Finite Group Databasebeta

← #27417762

Subgroups — #27417762

Order 13568.

1138 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810951C211yesmaximal, characteristic, complemented
2C22×#13133148C211yesmaximal, characteristic, complemented
3#20279023C211yesmaximal, characteristic
4#20276045C212yesmaximal, complemented
5#20275908C212yesmaximal, complemented
6#20275139C211yesmaximal, characteristic, complemented
7#20275008C211yesmaximal, characteristic
8#20274983C212yesmaximal, complemented
9#20274908C212yesmaximal, complemented
10#20274805C211yesmaximal, characteristic, complemented
11C2×C53×#554C211yesmaximal, characteristic
12C23×#88322C2211yescharacteristic
13C2×#13133198—21no—
14C22×C4⋊Dic53C2211yescharacteristic
15#16810951—24no—
16C2×#13133148—24no—
17C2×#13133148C2212yes—
18C2×#13133148C2212yes—
19C2×#13133148C2212yes—
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2212yes—
24#16810553—21no—
25#16810548—21no—
26#16810521C2212yes—
27#16810484—21no—
28#16810483C2211yescharacteristic
29#16810483C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C53×#554—24no—
35C53×#554—24no—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×#125×C53—21no—
38C2×C22⋊Q8×C53—21no—
39C2×C22⋊Q8×C53—21no—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×C22⋊Q8×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4×D4×C53—21no—
44C2×C4×D4×C53C2211yescharacteristic
45C22×C4⋊C4×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C23×Dic53—21no—
48C22×#88322—24no—
49C22×#88322—22no—
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52#13133198—44no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53C2312yes—
57C2×C4⋊Dic53C2312yes—
58C2×#11719919—22no—
59C2×C22⋊Dic53—22no—
60C2×C22⋊Dic53—21no—
61#13133148—24no—
62#13133148—24no—
63#13133148—24no—
64#13133141—24no—
65#13133141—24no—
66C22⋊#88322—24no—
67C22⋊#88322—22no—
68#13133124—22no—
69#13133124—21no—
70#13133124—22no—
71C22×Q8×C53—21no—
72C22×Q8×C53C2311yescharacteristic
73C22×D4×C53D411yescharacteristic
74C23×C4×C53—21no—
75C23×C4×C53—21no—
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2311yescharacteristic
78C4⋊Q8×C53—22no—
79C4⋊Q8×C53—22no—
80#125×C53—28no—
81C22⋊Q8×C53—28no—
82C22⋊Q8×C53—28no—
83C22⋊Q8×C53—22no—
84C22⋊Q8×C53—22no—
85C22⋊Q8×C53—22no—
86C22⋊Q8×C53D414yes—
87C4×Q8×C53—22no—
88C4×Q8×C53—22no—
89C4×D4×C53—28no—
90C4×D4×C53—22no—
91C4×D4×C53—22no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53D414yes—
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53D411yescharacteristic
109C2×C22⋊C4×C53D411yescharacteristic
110C2×C42×C53—21no—
111C2×C42×C53C2311yescharacteristic
112C24×C53C2×D411yescharacteristic
113C24×C53C2×D411yescharacteristic
114C2×Q8×C53—24no—
115C2×Q8×C53—21no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53—82no—
119C2×Q8×C53C2×D412yes—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53SD1614yes—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—24no—
130C22×C4×C53—24no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—82no—
136C22×C4×C53—82no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C4⋊C4×C53—24no—
150C4⋊C4×C53—24no—
151C4⋊C4×C53—24no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53C2×D412yes—
160C4⋊C4×C53C2×D412yes—
161C4⋊C4×C53C2×D412yes—
162C22⋊C4×C53—24no—
163C22⋊C4×C53—24no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—22no—
166C42×C53—24no—
167C42×C53—21no—
168C42×C53—21no—
169C22⋊Dic53—44no—
170C22⋊Dic53—42no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—42no—
176#11719919—44no—
177C2×#88322—28no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322#4114yes—
184C23×C53—22no—
185C23×C53—22no—
186C23×C53—21no—
187C23×C53—22no—
188C23×C53—21no—
189C23×C53—22no—
190C23×C53#12012yes—
191C23×C53C2×SD1612yes—
192C23×C53C2×SD1612yes—
193C23×C53#13712yes—
194C23×C53C22×D411yescharacteristic
195Q8×C53—24no—
196Q8×C53—41no—
197Q8×C53—41no—
198Q8×C53—84no—
199D4×C53—28no—
200C2×C4×C53—21no—
201C2×C4×C53—22no—
202C2×C4×C53—21no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—24no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—21no—
210C2×C4×C53—22no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—42no—
221C2×C4×C53—84no—
222C2×C4×C53—84no—
223C2×C4×C53#12012yes—
224C2×C4×C53C2×SD1612yes—
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C2×#4112yes—
228C2×C4×C532−1+412yes—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—84no—
236#88322—24no—
237#88322—42no—
238#24097—531nomaximal, Sylow
239C22×C53—24no—
240C22×C53—24no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—24no—
244C22×C53—22no—
245C22×C53—42no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53#44814yes—
249C22×C53C2×#12011yescharacteristic
250C22×C53#54712yes—
251C22×C53C22×SD1611yescharacteristic
252C22×C53C2×#13711yescharacteristic
253C22×C53#57712yes—
254C4×C53—22no—
255C4×C53—22no—
256C4×C53—41no—
257C4×C53—42no—
258C4×C53—41no—
259C4×C53—42no—
260C4×C53—42no—
261C4×C53—42no—
262C4×C53#52511yescharacteristic
263C4×C53#53611yescharacteristic
264C4×C53#54712yes—
265Dic53—82no—
266Dic53—82no—
267C2×#554D5311yescharacteristic
268C2×#483—531no—
269C22×#106—531no—
270#2486—531no—
271#2058—532no—
272#2035—532no—
273#1892—531no—
274#1869—531no—
275#1865—532no—
276#1853—532no—
277#1824—531no—
278C2×C53—24no—
279C2×C53—42no—
280C2×C53#240611yescharacteristic
281C2×C53C2×#44811yescharacteristic
282C2×C53#300911yescharacteristic
283C2×C53#333512yes—
284C2×C53#334412yes—
285C23×C8—531no—
286#554—24no—
287#554—24no—
288C2×C4⋊Q8C2×D5311yescharacteristic
289C2×#125—21no—
290C2×#123—1061no—
291C2×C22⋊Q8—21no—
292C2×C22⋊Q8—21no—
293C2×C22⋊Q8C2×D5311yescharacteristic
294C2×C22⋊Q8C2×D5311yescharacteristic
295C2×C4×Q8C2×D5311yescharacteristic
296C2×C4×D4—21no—
297C2×C4×D4C2×D5311yescharacteristic
298C22×C4⋊C4—531no—
299C22×C4⋊C4C2×D5311yescharacteristic
300C22×C4⋊C4C2×D5311yescharacteristic
301#483—1064no—
302C2×#106—532no—
303C2×#106—532no—
304C2×#106—532no—
305C2×#106—1064no—
306C2×#103—532no—
307C2×#103—532no—
308C2×C22⋊C8—531no—
309C2×C22⋊C8—532no—
310#398—1061no—
311#397—1061no—
312#389—532no—
313#381—1061no—
314#380—531no—
315#380—532no—
316#340—532no—
317#340—532no—
318#340—532no—
319#336—532no—
320C53#2409711yescharacteristic, complemented, Sylow
321C22×Q8—21no—
322C22×Q8C22×D5311yescharacteristic
323C22×D4C22⋊D5311yescharacteristic
324C23×C4—21no—
325C23×C4—21no—
326C23×C4—1061no—
327C23×C4C22×D5311yescharacteristic
328C23×C4C22×D5311yescharacteristic
329C22×C8—532no—
330C22×C8—532no—
331C22×C8—1064no—
332C22×C8—1062no—
333C4⋊Q8—22no—
334C4⋊Q8—22no—
335#125—28no—
336#123—2124no—
337C22⋊Q8—28no—
338C22⋊Q8—28no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8C22⋊D5314yes—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×D4—28no—
346C4×D4—22no—
347C4×D4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—532no—
358C2×C4⋊C4—532no—
359C2×C4⋊C4—1062no—
360C2×C4⋊C4—1062no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4—1062no—
363C2×C4⋊C4C22⋊D5311yescharacteristic
364C2×C4⋊C4C22⋊D5314yes—
365C2×C4⋊C4C22×D5311yescharacteristic
366C2×C4⋊C4C22×D5311yescharacteristic
367C2×C4⋊C4C22×D5311yescharacteristic
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—1062no—
372C2×C22⋊C4C22⋊D5311yescharacteristic
373C2×C22⋊C4C22⋊D5311yescharacteristic
374C2×C42—21no—
375C2×C42C22×D5311yescharacteristic
376#106—1064no—
377#106—1064no—
378#106—1064no—
379#103—1064no—
380#103—1064no—
381C22⋊C8—1064no—
382C22⋊C8—1062no—
383#95—1062no—
384#95—1061no—
385#95—1062no—
386C24C2×C22⋊D5311yescharacteristic
387C24C2×C22⋊D5311yescharacteristic
388C2×Q8—24no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—82no—
393C2×Q8C2×C22⋊D5312yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4#1171992414yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—1062no—
412C22×C4—1062no—
413C22×C4—2122no—
414C22×C4—2122no—
415C22×C4—2122no—
416C22×C4C2×C22⋊D5311yescharacteristic
417C22×C4C2×C22⋊D5311yescharacteristic
418C22×C4C2×C22⋊D5311yescharacteristic
419C22×C4C2×C22⋊D5312yes—
420C22×C4C2×C22⋊D5312yes—
421C22×C4C2×C22⋊D5311yescharacteristic
422C22×C4C23×D5311yescharacteristic
423C2×C8—534no—
424C2×C8—1068no—
425C2×C8—1062no—
426C2×C8—1062no—
427C2×C8—1062no—
428C2×C8—1062no—
429C2×C8—1062no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—1062no—
441C4⋊C4—1062no—
442C4⋊C4—1062no—
443C4⋊C4—1062no—
444C4⋊C4—2122no—
445C4⋊C4—2124no—
446C4⋊C4C2×C22⋊D5312yes—
447C4⋊C4C2×C22⋊D5312yes—
448C4⋊C4C2×C22⋊D5312yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—2122no—
454C22⋊C4—2124no—
455C42—24no—
456C42—21no—
457C42—21no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23#1313318612yes—
465C23C2×#1171992412yes—
466C23C2×#1171992412yes—
467C23#1313324812yes—
468C23C22×C22⋊D5311yescharacteristic
469Q8—24no—
470Q8—41no—
471Q8—41no—
472Q8—84no—
473D4—28no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—4244no—
498C2×C4—2122no—
499C2×C4—2122no—
500C2×C4—2122no—
501C2×C4—2122no—
502C2×C4—2122no—
503C2×C4—2122no—
504C2×C4#1313318612yes—
505C2×C4C2×#1171992412yes—
506C2×C4C22×C22⋊D5311yescharacteristic
507C2×C4C22×C22⋊D5311yescharacteristic
508C2×C4#41×D5312yes—
509C2×C4#1313326412yes—
510C8—1064no—
511C8—2122no—
512C22—22no—
513C22—22no—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—22no—
518C22—41no—
519C22—41no—
520C22—42no—
521C22#1681076214yes—
522C22C2×#1313318611yescharacteristic
523C22#1681127212yes—
524C22C22×#1171992411yescharacteristic
525C22C2×#1313324811yescharacteristic
526C22#1681143412yes—
527C4—22no—
528C4—22no—
529C4—41no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—4242no—
536C4—4242no—
537C4#1681113011yescharacteristic
538C4#1681119411yescharacteristic
539C4#1681127212yes—
540C2—24no—
541C2—42no—
542C2#2027804611yescharacteristic
543C2C2×#1681076211yescharacteristic
544C2#2028399211yescharacteristic
545C2#2028706612yes—
546C2#2028716412yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#554C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenterC2×#554D53
hypocenterC53#24097
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16810951C2semidirect
2. C22×#13133148C2semidirect
4. #20276045C2semidirect
5. #20275908C2semidirect
6. #20275139C2semidirect
8. #20274983C2semidirect
9. #20274908C2semidirect
10. #20274805C2semidirect
320. C53#24097semidirect

Maximal subgroups

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

subgroupindex
C2×#168109512
C22×#131331482
#202790232
#202760452
#202759082
#202751392
#202750082
#202749832
#202749082
#202748052
C2×C53×#5542
#2409753