Finite Group Databasebeta

← #27411584

Subgroups — #27411584

Order 13568.

1902 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133191C211yesmaximal, characteristic
2C2×#16810666C211yesmaximal, characteristic, complemented
3C2×#16810627C211yesmaximal, characteristic, complemented
4C2×#16810627C211yesmaximal, characteristic, complemented
5C2×#16810587C211yesmaximal, characteristic, complemented
6#20275904C214yesmaximal, complemented
7#20275897C214yesmaximal, complemented
8#20274436C211yesmaximal, characteristic, complemented
9C53×#2280C211yesmaximal, characteristic, complemented
10C23×D212C2211yescharacteristic
11C23×C22⋊D53—21no—
12C22×#11719917C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14C2×#13133191—24no—
15C2×#13133191—24no—
16C2×#13133191—22no—
17C2×#13133191C2214yes—
18C22×C4⋊Dic53C2211yescharacteristic
19C22×#11719921—21no—
20#16810666—24no—
21C2×#13133150C2212yescomplemented
22C2×#13133148C2212yescomplemented
23#16810627—24no—
24#16810627—24no—
25C2×#13133137C2212yescomplemented
26C2×#13133137C2214yescomplemented
27C2×#13133137C2212yescomplemented
28#16810587—24no—
29C2×C22⋊#88322C2214yescomplemented
30#16810385C2214yescomplemented
31C24×C4×C53C2211yescharacteristic
32C2×#117×C53C2211yescharacteristic
33C2×#117×C53C2211yescharacteristic
34C53×#381—21no—
35C53×#380C2212yescomplemented
36C53×#380C2212yescomplemented
37C53×#379—21no—
38C4×C22⋊C4×C53—22no—
39C24×D53—21no—
40C22×D212—22no—
41C22×D212—22no—
42C22×D212—22no—
43C22×D212—22no—
44C22×D212—21no—
45C22×D212—22no—
46C22×D212D414yes—
47C22×D212C2312yes—
48C22×C22⋊D53—28no—
49C22×C22⋊D53—24no—
50C22×C22⋊D53—48no—
51C23×Dic53—21no—
52C2×#11719917—22no—
53C2×#11719917—22no—
54C2×#11719917—22no—
55C2×#11719917—22no—
56C2×#11719917—22no—
57C2×#11719917D414yescomplemented
58C22×#88322C2312yes—
59C22×#88322C2312yes—
60#13133191—216no—
61#13133191—28no—
62#13133191—28no—
63C2×C4⋊Dic53—22no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53D414yes—
66C2×C4⋊Dic53C2312yes—
67C2×#11719921—28no—
68C2×#11719921—24no—
69#13133150—24no—
70#13133148—24no—
71#13133137—28no—
72#13133137—24no—
73#13133137—24no—
74C22⋊#88322—28no—
75C25×C53D411yescharacteristic
76C23×C4×C53—22no—
77C23×C4×C53—22no—
78C23×C4×C53—22no—
79C23×C4×C53—22no—
80C23×C4×C53—22no—
81C23×C4×C53—22no—
82C23×C4×C53D411yescharacteristic
83C23×C4×C53D412yes—
84C23×C4×C53C2312yes—
85C23×C4×C53C2311yescharacteristic
86#117×C53—22no—
87#117×C53—22no—
88#117×C53—22no—
89#117×C53—22no—
90C2×C4⋊C4×C53C2312yes—
91C2×C4⋊C4×C53C2311yescharacteristic
92C2×C4⋊C4×C53C2311yescharacteristic
93C2×C22⋊C4×C53—21no—
94C2×C22⋊C4×C53—21no—
95C2×C22⋊C4×C53—22no—
96C2×C22⋊C4×C53—22no—
97C2×C42×C53—21no—
98C2×C42×C53—21no—
99#95×C53—22no—
100#95×C53—22no—
101C24×C53—22no—
102C24×C53—24no—
103C24×C53—22no—
104C24×C53—22no—
105C24×C53—42no—
106C24×C53C2×D412yes—
107C24×C53C2×D411yescharacteristic
108C23×D53—24no—
109C23×D53—22no—
110C23×D53—44no—
111C23×D53—44no—
112C23×D53—42no—
113C23×D53—42no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—24no—
118C22×C4×C53—24no—
119C22×C4×C53—24no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—22no—
129C22×C4×C53—44no—
130C22×C4×C53—44no—
131C22×C4×C53—44no—
132C22×C4×C53—44no—
133C22×C4×C53C2×D412yes—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D412yes—
138C22×C4×C53#4112yes—
139C22×C4×C53#4112yes—
140C22×C4×C53#4112yes—
141C22×C4×C53C2411yescharacteristic
142C2×C22⋊D53—216no—
143C2×C22⋊D53—432no—
144C2×C22⋊D53—48no—
145C2×C22⋊D53—48no—
146C22×Dic53—24no—
147C22×Dic53—22no—
148C22×Dic53—44no—
149C2×D212—28no—
150C2×D212—28no—
151C2×D212—22no—
152C2×D212—22no—
153C2×D212—22no—
154C2×D212—22no—
155C2×D212—24no—
156C2×D212—22no—
157C2×D212—44no—
158C2×D212—42no—
159C2×D212—42no—
160C2×D212—44no—
161C2×D212C2×D414yes—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—22no—
168C22⋊C4×C53—44no—
169C22⋊C4×C53—42no—
170C22⋊C4×C53—42no—
171C22⋊C4×C53—44no—
172C42×C53—42no—
173C42×C53—42no—
174#11719921—216no—
175C4⋊Dic53—24no—
176C4⋊Dic53—22no—
177C4⋊Dic53C2×D414yes—
178#11719917—28no—
179#11719917—44no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322C2×D414yescomplemented
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—28no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—48no—
196C23×C53—48no—
197C23×C53—44no—
198C23×C53—44no—
199C23×C53#12012yes—
200C23×C53#12112yes—
201C23×C53#12312yes—
202C23×C53AGL(1,ℤ/8)14yes—
203C23×C53C22×D411yescharacteristic
204C22×D53—24no—
205C22×D53—48no—
206C22×D53—48no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—44no—
210C22×D53—44no—
211C22×D53—48no—
212C22×D53—88no—
213C22×D53—88no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—44no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—48no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53#12012yes—
231C2×C4×C53#12112yes—
232C2×C4×C53#12114yes—
233C2×C4×C53#12114yes—
234C2×C4×C53#12314yes—
235C2×C4×C53#12312yes—
236C2×C4×C53#12414yes—
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C22⋊D53—432no—
244C2×Dic53—24no—
245C2×Dic53—48no—
246C2×Dic53—44no—
247D212—24no—
248D212—24no—
249D212—42no—
250D212—42no—
251D212—48no—
252#88322—24no—
253#88322—42no—
254#23480—531nomaximal, Sylow
255C22×C53—24no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—24no—
259C22×C53—24no—
260C22×C53—24no—
261C22×C53—24no—
262C22×C53—416no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53#39414yes—
268C22×C53C2×#12011yescharacteristic
269C22×C53C2×#12111yescharacteristic
270C22×C53C2×#12311yescharacteristic
271C22×C53C2×AGL(1,ℤ/8)12yes—
272C22×C53#57512yes—
273C2×D53—44no—
274C2×D53—44no—
275C2×D53—44no—
276C2×D53—816no—
277C2×D53—84no—
278C2×D53—84no—
279C4×C53—42no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53—82no—
283C4×C53#39414yes—
284C4×C53C2×#12111yescharacteristic
285C4×C53C2×#12111yescharacteristic
286C4×C53C2×#12311yescharacteristic
287C4×C53C2×#12411yescharacteristic
288Dic53—44no—
289C22×#121—531no—
290C2×#428—531no—
291C2×#418—531no—
292C2×#418—531no—
293C2×#407—531no—
294#2280D5311yescharacteristic, complemented
295#2034—534no—
296#2033—534no—
297#1736—531no—
298C2×C53—24no—
299C2×C53—44no—
300C2×C53C2×#39411yescharacteristic
301C2×C53#300311yescharacteristic
302C2×C53#300511yescharacteristic
303C2×C53#305512yes—
304C2×C53#309412yes—
305D53—84no—
306C23×D4—531no—
307C23×D4—1061no—
308C24×C4C2×D5311yescharacteristic
309C22×M16—531no—
310C22×M16—531no—
311C2×#121—534no—
312C2×#121—1064no—
313C2×#121—1064no—
314C2×#121—1062no—
315C2×#117C2×D5311yescharacteristic
316C2×#117C2×D5311yescharacteristic
317C22×C4⋊C4—531no—
318C22×C22⋊C4—1061no—
319#428—1064no—
320C2×#107—532no—
321C2×#106—532no—
322#418—1064no—
323#418—1064no—
324C2×#102—532no—
325C2×#102—534no—
326C2×#102—532no—
327#407—1064no—
328C2×C22⋊C8—534no—
329#381—21no—
330#380C2×D5312yescomplemented
331#380C2×D5312yescomplemented
332#379—21no—
333C4×C22⋊C4—22no—
334#340—534no—
335C53#2348011yescharacteristic, complemented, Sylow
336C25—1061no—
337C25C22⋊D5311yescharacteristic
338C22×D4—534no—
339C22×D4—532no—
340C22×D4—1068no—
341C22×D4—1064no—
342C22×D4—1062no—
343C22×D4—1062no—
344C22×D4—1062no—
345C22×D4—1062no—
346C22×D4—1062no—
347C22×D4—1061no—
348C22×D4—2128no—
349C23×C4—22no—
350C23×C4—22no—
351C23×C4—22no—
352C23×C4—22no—
353C23×C4—22no—
354C23×C4—22no—
355C23×C4—1061no—
356C23×C4C22⋊D5311yescharacteristic
357C23×C4C22⋊D5312yes—
358C23×C4C22×D5312yes—
359C23×C4C22×D5311yescharacteristic
360C2×M16—534no—
361C2×M16—1062no—
362C2×M16—1062no—
363C2×M16—1062no—
364C2×M16—1062no—
365C2×M16—1062no—
366C22×C8—532no—
367C22×C8—532no—
368#121—1068no—
369#121—1068no—
370#121—10616no—
371#117—22no—
372#117—22no—
373#117—22no—
374#117—22no—
375C2×C4⋊C4—534no—
376C2×C4⋊C4—532no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4C22×D5312yes—
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—1068no—
387C2×C22⋊C4—1064no—
388C2×C42—21no—
389C2×C42—21no—
390#107—1064no—
391#106—1064no—
392#102—1068no—
393#102—1064no—
394#102—1064no—
395C22⋊C8—1068no—
396#95—22no—
397#95—22no—
398C24—22no—
399C24—24no—
400C24—22no—
401C24—22no—
402C24—42no—
403C24—1064no—
404C24—1062no—
405C24—2124no—
406C24—2124no—
407C24—2122no—
408C24—2122no—
409C24C2×C22⋊D5312yes—
410C24C2×C22⋊D5311yescharacteristic
411C2×D4—534no—
412C2×D4—10616no—
413C2×D4—1064no—
414C2×D4—1068no—
415C2×D4—1068no—
416C2×D4—1062no—
417C2×D4—1062no—
418C2×D4—1062no—
419C2×D4—1062no—
420C2×D4—1062no—
421C2×D4—21232no—
422C2×D4—2128no—
423C2×D4—2128no—
424C2×D4—2122no—
425C2×D4—2124no—
426C2×D4—2122no—
427C2×D4—2124no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—1064no—
448C22×C4—1062no—
449C22×C4—2124no—
450C22×C4#1171994512yes—
451C22×C4#1171994512yes—
452C22×C4D4×D5312yes—
453C22×C4D4×D5312yes—
454C22×C4D4×D5312yes—
455C22×C4#1171994712yes—
456C22×C4C2×C22⋊D5312yes—
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C23×D5311yescharacteristic
459M16—1068no—
460M16—2124no—
461C2×C8—534no—
462C2×C8—1062no—
463C2×C8—1062no—
464C2×C8—1062no—
465C2×C8—1062no—
466C2×C8—1062no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—534no—
474C4⋊C4—1064no—
475C4⋊C4—1062no—
476C22⋊C4—44no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—44no—
480C22⋊C4—10616no—
481C42—42no—
482C42—42no—
483C23—22no—
484C23—22no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—48no—
493C23—48no—
494C23—44no—
495C23—44no—
496C23—1064no—
497C23—4248no—
498C23—4248no—
499C23—2128no—
500C23—2128no—
501C23—2128no—
502C23—2122no—
503C23—2122no—
504C23—2124no—
505C23—2124no—
506C23#1313318512yes—
507C23#1313318912yes—
508C23#1313320012yes—
509C23#1313324214yes—
510C23C22×C22⋊D5311yescharacteristic
511D4—1064no—
512D4—1064no—
513D4—21232no—
514D4—2128no—
515D4—2122no—
516D4—2122no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—1064no—
534C2×C4—2128no—
535C2×C4—2124no—
536C2×C4#1313318512yes—
537C2×C4#1313318912yes—
538C2×C4#1313319014yes—
539C2×C4#1313319014yes—
540C2×C4#1313319914yes—
541C2×C4#1313320012yes—
542C2×C4#1313320314yes—
543C2×C4C2×D4×D5311yescharacteristic
544C2×C4C2×D4×D5311yescharacteristic
545C2×C4C2×D4×D5311yescharacteristic
546C2×C4C2×#1171994711yescharacteristic
547C2×C4C2×#1171994511yescharacteristic
548C2×C4C2×#1171994511yescharacteristic
549C8—1064no—
550C8—2122no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—416no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42416no—
564C22—4244no—
565C22—4244no—
566C22—2124no—
567C22—2124no—
568C22—2124no—
569C22#1681053814yes—
570C22C2×#1313318511yescharacteristic
571C22C2×#1313318911yescharacteristic
572C22C2×#1313320011yescharacteristic
573C22C2×#1313324212yes—
574C22#1681142112yes—
575C4—42no—
576C4—81no—
577C4—81no—
578C4—82no—
579C4—2124no—
580C4#1681053814yes—
581C4C2×#1313319011yescharacteristic
582C4C2×#1313319011yescharacteristic
583C4C2×#1313319911yescharacteristic
584C4C2×#1313320311yescharacteristic
585C2—24no—
586C2—44no—
587C2—4244no—
588C2C2×#1681053811yescharacteristic
589C2#2028394111yescharacteristic
590C2#2028395511yescharacteristic
591C2#2028433612yes—
592C2#2028465512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2280C2
socleC23×C53C22×D4
prime radicalC22×#13133191C2
hypercenter#2280D53
hypocenterC53#23480
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. C2×#16810666C2semidirect
3. C2×#16810627C2semidirect
4. C2×#16810627C2semidirect
5. C2×#16810587C2semidirect
6. #20275904C2semidirect
7. #20275897C2semidirect
8. #20274436C2semidirect
9. C53×#2280C2semidirect
21. C2×#13133150C22semidirect
22. C2×#13133148C22semidirect
25. C2×#13133137C22semidirect
26. C2×#13133137C22semidirect
27. C2×#13133137C22semidirect
29. C2×C22⋊#88322C22semidirect
30. #16810385C22semidirect
35. C53×#380C22semidirect
36. C53×#380C22semidirect
57. C2×#11719917D4semidirect
185. C2×#88322C2×D4semidirect
294. #2280D53semidirect
330. #380C2×D53semidirect
331. #380C2×D53semidirect
335. C53#23480semidirect

Maximal subgroups

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

subgroupindex
C22×#131331912
C2×#168106662
C2×#168106272
C2×#168105872
#202759042
#202758972
#202744362
C53×#22802
#2348053