Finite Group Databasebeta

← #27310468

Subgroups — #27310468

Order 13568.

2142 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133193C211yesmaximal, characteristic
2C22×#13133174C211yesmaximal, characteristic
3C2×#16810530C211yesmaximal, characteristic, complemented
4C2×#16810480C211yesmaximal, characteristic
5#20272966C214yesmaximal, complemented
6#20272963C214yesmaximal
7#20272945C211yesmaximal, characteristic, complemented
8#20272944C211yesmaximal, characteristic, complemented
9C2×C53×#391C211yesmaximal, characteristic, complemented
10C23×C4×D53C2211yescharacteristic
11C2×#13133193—24no—
12C2×#13133193—24no—
13C2×#13133193C418yescomplemented
14C2×#13133174—24no—
15C2×#13133174—24no—
16C2×#13133174—24no—
17C22×C4⋊Dic53C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C22×#11719919C2212yes—
20C22×#11719919C2212yes—
21C22×#11719921C2212yes—
22C22×#11719921C2212yes—
23C22×#11719921C2212yes—
24C22×C4×Dic53C2211yescharacteristic
25#16810530—24no—
26#16810480—24no—
27C2×#13133124C2214yes—
28C2×#13133124C2212yes—
29C2×#13133124C2214yes—
30C2×#13133124C2214yes—
31C23×Q8×C53C2211yescharacteristic
32C22×C4⋊C4×C53C2211yescharacteristic
33C22×C42×C53C2211yescharacteristic
34C53×#391—24no—
35C2×#95×C53C2214yes—
36C24×D53C2311yescharacteristic
37C23×Dic53C2312yes—
38C23×Dic53C2312yes—
39C23×Dic53C2312yes—
40C23×Dic53C2311yescharacteristic
41C22×C4×D53—22no—
42C22×C4×D53—22no—
43C22×C4×D53—22no—
44C22×C4×D53—22no—
45C22×C4×D53C2×C414yes—
46C22×C4×D53D414yes—
47C22×C4×D53D414yes—
48#13133193—232no—
49#13133174—232no—
50C2×C4⋊Dic53—22no—
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53C2×C414yes—
56C2×#11719919—24no—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919—24no—
60C2×#11719919—24no—
61C2×#11719919C2×C418yes—
62C2×#11719921—24no—
63C2×#11719921—24no—
64C2×#11719921—24no—
65C2×#11719921—24no—
66C2×#11719921—24no—
67C2×#11719921—24no—
68C2×#11719921—44no—
69C2×#11719921C2×C418yes—
70C2×C4×Dic53—22no—
71C2×C4×Dic53—22no—
72C2×C4×Dic53D414yes—
73#13133124—28no—
74#13133124—24no—
75#13133124—44no—
76#13133124—44no—
77C22×Q8×C53—22no—
78C22×Q8×C53—22no—
79C22×Q8×C53—22no—
80C22×Q8×C53—22no—
81C22×Q8×C53—22no—
82C22×Q8×C53—22no—
83C22×Q8×C53C2×C414yes—
84C23×C4×C53C2312yes—
85C23×C4×C53C2312yes—
86C23×C4×C53C2312yes—
87C23×C4×C53C2311yescharacteristic
88C2×C4⋊C4×C53—22no—
89C2×C4⋊C4×C53—22no—
90C2×C4⋊C4×C53—22no—
91C2×C42×C53—22no—
92C2×C42×C53—22no—
93C2×C42×C53D414yes—
94#95×C53—28no—
95C24×C53C2411yescharacteristic
96C23×D53—21no—
97C23×D53—21no—
98C23×D53—21no—
99C23×D53—21no—
100C23×D53—22no—
101C23×D53—22no—
102C23×D53—22no—
103C23×D53—22no—
104C23×D53C22×C412yes—
105C23×D53C2×D412yes—
106C23×D53C2×D412yes—
107C2×Q8×C53—28no—
108C2×Q8×C53—28no—
109C2×Q8×C53—28no—
110C2×Q8×C53—44no—
111C2×Q8×C53—44no—
112C2×Q8×C53—44no—
113C2×Q8×C53—44no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
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—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—24no—
130C22×C4×C53—24no—
131C22×C4×C53C22×C414yes—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C2×D414yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D412yes—
138C22×C4×C53#4112yes—
139C22×C4×C53#4112yes—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—24no—
149C22×Dic53—22no—
150C22×Dic53—24no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—24no—
158C22×Dic53—24no—
159C22×Dic53C22×C414yes—
160C22×Dic53C22×C412yes—
161C22×Dic53C2×D414yes—
162C22×Dic53C2×D412yes—
163C22×Dic53C2×D412yes—
164C2×C4×D53—28no—
165C2×C4×D53—28no—
166C2×C4×D53—28no—
167C2×C4×D53—28no—
168C2×C4×D53—28no—
169C2×C4×D53—28no—
170C2×C4×D53C22⋊C4116yes—
171C4⋊C4×C53—44no—
172C42×C53—28no—
173#11719921—216no—
174#11719921—216no—
175#11719921—48no—
176C4⋊Dic53—28no—
177C4⋊Dic53—44no—
178#11719919—216no—
179#11719919—216no—
180C4×Dic53—28no—
181C23×C53C23×C411yescharacteristic
182C23×C53C22×D411yescharacteristic
183C23×C53C22×D411yescharacteristic
184C23×C53C22×D411yescharacteristic
185C23×C53C22×D411yescharacteristic
186C23×C53C2×#4111yescharacteristic
187C23×C53C2×#4111yescharacteristic
188C23×C53H(5,𝔽2)12yes—
189C23×C532−1+412yes—
190C23×C532−1+412yes—
191C23×C532−1+412yes—
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—24no—
195C22×D53—24no—
196C22×D53—22no—
197C22×D53—22no—
198C22×D53—22no—
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—22no—
202C22×D53—24no—
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—24no—
206C22×D53—42no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53C2×C22⋊C414yes—
215Q8×C53—416no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—44no—
234C2×C4×C53—44no—
235C2×C4×C53C2×C22⋊C414yes—
236C2×C4×C53C4×D414yes—
237C2×C4×C53C4×D414yes—
238C2×C4×C53#12014yes—
239C2×C4×C53#12114yes—
240C2×C4×C53#12114yes—
241C2×C4×C53#12314yes—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—24no—
249C2×Dic53—28no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—44no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—44no—
264C2×Dic53—44no—
265C2×Dic53C2×C22⋊C414yes—
266C4×D53—232no—
267#14258—531nomaximal, Sylow
268C22×C53C22×C22⋊C411yescharacteristic
269C22×C53C2×C4×D411yescharacteristic
270C22×C53C2×C4×D411yescharacteristic
271C22×C53#51912yes—
272C22×C53#52012yes—
273C22×C53C2×#12011yescharacteristic
274C22×C53C2×#12111yescharacteristic
275C22×C53C2×#12111yescharacteristic
276C22×C53C2×#12311yescharacteristic
277C22×C53#53511yescharacteristic
278C22×C53#53611yescharacteristic
279C22×C53#53612yes—
280C22×C53#53612yes—
281C22×C53#53612yes—
282C22×C53#53611yescharacteristic
283C22×C53#53611yescharacteristic
284C22×C53#53712yes—
285C22×C53#53712yes—
286C22×C53#53712yes—
287C22×C53#54112yes—
288C22×C53#54112yes—
289C22×C53#54212yes—
290C22×C53#54212yes—
291C2×D53—24no—
292C2×D53—24no—
293C2×D53—24no—
294C2×D53—44no—
295C2×D53—44no—
296C2×D53—44no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—44no—
302C4×C53—44no—
303C4×C53—44no—
304C4×C53—82no—
305C4×C53#38618yes—
306Dic53—24no—
307Dic53—44no—
308Dic53—44no—
309Dic53—82no—
310C22×C22⋊Q8—531no—
311C22×#117—531no—
312C2×#391—531no—
313C2×#391D5311yescharacteristic, complemented
314C2×#380—531no—
315#1441—534no—
316#1440—534no—
317C4⋊#95—531no—
318C4⋊#95—531no—
319C2×C53C2×#38611yescharacteristic
320C2×C53#231011yescharacteristic
321C2×C53#231111yescharacteristic
322C2×C53#232312yes—
323C2×C53#232312yes—
324C2×C53#261612yes—
325C2×C53#262011yescharacteristic
326C2×C53#262011yescharacteristic
327C2×C53#262111yescharacteristic
328C2×C53#262111yescharacteristic
329C2×C53#262512yes—
330D53—44no—
331C23×Q8C2×D5311yescharacteristic
332C24×C4—531no—
333C2×C22⋊Q8—538no—
334C2×C22⋊Q8—1064no—
335C2×C22⋊Q8—1064no—
336C2×#117—1064no—
337C2×#117—1064no—
338C2×#117—1064no—
339C22×C4⋊C4—532no—
340C22×C4⋊C4—531no—
341C22×C4⋊C4—532no—
342C22×C4⋊C4—531no—
343C22×C4⋊C4C2×D5311yescharacteristic
344C22×C22⋊C4—532no—
345C22×C22⋊C4—532no—
346C22×C22⋊C4—532no—
347C22×C42—531no—
348C22×C42C2×D5311yescharacteristic
349#391—24no—
350#391—1064no—
351#380—1064no—
352C2×#95—534no—
353C2×#95—532no—
354C2×#95—534no—
355C2×#95—534no—
356C2×#95C2×D5314yes—
357C53#1425811yescharacteristic, complemented, Sylow
358C25—531no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8—22no—
362C22×Q8—22no—
363C22×Q8—22no—
364C22×Q8—22no—
365C22×Q8C4×D5314yes—
366C23×C4—531no—
367C23×C4—534no—
368C23×C4—534no—
369C23×C4—534no—
370C23×C4—532no—
371C23×C4—532no—
372C23×C4—532no—
373C23×C4—1062no—
374C23×C4—1062no—
375C23×C4—1062no—
376C23×C4—1062no—
377C23×C4C22×D5312yes—
378C23×C4C22×D5312yes—
379C23×C4C22×D5312yes—
380C23×C4C22×D5311yescharacteristic
381C22⋊Q8—10632no—
382#117—10632no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—538no—
387C2×C4⋊C4—534no—
388C2×C4⋊C4—1064no—
389C2×C4⋊C4—1064no—
390C2×C4⋊C4—1064no—
391C2×C4⋊C4—1064no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1064no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4—1062no—
398C2×C22⋊C4—538no—
399C2×C22⋊C4—1064no—
400C2×C22⋊C4—1064no—
401C2×C22⋊C4—1064no—
402C2×C22⋊C4—1064no—
403C2×C22⋊C4—1064no—
404C2×C22⋊C4—1064no—
405C2×C22⋊C4—2124no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—534no—
409C2×C42—1062no—
410C2×C42—1062no—
411C2×C42C22⋊D5314yes—
412#95—28no—
413#95—1064no—
414#95—1068no—
415#95—2124no—
416#95—2124no—
417C24—532no—
418C24—532no—
419C24—532no—
420C24—1061no—
421C24—1061no—
422C24—1061no—
423C24—1061no—
424C24—1062no—
425C24—1062no—
426C24—1062no—
427C24—1062no—
428C24C23×D5311yescharacteristic
429C2×Q8—28no—
430C2×Q8—28no—
431C2×Q8—28no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—532no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—5316no—
457C22×C4—534no—
458C22×C4—534no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1068no—
464C22×C4—1068no—
465C22×C4—1068no—
466C22×C4—1068no—
467C22×C4—1068no—
468C22×C4—1068no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1064no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1064no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1064no—
483C22×C4—1064no—
484C22×C4C2×C4×D5314yes—
485C22×C4C2×C4×D5312yes—
486C22×C4#1171994512yes—
487C22×C4D4×D5312yes—
488C22×C4D4×D5312yes—
489C22×C4D4×D5312yes—
490C22×C4#1171994712yes—
491C22×C4C2×C22⋊D5314yes—
492C22×C4C2×C22⋊D5312yes—
493C4⋊C4—44no—
494C4⋊C4—10616no—
495C4⋊C4—10616no—
496C4⋊C4—1068no—
497C4⋊C4—2124no—
498C22⋊C4—10616no—
499C22⋊C4—10616no—
500C22⋊C4—2128no—
501C42—28no—
502C42—1068no—
503C23—534no—
504C23—1062no—
505C23—1062no—
506C23—1064no—
507C23—1064no—
508C23—1062no—
509C23—1062no—
510C23—1062no—
511C23—1062no—
512C23—1062no—
513C23—1062no—
514C23—1064no—
515C23—1062no—
516C23—1062no—
517C23—1064no—
518C23—2122no—
519C23—2122no—
520C23—2122no—
521C23—2122no—
522C23—2122no—
523C23—2122no—
524C23—2122no—
525C23—2122no—
526C23C22×C4×D5311yescharacteristic
527C23C2×D4×D5311yescharacteristic
528C23C2×D4×D5311yescharacteristic
529C23C2×D4×D5311yescharacteristic
530C23C22×C22⋊D5311yescharacteristic
531C23C2×#1171994711yescharacteristic
532C23C2×#1171994511yescharacteristic
533C23#1313326312yes—
534C23#1313326412yes—
535C23#1313326512yes—
536C23#1313326512yes—
537Q8—416no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—534no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—10632no—
565C2×C4—1064no—
566C2×C4—1068no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2124no—
574C2×C4—2124no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2124no—
581C2×C4—2122no—
582C2×C4C22⋊C4×D5314yes—
583C2×C4#1313317714yes—
584C2×C4C4×C22⋊D5314yes—
585C2×C4#1313318514yes—
586C2×C4#1313318814yes—
587C2×C4#1313319014yes—
588C2×C4#1313319914yes—
589C22—1064no—
590C22—1064no—
591C22—1064no—
592C22—2124no—
593C22—2124no—
594C22—2124no—
595C22—2122no—
596C22—2122no—
597C22—2122no—
598C22—2122no—
599C22—2124no—
600C22C2×C22⋊C4×D5311yescharacteristic
601C22C2×#1313317711yescharacteristic
602C22C2×C4×C22⋊D5311yescharacteristic
603C22#1681109912yes—
604C22#1681110512yes—
605C22C2×#1313318511yescharacteristic
606C22C2×#1313318811yescharacteristic
607C22C2×#1313319011yescharacteristic
608C22C2×#1313319911yescharacteristic
609C22#1681119111yescharacteristic
610C22#1681119311yescharacteristic
611C22#1681119612yes—
612C22#1681119612yes—
613C22#1681119712yes—
614C22#1681119811yescharacteristic
615C22#1681119811yescharacteristic
616C22#1681120112yes—
617C22#1681120312yes—
618C22#1681120312yes—
619C22#1681123112yes—
620C22#1681123412yes—
621C22#1681124212yes—
622C22#1681124512yes—
623C4—44no—
624C4—44no—
625C4—82no—
626C4—1064no—
627C4—4242no—
628C4—2124no—
629C4—2124no—
630C4#1681049918yes—
631C2—2124no—
632C2C2×#1681049911yescharacteristic
633C2#2027729811yescharacteristic
634C2#2027730611yescharacteristic
635C2#2027739112yes—
636C2#2027740012yes—
637C2#2028044512yes—
638C2#2028047311yescharacteristic
639C2#2028047911yescharacteristic
640C2#2028048211yescharacteristic
641C2#2028048611yescharacteristic
642C2#2028051512yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#391C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC2×#391D53
hypocenterC53#14258
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. C2×#16810530C2semidirect
5. #20272966C2semidirect
7. #20272945C2semidirect
8. #20272944C2semidirect
9. C2×C53×#391C2semidirect
13. C2×#13133193C4semidirect
313. C2×#391D53semidirect
357. C53#14258semidirect

Maximal subgroups

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

subgroupindex
C22×#131331932
C22×#131331742
C2×#168105302
C2×#168104802
#202729662
#202729632
#202729452
#202729442
C2×C53×#3912
#1425853