Finite Group Databasebeta

← #27389269

Subgroups — #27389269

Order 13568.

1518 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811168C211yesmaximal, characteristic
2C22×C8×Dic53C211yesmaximal, characteristic, complemented
3#20277117C211yesmaximal, characteristic, complemented
4#20275674C212yesmaximal, complemented
5#20275519C212yesmaximal, complemented
6#20275144C211yesmaximal, characteristic, complemented
7#20275143C211yesmaximal, characteristic, complemented
8#20275141C211yesmaximal, characteristic, complemented
9#102×Dic53C212yesmaximal, complemented
10#20274529C212yesmaximal, complemented
11C53×#1893C211yesmaximal, characteristic, complemented
12C22×#11719947—21no—
13C23×#88322C2211yescharacteristic
14#16811168—24no—
15#16811168C418yescomplemented
16C2×#13133216C2211yescharacteristic
17C2×#13133213C2211yescharacteristic
18C2×#13133203—21no—
19C2×#13133191C2211yescharacteristic
20C2×#13133189—21no—
21C2×D4×Dic53C2211yescharacteristic
22C2×#13133179C2211yescharacteristic
23C2×#13133177—21no—
24C2×#13133173—21no—
25C22×C4×Dic53C2211yescharacteristic
26C2×#13133139C2212yes—
27C2×#13133138C2212yes—
28C2×#13133137C2212yes—
29C2×C8×Dic53—24no—
30C2×C8×Dic53C2212yes—
31C2×C8×Dic53C2212yes—
32#16810492C2212yes—
33#16810487—21no—
34#16810483C2211yescharacteristic
35#16810481—21no—
36C4⋊C4×Dic53C2212yes—
37#16810474—21no—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C23×C8×C53C2211yescharacteristic
42C2×#121×C53C2211yescharacteristic
43C2×#102×C53C2212yes—
44C53×#380C2211yescharacteristic
45C53×#340C2212yes—
46C2×#11719947—28no—
47C2×#11719947—44no—
48C2×#11719947—44no—
49C22×Dic106—21no—
50C22×D212C2311yescharacteristic
51C22×C22⋊D53—21no—
52C22×C22⋊D53—21no—
53C22×C22⋊D53—21no—
54C23×Dic53—21no—
55C23×Dic53D411yescharacteristic
56C23×Dic53D411yescharacteristic
57C22×C4×D53—21no—
58C22×#88322—24no—
59C22×#88322—22no—
60C22×#88322C2312yes—
61#13133216—22no—
62#13133216C2×C414yes—
63#13133213—22no—
64#13133213C2×C414yes—
65#13133203—28no—
66#13133191—22no—
67#13133191C2×C414yes—
68#13133189—28no—
69D4×Dic53—22no—
70D4×Dic53C2×C414yes—
71#13133179—22no—
72#13133179C2×C414yes—
73#13133177—28no—
74#13133173—44no—
75C2×C4⋊Dic53C2312yes—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—22no—
79C2×#11719919—22no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×C4×Dic53—22no—
85C2×C4×Dic53C2×C414yes—
86C2×C4×Dic53D414yes—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133139—24no—
92#13133138—24no—
93#13133137—24no—
94C8×Dic53—24no—
95C8×Dic53—24no—
96#13133124—21no—
97#13133124—22no—
98C22×D4×C53—21no—
99C22×D4×C53C2311yescharacteristic
100C23×C4×C53C2311yescharacteristic
101C22×C8×C53—24no—
102C22×C8×C53—22no—
103C22×C8×C53C2312yes—
104#121×C53—22no—
105#121×C53C2×C414yes—
106C2×C4⋊C4×C53C2312yes—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110#102×C53—24no—
111#95×C53—21no—
112C24×C53—21no—
113C24×C53C2×D411yescharacteristic
114C23×D53—21no—
115C2×D4×C53—24no—
116C2×D4×C53—21no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—42no—
120C2×D4×C53—42no—
121C2×D4×C53C22×C412yes—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2411yescharacteristic
130C2×C22⋊D53—24no—
131C2×C22⋊D53—24no—
132C2×C22⋊D53—24no—
133C2×C22⋊D53—42no—
134C2×C22⋊D53—42no—
135C2×C22⋊D53—42no—
136C2×C22⋊D53—42no—
137C2×C22⋊D53—44no—
138C22×Dic53—24no—
139C22×Dic53—21no—
140C22×Dic53—24no—
141C22×Dic53—22no—
142C22×Dic53—42no—
143C22×Dic53—42no—
144C22×Dic53C22⋊C414yes—
145C22×Dic53C22⋊C414yes—
146C22×Dic53D814yes—
147C22×Dic53SD1614yes—
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152#11719947—432no—
153C2×D212—22no—
154C2×D212—22no—
155C2×D212—21no—
156C2×D212C22×C412yes—
157C2×C4×D53—24no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×Dic106—24no—
161C2×Dic106—42no—
162C2×Dic106—42no—
163C2×C8×C53—28no—
164C2×C8×C53—22no—
165C2×C8×C53—22no—
166C2×C8×C53#4114yescomplemented
167C4⋊C4×C53—21no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53C22×C412yes—
171C22⋊C4×C53—24no—
172C22⋊C4×C53—42no—
173C22⋊Dic53—24no—
174C22⋊Dic53—42no—
175#11719921—24no—
176#11719921—24no—
177C4⋊Dic53—21no—
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53C22×C412yes—
181#11719919—24no—
182#11719919—44no—
183#11719919—44no—
184C4×Dic53—44no—
185C4×Dic53C22⋊C414yes—
186C4×Dic53C22⋊C414yes—
187C4×Dic53C22×C412yes—
188C4×Dic53C22×C412yes—
189C4×Dic53C2×D412yes—
190C4×Dic53C2×D412yes—
191C2×#88322—28no—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322#4114yescomplemented
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53—42no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—42no—
202C23×C53C2×C22⋊C412yes—
203C23×C53C2×D812yes—
204C23×C53C2×SD1612yes—
205C23×C53C22×D411yescharacteristic
206C22×D53—22no—
207C22×D53—42no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—42no—
211D4×C53—24no—
212D4×C53—48no—
213D4×C53—41no—
214D4×C53—41no—
215C2×C4×C53—22no—
216C2×C4×C53—24no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53C2×C22⋊C412yes—
226C2×C4×C53C23×C411yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C2×#4112yes—
229C2×C4×C53C2×#4112yes—
230C22⋊D53—48no—
231C22⋊D53—48no—
232C22⋊D53—48no—
233C2×Dic53—22no—
234C2×Dic53—28no—
235C2×Dic53—22no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—48no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53#102116yescomplemented
243C2×Dic53C2×C22⋊C412yes—
244C2×Dic53C2×C22⋊C412yes—
245C2×Dic53C2×C22⋊C412yes—
246C2×Dic53C2×C22⋊C412yes—
247C2×Dic53C2×D812yes—
248C2×Dic53C2×D812yes—
249C2×Dic53C2×SD1612yes—
250C2×Dic53C2×SD1612yes—
251C2×Dic53#13514yes—
252D212—24no—
253D212—41no—
254D212—41no—
255C4×D53—48no—
256Dic106—48no—
257C8×C53—24no—
258#88322—24no—
259#21401—531nomaximal, Sylow
260C22×C53—22no—
261C22×C53—24no—
262C22×C53—24no—
263C22×C53—22no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—82no—
268C22×C53—82no—
269C22×C53C2×#10214yes—
270C22×C53C22×C22⋊C411yescharacteristic
271C22×C53#53212yes—
272C22×C53C22×D811yescharacteristic
273C22×C53C22×SD1611yescharacteristic
274C22×C53C2×#13512yes—
275C2×D53—42no—
276C2×D53—42no—
277C2×D53—42no—
278C2×D53—82no—
279C2×D53—82no—
280C4×C53—22no—
281C4×C53—42no—
282C4×C53—82no—
283C4×C53C4×#4112yes—
284C4×C53#52511yescharacteristic
285C4×C53#52511yescharacteristic
286Dic53—22no—
287Dic53—42no—
288Dic53—82no—
289Dic53C2×#10214yes—
290Dic53C2×#10214yes—
291Dic53#41614yes—
292C2×#532—531no—
293C22×C4×C8—531no—
294#2285—531no—
295#1990—532no—
296#1957—532no—
297#1893—531no—
298#1893—531no—
299#1893—531no—
300#1893D5311yescharacteristic, complemented
301C4×#102—532no—
302C4×#102—532no—
303C2×C53—24no—
304C2×C53—82no—
305C2×C53#229711yescharacteristic
306C2×C53C22×#10211yescharacteristic
307C2×C53C2×#41611yescharacteristic
308C2×C53#316712yes—
309C2×C53#316912yes—
310D53—82no—
311C22×#41—1061no—
312C23×C8—531no—
313C23×C8C2×D5311yescharacteristic
314#532—538no—
315#532—1064no—
316C2×C4⋊Q8—531no—
317C2×#127—531no—
318C2×#124—1061no—
319C2×#121—531no—
320C2×#121—1061no—
321C2×#121C2×D5311yescharacteristic
322C2×C4×D4—531no—
323C2×C4×D4—531no—
324C2×C4×D4—1061no—
325C2×#117—1061no—
326C22×C42—531no—
327C2×#102—532no—
328C2×#102—532no—
329C2×#102—532no—
330C2×#102C2×D5312yes—
331C2×C4×C8—532no—
332C2×C4×C8—532no—
333C2×C4×C8—1064no—
334#384—532no—
335#382—1061no—
336#380—531no—
337#380—1061no—
338#380C2×D5311yescharacteristic
339C4×C4⋊C4—532no—
340C4×C4⋊C4—1061no—
341#340—532no—
342#340—532no—
343#340—532no—
344#340C2×D5312yes—
345C53#2140111yescharacteristic, complemented, Sylow
346C2×#41—1068no—
347C2×#41—2124no—
348C2×#41—2124no—
349C22×Q8—1061no—
350C22×D4—21no—
351C22×D4—531no—
352C22×D4—1061no—
353C22×D4—1061no—
354C22×D4—1061no—
355C22×D4C22×D5311yescharacteristic
356C23×C4—531no—
357C23×C4—531no—
358C23×C4—1061no—
359C23×C4—1061no—
360C23×C4C22×D5311yescharacteristic
361C22×C8—22no—
362C22×C8—24no—
363C22×C8—532no—
364C22×C8—1064no—
365C22×C8—1062no—
366C22×C8C22×D5312yes—
367C4⋊Q8—534no—
368C4⋊Q8—1062no—
369#127—534no—
370#127—1062no—
371#124—1068no—
372#121—22no—
373#121—534no—
374#121—1068no—
375#121—1062no—
376#121C4×D5314yes—
377C4×D4—534no—
378C4×D4—534no—
379C4×D4—1068no—
380C4×D4—1062no—
381C4×D4—1062no—
382#117—2124no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—532no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1062no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4C22×D5312yes—
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C42—534no—
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—534no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—1062no—
403#102—24no—
404#102—1064no—
405#102—1064no—
406#102—1064no—
407C4×C8—1064no—
408C4×C8—1064no—
409#95—21no—
410#95—1062no—
411#95—1061no—
412C24—21no—
413C24—1061no—
414C24D4×D5311yescharacteristic
415#41—21232no—
416C2×Q8—1064no—
417C2×Q8—2122no—
418C2×Q8—2122no—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—532no—
426C2×D4—1064no—
427C2×D4—1064no—
428C2×D4—1064no—
429C2×D4—1061no—
430C2×D4—1062no—
431C2×D4—1062no—
432C2×D4—2124no—
433C2×D4—2122no—
434C2×D4—2122no—
435C2×D4—2122no—
436C2×D4—2122no—
437C2×D4C2×C4×D5312yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—534no—
443C22×C4—534no—
444C22×C4—534no—
445C22×C4—531no—
446C22×C4—531no—
447C22×C4—531no—
448C22×C4—534no—
449C22×C4—531no—
450C22×C4—1064no—
451C22×C4—1064no—
452C22×C4—1064no—
453C22×C4—1061no—
454C22×C4—1062no—
455C22×C4—2122no—
456C22×C4—2122no—
457C22×C4—2122no—
458C22×C4—2122no—
459C22×C4C2×C4×D5312yes—
460C22×C4D4×D5311yescharacteristic
461C22×C4D4×D5312yes—
462C22×C4C23×D5311yescharacteristic
463C2×C8—28no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—534no—
467C2×C8—1068no—
468C2×C8—1062no—
469C2×C8—1062no—
470C2×C8#1171994714yescomplemented
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—532no—
475C4⋊C4—1064no—
476C4⋊C4—1061no—
477C4⋊C4—1062no—
478C4⋊C4—1062no—
479C4⋊C4—2124no—
480C4⋊C4—2124no—
481C4⋊C4C2×C4×D5312yes—
482C22⋊C4—24no—
483C22⋊C4—42no—
484C22⋊C4—1064no—
485C22⋊C4—1064no—
486C22⋊C4—1064no—
487C22⋊C4—2122no—
488C42—534no—
489C42—534no—
490C42—532no—
491C42—532no—
492C42—532no—
493C42—532no—
494C42—2124no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—1062no—
503C23—2121no—
504C23—2121no—
505C23—2122no—
506C23—2122no—
507C23C22⋊C4×D5312yes—
508C23D8×D5312yes—
509C23SD16×D5312yes—
510C23C2×D4×D5311yescharacteristic
511Q8—2128no—
512D4—24no—
513D4—48no—
514D4—41no—
515D4—41no—
516D4—1064no—
517D4—2128no—
518D4—2128no—
519D4—2128no—
520D4—2121no—
521D4—2121no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—532no—
533C2×C4—532no—
534C2×C4—532no—
535C2×C4—5316no—
536C2×C4—532no—
537C2×C4—532no—
538C2×C4—532no—
539C2×C4—532no—
540C2×C4—534no—
541C2×C4—532no—
542C2×C4—1068no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—2128no—
546C2×C4—2128no—
547C2×C4—2122no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4C22⋊C4×D5312yes—
553C2×C4C22×C4×D5311yescharacteristic
554C2×C4C2×D4×D5311yescharacteristic
555C2×C4#41×D5312yes—
556C2×C4C2×#1171994712yes—
557C8—24no—
558C8—1064no—
559C22—22no—
560C22—24no—
561C22—24no—
562C22—22no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—82no—
567C22—82no—
568C22—4242no—
569C22—4242no—
570C22—2122no—
571C22—2122no—
572C22—2122no—
573C22#102×D5314yes—
574C22C2×C22⋊C4×D5311yescharacteristic
575C22#1681116812yes—
576C22C2×D8×D5311yescharacteristic
577C22C2×SD16×D5311yescharacteristic
578C22#135×D5312yes—
579C4—22no—
580C4—42no—
581C4—82no—
582C4—534no—
583C4—534no—
584C4—534no—
585C4—1062no—
586C4—4242no—
587C4—2122no—
588C4C4×#1171994712yes—
589C4#1681112911yescharacteristic
590C4#1681113111yescharacteristic
591C2—24no—
592C2—82no—
593C2—4242no—
594C2#2027720011yescharacteristic
595C2C2×#102×D5311yescharacteristic
596C2#416×D5311yescharacteristic
597C2#2028526912yes—
598C2#2028528912yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1893C2
socleC23×C53C22×D4
prime radicalC2×#16811168C2
cosocleC22×C4×C53C24
hypercenter#1893D53
hypocenterC53#21401
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
2. C22×C8×Dic53C2semidirect
3. #20277117C2semidirect
4. #20275674C2semidirect
5. #20275519C2semidirect
6. #20275144C2semidirect
7. #20275143C2semidirect
8. #20275141C2semidirect
9. #102×Dic53C2semidirect
10. #20274529C2semidirect
11. C53×#1893C2semidirect
15. #16811168C4semidirect
166. C2×C8×C53#41semidirect
194. C2×#88322#41semidirect
242. C2×Dic53#102semidirect
300. #1893D53semidirect
345. C53#21401semidirect
470. C2×C8#11719947semidirect

Maximal subgroups

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

subgroupindex
C2×#168111682
C22×C8×Dic532
#202771172
#202756742
#202755192
#202751442
#202751432
#202751412
#102×Dic532
#202745292
C53×#18932
#2140153