Finite Group Databasebeta

← #27393828

Subgroups — #27393828

Order 13568.

1234 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811152C211yesmaximal, characteristic
2C2×#16810678C211yesmaximal, characteristic, complemented
3#384×D53C211yesmaximal, characteristic, complemented
4#20275684C211yesmaximal, characteristic, complemented
5#20275683C211yesmaximal, characteristic, complemented
6#20275532C211yesmaximal, characteristic, complemented
7#20275029C212yesmaximal
8#20275019C212yesmaximal, complemented
9#20274718C212yesmaximal, complemented
10#20274713C212yesmaximal, complemented
11C53×#1959C211yesmaximal, characteristic, complemented
12#16811152—24no—
13#16811152C418yescomplemented
14C2×#13133210—21no—
15C2×#13133209C2211yescharacteristic
16C2×#13133199—21no—
17C2×#13133193C2211yescharacteristic
18C2×#13133188C2211yescharacteristic
19C2×Q8×Dic53C2211yescharacteristic
20C2×D4×Dic53C2211yescharacteristic
21C2×#13133177—21no—
22C2×#13133173—21no—
23C2×C4⋊C4×D53—21no—
24C2×C4⋊C4×D53—21no—
25C2×C4⋊C4×D53C2212yes—
26C2×C42×D53C2211yescharacteristic
27#16810678—24no—
28C2×#13133146C2212yes—
29C2×#13133141C2212yes—
30C2×#13133138C2212yes—
31C2×#13133129C2212yes—
32C2×#13133128C2211yescharacteristic
33#16810493C2211yescharacteristic
34#16810492—21no—
35#16810492—21no—
36#16810492C2212yes—
37#16810386C2212yes—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×#124×C53C2211yescharacteristic
42C2×#103×C53C2212yes—
43C2×#102×C53C2212yes—
44C2×#97×C53C2211yescharacteristic
45C53×#384C2211yescharacteristic
46C22×C22⋊D53—21no—
47C23×Dic53—21no—
48C22×C4×D53—22no—
49C22×C4×D53D411yescharacteristic
50C22×C4×D53D411yescharacteristic
51C22×C4×D53C2311yescharacteristic
52C22×#88322C2312yes—
53#13133210—28no—
54#13133209—22no—
55#13133209C2×C414yescomplemented
56#13133199—28no—
57#13133193—22no—
58#13133193C2×C414yes—
59#13133188—22no—
60#13133188C2×C414yes—
61Q8×Dic53—22no—
62Q8×Dic53C2×C414yescomplemented
63D4×Dic53—22no—
64D4×Dic53C2×C414yescomplemented
65#13133177—28no—
66#13133173—28no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2312yes—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—22no—
77C2×#11719919—22no—
78C2×#11719919—22no—
79C4⋊C4×D53—42no—
80C4⋊C4×D53—42no—
81C4⋊C4×D53—44no—
82C4⋊C4×D53—44no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×#11719921—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C42×D53C2×C414yes—
91C42×D53D414yes—
92#13133146—24no—
93#13133141—42no—
94#13133138—42no—
95#13133129—24no—
96#13133128—22no—
97C22×Q8×C53C2311yescharacteristic
98C22×D4×C53C2311yescharacteristic
99C22×C8×C53C2312yes—
100#124×C53—22no—
101#124×C53C2×C414yescomplemented
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53C2312yes—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C42×C53C2311yescharacteristic
108#103×C53—42no—
109#102×C53—42no—
110#97×C53—22no—
111C24×C53—21no—
112C23×D53C2×D411yescharacteristic
113C2×Q8×C53—21no—
114C2×Q8×C53—22no—
115C2×Q8×C53—22no—
116C2×Q8×C53C22×C412yes—
117C2×D4×C53—21no—
118C2×D4×C53—22no—
119C2×D4×C53—22no—
120C2×D4×C53C22×C412yes—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2411yescharacteristic
126C2×C22⋊D53—24no—
127C2×C22⋊D53—42no—
128C2×C22⋊D53—42no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—24no—
132C22×Dic53—22no—
133C22×Dic53—42no—
134C22×Dic53—42no—
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C2×C4×D53—22no—
139C2×C4×D53—22no—
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—44no—
145C2×C4×D53—44no—
146C2×C4×D53—44no—
147C2×C4×D53C22⋊C414yes—
148C2×C4×D53C22⋊C414yes—
149C2×C4×D53C22×C412yes—
150C2×C4×D53C2×D412yes—
151C2×C8×C53—22no—
152C2×C8×C53—22no—
153C2×C8×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—42no—
157C4⋊C4×C53—42no—
158C22⋊C4×C53—24no—
159C22⋊C4×C53—24no—
160C42×C53C22×C412yes—
161C42×C53C2×D412yes—
162C22⋊Dic53—24no—
163C22⋊Dic53—24no—
164#11719921—24no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—24no—
168C4⋊Dic53—22no—
169C4⋊Dic53—22no—
170C4⋊Dic53—42no—
171C4⋊Dic53—42no—
172C4⋊Dic53C22×C412yes—
173C4⋊Dic53C22×C412yes—
174#11719919—24no—
175#11719919—24no—
176#11719919—24no—
177#11719919—44no—
178#11719919—44no—
179#11719919—44no—
180C4×Dic53—24no—
181C4×Dic53C22⋊C414yes—
182C4×Dic53C22⋊C414yes—
183C4×Dic53C22×C412yes—
184C4×Dic53C2×D412yes—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C23×C53—22no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53C22×D411yescharacteristic
196C22×D53—21no—
197C22×D53—21no—
198C22×D53—21no—
199C22×D53—21no—
200C22×D53C2×C22⋊C412yes—
201C22×D53AGL(1,ℤ/8)12yes—
202C22×D53#13712yes—
203Q8×C53—24no—
204Q8×C53—42no—
205D4×C53—24no—
206D4×C53—42no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—22no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53C2×C22⋊C412yes—
218C2×C4×C53C2×C22⋊C412yes—
219C2×C4×C53C23×C411yescharacteristic
220C2×C4×C53C22×D411yescharacteristic
221C2×C4×C53H(5,𝔽2)12yes—
222C2×C4×C532−1+412yes—
223C22⋊D53—48no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—48no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53C2×C22⋊C412yes—
239C2×Dic53C2×C22⋊C412yes—
240C2×Dic53C2×C22⋊C412yes—
241C2×Dic53AGL(1,ℤ/8)12yes—
242C2×Dic53#13712yes—
243C4×D53—28no—
244C4×D53—24no—
245C4×D53—24no—
246C4×D53—28no—
247C4×D53—88no—
248C8×C53—42no—
249#88322—42no—
250#21867—531nomaximal, Sylow
251C22×C53—42no—
252C22×C53—42no—
253C22×C53—42no—
254C22×C53—81no—
255C22×C53—81no—
256C22×C53—81no—
257C22×C53—81no—
258C22×C53C22×C22⋊C411yescharacteristic
259C22×C53#53511yescharacteristic
260C22×C53#53611yescharacteristic
261C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C53C2×#13711yescharacteristic
264C22×C53C2×#13711yescharacteristic
265C2×D53—22no—
266C2×D53—22no—
267C2×D53—22no—
268C2×D53—22no—
269C2×D53—22no—
270C2×D53—22no—
271C2×D53#41714yes—
272C4×C53—22no—
273C4×C53—22no—
274C4×C53—42no—
275C4×C53—82no—
276C4×C53#52012yes—
277C4×C53#53712yes—
278Dic53—22no—
279Dic53—22no—
280Dic53—42no—
281Dic53—42no—
282Dic53—82no—
283Dic53#41714yes—
284C2×#529—531no—
285C2×#432—531no—
286C2×#384—531no—
287#1992—531no—
288#1992—531no—
289#1959—531no—
290#1959D5311yescharacteristic, complemented
291#1874—532no—
292#1872—532no—
293#1803—532no—
294#1802—532no—
295C2×C53—82no—
296C2×C53#231111yescharacteristic
297C2×C53C2×#41711yescharacteristic
298C2×C53C2×#41711yescharacteristic
299C2×C53#321711yescharacteristic
300C2×C53#321811yescharacteristic
301C2×C53#323811yescharacteristic
302C2×C53#323911yescharacteristic
303D53—24no—
304#529—538no—
305#529—1064no—
306C2×#126—1061no—
307C2×#125—531no—
308C2×#124C2×D5311yescharacteristic
309C2×#123—1061no—
310C2×C22⋊Q8—531no—
311C2×#121—531no—
312C2×C4×Q8—531no—
313C2×C4×D4—531no—
314C2×C4×D4—1061no—
315C2×#117—1061no—
316C22×C4⋊C4—532no—
317C22×C4⋊C4—1061no—
318C22×C4⋊C4—1061no—
319C22×C42—531no—
320#432—1064no—
321C2×C4⋊C8—532no—
322C2×#103—532no—
323C2×#103C2×D5312yes—
324C2×#102—532no—
325C2×#102C2×D5312yes—
326C2×C22⋊C8—532no—
327C2×#97—531no—
328C2×#97C2×D5311yescharacteristic
329#384—532no—
330#384—531no—
331#384—1061no—
332#384—1061no—
333#384C2×D5311yescharacteristic
334#340—532no—
335#340—532no—
336#340—532no—
337#340—532no—
338C53#2186711yescharacteristic, complemented, Sylow
339C22×Q8C22×D5311yescharacteristic
340C22×D4—1061no—
341C22×D4C22×D5311yescharacteristic
342C23×C4—531no—
343C23×C4—531no—
344C23×C4—531no—
345C23×C4—1062no—
346C23×C4—1061no—
347C22×C8—532no—
348C22×C8C22×D5312yes—
349#126—1068no—
350#125—534no—
351#125—1062no—
352#124—22no—
353#124C4×D5314yescomplemented
354#123—1068no—
355C22⋊Q8—534no—
356C22⋊Q8—1062no—
357#121—534no—
358#121—1062no—
359C4×Q8—534no—
360C4×Q8—1062no—
361C4×D4—534no—
362C4×D4—1062no—
363C4×D4—1068no—
364#117—1068no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—532no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4—1062no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—2122no—
380C2×C4⋊C4—2122no—
381C2×C4⋊C4—2124no—
382C2×C4⋊C4—2124no—
383C2×C4⋊C4C22×D5312yes—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4—1061no—
389C2×C42—531no—
390C2×C42—531no—
391C2×C42—531no—
392C2×C42—534no—
393C2×C42—534no—
394C2×C42—1061no—
395C2×C42C22×D5311yescharacteristic
396C4⋊C8—1064no—
397#103—42no—
398#103—2122no—
399#102—42no—
400#102—2122no—
401C22⋊C8—1064no—
402#97—22no—
403#97—1062no—
404C24—21no—
405C24—531no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8C2×C4×D5312yes—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—1064no—
414C2×D4—2122no—
415C2×D4—2122no—
416C2×D4C2×C4×D5312yes—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—531no—
420C22×C4—531no—
421C22×C4—531no—
422C22×C4—534no—
423C22×C4—534no—
424C22×C4—532no—
425C22×C4—532no—
426C22×C4—1062no—
427C22×C4—1062no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—1064no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1062no—
436C22×C4—2124no—
437C22×C4—2124no—
438C22×C4—2124no—
439C22×C4—2122no—
440C22×C4—2122no—
441C22×C4D4×D5311yescharacteristic
442C22×C4D4×D5311yescharacteristic
443C22×C4C23×D5311yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—1062no—
448C2×C8—1062no—
449C2×C8—1062no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—532no—
455C4⋊C4—532no—
456C4⋊C4—1064no—
457C4⋊C4—1064no—
458C4⋊C4—1064no—
459C4⋊C4—1064no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1062no—
463C4⋊C4—1062no—
464C4⋊C4—2122no—
465C4⋊C4—2122no—
466C4⋊C4—2124no—
467C4⋊C4—2124no—
468C4⋊C4—2124no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—1064no—
472C22⋊C4—1064no—
473C22⋊C4—1064no—
474C42—534no—
475C42—534no—
476C42—532no—
477C42—532no—
478C42—1064no—
479C42C2×C4×D5312yes—
480C42D4×D5312yes—
481C23—22no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—532no—
489C23—532no—
490C23—532no—
491C23—1061no—
492C23—1061no—
493C23—1061no—
494C23—1061no—
495C23C2×D4×D5311yescharacteristic
496Q8—24no—
497Q8—42no—
498D4—24no—
499D4—42no—
500D4—2128no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—532no—
512C2×C4—532no—
513C2×C4—532no—
514C2×C4—532no—
515C2×C4—532no—
516C2×C4—1068no—
517C2×C4—1068no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1064no—
523C2×C4—1064no—
524C2×C4—1062no—
525C2×C4—1062no—
526C2×C4—4248no—
527C2×C4—2128no—
528C2×C4—2122no—
529C2×C4—2122no—
530C2×C4—2122no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4C22⋊C4×D5312yes—
536C2×C4C22⋊C4×D5312yes—
537C2×C4C22×C4×D5311yescharacteristic
538C2×C4C2×D4×D5311yescharacteristic
539C2×C4#1313326212yes—
540C2×C4#1313326412yes—
541C8—42no—
542C8—2122no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—534no—
551C22—1062no—
552C22—1062no—
553C22—1062no—
554C22—1062no—
555C22—1062no—
556C22—1062no—
557C22C2×C22⋊C4×D5311yescharacteristic
558C22#1681118611yescharacteristic
559C22#1681119311yescharacteristic
560C22AGL(1,ℤ/8)×D5311yescharacteristic
561C22AGL(1,ℤ/8)×D5311yescharacteristic
562C22#137×D5311yescharacteristic
563C22#137×D5311yescharacteristic
564C4—22no—
565C4—22no—
566C4—42no—
567C4—82no—
568C4—534no—
569C4—1062no—
570C4—1062no—
571C4—4242no—
572C4—2122no—
573C4—2122no—
574C4#1681110212yes—
575C4#1681120212yes—
576C2—82no—
577C2—1064no—
578C2#2027730511yescharacteristic
579C2#417×D5311yescharacteristic
580C2#417×D5311yescharacteristic
581C2#2028572911yescharacteristic
582C2#2028573811yescharacteristic
583C2#2028593211yescharacteristic
584C2#2028594111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1959C2
socleC23×C53C22×D4
prime radicalC2×#13133188C22
cosocleC22×C4×C53C24
hypercenter#1959D53
hypocenterC53#21867
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
2. C2×#16810678C2semidirect
3. #384×D53C2semidirect
4. #20275684C2semidirect
5. #20275683C2semidirect
6. #20275532C2semidirect
8. #20275019C2semidirect
9. #20274718C2semidirect
10. #20274713C2semidirect
11. C53×#1959C2semidirect
13. #16811152C4semidirect
55. #13133209C2×C4semidirect
62. Q8×Dic53C2×C4semidirect
64. D4×Dic53C2×C4semidirect
101. #124×C53C2×C4semidirect
290. #1959D53semidirect
338. C53#21867semidirect
353. #124C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111522
C2×#168106782
#384×D532
#202756842
#202756832
#202755322
#202750292
#202750192
#202747182
#202747132
C53×#19592
#2186753