Finite Group Databasebeta

← #27383624

Subgroups — #27383624

Order 13568.

1462 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×#16811104C211yesmaximal, characteristic
2C2×#16810649C211yesmaximal, characteristic, complemented
3C2×#16810622C211yesmaximal, characteristic
4C2×#16810620C211yesmaximal, characteristic
5#20277339C211yesmaximal, characteristic, complemented
6#20274729C212yesmaximal, complemented
7#20274724C212yesmaximal, complemented
8#20274719C212yesmaximal, complemented
9#20274714C212yesmaximal, complemented
10#20274434C211yesmaximal, characteristic, complemented
11C2×C53×#340C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14#16811104—24no—
15#16811104C418yescomplemented
16C2×#13133182C2212yes—
17C2×#13133179C2212yes—
18C2×C4×C22⋊D53—22no—
19C2×#13133176C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21C2×#13133175—21no—
22C2×C4⋊C4×D53—21no—
23#16810649—24no—
24C2×#13133146C2212yes—
25C2×#13133146C2212yes—
26#16810622—24no—
27#16810620—24no—
28C2×#13133142C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133140C2212yes—
31C2×#13133139C2211yescharacteristic
32C2×#13133139C2211yescharacteristic
33C2×#13133137C2212yes—
34#16810519—21no—
35#16810519—21no—
36#16810496—22no—
37#16810492C2212yes—
38#16810488C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C23×C8×C53C2211yescharacteristic
42C22×C4⋊C4×C53C2211yescharacteristic
43C22×C4⋊C4×C53C2211yescharacteristic
44C53×#340C2212yes—
45C53×#340C2212yes—
46C2×#11719945—22no—
47C2×#11719945—22no—
48C2×#11719945—22no—
49C2×#11719945C2×C414yes—
50C2×#11719945D414yes—
51C2×#11719945Q814yes—
52C22×Dic106C2311yescharacteristic
53C22×D212C2311yescharacteristic
54C22×C22⋊D53—21no—
55C22×C4×D53—21no—
56C22×C4×D53—22no—
57C2×#11719917—22no—
58C2×#11719917—22no—
59C2×#11719917—22no—
60C22×#88322C2312yes—
61#13133182—24no—
62#13133182C2×C418yescomplemented
63#13133179—24no—
64#13133179C2×C418yescomplemented
65C4×C22⋊D53—216no—
66#13133176—22no—
67#13133176—22no—
68#13133176C2×C414yes—
69#13133176D414yes—
70#13133175—28no—
71C2×C4⋊Dic53D411yescharacteristic
72C2×C4⋊Dic53D411yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—22no—
76C2×#11719919—22no—
77C2×#11719919—22no—
78C4⋊C4×D53—28no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53D412yes—
81C2×#11719921—22no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—22no—
84C2×C4×Dic53C2312yes—
85#13133146—24no—
86#13133146—24no—
87#13133142—22no—
88#13133142—22no—
89#13133140—24no—
90#13133139—22no—
91#13133139—22no—
92#13133137—24no—
93#13133124—22no—
94#13133124—22no—
95C23×C4×C53—21no—
96C23×C4×C53—21no—
97C23×C4×C53C2311yescharacteristic
98C22×C8×C53—22no—
99C22×C8×C53—22no—
100C22×C8×C53—22no—
101C22×C8×C53C2311yescharacteristic
102C22×C8×C53C2311yescharacteristic
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—22no—
107C2×C4⋊C4×C53—22no—
108C2×C4⋊C4×C53C2×C414yes—
109C2×C4⋊C4×C53C2312yes—
110C2×C4⋊C4×C53C2312yes—
111C24×C53C2×D411yescharacteristic
112C23×D53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—24no—
115C22×C4×C53—22no—
116C22×C4×C53—42no—
117C22×C4×C53—81no—
118C22×C4×C53—81no—
119C22×C4×C53—81no—
120C22×C4×C53—81no—
121C22×C4×C53C22×C412yes—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D412yes—
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—21no—
134C22×Dic53—21no—
135C22×Dic53C2×D412yes—
136C22×Dic53C2×D412yes—
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—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—44no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53C22×C414yes—
172C22⋊Dic53—42no—
173C22⋊Dic53C22⋊C418yes—
174#11719921—28no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53C22⋊C414yes—
180C4⋊Dic53C22⋊C414yes—
181#11719919—28no—
182#11719919—82no—
183#11719919—82no—
184C4×Dic53—28no—
185C4×Dic53—42no—
186C4×Dic53C22×C414yes—
187C4×Dic53C2×D414yes—
188#11719917—44no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53C2×C22⋊C412yes—
197C23×C53#12112yes—
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—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—84no—
214C2×C4×C53—84no—
215C2×C4×C53C2×C22⋊C412yes—
216C2×C4×C53C2×C4⋊C414yes—
217C2×C4×C53#11714yes—
218C2×C4×C53#12114yes—
219C2×C4×C53#12112yes—
220C2×C4×C53C22⋊Q814yes—
221C2×C4×C53C22⋊Q812yes—
222C2×C4×C53C23×C411yescharacteristic
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53C22×D411yescharacteristic
225C2×C4×C53C22×Q811yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C22⋊D53—216no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53C2×C22⋊C414yes—
243C2×Dic53C2×C22⋊C414yes—
244D212—22no—
245D212—22no—
246D212—22no—
247D212C2×C4⋊C414yes—
248C4×D53—216no—
249C4×D53—88no—
250Dic106—22no—
251Dic106—22no—
252Dic106—22no—
253Dic106C2×C4⋊C414yes—
254C8×C53—41no—
255C8×C53—41no—
256#88322—42no—
257#20652—531nomaximal, Sylow
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53#38014yes—
265C22×C53C22×C22⋊C411yescharacteristic
266C22×C53C2×#12111yescharacteristic
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—82no—
281C4×C53#38014yes—
282C4×C53C22×C4⋊C411yescharacteristic
283C4×C53C2×#11711yescharacteristic
284C4×C53C2×#12111yescharacteristic
285C4×C53C2×C22⋊Q811yescharacteristic
286Dic53—24no—
287Dic53—24no—
288Dic53—24no—
289Dic53—81no—
290Dic53—81no—
291C2×#520—531no—
292C2×#424—531no—
293C2×#417—531no—
294C2×#416—531no—
295#2317—531no—
296#1805—532no—
297#1804—532no—
298#1803—532no—
299#1802—532no—
300#1736—531no—
301C2×#340D5311yescharacteristic, complemented
302C2×C53—24no—
303C2×C53C2×#38011yescharacteristic
304C2×C53#289711yescharacteristic
305C2×C53#289811yescharacteristic
306C2×C53#304511yescharacteristic
307C2×C53#304611yescharacteristic
308C2×C53#308311yescharacteristic
309C2×C53#308411yescharacteristic
310D53—44no—
311C22×#41—531no—
312C22×M16—531no—
313C23×C8C2×D5311yescharacteristic
314#520—538no—
315#520—1064no—
316C2×C4×Q8—532no—
317C2×C4×D4—532no—
318C2×C4×D4—1062no—
319C2×#117—531no—
320C2×#117—531no—
321C2×#117—1061no—
322C22×C4⋊C4—1061no—
323C22×C4⋊C4C2×D5311yescharacteristic
324C22×C4⋊C4C2×D5311yescharacteristic
325#424—1064no—
326C2×C4⋊C8—532no—
327C2×C4⋊C8—532no—
328#417—1064no—
329#416—1064no—
330C2×#103—531no—
331C2×#103—531no—
332C2×#103—532no—
333C2×#102—531no—
334C2×#102—531no—
335C2×#102—532no—
336#389—1061no—
337#389—1061no—
338#385—1062no—
339#384—532no—
340#382—532no—
341#340—532no—
342#340—532no—
343#340C2×D5312yes—
344#340C2×D5312yes—
345C53#2065211yescharacteristic, 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—21no—
357C23×C4—1062no—
358C23×C4—1061no—
359C23×C4C22×D5311yescharacteristic
360C2×M16—1062no—
361C2×M16—1062no—
362C2×M16—1062no—
363C22×C8—22no—
364C22×C8—22no—
365C22×C8—22no—
366C22×C8—532no—
367C22×C8C22×D5311yescharacteristic
368C22×C8C22×D5311yescharacteristic
369C4×Q8—538no—
370C4×Q8—1064no—
371C4×D4—538no—
372C4×D4—1064no—
373C4×D4—10616no—
374#117—534no—
375#117—534no—
376#117—1068no—
377#117—1062no—
378#117—1062no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—1068no—
389C2×C4⋊C4—1062no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4C4×D5314yes—
393C2×C4⋊C4C22×D5312yes—
394C2×C4⋊C4C22×D5312yes—
395C2×C22⋊C4—532no—
396C2×C22⋊C4—1062no—
397C2×C22⋊C4—1061no—
398C2×C42—532no—
399C2×C42—1062no—
400C2×C42—1061no—
401C4⋊C8—1064no—
402C4⋊C8—1064no—
403#103—1064no—
404#103—1062no—
405#103—1062no—
406#102—1064no—
407#102—1062no—
408#102—1062no—
409#95—1062no—
410#95—1062no—
411C24—1061no—
412C24D4×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—22no—
435C22×C4—42no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—532no—
441C22×C4—532no—
442C22×C4—1068no—
443C22×C4—1061no—
444C22×C4—1064no—
445C22×C4—1064no—
446C22×C4—1064no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—4242no—
450C22×C4—4242no—
451C22×C4C2×C4×D5312yes—
452C22×C4#1171994512yes—
453C22×C4#1171994512yes—
454C22×C4D4×D5311yescharacteristic
455C22×C4D4×D5312yes—
456C22×C4D4×D5312yes—
457C22×C4Q8×D5312yes—
458C22×C4C23×D5311yescharacteristic
459M16—2124no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—44no—
467C2×C8—1062no—
468C2×C8—1062no—
469C2×C8—1062no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—42no—
475C4⋊C4—534no—
476C4⋊C4—534no—
477C4⋊C4—1068no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—4242no—
483C4⋊C4—4242no—
484C4⋊C4C2×C4×D5314yes—
485C22⋊C4—538no—
486C22⋊C4—1068no—
487C22⋊C4—2122no—
488C42—534no—
489C42—534no—
490C42—1068no—
491C42—2122no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—1062no—
497C23—1062no—
498C23—1062no—
499C23—2121no—
500C23—2121no—
501C23—2121no—
502C23—2121no—
503C23C22⋊C4×D5312yes—
504C23#1313319012yes—
505C23#1313319512yes—
506C23C2×D4×D5311yescharacteristic
507Q8—534no—
508Q8—1062no—
509Q8—1062no—
510Q8—1062no—
511D4—534no—
512D4—10616no—
513D4—1062no—
514D4—1062no—
515D4—1062no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—534no—
525C2×C4—534no—
526C2×C4—1062no—
527C2×C4—1062no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—10616no—
534C2×C4—4248no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4C22⋊C4×D5312yes—
542C2×C4C4⋊C4×D5314yes—
543C2×C4#1313317414yes—
544C2×C4#1313319014yes—
545C2×C4#1313319012yes—
546C2×C4#1313319514yes—
547C2×C4#1313319512yes—
548C2×C4C22×C4×D5311yescharacteristic
549C2×C4C2×D4×D5311yescharacteristic
550C2×C4C2×D4×D5311yescharacteristic
551C2×C4C2×Q8×D5311yescharacteristic
552C2×C4C2×#1171994511yescharacteristic
553C2×C4C2×#1171994511yescharacteristic
554C8—41no—
555C8—41no—
556C8—2122no—
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#1681048214yes—
571C22C2×C22⋊C4×D5311yescharacteristic
572C22C2×#1313319011yescharacteristic
573C22C2×#1313319511yescharacteristic
574C22#1681142411yescharacteristic
575C22#1681142911yescharacteristic
576C22#1681142911yescharacteristic
577C22#1681143611yescharacteristic
578C4—44no—
579C4—82no—
580C4—1064no—
581C4—1064no—
582C4—1064no—
583C4—4241no—
584C4—4241no—
585C4#1681048214yes—
586C4C2×C4⋊C4×D5311yescharacteristic
587C4C2×#1313317411yescharacteristic
588C4C2×#1313319011yescharacteristic
589C4C2×#1313319511yescharacteristic
590C2—24no—
591C2—2124no—
592C2C2×#1681048211yescharacteristic
593C2#2028323611yescharacteristic
594C2#2028324311yescharacteristic
595C2#2028425911yescharacteristic
596C2#2028427011yescharacteristic
597C2#2028457111yescharacteristic
598C2#2028458211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#340C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×#340D53
hypocenterC53#20652
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C2×#16810649C2semidirect
5. #20277339C2semidirect
6. #20274729C2semidirect
7. #20274724C2semidirect
8. #20274719C2semidirect
9. #20274714C2semidirect
10. #20274434C2semidirect
11. C2×C53×#340C2semidirect
15. #16811104C4semidirect
62. #13133182C2×C4semidirect
64. #13133179C2×C4semidirect
140. #11719945C4⋊C4semidirect
301. C2×#340D53semidirect
345. C53#20652semidirect

Maximal subgroups

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

subgroupindex
C2×#168111042
C2×#168106492
C2×#168106222
C2×#168106202
#202773392
#202747292
#202747242
#202747192
#202747142
#202744342
C2×C53×#3402
#2065253