Finite Group Databasebeta

← #27312551

Subgroups — #27312551

Order 13568.

1658 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810525C211yesmaximal, characteristic
2C2×#16810513C212yesmaximal, complemented
3#20272963C212yesmaximal, complemented
4#20272957C212yesmaximal, complemented
5#20272955C212yesmaximal
6#20272951C212yesmaximal, complemented
7#20272949C212yesmaximal, complemented
8#20272942C211yesmaximal, characteristic, complemented
9C53×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×D212C2211yescharacteristic
11C22×C4⋊Dic53C2211yescharacteristic
12C22×#11719921C2212yes—
13C22×#11719921C2212yes—
14C22×#11719921C2212yes—
15C22×#11719921C2212yes—
16C22×#11719921C2212yes—
17C22×#11719921C2212yes—
18C22×C4×Dic53C2212yes—
19#16810525C418yescomplemented
20#16810513—28no—
21C2×#13133124C2212yes—
22C2×#13133124C2211yescharacteristic
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2211yescharacteristic
28C22×C4⋊C4×C53C2212yes—
29C22×C42×C53C2211yescharacteristic
30C2×#95×C53C2212yes—
31C2×#95×C53C2211yescharacteristic
32C2×#95×C53C2211yescharacteristic
33C24×D53C2312yes—
34C22×D212—22no—
35C22×D212—22no—
36C22×D212—42no—
37C22×D212—42no—
38C22×D212C2×C414yes—
39C23×Dic53C2312yes—
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C2×C4⋊Dic53—42no—
43C2×C4⋊Dic53C2×C414yes—
44C2×#11719921—24no—
45C2×#11719921—24no—
46C2×#11719921—24no—
47C2×#11719921—24no—
48C2×#11719921—24no—
49C2×#11719921—24no—
50C2×#11719921—42no—
51C2×#11719921—42no—
52C2×#11719921—42no—
53C2×#11719921—42no—
54C2×#11719921—42no—
55C2×#11719921—42no—
56C2×#11719921—42no—
57C2×#11719921—42no—
58C2×#11719921—42no—
59C2×#11719921—42no—
60C2×#11719921C2×C418yes—
61C2×#11719921C2×C418yes—
62C2×C4×Dic53—24no—
63C2×C4×Dic53—42no—
64C2×C4×Dic53—42no—
65#13133124—24no—
66#13133124—22no—
67#13133124—24no—
68#13133124—42no—
69#13133124—42no—
70#13133124—42no—
71#13133124D414yescomplemented
72C23×C4×C53C2312yes—
73C23×C4×C53C2311yescharacteristic
74C23×C4×C53C2312yes—
75C23×C4×C53C2311yescharacteristic
76C23×C4×C53C2311yescharacteristic
77C2×C4⋊C4×C53—24no—
78C2×C4⋊C4×C53—24no—
79C2×C4⋊C4×C53—42no—
80C2×C42×C53—42no—
81C2×C42×C53C2×C414yes—
82#95×C53—24no—
83#95×C53—41no—
84#95×C53—41no—
85C24×C53C2411yescharacteristic
86C23×D53—22no—
87C23×D53—22no—
88C23×D53—22no—
89C23×D53—22no—
90C23×D53—22no—
91C23×D53—22no—
92C23×D53—22no—
93C23×D53—22no—
94C23×D53—22no—
95C23×D53—22no—
96C23×D53—22no—
97C23×D53—22no—
98C23×D53—22no—
99C23×D53—22no—
100C23×D53C22×C414yes—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
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—21no—
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×C53C22×C412yes—
123C22×C4×C53C22×C412yes—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C2×D412yes—
126C22×C4×C53#4114yes—
127C22×C4×C53#4112yes—
128C22×C4×C53#4112yes—
129C22×C4×C53#4112yes—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
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—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53C22×C414yes—
150C22×Dic53C2×D414yes—
151C2×D212—28no—
152C2×D212—44no—
153C2×D212—44no—
154C2×D212—48no—
155C2×D212—48no—
156C4⋊C4×C53—48no—
157C42×C53—44no—
158#11719921—216no—
159#11719921—48no—
160#11719921—48no—
161#11719921—48no—
162#11719921—48no—
163#11719921—48no—
164C4⋊Dic53—44no—
165C4×Dic53—48no—
166C23×C53C23×C411yescharacteristic
167C23×C53C22×D411yescharacteristic
168C23×C53C2×#4111yescharacteristic
169C23×C53C2×#4112yes—
170C23×C53C2×#4111yescharacteristic
171C23×C53C2×#4111yescharacteristic
172C23×C53H(5,𝔽2)12yes—
173C23×C53H(5,𝔽2)11yescharacteristic
174C23×C53H(5,𝔽2)12yes—
175C23×C53H(5,𝔽2)12yes—
176C23×C532−1+411yescharacteristic
177C22×D53—24no—
178C22×D53—24no—
179C22×D53—24no—
180C22×D53—24no—
181C22×D53—24no—
182C22×D53—24no—
183C22×D53—24no—
184C22×D53—42no—
185C22×D53—42no—
186C22×D53—42no—
187C22×D53—42no—
188C22×D53—42no—
189C22×D53—42no—
190C22×D53—42no—
191C22×D53—42no—
192C22×D53—42no—
193C22×D53—42no—
194C22×D53—42no—
195C22×D53—42no—
196C22×D53—42no—
197C22×D53—42no—
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—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53—42no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
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×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53#11714yes—
240C2×C4×C53C4×D414yes—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
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—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262D212—416no—
263#14444—531nomaximal, Sylow
264C22×C53C2×#11711yescharacteristic
265C22×C53C2×C4×D411yescharacteristic
266C22×C53C4×#4111yescharacteristic
267C22×C53#51811yescharacteristic
268C22×C53#51811yescharacteristic
269C22×C53#51811yescharacteristic
270C22×C53#52011yescharacteristic
271C22×C53#53512yes—
272C22×C53#53812yes—
273C22×C53#53812yes—
274C22×C53#53812yes—
275C22×C53#54012yes—
276C22×C53#54212yes—
277C22×C53#54612yes—
278C22×C53#54711yescharacteristic
279C22×C53#55011yescharacteristic
280C22×C53#55312yes—
281C22×C53#55312yes—
282C22×C53#55511yescharacteristic
283C22×C53#55611yescharacteristic
284C22×C53#55912yes—
285C22×C53#56012yes—
286C22×C53#56212yes—
287C2×D53—44no—
288C2×D53—44no—
289C2×D53—44no—
290C2×D53—44no—
291C2×D53—44no—
292C2×D53—44no—
293C2×D53—44no—
294C2×D53—82no—
295C2×D53—82no—
296C2×D53—82no—
297C2×D53—82no—
298C2×D53—82no—
299C2×D53—82no—
300C2×D53—82no—
301C2×D53—82no—
302C4×C53—24no—
303C4×C53—24no—
304C4×C53—42no—
305C4×C53—44no—
306C4×C53—82no—
307Dic53—44no—
308Dic53—44no—
309Dic53—82no—
310C2×#390—531no—
311C2×#388—532no—
312#1440—532no—
313#1439—532no—
314#1438—532no—
315#1437—532no—
316#1436—532no—
317C4⋊#95D5311yescharacteristic, complemented
318#1434—531no—
319C2×C53#231211yescharacteristic
320C2×C53#232811yescharacteristic
321C2×C53#233711yescharacteristic
322C2×C53#235411yescharacteristic
323C2×C53#235711yescharacteristic
324C2×C53#236411yescharacteristic
325C2×C53#237711yescharacteristic
326C2×C53#271012yes—
327C2×C53#271412yes—
328C2×C53#280012yes—
329C2×C53#280312yes—
330D53—84no—
331C23×D4—531no—
332C22×C4⋊C4—531no—
333C22×C4⋊C4C2×D5312yes—
334C22×C22⋊C4—532no—
335C22×C22⋊C4—532no—
336C22×C22⋊C4—532no—
337C22×C22⋊C4—532no—
338C22×C22⋊C4—532no—
339C22×C22⋊C4—532no—
340C22×C42—532no—
341C22×C42C2×D5311yescharacteristic
342#390—538no—
343#388—1068no—
344C2×#95—531no—
345C2×#95—532no—
346C2×#95—531no—
347C2×#95—532no—
348C2×#95—532no—
349C2×#95—532no—
350C2×#95—532no—
351C2×#95C2×D5312yes—
352C2×#95C2×D5311yescharacteristic
353C2×#95C2×D5311yescharacteristic
354C53#1444411yescharacteristic, complemented, Sylow
355C25—532no—
356C22×D4—534no—
357C22×D4—1062no—
358C22×D4—1062no—
359C22×D4—2122no—
360C22×D4—2122no—
361C23×C4—532no—
362C23×C4—532no—
363C23×C4—532no—
364C23×C4C22×D5312yes—
365C23×C4C22×D5311yescharacteristic
366C23×C4C22×D5312yes—
367C23×C4C22×D5311yescharacteristic
368C23×C4C22×D5311yescharacteristic
369C2×C4⋊C4—24no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—42no—
372C2×C4⋊C4—534no—
373C2×C4⋊C4—2122no—
374C2×C22⋊C4—538no—
375C2×C22⋊C4—538no—
376C2×C22⋊C4—1064no—
377C2×C22⋊C4—1064no—
378C2×C22⋊C4—1064no—
379C2×C22⋊C4—1064no—
380C2×C22⋊C4—1064no—
381C2×C22⋊C4—1064no—
382C2×C22⋊C4—2122no—
383C2×C22⋊C4—2122no—
384C2×C22⋊C4—2122no—
385C2×C22⋊C4—2122no—
386C2×C22⋊C4—2122no—
387C2×C22⋊C4—2122no—
388C2×C22⋊C4—2122no—
389C2×C22⋊C4—2122no—
390C2×C22⋊C4—2122no—
391C2×C22⋊C4—2122no—
392C2×C42—42no—
393C2×C42—1064no—
394C2×C42—2122no—
395C2×C42—2122no—
396C2×C42C4×D5314yes—
397#95—24no—
398#95—41no—
399#95—41no—
400#95—534no—
401#95—1064no—
402#95—1064no—
403#95—1062no—
404#95—2122no—
405#95—2122no—
406#95—2122no—
407C24—534no—
408C24—1062no—
409C24—1062no—
410C24—1062no—
411C24—1062no—
412C24—1062no—
413C24—1062no—
414C24—1062no—
415C24—1062no—
416C24—1062no—
417C24—1062no—
418C24—1062no—
419C24—1062no—
420C24—1062no—
421C24—1062no—
422C24C23×D5311yescharacteristic
423C2×D4—1068no—
424C2×D4—2124no—
425C2×D4—2124no—
426C2×D4—2128no—
427C2×D4—2128no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—534no—
450C22×C4—534no—
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—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4C2×C4×D5312yes—
471C22×C4C2×C4×D5312yes—
472C22×C4C2×C4×D5312yes—
473C22×C4#1171994514yes—
474C22×C4D4×D5312yes—
475C22×C4#1171994712yes—
476C22×C4#1171994912yes—
477C22×C4#1171994912yes—
478C4⋊C4—48no—
479C4⋊C4—2124no—
480C22⋊C4—10616no—
481C22⋊C4—2128no—
482C22⋊C4—2128no—
483C22⋊C4—2128no—
484C22⋊C4—2128no—
485C22⋊C4—2128no—
486C42—44no—
487C42—2128no—
488C23—1064no—
489C23—1064no—
490C23—1064no—
491C23—1064no—
492C23—1064no—
493C23—1064no—
494C23—1064no—
495C23—2122no—
496C23—2122no—
497C23—2122no—
498C23—2122no—
499C23—2122no—
500C23—2122no—
501C23—2122no—
502C23—2122no—
503C23—2122no—
504C23—2122no—
505C23—2122no—
506C23—2122no—
507C23—2122no—
508C23—2122no—
509C23—2122no—
510C23—2122no—
511C23—2122no—
512C23—2122no—
513C23—2122no—
514C23—2122no—
515C23—2122no—
516C23—2122no—
517C23—2122no—
518C23—2122no—
519C23—2122no—
520C23—2122no—
521C23—2122no—
522C23—2122no—
523C23C22×C4×D5311yescharacteristic
524C23C2×D4×D5311yescharacteristic
525C23C2×#1171994711yescharacteristic
526C23C2×#1171994512yes—
527C23C2×#1171994911yescharacteristic
528C23C2×#1171994911yescharacteristic
529C23#1313326211yescharacteristic
530C23#1313326212yes—
531C23#1313326312yes—
532C23#1313326312yes—
533C23#1313326511yescharacteristic
534D4—21216no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
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—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—1064no—
563C2×C4—1064no—
564C2×C4—1064no—
565C2×C4—1064no—
566C2×C4—1064no—
567C2×C4—1064no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4#1313317514yes—
584C2×C4#1313317914yes—
585C22—4242no—
586C22—4242no—
587C22—4242no—
588C22—4242no—
589C22—4242no—
590C22—4242no—
591C22—4242no—
592C22—4242no—
593C22—2124no—
594C22—2124no—
595C22—2124no—
596C22—2124no—
597C22—2124no—
598C22—2124no—
599C22—2124no—
600C22C2×#1313317511yescharacteristic
601C22C2×#1313317911yescharacteristic
602C22C4×#1171994511yescharacteristic
603C22#1681109611yescharacteristic
604C22#1681109711yescharacteristic
605C22#1681109711yescharacteristic
606C22#1681110411yescharacteristic
607C22#1681119112yes—
608C22#1681120612yes—
609C22#1681121012yes—
610C22#1681121312yes—
611C22#1681122912yes—
612C22#1681124512yes—
613C22#1681126312yes—
614C22#1681127611yescharacteristic
615C22#1681129311yescharacteristic
616C22#1681131912yes—
617C22#1681132112yes—
618C22#1681133911yescharacteristic
619C22#1681134711yescharacteristic
620C22#1681135812yes—
621C22#1681136112yes—
622C22#1681136912yes—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—82no—
628C4—4242no—
629C4—2124no—
630C4—2124no—
631C2—4244no—
632C2#2027731211yescharacteristic
633C2#2027743811yescharacteristic
634C2#2027752811yescharacteristic
635C2#2027766611yescharacteristic
636C2#2027768611yescharacteristic
637C2#2027776211yescharacteristic
638C2#2027786911yescharacteristic
639C2#2028145212yes—
640C2#2028150612yes—
641C2#2028260512yes—
642C2#2028264212yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C4⋊#95C2
socle, cosocleC24×C53C24
prime radicalC23×D212C22
hypercenterC4⋊#95D53
hypocenterC53#14444
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#16810513C2semidirect
3. #20272963C2semidirect
4. #20272957C2semidirect
6. #20272951C2semidirect
7. #20272949C2semidirect
8. #20272942C2semidirect
9. C53×C4⋊#95C2semidirect
19. #16810525C4semidirect
71. #13133124D4semidirect
317. C4⋊#95D53semidirect
354. C53#14444semidirect

Maximal subgroups

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

subgroupindex
C2×#168105252
C2×#168105132
#202729632
#202729572
#202729552
#202729512
#202729492
#202729422
C53×C4⋊#952
#1444453