Finite Group Databasebeta

← #27320479

Subgroups — #27320479

Order 13568.

1334 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810514C211yesmaximal, characteristic
2#20272976C212yesmaximal, complemented
3#20272974C211yesmaximal, characteristic, complemented
4#20272974C211yesmaximal, characteristic, complemented
5#20272963C212yesmaximal
6#20272957C211yesmaximal, characteristic
7#20272957C211yesmaximal, characteristic
8#20272953C212yesmaximal
9#20272948C212yesmaximal, complemented
10#20272942C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C53C211yesmaximal, characteristic, complemented
12C22×#11719919C2211yescharacteristic
13C22×#11719921C2211yescharacteristic
14C22×#11719921C2211yescharacteristic
15C22×#11719921C2211yescharacteristic
16C22×#11719921C2212yes—
17C22×#11719921C2212yes—
18#16810514—24no—
19C2×#13133124C2212yes—
20C2×#13133124C2211yescharacteristic
21C2×#13133124C2211yescharacteristic
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×C4⋊C4×C53C2211yescharacteristic
33C22×C42×C53C2211yescharacteristic
34C22×C42×C53C2212yes—
35C4×C4⋊C4×C53—24no—
36C2×#95×C53C2212yes—
37C24×D53C2311yescharacteristic
38C23×Dic53C2311yescharacteristic
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C2×#11719919—22no—
44C2×#11719919—24no—
45C2×#11719921—22no—
46C2×#11719921—24no—
47C2×#11719921—22no—
48C2×#11719921—22no—
49C2×#11719921—22no—
50C2×#11719921—24no—
51C2×#11719921—24no—
52C2×#11719921—22no—
53C2×#11719921—22no—
54C2×#11719921—24no—
55C2×#11719921—22no—
56C2×#11719921—42no—
57C2×#11719921—42no—
58C2×#11719921—42no—
59#13133124—22no—
60#13133124—22no—
61#13133124—24no—
62#13133124—24no—
63#13133124—22no—
64#13133124—22no—
65#13133124—24no—
66#13133124—24no—
67#13133124—42no—
68#13133124—42no—
69#13133124—42no—
70#13133124—42no—
71C23×C4×C53C2311yescharacteristic
72C23×C4×C53C2312yes—
73C23×C4×C53C2312yes—
74C23×C4×C53C2311yescharacteristic
75C23×C4×C53C2311yescharacteristic
76C2×C4⋊C4×C53—22no—
77C2×C4⋊C4×C53—22no—
78C2×C4⋊C4×C53—24no—
79C2×C4⋊C4×C53—24no—
80C2×C42×C53—24no—
81C2×C42×C53—22no—
82C2×C42×C53—22no—
83C2×C42×C53—42no—
84C2×C42×C53—42no—
85C2×C42×C53D414yes—
86#95×C53—24no—
87C24×C53C2411yescharacteristic
88C23×D53—21no—
89C23×D53—21no—
90C23×D53—22no—
91C23×D53—22no—
92C23×D53—22no—
93C23×D53—21no—
94C23×D53—21no—
95C23×D53—22no—
96C23×D53—21no—
97C23×D53—21no—
98C23×D53#4112yes—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53—22no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53C2×D412yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53C2×D412yes—
125C22×C4×C53#4114yes—
126C22×C4×C53#4114yes—
127C22×C4×C53#4112yes—
128C22×C4×C53#4112yes—
129C22×C4×C53#4114yes—
130C22×Dic53—21no—
131C22×Dic53—22no—
132C22×Dic53—21no—
133C22×Dic53—22no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—21no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—21no—
158C22×Dic53#4112yes—
159C4⋊C4×C53—28no—
160C4⋊C4×C53—28no—
161C42×C53—28no—
162C42×C53—48no—
163#11719921—28no—
164#11719921—48no—
165#11719921—48no—
166#11719921—44no—
167#11719921—44no—
168#11719919—44no—
169C23×C53C22×D411yescharacteristic
170C23×C53C2×#4111yescharacteristic
171C23×C53C2×#4111yescharacteristic
172C23×C53C2×#4111yescharacteristic
173C23×C53C2×#4111yescharacteristic
174C23×C53C2×#4112yes—
175C23×C53C2×#4112yes—
176C23×C53C2×#4112yes—
177C23×C53H(5,𝔽2)12yes—
178C23×C53H(5,𝔽2)11yescharacteristic
179C23×C532−1+411yescharacteristic
180C22×D53—22no—
181C22×D53—24no—
182C22×D53—24no—
183C22×D53—22no—
184C22×D53—22no—
185C22×D53—41no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—42no—
190C22×D53—42no—
191C22×D53—42no—
192C22×D53—41no—
193C22×D53—41no—
194C22×D53—42no—
195C22×D53—42no—
196C22×D53—42no—
197C22×D53—42no—
198C22×D53—42no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—42no—
202C22×D53—42no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—24no—
207C2×C4×C53—24no—
208C2×C4×C53—22no—
209C2×C4×C53—24no—
210C2×C4×C53—24no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—22no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
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—41no—
228C2×C4×C53—41no—
229C2×C4×C53—42no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53#12314yes—
233C2×C4×C53#12414yes—
234C2×Dic53—22no—
235C2×Dic53—24no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—42no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
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—
263#15110—531nomaximal, Sylow
264C22×C53C2×#12311yescharacteristic
265C22×C53#52511yescharacteristic
266C22×C53#52511yescharacteristic
267C22×C53#52511yescharacteristic
268C22×C53C2×#12411yescharacteristic
269C22×C53#52912yes—
270C22×C53#52912yes—
271C22×C53#52912yes—
272C22×C53#52912yes—
273C22×C53#53511yescharacteristic
274C22×C53#53711yescharacteristic
275C22×C53#53812yes—
276C22×C53#53912yes—
277C22×C53#55112yes—
278C22×C53#55211yescharacteristic
279C22×C53#55311yescharacteristic
280C22×C53#55312yes—
281C22×C53#55312yes—
282C22×C53#55312yes—
283C22×C53#55511yescharacteristic
284C22×C53#55611yescharacteristic
285C22×C53#55612yes—
286C22×C53#55912yes—
287C2×D53—44no—
288C2×D53—44no—
289C2×D53—42no—
290C2×D53—42no—
291C2×D53—42no—
292C2×D53—81no—
293C2×D53—82no—
294C2×D53—82no—
295C2×D53—81no—
296C2×D53—81no—
297C2×D53—81no—
298C4×C53—24no—
299C4×C53—24no—
300C4×C53—44no—
301C4×C53—44no—
302C4×C53—42no—
303Dic53—42no—
304Dic53—81no—
305Dic53—82no—
306Dic53—82no—
307Dic53—81no—
308C2×#388—531no—
309C2×C4×C4⋊C4D5311yescharacteristic, complemented
310#1443—532no—
311#1442—531no—
312#1442—531no—
313#1440—532no—
314#1439—531no—
315#1439—531no—
316#1437—532no—
317#1436—532no—
318#1434—531no—
319C2×C53#239811yescharacteristic
320C2×C53#248511yescharacteristic
321C2×C53#250411yescharacteristic
322C2×C53#257312yes—
323C2×C53#259211yescharacteristic
324C2×C53#259312yes—
325C2×C53#259412yes—
326C2×C53#259512yes—
327C2×C53#259811yescharacteristic
328C2×C53#263611yescharacteristic
329C2×C53#285311yescharacteristic
330D53—82no—
331C22×C4⋊C4—531no—
332C22×C4⋊C4C2×D5311yescharacteristic
333C22×C4⋊C4C2×D5311yescharacteristic
334C22×C22⋊C4—531no—
335C22×C22⋊C4—531no—
336C22×C22⋊C4—531no—
337C22×C22⋊C4—532no—
338C22×C22⋊C4—532no—
339C22×C42C2×D5311yescharacteristic
340C22×C42C2×D5312yes—
341#388—1064no—
342C4×C4⋊C4—24no—
343C2×#95—532no—
344C2×#95—531no—
345C2×#95—531no—
346C2×#95—532no—
347C2×#95—532no—
348C2×#95—532no—
349C2×#95—531no—
350C2×#95—531no—
351C2×#95—532no—
352C2×#95—532no—
353C2×#95—532no—
354C2×#95—532no—
355C2×#95C2×D5312yes—
356C53#1511011yescharacteristic, complemented, Sylow
357C25—531no—
358C23×C4—531no—
359C23×C4—531no—
360C23×C4—531no—
361C23×C4—532no—
362C23×C4—532no—
363C23×C4C22×D5312yes—
364C23×C4C22×D5312yes—
365C23×C4C22×D5311yescharacteristic
366C23×C4C22×D5311yescharacteristic
367C23×C4C22×D5311yescharacteristic
368C2×C4⋊C4—24no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—1064no—
373C2×C4⋊C4—1062no—
374C2×C22⋊C4—1062no—
375C2×C22⋊C4—1062no—
376C2×C22⋊C4—1062no—
377C2×C22⋊C4—1064no—
378C2×C22⋊C4—1064no—
379C2×C22⋊C4—1064no—
380C2×C22⋊C4—1062no—
381C2×C22⋊C4—1062no—
382C2×C22⋊C4—1064no—
383C2×C22⋊C4—1062no—
384C2×C22⋊C4—1062no—
385C2×C22⋊C4—2122no—
386C2×C22⋊C4—2122no—
387C2×C22⋊C4—2122no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—24no—
391C2×C42—42no—
392C2×C42—42no—
393C2×C42C22⋊D5314yes—
394#95—24no—
395#95—1064no—
396#95—1064no—
397#95—1064no—
398#95—1062no—
399#95—1062no—
400#95—1064no—
401#95—1062no—
402#95—1062no—
403#95—2122no—
404#95—2122no—
405#95—2122no—
406#95—2122no—
407C24—532no—
408C24—1061no—
409C24—1061no—
410C24—1062no—
411C24—1061no—
412C24—1061no—
413C24—1062no—
414C24—1062no—
415C24—1062no—
416C24—1061no—
417C24—1061no—
418C24C23×D5311yescharacteristic
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—532no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1061no—
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—1061no—
469C22×C4—1062no—
470C22×C4—1061no—
471C22×C4#1171994514yes—
472C22×C4#1171994514yes—
473C22×C4#1171994512yes—
474C22×C4#1171994512yes—
475C22×C4#1171994914yes—
476C22×C4C2×C22⋊D5312yes—
477C22×C4C2×C22⋊D5312yes—
478C22×C4C2×C22⋊D5312yes—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—2124no—
482C22⋊C4—1068no—
483C22⋊C4—2128no—
484C22⋊C4—2128no—
485C22⋊C4—2124no—
486C22⋊C4—2124no—
487C42—28no—
488C42—48no—
489C23—1064no—
490C23—1062no—
491C23—1062no—
492C23—1062no—
493C23—1064no—
494C23—2122no—
495C23—2122no—
496C23—2122no—
497C23—2121no—
498C23—2121no—
499C23—2122no—
500C23—2122no—
501C23—2121no—
502C23—2121no—
503C23—2121no—
504C23—2121no—
505C23—2122no—
506C23—2122no—
507C23—2122no—
508C23—2121no—
509C23—2121no—
510C23—2122no—
511C23—2122no—
512C23C22×C22⋊D5311yescharacteristic
513C23#41×D5311yescharacteristic
514C23#41×D5311yescharacteristic
515C23C2×#1171994512yes—
516C23C2×#1171994511yescharacteristic
517C23C2×#1171994511yescharacteristic
518C23C2×#1171994512yes—
519C23C2×#1171994912yes—
520C23#1313326212yes—
521C23#1313326311yescharacteristic
522C23#1313326411yescharacteristic
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—1062no—
553C2×C4—1064no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2122no—
581C2×C4#1313320014yes—
582C2×C4#1313320614yes—
583C22—4241no—
584C22—4241no—
585C22—4242no—
586C22—4242no—
587C22—4241no—
588C22—4241no—
589C22—2124no—
590C22—2124no—
591C22—2122no—
592C22—2122no—
593C22—2122no—
594C22C2×#1313320011yescharacteristic
595C22#1681113011yescharacteristic
596C22#1681113411yescharacteristic
597C22#1681113411yescharacteristic
598C22C2×#1313320611yescharacteristic
599C22#1681115712yes—
600C22#1681115712yes—
601C22#1681116012yes—
602C22#1681116012yes—
603C22#1681119011yescharacteristic
604C22#1681120111yescharacteristic
605C22#1681120612yes—
606C22#1681121512yes—
607C22#1681130212yes—
608C22#1681131211yescharacteristic
609C22#1681131711yescharacteristic
610C22#1681132112yes—
611C22#1681132112yes—
612C22#1681132212yes—
613C22#1681133611yescharacteristic
614C22#1681134111yescharacteristic
615C22#1681134612yes—
616C22#1681135812yes—
617C4—24no—
618C4—24no—
619C4—44no—
620C4—44no—
621C4—42no—
622C4—4241no—
623C4—4242no—
624C4—4241no—
625C4—4242no—
626C4—2122no—
627C2—4242no—
628C2#2027799811yescharacteristic
629C2#2027900611yescharacteristic
630C2#2027920311yescharacteristic
631C2#2027996812yes—
632C2#2028020311yescharacteristic
633C2#2028020612yes—
634C2#2028022112yes—
635C2#2028023612yes—
636C2#2028027511yescharacteristic
637C2#2028061011yescharacteristic
638C2#2028303711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
FittingC2×C4×C4⋊C4×C53C2
prime radicalC24×D53C23
hypercenterC2×C4×C4⋊C4D53
hypocenterC53#15110
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20272976C2semidirect
3. #20272974C2semidirect
4. #20272974C2semidirect
9. #20272948C2semidirect
10. #20272942C2semidirect
11. C2×C4×C4⋊C4×C53C2semidirect
309. C2×C4×C4⋊C4D53semidirect
356. C53#15110semidirect

Maximal subgroups

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

subgroupindex
C2×#168105142
#202729762
#202729742
#202729632
#202729572
#202729532
#202729482
#202729422
C2×C4×C4⋊C4×C532
#1511053