Finite Group Databasebeta

← #27469529

Subgroups — #27469529

Order 13568.

888 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810941C211yesmaximal, characteristic
2C2×#16810964C211yesmaximal, characteristic, complemented
3C2×#16810869C211yesmaximal, characteristic
4#20279196C211yesmaximal, characteristic, complemented
5#20279173C211yesmaximal, characteristic
6#20276260C211yesmaximal, characteristic, complemented
7#20276255C211yesmaximal, characteristic, complemented
8#20276237C211yesmaximal, characteristic, complemented
9#20276233C211yesmaximal, characteristic, complemented
10#20275594C211yesmaximal, characteristic, complemented
11#20275567C211yesmaximal, characteristic, complemented
12#20274880C211yesmaximal, characteristic, complemented
13#20274878C211yesmaximal, characteristic, complemented
14#20274496C211yesmaximal, characteristic, complemented
15C2×C53×#428C211yesmaximal, characteristic, complemented
16C22×#11719913C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133191C2211yescharacteristic
19#16810964—24no—
20#16810941—24no—
21#16810869—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133139C2211yescharacteristic
31C2×#13133137C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×#13133127C2211yescharacteristic
35C2×C8×Dic53C2211yescharacteristic
36#16810561—22no—
37#16810538—22no—
38#16810528C2211yescharacteristic
39#16810528C2211yescharacteristic
40#16810524C2211yescharacteristic
41#16810524C2211yescharacteristic
42#16810514—22no—
43#16810492C2211yescharacteristic
44#16810488C2211yescharacteristic
45#16810488C2211yescharacteristic
46#16810488C2211yescharacteristic
47#16810486—22no—
48C4×C22⋊Dic53C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51C22×M16×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C53×#428—24no—
55C2×#107×C53C2211yescharacteristic
56C2×#107×C53C2211yescharacteristic
57C2×#106×C53C2211yescharacteristic
58C2×#106×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C2×#11719913—22no—
63C2×#11719913—22no—
64C2×#11719913—22no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133196—22no—
68#13133196—22no—
69#13133191—22no—
70#13133191—22no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—22no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53D412yes—
77C2×#11719921—21no—
78C2×#11719921—22no—
79C2×#11719921—22no—
80C2×C4×Dic53C2311yescharacteristic
81C2×C4×Dic53C2311yescharacteristic
82#13133151—22no—
83#13133150—22no—
84#13133148—22no—
85#13133148—22no—
86#13133148—22no—
87#13133142—22no—
88#13133140—22no—
89#13133140—22no—
90#13133139—22no—
91#13133137—22no—
92#13133137—22no—
93C22⋊#88322D414yes—
94#13133127—22no—
95C8×Dic53—22no—
96#13133124—22no—
97#13133124—22no—
98#13133124—22no—
99#13133124—22no—
100#13133124—22no—
101#13133124D412yes—
102C23×C4×C53—21no—
103C23×C4×C53C2311yescharacteristic
104C2×M16×C53—22no—
105C2×M16×C53—22no—
106C2×M16×C53D414yes—
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109#117×C53—22no—
110#117×C53—22no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—22no—
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C42×C53—21no—
120#107×C53—22no—
121#107×C53—22no—
122#106×C53—22no—
123#106×C53—22no—
124C24×C53C2×D411yescharacteristic
125C23×D53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—44no—
135C22×C4×C53—42no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D412yes—
139C22×C4×C53C2411yescharacteristic
140C2×C22⋊D53—42no—
141C2×C22⋊D53—82no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53C2×D412yes—
146C22×Dic53C2×D412yes—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159M16×C53—28no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53C2×D412yes—
163C2×C8×C53C2×D412yes—
164C2×C8×C53#4112yes—
165C2×C8×C53#4112yes—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—42no—
175C22⋊C4×C53—42no—
176C42×C53—42no—
177C22⋊Dic53—42no—
178C22⋊Dic53—42no—
179#11719921—44no—
180#11719921—42no—
181#11719921—82no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186#11719919—42no—
187#11719919—82no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322C2×D412yes—
197C2×#88322C2×D412yes—
198#11719913—44no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53#12712yes—
204C23×C53C22×D411yescharacteristic
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—42no—
209C22×D53—42no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—84no—
225C2×C4×C53#12712yes—
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C532−1+411yescharacteristic
231C2×C4×C532−1+411yescharacteristic
232C22⋊D53—84no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246D212—41no—
247D212—41no—
248D212—41no—
249D212—41no—
250Dic106—41no—
251Dic106—41no—
252Dic106—41no—
253Dic106—41no—
254C8×C53—22no—
255C8×C53#12314yes—
256#88322—22no—
257#88322—22no—
258#27946—531nomaximal, Sylow
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53C2×#12711yescharacteristic
265C22×C53#54711yescharacteristic
266C22×C53#54711yescharacteristic
267C22×C53#57512yes—
268C22×C53#57712yes—
269C2×D53—82no—
270C2×D53—82no—
271C2×D53—81no—
272C2×D53—81no—
273C2×D53—81no—
274C4×C53—22no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53C2×#12311yescharacteristic
280C4×C53#53611yescharacteristic
281C4×C53#54711yescharacteristic
282C4×C53#54711yescharacteristic
283Dic53—42no—
284Dic53—42no—
285Dic53—81no—
286Dic53—81no—
287Dic53—81no—
288C2×#481—531no—
289C2×#485—531no—
290C2×#468—531no—
291C2×#428D5311yescharacteristic, complemented
292#2503—531no—
293#2500—531no—
294#2097—531no—
295#2096—531no—
296#2092—531no—
297#2092—531no—
298#1973—531no—
299#1968—531no—
300#1847—531no—
301#1846—531no—
302#1753—531no—
303C2×C53—42no—
304C2×C53#250111yescharacteristic
305C2×C53#315311yescharacteristic
306C2×C53#315411yescharacteristic
307C2×C53#334812yes—
308C2×C53#335112yes—
309D53—161no—
310C22×SD16—531no—
311C22×M16C2×D5311yescharacteristic
312C2×C22⋊Q8—531no—
313C2×#121—531no—
314C2×#117C2×D5311yescharacteristic
315C22×C4⋊C4C2×D5311yescharacteristic
316#485—1064no—
317#481—1064no—
318#468—1064no—
319#428—24no—
320C2×#107—531no—
321C2×#107—531no—
322C2×#107C2×D5311yescharacteristic
323C2×#107C2×D5311yescharacteristic
324C2×#106—531no—
325C2×#106—531no—
326C2×#106—531no—
327C2×#106C2×D5311yescharacteristic
328C2×#106C2×D5311yescharacteristic
329C2×#103—531no—
330C2×#103—531no—
331C2×#103—531no—
332C2×#102—531no—
333C2×#102—531no—
334C2×#102—531no—
335C2×C22⋊C8—531no—
336C2×#97—531no—
337C2×C4×C8—531no—
338#400—1062no—
339#394—1062no—
340#391—531no—
341#391—531no—
342#390—531no—
343#390—531no—
344#388—1062no—
345#384—531no—
346#382—531no—
347#382—531no—
348#382—531no—
349#381—1062no—
350C4×C22⋊C4—531no—
351#336—531no—
352#336—531no—
353C53#2794611yescharacteristic, complemented, Sylow
354C22×Q8—531no—
355C22×D4—531no—
356C22×D4—1061no—
357C23×C4—21no—
358C23×C4C22×D5311yescharacteristic
359C2×SD16—1062no—
360C2×SD16—1062no—
361C2×SD16—1062no—
362C2×M16—22no—
363C2×M16—22no—
364C2×M16C22⋊D5314yes—
365C22×C8—531no—
366C22×C8—531no—
367C22×C8C22×D5311yescharacteristic
368C22×C8C22×D5311yescharacteristic
369C22⋊Q8—1062no—
370C22⋊Q8—1062no—
371#121—1062no—
372#121—1062no—
373#117—22no—
374#117—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—1062no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—532no—
388C2×C22⋊C4—1062no—
389C2×C22⋊C4—1062no—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C42—21no—
393C2×C42—531no—
394C2×C42—531no—
395#107—22no—
396#107—22no—
397#107—1062no—
398#107—1062no—
399#106—22no—
400#106—22no—
401#106—1062no—
402#106—1062no—
403#106—1062no—
404#103—1062no—
405#103—1062no—
406#103—1062no—
407#102—1062no—
408#102—1062no—
409#102—1062no—
410C22⋊C8—534no—
411#97—1062no—
412C4×C8—1062no—
413#95—532no—
414#95—1062no—
415#95—1062no—
416#95—1062no—
417#95—1062no—
418#95—1062no—
419C24—1061no—
420C24D4×D5311yescharacteristic
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×Q8—1061no—
427C2×D4—1061no—
428C2×D4—1061no—
429C2×D4—1061no—
430C2×D4—1061no—
431C2×D4—1061no—
432C2×D4—1061no—
433C2×D4—4242no—
434C2×D4—2122no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—532no—
446C22×C4—532no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4D4×D5311yescharacteristic
451C22×C4D4×D5312yes—
452C22×C4C2×C22⋊D5312yes—
453C22×C4C23×D5311yescharacteristic
454SD16—2124no—
455M16—28no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—532no—
459C2×C8—532no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8#1171994512yes—
465C2×C8#1171994512yes—
466C2×C8C2×C22⋊D5312yes—
467C2×C8C2×C22⋊D5312yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—4242no—
482C4⋊C4—2122no—
483C22⋊C4—42no—
484C22⋊C4—4242no—
485C22⋊C4—2124no—
486C22⋊C4—2122no—
487C22⋊C4—2122no—
488C22⋊C4—2122no—
489C42—42no—
490C42—1061no—
491C42—1061no—
492C42—1061no—
493C42—1061no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—2121no—
499C23—2121no—
500C23—2121no—
501C23—2122no—
502C23—2122no—
503C23#1313321312yes—
504C23C2×D4×D5311yescharacteristic
505Q8—2121no—
506Q8—2121no—
507Q8—2121no—
508Q8—2121no—
509D4—4244no—
510D4—2121no—
511D4—2121no—
512D4—2121no—
513D4—2121no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—84no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4#1313321312yes—
543C2×C4C2×D4×D5311yescharacteristic
544C2×C4C22×C22⋊D5311yescharacteristic
545C2×C4C2×#1171994511yescharacteristic
546C2×C4C2×#1171994511yescharacteristic
547C2×C4#1313326511yescharacteristic
548C2×C4#1313326511yescharacteristic
549C8—22no—
550C8—1062no—
551C8—1062no—
552C8#1313320014yes—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—42no—
557C22—42no—
558C22—4241no—
559C22—4242no—
560C22—4241no—
561C22—4241no—
562C22—4242no—
563C22C2×#1313321311yescharacteristic
564C22#1681127411yescharacteristic
565C22#1681127411yescharacteristic
566C22#1681141912yes—
567C22#1681143212yes—
568C4—22no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—4241no—
574C4—4241no—
575C4—4241no—
576C4—2122no—
577C4—2122no—
578C4C2×#1313320011yescharacteristic
579C4#1681119711yescharacteristic
580C4#1681127411yescharacteristic
581C4#1681127411yescharacteristic
582C2—42no—
583C2—8481no—
584C2#2027918211yescharacteristic
585C2#2028517711yescharacteristic
586C2#2028519411yescharacteristic
587C2#2028723112yes—
588C2#2028727112yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#428C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#428D53
hypocenterC53#27946
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#16810964C2semidirect
4. #20279196C2semidirect
6. #20276260C2semidirect
7. #20276255C2semidirect
8. #20276237C2semidirect
9. #20276233C2semidirect
10. #20275594C2semidirect
11. #20275567C2semidirect
12. #20274880C2semidirect
13. #20274878C2semidirect
14. #20274496C2semidirect
15. C2×C53×#428C2semidirect
291. C2×#428D53semidirect
353. C53#27946semidirect

Maximal subgroups

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

subgroupindex
C2×#168109412
C2×#168109642
C2×#168108692
#202791962
#202791732
#202762602
#202762552
#202762372
#202762332
#202755942
#202755672
#202748802
#202748782
#202744962
C2×C53×#4282
#2794653