Finite Group Databasebeta

← #27442321

Subgroups — #27442321

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#20279736C211yesmaximal, characteristic, complemented
2#20277662C211yesmaximal, characteristic
3#20276088C211yesmaximal, characteristic
4#20276088C211yesmaximal, characteristic
5#20275862C211yesmaximal, characteristic
6#20275852C211yesmaximal, characteristic
7#20275670C211yesmaximal, characteristic, complemented
8#20275666C211yesmaximal, characteristic, complemented
9#20275479C211yesmaximal, characteristic, complemented
10#20275301C211yesmaximal, characteristic, complemented
11#20275294C211yesmaximal, characteristic, complemented
12#20274849C211yesmaximal, characteristic, complemented
13#20274726C211yesmaximal, characteristic
14#20274541C211yesmaximal, characteristic
15C53×#1940C211yesmaximal, characteristic, complemented
16C2×#13133217C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133175—21no—
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133139C2211yescharacteristic
23C2×#13133139C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26#16810556—21no—
27#16810543—21no—
28#16810527—21no—
29#16810525C2211yescharacteristic
30#16810525C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810520—21no—
34#16810515—21no—
35#16810515—21no—
36#16810493C2211yescharacteristic
37#16810489C2211yescharacteristic
38#16810480—21no—
39C4×C4⋊Dic53C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×C4⋊C8×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×C4⋊C8×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106—21no—
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62#13133217—22no—
63#13133217—41no—
64#13133209—22no—
65#13133209—41no—
66C4×D212—22no—
67C4×D212Q814yescomplemented
68#13133175—44no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91#13133149—22no—
92#13133149—22no—
93#13133139—22no—
94#13133139—22no—
95#13133139—41no—
96#13133139—41no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C22×C8×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112C2×C42×C53C2311yescharacteristic
113C4⋊C8×C53—22no—
114C4⋊C8×C53—41no—
115C4⋊C8×C53—41no—
116#97×C53—22no—
117#95×C53D411yescharacteristic
118#95×C53D411yescharacteristic
119C23×D53—21no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2411yescharacteristic
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C2×D212—21no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212C2×Q812yes—
140C2×C4×D53—24no—
141C2×C4×D53—42no—
142C2×C4×D53—42no—
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—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—41no—
160C42×C53—21no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53C2×Q812yes—
165C42×C53#4112yes—
166#11719921—24no—
167#11719921—42no—
168#11719921—42no—
169#11719921—81no—
170#11719921—81no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—42no—
183C4⋊Dic53—42no—
184C4⋊Dic53—81no—
185C4⋊Dic53C2×Q812yes—
186C4⋊Dic53#4112yes—
187C4⋊Dic53#4112yes—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191#11719919—81no—
192C4×Dic53—42no—
193C4×Dic53—42no—
194C4×Dic53—81no—
195C23×C53C22×D411yescharacteristic
196C22×D53—22no—
197C22×D53—41no—
198C22×D53—41no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53C22⋊Q812yes—
219C2×C4×C53C22⋊Q812yes—
220C2×C4×C53AGL(1,ℤ/8)12yes—
221C2×C4×C53C22×Q811yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53H(5,𝔽2)11yescharacteristic
226C2×C4×C532−1+411yescharacteristic
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248D212—22no—
249D212—22no—
250D212—41no—
251D212—41no—
252C4×D53—48no—
253Dic106—84no—
254C8×C53—41no—
255C8×C53—41no—
256C8×C53—41no—
257C8×C53—41no—
258#26032—531nomaximal, Sylow
259C22×C53C2×C22⋊Q811yescharacteristic
260C22×C53#54611yescharacteristic
261C22×C53#54811yescharacteristic
262C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C53#57511yescharacteristic
264C22×C53#57711yescharacteristic
265C22×C53#57711yescharacteristic
266C2×D53—42no—
267C2×D53—42no—
268C2×D53—42no—
269C2×D53—81no—
270C2×D53—81no—
271C2×D53—81no—
272C2×D53—81no—
273C4×C53—22no—
274C4×C53—22no—
275C4×C53—22no—
276C4×C53—41no—
277C4×C53—41no—
278C4×C53—41no—
279C4×C53#53311yescharacteristic
280C4×C53#54411yescharacteristic
281C4×C53#55511yescharacteristic
282C4×C53#55611yescharacteristic
283Dic53—42no—
284Dic53—42no—
285Dic53—42no—
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#1940D5311yescharacteristic, complemented
300#1920—531no—
301#1919—531no—
302#1836—531no—
303#1804—531no—
304#1762—531no—
305C2×C53#243911yescharacteristic
306C2×C53#307711yescharacteristic
307C2×C53#308411yescharacteristic
308C2×C53#330611yescharacteristic
309C2×C53#332111yescharacteristic
310C2×C53#336811yescharacteristic
311C2×C53#337611yescharacteristic
312D53—82no—
313C2×C4⋊Q8—531no—
314C2×#125—531no—
315C2×C4×D4—531no—
316C2×#117—1061no—
317C2×#106—531no—
318C2×#106—531no—
319C2×C4⋊C8C2×D5311yescharacteristic
320C2×C4⋊C8C2×D5311yescharacteristic
321C2×C4⋊C8C2×D5311yescharacteristic
322C2×#102—531no—
323C2×#102—531no—
324C2×#102—531no—
325C2×#102—531no—
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#340—531no—
353#340—531no—
354#336C2×D5311yescharacteristic
355#336C2×D5311yescharacteristic
356C53#2603211yescharacteristic, complemented, Sylow
357C22×Q8—1061no—
358C22×D4—531no—
359C23×C4—1061no—
360C22×C8C22×D5311yescharacteristic
361C22×C8C22×D5311yescharacteristic
362C22×C8C22×D5311yescharacteristic
363C22×C8C22×D5311yescharacteristic
364C4⋊Q8—1062no—
365C4⋊Q8—2121no—
366#125—1062no—
367#125—2121no—
368C4×D4—534no—
369C4×D4—1062no—
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—531no—
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⋊C4—1061no—
385C2×C4⋊C4D21211yescharacteristic
386C2×C4⋊C4D21211yescharacteristic
387C2×C22⋊C4—1061no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C42—1061no—
393C2×C42—1061no—
394C2×C42—1061no—
395C2×C42C22×D5311yescharacteristic
396C2×C42C22×D5311yescharacteristic
397C2×C42C22×D5311yescharacteristic
398#106—1062no—
399#106—1062no—
400C4⋊C8—22no—
401C4⋊C8—41no—
402C4⋊C8—41no—
403#102—1062no—
404#102—1062no—
405#102—2121no—
406#102—2121no—
407#97—22no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95D21211yescharacteristic
416#95D21211yescharacteristic
417C24—1061no—
418C2×Q8—4241no—
419C2×Q8—2122no—
420C2×Q8—2122no—
421C2×D4—532no—
422C2×D4—1061no—
423C2×D4—1061no—
424C2×D4—1061no—
425C2×D4—1061no—
426C2×D4—1061no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1064no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—2122no—
436C22×C4—2122no—
437C22×C4C2×D21211yescharacteristic
438C22×C4C2×D21211yescharacteristic
439C22×C4C2×D21211yescharacteristic
440C22×C4C2×D21211yescharacteristic
441C22×C4C2×D21211yescharacteristic
442C22×C4C2×D21211yescharacteristic
443C22×C4C23×D5311yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C4⋊C4—22no—
457C4⋊C4—41no—
458C4⋊C4—532no—
459C4⋊C4—532no—
460C4⋊C4—532no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—4241no—
473C4⋊C4—4241no—
474C4⋊C4—2122no—
475C4⋊C4—2122no—
476C4⋊C4—2122no—
477C4⋊C4—2122no—
478C4⋊C4—2122no—
479C22⋊C4—1064no—
480C22⋊C4—4241no—
481C22⋊C4—4241no—
482C22⋊C4—2122no—
483C22⋊C4—2122no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—4241no—
489C42—2122no—
490C42—2122no—
491C42#1171994712yes—
492C42Q8×D5312yes—
493C23—1062no—
494C23—2121no—
495C23—2121no—
496C23—2121no—
497C23—2121no—
498C23—2121no—
499C23—2121no—
500C23C22×D21211yescharacteristic
501Q8—4244no—
502D4—1062no—
503D4—1062no—
504D4—2121no—
505D4—2121no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—1062no—
522C2×C4—1062no—
523C2×C4—1062no—
524C2×C4—2128no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
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#1313319712yes—
544C2×C4#1313319712yes—
545C2×C4#1313324512yes—
546C2×C4C2×Q8×D5311yescharacteristic
547C2×C4#41×D5311yescharacteristic
548C2×C4#41×D5311yescharacteristic
549C2×C4C2×#1171994711yescharacteristic
550C2×C4#1313326211yescharacteristic
551C2×C4#1313326511yescharacteristic
552C8—41no—
553C8—41no—
554C8—41no—
555C8—41no—
556C22—4241no—
557C22—4241no—
558C22—4241no—
559C22—4241no—
560C22—2122no—
561C22—2122no—
562C22—2122no—
563C22C2×#1313319711yescharacteristic
564C22#1681126811yescharacteristic
565C22#1681128511yescharacteristic
566C22C2×#1313324511yescharacteristic
567C22#1681142311yescharacteristic
568C22#1681143511yescharacteristic
569C22#1681143511yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—4241no—
577C4—4241no—
578C4—4241no—
579C4—4241no—
580C4—2122no—
581C4—2122no—
582C4—2122no—
583C4#1681117911yescharacteristic
584C4#1681125011yescharacteristic
585C4#1681133711yescharacteristic
586C4#1681134411yescharacteristic
587C2—4242no—
588C2#2027840411yescharacteristic
589C2#2028451411yescharacteristic
590C2#2028457911yescharacteristic
591C2#2028667811yescharacteristic
592C2#2028690311yescharacteristic
593C2#2028744611yescharacteristic
594C2#2028754411yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1940C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#1940D53
hypocenterC53#26032
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
1. #20279736C2semidirect
7. #20275670C2semidirect
8. #20275666C2semidirect
9. #20275479C2semidirect
10. #20275301C2semidirect
11. #20275294C2semidirect
12. #20274849C2semidirect
15. C53×#1940C2semidirect
67. C4×D212Q8semidirect
299. #1940D53semidirect
356. C53#26032semidirect

Maximal subgroups

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

subgroupindex
#202797362
#202776622
#202760882
#202758622
#202758522
#202756702
#202756662
#202754792
#202753012
#202752942
#202748492
#202747262
#202745412
C53×#19402
#2603253