Finite Group Databasebeta

← #27385337

Subgroups — #27385337

Order 13568.

2278 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133188C211yesmaximal, characteristic
2C2×#117×D53C211yesmaximal, characteristic, complemented
3C2×#16810628C211yesmaximal, characteristic, complemented
4C2×#16810586C211yesmaximal, characteristic, complemented
5#20275034C214yesmaximal, complemented
6#20275029C214yesmaximal, complemented
7#20274436C211yesmaximal, characteristic, complemented
8#20274435C211yesmaximal, characteristic, complemented
9C2×C53×#418C211yesmaximal, characteristic, complemented
10C23×C22⋊D53—21no—
11C23×C4×D53C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13C2×#13133188—24no—
14C2×#13133188—21no—
15C2×#13133188—21no—
16C2×#13133188C418yescomplemented
17C2×#13133188C2212yes—
18C2×#13133188C2212yes—
19C2×#13133176C2211yescharacteristic
20C2×#13133175C2212yescomplemented
21C2×#13133174—21no—
22C2×#13133173—21no—
23#117×D53—24no—
24#117×D53—24no—
25C22×C4⋊Dic53C2211yescharacteristic
26C2×C4⋊C4×D53C2212yescomplemented
27C22×C22⋊Dic53—21no—
28C2×C22⋊C4×D53—21no—
29C2×C42×D53—21no—
30#16810628—24no—
31C2×#13133138C2214yescomplemented
32#16810586—24no—
33C2×#13133129C2212yescomplemented
34C2×#13133129C2212yescomplemented
35#16810386C2214yescomplemented
36#16810385C2214yescomplemented
37C23×D4×C53C2211yescharacteristic
38C22×M16×C53C2211yescharacteristic
39C2×#117×C53C2211yescharacteristic
40C53×#418—24no—
41C2×#102×C53C2214yescomplemented
42C24×D53D411yescharacteristic
43C22×C22⋊D53—28no—
44C22×C22⋊D53—22no—
45C22×C22⋊D53—22no—
46C22×C22⋊D53—48no—
47C23×Dic53—21no—
48C23×Dic53D411yescharacteristic
49C22×C4×D53—22no—
50C22×C4×D53—22no—
51C22×C4×D53—22no—
52C22×C4×D53—22no—
53C22×C4×D53—22no—
54C22×C4×D53C2×C414yes—
55C22×C4×D53D411yescharacteristic
56C22×C4×D53D411yescharacteristic
57C22×C4×D53D414yes—
58C22×C4×D53C2311yescharacteristic
59C22×C4×D53C2311yescharacteristic
60C2×#11719917—22no—
61C2×#11719917—22no—
62C2×#11719917—22no—
63C22×#88322C2312yes—
64#13133188—28no—
65#13133188—28no—
66#13133188—24no—
67#13133188—24no—
68#13133188C2×C418yes—
69#13133188C2×C418yescomplemented
70#13133176—22no—
71#13133176D414yescomplemented
72#13133175—24no—
73#13133175—24no—
74#13133174—28no—
75#13133173—28no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53C2×C414yescomplemented
79C2×C4⋊Dic53C2312yes—
80C2×C4⋊Dic53C2312yes—
81C2×#11719919—22no—
82C4⋊C4×D53—24no—
83C4⋊C4×D53—24no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—28no—
86C2×C22⋊Dic53—22no—
87C2×C22⋊Dic53—22no—
88C2×#11719921—22no—
89C22⋊C4×D53—28no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—22no—
92C42×D53—28no—
93#13133138—28no—
94#13133129—24no—
95#13133129—24no—
96C25×C53—21no—
97C22×D4×C53—22no—
98C22×D4×C53—22no—
99C22×D4×C53—21no—
100C22×D4×C53—22no—
101C22×D4×C53—22no—
102C22×D4×C53—22no—
103C22×D4×C53C2×C414yescomplemented
104C22×D4×C53C2312yes—
105C23×C4×C53C2311yescharacteristic
106C2×M16×C53—22no—
107C2×M16×C53—22no—
108C2×M16×C53—22no—
109C22×C8×C53C2312yes—
110#117×C53—22no—
111#117×C53D414yescomplemented
112C2×C4⋊C4×C53C2312yes—
113C2×C22⋊C4×C53—21no—
114C2×C42×C53—21no—
115#102×C53—28no—
116C24×C53—24no—
117C24×C53—22no—
118C24×C53—44no—
119C24×C53—44no—
120C24×C53—42no—
121C24×C53—42no—
122C24×C53C2×D411yescharacteristic
123C23×D53—22no—
124C23×D53—22no—
125C23×D53—24no—
126C23×D53—42no—
127C23×D53C22⋊C414yes—
128C23×D53C2×D411yescharacteristic
129C23×D53C2×D411yescharacteristic
130C2×D4×C53—24no—
131C2×D4×C53—28no—
132C2×D4×C53—28no—
133C2×D4×C53—22no—
134C2×D4×C53—22no—
135C2×D4×C53—22no—
136C2×D4×C53—22no—
137C2×D4×C53—22no—
138C2×D4×C53—42no—
139C2×D4×C53—44no—
140C2×D4×C53—44no—
141C2×D4×C53—42no—
142C2×D4×C53C22×C414yes—
143C22×C4×C53—22no—
144C22×C4×C53C22×C412yes—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D412yes—
147C22×C4×C53C2×D412yes—
148C22×C4×C53C2×D412yes—
149C22×C4×C53#4112yes—
150C22×C4×C53#4112yes—
151C22×C4×C53C2411yescharacteristic
152C2×C22⋊D53—28no—
153C2×C22⋊D53—28no—
154C2×C22⋊D53—432no—
155C2×C22⋊D53—48no—
156C2×C22⋊D53—48no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—24no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—44no—
163C22×Dic53C22⋊C414yes—
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166C2×C4×D53—28no—
167C2×C4×D53—28no—
168C2×C4×D53—22no—
169C2×C4×D53—22no—
170C2×C4×D53—28no—
171C2×C4×D53—28no—
172C2×C4×D53—22no—
173C2×C4×D53—22no—
174C2×C4×D53—21no—
175C2×C4×D53—21no—
176C2×C4×D53—21no—
177C2×C4×D53—21no—
178C2×C4×D53—21no—
179C2×C4×D53—21no—
180C2×C4×D53—21no—
181C2×C4×D53—21no—
182C2×C4×D53—28no—
183C2×C4×D53—44no—
184C2×C4×D53—44no—
185C2×C4×D53C22⋊C414yes—
186C2×C4×D53C22⋊C414yes—
187C2×C4×D53C22×C412yes—
188C2×C4×D53C22×C412yes—
189C2×C4×D53C2×D412yes—
190C2×C4×D53C2×D412yes—
191M16×C53—44no—
192C2×C8×C53—22no—
193C2×C8×C53—22no—
194C2×C8×C53—22no—
195C4⋊C4×C53—22no—
196C4⋊C4×C53C2×D414yescomplemented
197C22⋊C4×C53—24no—
198C42×C53—24no—
199C22⋊Dic53—24no—
200C22⋊Dic53—28no—
201C22⋊Dic53—28no—
202#11719921—28no—
203C4⋊Dic53—24no—
204C4⋊Dic53—22no—
205C4⋊Dic53—22no—
206C4⋊Dic53C22×C414yes—
207C4⋊Dic53C2×D414yescomplemented
208#11719919—28no—
209C4×Dic53—28no—
210C4×Dic53—24no—
211#11719917—44no—
212C2×#88322—22no—
213C2×#88322—22no—
214C2×#88322—22no—
215C23×C53—22no—
216C23×C53—22no—
217C23×C53—24no—
218C23×C53—48no—
219C23×C53—48no—
220C23×C53—42no—
221C23×C53—42no—
222C23×C53—44no—
223C23×C53—44no—
224C23×C53—48no—
225C23×C53—88no—
226C23×C53—88no—
227C23×C53C2×C22⋊C412yes—
228C23×C53#12012yes—
229C23×C53#12312yes—
230C23×C53C22×D411yescharacteristic
231C22×D53—21no—
232C22×D53—21no—
233C22×D53—22no—
234C22×D53—21no—
235C22×D53—21no—
236C22×D53—22no—
237C22×D53—22no—
238C22×D53—28no—
239C22×D53—28no—
240C22×D53—44no—
241C22×D53—48no—
242C22×D53—48no—
243C22×D53C2×C22⋊C412yes—
244C22×D53C2×C22⋊C412yes—
245C22×D53AGL(1,ℤ/8)14yes—
246D4×C53—24no—
247D4×C53—24no—
248D4×C53—42no—
249D4×C53—42no—
250D4×C53—48no—
251C2×C4×C53—24no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53C2×C22⋊C412yes—
255C2×C4×C53C4×D414yes—
256C2×C4×C53C4×D414yes—
257C2×C4×C53#12012yes—
258C2×C4×C53#12114yes—
259C2×C4×C53#12114yes—
260C2×C4×C53#12312yes—
261C2×C4×C53C23×C411yescharacteristic
262C2×C4×C53C22×D411yescharacteristic
263C2×C4×C53C22×D411yescharacteristic
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C22⋊D53—416no—
268C22⋊D53—416no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—28no—
274C2×Dic53—24no—
275C2×Dic53—24no—
276C2×Dic53—48no—
277C2×Dic53—42no—
278C2×Dic53—44no—
279C2×Dic53—42no—
280C2×Dic53C2×C22⋊C412yes—
281C2×Dic53C2×C22⋊C412yes—
282C4×D53—22no—
283C4×D53—22no—
284C4×D53—22no—
285C4×D53—22no—
286C4×D53—22no—
287C4×D53—22no—
288C4×D53—22no—
289C4×D53—22no—
290C4×D53—28no—
291C4×D53—28no—
292C4×D53—416no—
293C8×C53—42no—
294#88322—42no—
295#20912—531nomaximal, Sylow
296C22×C53—24no—
297C22×C53—22no—
298C22×C53—22no—
299C22×C53—24no—
300C22×C53—44no—
301C22×C53—44no—
302C22×C53—44no—
303C22×C53—816no—
304C22×C53—84no—
305C22×C53—84no—
306C22×C53#38614yes—
307C22×C53C22×C22⋊C411yescharacteristic
308C22×C53C2×#12011yescharacteristic
309C22×C53C2×#12311yescharacteristic
310C22×C53C2×AGL(1,ℤ/8)12yes—
311C22×C53#57512yes—
312C2×D53—24no—
313C2×D53—22no—
314C2×D53—24no—
315C2×D53—22no—
316C2×D53—22no—
317C2×D53—22no—
318C2×D53—22no—
319C2×D53—416no—
320C2×D53—42no—
321C2×D53—42no—
322C2×D53#41814yescomplemented
323C4×C53—44no—
324C4×C53#38614yes—
325C4×C53C2×C4×D411yescharacteristic
326C4×C53C2×C4×D411yescharacteristic
327C4×C53C2×#12111yescharacteristic
328C4×C53C2×#12111yescharacteristic
329Dic53—22no—
330Dic53—22no—
331Dic53—44no—
332Dic53—44no—
333C22×#121—531no—
334C22×#117—531no—
335C2×#418—531no—
336C2×#418D5311yescharacteristic, complemented
337C2×#407—531no—
338#1875—534no—
339#1874—534no—
340#1736—531no—
341#1736—531no—
342C2×C53—24no—
343C2×C53—84no—
344C2×C53C2×#38611yescharacteristic
345C2×C53C2×#41811yescharacteristic
346C2×C53#289611yescharacteristic
347C2×C53#300612yes—
348C2×C53#309412yes—
349D53—24no—
350D53—42no—
351C23×D4—1061no—
352C23×D4C2×D5311yescharacteristic
353C24×C4—531no—
354C22×M16—531no—
355C22×M16C2×D5311yescharacteristic
356C2×#121—538no—
357C2×#121—532no—
358C2×#121—532no—
359C2×#121—1061no—
360C2×#121—1064no—
361C2×#121—1061no—
362C2×#117—531no—
363C2×#117—532no—
364C2×#117—1061no—
365C2×#117—1061no—
366C2×#117—1064no—
367C2×#117—1064no—
368C2×#117C2×D5311yescharacteristic
369C22×C4⋊C4—531no—
370C22×C4⋊C4—532no—
371C22×C22⋊C4—1061no—
372C22×C22⋊C4—1061no—
373C22×C42—1061no—
374#418—24no—
375#418—1064no—
376C2×#102—534no—
377C2×#102C2×D5314yescomplemented
378#407—1064no—
379C2×C22⋊C8—532no—
380C2×C22⋊C8—532no—
381#340—534no—
382#340—534no—
383C53#2091211yescharacteristic, complemented, Sylow
384C25—21no—
385C25—531no—
386C22×D4—22no—
387C22×D4—22no—
388C22×D4—21no—
389C22×D4—22no—
390C22×D4—22no—
391C22×D4—22no—
392C22×D4—1068no—
393C22×D4—1062no—
394C22×D4—1062no—
395C22×D4—2128no—
396C22×D4C4×D5314yescomplemented
397C22×D4C22×D5312yes—
398C23×C4—531no—
399C23×C4—534no—
400C23×C4—531no—
401C23×C4—534no—
402C23×C4—531no—
403C23×C4—531no—
404C23×C4—531no—
405C23×C4—1062no—
406C23×C4—1062no—
407C23×C4—1062no—
408C23×C4—1062no—
409C23×C4—1062no—
410C23×C4—1061no—
411C23×C4C22×D5311yescharacteristic
412C2×M16—22no—
413C2×M16—22no—
414C2×M16—22no—
415C2×M16—1062no—
416C2×M16—1062no—
417C2×M16—1062no—
418C22×C8—532no—
419C22×C8C22×D5312yes—
420#121—538no—
421#121—538no—
422#121—1068no—
423#121—1068no—
424#121—1064no—
425#121—1064no—
426#117—22no—
427#117—534no—
428#117—1068no—
429#117—1068no—
430#117—1064no—
431#117—1062no—
432#117—1064no—
433#117C22⋊D5314yescomplemented
434C2×C4⋊C4—534no—
435C2×C4⋊C4—532no—
436C2×C4⋊C4—532no—
437C2×C4⋊C4—1064no—
438C2×C4⋊C4—1062no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1062no—
441C2×C4⋊C4—1064no—
442C2×C4⋊C4C22×D5312yes—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—1068no—
445C2×C22⋊C4—1062no—
446C2×C22⋊C4—1068no—
447C2×C22⋊C4—1062no—
448C2×C22⋊C4—1062no—
449C2×C22⋊C4—1061no—
450C2×C42—21no—
451C2×C42—1068no—
452C2×C42—1062no—
453C2×C42—1061no—
454#102—28no—
455#102—1068no—
456C22⋊C8—1064no—
457C22⋊C8—1064no—
458C24—24no—
459C24—22no—
460C24—44no—
461C24—44no—
462C24—42no—
463C24—42no—
464C24—531no—
465C24—531no—
466C24—534no—
467C24—1064no—
468C24—1062no—
469C24—1062no—
470C24—2122no—
471C24D4×D5311yescharacteristic
472C2×D4—24no—
473C2×D4—28no—
474C2×D4—28no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—42no—
481C2×D4—44no—
482C2×D4—44no—
483C2×D4—42no—
484C2×D4—1068no—
485C2×D4—1068no—
486C2×D4—2128no—
487C2×D4—21232no—
488C2×D4—2128no—
489C2×D4C2×C4×D5314yes—
490C22×C4—22no—
491C22×C4—534no—
492C22×C4—531no—
493C22×C4—531no—
494C22×C4—534no—
495C22×C4—534no—
496C22×C4—532no—
497C22×C4—532no—
498C22×C4—532no—
499C22×C4—532no—
500C22×C4—1068no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1068no—
504C22×C4—1068no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1068no—
508C22×C4—1068no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—1061no—
514C22×C4—1061no—
515C22×C4—1061no—
516C22×C4—1061no—
517C22×C4—1061no—
518C22×C4—1061no—
519C22×C4—1064no—
520C22×C4—1062no—
521C22×C4—1062no—
522C22×C4—2124no—
523C22×C4—2124no—
524C22×C4—2124no—
525C22×C4C2×C4×D5312yes—
526C22×C4#1171994512yes—
527C22×C4D4×D5311yescharacteristic
528C22×C4D4×D5312yes—
529C22×C4D4×D5312yes—
530C22×C4#1171994712yes—
531C22×C4C2×C22⋊D5312yes—
532C22×C4C23×D5311yescharacteristic
533M16—44no—
534M16—2124no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—1062no—
539C2×C8—1062no—
540C2×C8—1062no—
541C4⋊C4—22no—
542C4⋊C4—534no—
543C4⋊C4—534no—
544C4⋊C4—1068no—
545C4⋊C4—1064no—
546C4⋊C4—1062no—
547C4⋊C4—1062no—
548C4⋊C4C2×C22⋊D5314yescomplemented
549C22⋊C4—24no—
550C22⋊C4—1068no—
551C22⋊C4—1068no—
552C22⋊C4—1068no—
553C22⋊C4—1064no—
554C42—24no—
555C42—1068no—
556C42—1064no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—48no—
561C23—48no—
562C23—48no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—88no—
568C23—88no—
569C23—532no—
570C23—534no—
571C23—532no—
572C23—1068no—
573C23—1061no—
574C23—1061no—
575C23—1062no—
576C23—1068no—
577C23—1061no—
578C23—1061no—
579C23—1062no—
580C23—1062no—
581C23—2124no—
582C23—2128no—
583C23—2128no—
584C23C22⋊C4×D5312yes—
585C23#1313318512yes—
586C23#1313319912yes—
587C23C2×D4×D5311yescharacteristic
588D4—24no—
589D4—24no—
590D4—48no—
591D4—42no—
592D4—42no—
593D4—21216no—
594D4—21216no—
595C2×C4—24no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—532no—
599C2×C4—532no—
600C2×C4—1068no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1068no—
606C2×C4—1068no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—1062no—
615C2×C4—1064no—
616C2×C4—1064no—
617C2×C4—21216no—
618C2×C4—2128no—
619C2×C4—2122no—
620C2×C4—2124no—
621C2×C4—2122no—
622C2×C4C22⋊C4×D5312yes—
623C2×C4#1313317714yes—
624C2×C4C4×C22⋊D5314yes—
625C2×C4#1313318512yes—
626C2×C4#1313318814yes—
627C2×C4#1313319014yes—
628C2×C4#1313319912yes—
629C2×C4C22×C4×D5311yescharacteristic
630C2×C4C2×D4×D5311yescharacteristic
631C2×C4C2×D4×D5311yescharacteristic
632C2×C4C22×C22⋊D5311yescharacteristic
633C2×C4C2×#1171994711yescharacteristic
634C2×C4C2×#1171994511yescharacteristic
635C8—42no—
636C8—2122no—
637C22—24no—
638C22—22no—
639C22—22no—
640C22—24no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—816no—
645C22—84no—
646C22—84no—
647C22—534no—
648C22—1064no—
649C22—1062no—
650C22—1062no—
651C22—1064no—
652C22—1062no—
653C22—1062no—
654C22—1062no—
655C22—21216no—
656C22—2122no—
657C22—2122no—
658C22#1681049914yes—
659C22C2×C22⋊C4×D5311yescharacteristic
660C22C2×#1313318511yescharacteristic
661C22C2×#1313319911yescharacteristic
662C22AGL(1,ℤ/8)×D5312yes—
663C22#1681141912yes—
664C4—44no—
665C4—1062no—
666C4—1062no—
667C4—2124no—
668C4—2124no—
669C4#1681049914yes—
670C4C2×#1313317711yescharacteristic
671C4C2×C4×C22⋊D5311yescharacteristic
672C4C2×#1313318811yescharacteristic
673C4C2×#1313319011yescharacteristic
674C2—24no—
675C2—84no—
676C2—1064no—
677C2—2122no—
678C2C2×#1681049911yescharacteristic
679C2#418×D5311yescharacteristic
680C2#2028322911yescharacteristic
681C2#2028396312yes—
682C2#2028465312yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#418C2
socleC23×C53C22×D4
prime radicalC22×#13133188C2
cosocleC22×C4×C53C24
hypercenterC2×#418D53
hypocenterC53#20912
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×#117×D53C2semidirect
3. C2×#16810628C2semidirect
4. C2×#16810586C2semidirect
5. #20275034C2semidirect
6. #20275029C2semidirect
7. #20274436C2semidirect
8. #20274435C2semidirect
9. C2×C53×#418C2semidirect
16. C2×#13133188C4semidirect
20. C2×#13133175C22semidirect
26. C2×C4⋊C4×D53C22semidirect
31. C2×#13133138C22semidirect
33. C2×#13133129C22semidirect
34. C2×#13133129C22semidirect
35. #16810386C22semidirect
36. #16810385C22semidirect
41. C2×#102×C53C22semidirect
69. #13133188C2×C4semidirect
71. #13133176D4semidirect
78. C2×C4⋊Dic53C2×C4semidirect
103. C22×D4×C53C2×C4semidirect
111. #117×C53D4semidirect
196. C4⋊C4×C53C2×D4semidirect
207. C4⋊Dic53C2×D4semidirect
322. C2×D53#418semidirect
336. C2×#418D53semidirect
377. C2×#102C2×D53semidirect
383. C53#20912semidirect
396. C22×D4C4×D53semidirect
433. #117C22⋊D53semidirect
548. C4⋊C4C2×C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C22×#131331882
C2×#117×D532
C2×#168106282
C2×#168105862
#202750342
#202750292
#202744362
#202744352
C2×C53×#4182
#2091253