Finite Group Databasebeta

← #27394045

Subgroups — #27394045

Order 13568.

1130 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811177C211yesmaximal, characteristic
2C2×#16810645C211yesmaximal, characteristic
3#20277120C211yesmaximal, characteristic
4#20275665C212yesmaximal, complemented
5#20275636C212yesmaximal, complemented
6#20275147C212yesmaximal
7#20275145C212yesmaximal
8#20274721C211yesmaximal, characteristic, complemented
9#20274720C211yesmaximal, characteristic, complemented
10#20274717C211yesmaximal, characteristic, complemented
11C53×#1803C211yesmaximal, characteristic, complemented
12C22×#11719912C2212yes—
13#16811177—24no—
14#16811177C418yescomplemented
15C2×#13133216C2211yescharacteristic
16C2×#13133215C2211yescharacteristic
17C2×#13133207—21no—
18C2×#13133197C2211yescharacteristic
19C2×#13133195—21no—
20C2×#13133193C2211yescharacteristic
21C2×C4×Dic106—21no—
22C2×C4×Dic106C2211yescharacteristic
23C2×#13133175—21no—
24C2×#13133174—21no—
25C2×C42×D53C2211yescharacteristic
26#16810645—24no—
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133146C2212yes—
29C2×#13133142C2212yes—
30C2×#13133141C2212yes—
31C2×#13133140C2212yes—
32#16810493C2211yescharacteristic
33#16810488—21no—
34#16810480—21no—
35#16810480C2212yes—
36C4⋊C4×Dic53—21no—
37C4⋊C4×Dic53C2212yes—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C2×C4⋊C8×C53C2211yescharacteristic
43C2×#103×C53C2212yes—
44C53×#384C2211yescharacteristic
45C53×#340C2212yes—
46C22×Dic106—21no—
47C22×Dic106C2311yescharacteristic
48C22×C4×D53D411yescharacteristic
49C22×C4×D53D411yescharacteristic
50C22×C4×D53C2311yescharacteristic
51C2×#11719912—24no—
52C2×#11719912—42no—
53C2×#11719912—42no—
54C22×#88322C2312yes—
55#13133216—22no—
56#13133216C2×C414yes—
57#13133215—22no—
58#13133215C2×C414yes—
59#13133207—28no—
60#13133197—22no—
61#13133197C2×C414yes—
62#13133195—28no—
63#13133193—22no—
64#13133193C2×C414yes—
65C4×Dic106—22no—
66C4×Dic106—28no—
67C4×Dic106C2×C414yes—
68#13133175—44no—
69#13133174—28no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2312yes—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—22no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—22no—
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88C42×D53C2×C414yes—
89C42×D53D414yes—
90C4⋊#88322—22no—
91#13133146—24no—
92#13133142—42no—
93#13133141—42no—
94#13133140—42no—
95#13133124—22no—
96#13133124—22no—
97C22×Q8×C53C2311yescharacteristic
98C22×C8×C53C2312yes—
99C4×Q8×C53—22no—
100C4×Q8×C53C2×C414yes—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53C2312yes—
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C42×C53—21no—
107C2×C42×C53C2311yescharacteristic
108C4⋊C8×C53—22no—
109#103×C53—42no—
110C23×D53C2×D411yescharacteristic
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53C22×C412yes—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—22no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2411yescharacteristic
123C22×Dic53—21no—
124C22×Dic53—21no—
125C22×Dic53—22no—
126C22×Dic53C2×D411yescharacteristic
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C2×C4×D53—22no—
130C2×C4×D53—22no—
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—21no—
134C2×C4×D53—21no—
135C2×C4×D53—22no—
136C2×C4×D53—22no—
137C2×C4×D53C22⋊C414yes—
138C2×C4×D53C22⋊C414yes—
139C2×C4×D53C22×C412yes—
140C2×C4×D53C2×D412yes—
141C2×Dic106—21no—
142C2×Dic106—21no—
143C2×Dic106—21no—
144C2×Dic106—21no—
145C2×Dic106—21no—
146C2×Dic106—24no—
147C2×Dic106—42no—
148C2×Dic106—42no—
149C2×Dic106C22×C412yes—
150C2×C8×C53—22no—
151C2×C8×C53—22no—
152C2×C8×C53—22no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—42no—
158C4⋊C4×C53—42no—
159C4⋊C4×C53C22×C412yes—
160C42×C53—24no—
161C42×C53C22×C412yes—
162C42×C53C2×D412yes—
163#11719921—24no—
164#11719921—24no—
165#11719921—44no—
166C4⋊Dic53—21no—
167C4⋊Dic53—24no—
168C4⋊Dic53—22no—
169C4⋊Dic53—22no—
170C4⋊Dic53—42no—
171C4⋊Dic53—42no—
172C4⋊Dic53C22×C412yes—
173#11719919—24no—
174#11719919—24no—
175#11719919—24no—
176#11719919—24no—
177#11719919—24no—
178C4×Dic53—24no—
179C4×Dic53—44no—
180C4×Dic53C22⋊C414yes—
181C4×Dic53C22⋊C414yes—
182C4×Dic53C22×C412yes—
183C4×Dic53C2×D412yes—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187#11719912—48no—
188C23×C53C22×D411yescharacteristic
189C22×D53—21no—
190C22×D53—21no—
191C22×D53—21no—
192C22×D53—21no—
193C22×D53—21no—
194C22×D53—21no—
195C22×D53C2×C22⋊C412yes—
196Q8×C53—22no—
197Q8×C53—22no—
198Q8×C53—41no—
199Q8×C53—41no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53C2×C22⋊C412yes—
212C2×C4×C53C2×C22⋊C412yes—
213C2×C4×C53#13712yes—
214C2×C4×C53#13712yes—
215C2×C4×C53C23×C411yescharacteristic
216C2×C4×C53C22×D411yescharacteristic
217C2×C4×C53H(5,𝔽2)11yescharacteristic
218C2×C4×C53H(5,𝔽2)11yescharacteristic
219C2×C4×C532−1+411yescharacteristic
220C2×C4×C532−1+411yescharacteristic
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53C2×C22⋊C412yes—
235C2×Dic53C2×C22⋊C412yes—
236C2×Dic53C2×C22⋊C412yes—
237C2×Dic53AGL(1,ℤ/8)12yes—
238C2×Dic53#13712yes—
239C4×D53—22no—
240C4×D53—22no—
241C4×D53—22no—
242C4×D53—22no—
243C4×D53—28no—
244C4×D53—28no—
245Dic106—22no—
246Dic106—22no—
247Dic106—41no—
248Dic106—41no—
249Dic106—48no—
250C8×C53—42no—
251#88322—42no—
252#21894—531nomaximal, Sylow
253C22×C53C22×C22⋊C411yescharacteristic
254C22×C53#53711yescharacteristic
255C22×C53#53711yescharacteristic
256C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C53C2×#13711yescharacteristic
258C22×C53C2×#13711yescharacteristic
259C22×C53C2×#13711yescharacteristic
260C2×D53—22no—
261C2×D53—22no—
262C2×D53—22no—
263C2×D53—41no—
264C2×D53—41no—
265C2×D53—41no—
266C2×D53—41no—
267C4×C53—22no—
268C4×C53—42no—
269C4×C53—42no—
270C4×C53—82no—
271C4×C53#41914yes—
272C4×C53#52011yescharacteristic
273C4×C53#52011yescharacteristic
274C4×C53#53511yescharacteristic
275C4×C53#53611yescharacteristic
276Dic53—22no—
277Dic53—22no—
278Dic53—42no—
279Dic53—42no—
280Dic53—82no—
281Dic53#41714yes—
282C2×#533—531no—
283C2×#423—531no—
284#2285—531no—
285#1988—532no—
286#1981—532no—
287#1894—532no—
288#1894—532no—
289#1803—531no—
290#1803—531no—
291#1803—531no—
292#1803D5311yescharacteristic, complemented
293C2×C53#231111yescharacteristic
294C2×C53C2×#41911yescharacteristic
295C2×C53C2×#41711yescharacteristic
296C2×C53#325411yescharacteristic
297C2×C53#325411yescharacteristic
298C2×C53#325511yescharacteristic
299C2×C53#325511yescharacteristic
300D53—42no—
301C22×M16—532no—
302#533—538no—
303#533—1064no—
304C2×C4⋊Q8—531no—
305C2×C4⋊Q8—531no—
306C2×#125—1061no—
307C2×C22⋊Q8—531no—
308C2×C22⋊Q8—531no—
309C2×C22⋊Q8—1061no—
310C2×C4×Q8—531no—
311C2×C4×Q8—1061no—
312C2×C4×Q8C2×D5311yescharacteristic
313C2×#117—1061no—
314C2×#117—1061no—
315C22×C42—531no—
316#423—1064no—
317C2×C4⋊C8—532no—
318C2×C4⋊C8—531no—
319C2×C4⋊C8C2×D5311yescharacteristic
320C2×#103—532no—
321C2×#103—532no—
322C2×#103—532no—
323C2×#103C2×D5312yes—
324#384—531no—
325#384C2×D5311yescharacteristic
326#382—1061no—
327#380—532no—
328#380—1061no—
329C4×C4⋊C4—532no—
330C4×C4⋊C4—1061no—
331#340—532no—
332#340—532no—
333#340—532no—
334#340C2×D5312yes—
335C53#2189411yescharacteristic, complemented, Sylow
336C22×Q8—531no—
337C22×Q8—1061no—
338C22×Q8C22×D5311yescharacteristic
339C23×C4—531no—
340C23×C4—531no—
341C23×C4—531no—
342C2×M16—1064no—
343C2×M16—2122no—
344C2×M16—2122no—
345C22×C8—532no—
346C22×C8C22×D5312yes—
347C4⋊Q8—534no—
348C4⋊Q8—534no—
349C4⋊Q8—1062no—
350C4⋊Q8—1062no—
351#125—1068no—
352C22⋊Q8—534no—
353C22⋊Q8—534no—
354C22⋊Q8—1062no—
355C22⋊Q8—1068no—
356C22⋊Q8—1062no—
357C4×Q8—22no—
358C4×Q8—534no—
359C4×Q8—1068no—
360C4×Q8—1062no—
361C4×Q8C4×D5314yes—
362#117—1068no—
363#117—2124no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—532no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4C22×D5312yes—
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C22⋊C4—1061no—
380C2×C22⋊C4—1061no—
381C2×C22⋊C4—1062no—
382C2×C42—21no—
383C2×C42—531no—
384C2×C42—531no—
385C2×C42—531no—
386C2×C42—534no—
387C2×C42—534no—
388C2×C42—1061no—
389C2×C42—1062no—
390C2×C42C22×D5311yescharacteristic
391C4⋊C8—22no—
392C4⋊C8—1064no—
393C4⋊C8—1062no—
394#103—42no—
395#103—2122no—
396#103—2122no—
397#103—2122no—
398#95—1062no—
399#95—1062no—
400C24—531no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—532no—
407C2×Q8—1064no—
408C2×Q8—1061no—
409C2×Q8—1061no—
410C2×Q8—1061no—
411C2×Q8—1061no—
412C2×Q8—1061no—
413C2×Q8—2122no—
414C2×Q8—2122no—
415C2×Q8C2×C4×D5312yes—
416C22×C4—21no—
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—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1062no—
437C22×C4D4×D5311yescharacteristic
438C22×C4D4×D5311yescharacteristic
439C22×C4C23×D5311yescharacteristic
440M16—2128no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—1062no—
445C2×C8—1062no—
446C2×C8—1062no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—532no—
454C4⋊C4—1064no—
455C4⋊C4—1064no—
456C4⋊C4—1064no—
457C4⋊C4—1064no—
458C4⋊C4—1064no—
459C4⋊C4—1064no—
460C4⋊C4—1061no—
461C4⋊C4—1062no—
462C4⋊C4—1062no—
463C4⋊C4—2122no—
464C4⋊C4—2122no—
465C4⋊C4C2×C4×D5312yes—
466C22⋊C4—1064no—
467C22⋊C4—1064no—
468C22⋊C4—2124no—
469C42—24no—
470C42—534no—
471C42—534no—
472C42—532no—
473C42—532no—
474C42—1064no—
475C42—2124no—
476C42C2×C4×D5312yes—
477C42D4×D5312yes—
478C23—532no—
479C23—1061no—
480C23—1061no—
481C23—1061no—
482C23—1061no—
483C23—1061no—
484C23—1061no—
485C23C2×D4×D5311yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—41no—
489Q8—41no—
490Q8—1062no—
491Q8—1062no—
492Q8—2128no—
493Q8—2121no—
494Q8—2121no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—532no—
507C2×C4—532no—
508C2×C4—532no—
509C2×C4—532no—
510C2×C4—532no—
511C2×C4—1068no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1062no—
517C2×C4—1062no—
518C2×C4—1068no—
519C2×C4—1062no—
520C2×C4—1062no—
521C2×C4—1062no—
522C2×C4—1062no—
523C2×C4—2122no—
524C2×C4—2122no—
525C2×C4—2122no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4C22⋊C4×D5312yes—
531C2×C4C22⋊C4×D5312yes—
532C2×C4#1313324612yes—
533C2×C4#1313324912yes—
534C2×C4C22×C4×D5311yescharacteristic
535C2×C4C2×D4×D5311yescharacteristic
536C2×C4#1313326211yescharacteristic
537C2×C4#1313326311yescharacteristic
538C2×C4#1313326411yescharacteristic
539C2×C4#1313326511yescharacteristic
540C8—42no—
541C8—2122no—
542C22—1062no—
543C22—1062no—
544C22—1062no—
545C22—2121no—
546C22—2121no—
547C22—2121no—
548C22—2121no—
549C22C2×C22⋊C4×D5311yescharacteristic
550C22#1681120211yescharacteristic
551C22#1681120311yescharacteristic
552C22AGL(1,ℤ/8)×D5311yescharacteristic
553C22#137×D5311yescharacteristic
554C22C2×#1313324611yescharacteristic
555C22C2×#1313324911yescharacteristic
556C4—22no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—534no—
561C4—1062no—
562C4—1062no—
563C4—4242no—
564C4—2122no—
565C4—2122no—
566C4#1681063014yes—
567C4#1681110311yescharacteristic
568C4#1681110411yescharacteristic
569C4#1681118911yescharacteristic
570C4#1681119611yescharacteristic
571C2—2122no—
572C2#2027730411yescharacteristic
573C2C2×#1681063011yescharacteristic
574C2#417×D5311yescharacteristic
575C2#2028610011yescharacteristic
576C2#2028610111yescharacteristic
577C2#2028610711yescharacteristic
578C2#2028610811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1803C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1803D53
hypocenterC53#21894
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #20275665C2semidirect
5. #20275636C2semidirect
8. #20274721C2semidirect
9. #20274720C2semidirect
10. #20274717C2semidirect
11. C53×#1803C2semidirect
14. #16811177C4semidirect
292. #1803D53semidirect
335. C53#21894semidirect

Maximal subgroups

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

subgroupindex
C2×#168111772
C2×#168106452
#202771202
#202756652
#202756362
#202751472
#202751452
#202747212
#202747202
#202747172
C53×#18032
#2189453