Finite Group Databasebeta

← #27306662

Subgroups — #27306662

Order 13568.

1846 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133176C211yesmaximal, characteristic
2C2×#16810511C212yesmaximal
3#20272967C212yesmaximal, complemented
4#20272962C212yesmaximal
5#20272949C212yesmaximal, complemented
6#20272943C211yesmaximal, characteristic, complemented
7#20272941C211yesmaximal, characteristic, complemented
8#20272940C211yesmaximal, characteristic, complemented
9#20272938C211yesmaximal, characteristic
10#95×Dic53C211yesmaximal, characteristic, complemented
11C53×C22⋊#95C211yesmaximal, characteristic
12C2×#13133176—28no—
13C2×#13133176C418yescomplemented
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16C22×#11719919C2212yes—
17C22×C22⋊Dic53C2212yes—
18C22×C22⋊Dic53C2211yescharacteristic
19C22×C22⋊Dic53C2211yescharacteristic
20C22×C4×Dic53C2212yes—
21C22×C4×Dic53C2211yescharacteristic
22C22×C4×Dic53C2211yescharacteristic
23#16810511—28no—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2212yes—
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C24×C4×C53C2211yescharacteristic
34C22×C22⋊C4×C53C2212yes—
35C2×#95×C53C2211yescharacteristic
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C23×Dic53C2312yes—
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2311yescharacteristic
44#13133176—232no—
45C2×C4⋊Dic53—24no—
46C2×C4⋊Dic53—24no—
47C2×C4⋊Dic53C2×C414yes—
48C2×C4⋊Dic53C2×C414yes—
49C2×#11719919—24no—
50C2×#11719919—24no—
51C2×#11719919—42no—
52C2×C22⋊Dic53—24no—
53C2×C22⋊Dic53—24no—
54C2×C22⋊Dic53—24no—
55C2×C22⋊Dic53—42no—
56C2×C22⋊Dic53C2×C414yes—
57C2×C22⋊Dic53C2×C414yes—
58C2×C22⋊Dic53D418yes—
59C2×C4×Dic53—24no—
60C2×C4×Dic53—24no—
61C2×C4×Dic53—24no—
62C2×C4×Dic53—24no—
63C2×C4×Dic53—42no—
64C2×C4×Dic53C2×C414yes—
65C2×C4×Dic53C2×C414yes—
66#13133124—22no—
67#13133124—24no—
68#13133124—24no—
69#13133124—22no—
70#13133124—24no—
71#13133124—24no—
72#13133124—24no—
73#13133124—42no—
74#13133124—42no—
75C25×C53C2311yescharacteristic
76C23×C4×C53—24no—
77C23×C4×C53—22no—
78C23×C4×C53—22no—
79C23×C4×C53—24no—
80C23×C4×C53C2×C414yes—
81C23×C4×C53C2312yes—
82C23×C4×C53C2312yes—
83C23×C4×C53C2311yescharacteristic
84C23×C4×C53C2311yescharacteristic
85C2×C22⋊C4×C53—24no—
86C2×C22⋊C4×C53—24no—
87C2×C22⋊C4×C53—42no—
88#95×C53—22no—
89#95×C53—24no—
90#95×C53—22no—
91C24×C53—22no—
92C24×C53—21no—
93C24×C53—21no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—21no—
97C24×C53—21no—
98C24×C53—21no—
99C24×C53—21no—
100C24×C53C22×C412yes—
101C24×C53C2×D414yes—
102C24×C53C2411yescharacteristic
103C22×C4×C53—216no—
104C22×C4×C53—216no—
105C22×C4×C53—28no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—28no—
112C22×C4×C53—28no—
113C22×C4×C53—22no—
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×C53C22×C412yes—
127C22×C4×C53C22×C412yes—
128C22×C4×C53#4114yes—
129C22×C4×C53#4112yes—
130C22×C4×C53#4112yes—
131C22×C4×C53#4114yes—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53C22×C412yes—
160C22×Dic53C22×C412yes—
161C22×Dic53C22×C412yes—
162C22×Dic53C22×C412yes—
163C22×Dic53C2×D414yes—
164C22×Dic53C2×D414yes—
165C22×Dic53#4114yes—
166C22⋊C4×C53—48no—
167C22⋊Dic53—28no—
168C22⋊Dic53—48no—
169C22⋊Dic53C22⋊C4116yes—
170C4⋊Dic53—28no—
171C4⋊Dic53—28no—
172#11719919—48no—
173C4×Dic53—28no—
174C4×Dic53—28no—
175C4×Dic53—48no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—24no—
183C23×C53—24no—
184C23×C53—24no—
185C23×C53—24no—
186C23×C53—24no—
187C23×C53—24no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53C2×C22⋊C414yes—
199C23×C53C23×C411yescharacteristic
200C23×C53C22×D412yes—
201C23×C53C2×#4112yes—
202C23×C53C2×#4111yescharacteristic
203C23×C53C2×#4111yescharacteristic
204C23×C53C2×#4112yes—
205C23×C53C2×#4112yes—
206C23×C53H(5,𝔽2)11yescharacteristic
207C23×C53H(5,𝔽2)11yescharacteristic
208C23×C532−1+411yescharacteristic
209C23×C532−1+411yescharacteristic
210C2×C4×C53—232no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—24no—
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#11714yes—
234C2×C4×C53#11714yes—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—24no—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53C2×C22⋊C414yes—
268C2×Dic53C2×C22⋊C414yes—
269#13943—531nomaximal, Sylow
270C22×C53—24no—
271C22×C53—24no—
272C22×C53—24no—
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—44no—
277C22×C53—44no—
278C22×C53—44no—
279C22×C53—44no—
280C22×C53—42no—
281C22×C53C22×C22⋊C411yescharacteristic
282C22×C53C2×#11711yescharacteristic
283C22×C53C2×#11711yescharacteristic
284C22×C53C4×#4111yescharacteristic
285C22×C53C4×#4111yescharacteristic
286C22×C53#52011yescharacteristic
287C22×C53#52011yescharacteristic
288C22×C53#52512yes—
289C22×C53#52512yes—
290C22×C53#52912yes—
291C22×C53#52912yes—
292C22×C53#54011yescharacteristic
293C22×C53#54111yescharacteristic
294C22×C53#54211yescharacteristic
295C22×C53#54211yescharacteristic
296C22×C53#54612yes—
297C22×C53#54612yes—
298C22×C53#54712yes—
299C22×C53#54712yes—
300C22×C53#55212yes—
301C22×C53#55312yes—
302C22×C53#55512yes—
303C22×C53#55612yes—
304C4×C53—24no—
305C4×C53—24no—
306C4×C53—44no—
307C4×C53—82no—
308Dic53—24no—
309Dic53—24no—
310Dic53—42no—
311Dic53—42no—
312Dic53—44no—
313Dic53—82no—
314C22×#117—531no—
315C2×#388—532no—
316#1441—532no—
317#1440—532no—
318#1436—532no—
319C4⋊#95—531no—
320#1434—531no—
321#1434—531no—
322C22⋊#95—531no—
323C22⋊#95D5311yescharacteristic
324C4×#95—531no—
325C2×C53—44no—
326C2×C53#228011yescharacteristic
327C2×C53#229811yescharacteristic
328C2×C53#231411yescharacteristic
329C2×C53#233911yescharacteristic
330C2×C53#233911yescharacteristic
331C2×C53#234611yescharacteristic
332C2×C53#234711yescharacteristic
333C2×C53#247012yes—
334C2×C53#247412yes—
335C2×C53#247612yes—
336C2×C53#247812yes—
337C24×C4C2×D5311yescharacteristic
338C2×#117—538no—
339C2×#117—1068no—
340C22×C4⋊C4—531no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4—532no—
343C22×C22⋊C4—531no—
344C22×C22⋊C4—531no—
345C22×C22⋊C4—532no—
346C22×C22⋊C4C2×D5312yes—
347C22×C42—531no—
348C22×C42—531no—
349C22×C42—532no—
350#388—1068no—
351C2×#95—531no—
352C2×#95—532no—
353C2×#95—531no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95C2×D5311yescharacteristic
361C2×#95C2×D5311yescharacteristic
362C2×#95C2×D5312yes—
363C53#1394311yescharacteristic, complemented, Sylow
364C25C22×D5311yescharacteristic
365C23×C4—24no—
366C23×C4—24no—
367C23×C4—22no—
368C23×C4—22no—
369C23×C4—531no—
370C23×C4—531no—
371C23×C4—531no—
372C23×C4—531no—
373C23×C4—532no—
374C23×C4—532no—
375C23×C4C4×D5314yes—
376C23×C4C22×D5312yes—
377C23×C4C22×D5312yes—
378C23×C4C22×D5311yescharacteristic
379C23×C4C22×D5311yescharacteristic
380#117—10632no—
381C2×C4⋊C4—534no—
382C2×C4⋊C4—534no—
383C2×C4⋊C4—1064no—
384C2×C4⋊C4—1064no—
385C2×C4⋊C4—1064no—
386C2×C4⋊C4—1064no—
387C2×C4⋊C4—2122no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—42no—
391C2×C22⋊C4—534no—
392C2×C22⋊C4—538no—
393C2×C22⋊C4—534no—
394C2×C22⋊C4—1064no—
395C2×C22⋊C4—1064no—
396C2×C22⋊C4—1064no—
397C2×C22⋊C4—2122no—
398C2×C42—534no—
399C2×C42—534no—
400C2×C42—1064no—
401C2×C42—1064no—
402C2×C42—1064no—
403C2×C42—1064no—
404C2×C42—2122no—
405#95—22no—
406#95—24no—
407#95—22no—
408#95—1064no—
409#95—1064no—
410#95—1064no—
411#95—1064no—
412#95—1062no—
413#95—1064no—
414#95—1062no—
415#95—2122no—
416#95—2122no—
417C24—22no—
418C24—22no—
419C24—21no—
420C24—21no—
421C24—22no—
422C24—21no—
423C24—21no—
424C24—21no—
425C24—21no—
426C24C2×C4×D5312yes—
427C24D4×D5314yes—
428C24C23×D5311yescharacteristic
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—28no—
438C22×C4—28no—
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—22no—
452C22×C4—532no—
453C22×C4—532no—
454C22×C4—534no—
455C22×C4—534no—
456C22×C4—532no—
457C22×C4—532no—
458C22×C4—534no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1061no—
474C22×C4—1062no—
475C22×C4—1061no—
476C22×C4—1062no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4C2×C4×D5312yes—
487C22×C4C2×C4×D5312yes—
488C22×C4#1171994514yes—
489C22×C4#1171994714yes—
490C22×C4#1171994712yes—
491C22×C4#1171994712yes—
492C4⋊C4—1068no—
493C4⋊C4—1068no—
494C4⋊C4—2128no—
495C22⋊C4—48no—
496C22⋊C4—5316no—
497C22⋊C4—1068no—
498C22⋊C4—2128no—
499C42—1068no—
500C42—1068no—
501C42—2128no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23C22⋊C4×D5314yes—
525C23C22×C4×D5311yescharacteristic
526C23C2×D4×D5312yes—
527C23#41×D5312yes—
528C23C2×#1171994712yes—
529C23C2×#1171994711yescharacteristic
530C23C2×#1171994711yescharacteristic
531C23C2×#1171994512yes—
532C23#1313326211yescharacteristic
533C23#1313326311yescharacteristic
534C23#1313326411yescharacteristic
535C23#1313326511yescharacteristic
536C2×C4—232no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—534no—
560C2×C4—534no—
561C2×C4—1064no—
562C2×C4—1064no—
563C2×C4—1064no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—1064no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1062no—
576C2×C4—1062no—
577C2×C4—1064no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4#1313317314yes—
594C2×C4#1313317314yes—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—44no—
599C22—42no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22C2×C22⋊C4×D5311yescharacteristic
607C22C2×#1313317311yescharacteristic
608C22C2×#1313317311yescharacteristic
609C22#1681109111yescharacteristic
610C22C4×#1171994511yescharacteristic
611C22#1681110311yescharacteristic
612C22#1681110411yescharacteristic
613C22#1681112912yes—
614C22#1681112912yes—
615C22#1681115312yes—
616C22#1681115512yes—
617C22#1681122611yescharacteristic
618C22#1681123311yescharacteristic
619C22#1681124011yescharacteristic
620C22#1681124611yescharacteristic
621C22#1681126312yes—
622C22#1681126512yes—
623C22#1681127112yes—
624C22#1681127412yes—
625C22#1681131312yes—
626C22#1681132012yes—
627C22#1681133812yes—
628C22#1681134412yes—
629C4—24no—
630C4—24no—
631C4—44no—
632C4—82no—
633C4—1064no—
634C4—1064no—
635C4—4242no—
636C4—2122no—
637C4—2122no—
638C4—2124no—
639C2—44no—
640C2#2027709311yescharacteristic
641C2#2027721111yescharacteristic
642C2#2027732211yescharacteristic
643C2#2027753911yescharacteristic
644C2#2027754211yescharacteristic
645C2#2027759811yescharacteristic
646C2#2027761011yescharacteristic
647C2#2027880812yes—
648C2#2027886812yes—
649C2#2027890012yes—
650C2#2027893012yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C22⋊#95C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC22⋊#95D53
hypocenterC53#13943
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
3. #20272967C2semidirect
5. #20272949C2semidirect
6. #20272943C2semidirect
7. #20272941C2semidirect
8. #20272940C2semidirect
10. #95×Dic53C2semidirect
13. C2×#13133176C4semidirect
363. C53#13943semidirect

Maximal subgroups

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

subgroupindex
C22×#131331762
C2×#168105112
#202729672
#202729622
#202729492
#202729432
#202729412
#202729402
#202729382
#95×Dic532
C53×C22⋊#952
#1394353