Finite Group Databasebeta

← #27315500

Subgroups — #27315500

Order 13568.

2982 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11719919C211yesmaximal, characteristic, complemented
2C2×#16810537C212yesmaximal, complemented
3C2×#16810503C212yesmaximal, complemented
4C2×#16810503C212yesmaximal, complemented
5#20272965C212yesmaximal, complemented
6#20272958C214yesmaximal, complemented
7#20272938C211yesmaximal, characteristic, complemented
8C22×#121×C53C211yesmaximal, characteristic
9C24×Dic53C2211yescharacteristic
10C24×Dic53C2211yescharacteristic
11C22×#11719919—28no—
12C22×#11719919—24no—
13C22×#11719919C2212yes—
14C22×#11719919C2212yes—
15C22×C22⋊Dic53C2212yes—
16C22×C22⋊Dic53C2214yes—
17C22×C22⋊Dic53C2214yes—
18#16810537—28no—
19#16810503—28no—
20#16810503—28no—
21C2×#13133124C2212yes—
22C2×#13133124C2214yes—
23C2×#13133124C2214yes—
24C2×#13133124C2212yes—
25C23×D4×C53C2212yes—
26C23×D4×C53C2211yescharacteristic
27C24×C4×C53C2211yescharacteristic
28C2×#121×C53—28no—
29C2×#121×C53—24no—
30C22×C4⋊C4×C53C2211yescharacteristic
31C22×C22⋊C4×C53C2212yes—
32C23×Dic53—22no—
33C23×Dic53—22no—
34C23×Dic53—24no—
35C23×Dic53—24no—
36C23×Dic53—22no—
37C23×Dic53—24no—
38C23×Dic53—24no—
39C23×Dic53—42no—
40C23×Dic53D414yescomplemented
41C23×Dic53C2314yes—
42C23×Dic53C2312yes—
43C23×Dic53C2312yes—
44C2×#11719919—232no—
45C2×#11719919—24no—
46C2×#11719919—24no—
47C2×#11719919—24no—
48C2×#11719919—24no—
49C2×#11719919—24no—
50C2×#11719919D418yescomplemented
51C2×C22⋊Dic53—28no—
52C2×C22⋊Dic53—24no—
53C2×C22⋊Dic53—24no—
54C2×C22⋊Dic53—24no—
55C2×C22⋊Dic53—28no—
56C2×C22⋊Dic53—28no—
57C2×C22⋊Dic53—28no—
58C2×C22⋊Dic53—28no—
59C2×C22⋊Dic53—44no—
60#13133124—28no—
61#13133124—24no—
62#13133124—28no—
63#13133124—42no—
64C25×C53C2312yes—
65C25×C53C2311yescharacteristic
66C22×D4×C53—24no—
67C22×D4×C53—24no—
68C22×D4×C53—24no—
69C22×D4×C53—24no—
70C22×D4×C53—24no—
71C22×D4×C53—24no—
72C22×D4×C53—24no—
73C22×D4×C53—22no—
74C22×D4×C53—24no—
75C22×D4×C53—42no—
76C22×D4×C53D418yes—
77C23×C4×C53—24no—
78C23×C4×C53—24no—
79C23×C4×C53—42no—
80C23×C4×C53D414yes—
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2312yes—
83C23×C4×C53C2311yescharacteristic
84#121×C53—232no—
85C2×C4⋊C4×C53—24no—
86C2×C4⋊C4×C53—22no—
87C2×C22⋊C4×C53—24no—
88C2×C22⋊C4×C53—24no—
89C2×C22⋊C4×C53—24no—
90C24×C53—22no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53—22no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—24no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53—24no—
102C24×C53—24no—
103C24×C53—24no—
104C24×C53C2×D414yes—
105C24×C53C2×D414yes—
106C24×C53C2×D412yes—
107C24×C53C2×D412yes—
108C24×C53#4112yes—
109C24×C53#4114yes—
110C24×C53C2411yescharacteristic
111C2×D4×C53—28no—
112C2×D4×C53—216no—
113C2×D4×C53—216no—
114C2×D4×C53—216no—
115C2×D4×C53—416no—
116C2×D4×C53—44no—
117C2×D4×C53—48no—
118C2×D4×C53—48no—
119C2×D4×C53—48no—
120C2×D4×C53—48no—
121C2×D4×C53—44no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—28no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—28no—
135C22×C4×C53—44no—
136C22×C4×C53—416no—
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D414yes—
139C22×C4×C53C2×D414yes—
140C22×C4×C53C2×D412yes—
141C22×Dic53—28no—
142C22×Dic53—216no—
143C22×Dic53—28no—
144C22×Dic53—28no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—216no—
150C22×Dic53—28no—
151C22×Dic53—24no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—24no—
155C22×Dic53—24no—
156C22×Dic53—24no—
157C22×Dic53—24no—
158C22×Dic53—24no—
159C22×Dic53—24no—
160C22×Dic53—24no—
161C22×Dic53—24no—
162C22×Dic53—416no—
163C22×Dic53—44no—
164C22×Dic53—44no—
165C22×Dic53C2×D414yes—
166C22×Dic53C2×D414yes—
167C22×Dic53C2×D414yes—
168C22×Dic53#4114yes—
169C4⋊C4×C53—28no—
170C22⋊C4×C53—216no—
171C22⋊Dic53—416no—
172C22⋊Dic53—48no—
173C22⋊Dic53—416no—
174#11719919—216no—
175#11719919—216no—
176C23×C53—22no—
177C23×C53—24no—
178C23×C53—22no—
179C23×C53—24no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—24no—
183C23×C53—24no—
184C23×C53—28no—
185C23×C53—28no—
186C23×C53—28no—
187C23×C53—28no—
188C23×C53—24no—
189C23×C53—24no—
190C23×C53—24no—
191C23×C53—28no—
192C23×C53—28no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—44no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—44no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—44no—
204C23×C53—44no—
205C23×C53—44no—
206C23×C53—44no—
207C23×C53—44no—
208C23×C53—44no—
209C23×C53—44no—
210C23×C53—44no—
211C23×C53#12014yes—
212C23×C53#12114yes—
213C23×C53#12118yes—
214C23×C53#12318yes—
215C23×C53#12414yes—
216C23×C53C22×D412yes—
217C23×C53C22×D411yescharacteristic
218C23×C53C22×D411yescharacteristic
219C23×C53C22×D412yes—
220C23×C53C2×#4112yes—
221C23×C53C2×#4111yescharacteristic
222C23×C53C2×#4112yes—
223C23×C53H(5,𝔽2)12yes—
224C23×C532−1+412yes—
225D4×C53—416no—
226D4×C53—432no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—28no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—22no—
236C2×C4×C53—416no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—44no—
242C2×C4×C53#12014yes—
243C2×Dic53—232no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—28no—
252C2×Dic53—416no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—44no—
256C2×Dic53—44no—
257C2×Dic53—44no—
258C2×Dic53—44no—
259C2×Dic53—44no—
260C2×Dic53—44no—
261C2×Dic53—44no—
262C2×Dic53—44no—
263C2×Dic53#12118yescomplemented
264#14719—531nomaximal, Sylow
265C22×C53—24no—
266C22×C53—24no—
267C22×C53—24no—
268C22×C53—24no—
269C22×C53—24no—
270C22×C53—216no—
271C22×C53—24no—
272C22×C53—216no—
273C22×C53—44no—
274C22×C53—44no—
275C22×C53—44no—
276C22×C53—48no—
277C22×C53—48no—
278C22×C53—82no—
279C22×C53—82no—
280C22×C53—82no—
281C22×C53—84no—
282C22×C53—82no—
283C22×C53—84no—
284C22×C53#39418yescomplemented
285C22×C53C2×#12011yescharacteristic
286C22×C53C2×#12011yescharacteristic
287C22×C53C2×#12112yes—
288C22×C53C2×#12111yescharacteristic
289C22×C53C2×#12112yes—
290C22×C53C2×#12312yes—
291C22×C53#52511yescharacteristic
292C22×C53#52512yes—
293C22×C53C2×#12411yescharacteristic
294C22×C53#53411yescharacteristic
295C22×C53#53611yescharacteristic
296C22×C53#53712yes—
297C22×C53#53912yes—
298C22×C53D4214yes—
299C22×C53#54714yes—
300C22×C53#54814yes—
301C22×C53#55314yes—
302C4×C53—24no—
303C4×C53—44no—
304C4×C53—42no—
305Dic53—28no—
306Dic53—44no—
307Dic53—84no—
308C22×#121D5311yescharacteristic
309C23×C4⋊C4—531no—
310C2×#394—532no—
311C2×#386—532no—
312C2×#386—532no—
313#1441—532no—
314#1439—534no—
315C22⋊#95—531no—
316C2×C53—28no—
317C2×C53—84no—
318C2×C53C2×#39411yescharacteristic
319C2×C53#240311yescharacteristic
320C2×C53#240611yescharacteristic
321C2×C53#241614yes—
322C2×C53#243611yescharacteristic
323C2×C53#244314yes—
324C2×C53#259111yescharacteristic
325C2×C53#261811yescharacteristic
326C2×C53#284711yescharacteristic
327C23×D4C2×D5312yes—
328C23×D4C2×D5311yescharacteristic
329C24×C4—531no—
330C24×C4—531no—
331C24×C4C2×D5311yescharacteristic
332C2×#121—28no—
333C2×#121—24no—
334C22×C4⋊C4—532no—
335C22×C4⋊C4—532no—
336C22×C4⋊C4—1068no—
337C22×C4⋊C4—1064no—
338C22×C4⋊C4C2×D5311yescharacteristic
339C22×C22⋊C4—534no—
340C22×C22⋊C4—532no—
341C22×C22⋊C4—534no—
342C22×C22⋊C4C2×D5312yes—
343#394—1068no—
344#386—1068no—
345#386—1068no—
346C2×#95—532no—
347C2×#95—534no—
348C2×#95—534no—
349C2×#95—532no—
350C53#1471911yescharacteristic, complemented, Sylow
351C25C22×D5312yes—
352C25C22×D5311yescharacteristic
353C22×D4—24no—
354C22×D4—24no—
355C22×D4—24no—
356C22×D4—24no—
357C22×D4—24no—
358C22×D4—24no—
359C22×D4—24no—
360C22×D4—24no—
361C22×D4—22no—
362C22×D4—42no—
363C22×D4C22⋊D5318yes—
364C23×C4—24no—
365C23×C4—24no—
366C23×C4—42no—
367C23×C4—532no—
368C23×C4—534no—
369C23×C4—532no—
370C23×C4—534no—
371C23×C4—1064no—
372C23×C4—1064no—
373C23×C4—1064no—
374C23×C4—1062no—
375C23×C4—1062no—
376C23×C4—1062no—
377C23×C4—1064no—
378C23×C4—2122no—
379C23×C4C22⋊D5314yes—
380C23×C4C22×D5311yescharacteristic
381C23×C4C22×D5312yes—
382C23×C4C22×D5311yescharacteristic
383#121—232no—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—538no—
387C2×C4⋊C4—1064no—
388C2×C4⋊C4—1064no—
389C2×C4⋊C4—10632no—
390C2×C4⋊C4—1064no—
391C2×C4⋊C4—1064no—
392C2×C4⋊C4—1064no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—1068no—
397C2×C22⋊C4—1068no—
398C2×C22⋊C4—1068no—
399C2×C22⋊C4—1068no—
400C2×C22⋊C4—1064no—
401C2×C22⋊C4—1064no—
402C2×C22⋊C4—1068no—
403C2×C22⋊C4—1064no—
404C2×C22⋊C4—2124no—
405#95—1068no—
406#95—1064no—
407#95—1068no—
408#95—2122no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—24no—
415C24—22no—
416C24—22no—
417C24—24no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—24no—
422C24—24no—
423C24D4×D5312yes—
424C24#1171994712yes—
425C24#1171994714yes—
426C24C2×C22⋊D5314yes—
427C24C2×C22⋊D5312yes—
428C24C2×C22⋊D5314yes—
429C24C23×D5311yescharacteristic
430C2×D4—28no—
431C2×D4—216no—
432C2×D4—216no—
433C2×D4—216no—
434C2×D4—416no—
435C2×D4—44no—
436C2×D4—48no—
437C2×D4—48no—
438C2×D4—48no—
439C2×D4—48no—
440C2×D4—44no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—44no—
455C22×C4—416no—
456C22×C4—534no—
457C22×C4—534no—
458C22×C4—534no—
459C22×C4—534no—
460C22×C4—10616no—
461C22×C4—1068no—
462C22×C4—1068no—
463C22×C4—10616no—
464C22×C4—1068no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1068no—
470C22×C4—1062no—
471C22×C4—1064no—
472C22×C4—1062no—
473C22×C4—1064no—
474C22×C4—1064no—
475C22×C4—1064no—
476C22×C4—1064no—
477C22×C4—1064no—
478C22×C4—1064no—
479C22×C4—1064no—
480C22×C4—1064no—
481C22×C4—21216no—
482C22×C4—2124no—
483C22×C4—2124no—
484C22×C4D4×D5314yes—
485C22×C4C2×C22⋊D5314yes—
486C22×C4C2×C22⋊D5312yes—
487C22×C4C2×C22⋊D5312yes—
488C4⋊C4—28no—
489C4⋊C4—10616no—
490C4⋊C4—10616no—
491C22⋊C4—216no—
492C22⋊C4—21216no—
493C22⋊C4—21216no—
494C22⋊C4—2128no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—28no—
507C23—28no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23#1313318614yes—
531C23#1313318814yes—
532C23#1313318918yes—
533C23#1313319818yes—
534C23#1313320414yes—
535C23C2×D4×D5312yes—
536C23C2×D4×D5311yescharacteristic
537C23C22×C22⋊D5312yes—
538C23C22×C22⋊D5311yescharacteristic
539C23#41×D5312yes—
540C23C2×#1171994712yes—
541C23C2×#1171994711yescharacteristic
542C23#1313326212yes—
543C23#1313326412yes—
544D4—416no—
545D4—432no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—416no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—538no—
562C2×C4—10632no—
563C2×C4—1064no—
564C2×C4—1064no—
565C2×C4—1064no—
566C2×C4—1064no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—1068no—
571C2×C4—2124no—
572C2×C4—2124no—
573C2×C4—2124no—
574C2×C4—2124no—
575C2×C4—2124no—
576C2×C4—2124no—
577C2×C4—2124no—
578C2×C4—21216no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2124no—
582C2×C4#1313318514yes—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—216no—
589C22—24no—
590C22—216no—
591C22—44no—
592C22—44no—
593C22—48no—
594C22—44no—
595C22—48no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—84no—
600C22—82no—
601C22—84no—
602C22#1681053718yescomplemented
603C22C2×#1313318511yescharacteristic
604C22C2×#1313318611yescharacteristic
605C22#121×D5312yes—
606C22C2×#1313318811yescharacteristic
607C22C2×#1313318912yes—
608C22C2×#1313319812yes—
609C22#1681113011yescharacteristic
610C22#1681113312yes—
611C22C2×#1313320411yescharacteristic
612C22#1681118211yescharacteristic
613C22#1681119411yescharacteristic
614C22#1681120012yes—
615C22#1681121812yes—
616C22D4×C22⋊D5314yes—
617C22#1681127214yes—
618C22#1681127914yes—
619C22#1681131614yes—
620C4—24no—
621C4—42no—
622C4—44no—
623C4—1068no—
624C4—4244no—
625C4—2124no—
626C2—28no—
627C2—84no—
628C2C2×#1681053711yescharacteristic
629C2#2027802511yescharacteristic
630C2#2027804611yescharacteristic
631C2#2027815214yes—
632C2#2027837411yescharacteristic
633C2#2027844614yes—
634C2#2028019611yescharacteristic
635C2#2028045911yescharacteristic
636C2#2028299411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
Fitting, prime radicalC22×#121×C53C2
hypercenterC22×#121D53
hypocenterC53#14719
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. C23×#11719919C2semidirect
2. C2×#16810537C2semidirect
3. C2×#16810503C2semidirect
4. C2×#16810503C2semidirect
5. #20272965C2semidirect
6. #20272958C2semidirect
7. #20272938C2semidirect
40. C23×Dic53D4semidirect
50. C2×#11719919D4semidirect
263. C2×Dic53#121semidirect
284. C22×C53#394semidirect
350. C53#14719semidirect
602. C22#16810537semidirect

Maximal subgroups

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

subgroupindex
C23×#117199192
C2×#168105372
C2×#168105032
#202729652
#202729582
#202729382
C22×#121×C532
#1471953