—41no—
118C2×D4×C53C2×D411yescharacteristic
119C2×D4×C53C2×D411yescharacteristic
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—41no—
126C22×C4×C53—82no—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53#4111yescharacteristic
130C2×C22⋊D53—41no—
131#11719945—82no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212—21no—
135C2×C4×D53—41no—
136C2×Dic106—21no—
137SD16×C53—44no—
138SD16×C53—44no—
139C2×C8×C53—22no—
140C2×C8×C53—42no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53—41no—
146C4⋊C4×C53C2×D411yescharacteristic
147C4⋊C4×C53C2×D411yescharacteristic
148C22⋊C4×C53—21no—
149C22⋊C4×C53—21no—
150C22⋊C4×C53—21no—
151C22⋊C4×C53—21no—
152C42×C53C2×D411yescharacteristic
153C42×C53C2×D411yescharacteristic
154C42×C53#4111yescharacteristic
155C42×C53C2411yescharacteristic
156#11719922—42no—
157#11719922—42no—
158#11719921—41no—
159#11719921—41no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162#11719917—82no—
163C2×#88322—22no—
164C2×#88322—21no—
165C2×#88322—21no—
166C2×#88322—82no—
167C2×#88322C2×D412yes—
168C2×#88322C2×D412yes—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—42no—
172C23×C53—42no—
173C23×C53—81no—
174C23×C53#12111yescharacteristic
175C22×D53—41no—
176Q8×C53—42no—
177Q8×C53—42no—
178Q8×C53—82no—
179D4×C53—21no—
180D4×C53—21no—
181D4×C53—21no—
182D4×C53—42no—
183D4×C53AGL(1,ℤ/8)12yes—
184C2×C4×C53—22no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—22no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—42no—
193C2×C4×C53—81no—
194C2×C4×C53—81no—
195C2×C4×C53—81no—
196C2×C4×C53—82no—
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53C22×D411yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C2×#4111yescharacteristic
203C22⋊D53—82no—
204C2×Dic53—41no—
205D212—41no—
206D212—41no—
207D212—42no—
208D212—81no—
209C4×D53—82no—
210Dic106—81no—
211C8×C53—42no—
212#88322—22no—
213#88322—22no—
214#88322—41no—
215#88322—41no—
216#88322—42no—
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×C53—82no—
239C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
240C4×C53#57511yescharacteristic
241C4×C53#57712yes—
242Dic53—81no—
243#3227—531no—
244#3109—531no—
245C2×#461D5311yescharacteristic, complemented
246#2974—531no—
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—21no—
262#532—531no—
263C2×C4⋊Q8C2×D5311yescharacteristic
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—21no—
274#461—21no—
275#461C2×D5312yes—
276#461C2×D5312yes—
277#459—531no—
278#459—531no—
279#447—1061no—
280#442—1061no—
281#442—1061no—
282#436—532no—
283#435—1062no—
284C8×D4—532no—
285#432—531no—
286#432—531no—
287#431—531no—
288C2×C4⋊C8C2×D5311yescharacteristic
289C2×#103—21no—
290#407—1061no—
291C2×C22⋊C8—1061no—
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—21no—
301C22×D4C22⋊D5311yescharacteristic
302C23×C4—21no—
303C2×SD16—22no—
304C2×SD16—22no—
305C2×SD16—44no—
306C2×D8—1061no—
307C2×D8—1061no—
308C2×M16—1062no—
309C22×C8—21no—
310C22×C8—1062no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—531no—
315C4⋊Q8C22×D5312yes—
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×D4C22⋊D5311yescharacteristic
325C4×D4C22⋊D5311yescharacteristic
326C4×D4C22×D5311yescharacteristic
327C4×D4C22×D5311yescharacteristic
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4C22⋊D5311yescharacteristic
331C2×C22⋊C4—21no—
332C2×C42C22×D5311yescharacteristic
333#106—1062no—
334#106—1062no—
335#106—1061no—
336#106—1061no—
337C4⋊C8—21no—
338C4⋊C8—532no—
339C4⋊C8—532no—
340C4⋊C8—531no—
341C4⋊C8—531no—
342C4⋊C8C22×D5312yes—
343#103—22no—
344#103—22no—
345#102—1062no—
346#102—1062no—
347#102—1061no—
348#102—1061no—
349#102—1061no—
350#102—1061no—
351C22⋊C8—531no—
352C22⋊C8—531no—
353C22⋊C8—1062no—
354C22⋊C8—1062no—
355C22⋊C8—1061no—
356C22⋊C8—1061no—
357#97—1061no—
358#97—1061no—
359C4×C8—532no—
360C24—21no—
361#41—4242no—
362C2×Q8—22no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—41no—
366C2×Q8—1061no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—1061no—
371C2×D4—1061no—
372C2×D4—1061no—
373C2×D4—2121no—
374C2×D4C2×C22⋊D5311yescharacteristic
375C2×D4C2×C22⋊D5311yescharacteristic
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—41no—
382C22×C4—82no—
383C22×C4—2121no—
384C22×C4D4×D5311yescharacteristic
385C22×C4#1171994711yescharacteristic
386C22×C4C2×C22⋊D5311yescharacteristic
387SD16—44no—
388SD16—44no—
389D8—2122no—
390D8—2122no—
391M16—4242no—
392C2×C8—22no—
393C2×C8—42no—
394C2×C8—532no—
395C2×C8—532no—
396C2×C8—1062no—
397C2×C8—1061no—
398C2×C8—1061no—
399C2×C8—4242no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—1061no—
406C4⋊C4—1061no—
407C4⋊C4C2×C22⋊D5311yescharacteristic
408C4⋊C4C2×C22⋊D5311yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—2121no—
414C22⋊C4—2121no—
415C42D4×D5311yescharacteristic
416C42#1171994711yescharacteristic
417C42C2×C22⋊D5311yescharacteristic
418C42C23×D5311yescharacteristic
419C23—21no—
420C23—21no—
421C23—42no—
422C23—42no—
423C23—81no—
424C23—2121no—
425C23#1313318911yescharacteristic
426Q8—42no—
427Q8—42no—
428Q8—82no—
429Q8—4241no—
430D4—21no—
431D4—21no—
432D4—21no—
433D4—42no—
434D4—4241no—
435D4—4242no—
436D4—2121no—
437D4—2121no—
438D4—2122no—
439D4#1313324312yes—
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—42no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—4242no—
454C2×C4—2121no—
455C2×C4#1313318911yescharacteristic
456C2×C4#1313318911yescharacteristic
457C2×C4#1313318911yescharacteristic
458C2×C4C2×D4×D5311yescharacteristic
459C2×C4C22×C22⋊D5311yescharacteristic
460C2×C4C2×#1171994711yescharacteristic
461C8—42no—
462C8—1062no—
463C8—1062no—
464C8—2121no—
465C8—2121no—
466C8—2122no—
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#1681088812yes—
480C22C2×#1313318911yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—41no—
486C4—41no—
487C4—82no—
488C4—4241no—
489C4C2×#1313324311yescharacteristic
490C4#1681142111yescharacteristic
491C4#1681142912yes—
492C2—22no—
493C2—42no—
494C2—81no—
495C2—8481no—
496C2#2028422411yescharacteristic
497C2#2028433011yescharacteristic
498C2C2×#1681088811yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#461C2
socleC22×C53C2×#121
prime radical#20284806C2
hypercenterC2×#461D53
hypocenterC53#19462
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. #20285843C2semidirect
3. #20283750C2semidirect
4. #20274278C2semidirect
5. #20274212C2semidirect
6. #20274154C2semidirect
7. #20274088C2semidirect
8. #20273498C2semidirect
9. #20273432C2semidirect
10. #20273408C2semidirect
11. C2×C53×#461C2semidirect
13. #16811019C22semidirect
14. #16810971C22semidirect
34. #16810375C22semidirect
35. #16810373C22semidirect
36. #16810371C22semidirect
37. #16810369C22semidirect
38. #16810369C22semidirect
39. #16810366C22semidirect
245. C2×#461D53semidirect
298. C53#19462semidirect

Maximal subgroups

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

subgroupindex
#202858432
#202848062
#202837502
#202742782
#202742122
#202741542
#202740882
#202734982
#202734322
#202734082
C2×C53×#4612
#1946253