Finite Group Databasebeta

← #27438436

Subgroups — #27438436

Order 13568.

1114 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811033C211yesmaximal, characteristic, complemented
2C2×#16810982C211yesmaximal, characteristic, complemented
3#20283067C211yesmaximal, characteristic, complemented
4#20275908C212yesmaximal, complemented
5#20275894C212yesmaximal
6#20275685C211yesmaximal, characteristic, complemented
7#20275532C211yesmaximal, characteristic, complemented
8#20274976C212yesmaximal, complemented
9#20274969C212yesmaximal, complemented
10#20274477C211yesmaximal, characteristic, complemented
11C2×C53×#529C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13C2×#13133196—21no—
14#16811033—24no—
15#16810982—24no—
16C2×#13133150C2212yes—
17C2×#13133148C2212yes—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×C22⋊#88322C2212yes—
23C2×#13133128C2211yescharacteristic
24C2×#13133128C2211yescharacteristic
25#16810529—21no—
26#16810521—21no—
27#16810521C2212yes—
28#16810493C2211yescharacteristic
29#16810483—21no—
30#16810483C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C53×#529—24no—
35C53×#529—24no—
36C2×#126×C53—21no—
37C2×#125×C53C2211yescharacteristic
38C2×#124×C53C2211yescharacteristic
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×#121×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4×D4×C53—21no—
44C2×C4×D4×C53C2211yescharacteristic
45C2×#117×C53—21no—
46C22×C42×C53C2211yescharacteristic
47C22×Dic106—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133217—22no—
51#13133217—22no—
52#13133196—44no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53C2312yes—
57C2×C4⋊Dic53C2312yes—
58C2×#11719919—22no—
59C2×C22⋊Dic53—22no—
60#13133150—24no—
61#13133148—24no—
62#13133141—24no—
63#13133141—24no—
64#13133138—24no—
65#13133138—24no—
66C22⋊#88322—24no—
67#13133128—22no—
68#13133128—22no—
69#13133124—22no—
70#13133124—22no—
71C22×Q8×C53C2311yescharacteristic
72C22×D4×C53—21no—
73C22×D4×C53C2311yescharacteristic
74C23×C4×C53—21no—
75C23×C4×C53D411yescharacteristic
76C23×C4×C53D411yescharacteristic
77C23×C4×C53C2311yescharacteristic
78#126×C53—28no—
79#125×C53—22no—
80#125×C53D414yes—
81#124×C53—22no—
82#124×C53D414yes—
83#123×C53—28no—
84C22⋊Q8×C53—22no—
85C22⋊Q8×C53—22no—
86#121×C53—22no—
87#121×C53—22no—
88C4×Q8×C53—22no—
89C4×Q8×C53—22no—
90C4×D4×C53—28no—
91C4×D4×C53—22no—
92C4×D4×C53—22no—
93#117×C53—28no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C22⋊C4×C53—21no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C42×C53—22no—
106C2×C42×C53—22no—
107C2×C42×C53—21no—
108C2×C42×C53D411yescharacteristic
109C2×C42×C53D411yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112C24×C53—21no—
113C24×C53C2×D411yescharacteristic
114C2×Q8×C53—21no—
115C2×Q8×C53—22no—
116C2×Q8×C53—22no—
117C2×Q8×C53C2×D412yes—
118C2×D4×C53—24no—
119C2×D4×C53—21no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—42no—
123C2×D4×C53—42no—
124C2×D4×C53C2×D412yes—
125C22×C4×C53—24no—
126C22×C4×C53—22no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—41no—
138C22×C4×C53—41no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—22no—
148C22×Dic53—22no—
149C2×Dic106—42no—
150C2×Dic106—42no—
151C2×Dic106—42no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—24no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53C2×D412yes—
159C4⋊C4×C53C2×D412yes—
160C22⋊C4×C53—24no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—24no—
163C22⋊C4×C53—24no—
164C22⋊C4×C53—24no—
165C42×C53—24no—
166C42×C53—22no—
167C42×C53—22no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53C2×D412yes—
171C42×C53C2×D412yes—
172C22⋊Dic53—44no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—22no—
176C4⋊Dic53—22no—
177C4⋊Dic53—42no—
178C4⋊Dic53—42no—
179C4⋊Dic53—42no—
180#11719919—44no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322#4114yes—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—22no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53AGL(1,ℤ/8)12yes—
201C23×C53C22×D411yescharacteristic
202Q8×C53—24no—
203Q8×C53—42no—
204D4×C53—24no—
205D4×C53—48no—
206D4×C53—42no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—22no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—44no—
223C2×C4×C53—44no—
224C2×C4×C53—48no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53#12012yes—
234C2×C4×C53#12012yes—
235C2×C4×C53#13712yes—
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C2×#4112yes—
239C2×C4×C532−1+412yes—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246Dic106—84no—
247#88322—24no—
248#88322—42no—
249#25679—531nomaximal, Sylow
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53C2×#12011yescharacteristic
265C22×C53#53211yescharacteristic
266C22×C53#53611yescharacteristic
267C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C53C2×#13711yescharacteristic
269C22×C53#57511yescharacteristic
270C22×C53#57511yescharacteristic
271C4×C53—22no—
272C4×C53—22no—
273C4×C53—22no—
274C4×C53—41no—
275C4×C53—42no—
276C4×C53—41no—
277C4×C53—42no—
278C4×C53—42no—
279C4×C53#54712yes—
280C4×C53#54812yes—
281Dic53—82no—
282Dic53—82no—
283C2×#529D5311yescharacteristic
284C2×#501—531no—
285C2×#489—531no—
286#2857—531no—
287#2035—532no—
288#2033—532no—
289#1992—531no—
290#1959—531no—
291#1864—532no—
292#1863—532no—
293#1749—531no—
294C2×C53—42no—
295C2×C53—82no—
296C2×C53#243211yescharacteristic
297C2×C53#300611yescharacteristic
298C2×C53#300711yescharacteristic
299C2×C53#317211yescharacteristic
300C2×C53#318011yescharacteristic
301C2×C53#323811yescharacteristic
302C2×C53#324011yescharacteristic
303C2×C4⋊Q8—531no—
304#529—24no—
305#529—24no—
306C2×#126—21no—
307C2×#125C2×D5311yescharacteristic
308C2×#124C2×D5311yescharacteristic
309C2×#123—21no—
310C2×C22⋊Q8—1061no—
311C2×C22⋊Q8C2×D5311yescharacteristic
312C2×#121C2×D5311yescharacteristic
313C2×C4×Q8C2×D5311yescharacteristic
314C2×C4×D4—21no—
315C2×C4×D4C2×D5311yescharacteristic
316C2×#117—21no—
317C22×C42C2×D5311yescharacteristic
318#501—1064no—
319#489—1064no—
320C2×#107—532no—
321C2×#106—532no—
322C2×#103—532no—
323C2×#103—532no—
324C2×#102—532no—
325C2×#102—532no—
326C2×C22⋊C8—532no—
327C2×#97—531no—
328C2×#97—531no—
329#391—1061no—
330#389—532no—
331#389—1061no—
332#384—531no—
333#380—532no—
334#380—1061no—
335#340—532no—
336#340—532no—
337#336—532no—
338C53#2567911yescharacteristic, complemented, Sylow
339C22×Q8—1061no—
340C22×Q8C22×D5311yescharacteristic
341C22×D4—21no—
342C22×D4C22×D5311yescharacteristic
343C23×C4—21no—
344C23×C4C22⋊D5311yescharacteristic
345C23×C4C22⋊D5311yescharacteristic
346C23×C4C22×D5311yescharacteristic
347C22×C8—532no—
348C22×C8—532no—
349C4⋊Q8—1062no—
350C4⋊Q8—1062no—
351#126—28no—
352#125—22no—
353#125C22⋊D5314yes—
354#124—22no—
355#124C22⋊D5314yes—
356#123—28no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—2124no—
360#121—22no—
361#121—22no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×D4—28no—
365C4×D4—22no—
366C4×D4—22no—
367#117—28no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—532no—
373C2×C4⋊C4—532no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—1062no—
386C2×C42—21no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42C22⋊D5311yescharacteristic
390C2×C42C22⋊D5311yescharacteristic
391C2×C42C22×D5311yescharacteristic
392C2×C42C22×D5311yescharacteristic
393#107—1064no—
394#106—1064no—
395#103—1064no—
396#103—1064no—
397#102—1064no—
398#102—1064no—
399C22⋊C8—1064no—
400#97—1062no—
401#97—1062no—
402#95—1062no—
403#95—1062no—
404C24—21no—
405C24C2×C22⋊D5311yescharacteristic
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—2122no—
410C2×Q8—2122no—
411C2×Q8—2122no—
412C2×Q8C2×C22⋊D5312yes—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4C2×C22⋊D5312yes—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4C2×C22⋊D5311yescharacteristic
439C22×C4C2×C22⋊D5311yescharacteristic
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C2×C22⋊D5311yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C23×D5311yescharacteristic
444C2×C8—534no—
445C2×C8—1062no—
446C2×C8—1062no—
447C2×C8—1062no—
448C2×C8—1062no—
449C2×C8—1062no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—1062no—
457C4⋊C4—1062no—
458C4⋊C4—1062no—
459C4⋊C4—1062no—
460C4⋊C4—2124no—
461C4⋊C4—2122no—
462C4⋊C4—2122no—
463C4⋊C4—2122no—
464C4⋊C4C2×C22⋊D5312yes—
465C4⋊C4C2×C22⋊D5312yes—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—2124no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—21no—
476C42—21no—
477C42C2×C22⋊D5312yes—
478C42C2×C22⋊D5312yes—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23#1313324212yes—
493C23C22×C22⋊D5311yescharacteristic
494Q8—24no—
495Q8—42no—
496Q8—4244no—
497D4—24no—
498D4—48no—
499D4—42no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—48no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—2122no—
527C2×C4—2122no—
528C2×C4—2122no—
529C2×C4—2122no—
530C2×C4—2122no—
531C2×C4—2122no—
532C2×C4#1313318612yes—
533C2×C4#1313318612yes—
534C2×C4#1313324712yes—
535C2×C4C22×C22⋊D5311yescharacteristic
536C2×C4C22×C22⋊D5311yescharacteristic
537C2×C4#41×D5312yes—
538C2×C4#1313326412yes—
539C8—1064no—
540C8—2122no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C2×#1313318611yescharacteristic
556C22#1681116911yescharacteristic
557C22#1681119411yescharacteristic
558C22C2×#1313324211yescharacteristic
559C22C2×#1313324711yescharacteristic
560C22#1681142111yescharacteristic
561C22#1681142111yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—41no—
566C4—42no—
567C4—41no—
568C4—42no—
569C4—42no—
570C4—4242no—
571C4—4242no—
572C4#1681127212yes—
573C4#1681128012yes—
574C2—42no—
575C2—82no—
576C2#2027833911yescharacteristic
577C2#2028396711yescharacteristic
578C2#2028397511yescharacteristic
579C2#2028532211yescharacteristic
580C2#2028541411yescharacteristic
581C2#2028593311yescharacteristic
582C2#2028595111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#529C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC2×#529D53
hypocenterC53#25679
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
1. C2×#16811033C2semidirect
2. C2×#16810982C2semidirect
3. #20283067C2semidirect
4. #20275908C2semidirect
6. #20275685C2semidirect
7. #20275532C2semidirect
8. #20274976C2semidirect
9. #20274969C2semidirect
10. #20274477C2semidirect
338. C53#25679semidirect

Maximal subgroups

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

subgroupindex
C2×#168110332
C2×#168109822
#202830672
#202759082
#202758942
#202756852
#202755322
#202749762
#202749692
#202744772
C2×C53×#5292
#2567953