Finite Group Databasebeta

← #27385619

Subgroups — #27385619

Order 13568.

1702 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133248C211yesmaximal, characteristic
2#20287916C211yesmaximal, characteristic
3C2×#16810641C211yesmaximal, characteristic
4C2×#16810632C211yesmaximal, characteristic, complemented
5#20275087C218yesmaximal, complemented
6#20274454C211yesmaximal, characteristic, complemented
7#20274425C211yesmaximal, characteristic, complemented
8C53×#2310C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C2×#13133248—24no—
11C2×#13133248—22no—
12C2×#13133248C418yescomplemented
13C2×#13133248C2214yes—
14C22×#11719925—21no—
15C22×#11719924—21no—
16C22×#11719917C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18#16811104C2214yes—
19#16811103—22no—
20#16811100—21no—
21#16811097—21no—
22#16811096—22no—
23C4×#11719945—22no—
24#16811091—22no—
25C2×#13133176C2211yescharacteristic
26#117×D53—22no—
27#16810641—24no—
28C2×#13133145C2214yes—
29#16810632—22no—
30#16810632C2214yescomplemented
31C2×#13133141—22no—
32#16810392C2214yescomplemented
33#16810383C2214yescomplemented
34C23×Q8×C53C2211yescharacteristic
35C2×#117×C53C2211yescharacteristic
36C2×#117×C53C2211yescharacteristic
37C53×#391—22no—
38C53×#391C2214yescomplemented
39C53×#381—22no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—22no—
43C2×#11719945—22no—
44C2×#11719945C2×C414yes—
45C2×#11719945D414yes—
46C2×#11719945D412yes—
47C2×#11719945C2312yes—
48C22×Dic106D411yescharacteristic
49C22×D212D411yescharacteristic
50C22×C22⋊D53—21no—
51C22×C4×D53—21no—
52#13133248—216no—
53#13133248—28no—
54#13133248C2×C4116yescomplemented
55C2×#11719925—28no—
56C2×#11719925—24no—
57C2×#11719924—28no—
58C2×#11719924—24no—
59C2×#11719917—21no—
60C2×#11719917—22no—
61C2×#11719917—21no—
62C2×#11719917—22no—
63C2×#11719917—22no—
64C2×#11719917C2×C414yescomplemented
65C2×#11719917C2312yes—
66C2×#11719917C2312yes—
67C22×#88322—21no—
68C22×#88322—21no—
69C4×Dic106—24no—
70#13133182—24no—
71C4×D212—24no—
72#13133179—24no—
73C4×C22⋊D53—28no—
74#13133177—28no—
75#13133176—21no—
76#13133176—21no—
77#13133176D412yes—
78#13133176C2312yes—
79#13133175—24no—
80#13133174—24no—
81#13133173—24no—
82C2×C4⋊Dic53D412yes—
83C4⋊C4×D53—24no—
84C2×C22⋊Dic53—21no—
85C22⋊C4×D53—24no—
86C2×C4×Dic53—21no—
87C42×D53—24no—
88#13133145—28no—
89#13133141—28no—
90C22×Q8×C53—21no—
91C22×Q8×C53—22no—
92C22×Q8×C53—21no—
93C22×Q8×C53—21no—
94C22×Q8×C53—21no—
95C22×Q8×C53—21no—
96C22×Q8×C53—22no—
97C22×Q8×C53—22no—
98C22×Q8×C53C2×C414yescomplemented
99C22×Q8×C53C2312yes—
100C23×C4×C53—21no—
101C23×C4×C53C2311yescharacteristic
102#117×C53—21no—
103#117×C53—21no—
104#117×C53—21no—
105#117×C53—21no—
106#117×C53—22no—
107#117×C53—22no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53C2312yes—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53D412yes—
112C2×C42×C53D412yes—
113C2×C42×C53C2312yes—
114#95×C53—28no—
115C24×C53C2×D411yescharacteristic
116C23×D53—21no—
117C2×Q8×C53—24no—
118C2×Q8×C53—22no—
119C2×Q8×C53—24no—
120C2×Q8×C53—24no—
121C2×Q8×C53—24no—
122C2×Q8×C53—24no—
123C2×Q8×C53—22no—
124C2×Q8×C53—22no—
125C2×Q8×C53—22no—
126C2×Q8×C53—22no—
127C2×Q8×C53—41no—
128C2×Q8×C53—41no—
129C2×Q8×C53—42no—
130C2×Q8×C53—42no—
131C2×Q8×C53—42no—
132C2×Q8×C53—42no—
133C2×Q8×C53—42no—
134C2×Q8×C53C22×C414yes—
135C22×C4×C53—24no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—41no—
139C22×C4×C53—41no—
140C22×C4×C53—42no—
141C22×C4×C53C22×C412yes—
142C22×C4×C53C2×D414yes—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53#4112yes—
150C22×C4×C53#4112yes—
151C22×C4×C53C2411yescharacteristic
152C2×C22⋊D53—24no—
153C2×C22⋊D53—22no—
154C2×C22⋊D53—24no—
155C2×C22⋊D53—22no—
156C22×Dic53—21no—
157C22×Dic53—22no—
158#11719945—24no—
159#11719945—24no—
160#11719945—24no—
161#11719945—24no—
162#11719945—24no—
163#11719945—22no—
164#11719945—22no—
165#11719945C22⋊C418yes—
166#11719945C22×C414yes—
167#11719945C2×D414yes—
168C2×D212—21no—
169C2×D212—22no—
170C2×D212C22⋊C414yes—
171C2×D212C2×D412yes—
172C2×C4×D53—24no—
173C2×C4×D53—22no—
174C2×C4×D53—24no—
175C2×C4×D53—22no—
176C2×C4×D53—28no—
177C2×Dic106—21no—
178C2×Dic106—22no—
179C2×Dic106C22⋊C414yes—
180C2×Dic106C2×D412yes—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53—42no—
184C22⋊C4×C53—22no—
185C22⋊C4×C53—22no—
186C22⋊C4×C53—42no—
187C42×C53—22no—
188C42×C53—22no—
189C42×C53—22no—
190C42×C53—22no—
191C22⋊Dic53—22no—
192C22⋊Dic53—22no—
193#11719925—216no—
194#11719924—216no—
195#11719921—28no—
196C4⋊Dic53—22no—
197C4⋊Dic53C2×D414yes—
198#11719919—28no—
199C4×Dic53—28no—
200C4×Dic53—22no—
201C4×Dic53—22no—
202#11719917—24no—
203#11719917—22no—
204#11719917—22no—
205#11719917—22no—
206#11719917—42no—
207#11719917C22×C414yes—
208C2×#88322—24no—
209C2×#88322—22no—
210C2×#88322—22no—
211C2×#88322—42no—
212C23×C53—42no—
213C23×C53C4×D412yes—
214C23×C53#12011yescharacteristic
215C23×C53#12011yescharacteristic
216C23×C53#12112yes—
217C23×C53C22×D411yescharacteristic
218C22×D53—22no—
219C22×D53—24no—
220C22×D53—82no—
221Q8×C53—24no—
222Q8×C53—24no—
223Q8×C53—42no—
224Q8×C53—42no—
225Q8×C53—44no—
226Q8×C53—44no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—44no—
232C2×C4×C53—44no—
233C2×C4×C53—44no—
234C2×C4×C53—44no—
235C2×C4×C53—42no—
236C2×C4×C53C2×C22⋊C412yes—
237C2×C4×C53C2×C22⋊C412yes—
238C2×C4×C53C4×D412yes—
239C2×C4×C53C4×D412yes—
240C2×C4×C53C4×D412yes—
241C2×C4×C53#12011yescharacteristic
242C2×C4×C53#12011yescharacteristic
243C2×C4×C53#12112yes—
244C2×C4×C53#12112yes—
245C2×C4×C53#12112yes—
246C2×C4×C53#12312yes—
247C2×C4×C53#12312yes—
248C2×C4×C53C23×C411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C22×D411yescharacteristic
251C2×C4×C53C22×D411yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C22⋊D53—24no—
255C22⋊D53—24no—
256C22⋊D53—28no—
257C2×Dic53—22no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—82no—
261D212—22no—
262D212—24no—
263D212C2×C22⋊C414yes—
264C4×D53—24no—
265C4×D53—24no—
266C4×D53—28no—
267C4×D53—88no—
268Dic106—22no—
269Dic106—24no—
270Dic106C2×C22⋊C414yes—
271#88322—24no—
272#88322—42no—
273#20944—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—42no—
278C22×C53—42no—
279C22×C53#38612yes—
280C22×C53#38612yes—
281C22×C53C2×C4×D411yescharacteristic
282C22×C53C2×#12011yescharacteristic
283C22×C53C2×#12111yescharacteristic
284C2×D53—24no—
285C2×D53—44no—
286C2×D53—84no—
287C4×C53—44no—
288C4×C53—44no—
289C4×C53—82no—
290C4×C53#38612yes—
291C4×C53#38612yes—
292C4×C53C22×C22⋊C411yescharacteristic
293C4×C53C2×C4×D411yescharacteristic
294C4×C53C2×#12111yescharacteristic
295C4×C53C2×#12311yescharacteristic
296Dic53—24no—
297Dic53—44no—
298C22×#137—531no—
299#3428—531no—
300C2×#421—531no—
301C2×#419—531no—
302#2310D5311yescharacteristic, complemented
303#1882—538no—
304#1742—531no—
305#1733—531no—
306C2×C53—21no—
307C2×C53—22no—
308C2×C53—21no—
309C2×C53—22no—
310C2×C53C2×#38611yescharacteristic
311C2×C53#302212yes—
312D53—44no—
313C22×#41—531no—
314C23×Q8C2×D5311yescharacteristic
315C2×#137—538no—
316C2×#137—534no—
317C2×#137—1062no—
318C2×#137—1064no—
319C22×Q16—1061no—
320C22×SD16—1061no—
321C22×M16—531no—
322C22×M16—531no—
323#520—534no—
324#520—1062no—
325#519—1061no—
326#518—1061no—
327#518—1062no—
328C4×#41—1062no—
329C4×#41—1062no—
330C2×#117—531no—
331C2×#117—1062no—
332C2×#117C2×D5311yescharacteristic
333C2×#117C2×D5311yescharacteristic
334#421—1064no—
335C2×#104—534no—
336#419—534no—
337#419—1062no—
338C2×#103—1062no—
339#391—22no—
340#391C2×D5314yescomplemented
341#381—22no—
342#343—534no—
343#339—534no—
344C53#2094411yescharacteristic, complemented, Sylow
345C2×#41—534no—
346C2×#41—532no—
347C2×#41—534no—
348C2×#41—532no—
349C2×#41—1062no—
350C2×#41—1061no—
351C2×#41—1062no—
352C2×#41—1061no—
353C22×Q8—22no—
354C22×Q8—21no—
355C22×Q8—21no—
356C22×Q8—21no—
357C22×Q8—22no—
358C22×Q8—21no—
359C22×Q8—21no—
360C22×Q8—22no—
361C22×Q8—531no—
362C22×Q8C4×D5314yescomplemented
363C22×Q8C22×D5312yes—
364C22×D4—531no—
365C22×D4—1061no—
366C23×C4—21no—
367C23×C4—1061no—
368C23×C4C22×D5311yescharacteristic
369#137—5316no—
370#137—10616no—
371#137—1068no—
372C2×Q16—1068no—
373C2×Q16—1064no—
374C2×SD16—1068no—
375C2×SD16—1064no—
376C2×M16—534no—
377C2×M16—532no—
378C2×M16—532no—
379C2×M16—1062no—
380C2×M16—1061no—
381C2×M16—1061no—
382C2×M16—1062no—
383C2×M16—1062no—
384C22×C8—1061no—
385C22×C8—1061no—
386C4×Q8—1064no—
387C4×Q8—1064no—
388C4×D4—1064no—
389C4×D4—1068no—
390C4×D4—1064no—
391C4×D4—1068no—
392#117—21no—
393#117—21no—
394#117—21no—
395#117—21no—
396#117—22no—
397#117—22no—
398#117—532no—
399#117—532no—
400#117—1064no—
401#117—1064no—
402#117—1064no—
403#117—1061no—
404#117—1061no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—532no—
407C2×C4⋊C4—1064no—
408C2×C4⋊C4C22×D5312yes—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1064no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4C22⋊D5312yes—
413C2×C42—1064no—
414C2×C42—1061no—
415C2×C42C22⋊D5312yes—
416C2×C42C22×D5312yes—
417#104—1068no—
418#103—1068no—
419#95—28no—
420C24—1061no—
421C24C2×C22⋊D5311yescharacteristic
422#41—538no—
423#41—534no—
424#41—534no—
425#41—1064no—
426#41—1064no—
427#41—1064no—
428#41—1064no—
429#41—1064no—
430#41—1062no—
431#41—1062no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—24no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8—41no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—534no—
450C2×Q8—532no—
451C2×Q8—1061no—
452C2×Q8—1062no—
453C2×Q8C2×C4×D5314yes—
454C2×D4—534no—
455C2×D4—532no—
456C2×D4—1064no—
457C2×D4—1062no—
458C2×D4—1064no—
459C2×D4—1062no—
460C2×D4—1061no—
461C2×D4—1062no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—1068no—
469C22×C4—1061no—
470C22×C4—1064no—
471C22×C4—1062no—
472C22×C4—1064no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4C2×C4×D5312yes—
476C22×C4#1171994512yes—
477C22×C4D4×D5311yescharacteristic
478C22×C4D4×D5311yescharacteristic
479C22×C4D4×D5312yes—
480C22×C4D4×D5311yescharacteristic
481C22×C4D4×D5311yescharacteristic
482C22×C4#1171994712yes—
483C22×C4C2×C22⋊D5314yes—
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C23×D5311yescharacteristic
486Q16—10616no—
487SD16—10616no—
488M16—534no—
489M16—1064no—
490M16—1062no—
491M16—1062no—
492M16—1062no—
493M16—2122no—
494C2×C8—1064no—
495C2×C8—1062no—
496C2×C8—1062no—
497C2×C8—2122no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—534no—
502C4⋊C4—1068no—
503C4⋊C4—1062no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—42no—
507C22⋊C4—1068no—
508C22⋊C4—1062no—
509C22⋊C4—1062no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—1068no—
515C42—1062no—
516C42—1062no—
517C23—42no—
518C23—1064no—
519C23—1062no—
520C23—4242no—
521C23C4×C22⋊D5312yes—
522C23#1313318511yescharacteristic
523C23#1313318511yescharacteristic
524C23#1313318812yes—
525C23C22×C22⋊D5311yescharacteristic
526Q8—24no—
527Q8—24no—
528Q8—44no—
529Q8—44no—
530Q8—42no—
531Q8—42no—
532Q8—534no—
533Q8—1064no—
534Q8—1062no—
535D4—534no—
536D4—1068no—
537D4—1064no—
538D4—1064no—
539D4—1064no—
540D4—1062no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—1064no—
551C2×C4—1062no—
552C2×C4—1068no—
553C2×C4—1064no—
554C2×C4—1064no—
555C2×C4—1064no—
556C2×C4—4248no—
557C2×C4—4242no—
558C2×C4C22⋊C4×D5312yes—
559C2×C4C22⋊C4×D5312yes—
560C2×C4#1313317712yes—
561C2×C4#1313317712yes—
562C2×C4C4×C22⋊D5312yes—
563C2×C4#1313318511yescharacteristic
564C2×C4#1313318511yescharacteristic
565C2×C4#1313318812yes—
566C2×C4#1313319012yes—
567C2×C4#1313319012yes—
568C2×C4#1313319912yes—
569C2×C4#1313319912yes—
570C2×C4C22×C4×D5311yescharacteristic
571C2×C4C2×D4×D5311yescharacteristic
572C2×C4C2×D4×D5311yescharacteristic
573C2×C4C2×D4×D5311yescharacteristic
574C2×C4C2×#1171994711yescharacteristic
575C2×C4C2×#1171994511yescharacteristic
576C8—1064no—
577C8—2122no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—1064no—
584C22—4244no—
585C22—2124no—
586C22#1681049912yes—
587C22#1681049912yes—
588C22C2×C4×C22⋊D5311yescharacteristic
589C22C2×#1313318511yescharacteristic
590C22C2×#1313318811yescharacteristic
591C4—44no—
592C4—44no—
593C4—82no—
594C4—1064no—
595C4—2124no—
596C4#1681049912yes—
597C4#1681049912yes—
598C4C2×C22⋊C4×D5311yescharacteristic
599C4C2×#1313317711yescharacteristic
600C4C2×#1313319011yescharacteristic
601C4C2×#1313319911yescharacteristic
602C2—21no—
603C2—22no—
604C2—22no—
605C2—21no—
606C2—2124no—
607C2C2×#1681049911yescharacteristic
608C2#2028409312yes—
subgroupG/H
centerC22C2×#13133185
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2310C2
socleC22×C53C2×#120
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenter#2310D53
hypocenterC53#20944
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
4. C2×#16810632C2semidirect
5. #20275087C2semidirect
6. #20274454C2semidirect
7. #20274425C2semidirect
8. C53×#2310C2semidirect
12. C2×#13133248C4semidirect
30. #16810632C22semidirect
32. #16810392C22semidirect
33. #16810383C22semidirect
38. C53×#391C22semidirect
54. #13133248C2×C4semidirect
64. C2×#11719917C2×C4semidirect
98. C22×Q8×C53C2×C4semidirect
302. #2310D53semidirect
340. #391C2×D53semidirect
344. C53#20944semidirect
362. C22×Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C22×#131332482
#202879162
C2×#168106412
C2×#168106322
#202750872
#202744542
#202744252
C53×#23102
#2094453