Finite Group Databasebeta

← #27393851

Subgroups — #27393851

Order 13568.

1086 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811177C211yesmaximal, characteristic
2C2×#16810679C211yesmaximal, characteristic
3#20277120C211yesmaximal, characteristic
4#20275686C212yesmaximal, complemented
5#20275533C212yesmaximal, complemented
6#20275026C212yesmaximal
7#20275024C212yesmaximal
8#20274721C211yesmaximal, characteristic, complemented
9#20274720C211yesmaximal, characteristic, complemented
10#20274717C211yesmaximal, characteristic, complemented
11C53×#1803C211yesmaximal, characteristic, complemented
12#16811177—24no—
13#16811177C418yescomplemented
14C2×#13133216C2211yescharacteristic
15C2×#13133215C2211yescharacteristic
16C2×#13133207—21no—
17C2×#13133197C2211yescharacteristic
18C2×#13133195—21no—
19C2×#13133193C2211yescharacteristic
20C2×C4×Dic106—21no—
21C2×C4×Dic106C2211yescharacteristic
22C2×#13133175C2212yes—
23C2×#13133174—21no—
24C2×C42×D53C2211yescharacteristic
25#16810679—24no—
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133142C2212yes—
28C2×#13133141C2212yes—
29C2×#13133140C2212yes—
30C2×#13133129C2212yes—
31C2×#13133127C2212yes—
32#16810493C2211yescharacteristic
33#16810488C2212yes—
34#16810480—21no—
35#16810480—21no—
36C4⋊C4×Dic53—21no—
37C4⋊C4×Dic53—21no—
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
51C22×#88322C2312yes—
52#13133216—22no—
53#13133216C2×C414yes—
54#13133215—22no—
55#13133215C2×C414yes—
56#13133207—28no—
57#13133197—22no—
58#13133197C2×C414yes—
59#13133195—28no—
60#13133193—22no—
61#13133193C2×C414yes—
62C4×Dic106—22no—
63C4×Dic106—28no—
64C4×Dic106C2×C414yes—
65#13133175—42no—
66#13133175—42no—
67#13133174—28no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2312yes—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—22no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86C42×D53C2×C414yes—
87C42×D53D414yes—
88C4⋊#88322—22no—
89#13133142—42no—
90#13133141—42no—
91#13133140—42no—
92#13133129—24no—
93#13133127—24no—
94#13133124—22no—
95#13133124—22no—
96C22×Q8×C53C2311yescharacteristic
97C22×C8×C53C2312yes—
98C4×Q8×C53—22no—
99C4×Q8×C53C2×C414yes—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53C2312yes—
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C42×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C4⋊C8×C53—22no—
108#103×C53—42no—
109C23×D53C2×D411yescharacteristic
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53C22×C412yes—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2411yescharacteristic
122C22×Dic53—21no—
123C22×Dic53—21no—
124C22×Dic53—22no—
125C22×Dic53C2×D411yescharacteristic
126C22×Dic53C2×D411yescharacteristic
127C22×Dic53C2×D411yescharacteristic
128C2×C4×D53—22no—
129C2×C4×D53—22no—
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—21no—
134C2×C4×D53—41no—
135C2×C4×D53—41no—
136C2×C4×D53C22⋊C414yes—
137C2×C4×D53C22⋊C414yes—
138C2×C4×D53C22×C412yes—
139C2×C4×D53C2×D412yes—
140C2×Dic106—21no—
141C2×Dic106—21no—
142C2×Dic106—21no—
143C2×Dic106—21no—
144C2×Dic106—21no—
145C2×Dic106—24no—
146C2×Dic106—42no—
147C2×Dic106—42no—
148C2×Dic106C22×C412yes—
149C2×C8×C53—22no—
150C2×C8×C53—22no—
151C2×C8×C53—22no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—42no—
157C4⋊C4×C53—42no—
158C4⋊C4×C53C22×C412yes—
159C42×C53—24no—
160C42×C53C22×C412yes—
161C42×C53C2×D412yes—
162#11719921—24no—
163#11719921—24no—
164#11719921—44no—
165C4⋊Dic53—21no—
166C4⋊Dic53—24no—
167C4⋊Dic53—22no—
168C4⋊Dic53—22no—
169C4⋊Dic53—42no—
170C4⋊Dic53—42no—
171C4⋊Dic53C22×C412yes—
172#11719919—24no—
173#11719919—24no—
174#11719919—24no—
175#11719919—24no—
176#11719919—24no—
177C4×Dic53—24no—
178C4×Dic53—44no—
179C4×Dic53C22⋊C414yes—
180C4×Dic53C22⋊C414yes—
181C4×Dic53C22×C412yes—
182C4×Dic53C2×D412yes—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C23×C53C22×D411yescharacteristic
187C22×D53—21no—
188C22×D53—21no—
189C22×D53—21no—
190C22×D53—21no—
191C22×D53—21no—
192C22×D53—21no—
193C22×D53C2×C22⋊C412yes—
194Q8×C53—22no—
195Q8×C53—22no—
196Q8×C53—41no—
197Q8×C53—41no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53C2×C22⋊C412yes—
210C2×C4×C53C2×C22⋊C412yes—
211C2×C4×C53#13712yes—
212C2×C4×C53#13712yes—
213C2×C4×C53C23×C411yescharacteristic
214C2×C4×C53C22×D411yescharacteristic
215C2×C4×C53H(5,𝔽2)11yescharacteristic
216C2×C4×C53H(5,𝔽2)11yescharacteristic
217C2×C4×C532−1+411yescharacteristic
218C2×C4×C532−1+411yescharacteristic
219C2×Dic53—21no—
220C2×Dic53—21no—
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—
237C4×D53—22no—
238C4×D53—22no—
239C4×D53—22no—
240C4×D53—22no—
241C4×D53—28no—
242C4×D53—44no—
243Dic106—22no—
244Dic106—22no—
245Dic106—48no—
246Dic106—41no—
247Dic106—41no—
248C8×C53—42no—
249#88322—42no—
250#21870—531nomaximal, Sylow
251C22×C53C22×C22⋊C411yescharacteristic
252C22×C53#53511yescharacteristic
253C22×C53#53611yescharacteristic
254C22×C53C2×#13711yescharacteristic
255C22×C53C2×#13711yescharacteristic
256C22×C53#57511yescharacteristic
257C22×C53#57511yescharacteristic
258C2×D53—22no—
259C2×D53—22no—
260C2×D53—22no—
261C2×D53—41no—
262C2×D53—41no—
263C2×D53—41no—
264C2×D53—41no—
265C4×C53—22no—
266C4×C53—42no—
267C4×C53—42no—
268C4×C53—82no—
269C4×C53#41914yes—
270C4×C53#52011yescharacteristic
271C4×C53#52011yescharacteristic
272C4×C53#53711yescharacteristic
273C4×C53#53711yescharacteristic
274Dic53—22no—
275Dic53—22no—
276Dic53—22no—
277Dic53—42no—
278Dic53—42no—
279Dic53—82no—
280C2×#533—531no—
281C2×#432—531no—
282#2285—531no—
283#1993—532no—
284#1960—532no—
285#1873—532no—
286#1873—532no—
287#1803—531no—
288#1803—531no—
289#1803—531no—
290#1803D5311yescharacteristic, complemented
291C2×C53#231111yescharacteristic
292C2×C53C2×#41911yescharacteristic
293C2×C53#289611yescharacteristic
294C2×C53#321611yescharacteristic
295C2×C53#321611yescharacteristic
296C2×C53#323711yescharacteristic
297C2×C53#323711yescharacteristic
298D53—42no—
299#533—538no—
300#533—1064no—
301C2×C4⋊Q8—531no—
302C2×C4⋊Q8—531no—
303C2×#125—1061no—
304C2×C22⋊Q8—531no—
305C2×C22⋊Q8—531no—
306C2×C22⋊Q8—1061no—
307C2×C4×Q8—531no—
308C2×C4×Q8—1061no—
309C2×C4×Q8C2×D5311yescharacteristic
310C2×#117—532no—
311C2×#117—1061no—
312C22×C42—531no—
313#432—1064no—
314C2×C4⋊C8—531no—
315C2×C4⋊C8C2×D5311yescharacteristic
316C2×#103—532no—
317C2×#103—532no—
318C2×#103—532no—
319C2×#103C2×D5312yes—
320C2×C22⋊C8—532no—
321C2×#97—532no—
322#384—531no—
323#384C2×D5311yescharacteristic
324#382—532no—
325#380—1061no—
326#380—1061no—
327C4×C4⋊C4—1061no—
328C4×C4⋊C4—1061no—
329#340—532no—
330#340—532no—
331#340—532no—
332#340C2×D5312yes—
333C53#2187011yescharacteristic, complemented, Sylow
334C22×Q8—531no—
335C22×Q8—1061no—
336C22×Q8C22×D5311yescharacteristic
337C23×C4—531no—
338C23×C4—531no—
339C23×C4—531no—
340C22×C8—532no—
341C22×C8C22×D5312yes—
342C4⋊Q8—534no—
343C4⋊Q8—534no—
344C4⋊Q8—1062no—
345C4⋊Q8—1062no—
346#125—1068no—
347C22⋊Q8—534no—
348C22⋊Q8—534no—
349C22⋊Q8—1062no—
350C22⋊Q8—1068no—
351C22⋊Q8—1062no—
352C4×Q8—22no—
353C4×Q8—534no—
354C4×Q8—1068no—
355C4×Q8—1062no—
356C4×Q8C4×D5314yes—
357#117—1068no—
358#117—2122no—
359#117—2122no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—531no—
364C2×C4⋊C4—532no—
365C2×C4⋊C4—1061no—
366C2×C4⋊C4—1061no—
367C2×C4⋊C4—1061no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4C22×D5312yes—
374C2×C4⋊C4C22×D5311yescharacteristic
375C2×C22⋊C4—1061no—
376C2×C22⋊C4—1061no—
377C2×C22⋊C4—1062no—
378C2×C42—21no—
379C2×C42—531no—
380C2×C42—531no—
381C2×C42—531no—
382C2×C42—534no—
383C2×C42—534no—
384C2×C42—1061no—
385C2×C42—1062no—
386C2×C42C22×D5311yescharacteristic
387C4⋊C8—22no—
388C4⋊C8—1062no—
389#103—42no—
390#103—2122no—
391#103—2122no—
392#103—2122no—
393C22⋊C8—1064no—
394#97—1064no—
395#95—1062no—
396#95—1062no—
397C24—531no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—532no—
404C2×Q8—1064no—
405C2×Q8—1061no—
406C2×Q8—1061no—
407C2×Q8—1061no—
408C2×Q8—1061no—
409C2×Q8—1061no—
410C2×Q8—2122no—
411C2×Q8—2122no—
412C2×Q8C2×C4×D5312yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—531no—
417C22×C4—531no—
418C22×C4—531no—
419C22×C4—534no—
420C22×C4—532no—
421C22×C4—532no—
422C22×C4—534no—
423C22×C4—1062no—
424C22×C4—1062no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1062no—
432C22×C4—2121no—
433C22×C4—2121no—
434C22×C4D4×D5311yescharacteristic
435C22×C4D4×D5311yescharacteristic
436C22×C4C23×D5311yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—1062no—
441C2×C8—1062no—
442C2×C8—1062no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—532no—
450C4⋊C4—1064no—
451C4⋊C4—1064no—
452C4⋊C4—1064no—
453C4⋊C4—1064no—
454C4⋊C4—1064no—
455C4⋊C4—1064no—
456C4⋊C4—1061no—
457C4⋊C4—1062no—
458C4⋊C4—1062no—
459C4⋊C4—2122no—
460C4⋊C4—2122no—
461C4⋊C4C2×C4×D5312yes—
462C22⋊C4—1064no—
463C22⋊C4—1064no—
464C22⋊C4—2124no—
465C42—24no—
466C42—534no—
467C42—534no—
468C42—532no—
469C42—532no—
470C42—1064no—
471C42—2124no—
472C42C2×C4×D5312yes—
473C42D4×D5312yes—
474C23—532no—
475C23—1061no—
476C23—1061no—
477C23—1061no—
478C23—1061no—
479C23—1061no—
480C23—1061no—
481C23C2×D4×D5311yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—41no—
485Q8—41no—
486Q8—1062no—
487Q8—1062no—
488Q8—2128no—
489Q8—2121no—
490Q8—2121no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—532no—
503C2×C4—532no—
504C2×C4—532no—
505C2×C4—1061no—
506C2×C4—1061no—
507C2×C4—1061no—
508C2×C4—1061no—
509C2×C4—1061no—
510C2×C4—1061no—
511C2×C4—1062no—
512C2×C4—1062no—
513C2×C4—1068no—
514C2×C4—1062no—
515C2×C4—1062no—
516C2×C4—1062no—
517C2×C4—1062no—
518C2×C4—2124no—
519C2×C4—2122no—
520C2×C4—2122no—
521C2×C4—2122no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4C22⋊C4×D5312yes—
527C2×C4C22⋊C4×D5312yes—
528C2×C4#1313324612yes—
529C2×C4#1313324912yes—
530C2×C4C22×C4×D5311yescharacteristic
531C2×C4C2×D4×D5311yescharacteristic
532C2×C4#1313326211yescharacteristic
533C2×C4#1313326311yescharacteristic
534C2×C4#1313326411yescharacteristic
535C2×C4#1313326511yescharacteristic
536C8—42no—
537C8—2122no—
538C22—1062no—
539C22—1062no—
540C22—1062no—
541C22—2121no—
542C22—2121no—
543C22—2121no—
544C22—2121no—
545C22C2×C22⋊C4×D5311yescharacteristic
546C22#1681118911yescharacteristic
547C22#1681119611yescharacteristic
548C22C2×#1313324611yescharacteristic
549C22C2×#1313324911yescharacteristic
550C22#1681141911yescharacteristic
551C22#1681142211yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—1062no—
557C4—1062no—
558C4—1062no—
559C4—4242no—
560C4—2122no—
561C4—2122no—
562C4#1681063014yes—
563C4#1681110311yescharacteristic
564C4#1681110411yescharacteristic
565C4#1681120211yescharacteristic
566C4#1681120311yescharacteristic
567C2—2122no—
568C2#2027730411yescharacteristic
569C2C2×#1681063011yescharacteristic
570C2#2028323111yescharacteristic
571C2#2028572411yescharacteristic
572C2#2028572511yescharacteristic
573C2#2028592711yescharacteristic
574C2#2028592811yescharacteristic
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#21870
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. #20275686C2semidirect
5. #20275533C2semidirect
8. #20274721C2semidirect
9. #20274720C2semidirect
10. #20274717C2semidirect
11. C53×#1803C2semidirect
13. #16811177C4semidirect
290. #1803D53semidirect
333. C53#21870semidirect

Maximal subgroups

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

subgroupindex
C2×#168111772
C2×#168106792
#202771202
#202756862
#202755332
#202750262
#202750242
#202747212
#202747202
#202747172
C53×#18032
#2187053