Finite Group Databasebeta

← #27381894

Subgroups — #27381894

Order 13568.

1598 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×#16810625C211yesmaximal, characteristic
3C2×#16810617C211yesmaximal, characteristic
4C2×#16810581C211yesmaximal, characteristic, complemented
5#20277071C211yesmaximal, characteristic, complemented
6#20274544C212yesmaximal, complemented
7#20274539C212yesmaximal, complemented
8#20274534C212yesmaximal, complemented
9#20274529C212yesmaximal, complemented
10C2×#16810385C211yesmaximal, characteristic, complemented
11C53×#1736C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C23×#88322C2211yescharacteristic
14#16811104—24no—
15#16811104C418yescomplemented
16C2×#13133182C2212yes—
17C2×#13133179C2212yes—
18C2×C4×C22⋊D53—22no—
19C2×#13133176C2211yescharacteristic
20C2×#13133175—21no—
21C22×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊C4×D53—21no—
23#16810625C418yescomplemented
24#16810617C418yescomplemented
25C2×#13133142C2212yes—
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133139C2212yes—
29C2×#13133137C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31#16810581—24no—
32C2×#13133127C2212yes—
33C2×C8×Dic53C2212yes—
34C4⋊C4×Dic53C2212yes—
35C4⋊C4×Dic53C2212yes—
36C4×C4⋊Dic53—21no—
37C4×C4⋊Dic53—21no—
38C22⋊C4×Dic53—22no—
39#16810385C2212yes—
40#16810385C2212yes—
41C22×M16×C53C2211yescharacteristic
42C2×#117×C53C2211yescharacteristic
43C22×C4⋊C4×C53C2211yescharacteristic
44C53×#340C2212yes—
45C53×#340C2212yes—
46C2×#11719945—22no—
47C2×#11719945—22no—
48C2×#11719945—22no—
49C2×#11719945C2×C414yes—
50C2×#11719945C2×C414yes—
51C2×#11719945C2×C414yes—
52C22×Dic106C2311yescharacteristic
53C22×D212C2311yescharacteristic
54C22×C22⋊D53—21no—
55C23×Dic53—21no—
56C22×C4×D53—21no—
57C22×C4×D53—22no—
58C22×#88322—22no—
59C22×#88322—22no—
60C22×#88322C2×C414yescomplemented
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133182—24no—
64#13133182C2×C418yescomplemented
65#13133179—24no—
66#13133179C2×C418yescomplemented
67C4×C22⋊D53—216no—
68#13133176C2×C414yes—
69#13133176D414yes—
70#13133175—28no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53C2×C414yescomplemented
74C2×C4⋊Dic53D411yescharacteristic
75C2×C4⋊Dic53D411yescharacteristic
76C2×C4⋊Dic53C2312yes—
77C2×#11719919—22no—
78C4⋊C4×D53—28no—
79C2×C22⋊Dic53D412yes—
80C2×#11719921—22no—
81C2×C4×Dic53—22no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53—22no—
84C2×C4×Dic53C2312yes—
85#13133142C2×C418yescomplemented
86#13133140C2×C414yescomplemented
87#13133140C2×C414yescomplemented
88#13133139C2×C418yescomplemented
89#13133137C2×C414yescomplemented
90#13133137C2×C414yescomplemented
91#13133127—24no—
92C8×Dic53—24no—
93#13133124—22no—
94#13133124—22no—
95C23×C4×C53—21no—
96C23×C4×C53C2311yescharacteristic
97C2×M16×C53—22no—
98C2×M16×C53—22no—
99C2×M16×C53C2×C414yescomplemented
100C22×C8×C53C2312yes—
101#117×C53—22no—
102#117×C53C2×C414yescomplemented
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—22no—
105C2×C4⋊C4×C53C2×C414yes—
106C2×C4⋊C4×C53C2312yes—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C22⋊C4×C53—21no—
110C2×C42×C53—21no—
111C24×C53C2×D411yescharacteristic
112C23×D53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—24no—
117C22×C4×C53—81no—
118C22×C4×C53—81no—
119C22×C4×C53C22×C412yes—
120C22×C4×C53C22×C412yes—
121C22×C4×C53C22×C412yes—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D412yes—
124C22×C4×C53#4112yes—
125C22×C4×C53#4112yes—
126C22×C4×C53C2411yescharacteristic
127C2×C22⋊D53—24no—
128C2×C22⋊D53—24no—
129C2×C22⋊D53—24no—
130C22×Dic53—21no—
131C22×Dic53—22no—
132C22×Dic53—24no—
133C22×Dic53—81no—
134C22×Dic53—81no—
135C22×Dic53C2×D412yes—
136C22×Dic53C2×D412yes—
137#11719945—28no—
138#11719945—28no—
139#11719945—28no—
140#11719945C42116yescomplemented
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212C22×C412yes—
145C2×D212C22×C412yes—
146C2×D212C22×C412yes—
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×Dic106C22×C412yes—
158C2×Dic106C22×C412yes—
159M16×C53—28no—
160C2×C8×C53—22no—
161C2×C8×C53—22no—
162C2×C8×C53C22×C414yes—
163C4⋊C4×C53—24no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53C22×C414yes—
168C4⋊C4×C53C22×C412yes—
169C4⋊C4×C53C22×C412yes—
170C22⋊C4×C53—24no—
171C42×C53—24no—
172C22⋊Dic53C22⋊C418yes—
173#11719921—28no—
174C4⋊Dic53—24no—
175C4⋊Dic53—22no—
176C4⋊Dic53C22⋊C414yes—
177C4⋊Dic53C22⋊C414yes—
178C4⋊Dic53C22×C414yes—
179#11719919—28no—
180C4×Dic53—28no—
181C4×Dic53—82no—
182C4×Dic53—82no—
183C4×Dic53C22×C414yes—
184C4×Dic53C2×D414yes—
185C2×#88322—28no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322C22×C412yes—
191C2×#88322C22×C412yes—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53C2×C22⋊C412yes—
197C23×C53C4×D412yes—
198C23×C53C4×D412yes—
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—42no—
211C2×C4×C53—41no—
212C2×C4×C53—42no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—84no—
217C2×C4×C53C2×C4214yes—
218C2×C4×C53C2×C22⋊C412yes—
219C2×C4×C53#11714yes—
220C2×C4×C53C4×D414yes—
221C2×C4×C53C4×D412yes—
222C2×C4×C53C4×D414yes—
223C2×C4×C53C4×D412yes—
224C2×C4×C53C23×C411yescharacteristic
225C2×C4×C53C23×C411yescharacteristic
226C2×C4×C53C23×C411yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C22⋊D53—216no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—24no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—84no—
242C2×Dic53C2×C22⋊C414yes—
243C2×Dic53C2×C22⋊C414yes—
244D212—22no—
245D212—22no—
246D212—22no—
247D212C2×C4214yes—
248C4×D53—216no—
249C4×D53—88no—
250Dic106—22no—
251Dic106—22no—
252Dic106—22no—
253Dic106C2×C4214yes—
254C8×C53—24no—
255#88322—22no—
256#88322—22no—
257#20415—531nomaximal, Sylow
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53C4×C22⋊C414yes—
265C22×C53C22×C22⋊C411yescharacteristic
266C22×C53C2×C4×D411yescharacteristic
267C22×C53C2×C4×D411yescharacteristic
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×C53C4×C22⋊C414yes—
283C4×C53C22×C4211yescharacteristic
284C4×C53C2×#11711yescharacteristic
285C4×C53C2×C4×D411yescharacteristic
286C4×C53C2×C4×D411yescharacteristic
287Dic53—24no—
288Dic53—24no—
289Dic53—24no—
290Dic53—44no—
291C2×#520—531no—
292C2×#417—531no—
293C2×#416—531no—
294C2×#405—531no—
295#2274—531no—
296#1763—532no—
297#1762—532no—
298C4×#103—532no—
299C4×#102—532no—
300#1736D5311yescharacteristic, complemented
301C2×#340—531no—
302C2×C53—24no—
303C2×C53C2×C4×C22⋊C411yescharacteristic
304C2×C53#289711yescharacteristic
305C2×C53#289811yescharacteristic
306C2×C53#294611yescharacteristic
307C2×C53#294711yescharacteristic
308C2×C53#295111yescharacteristic
309C2×C53#295211yescharacteristic
310D53—44no—
311C22×#41—531no—
312C22×M16C2×D5311yescharacteristic
313C23×C8—531no—
314#520—538no—
315#520—1064no—
316C2×C4×Q8—532no—
317C2×C4×D4—532no—
318C2×C4×D4—1062no—
319C2×#117—531no—
320C2×#117—1061no—
321C2×#117C2×D5311yescharacteristic
322C22×C4⋊C4—531no—
323C22×C4⋊C4—1061no—
324C22×C4⋊C4C2×D5311yescharacteristic
325#417—538no—
326#416—538no—
327C2×#103—532no—
328C2×#103—531no—
329C2×#103—531no—
330C2×#102—532no—
331C2×#102—531no—
332C2×#102—531no—
333#405—1064no—
334C2×#97—532no—
335C2×C4×C8—532no—
336C4×C4⋊C4—532no—
337C4×C4⋊C4—532no—
338C4×C4⋊C4—1061no—
339C4×C4⋊C4—1061no—
340C4×C22⋊C4—1062no—
341#340—532no—
342#340—532no—
343#340C2×D5312yes—
344#340C2×D5312yes—
345C53#2041511yescharacteristic, 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—1062no—
357C23×C4—1061no—
358C23×C4—1061no—
359C23×C4C22×D5311yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16C4×D5314yescomplemented
363C22×C8—534no—
364C22×C8—531no—
365C22×C8—531no—
366C22×C8—1062no—
367C22×C8—1062no—
368C22×C8C22×D5312yes—
369C4×Q8—538no—
370C4×Q8—1064no—
371C4×D4—538no—
372C4×D4—1064no—
373C4×D4—10616no—
374#117—22no—
375#117—534no—
376#117—534no—
377#117—1068no—
378#117C4×D5314yescomplemented
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
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×D5314yes—
390C2×C4⋊C4C22×D5312yes—
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—532no—
395C2×C22⋊C4—1062no—
396C2×C42—21no—
397C2×C42—532no—
398C2×C42—1062no—
399C2×C42—1062no—
400C2×C42—1062no—
401#103—538no—
402#103—534no—
403#103—534no—
404#102—538no—
405#102—534no—
406#102—534no—
407#97—1064no—
408C4×C8—1064no—
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—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—532no—
439C22×C4—532no—
440C22×C4—1068no—
441C22×C4—1061no—
442C22×C4—1064no—
443C22×C4—1064no—
444C22×C4—1064no—
445C22×C4—1064no—
446C22×C4—1062no—
447C22×C4—4242no—
448C22×C4—4242no—
449C22×C4—4241no—
450C22×C4—4241no—
451C22×C4C2×C4×D5312yes—
452C22×C4C2×C4×D5312yes—
453C22×C4C2×C4×D5312yes—
454C22×C4D4×D5311yescharacteristic
455C22×C4D4×D5312yes—
456C22×C4#1171994712yes—
457C22×C4#1171994912yes—
458C22×C4C23×D5311yescharacteristic
459M16—28no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—532no—
463C2×C8—532no—
464C2×C8—1068no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8C2×C4×D5314yes—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—534no—
475C4⋊C4—534no—
476C4⋊C4—534no—
477C4⋊C4—1068no—
478C4⋊C4—1064no—
479C4⋊C4—1062no—
480C4⋊C4C2×C4×D5312yes—
481C4⋊C4C2×C4×D5312yes—
482C4⋊C4C2×C4×D5314yes—
483C22⋊C4—24no—
484C22⋊C4—538no—
485C22⋊C4—1068no—
486C42—24no—
487C42—534no—
488C42—534no—
489C42—1068no—
490C42—4242no—
491C42—4242no—
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#1313317712yes—
505C23#1313317912yes—
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—22no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—534no—
527C2×C4—534no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
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×C4C42×D5314yes—
542C2×C4C22⋊C4×D5312yes—
543C2×C4#1313317514yes—
544C2×C4#1313317714yes—
545C2×C4#1313317712yes—
546C2×C4#1313317914yes—
547C2×C4#1313317912yes—
548C2×C4C22×C4×D5311yescharacteristic
549C2×C4C22×C4×D5311yescharacteristic
550C2×C4C22×C4×D5311yescharacteristic
551C2×C4C2×D4×D5311yescharacteristic
552C2×C4C2×#1171994711yescharacteristic
553C2×C4C2×#1171994911yescharacteristic
554C8—24no—
555C8—1062no—
556C8—1062no—
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#1681047014yes—
571C22C2×C22⋊C4×D5311yescharacteristic
572C22C2×#1313317711yescharacteristic
573C22C2×#1313317911yescharacteristic
574C22#1681142711yescharacteristic
575C22#1681143211yescharacteristic
576C22#1681143311yescharacteristic
577C22#1681143911yescharacteristic
578C4—44no—
579C4—42no—
580C4—42no—
581C4—1064no—
582C4—1064no—
583C4—1064no—
584C4—2124no—
585C4#1681047014yes—
586C4C2×C42×D5311yescharacteristic
587C4C2×#1313317511yescharacteristic
588C4C2×#1313317711yescharacteristic
589C4C2×#1313317911yescharacteristic
590C2—24no—
591C2—2124no—
592C2C2×#1681047011yescharacteristic
593C2#2028324011yescharacteristic
594C2#2028324711yescharacteristic
595C2#2028354511yescharacteristic
596C2#2028355611yescharacteristic
597C2#2028358611yescharacteristic
598C2#2028359411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC4×C53C22×C42
FrattiniC22×C4C23×D53
FittingC53×#1736C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenter#1736D53
hypocenterC53#20415
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. C2×#16810581C2semidirect
5. #20277071C2semidirect
6. #20274544C2semidirect
7. #20274539C2semidirect
8. #20274534C2semidirect
9. #20274529C2semidirect
10. C2×#16810385C2semidirect
11. C53×#1736C2semidirect
15. #16811104C4semidirect
23. #16810625C4semidirect
24. #16810617C4semidirect
60. C22×#88322C2×C4semidirect
64. #13133182C2×C4semidirect
66. #13133179C2×C4semidirect
73. C2×C4⋊Dic53C2×C4semidirect
85. #13133142C2×C4semidirect
86. #13133140C2×C4semidirect
87. #13133140C2×C4semidirect
88. #13133139C2×C4semidirect
89. #13133137C2×C4semidirect
90. #13133137C2×C4semidirect
99. C2×M16×C53C2×C4semidirect
102. #117×C53C2×C4semidirect
140. #11719945C42semidirect
300. #1736D53semidirect
345. C53#20415semidirect
362. C2×M16C4×D53semidirect
378. #117C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111042
C2×#168106252
C2×#168106172
C2×#168105812
#202770712
#202745442
#202745392
#202745342
#202745292
C2×#168103852
C53×#17362
#2041553