Finite Group Databasebeta

← #27442323

Subgroups — #27442323

Order 13568.

736 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279730C211yesmaximal, characteristic, complemented
2#20277664C211yesmaximal, characteristic
3#20276085C211yesmaximal, characteristic
4#20276085C211yesmaximal, characteristic
5#20275859C211yesmaximal, characteristic
6#20275849C211yesmaximal, characteristic
7#20275669C211yesmaximal, characteristic, complemented
8#20275665C211yesmaximal, characteristic, complemented
9#20275478C211yesmaximal, characteristic, complemented
10#20275438C211yesmaximal, characteristic, complemented
11#20275303C211yesmaximal, characteristic, complemented
12#20275296C211yesmaximal, characteristic, complemented
13#20274849C211yesmaximal, characteristic, complemented
14#20274725C211yesmaximal, characteristic
15C53×#1762C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×#13133176—21no—
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26#16810555—21no—
27#16810544—21no—
28#16810529—21no—
29#16810526C2211yescharacteristic
30#16810526C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810521—21no—
34#16810516—21no—
35#16810516—21no—
36#16810493C2211yescharacteristic
37#16810489C2211yescharacteristic
38#16810483—21no—
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C4×D4×C53C2211yescharacteristic
53C2×#102×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133216—22no—
63#13133216—41no—
64#13133208—22no—
65#13133208—41no—
66#13133176—44no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53D411yescharacteristic
73C2×C4⋊Dic53D411yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133148—22no—
89#13133148—22no—
90C4⋊#88322—22no—
91#13133146—41no—
92#13133146—41no—
93#13133138—41no—
94#13133138—41no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124D411yescharacteristic
102#13133124D411yescharacteristic
103C22×D4×C53C2311yescharacteristic
104C23×C4×C53—21no—
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C4×D4×C53—22no—
108C4×D4×C53Q814yescomplemented
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C22⋊C4×C53—21no—
113C2×C42×C53—21no—
114C2×C42×C53C2311yescharacteristic
115#102×C53—22no—
116#102×C53—22no—
117#97×C53—22no—
118#95×C53—21no—
119C24×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53C2×Q812yes—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—24no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2411yescharacteristic
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C2×Dic106—42no—
144C2×Dic106—42no—
145C2×Dic106—81no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53C2×Q812yes—
156C4⋊C4×C53#4112yes—
157C4⋊C4×C53#4112yes—
158C22⋊C4×C53—24no—
159C42×C53—81no—
160C42×C53C2×Q812yes—
161C42×C53#4112yes—
162C22⋊Dic53—42no—
163C22⋊Dic53—42no—
164C22⋊Dic53—81no—
165C22⋊Dic53—81no—
166C4⋊Dic53—22no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—41no—
176C4⋊Dic53—42no—
177C4⋊Dic53—42no—
178C4⋊Dic53—42no—
179C4⋊Dic53—81no—
180C4⋊Dic53—81no—
181#11719919—42no—
182#11719919—42no—
183C4×Dic53—21no—
184C4×Dic53—21no—
185C4×Dic53—21no—
186C4×Dic53—21no—
187C4×Dic53—42no—
188C4×Dic53—42no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C23×C53—22no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53C22×D411yescharacteristic
203D4×C53—22no—
204D4×C53—22no—
205D4×C53—41no—
206D4×C53—41no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—48no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53C22⋊Q812yes—
222C2×C4×C53C22⋊Q812yes—
223C2×C4×C53C22×Q811yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53H(5,𝔽2)11yescharacteristic
228C2×C4×C532−1+411yescharacteristic
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53AGL(1,ℤ/8)12yes—
253Dic106—84no—
254C8×C53—41no—
255C8×C53—41no—
256#88322—41no—
257#88322—41no—
258#26032—531nomaximal, Sylow
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53—81no—
266C22×C53C2×C22⋊Q811yescharacteristic
267C22×C53#54611yescharacteristic
268C22×C53#54811yescharacteristic
269C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C53#57511yescharacteristic
271C22×C53#57711yescharacteristic
272C22×C53#57711yescharacteristic
273C4×C53—22no—
274C4×C53—22no—
275C4×C53—42no—
276C4×C53—42no—
277C4×C53—42no—
278C4×C53#53311yescharacteristic
279C4×C53#54411yescharacteristic
280C4×C53#55511yescharacteristic
281C4×C53#55611yescharacteristic
282Dic53—22no—
283Dic53—41no—
284Dic53—41no—
285Dic53—41no—
286Dic53—81no—
287Dic53—81no—
288Dic53—81no—
289Dic53—81no—
290#2550—531no—
291#2354—531no—
292#2065—531no—
293#2065—531no—
294#2025—531no—
295#2023—531no—
296#1989—531no—
297#1988—531no—
298#1948—531no—
299#1940—531no—
300#1920—531no—
301#1919—531no—
302#1836—531no—
303#1804—531no—
304#1762D5311yescharacteristic
305C2×C53—82no—
306C2×C53#243911yescharacteristic
307C2×C53#307711yescharacteristic
308C2×C53#308411yescharacteristic
309C2×C53#330611yescharacteristic
310C2×C53#332111yescharacteristic
311C2×C53#336811yescharacteristic
312C2×C53#337611yescharacteristic
313C2×C4⋊Q8—531no—
314C2×#125—531no—
315C2×C4×D4C2×D5311yescharacteristic
316C2×#117—1061no—
317C2×#106—531no—
318C2×#106—531no—
319C2×C4⋊C8—531no—
320C2×C4⋊C8—531no—
321C2×C4⋊C8—531no—
322C2×#102—531no—
323C2×#102—531no—
324C2×#102C2×D5311yescharacteristic
325C2×#102C2×D5311yescharacteristic
326C2×#97C2×D5311yescharacteristic
327#398—1061no—
328#395—1061no—
329#391—1061no—
330#390—531no—
331#390—531no—
332#389—531no—
333#389—531no—
334#389—1061no—
335#388—1061no—
336#388—1061no—
337#384—531no—
338#382—531no—
339#380—1061no—
340C4×C4⋊C4—531no—
341C4×C4⋊C4—531no—
342C4×C4⋊C4—531no—
343C4×C4⋊C4C2×D5311yescharacteristic
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340C2×D5311yescharacteristic
353#340C2×D5311yescharacteristic
354#336—531no—
355#336—531no—
356C53#2603211yescharacteristic, complemented, Sylow
357C22×Q8—1061no—
358C22×D4C22×D5311yescharacteristic
359C23×C4—21no—
360C22×C8—531no—
361C22×C8—531no—
362C22×C8C22×D5311yescharacteristic
363C22×C8C22×D5311yescharacteristic
364C4⋊Q8—1062no—
365C4⋊Q8—2121no—
366#125—1062no—
367#125—2121no—
368C4×D4—22no—
369C4×D4Dic10614yescomplemented
370#117—2124no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C42—21no—
393C2×C42—531no—
394C2×C42—531no—
395C2×C42—1061no—
396C2×C42—1061no—
397C2×C42C22×D5311yescharacteristic
398#106—1062no—
399#106—1062no—
400C4⋊C8—1062no—
401C4⋊C8—2121no—
402C4⋊C8—2121no—
403#102—22no—
404#102—22no—
405#102—2121no—
406#102—2121no—
407#97—22no—
408#95—21no—
409#95—531no—
410#95—531no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417C24—21no—
418C2×Q8—4241no—
419C2×Q8—2122no—
420C2×Q8—2122no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4C2×Dic10612yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—531no—
434C22×C4—531no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4D4×D5311yescharacteristic
442C22×C4D4×D5311yescharacteristic
443C22×C4C23×D5311yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—1062no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—4241no—
469C4⋊C4—4241no—
470C4⋊C4—2121no—
471C4⋊C4—2122no—
472C4⋊C4—2122no—
473C4⋊C4—2122no—
474C4⋊C4—2122no—
475C4⋊C4—2122no—
476C4⋊C4C2×Dic10612yes—
477C4⋊C4#1171994512yes—
478C4⋊C4#1171994512yes—
479C22⋊C4—24no—
480C22⋊C4—4241no—
481C22⋊C4—4241no—
482C22⋊C4—2122no—
483C22⋊C4—2122no—
484C42—81no—
485C42—1061no—
486C42—1061no—
487C42—1061no—
488C42—1061no—
489C42—2122no—
490C42—2122no—
491C42C2×Dic10612yes—
492C42#1171994712yes—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23C2×D4×D5311yescharacteristic
501Q8—4244no—
502D4—22no—
503D4—22no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—48no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—532no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4#1313319412yes—
545C2×C4#1313319412yes—
546C2×C4C22×Dic10611yescharacteristic
547C2×C4C2×#1171994711yescharacteristic
548C2×C4C2×#1171994511yescharacteristic
549C2×C4C2×#1171994511yescharacteristic
550C2×C4#1313326311yescharacteristic
551C2×C4#1313326411yescharacteristic
552C8—41no—
553C8—41no—
554C8—2121no—
555C8—2121no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22C2×#1313319411yescharacteristic
564C22#1681126911yescharacteristic
565C22#1681128211yescharacteristic
566C22AGL(1,ℤ/8)×D5311yescharacteristic
567C22#1681141911yescharacteristic
568C22#1681143211yescharacteristic
569C22#1681143311yescharacteristic
570C4—22no—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—1062no—
576C4—4241no—
577C4—4241no—
578C4—4241no—
579C4—4241no—
580C4—2121no—
581C4—2121no—
582C4—2121no—
583C4#1681118011yescharacteristic
584C4#1681125111yescharacteristic
585C4#1681133811yescharacteristic
586C4#1681134311yescharacteristic
587C2—82no—
588C2#2027840611yescharacteristic
589C2#2028450911yescharacteristic
590C2#2028458111yescharacteristic
591C2#2028668111yescharacteristic
592C2#2028690611yescharacteristic
593C2#2028744511yescharacteristic
594C2#2028754311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1762C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#1762D53
hypocenterC53#26032
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
1. #20279730C2semidirect
7. #20275669C2semidirect
8. #20275665C2semidirect
9. #20275478C2semidirect
10. #20275438C2semidirect
11. #20275303C2semidirect
12. #20275296C2semidirect
13. #20274849C2semidirect
108. C4×D4×C53Q8semidirect
356. C53#26032semidirect
369. C4×D4Dic106semidirect

Maximal subgroups

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

subgroupindex
#202797302
#202776642
#202760852
#202758592
#202758492
#202756692
#202756652
#202754782
#202754382
#202753032
#202752962
#202748492
#202747252
C53×#17622
#2603253