Finite Group Databasebeta

← #27385095

Subgroups — #27385095

Order 13568.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811103C211yesmaximal, characteristic
2C2×#16810666C211yesmaximal, characteristic, complemented
3C2×#16810622C211yesmaximal, characteristic
4C2×#16810617C211yesmaximal, characteristic
5#20274982C212yesmaximal, complemented
6#20274975C212yesmaximal, complemented
7#20274968C212yesmaximal, complemented
8#20274961C212yesmaximal, complemented
9C2×#16810385C211yesmaximal, characteristic, complemented
10#20274412C211yesmaximal, characteristic, complemented
11C53×#2316C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14C23×#88322C2211yescharacteristic
15#16811103—24no—
16#16811103—24no—
17C2×C4×Dic106C2212yes—
18C2×C4×D212C2212yes—
19C2×#13133177—22no—
20C2×#13133175—21no—
21C22×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊C4×D53—21no—
23#16810666—24no—
24C2×#13133150C2212yes—
25C2×#13133148C2212yes—
26#16810622C418yescomplemented
27#16810617—24no—
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133140C2212yes—
31C2×#13133137C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×#13133137C2212yes—
34#16810385C2212yes—
35#16810385C2212yes—
36#16810378C2212yes—
37#16810378C2212yes—
38C2×#117×C53C2211yescharacteristic
39C2×#117×C53C2211yescharacteristic
40C22×C4⋊C4×C53C2211yescharacteristic
41C53×#389—21no—
42C53×#389—21no—
43C53×#389C2212yes—
44C53×#389C2212yes—
45C53×#380—22no—
46C2×#11719945—22no—
47C2×#11719945—22no—
48C2×#11719945—22no—
49C2×#11719945C2×C414yes—
50C2×#11719945D414yes—
51C2×#11719945Q814yes—
52C22×Dic106C2311yescharacteristic
53C22×D212C2311yescharacteristic
54C22×C22⋊D53—21no—
55C23×Dic53—21no—
56C22×C4×D53—21no—
57C22×C4×D53—22no—
58C2×#11719917—22no—
59C2×#11719917—22no—
60C2×#11719917C2×C414yescomplemented
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322—22no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2312yes—
67C4×Dic106—24no—
68C4×Dic106—24no—
69C4×D212—24no—
70C4×D212—24no—
71#13133177—216no—
72#13133175—28no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53D414yes—
76C2×C4⋊Dic53C2312yes—
77C2×#11719919—22no—
78C4⋊C4×D53—28no—
79C2×#11719921—22no—
80C2×C4×Dic53—22no—
81#13133150—24no—
82#13133148—24no—
83#13133140—22no—
84#13133140—22no—
85#13133140C2×C418yescomplemented
86#13133137—22no—
87#13133137—22no—
88#13133137C2×C418yescomplemented
89C23×C4×C53—21no—
90C23×C4×C53C2311yescharacteristic
91#117×C53—22no—
92#117×C53—22no—
93#117×C53—22no—
94#117×C53—22no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53C2×C414yescomplemented
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2312yes—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53D412yes—
107C2×C42×C53—21no—
108C2×C42×C53C2312yes—
109#95×C53—22no—
110#95×C53—22no—
111C24×C53C2×D411yescharacteristic
112C23×D53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—24no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—81no—
118C22×C4×C53—81no—
119C22×C4×C53C22×C412yes—
120C22×C4×C53C2×D412yes—
121C22×C4×C53C2×D412yes—
122C22×C4×C53C2×D412yes—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D412yes—
125C22×C4×C53C2×Q812yes—
126C22×C4×C53#4112yes—
127C22×C4×C53#4112yes—
128C22×C4×C53C2411yescharacteristic
129C2×C22⋊D53—24no—
130C2×C22⋊D53—24no—
131C2×C22⋊D53—24no—
132C22×Dic53—21no—
133C22×Dic53—22no—
134C22×Dic53—24no—
135C22×Dic53—81no—
136C22×Dic53—81no—
137#11719945—28no—
138#11719945—28no—
139#11719945—28no—
140#11719945C4⋊C4116yescomplemented
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212C22×C412yes—
145C2×D212C2×D412yes—
146C2×D212C2×Q812yes—
147C2×C4×D53—24no—
148C2×C4×D53—24no—
149C2×C4×D53—24no—
150C2×C4×D53—28no—
151C2×C4×D53—82no—
152C2×C4×D53—82no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106C22×C412yes—
157C2×Dic106C2×D412yes—
158C2×Dic106C2×Q812yes—
159C4⋊C4×C53—24no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—82no—
168C4⋊C4×C53—82no—
169C4⋊C4×C53C22×C414yes—
170C22⋊C4×C53—24no—
171C22⋊C4×C53—42no—
172C42×C53—22no—
173C42×C53—22no—
174C42×C53—42no—
175#11719921—28no—
176C4⋊Dic53—24no—
177C4⋊Dic53—22no—
178C4⋊Dic53C2×D414yes—
179#11719919—28no—
180C4×Dic53—28no—
181#11719917—28no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—44no—
191C2×#88322C22×C414yes—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53C4×D412yes—
197C23×C53#12012yes—
198C23×C53C22⋊Q812yes—
199C23×C53C22×D411yescharacteristic
200C22×D53—22no—
201C22×D53—22no—
202C22×D53—22no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—41no—
207C2×C4×C53—24no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—84no—
223C2×C4×C53C2×C4⋊C414yes—
224C2×C4×C53C4×D412yes—
225C2×C4×C53C4×D414yes—
226C2×C4×C53#12012yes—
227C2×C4×C53C22⋊Q812yes—
228C2×C4×C53C22⋊Q814yes—
229C2×C4×C53#12314yes—
230C2×C4×C53C23×C411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C22×Q811yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C22⋊D53—216no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—24no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—84no—
244D212—22no—
245D212—22no—
246D212—22no—
247D212C2×C4⋊C414yes—
248C4×D53—216no—
249C4×D53—88no—
250Dic106—22no—
251Dic106—22no—
252Dic106—22no—
253Dic106C2×C4⋊C414yes—
254#88322—24no—
255#88322—41no—
256#88322—41no—
257#20878—531nomaximal, Sylow
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53#38514yes—
265C22×C53C2×C4×D411yescharacteristic
266C22×C53C2×#12011yescharacteristic
267C22×C53C2×C22⋊Q811yescharacteristic
268C22×C53#57611yescharacteristic
269C22×C53#57711yescharacteristic
270C22×C53#57711yescharacteristic
271C22×C53#57811yescharacteristic
272C2×D53—24no—
273C2×D53—42no—
274C2×D53—42no—
275C2×D53—42no—
276C2×D53—42no—
277C2×D53—42no—
278C2×D53—42no—
279C4×C53—44no—
280C4×C53—42no—
281C4×C53—42no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53#38514yes—
285C4×C53C22×C4⋊C411yescharacteristic
286C4×C53C2×C4×D411yescharacteristic
287C4×C53C2×C22⋊Q811yescharacteristic
288C4×C53C2×#12311yescharacteristic
289Dic53—24no—
290Dic53—44no—
291C2×#520—531no—
292C2×#428—531no—
293C2×#417—531no—
294C2×#416—531no—
295#2316D5311yescharacteristic, complemented
296#1865—532no—
297#1864—532no—
298#1863—532no—
299#1862—532no—
300C2×#340—531no—
301#1729—531no—
302C2×C53—24no—
303C2×C53C2×#38511yescharacteristic
304C2×C53#294611yescharacteristic
305C2×C53#294711yescharacteristic
306C2×C53#301211yescharacteristic
307C2×C53#301311yescharacteristic
308C2×C53#308311yescharacteristic
309C2×C53#308411yescharacteristic
310D53—44no—
311C22×#41—531no—
312C22×M16—531no—
313C23×C8—531no—
314#520—1064no—
315#520—1064no—
316C2×C4×Q8—532no—
317C2×C4×D4—532no—
318C2×C4×D4—1062no—
319C2×#117—1061no—
320C2×#117C2×D5311yescharacteristic
321C2×#117C2×D5311yescharacteristic
322C22×C4⋊C4—531no—
323C22×C4⋊C4—1061no—
324C22×C4⋊C4C2×D5311yescharacteristic
325#428—1064no—
326C2×#107—532no—
327C2×#106—532no—
328#417—538no—
329#416—1064no—
330C2×#103—532no—
331C2×#103—531no—
332C2×#103—531no—
333C2×#102—532no—
334C2×#102—531no—
335C2×#102—531no—
336#389—21no—
337#389—21no—
338#389C2×D5312yes—
339#389C2×D5312yes—
340#380—22no—
341#340—532no—
342#340—532no—
343#336—532no—
344#336—532no—
345C53#2087811yescharacteristic, complemented, Sylow
346C2×#41—534no—
347C2×#41—534no—
348C2×#41—534no—
349C2×#41—1062no—
350C2×#41—1062no—
351C2×#41—1062no—
352C22×Q8—531no—
353C22×D4—531no—
354C22×D4—1061no—
355C23×C4—21no—
356C23×C4—1061no—
357C23×C4—1061no—
358C23×C4—1062no—
359C23×C4C22×D5311yescharacteristic
360C2×M16—534no—
361C2×M16—1062no—
362C2×M16—1062no—
363C22×C8—532no—
364C22×C8—531no—
365C22×C8—531no—
366C22×C8—1062no—
367C22×C8—1062no—
368C22×C8—1062no—
369C4×Q8—1064no—
370C4×Q8—1064no—
371C4×D4—1064no—
372C4×D4—1064no—
373C4×D4—10616no—
374#117—22no—
375#117—22no—
376#117—22no—
377#117—22no—
378#117—1068no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—534no—
384C2×C4⋊C4—532no—
385C2×C4⋊C4—1068no—
386C2×C4⋊C4—1062no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4C4×D5314yescomplemented
390C2×C4⋊C4C22⋊D5311yescharacteristic
391C2×C4⋊C4C22⋊D5311yescharacteristic
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5312yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1062no—
397C2×C22⋊C4C22⋊D5312yes—
398C2×C42—21no—
399C2×C42—1062no—
400C2×C42C22×D5312yes—
401#107—1064no—
402#106—1064no—
403#103—538no—
404#103—1062no—
405#103—1062no—
406#102—538no—
407#102—1062no—
408#102—1062no—
409#95—22no—
410#95—22no—
411C24—1061no—
412C24C2×C22⋊D5311yescharacteristic
413#41—5316no—
414#41—1068no—
415#41—1068no—
416#41—1068no—
417C2×Q8—532no—
418C2×Q8—532no—
419C2×Q8—532no—
420C2×Q8—1061no—
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×D4—532no—
424C2×D4—532no—
425C2×D4—532no—
426C2×D4—1064no—
427C2×D4—1064no—
428C2×D4—1064no—
429C2×D4—1061no—
430C2×D4—1061no—
431C2×D4—1061no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—1068no—
439C22×C4—1061no—
440C22×C4—1064no—
441C22×C4—1064no—
442C22×C4—1064no—
443C22×C4—1064no—
444C22×C4—1062no—
445C22×C4—4242no—
446C22×C4—4242no—
447C22×C4—4241no—
448C22×C4—4241no—
449C22×C4C2×C4×D5312yes—
450C22×C4#1171994512yes—
451C22×C4D4×D5312yes—
452C22×C4D4×D5312yes—
453C22×C4Q8×D5312yes—
454C22×C4#1171994912yes—
455C22×C4C2×C22⋊D5312yes—
456C22×C4C2×C22⋊D5312yes—
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C23×D5311yescharacteristic
459M16—1068no—
460C2×C8—534no—
461C2×C8—1062no—
462C2×C8—1062no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—2124no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—534no—
481C4⋊C4—1068no—
482C4⋊C4—1064no—
483C4⋊C4—1062no—
484C4⋊C4C2×C4×D5314yes—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—1068no—
488C42—22no—
489C42—22no—
490C42—42no—
491C42—1068no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—1062no—
497C23—1062no—
498C23—1062no—
499C23—2121no—
500C23—2121no—
501C23—2121no—
502C23—2121no—
503C23C4×C22⋊D5312yes—
504C23#1313318512yes—
505C23#1313319312yes—
506C23C22×C22⋊D5311yescharacteristic
507Q8—534no—
508Q8—1062no—
509Q8—1062no—
510Q8—1062no—
511D4—534no—
512D4—10616no—
513D4—1062no—
514D4—1062no—
515D4—1062no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—10616no—
536C2×C4—1064no—
537C2×C4—4248no—
538C2×C4—4244no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4C4⋊C4×D5314yes—
542C2×C4C4×C22⋊D5312yes—
543C2×C4#1313317914yes—
544C2×C4#1313318512yes—
545C2×C4#1313319312yes—
546C2×C4#1313319514yes—
547C2×C4#1313320214yes—
548C2×C4C22×C4×D5311yescharacteristic
549C2×C4C2×D4×D5311yescharacteristic
550C2×C4C2×D4×D5311yescharacteristic
551C2×C4C2×Q8×D5311yescharacteristic
552C2×C4C2×#1171994511yescharacteristic
553C2×C4C2×#1171994911yescharacteristic
554C8—1064no—
555C8—2121no—
556C8—2121no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—1064no—
564C22—2122no—
565C22—2122no—
566C22—2122no—
567C22—2122no—
568C22—2122no—
569C22—2122no—
570C22#1681049414yes—
571C22C2×C4×C22⋊D5311yescharacteristic
572C22C2×#1313318511yescharacteristic
573C22C2×#1313319311yescharacteristic
574C22#1681142611yescharacteristic
575C22#1681143111yescharacteristic
576C22#1681143411yescharacteristic
577C22#1681143811yescharacteristic
578C4—42no—
579C4—42no—
580C4—44no—
581C4—81no—
582C4—81no—
583C4—1064no—
584C4—2124no—
585C4#1681049414yes—
586C4C2×C4⋊C4×D5311yescharacteristic
587C4C2×#1313317911yescharacteristic
588C4C2×#1313319511yescharacteristic
589C4C2×#1313320211yescharacteristic
590C2—24no—
591C2—2124no—
592C2C2×#1681049411yescharacteristic
593C2#2028354711yescharacteristic
594C2#2028355811yescharacteristic
595C2#2028401611yescharacteristic
596C2#2028402711yescharacteristic
597C2#2028456211yescharacteristic
598C2#2028457311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2316C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenter#2316D53
hypocenterC53#20878
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2316

Complements

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

NH ≅ G/Ntype
2. C2×#16810666C2semidirect
5. #20274982C2semidirect
6. #20274975C2semidirect
7. #20274968C2semidirect
8. #20274961C2semidirect
9. C2×#16810385C2semidirect
10. #20274412C2semidirect
11. C53×#2316C2semidirect
26. #16810622C4semidirect
60. C2×#11719917C2×C4semidirect
85. #13133140C2×C4semidirect
88. #13133137C2×C4semidirect
99. C2×C4⋊C4×C53C2×C4semidirect
140. #11719945C4⋊C4semidirect
295. #2316D53semidirect
345. C53#20878semidirect
389. C2×C4⋊C4C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111032
C2×#168106662
C2×#168106222
C2×#168106172
#202749822
#202749752
#202749682
#202749612
C2×#168103852
#202744122
C53×#23162
#2087853