Finite Group Databasebeta

← #27310774

Subgroups — #27310774

Order 13568.

1782 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133195C211yesmaximal, characteristic
2C2×#16810485C211yesmaximal, characteristic
3C2×C4×#11719921C211yesmaximal, characteristic
4#20272985C212yesmaximal, complemented
5#20272981C212yesmaximal, complemented
6#20272963C212yesmaximal
7#20272963C212yesmaximal
8#20272950C211yesmaximal, characteristic, complemented
9#20272948C211yesmaximal, characteristic, complemented
10#20272947C211yesmaximal, characteristic, complemented
11C53×#1436C211yesmaximal, characteristic, complemented
12C23×Dic106C2211yescharacteristic
13C23×C4×D53C2211yescharacteristic
14C2×#13133195—24no—
15C2×#13133195—24no—
16C2×#13133195C418yescomplemented
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2211yescharacteristic
19C22×#11719921C2212yes—
20C22×#11719921C2211yescharacteristic
21C22×#11719921C2211yescharacteristic
22C22×#11719921C2212yes—
23C22×C4×Dic53C2211yescharacteristic
24#16810485—24no—
25C4×#11719921—24no—
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C42×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2212yes—
40C24×D53C2311yescharacteristic
41C22×Dic106—22no—
42C22×Dic106—22no—
43C22×Dic106—42no—
44C22×Dic106—42no—
45C22×Dic106C2×C414yes—
46C23×Dic53C2312yes—
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2312yes—
50C23×Dic53C2311yescharacteristic
51C22×C4×D53—42no—
52C22×C4×D53—42no—
53C22×C4×D53C2×C414yes—
54C22×C4×D53D414yes—
55C22×C4×D53D414yes—
56#13133195—232no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919—22no—
61C2×#11719919C2×C414yes—
62C2×#11719919C2×C414yes—
63C2×#11719921—22no—
64C2×#11719921—22no—
65C2×#11719921—22no—
66C2×#11719921—22no—
67C2×#11719921—24no—
68C2×#11719921—24no—
69C2×#11719921—24no—
70C2×#11719921—24no—
71C2×#11719921—42no—
72C2×#11719921—42no—
73C2×#11719921C2×C414yes—
74C2×#11719921C2×C414yes—
75C2×C4×Dic53—22no—
76C2×C4×Dic53—42no—
77#13133124—22no—
78#13133124—24no—
79#13133124—24no—
80#13133124—24no—
81#13133124—24no—
82#13133124—24no—
83#13133124—42no—
84#13133124—42no—
85#13133124—42no—
86C23×C4×C53C2312yes—
87C23×C4×C53C2311yescharacteristic
88C23×C4×C53C2311yescharacteristic
89C23×C4×C53C2312yes—
90C23×C4×C53C2311yescharacteristic
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53C2×C414yes—
94C2×C42×C53—22no—
95C2×C42×C53—42no—
96#95×C53—22no—
97#95×C53—24no—
98#95×C53—42no—
99C24×C53C2411yescharacteristic
100C23×D53—22no—
101C23×D53—22no—
102C23×D53—22no—
103C23×D53—22no—
104C23×D53—22no—
105C23×D53—22no—
106C23×D53C22×C412yes—
107C23×D53C2×D412yes—
108C23×D53C2×D412yes—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
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—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C22×C412yes—
134C22×C4×C53C22×C412yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53#4114yes—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53C22×C412yes—
162C22×Dic53C22×C412yes—
163C22×Dic53C22×C412yes—
164C22×Dic53C2×D412yes—
165C22×Dic53C2×D412yes—
166C22×Dic53#4114yes—
167C2×C4×D53—48no—
168C2×C4×D53—48no—
169C2×C4×D53—44no—
170C2×C4×D53—44no—
171C2×C4×D53C22⋊C4116yes—
172C2×Dic106—28no—
173C2×Dic106—48no—
174C2×Dic106—48no—
175C2×Dic106—44no—
176C2×Dic106—44no—
177C4⋊C4×C53—28no—
178C42×C53—44no—
179#11719921—216no—
180#11719921—28no—
181#11719921—28no—
182#11719921—48no—
183#11719919—28no—
184#11719919—28no—
185C4×Dic53—44no—
186C23×C53C23×C411yescharacteristic
187C23×C53C22×D411yescharacteristic
188C23×C53C22×D411yescharacteristic
189C23×C53C2×#4112yes—
190C23×C53C2×#4112yes—
191C23×C53H(5,𝔽2)12yes—
192C23×C53H(5,𝔽2)12yes—
193C23×C532−1+412yes—
194C23×C532−1+412yes—
195C22×D53—22no—
196C22×D53—22no—
197C22×D53—24no—
198C22×D53—24no—
199C22×D53—24no—
200C22×D53—24no—
201C22×D53—22no—
202C22×D53—22no—
203C22×D53—24no—
204C22×D53—22no—
205C22×D53—24no—
206C22×D53—22no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53C2×C22⋊C414yes—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53C2×C22⋊C414yes—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—24no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53C2×C22⋊C414yes—
274C4×D53—416no—
275Dic106—416no—
276#14292—531nomaximal, Sylow
277C22×C53C22×C22⋊C411yescharacteristic
278C22×C53C4×#4111yescharacteristic
279C22×C53C4×#4111yescharacteristic
280C22×C53#51812yes—
281C22×C53#51912yes—
282C22×C53#52512yes—
283C22×C53#53212yes—
284C22×C53#53412yes—
285C22×C53#53411yescharacteristic
286C22×C53#53411yescharacteristic
287C22×C53#53611yescharacteristic
288C22×C53#53612yes—
289C22×C53#53611yescharacteristic
290C22×C53#53912yes—
291C22×C53#53912yes—
292C22×C53#53912yes—
293C22×C53#53912yes—
294C22×C53#54212yes—
295C22×C53#54212yes—
296C22×C53#54212yes—
297C22×C53#54212yes—
298C2×D53—24no—
299C2×D53—24no—
300C2×D53—24no—
301C2×D53—44no—
302C2×D53—44no—
303C2×D53—44no—
304C2×D53—44no—
305C2×D53—44no—
306C2×D53—44no—
307C4×C53—24no—
308C4×C53—42no—
309C4×C53—44no—
310C4×C53—42no—
311C4×C53—82no—
312Dic53—24no—
313Dic53—42no—
314Dic53—44no—
315Dic53—42no—
316Dic53—82no—
317C22×C22⋊Q8—531no—
318C2×#381—531no—
319C2×C4×C22⋊C4—531no—
320#1446—532no—
321#1445—532no—
322#1440—532no—
323#1440—532no—
324#1436—531no—
325#1436—531no—
326#1436—531no—
327#1436D5311yescharacteristic, complemented
328C2×C53#229711yescharacteristic
329C2×C53#230811yescharacteristic
330C2×C53#231011yescharacteristic
331C2×C53#233212yes—
332C2×C53#233212yes—
333C2×C53#263812yes—
334C2×C53#263812yes—
335C2×C53#266312yes—
336C2×C53#266312yes—
337D53—44no—
338C23×Q8—531no—
339C24×C4—531no—
340C2×C22⋊Q8—538no—
341C2×C22⋊Q8—1064no—
342C2×C22⋊Q8—1064no—
343C22×C4⋊C4—531no—
344C22×C4⋊C4—531no—
345C22×C4⋊C4C2×D5311yescharacteristic
346C22×C22⋊C4—531no—
347C22×C22⋊C4—531no—
348C22×C22⋊C4—532no—
349C22×C22⋊C4—532no—
350C22×C42—531no—
351C22×C42C2×D5311yescharacteristic
352#381—1064no—
353C4×C22⋊C4—1064no—
354C2×#95—531no—
355C2×#95—532no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95C2×D5311yescharacteristic
364C2×#95C2×D5312yes—
365C2×#95C2×D5312yes—
366C53#1429211yescharacteristic, complemented, Sylow
367C25—531no—
368C22×Q8—534no—
369C22×Q8—1062no—
370C22×Q8—1062no—
371C22×Q8—2122no—
372C22×Q8—2122no—
373C23×C4—531no—
374C23×C4—534no—
375C23×C4—534no—
376C23×C4—534no—
377C23×C4—531no—
378C23×C4—531no—
379C23×C4—532no—
380C23×C4—532no—
381C23×C4—2122no—
382C23×C4—2122no—
383C23×C4C22×D5312yes—
384C23×C4C22×D5311yescharacteristic
385C23×C4C22×D5311yescharacteristic
386C23×C4C22×D5312yes—
387C23×C4C22×D5311yescharacteristic
388C22⋊Q8—10632no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—534no—
392C2×C4⋊C4—534no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4C4×D5314yes—
398C2×C22⋊C4—534no—
399C2×C22⋊C4—534no—
400C2×C22⋊C4—1064no—
401C2×C22⋊C4—1062no—
402C2×C22⋊C4—1062no—
403C2×C22⋊C4—1062no—
404C2×C22⋊C4—1062no—
405C2×C22⋊C4—1064no—
406C2×C22⋊C4—1064no—
407C2×C22⋊C4—1064no—
408C2×C22⋊C4—2122no—
409C2×C22⋊C4—2122no—
410C2×C42—22no—
411C2×C42—42no—
412C2×C42—1062no—
413C2×C42—2122no—
414#95—24no—
415#95—22no—
416#95—42no—
417#95—1064no—
418#95—1064no—
419#95—1064no—
420#95—1064no—
421#95—1064no—
422#95—1062no—
423#95—2122no—
424#95—2122no—
425#95—2122no—
426C24—532no—
427C24—532no—
428C24—532no—
429C24—1062no—
430C24—1062no—
431C24—1062no—
432C24—1062no—
433C24—1062no—
434C24—1062no—
435C24C23×D5311yescharacteristic
436C2×Q8—1068no—
437C2×Q8—2128no—
438C2×Q8—2128no—
439C2×Q8—2124no—
440C2×Q8—2124no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—532no—
465C22×C4—532no—
466C22×C4—532no—
467C22×C4—5316no—
468C22×C4—532no—
469C22×C4—532no—
470C22×C4—534no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—2128no—
495C22×C4—2128no—
496C22×C4—2124no—
497C22×C4—2124no—
498C22×C4C2×C4×D5312yes—
499C22×C4C2×C4×D5312yes—
500C22×C4C2×C4×D5312yes—
501C22×C4#1171994514yes—
502C22×C4D4×D5312yes—
503C22×C4D4×D5312yes—
504C4⋊C4—28no—
505C4⋊C4—1068no—
506C4⋊C4—1068no—
507C22⋊C4—1068no—
508C22⋊C4—1068no—
509C22⋊C4—10616no—
510C22⋊C4—2128no—
511C42—44no—
512C42—2124no—
513C23—534no—
514C23—1062no—
515C23—1062no—
516C23—1064no—
517C23—1064no—
518C23—1064no—
519C23—1064no—
520C23—1064no—
521C23—1062no—
522C23—1062no—
523C23—1062no—
524C23—1064no—
525C23—1062no—
526C23—2122no—
527C23—2122no—
528C23—2122no—
529C23—2122no—
530C23—2122no—
531C23—2122no—
532C23—2122no—
533C23—2122no—
534C23C22×C4×D5311yescharacteristic
535C23C2×D4×D5311yescharacteristic
536C23C2×D4×D5311yescharacteristic
537C23#41×D5312yes—
538C23C2×#1171994512yes—
539C23#1313326212yes—
540C23#1313326312yes—
541C23#1313326412yes—
542C23#1313326512yes—
543Q8—21216no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—534no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—1062no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1064no—
582C2×C4—1064no—
583C2×C4—1064no—
584C2×C4—1064no—
585C2×C4—21216no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2121no—
597C2×C4—2122no—
598C2×C4—2121no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4C22⋊C4×D5314yes—
603C22—1064no—
604C22—1064no—
605C22—1064no—
606C22—2124no—
607C22—2124no—
608C22—2124no—
609C22—2124no—
610C22—2124no—
611C22—2124no—
612C22C2×C22⋊C4×D5311yescharacteristic
613C22#1681109111yescharacteristic
614C22C4×#1171994511yescharacteristic
615C22#1681109512yes—
616C22#1681109912yes—
617C22#1681113212yes—
618C22#1681117012yes—
619C22#1681118112yes—
620C22#1681118311yescharacteristic
621C22#1681118411yescharacteristic
622C22#1681119311yescharacteristic
623C22#1681119612yes—
624C22#1681119811yescharacteristic
625C22#1681121412yes—
626C22#1681121512yes—
627C22#1681122012yes—
628C22#1681122112yes—
629C22#1681124112yes—
630C22#1681124312yes—
631C22#1681124412yes—
632C22#1681124512yes—
633C4—24no—
634C4—42no—
635C4—44no—
636C4—42no—
637C4—82no—
638C4—1064no—
639C4—4242no—
640C4—2122no—
641C4—2124no—
642C4—2122no—
643C2—2124no—
644C2#2027720111yescharacteristic
645C2#2027728911yescharacteristic
646C2#2027729811yescharacteristic
647C2#2027746612yes—
648C2#2027746912yes—
649C2#2028063612yes—
650C2#2028063912yes—
651C2#2028086712yes—
652C2#2028086912yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1436C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1436D53
hypocenterC53#14292
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
4. #20272985C2semidirect
5. #20272981C2semidirect
8. #20272950C2semidirect
9. #20272948C2semidirect
10. #20272947C2semidirect
11. C53×#1436C2semidirect
16. C2×#13133195C4semidirect
327. #1436D53semidirect
366. C53#14292semidirect

Maximal subgroups

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

subgroupindex
C22×#131331952
C2×#168104852
C2×C4×#117199212
#202729852
#202729812
#202729632
#202729502
#202729482
#202729472
C53×#14362
#1429253