Finite Group Databasebeta

← #27306619

Subgroups — #27306619

Order 13568.

1862 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×#13133174C211yesmaximal, characteristic
2C2×#16810490C211yesmaximal, characteristic, complemented
3C2×#16810474C211yesmaximal, characteristic, complemented
4#20272966C212yesmaximal, complemented
5#20272959C212yesmaximal
6#20272951C212yesmaximal
7#20272946C212yesmaximal, complemented
8#20272942C211yesmaximal, characteristic, complemented
9#20272936C211yesmaximal, characteristic
10#20272936C211yesmaximal, characteristic
11C53×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C2×#13133174—24no—
14C2×#13133174—24no—
15C2×#13133174C418yescomplemented
16C22×#11719919C2212yes—
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2211yescharacteristic
19C22×#11719921C2212yes—
20C22×#11719921C2212yes—
21C22×#11719921C2211yescharacteristic
22C22×#11719921C2211yescharacteristic
23C22×C4×Dic53C2212yes—
24C22×C4×Dic53C2211yescharacteristic
25#16810490—24no—
26#16810474—24no—
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C42×C53C2211yescharacteristic
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2212yes—
40C24×D53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2311yescharacteristic
46C22×C4×D53—24no—
47C22×C4×D53—22no—
48C22×C4×D53—22no—
49C22×C4×D53—24no—
50C22×C4×D53C2×C414yes—
51#13133174—232no—
52C2×#11719919—24no—
53C2×#11719919—22no—
54C2×#11719919—22no—
55C2×#11719919—22no—
56C2×#11719919—22no—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919C2×C414yes—
60C2×#11719919C2×C414yes—
61C2×#11719921—24no—
62C2×#11719921—24no—
63C2×#11719921—24no—
64C2×#11719921—22no—
65C2×#11719921—22no—
66C2×#11719921—22no—
67C2×#11719921—22no—
68C2×#11719921—24no—
69C2×#11719921—42no—
70C2×#11719921—42no—
71C2×#11719921C2×C414yes—
72C2×#11719921C2×C414yes—
73C2×C4×Dic53—24no—
74C2×C4×Dic53—42no—
75C2×C4×Dic53—42no—
76C2×C4×Dic53C2×C414yes—
77C2×C4×Dic53D414yes—
78C2×C4×Dic53D414yes—
79#13133124—22no—
80#13133124—22no—
81#13133124—24no—
82#13133124—24no—
83#13133124—24no—
84#13133124—24no—
85#13133124—24no—
86#13133124—42no—
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2312yes—
91C23×C4×C53C2311yescharacteristic
92C2×C42×C53—22no—
93C2×C42×C53—22no—
94C2×C42×C53C2×C414yes—
95#95×C53—22no—
96#95×C53—22no—
97#95×C53—24no—
98#95×C53—42no—
99C24×C53C2411yescharacteristic
100C23×D53—21no—
101C23×D53—21no—
102C23×D53—22no—
103C23×D53—22no—
104C23×D53—22no—
105C23×D53—21no—
106C23×D53—21no—
107C23×D53—21no—
108C23×D53—21no—
109C23×D53C22×C412yes—
110C23×D53#4114yes—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53—22no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53C22×C412yes—
137C22×C4×C53C22×C412yes—
138C22×C4×C53C22×C412yes—
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2×D412yes—
141C22×C4×C53#4114yes—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
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×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53C22×C412yes—
165C22×Dic53C22×C412yes—
166C22×Dic53C22×C412yes—
167C22×Dic53C2×D412yes—
168C22×Dic53C2×D412yes—
169C22×Dic53C2×D412yes—
170C22×Dic53C2×D412yes—
171C22×Dic53#4114yes—
172C22×Dic53#4114yes—
173C2×C4×D53—216no—
174C2×C4×D53—216no—
175C2×C4×D53—28no—
176C2×C4×D53—28no—
177C2×C4×D53—28no—
178C42×C53—28no—
179#11719921—28no—
180#11719921—28no—
181#11719921—48no—
182#11719921—48no—
183#11719919—216no—
184#11719919—28no—
185#11719919—28no—
186C4×Dic53—48no—
187C4×Dic53C22⋊C4116yescomplemented
188C23×C53C23×C411yescharacteristic
189C23×C53C22×D411yescharacteristic
190C23×C53C22×D411yescharacteristic
191C23×C53C2×#4112yes—
192C23×C53C2×#4112yes—
193C23×C53C2×#4112yes—
194C23×C53C2×#4112yes—
195C23×C53H(5,𝔽2)11yescharacteristic
196C23×C53H(5,𝔽2)11yescharacteristic
197C23×C532−1+411yescharacteristic
198C23×C532−1+411yescharacteristic
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—22no—
202C22×D53—22no—
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—24no—
206C22×D53—24no—
207C22×D53—24no—
208C22×D53—24no—
209C22×D53—24no—
210C22×D53—24no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53#11714yes—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—41no—
239C2×C4×C53—42no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53C2×C22⋊C414yes—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53C2×C22⋊C414yes—
282C2×Dic53C2×C22⋊C414yes—
283C2×Dic53#11714yes—
284C4×D53—232no—
285#13938—531nomaximal, Sylow
286C22×C53C22×C22⋊C411yescharacteristic
287C22×C53C2×#11711yescharacteristic
288C22×C53C2×#11711yescharacteristic
289C22×C53C4×#4111yescharacteristic
290C22×C53C4×#4111yescharacteristic
291C22×C53#52011yescharacteristic
292C22×C53#52011yescharacteristic
293C22×C53#52512yes—
294C22×C53#52512yes—
295C22×C53#52512yes—
296C22×C53#53212yes—
297C22×C53#53511yescharacteristic
298C22×C53#53611yescharacteristic
299C22×C53#53711yescharacteristic
300C22×C53#53711yescharacteristic
301C22×C53#55112yes—
302C22×C53#55112yes—
303C22×C53#55212yes—
304C22×C53#55212yes—
305C22×C53#55312yes—
306C22×C53#55312yes—
307C22×C53#55612yes—
308C22×C53#55612yes—
309C2×D53—24no—
310C2×D53—24no—
311C2×D53—24no—
312C2×D53—42no—
313C2×D53—44no—
314C2×D53—44no—
315C2×D53—44no—
316C2×D53—44no—
317C2×D53—42no—
318C2×D53—42no—
319C2×D53—42no—
320C4×C53—24no—
321C4×C53—42no—
322C4×C53—44no—
323C4×C53—42no—
324C4×C53—82no—
325Dic53—24no—
326Dic53—24no—
327Dic53—24no—
328Dic53—44no—
329Dic53—82no—
330C22×#117—531no—
331C2×#383—531no—
332C2×C4×C4⋊C4—531no—
333#1441—532no—
334#1439—532no—
335#1437—532no—
336#1436—532no—
337#1434—531no—
338#1434D5311yescharacteristic, complemented
339C22⋊#95—531no—
340C22⋊#95—531no—
341C2×C53#228011yescharacteristic
342C2×C53#229711yescharacteristic
343C2×C53#231111yescharacteristic
344C2×C53#234311yescharacteristic
345C2×C53#234311yescharacteristic
346C2×C53#234711yescharacteristic
347C2×C53#234711yescharacteristic
348C2×C53#259812yes—
349C2×C53#259912yes—
350C2×C53#260112yes—
351C2×C53#260112yes—
352D53—44no—
353C24×C4—531no—
354C2×#117—538no—
355C2×#117—1064no—
356C2×#117—1064no—
357C22×C4⋊C4—531no—
358C22×C4⋊C4—531no—
359C22×C4⋊C4—532no—
360C22×C22⋊C4—531no—
361C22×C22⋊C4—531no—
362C22×C22⋊C4—532no—
363C22×C22⋊C4—532no—
364C22×C42—531no—
365C22×C42—532no—
366C22×C42C2×D5311yescharacteristic
367#383—1064no—
368C4×C4⋊C4—1064no—
369C2×#95—531no—
370C2×#95—531no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95—532no—
374C2×#95—532no—
375C2×#95—532no—
376C2×#95—532no—
377C2×#95C2×D5311yescharacteristic
378C2×#95C2×D5311yescharacteristic
379C2×#95C2×D5312yes—
380C2×#95C2×D5312yes—
381C53#1393811yescharacteristic, complemented, Sylow
382C25—531no—
383C23×C4—531no—
384C23×C4—531no—
385C23×C4—531no—
386C23×C4—532no—
387C23×C4—534no—
388C23×C4—532no—
389C23×C4—1064no—
390C23×C4—1064no—
391C23×C4—1062no—
392C23×C4—1062no—
393C23×C4C22×D5312yes—
394C23×C4C22×D5311yescharacteristic
395C23×C4C22×D5311yescharacteristic
396C23×C4C22×D5312yes—
397C23×C4C22×D5311yescharacteristic
398#117—10632no—
399C2×C4⋊C4—534no—
400C2×C4⋊C4—534no—
401C2×C4⋊C4—1062no—
402C2×C4⋊C4—1062no—
403C2×C4⋊C4—1062no—
404C2×C4⋊C4—1062no—
405C2×C4⋊C4—1064no—
406C2×C4⋊C4—1064no—
407C2×C4⋊C4—1064no—
408C2×C22⋊C4—534no—
409C2×C22⋊C4—534no—
410C2×C22⋊C4—1062no—
411C2×C22⋊C4—1062no—
412C2×C22⋊C4—1062no—
413C2×C22⋊C4—1062no—
414C2×C22⋊C4—1064no—
415C2×C22⋊C4—1064no—
416C2×C22⋊C4—1064no—
417C2×C22⋊C4—1064no—
418C2×C22⋊C4—2122no—
419C2×C22⋊C4—2122no—
420C2×C42—22no—
421C2×C42—22no—
422C2×C42—534no—
423C2×C42—534no—
424C2×C42—534no—
425C2×C42—1064no—
426C2×C42—2122no—
427C2×C42—2122no—
428C2×C42C4×D5314yes—
429#95—24no—
430#95—22no—
431#95—22no—
432#95—42no—
433#95—1064no—
434#95—1064no—
435#95—1064no—
436#95—1064no—
437#95—1064no—
438#95—1062no—
439#95—1062no—
440#95—2122no—
441C24—532no—
442C24—534no—
443C24—1061no—
444C24—1061no—
445C24—1062no—
446C24—1062no—
447C24—1062no—
448C24—1061no—
449C24—1061no—
450C24—1061no—
451C24—1061no—
452C24C23×D5311yescharacteristic
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—532no—
479C22×C4—532no—
480C22×C4—532no—
481C22×C4—532no—
482C22×C4—532no—
483C22×C4—532no—
484C22×C4—532no—
485C22×C4—534no—
486C22×C4—534no—
487C22×C4—1068no—
488C22×C4—1068no—
489C22×C4—10616no—
490C22×C4—10616no—
491C22×C4—1061no—
492C22×C4—1061no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1068no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1062no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4C2×C4×D5312yes—
515C22×C4C2×C4×D5312yes—
516C22×C4C2×C4×D5312yes—
517C22×C4#1171994514yes—
518C22×C4D4×D5312yes—
519C22×C4D4×D5312yes—
520C4⋊C4—1068no—
521C4⋊C4—1068no—
522C4⋊C4—10616no—
523C22⋊C4—1068no—
524C22⋊C4—1068no—
525C22⋊C4—2128no—
526C22⋊C4—2128no—
527C42—28no—
528C42—5316no—
529C42—2128no—
530C23—534no—
531C23—1064no—
532C23—1064no—
533C23—1064no—
534C23—1064no—
535C23—1062no—
536C23—1062no—
537C23—1062no—
538C23—1062no—
539C23—1062no—
540C23—1062no—
541C23—1064no—
542C23—1064no—
543C23—2122no—
544C23—2122no—
545C23—2122no—
546C23—2122no—
547C23—2122no—
548C23—2122no—
549C23—2121no—
550C23—2121no—
551C23—2121no—
552C23—2121no—
553C23C22×C4×D5311yescharacteristic
554C23C2×D4×D5311yescharacteristic
555C23C2×D4×D5311yescharacteristic
556C23#41×D5312yes—
557C23#41×D5312yes—
558C23#41×D5312yes—
559C23C2×#1171994512yes—
560C23#1313326211yescharacteristic
561C23#1313326311yescharacteristic
562C23#1313326411yescharacteristic
563C23#1313326511yescharacteristic
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—534no—
595C2×C4—534no—
596C2×C4—534no—
597C2×C4—10632no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1062no—
602C2×C4—1062no—
603C2×C4—1062no—
604C2×C4—1064no—
605C2×C4—1064no—
606C2×C4—1064no—
607C2×C4—1064no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—1064no—
613C2×C4—1064no—
614C2×C4—1064no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4—2122no—
621C2×C4—2122no—
622C2×C4—2122no—
623C2×C4—2122no—
624C2×C4—2122no—
625C2×C4—2122no—
626C2×C4C22⋊C4×D5314yes—
627C22—1064no—
628C22—1064no—
629C22—1064no—
630C22—2122no—
631C22—2122no—
632C22—2122no—
633C22—2122no—
634C22—2124no—
635C22—2124no—
636C22—2124no—
637C22—2124no—
638C22C2×C22⋊C4×D5311yescharacteristic
639C22#117×D5311yescharacteristic
640C22#117×D5311yescharacteristic
641C22#1681109111yescharacteristic
642C22C4×#1171994511yescharacteristic
643C22#1681110311yescharacteristic
644C22#1681110411yescharacteristic
645C22#1681113112yes—
646C22#1681113112yes—
647C22#1681113212yes—
648C22#1681117012yes—
649C22#1681118911yescharacteristic
650C22#1681119611yescharacteristic
651C22#1681120211yescharacteristic
652C22#1681120311yescharacteristic
653C22#1681129712yes—
654C22#1681130312yes—
655C22#1681130512yes—
656C22#1681131212yes—
657C22#1681132212yes—
658C22#1681132412yes—
659C22#1681134012yes—
660C22#1681134512yes—
661C4—24no—
662C4—42no—
663C4—42no—
664C4—44no—
665C4—82no—
666C4—1064no—
667C4—1064no—
668C4—1064no—
669C4—4242no—
670C4—2124no—
671C2—2124no—
672C2#2027709511yescharacteristic
673C2#2027720111yescharacteristic
674C2#2027730411yescharacteristic
675C2#2027756811yescharacteristic
676C2#2027757611yescharacteristic
677C2#2027760311yescharacteristic
678C2#2027760911yescharacteristic
679C2#2028027912yes—
680C2#2028029012yes—
681C2#2028031012yes—
682C2#2028031812yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1434C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1434D53
hypocenterC53#13938
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C2×#16810490C2semidirect
3. C2×#16810474C2semidirect
4. #20272966C2semidirect
7. #20272946C2semidirect
8. #20272942C2semidirect
11. C53×#1434C2semidirect
15. C2×#13133174C4semidirect
187. C4×Dic53C22⋊C4semidirect
338. #1434D53semidirect
381. C53#13938semidirect

Maximal subgroups

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

subgroupindex
C22×#131331742
C2×#168104902
C2×#168104742
#202729662
#202729592
#202729512
#202729462
#202729422
#202729362
C53×#14342
#1393853