Finite Group Databasebeta

← #27385613

Subgroups — #27385613

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×#16810639C211yesmaximal, characteristic
4#20277301C211yesmaximal, characteristic, complemented
5#20275081C218yesmaximal, complemented
6#20274453C211yesmaximal, characteristic, complemented
7#20274424C211yesmaximal, characteristic, complemented
8C2×C53×#419C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C2×#13133248—24no—
11C2×#13133248—22no—
12C2×#13133248C418yescomplemented
13C2×#13133248C2214yes—
14C22×#11719925—21no—
15C22×#11719924—21no—
16C22×#11719917C2211yescharacteristic
17#16811104—22no—
18#16811103C2214yes—
19#16811100—21no—
20#16811097—21no—
21#16811096—22no—
22C4×#11719945—22no—
23#16811091—22no—
24C2×#13133176C2211yescharacteristic
25C2×#13133176C2211yescharacteristic
26#117×D53—22no—
27#16810639—24no—
28C2×#13133143C2214yes—
29#16810530—22no—
30#16810530C2214yescomplemented
31#16810486—22no—
32#16810391C2214yescomplemented
33#16810382C2214yescomplemented
34C23×Q8×C53C2211yescharacteristic
35C22×M16×C53C2211yescharacteristic
36C2×#117×C53C2211yescharacteristic
37C53×#419—22no—
38C53×#419C2214yescomplemented
39C2×#103×C53—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×#11719917C2×C414yescomplemented
62C2×#11719917C2312yes—
63C22×#88322—21no—
64C4×Dic106—24no—
65#13133182—24no—
66C4×D212—24no—
67#13133179—24no—
68C4×C22⋊D53—28no—
69#13133177—28no—
70#13133176—21no—
71#13133176—21no—
72#13133176—21no—
73#13133176—21no—
74#13133176—22no—
75#13133176—22no—
76#13133175—24no—
77#13133174—24no—
78#13133173—24no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53C2312yes—
81C4⋊C4×D53—24no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53D412yes—
84C22⋊C4×D53—24no—
85C2×C4×Dic53D412yes—
86C2×C4×Dic53C2312yes—
87C42×D53—24no—
88#13133143—28no—
89#13133124—28no—
90C22×Q8×C53—21no—
91C22×Q8×C53—22no—
92C22×Q8×C53—21no—
93C22×Q8×C53—21no—
94C22×Q8×C53—22no—
95C22×Q8×C53—21no—
96C22×Q8×C53—21no—
97C22×Q8×C53—22no—
98C22×Q8×C53C2×C414yescomplemented
99C22×Q8×C53C2312yes—
100C23×C4×C53—21no—
101C23×C4×C53C2311yescharacteristic
102C2×M16×C53—21no—
103C2×M16×C53—22no—
104C2×M16×C53—22no—
105C2×M16×C53C2312yes—
106C22×C8×C53—21no—
107#117×C53—21no—
108#117×C53—21no—
109#117×C53D412yes—
110#117×C53C2312yes—
111C2×C4⋊C4×C53D412yes—
112C2×C22⋊C4×C53—21no—
113C2×C42×C53—21no—
114#103×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×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53#4112yes—
149C22×C4×C53#4112yes—
150C22×C4×C53C2411yescharacteristic
151C2×C22⋊D53—24no—
152C2×C22⋊D53—22no—
153C2×C22⋊D53—24no—
154C2×C22⋊D53—22no—
155C22×Dic53—21no—
156C22×Dic53—22no—
157C22×Dic53C2×D414yes—
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—
181M16×C53—22no—
182M16×C53—22no—
183M16×C53—42no—
184C2×C8×C53—22no—
185C2×C8×C53—42no—
186C4⋊C4×C53—22no—
187C4⋊C4×C53C2×D414yes—
188C22⋊C4×C53—22no—
189C22⋊C4×C53—22no—
190C42×C53—22no—
191C42×C53—22no—
192C22⋊Dic53—22no—
193C22⋊Dic53—22no—
194C22⋊Dic53—42no—
195#11719925—216no—
196#11719924—216no—
197#11719921—28no—
198C4⋊Dic53—22no—
199C4⋊Dic53—22no—
200C4⋊Dic53—42no—
201#11719919—28no—
202C4×Dic53—28no—
203C4×Dic53—22no—
204C4×Dic53—22no—
205C4×Dic53—22no—
206C4×Dic53—22no—
207#11719917—24no—
208#11719917—22no—
209#11719917C22×C414yes—
210C2×#88322—24no—
211C2×#88322—22no—
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—82no—
234C2×C4×C53C2×C22⋊C412yes—
235C2×C4×C53C2×C22⋊C412yes—
236C2×C4×C53C4×D412yes—
237C2×C4×C53C4×D412yes—
238C2×C4×C53C4×D412yes—
239C2×C4×C53#12011yescharacteristic
240C2×C4×C53#12011yescharacteristic
241C2×C4×C53#12112yes—
242C2×C4×C53#12112yes—
243C2×C4×C53#12112yes—
244C2×C4×C53#12312yes—
245C2×C4×C53#12312yes—
246C2×C4×C53C23×C411yescharacteristic
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C22⋊D53—24no—
253C22⋊D53—24no—
254C22⋊D53—28no—
255C2×Dic53—22no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—44no—
259C2×Dic53—44no—
260C2×Dic53—42no—
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—
271C8×C53—42no—
272#88322—24no—
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#38612yes—
290C4×C53#38612yes—
291C4×C53C22×C22⋊C411yescharacteristic
292C4×C53C2×C4×D411yescharacteristic
293C4×C53C2×#12111yescharacteristic
294C4×C53C2×#12311yescharacteristic
295Dic53—24no—
296Dic53—44no—
297Dic53—82no—
298C22×#137—531no—
299#3428—531no—
300C2×#421—531no—
301C2×#419D5311yescharacteristic, complemented
302#2310—531no—
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×M16C2×D5311yescharacteristic
323#520—534no—
324#520—1062no—
325#519—1061no—
326#518—1062no—
327#518—1061no—
328C4×#41—1062no—
329C4×#41—1062no—
330C2×#117—531no—
331C2×#117—531no—
332C2×#117—1062no—
333C2×#117C2×D5311yescharacteristic
334#421—1064no—
335C2×#104—534no—
336#419—22no—
337#419C2×D5314yescomplemented
338C2×#103—22no—
339#391—534no—
340#391—1062no—
341#381—1062no—
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—21no—
354C22×Q8—22no—
355C22×Q8—22no—
356C22×Q8—21no—
357C22×Q8—21no—
358C22×Q8—22no—
359C22×Q8—21no—
360C22×Q8—21no—
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—21no—
377C2×M16—22no—
378C2×M16—22no—
379C2×M16—534no—
380C2×M16—532no—
381C2×M16—1062no—
382C2×M16—1061no—
383C2×M16C22×D5312yes—
384C22×C8—21no—
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—1064no—
395#117—1064no—
396#117—1061no—
397#117—1061no—
398#117—1061no—
399#117—1061no—
400#117—1064no—
401#117—1062no—
402#117—1062no—
403#117C22⋊D5312yes—
404#117C22×D5312yes—
405C2×C4⋊C4—532no—
406C2×C4⋊C4—1064no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4C22⋊D5312yes—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—532no—
411C2×C22⋊C4—1064no—
412C2×C22⋊C4—1061no—
413C2×C42—21no—
414C2×C42—532no—
415C2×C42—532no—
416C2×C42—1064no—
417#104—1068no—
418#103—28no—
419#95—1068no—
420C24—1061no—
421C24D4×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—534no—
469C22×C4—1068no—
470C22×C4—1061no—
471C22×C4—1064no—
472C22×C4—1062no—
473C22×C4—1064no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4C2×C4×D5312yes—
477C22×C4#1171994512yes—
478C22×C4D4×D5311yescharacteristic
479C22×C4D4×D5311yescharacteristic
480C22×C4D4×D5312yes—
481C22×C4D4×D5311yescharacteristic
482C22×C4#1171994712yes—
483C22×C4C2×C22⋊D5311yescharacteristic
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C23×D5311yescharacteristic
486Q16—10616no—
487SD16—10616no—
488M16—22no—
489M16—22no—
490M16—42no—
491M16—534no—
492M16—1064no—
493M16—1062no—
494C2×C8—22no—
495C2×C8—42no—
496C2×C8—1064no—
497C2×C8—1062no—
498C4⋊C4—22no—
499C4⋊C4—1068no—
500C4⋊C4—1062no—
501C4⋊C4—1062no—
502C4⋊C4—2122no—
503C4⋊C4C2×C22⋊D5314yes—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—1068no—
507C22⋊C4—1062no—
508C22⋊C4—1062no—
509C22⋊C4—2122no—
510C42—22no—
511C42—22no—
512C42—1068no—
513C42—1062no—
514C42—1062no—
515C42—1062no—
516C42—1062no—
517C23—42no—
518C23—1064no—
519C23—1062no—
520C23—4242no—
521C23#1313317712yes—
522C23#1313318511yescharacteristic
523C23#1313318511yescharacteristic
524C23#1313319012yes—
525C23C2×D4×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—82no—
548C2×C4—1064no—
549C2×C4—1062no—
550C2×C4—1068no—
551C2×C4—1064no—
552C2×C4—1064no—
553C2×C4—1064no—
554C2×C4—4248no—
555C2×C4—2124no—
556C2×C4—2122no—
557C2×C4—2124no—
558C2×C4C22⋊C4×D5312yes—
559C2×C4C22⋊C4×D5312yes—
560C2×C4#1313317712yes—
561C2×C4C4×C22⋊D5312yes—
562C2×C4C4×C22⋊D5312yes—
563C2×C4#1313318511yescharacteristic
564C2×C4#1313318511yescharacteristic
565C2×C4#1313318812yes—
566C2×C4#1313318812yes—
567C2×C4#1313319012yes—
568C2×C4#1313319912yes—
569C2×C4#1313319912yes—
570C2×C4C22×C4×D5311yescharacteristic
571C2×C4C2×D4×D5311yescharacteristic
572C2×C4C2×D4×D5311yescharacteristic
573C2×C4C22×C22⋊D5311yescharacteristic
574C2×C4C2×#1171994711yescharacteristic
575C2×C4C2×#1171994511yescharacteristic
576C8—42no—
577C8—1064no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—1064no—
584C22—4244no—
585C22—2124no—
586C22#1681049912yes—
587C22#1681049912yes—
588C22C2×#1313317711yescharacteristic
589C22C2×#1313318511yescharacteristic
590C22C2×#1313319011yescharacteristic
591C4—44no—
592C4—44no—
593C4—1064no—
594C4—4242no—
595C4—2124no—
596C4#1681049912yes—
597C4#1681049912yes—
598C4C2×C22⋊C4×D5311yescharacteristic
599C4C2×C4×C22⋊D5311yescharacteristic
600C4C2×#1313318811yescharacteristic
601C4C2×#1313319911yescharacteristic
602C2—21no—
603C2—22no—
604C2—22no—
605C2—21no—
606C2—2124no—
607C2C2×#1681049911yescharacteristic
608C2#2028409112yes—
subgroupG/H
centerC22C2×#13133185
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#419C2
socleC22×C53C2×#120
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×#419D53
hypocenterC53#20944
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
4. #20277301C2semidirect
5. #20275081C2semidirect
6. #20274453C2semidirect
7. #20274424C2semidirect
8. C2×C53×#419C2semidirect
12. C2×#13133248C4semidirect
30. #16810530C22semidirect
32. #16810391C22semidirect
33. #16810382C22semidirect
38. C53×#419C22semidirect
54. #13133248C2×C4semidirect
61. C2×#11719917C2×C4semidirect
98. C22×Q8×C53C2×C4semidirect
301. C2×#419D53semidirect
337. #419C2×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×#168106392
#202773012
#202750812
#202744532
#202744242
C2×C53×#4192
#2094453