Finite Group Databasebeta

← #27311113

Subgroups — #27311113

Order 13568.

1454 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810534C211yesmaximal, characteristic
2C2×#16810514C211yesmaximal, characteristic
3#20272986C211yesmaximal, characteristic, complemented
4#20272976C212yesmaximal, complemented
5#20272974C211yesmaximal, characteristic, complemented
6#20272963C212yesmaximal
7#20272957C211yesmaximal, characteristic
8#20272951C212yesmaximal
9#20272950C212yesmaximal, complemented
10C4×#13133124C211yesmaximal, characteristic, complemented
11C2×C53×#391C211yesmaximal, characteristic, complemented
12C22×#11719919C2211yescharacteristic
13C22×#11719921C2211yescharacteristic
14C22×#11719921C2211yescharacteristic
15C22×#11719921C2212yes—
16C22×#11719921C2211yescharacteristic
17C22×#11719921C2212yes—
18C22×C4×Dic53C2212yes—
19#16810534C418yescomplemented
20#16810514—24no—
21C2×#13133124C2211yescharacteristic
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2212yes—
32C23×Q8×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C42×C53C2211yescharacteristic
35C53×#391—24no—
36C2×#95×C53C2212yes—
37C2×#95×C53C2212yes—
38C24×D53C2311yescharacteristic
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2312yes—
44C2×#11719919—22no—
45C2×#11719919—24no—
46C2×#11719921—22no—
47C2×#11719921—22no—
48C2×#11719921—22no—
49C2×#11719921—22no—
50C2×#11719921—24no—
51C2×#11719921—24no—
52C2×#11719921—42no—
53C2×#11719921—42no—
54C2×#11719921—42no—
55C2×#11719921—42no—
56C2×#11719921—42no—
57C2×#11719921C2×C414yes—
58C2×#11719921C2×C418yes—
59C2×C4×Dic53—24no—
60C2×C4×Dic53—42no—
61C2×C4×Dic53—42no—
62#13133124—22no—
63#13133124—22no—
64#13133124—22no—
65#13133124—24no—
66#13133124—24no—
67#13133124—42no—
68#13133124—42no—
69#13133124—42no—
70#13133124—42no—
71#13133124C2×C418yes—
72#13133124C2×C414yes—
73C22×Q8×C53—22no—
74C22×Q8×C53—22no—
75C22×Q8×C53—42no—
76C22×Q8×C53—42no—
77C22×Q8×C53C2×C414yes—
78C23×C4×C53C2312yes—
79C23×C4×C53C2312yes—
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2311yescharacteristic
83C2×C4⋊C4×C53—22no—
84C2×C4⋊C4×C53—42no—
85C2×C42×C53—42no—
86C2×C42×C53D414yes—
87#95×C53—24no—
88#95×C53—24no—
89C24×C53C2411yescharacteristic
90C23×D53—21no—
91C23×D53—21no—
92C23×D53—22no—
93C23×D53—22no—
94C23×D53—22no—
95C23×D53—22no—
96C23×D53—21no—
97C23×D53—21no—
98C23×D53—21no—
99C23×D53—21no—
100C23×D53C22×C412yes—
101C2×Q8×C53—28no—
102C2×Q8×C53—48no—
103C2×Q8×C53—44no—
104C2×Q8×C53—44no—
105C2×Q8×C53—48no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
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—21no—
120C22×C4×C53—21no—
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×C53C22×C414yes—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53#4112yes—
134C22×C4×C53#4114yes—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—21no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—21no—
157C22×Dic53—22no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53C22×C412yes—
162C22×Dic53C22×C414yes—
163C22×Dic53#4114yes—
164C4⋊C4×C53—44no—
165C42×C53—44no—
166#11719921—28no—
167#11719921—48no—
168#11719921—48no—
169#11719921—44no—
170#11719921—44no—
171#11719919—44no—
172C4×Dic53—48no—
173C23×C53C23×C411yescharacteristic
174C23×C53C22×D411yescharacteristic
175C23×C53C2×#4112yes—
176C23×C53C2×#4111yescharacteristic
177C23×C53C2×#4112yes—
178C23×C53H(5,𝔽2)12yes—
179C23×C53H(5,𝔽2)12yes—
180C23×C53H(5,𝔽2)11yescharacteristic
181C23×C532−1+411yescharacteristic
182C23×C532−1+411yescharacteristic
183C23×C532−1+411yescharacteristic
184C22×D53—22no—
185C22×D53—24no—
186C22×D53—24no—
187C22×D53—22no—
188C22×D53—22no—
189C22×D53—41no—
190C22×D53—42no—
191C22×D53—41no—
192C22×D53—41no—
193C22×D53—41no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—42no—
197C22×D53—42no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—42no—
201C22×D53—41no—
202C22×D53—42no—
203C22×D53—42no—
204C22×D53—41no—
205C22×D53—42no—
206C22×D53—42no—
207Q8×C53—416no—
208C2×C4×C53—24no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
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—42no—
234C2×C4×C53—42no—
235C2×C4×C53C4×D414yes—
236C2×C4×C53#12414yes—
237C2×Dic53—22no—
238C2×Dic53—24no—
239C2×Dic53—24no—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—41no—
246C2×Dic53—42no—
247C2×Dic53—41no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266#14323—531nomaximal, Sylow
267C22×C53C2×C4×D411yescharacteristic
268C22×C53C4×#4112yes—
269C22×C53#51812yes—
270C22×C53#51911yescharacteristic
271C22×C53#52011yescharacteristic
272C22×C53C2×#12411yescharacteristic
273C22×C53#52912yes—
274C22×C53#53211yescharacteristic
275C22×C53#53411yescharacteristic
276C22×C53#53511yescharacteristic
277C22×C53#53611yescharacteristic
278C22×C53#53711yescharacteristic
279C22×C53#53812yes—
280C22×C53#53812yes—
281C22×C53#53812yes—
282C22×C53#53912yes—
283C22×C53#53912yes—
284C22×C53#54111yescharacteristic
285C22×C53#54212yes—
286C22×C53#54212yes—
287C22×C53#54211yescharacteristic
288C22×C53#54212yes—
289C22×C53#54212yes—
290C2×D53—42no—
291C2×D53—42no—
292C2×D53—44no—
293C2×D53—44no—
294C2×D53—42no—
295C2×D53—81no—
296C2×D53—81no—
297C2×D53—82no—
298C2×D53—82no—
299C2×D53—81no—
300C2×D53—81no—
301C4×C53—24no—
302C4×C53—24no—
303C4×C53—44no—
304C4×C53—42no—
305C4×C53—82no—
306Dic53—44no—
307Dic53—44no—
308Dic53—42no—
309Dic53—81no—
310Dic53—81no—
311C2×#393—531no—
312C2×#391D5311yescharacteristic, complemented
313C2×#388—531no—
314#1446—531no—
315#1443—532no—
316#1442—531no—
317#1440—532no—
318#1439—531no—
319#1437—532no—
320#1436—532no—
321C4×#95—531no—
322C2×C53C4×#12411yescharacteristic
323C2×C53#232111yescharacteristic
324C2×C53#232311yescharacteristic
325C2×C53#233212yes—
326C2×C53#233712yes—
327C2×C53#263911yescharacteristic
328C2×C53#264011yescharacteristic
329C2×C53#265012yes—
330C2×C53#265312yes—
331C2×C53#266211yescharacteristic
332C2×C53#266311yescharacteristic
333D53—82no—
334C23×Q8C2×D5311yescharacteristic
335C22×C4⋊C4—531no—
336C22×C4⋊C4C2×D5311yescharacteristic
337C22×C22⋊C4—531no—
338C22×C22⋊C4—532no—
339C22×C22⋊C4—531no—
340C22×C22⋊C4—532no—
341C22×C22⋊C4—531no—
342C22×C42—532no—
343C22×C42C2×D5311yescharacteristic
344#393—538no—
345#391—24no—
346#388—1064no—
347C2×#95—531no—
348C2×#95—532no—
349C2×#95—531no—
350C2×#95—532no—
351C2×#95—532no—
352C2×#95—531no—
353C2×#95—531no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95C2×D5312yes—
359C2×#95C2×D5312yes—
360C53#1432311yescharacteristic, complemented, Sylow
361C25—531no—
362C22×Q8—22no—
363C22×Q8—22no—
364C22×Q8—42no—
365C22×Q8—42no—
366C22×Q8C4×D5314yes—
367C23×C4—531no—
368C23×C4—532no—
369C23×C4—532no—
370C23×C4—531no—
371C23×C4—531no—
372C23×C4C22×D5312yes—
373C23×C4C22×D5312yes—
374C23×C4C22×D5311yescharacteristic
375C23×C4C22×D5311yescharacteristic
376C23×C4C22×D5311yescharacteristic
377C2×C4⋊C4—22no—
378C2×C4⋊C4—42no—
379C2×C4⋊C4—1064no—
380C2×C4⋊C4—1062no—
381C2×C22⋊C4—534no—
382C2×C22⋊C4—538no—
383C2×C22⋊C4—1062no—
384C2×C22⋊C4—1062no—
385C2×C22⋊C4—1064no—
386C2×C22⋊C4—1064no—
387C2×C22⋊C4—1062no—
388C2×C22⋊C4—1062no—
389C2×C22⋊C4—2122no—
390C2×C22⋊C4—2122no—
391C2×C22⋊C4—2122no—
392C2×C22⋊C4—2122no—
393C2×C22⋊C4—2122no—
394C2×C42—42no—
395C2×C42—1064no—
396C2×C42—2122no—
397C2×C42—2122no—
398C2×C42C22⋊D5314yes—
399#95—24no—
400#95—24no—
401#95—538no—
402#95—534no—
403#95—1064no—
404#95—1064no—
405#95—1062no—
406#95—1062no—
407#95—1062no—
408#95—2122no—
409#95—2122no—
410#95—2122no—
411#95—2122no—
412C24—532no—
413C24—1061no—
414C24—1062no—
415C24—1061no—
416C24—1062no—
417C24—1062no—
418C24—1062no—
419C24—1061no—
420C24—1061no—
421C24—1061no—
422C24—1061no—
423C24C23×D5311yescharacteristic
424C2×Q8—28no—
425C2×Q8—48no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—48no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—532no—
452C22×C4—534no—
453C22×C4—534no—
454C22×C4—1061no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1061no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4C2×C4×D5314yes—
481C22×C4C2×C4×D5312yes—
482C22×C4#1171994512yes—
483C22×C4#1171994914yes—
484C22×C4C2×C22⋊D5312yes—
485C22×C4C2×C22⋊D5312yes—
486C22×C4C2×C22⋊D5312yes—
487C4⋊C4—44no—
488C4⋊C4—2124no—
489C22⋊C4—1068no—
490C22⋊C4—2128no—
491C22⋊C4—2124no—
492C22⋊C4—2128no—
493C22⋊C4—2124no—
494C42—44no—
495C42—2128no—
496C23—1062no—
497C23—1064no—
498C23—1064no—
499C23—1062no—
500C23—1062no—
501C23—2121no—
502C23—2122no—
503C23—2121no—
504C23—2121no—
505C23—2121no—
506C23—2121no—
507C23—2121no—
508C23—2122no—
509C23—2122no—
510C23—2122no—
511C23—2122no—
512C23—2122no—
513C23—2121no—
514C23—2122no—
515C23—2122no—
516C23—2121no—
517C23—2122no—
518C23—2122no—
519C23C22×C4×D5311yescharacteristic
520C23C22×C22⋊D5311yescharacteristic
521C23#41×D5312yes—
522C23C2×#1171994511yescharacteristic
523C23C2×#1171994912yes—
524C23#1313326212yes—
525C23#1313326212yes—
526C23#1313326311yescharacteristic
527C23#1313326411yescharacteristic
528C23#1313326511yescharacteristic
529C23#1313326511yescharacteristic
530Q8—416no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
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—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—1062no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—1064no—
562C2×C4—1064no—
563C2×C4—1064no—
564C2×C4—1064no—
565C2×C4—1064no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2121no—
577C2×C4—2122no—
578C2×C4—2121no—
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×C4C4×C22⋊D5314yes—
588C2×C4#1313320614yes—
589C22—4241no—
590C22—4241no—
591C22—4241no—
592C22—4241no—
593C22—4242no—
594C22—4242no—
595C22—2122no—
596C22—2124no—
597C22—2122no—
598C22—2122no—
599C22—2124no—
600C22C2×C4×C22⋊D5311yescharacteristic
601C22C4×#1171994912yes—
602C22#1681109612yes—
603C22#1681109911yescharacteristic
604C22#1681110511yescharacteristic
605C22C2×#1313320611yescharacteristic
606C22#1681116012yes—
607C22#1681116911yescharacteristic
608C22#1681118211yescharacteristic
609C22#1681119011yescharacteristic
610C22#1681119711yescharacteristic
611C22#1681120111yescharacteristic
612C22#1681120712yes—
613C22#1681121112yes—
614C22#1681121312yes—
615C22#1681121912yes—
616C22#1681122212yes—
617C22#1681123111yescharacteristic
618C22#1681123712yes—
619C22#1681124312yes—
620C22#1681124511yescharacteristic
621C22#1681124712yes—
622C22#1681124712yes—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—82no—
628C4—4241no—
629C4—4241no—
630C4—2122no—
631C4—2124no—
632C4—2124no—
633C2—4242no—
634C2C4×#1313320611yescharacteristic
635C2#2027737311yescharacteristic
636C2#2027739111yescharacteristic
637C2#2027747012yes—
638C2#2027752712yes—
639C2#2028064911yescharacteristic
640C2#2028065611yescharacteristic
641C2#2028073912yes—
642C2#2028078412yes—
643C2#2028085411yescharacteristic
644C2#2028086611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#391C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC2×#391D53
hypocenterC53#14323
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. #20272986C2semidirect
4. #20272976C2semidirect
5. #20272974C2semidirect
9. #20272950C2semidirect
10. C4×#13133124C2semidirect
11. C2×C53×#391C2semidirect
19. #16810534C4semidirect
312. C2×#391D53semidirect
360. C53#14323semidirect

Maximal subgroups

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

subgroupindex
C2×#168105342
C2×#168105142
#202729862
#202729762
#202729742
#202729632
#202729572
#202729512
#202729502
C4×#131331242
C2×C53×#3912
#1432353