Finite Group Databasebeta

← #27314775

Subgroups — #27314775

Order 13568.

1390 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×#16810536C211yesmaximal, characteristic
2C2×#16810529C211yesmaximal, characteristic, complemented
3C2×#16810514C211yesmaximal, characteristic
4#20272985C211yesmaximal, characteristic, complemented
5#20272980C211yesmaximal, characteristic, complemented
6#20272971C212yesmaximal, complemented
7#20272970C212yesmaximal, complemented
8#20272963C212yesmaximal
9#20272961C211yesmaximal, characteristic
10#20272961C212yesmaximal
11C53×#1434C211yesmaximal, characteristic, complemented
12C23×Dic106C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×C4⋊Dic53C2212yes—
15C22×#11719919C2211yescharacteristic
16C22×#11719921C2211yescharacteristic
17C22×#11719921C2211yescharacteristic
18C22×#11719921C2211yescharacteristic
19C22×#11719921C2212yes—
20C22×#11719921C2212yes—
21#16810536C418yescomplemented
22#16810529—24no—
23#16810514—24no—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C22×C42×C53C2211yescharacteristic
34C2×#95×C53C2211yescharacteristic
35C2×#95×C53C2212yes—
36C2×#95×C53C2212yes—
37C2×#95×C53C2211yescharacteristic
38C24×D53C2311yescharacteristic
39C22×Dic106—22no—
40C22×Dic106—22no—
41C22×Dic106—42no—
42C22×Dic106—42no—
43C22×Dic106C2×C414yes—
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2311yescharacteristic
46C23×Dic53C2312yes—
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2312yes—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—42no—
51C2×C4⋊Dic53—42no—
52C2×C4⋊Dic53—42no—
53C2×C4⋊Dic53—42no—
54C2×#11719919—22no—
55C2×#11719919—42no—
56C2×#11719921—22no—
57C2×#11719921—22no—
58C2×#11719921—22no—
59C2×#11719921—22no—
60C2×#11719921—42no—
61C2×#11719921—42no—
62C2×#11719921—42no—
63C2×#11719921—42no—
64C2×#11719921—42no—
65C2×#11719921—42no—
66C2×#11719921—42no—
67C2×#11719921C2×C414yes—
68C2×#11719921C2×C418yes—
69#13133124—22no—
70#13133124—22no—
71#13133124—24no—
72#13133124—24no—
73#13133124—42no—
74#13133124—42no—
75#13133124—42no—
76#13133124—42no—
77#13133124C2×C418yes—
78C23×C4×C53C2312yes—
79C23×C4×C53C2312yes—
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2311yescharacteristic
83C2×C42×C53—42no—
84C2×C42×C53D414yes—
85#95×C53—22no—
86#95×C53—42no—
87#95×C53—42no—
88#95×C53C2×C414yes—
89C24×C53C2411yescharacteristic
90C23×D53—21no—
91C23×D53—21no—
92C23×D53—22no—
93C23×D53—22no—
94C23×D53—22no—
95C23×D53—21no—
96C23×D53—21no—
97C23×D53—21no—
98C23×D53—21no—
99C23×D53—22no—
100C23×D53C22×C412yes—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—21no—
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×C53C22×C414yes—
125C22×C4×C53C22×C412yes—
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2×D412yes—
128C22×C4×C53C2×D412yes—
129C22×C4×C53#4112yes—
130C22×Dic53—22no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—21no—
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—22no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—22no—
157C22×Dic53C22×C412yes—
158C22×Dic53C22×C414yes—
159C2×Dic106—28no—
160C2×Dic106—48no—
161C2×Dic106—44no—
162C2×Dic106—44no—
163C2×Dic106—48no—
164C42×C53—44no—
165#11719921—28no—
166#11719921—48no—
167#11719921—44no—
168#11719921—44no—
169#11719921—48no—
170C4⋊Dic53—44no—
171C4⋊Dic53—48no—
172#11719919—44no—
173C23×C53C23×C411yescharacteristic
174C23×C53C22×D411yescharacteristic
175C23×C53C2×#4111yescharacteristic
176C23×C53H(5,𝔽2)12yes—
177C23×C53H(5,𝔽2)11yescharacteristic
178C23×C53H(5,𝔽2)12yes—
179C23×C53H(5,𝔽2)12yes—
180C23×C532−1+411yescharacteristic
181C23×C532−1+411yescharacteristic
182C23×C532−1+412yes—
183C23×C532−1+411yescharacteristic
184C22×D53—22no—
185C22×D53—24no—
186C22×D53—22no—
187C22×D53—22no—
188C22×D53—24no—
189C22×D53—41no—
190C22×D53—41no—
191C22×D53—41no—
192C22×D53—41no—
193C22×D53—41no—
194C22×D53—41no—
195C22×D53—42no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—42no—
201C22×D53—42no—
202C22×D53—42no—
203C22×D53—42no—
204C22×D53—42no—
205C22×D53—42no—
206C22×D53—42no—
207C2×C4×C53—24no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—24no—
211C2×C4×C53—24no—
212C2×C4×C53—24no—
213C2×C4×C53—24no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
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×Dic53—24no—
237C2×Dic53—22no—
238C2×Dic53—24no—
239C2×Dic53—24no—
240C2×Dic53—24no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—41no—
249C2×Dic53—42no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—42no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
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—
265Dic106—416no—
266#14625—531nomaximal, Sylow
267C22×C53C2×C4×D411yescharacteristic
268C22×C53#51812yes—
269C22×C53#51911yescharacteristic
270C22×C53#52011yescharacteristic
271C22×C53#52012yes—
272C22×C53#53411yescharacteristic
273C22×C53#53511yescharacteristic
274C22×C53#53511yescharacteristic
275C22×C53#53611yescharacteristic
276C22×C53#53611yescharacteristic
277C22×C53#53711yescharacteristic
278C22×C53#53812yes—
279C22×C53#54111yescharacteristic
280C22×C53#54211yescharacteristic
281C22×C53#54212yes—
282C22×C53H(3,𝔽4)12yes—
283C22×C53#56212yes—
284C22×C53#56212yes—
285C22×C53#56212yes—
286C22×C53#56212yes—
287C22×C53#56312yes—
288C22×C53#56312yes—
289C22×C53#56312yes—
290C2×D53—42no—
291C2×D53—42no—
292C2×D53—42no—
293C2×D53—44no—
294C2×D53—44no—
295C2×D53—81no—
296C2×D53—81no—
297C2×D53—81no—
298C2×D53—82no—
299C2×D53—82no—
300C2×D53—81no—
301C4×C53—24no—
302C4×C53—42no—
303C4×C53—44no—
304C4×C53—42no—
305C4×C53—82no—
306Dic53—44no—
307Dic53—42no—
308Dic53—81no—
309Dic53—82no—
310Dic53—81no—
311C2×#393—531no—
312C2×#391—531no—
313C2×#388—531no—
314#1446—531no—
315#1444—531no—
316#1442—532no—
317#1442—532no—
318#1440—532no—
319#1439—531no—
320#1439—532no—
321#1434D5311yescharacteristic, complemented
322C2×C53#232111yescharacteristic
323C2×C53#232311yescharacteristic
324C2×C53#232611yescharacteristic
325C2×C53#237812yes—
326C2×C53#237912yes—
327C2×C53#281411yescharacteristic
328C2×C53#281611yescharacteristic
329C2×C53#281911yescharacteristic
330C2×C53#282111yescharacteristic
331C2×C53#283112yes—
332C2×C53#283212yes—
333D53—82no—
334C23×Q8—531no—
335C22×C4⋊C4—531no—
336C22×C4⋊C4—531no—
337C22×C4⋊C4—532no—
338C22×C22⋊C4—531no—
339C22×C22⋊C4—532no—
340C22×C22⋊C4—531no—
341C22×C22⋊C4—531no—
342C22×C22⋊C4—532no—
343C22×C42C2×D5311yescharacteristic
344#393—538no—
345#391—1064no—
346#388—1064no—
347C2×#95—532no—
348C2×#95—532no—
349C2×#95—532no—
350C2×#95—531no—
351C2×#95—531no—
352C2×#95—532no—
353C2×#95—532no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95C2×D5311yescharacteristic
357C2×#95C2×D5312yes—
358C2×#95C2×D5312yes—
359C2×#95C2×D5311yescharacteristic
360C53#1462511yescharacteristic, complemented, Sylow
361C25—531no—
362C22×Q8—534no—
363C22×Q8—1062no—
364C22×Q8—1062no—
365C22×Q8—2122no—
366C22×Q8—2122no—
367C23×C4—531no—
368C23×C4—532no—
369C23×C4—531no—
370C23×C4—532no—
371C23×C4—531no—
372C23×C4C22×D5312yes—
373C23×C4C22×D5312yes—
374C23×C4C22×D5311yescharacteristic
375C23×C4C22×D5311yescharacteristic
376C23×C4C22×D5311yescharacteristic
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4—2122no—
380C2×C4⋊C4—2122no—
381C2×C4⋊C4—2122no—
382C2×C4⋊C4—2122no—
383C2×C4⋊C4—2122no—
384C2×C22⋊C4—534no—
385C2×C22⋊C4—538no—
386C2×C22⋊C4—1062no—
387C2×C22⋊C4—1062no—
388C2×C22⋊C4—1062no—
389C2×C22⋊C4—1062no—
390C2×C22⋊C4—2122no—
391C2×C22⋊C4—2122no—
392C2×C22⋊C4—2122no—
393C2×C22⋊C4—2122no—
394C2×C22⋊C4—2122no—
395C2×C22⋊C4—2122no—
396C2×C22⋊C4—2122no—
397C2×C42—42no—
398C2×C42D21214yes—
399#95—22no—
400#95—42no—
401#95—42no—
402#95—538no—
403#95—1062no—
404#95—1062no—
405#95—1064no—
406#95—1064no—
407#95—2122no—
408#95—2122no—
409#95—2122no—
410#95—2122no—
411#95C4×D5314yes—
412C24—532no—
413C24—1061no—
414C24—1061no—
415C24—1062no—
416C24—1061no—
417C24—1062no—
418C24—1062no—
419C24—1061no—
420C24—1061no—
421C24—1062no—
422C24—1061no—
423C24C23×D5311yescharacteristic
424C2×Q8—1068no—
425C2×Q8—2128no—
426C2×Q8—2124no—
427C2×Q8—2124no—
428C2×Q8—2128no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
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—21no—
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—534no—
454C22×C4—1062no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1061no—
465C22×C4—1062no—
466C22×C4—1061no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1061no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1062no—
480C22×C4—1061no—
481C22×C4C2×C4×D5314yes—
482C22×C4C2×C4×D5312yes—
483C22×C4C2×D21212yes—
484C22×C4C2×D21212yes—
485C22×C4C2×D21212yes—
486C22×C4#1171994512yes—
487C4⋊C4—2124no—
488C4⋊C4—2124no—
489C4⋊C4—2128no—
490C22⋊C4—1068no—
491C22⋊C4—2128no—
492C22⋊C4—2124no—
493C22⋊C4—2124no—
494C22⋊C4—2128no—
495C42—44no—
496C23—1062no—
497C23—1064no—
498C23—1062no—
499C23—1062no—
500C23—1064no—
501C23—2122no—
502C23—2122no—
503C23—2122no—
504C23—2122no—
505C23—2122no—
506C23—2122no—
507C23—2122no—
508C23—2122no—
509C23—2121no—
510C23—2121no—
511C23—2121no—
512C23—2121no—
513C23—2121no—
514C23—2121no—
515C23—2122no—
516C23—2121no—
517C23—2121no—
518C23—2122no—
519C23C22×C4×D5311yescharacteristic
520C23C22×D21211yescharacteristic
521C23C2×#1171994511yescharacteristic
522C23#1313326212yes—
523C23#1313326211yescharacteristic
524C23#1313326312yes—
525C23#1313326312yes—
526C23#1313326411yescharacteristic
527C23#1313326411yescharacteristic
528C23#1313326412yes—
529C23#1313326511yescharacteristic
530Q8—21216no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
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—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—1062no—
560C2×C4—1064no—
561C2×C4—1064no—
562C2×C4—1064no—
563C2×C4—1064no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4—2122no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2121no—
583C2×C4—2122no—
584C2×C4—2121no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4C4×D21214yes—
589C22—4242no—
590C22—4241no—
591C22—4241no—
592C22—4241no—
593C22—4241no—
594C22—4242no—
595C22—2124no—
596C22—2124no—
597C22—2122no—
598C22—2122no—
599C22—2122no—
600C22C2×C4×D21211yescharacteristic
601C22#1681109512yes—
602C22#1681109911yescharacteristic
603C22#1681110211yescharacteristic
604C22#1681110312yes—
605C22#1681118511yescharacteristic
606C22#1681119211yescharacteristic
607C22#1681119211yescharacteristic
608C22#1681119911yescharacteristic
609C22#1681119911yescharacteristic
610C22#1681120411yescharacteristic
611C22#1681120612yes—
612C22#1681123111yescharacteristic
613C22#1681123811yescharacteristic
614C22#1681124412yes—
615C22#1681136512yes—
616C22#1681136812yes—
617C22#1681137012yes—
618C22#1681137012yes—
619C22#1681137112yes—
620C22#1681137412yes—
621C22#1681137512yes—
622C22#1681137512yes—
623C4—24no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—82no—
628C4—4241no—
629C4—4242no—
630C4—4241no—
631C4—2124no—
632C4—2122no—
633C2—4242no—
634C2#2027737511yescharacteristic
635C2#2027739811yescharacteristic
636C2#2027742611yescharacteristic
637C2#2027788012yes—
638C2#2027788412yes—
639C2#2028273011yescharacteristic
640C2#2028274511yescharacteristic
641C2#2028276111yescharacteristic
642C2#2028277611yescharacteristic
643C2#2028286912yes—
644C2#2028288012yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1434C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1434D53
hypocenterC53#14625
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C2×#16810529C2semidirect
4. #20272985C2semidirect
5. #20272980C2semidirect
6. #20272971C2semidirect
7. #20272970C2semidirect
11. C53×#1434C2semidirect
21. #16810536C4semidirect
321. #1434D53semidirect
360. C53#14625semidirect

Maximal subgroups

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

subgroupindex
C2×#168105362
C2×#168105292
C2×#168105142
#202729852
#202729802
#202729712
#202729702
#202729632
#202729612
C53×#14342
#1462553