Finite Group Databasebeta

← #27311336

Subgroups — #27311336

Order 13568.

1422 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719921C211yesmaximal, characteristic
2#20272975C212yesmaximal, complemented
3#20272974C212yesmaximal, complemented
4#20272963C212yesmaximal
5#20272957C212yesmaximal
6#20272942C211yesmaximal, characteristic, complemented
7#20272940C211yesmaximal, characteristic, complemented
8#20272940C211yesmaximal, characteristic, complemented
9#20272936C211yesmaximal, characteristic
10#20272936C211yesmaximal, characteristic
11C53×C4⋊#95C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×#11719921C2211yescharacteristic
14C22×#11719921C2211yescharacteristic
15C22×#11719921C2212yes—
16C22×#11719921C2212yes—
17C22×C4×Dic53C2211yescharacteristic
18C4×#11719921C418yescomplemented
19C2×#13133124C2211yescharacteristic
20C2×#13133124C2211yescharacteristic
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2211yescharacteristic
31C22×C4⋊C4×C53C2212yes—
32C22×C42×C53C2211yescharacteristic
33C2×#95×C53C2211yescharacteristic
34C2×#95×C53C2211yescharacteristic
35C2×#95×C53C2212yes—
36C24×D53C2311yescharacteristic
37C23×Dic53C2312yes—
38C23×Dic53C2312yes—
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C22×C4×D53—22no—
43C22×C4×D53—22no—
44C22×C4×D53—42no—
45C22×C4×D53—42no—
46C22×C4×D53C2×C414yes—
47C2×#11719921—24no—
48C2×#11719921—24no—
49C2×#11719921—24no—
50C2×#11719921—24no—
51C2×#11719921—42no—
52C2×#11719921—42no—
53C2×#11719921—42no—
54C2×#11719921—42no—
55C2×#11719921C2×C414yes—
56C2×#11719921C2×C414yes—
57C2×C4×Dic53—42no—
58C2×C4×Dic53C2×C414yes—
59#13133124—24no—
60#13133124—22no—
61#13133124—22no—
62#13133124—42no—
63#13133124—42no—
64#13133124—42no—
65#13133124—42no—
66#13133124—42no—
67#13133124—42no—
68#13133124—42no—
69#13133124C2×C414yes—
70#13133124C2×C414yes—
71C23×C4×C53C2312yes—
72C23×C4×C53C2312yes—
73C23×C4×C53C2311yescharacteristic
74C23×C4×C53C2311yescharacteristic
75C23×C4×C53C2311yescharacteristic
76C2×C4⋊C4×C53—24no—
77C2×C4⋊C4×C53—42no—
78C2×C4⋊C4×C53—42no—
79C2×C42×C53—42no—
80C2×C42×C53C2×C414yes—
81#95×C53—22no—
82#95×C53—22no—
83#95×C53—42no—
84C24×C53C2411yescharacteristic
85C23×D53—22no—
86C23×D53—21no—
87C23×D53—21no—
88C23×D53—21no—
89C23×D53—21no—
90C23×D53—22no—
91C23×D53—22no—
92C23×D53—21no—
93C23×D53—21no—
94C23×D53C22×C412yes—
95C23×D53#4114yes—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—21no—
99C22×C4×C53—22no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53—21no—
103C22×C4×C53—22no—
104C22×C4×C53—21no—
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—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53C22×C412yes—
120C22×C4×C53C22×C412yes—
121C22×C4×C53C22×C412yes—
122C22×C4×C53#4114yes—
123C22×C4×C53#4112yes—
124C22×C4×C53#4112yes—
125C22×Dic53—22no—
126C22×Dic53—22no—
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—22no—
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—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53C22×C412yes—
152C22×Dic53C22×C412yes—
153C22×Dic53C22×C412yes—
154C2×C4×D53—28no—
155C2×C4×D53—44no—
156C2×C4×D53—44no—
157C2×C4×D53—48no—
158C2×C4×D53—48no—
159C4⋊C4×C53—48no—
160C42×C53—44no—
161#11719921—28no—
162#11719921—28no—
163#11719921—48no—
164#11719921—48no—
165C4×Dic53—44no—
166C23×C53C23×C411yescharacteristic
167C23×C53C2×#4112yes—
168C23×C53C2×#4111yescharacteristic
169C23×C53C2×#4112yes—
170C23×C53C2×#4111yescharacteristic
171C23×C53H(5,𝔽2)12yes—
172C23×C53H(5,𝔽2)11yescharacteristic
173C23×C53H(5,𝔽2)11yescharacteristic
174C23×C53H(5,𝔽2)12yes—
175C23×C532−1+411yescharacteristic
176C23×C532−1+411yescharacteristic
177C22×D53—22no—
178C22×D53—22no—
179C22×D53—22no—
180C22×D53—22no—
181C22×D53—22no—
182C22×D53—22no—
183C22×D53—24no—
184C22×D53—24no—
185C22×D53—24no—
186C22×D53—24no—
187C22×D53—24no—
188C22×D53—24no—
189C22×D53—42no—
190C22×D53—42no—
191C22×D53—42no—
192C22×D53—42no—
193C22×D53—42no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—42no—
197C22×D53—41no—
198C22×D53—41no—
199C22×D53#11714yes—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—24no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
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#11714yes—
228C2×C4×C53#11714yes—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
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—42no—
257C2×Dic53—42no—
258C4×D53—416no—
259#14351—531nomaximal, Sylow
260C22×C53C2×#11711yescharacteristic
261C22×C53C2×#11711yescharacteristic
262C22×C53C2×#11711yescharacteristic
263C22×C53#51811yescharacteristic
264C22×C53#51811yescharacteristic
265C22×C53#52011yescharacteristic
266C22×C53#52011yescharacteristic
267C22×C53#52912yes—
268C22×C53#52912yes—
269C22×C53#53812yes—
270C22×C53#53812yes—
271C22×C53#53812yes—
272C22×C53#53812yes—
273C22×C53#53812yes—
274C22×C53#53912yes—
275C22×C53#53912yes—
276C22×C53#53912yes—
277C22×C53#54012yes—
278C22×C53#54011yescharacteristic
279C22×C53#54111yescharacteristic
280C22×C53#54212yes—
281C22×C53#54211yescharacteristic
282C22×C53#54211yescharacteristic
283C2×D53—24no—
284C2×D53—24no—
285C2×D53—24no—
286C2×D53—42no—
287C2×D53—42no—
288C2×D53—42no—
289C2×D53—42no—
290C2×D53—44no—
291C2×D53—44no—
292C2×D53—44no—
293C2×D53—44no—
294C4×C53—24no—
295C4×C53—24no—
296C4×C53—42no—
297C4×C53—82no—
298C4×C53—82no—
299Dic53—42no—
300Dic53—42no—
301Dic53—42no—
302Dic53—82no—
303Dic53—82no—
304C2×C4×C22⋊C4—531no—
305#1443—532no—
306#1442—532no—
307#1440—532no—
308#1439—532no—
309C4⋊#95D5311yescharacteristic, complemented
310#1434—531no—
311#1434—531no—
312#1434—531no—
313C22⋊#95—531no—
314C22⋊#95—531no—
315C2×C53#228311yescharacteristic
316C2×C53#231211yescharacteristic
317C2×C53#231411yescharacteristic
318C2×C53#231811yescharacteristic
319C2×C53#231811yescharacteristic
320C2×C53#231811yescharacteristic
321C2×C53#231911yescharacteristic
322C2×C53#264812yes—
323C2×C53#264912yes—
324C2×C53#265212yes—
325C2×C53#265312yes—
326D53—44no—
327C24×C4—531no—
328C22×C4⋊C4C2×D5312yes—
329C22×C22⋊C4—531no—
330C22×C22⋊C4—531no—
331C22×C22⋊C4—532no—
332C22×C22⋊C4—532no—
333C22×C42—531no—
334C22×C42C2×D5311yescharacteristic
335C4×C22⋊C4—538no—
336C2×#95—531no—
337C2×#95—531no—
338C2×#95—531no—
339C2×#95—531no—
340C2×#95—532no—
341C2×#95—532no—
342C2×#95—532no—
343C2×#95—532no—
344C2×#95—532no—
345C2×#95—532no—
346C2×#95—532no—
347C2×#95—532no—
348C2×#95C2×D5311yescharacteristic
349C2×#95C2×D5311yescharacteristic
350C2×#95C2×D5312yes—
351C53#1435111yescharacteristic, complemented, Sylow
352C25—531no—
353C23×C4—531no—
354C23×C4—531no—
355C23×C4—531no—
356C23×C4—534no—
357C23×C4—532no—
358C23×C4—532no—
359C23×C4—1062no—
360C23×C4—1062no—
361C23×C4—2122no—
362C23×C4—2122no—
363C23×C4C22×D5312yes—
364C23×C4C22×D5312yes—
365C23×C4C22×D5311yescharacteristic
366C23×C4C22×D5311yescharacteristic
367C23×C4C22×D5311yescharacteristic
368C2×C4⋊C4—24no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4—42no—
371C2×C22⋊C4—534no—
372C2×C22⋊C4—534no—
373C2×C22⋊C4—1064no—
374C2×C22⋊C4—1064no—
375C2×C22⋊C4—1064no—
376C2×C22⋊C4—1064no—
377C2×C22⋊C4—2122no—
378C2×C22⋊C4—2122no—
379C2×C22⋊C4—2122no—
380C2×C22⋊C4—2122no—
381C2×C42—42no—
382C2×C42—534no—
383C2×C42—2122no—
384C2×C42C4×D5314yes—
385#95—22no—
386#95—22no—
387#95—42no—
388#95—534no—
389#95—534no—
390#95—1064no—
391#95—1062no—
392#95—1062no—
393#95—2122no—
394#95—2122no—
395#95—2122no—
396#95—2122no—
397#95—2122no—
398#95—2122no—
399#95—2122no—
400C24—532no—
401C24—534no—
402C24—1061no—
403C24—1062no—
404C24—1061no—
405C24—1061no—
406C24—1061no—
407C24—1062no—
408C24—1062no—
409C24—1061no—
410C24—1061no—
411C24C23×D5311yescharacteristic
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
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—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—532no—
436C22×C4—532no—
437C22×C4—532no—
438C22×C4—1062no—
439C22×C4—1061no—
440C22×C4—1062no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1068no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—2124no—
466C22×C4—2124no—
467C22×C4—2128no—
468C22×C4—2128no—
469C22×C4C2×C4×D5312yes—
470C22×C4C2×C4×D5312yes—
471C22×C4C2×C4×D5312yes—
472C22×C4#1171994514yes—
473C22×C4#1171994712yes—
474C22×C4#1171994912yes—
475C4⋊C4—48no—
476C22⋊C4—1068no—
477C22⋊C4—1068no—
478C22⋊C4—2128no—
479C22⋊C4—2128no—
480C42—44no—
481C42—2124no—
482C23—534no—
483C23—1062no—
484C23—1062no—
485C23—1062no—
486C23—1062no—
487C23—1062no—
488C23—1062no—
489C23—1064no—
490C23—1064no—
491C23—1064no—
492C23—1064no—
493C23—1064no—
494C23—1064no—
495C23—2122no—
496C23—2122no—
497C23—2122no—
498C23—2122no—
499C23—2122no—
500C23—2121no—
501C23—2121no—
502C23—2122no—
503C23—2121no—
504C23—2121no—
505C23C22×C4×D5311yescharacteristic
506C23#41×D5312yes—
507C23C2×#1171994711yescharacteristic
508C23C2×#1171994512yes—
509C23C2×#1171994911yescharacteristic
510C23#1313326212yes—
511C23#1313326211yescharacteristic
512C23#1313326211yescharacteristic
513C23#1313326312yes—
514C23#1313326511yescharacteristic
515C23#1313326511yescharacteristic
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—21216no—
553C2×C4—2122no—
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—2122no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4#1313317414yes—
574C2×C4#1313317514yes—
575C22—1064no—
576C22—1064no—
577C22—1064no—
578C22—2122no—
579C22—2122no—
580C22—2122no—
581C22—2122no—
582C22—2124no—
583C22—2124no—
584C22—2124no—
585C22—2124no—
586C22#117×D5311yescharacteristic
587C22C2×#1313317411yescharacteristic
588C22C2×#1313317511yescharacteristic
589C22#1681109611yescharacteristic
590C22#1681109711yescharacteristic
591C22#1681110411yescharacteristic
592C22#1681110411yescharacteristic
593C22#1681115512yes—
594C22#1681116012yes—
595C22#1681120512yes—
596C22#1681120712yes—
597C22#1681121012yes—
598C22#1681121112yes—
599C22#1681121312yes—
600C22#1681121912yes—
601C22#1681122012yes—
602C22#1681122112yes—
603C22#1681122412yes—
604C22#1681122711yescharacteristic
605C22#1681123411yescharacteristic
606C22#1681124512yes—
607C22#1681124711yescharacteristic
608C22#1681124711yescharacteristic
609C4—24no—
610C4—24no—
611C4—42no—
612C4—82no—
613C4—82no—
614C4—4242no—
615C4—4242no—
616C4—2122no—
617C4—2122no—
618C4—2122no—
619C2—2124no—
620C2#2027711111yescharacteristic
621C2#2027731211yescharacteristic
622C2#2027732311yescharacteristic
623C2#2027734911yescharacteristic
624C2#2027735011yescharacteristic
625C2#2027735111yescharacteristic
626C2#2027735311yescharacteristic
627C2#2028070712yes—
628C2#2028072412yes—
629C2#2028076712yes—
630C2#2028078412yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C4⋊#95C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC4⋊#95D53
hypocenterC53#14351
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. #20272975C2semidirect
3. #20272974C2semidirect
6. #20272942C2semidirect
7. #20272940C2semidirect
8. #20272940C2semidirect
11. C53×C4⋊#95C2semidirect
18. C4×#11719921C4semidirect
309. C4⋊#95D53semidirect
351. C53#14351semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199212
#202729752
#202729742
#202729632
#202729572
#202729422
#202729402
#202729362
C53×C4⋊#952
#1435153