Finite Group Databasebeta

← #27469407

Subgroups — #27469407

Order 13568.

880 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810948C211yesmaximal, characteristic
2C2×#16810854C211yesmaximal, characteristic
3C2×#16810659C211yesmaximal, characteristic, complemented
4#20279173C211yesmaximal, characteristic
5#20279169C211yesmaximal, characteristic, complemented
6#20276261C211yesmaximal, characteristic, complemented
7#20276256C211yesmaximal, characteristic, complemented
8#20276245C211yesmaximal, characteristic, complemented
9#20276233C211yesmaximal, characteristic, complemented
10#20275593C211yesmaximal, characteristic, complemented
11#20275593C211yesmaximal, characteristic, complemented
12#20274882C211yesmaximal, characteristic, complemented
13#20274880C211yesmaximal, characteristic, complemented
14#20274494C211yesmaximal, characteristic, complemented
15C2×C53×#485C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133191C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20#16810948—24no—
21#16810854—24no—
22#16810659—24no—
23C2×#13133151C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133148C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133139C2211yescharacteristic
32C2×#13133139C2211yescharacteristic
33C2×#13133137C2211yescharacteristic
34C2×#13133127C2211yescharacteristic
35C2×#13133127C2211yescharacteristic
36#16810551—22no—
37#16810538—22no—
38#16810530C2211yescharacteristic
39#16810530C2211yescharacteristic
40#16810524C2211yescharacteristic
41#16810524C2211yescharacteristic
42#16810514—22no—
43#16810504—22no—
44#16810492C2211yescharacteristic
45#16810492C2211yescharacteristic
46#16810488C2211yescharacteristic
47#16810488C2211yescharacteristic
48C4×C22⋊Dic53C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51C2×C22⋊Q8×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C53×#485—24no—
54C2×#107×C53C2211yescharacteristic
55C2×#106×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C2×#11719924—22no—
62C2×#11719924—22no—
63C2×#11719924—22no—
64C22×#88322—22no—
65C22×#88322—22no—
66C22×#88322D414yes—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133191—22no—
70#13133191—22no—
71#13133176—22no—
72#13133176—22no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—22no—
77C2×#11719919—22no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53D412yes—
80C2×#11719921—21no—
81C2×#11719921—22no—
82C2×#11719921—22no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53C2311yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86#13133151—22no—
87#13133150—22no—
88#13133150—22no—
89#13133149—22no—
90#13133148—22no—
91#13133148—22no—
92#13133148—22no—
93#13133141—22no—
94#13133139—22no—
95#13133139—22no—
96#13133137—22no—
97#13133127—22no—
98#13133127—22no—
99#13133124—22no—
100#13133124—22no—
101#13133124—22no—
102#13133124—22no—
103#13133124D412yes—
104C22×Q8×C53C2311yescharacteristic
105C23×C4×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22⋊Q8×C53—22no—
109C22⋊Q8×C53—22no—
110#117×C53—22no—
111#117×C53—22no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C42×C53—21no—
119#107×C53—22no—
120#106×C53—22no—
121#103×C53—22no—
122#103×C53—22no—
123C22⋊C8×C53D414yes—
124C24×C53C2×D411yescharacteristic
125C23×D53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2411yescharacteristic
144C2×C22⋊D53—42no—
145C2×C22⋊D53—82no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53C2×D412yes—
150C22×Dic53C2×D412yes—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53C2×D412yes—
162C2×C8×C53C2×D412yes—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—42no—
170C22⋊C4×C53—42no—
171C22⋊C4×C53—42no—
172C42×C53—42no—
173C22⋊Dic53—42no—
174C22⋊Dic53—42no—
175#11719924—44no—
176#11719921—44no—
177#11719921—42no—
178#11719921—82no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185#11719919—44no—
186#11719919—82no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—42no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—28no—
195C2×#88322C2×D412yes—
196C2×#88322C2×D412yes—
197C2×#88322#4112yes—
198C2×#88322#4112yes—
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—
210Q8×C53—41no—
211Q8×C53—41no—
212Q8×C53—41no—
213Q8×C53—41no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53#12712yes—
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C532−1+411yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C22⋊D53—84no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249D212—41no—
250D212—41no—
251D212—41no—
252D212—41no—
253C8×C53—22no—
254C8×C53—22no—
255#88322—22no—
256#88322#12314yes—
257#27937—531nomaximal, Sylow
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53C2×#12711yescharacteristic
264C22×C53#54711yescharacteristic
265C22×C53#54711yescharacteristic
266C22×C53#57512yes—
267C22×C53#57712yes—
268C2×D53—82no—
269C2×D53—82no—
270C2×D53—81no—
271C2×D53—81no—
272C2×D53—81no—
273C4×C53—22no—
274C4×C53—81no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53C2×#12311yescharacteristic
279C4×C53#53611yescharacteristic
280C4×C53#54711yescharacteristic
281C4×C53#54711yescharacteristic
282Dic53—42no—
283Dic53—42no—
284Dic53—81no—
285Dic53—81no—
286Dic53—81no—
287C2×#485D5311yescharacteristic, complemented
288C2×#482—531no—
289C2×#465—531no—
290C2×#427—531no—
291#2500—531no—
292#2499—531no—
293#2097—531no—
294#2096—531no—
295#2094—531no—
296#2092—531no—
297#1973—531no—
298#1973—531no—
299#1847—531no—
300#1847—531no—
301#1752—531no—
302C2×C53—42no—
303C2×C53#250111yescharacteristic
304C2×C53#315011yescharacteristic
305C2×C53#315411yescharacteristic
306C2×C53#334812yes—
307C2×C53#335012yes—
308D53—161no—
309C22×SD16—531no—
310C23×C8—531no—
311C2×C22⋊Q8C2×D5311yescharacteristic
312C2×#121—531no—
313C2×#117—531no—
314C2×#117C2×D5311yescharacteristic
315#485—24no—
316#482—1064no—
317#465—1064no—
318#427—1064no—
319C2×#107—531no—
320C2×#107—531no—
321C2×#107—531no—
322C2×#107C2×D5311yescharacteristic
323C2×#106—531no—
324C2×#106—531no—
325C2×#106—531no—
326C2×#106—531no—
327C2×#106C2×D5311yescharacteristic
328C2×#103—531no—
329C2×#103C2×D5311yescharacteristic
330C2×#103C2×D5311yescharacteristic
331C2×#102—531no—
332C2×#102—531no—
333C2×#102—531no—
334C2×C22⋊C8C2×D5311yescharacteristic
335C2×#97—531no—
336C2×#97—531no—
337#397—1062no—
338#394—1062no—
339#391—531no—
340#391—531no—
341#390—531no—
342#390—531no—
343#388—1062no—
344#387—1062no—
345#384—531no—
346#384—531no—
347#382—531no—
348#382—531no—
349C4×C22⋊C4—531no—
350#336—531no—
351#336—531no—
352C53#2793711yescharacteristic, complemented, Sylow
353C22×Q8C22×D5311yescharacteristic
354C22×D4—531no—
355C22×D4—1061no—
356C23×C4C22×D5311yescharacteristic
357C2×SD16—1062no—
358C2×SD16—1062no—
359C2×SD16—1062no—
360C22×C8—531no—
361C22×C8—534no—
362C22×C8—531no—
363C22×C8—1062no—
364C22×C8—1062no—
365C22×C8C22×D5311yescharacteristic
366C22×C8C22×D5311yescharacteristic
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369#121—1062no—
370#121—1062no—
371#117—22no—
372#117—22no—
373#117—1062no—
374#117—1062no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—1062no—
380C2×C4⋊C4—1062no—
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—532no—
387C2×C22⋊C4—1062no—
388C2×C22⋊C4—1062no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C42—21no—
392C2×C42—531no—
393C2×C42—531no—
394C2×C42—1061no—
395#107—22no—
396#107—1062no—
397#107—1062no—
398#107—1062no—
399#106—22no—
400#106—1062no—
401#106—1062no—
402#106—1062no—
403#106—1062no—
404#103—22no—
405#103—22no—
406#103—1062no—
407#102—1062no—
408#102—1062no—
409#102—1062no—
410C22⋊C8C22⋊D5314yes—
411#97—1062no—
412#97—1062no—
413#95—532no—
414#95—1062no—
415#95—1062no—
416#95—1062no—
417#95—1062no—
418C24—1061no—
419C24D4×D5311yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×D4—1061no—
427C2×D4—1061no—
428C2×D4—1061no—
429C2×D4—1061no—
430C2×D4—1061no—
431C2×D4—1061no—
432C2×D4—4242no—
433C2×D4—2122no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—532no—
443C22×C4—532no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4D4×D5311yescharacteristic
448C22×C4D4×D5312yes—
449C22×C4C2×C22⋊D5312yes—
450C22×C4C23×D5311yescharacteristic
451SD16—2124no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—532no—
457C2×C8—532no—
458C2×C8—532no—
459C2×C8—532no—
460C2×C8—1068no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8C2×C22⋊D5312yes—
464C2×C8C2×C22⋊D5312yes—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—4242no—
479C4⋊C4—2124no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—4242no—
483C22⋊C4—2124no—
484C22⋊C4—2122no—
485C22⋊C4—2122no—
486C22⋊C4—2122no—
487C42—42no—
488C42—1061no—
489C42—1061no—
490C42—1061no—
491C42—1061no—
492C42—2122no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23—2121no—
498C23—2121no—
499C23—2121no—
500C23—2122no—
501C23—2122no—
502C23#1313321312yes—
503C23C2×D4×D5311yescharacteristic
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508D4—4244no—
509D4—2121no—
510D4—2121no—
511D4—2121no—
512D4—2121no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4#1313321312yes—
541C2×C4C2×D4×D5311yescharacteristic
542C2×C4C22×C22⋊D5311yescharacteristic
543C2×C4#41×D5311yescharacteristic
544C2×C4#41×D5311yescharacteristic
545C2×C4#1313326411yescharacteristic
546C2×C4#1313326411yescharacteristic
547C8—22no—
548C8—22no—
549C8—534no—
550C8—1062no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—42no—
555C22—42no—
556C22—4242no—
557C22—4241no—
558C22—4242no—
559C22—4241no—
560C22—4241no—
561C22C2×#1313321311yescharacteristic
562C22#1681127111yescharacteristic
563C22#1681127111yescharacteristic
564C22#1681142012yes—
565C22#1681143212yes—
566C4—22no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—4241no—
572C4—4241no—
573C4—4241no—
574C4—2122no—
575C4—2122no—
576C4#123×D5311yescharacteristic
577C4#1681119511yescharacteristic
578C4#1681127211yescharacteristic
579C4#1681127211yescharacteristic
580C2—42no—
581C2—8481no—
582C2#2027918411yescharacteristic
583C2#2028515811yescharacteristic
584C2#2028519411yescharacteristic
585C2#2028723412yes—
586C2#2028725812yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#485C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#485D53
hypocenterC53#27937
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#485

Complements

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

NH ≅ G/Ntype
3. C2×#16810659C2semidirect
5. #20279169C2semidirect
6. #20276261C2semidirect
7. #20276256C2semidirect
8. #20276245C2semidirect
9. #20276233C2semidirect
10. #20275593C2semidirect
11. #20275593C2semidirect
12. #20274882C2semidirect
13. #20274880C2semidirect
14. #20274494C2semidirect
15. C2×C53×#485C2semidirect
287. C2×#485D53semidirect
352. C53#27937semidirect

Maximal subgroups

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

subgroupindex
C2×#168109482
C2×#168108542
C2×#168106592
#202791732
#202791692
#202762612
#202762562
#202762452
#202762332
#202755932
#202748822
#202748802
#202744942
C2×C53×#4852
#2793753