Finite Group Databasebeta

← #27425619

Subgroups — #27425619

Order 13568.

1222 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810815C211yesmaximal, characteristic, complemented
2C2×#16810815C212yesmaximal, complemented
3C2×C4×#11719922C211yesmaximal, characteristic, complemented
4C4⋊C4×D212C211yesmaximal, characteristic, complemented
5#20276103C212yesmaximal, complemented
6#20275946C212yesmaximal, complemented
7#20275522C211yesmaximal, characteristic, complemented
8#20275434C211yesmaximal, characteristic, complemented
9#20275066C212yesmaximal, complemented
10#20274715C211yesmaximal, characteristic, complemented
11C53×#2541C211yesmaximal, characteristic, complemented
12C22×#11719922C2212yes—
13C22×#11719922C2211yescharacteristic
14C2×C4×D212C2212yes—
15C2×C4×D212C2211yescharacteristic
16C2×C4⋊C4×D53—21no—
17#16810815—28no—
18#16810815—24no—
19C4×#11719922—24no—
20C2×#13133150C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×C4⋊#88322C2212yes—
23C2×#13133138C2212yes—
24C2×#13133138C2212yes—
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2212yes—
27C2×#13133138C2212yes—
28C2×#13133137C2211yescharacteristic
29C2×C4×#88322C2211yescharacteristic
30#16810521C2212yes—
31#16810497—22no—
32#16810493C2211yescharacteristic
33#16810480—21no—
34#16810385C2212yes—
35#16810378C2212yes—
36C2×#127×C53C2211yescharacteristic
37C2×#127×C53C2212yes—
38C2×#121×C53—21no—
39C2×#121×C53—21no—
40C2×C4×D4×C53C2211yescharacteristic
41C53×#392—22no—
42C53×#390C2212yes—
43C4×C4⋊C4×C53C2211yescharacteristic
44C22×D212C2311yescharacteristic
45C22×C4×D53—21no—
46C22×C4×D53—22no—
47C2×#11719922—22no—
48C2×#11719922—24no—
49C2×#11719922—24no—
50C2×#11719922—22no—
51C2×#11719922—24no—
52C2×#11719922D414yescomplemented
53C22×#88322C2311yescharacteristic
54C22×#88322C2312yes—
55C22×#88322C2311yescharacteristic
56C4×D212—22no—
57C4×D212—24no—
58C4×D212—42no—
59C4×D212D414yescomplemented
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53C2312yes—
63C2×C4⋊Dic53C2311yescharacteristic
64C2×#11719919—22no—
65C4⋊C4×D53—44no—
66C2×#11719921—21no—
67C2×#11719921—22no—
68#13133150—22no—
69C4⋊#88322—22no—
70C4⋊#88322—24no—
71#13133138—24no—
72#13133138—22no—
73#13133138—24no—
74#13133138—24no—
75#13133138—42no—
76#13133137—22no—
77C4×#88322—22no—
78#13133124—22no—
79C22×D4×C53—21no—
80C22×D4×C53—22no—
81C22×D4×C53—22no—
82C22×D4×C53—21no—
83C22×D4×C53C2312yes—
84C22×D4×C53C2311yescharacteristic
85C23×C4×C53—21no—
86#127×C53—22no—
87#127×C53—22no—
88#127×C53—24no—
89#127×C53—42no—
90#121×C53—44no—
91#121×C53—44no—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94C2×C4⋊C4×C53D411yescharacteristic
95C2×C4⋊C4×C53D411yescharacteristic
96C2×C4⋊C4×C53C2311yescharacteristic
97C2×C22⋊C4×C53—21no—
98C2×C22⋊C4×C53—22no—
99C2×C22⋊C4×C53—22no—
100C2×C42×C53C2312yes—
101C2×C42×C53C2311yescharacteristic
102#95×C53D412yes—
103C24×C53—21no—
104C24×C53—22no—
105C23×D53—21no—
106C2×D4×C53—21no—
107C2×D4×C53—22no—
108C2×D4×C53—22no—
109C2×D4×C53—21no—
110C2×D4×C53—21no—
111C2×D4×C53—22no—
112C2×D4×C53—22no—
113C2×D4×C53—21no—
114C2×D4×C53—22no—
115C2×D4×C53—21no—
116C2×D4×C53—22no—
117C2×D4×C53—44no—
118C2×D4×C53—44no—
119C2×D4×C53—44no—
120C2×D4×C53—42no—
121C2×D4×C53—42no—
122C2×D4×C53—44no—
123C2×D4×C53—42no—
124C2×D4×C53—42no—
125C2×D4×C53—82no—
126C2×D4×C53—81no—
127C2×D4×C53—81no—
128C2×D4×C53—82no—
129C2×D4×C53C2×D412yes—
130C22×C4×C53—21no—
131C22×C4×C53—42no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—22no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212C2×D412yes—
145C2×D212C2×D412yes—
146C2×D212#4112yes—
147C2×C4×D53—24no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C4×D53—44no—
151C2×C4×D53—44no—
152C2×C4×D53—82no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53D814yescomplemented
157C22⋊C4×C53—44no—
158C22⋊C4×C53—42no—
159C22⋊C4×C53—44no—
160C42×C53—22no—
161C42×C53—22no—
162C42×C53C2×D412yes—
163C42×C53#4112yes—
164#11719922—28no—
165#11719922—48no—
166#11719921—24no—
167#11719921—44no—
168C4⋊Dic53—21no—
169C4⋊Dic53—22no—
170C4⋊Dic53—22no—
171C4⋊Dic53—42no—
172C4⋊Dic53—81no—
173C4⋊Dic53C2×D412yes—
174#11719919—44no—
175C2×#88322—21no—
176C2×#88322—21no—
177C2×#88322—21no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322C2×D412yes—
184C23×C53—22no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53C22×D411yescharacteristic
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—22no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206D4×C53—22no—
207D4×C53—22no—
208D4×C53—42no—
209D4×C53—41no—
210D4×C53—42no—
211D4×C53—42no—
212D4×C53—41no—
213D4×C53—42no—
214D4×C53—88no—
215D4×C53—84no—
216D4×C53—88no—
217D4×C53—84no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—84no—
229C2×C4×C53#12112yes—
230C2×C4×C53#12112yes—
231C2×C4×C53C2×D814yes—
232C2×C4×C53C2×D812yes—
233C2×C4×C53AGL(1,ℤ/8)12yes—
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53H(5,𝔽2)11yescharacteristic
239C2×C4×C532−1+411yescharacteristic
240C2×Dic53—22no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246D212—22no—
247D212—22no—
248D212—22no—
249D212#12114yescomplemented
250C4×D53—48no—
251C4×D53—88no—
252#88322—22no—
253#88322—42no—
254#88322—41no—
255#24726—531nomaximal, Sylow
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—81no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—82no—
264C22×C53—82no—
265C22×C53—82no—
266C22×C53—82no—
267C22×C53—82no—
268C22×C53—82no—
269C22×C53—82no—
270C22×C53C2×#12111yescharacteristic
271C22×C53D4211yescharacteristic
272C22×C53#54811yescharacteristic
273C22×C53C22×D811yescharacteristic
274C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C53#57611yescharacteristic
276C22×C53#57711yescharacteristic
277C2×D53—24no—
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—42no—
281C2×D53—42no—
282C2×D53—42no—
283C2×D53—42no—
284C4×C53—24no—
285C4×C53—22no—
286C4×C53—42no—
287C4×C53—81no—
288C4×C53#45914yescomplemented
289C4×C53C2×#12111yescharacteristic
290C4×C53#53711yescharacteristic
291C4×C53#54711yescharacteristic
292C4×C53#55311yescharacteristic
293Dic53—42no—
294Dic53—82no—
295C2×#459—531no—
296C2×#459—532no—
297C2×C4×D8—531no—
298#2541D5311yescharacteristic, complemented
299D4×C4⋊C4—531no—
300#2067—532no—
301#2041—532no—
302#1957—531no—
303#1939—531no—
304#1879—532no—
305#1802—531no—
306C2×C53—82no—
307C2×C53—162no—
308C2×C53#241611yescharacteristic
309C2×C53C2×#45911yescharacteristic
310C2×C53#304511yescharacteristic
311C2×C53#327211yescharacteristic
312C2×C53D4×D811yescharacteristic
313C2×C53#335811yescharacteristic
314C2×C53#336611yescharacteristic
315D53—44no—
316C22×D8—531no—
317C22×D8—532no—
318C2×#127C2×D5311yescharacteristic
319C2×#127C2×D5312yes—
320C2×#121—21no—
321C2×#121—21no—
322C2×C4×D4—531no—
323C2×C4×D4—532no—
324C2×C4×D4C2×D5311yescharacteristic
325C22×C4⋊C4—1061no—
326#459—1064no—
327#459—1068no—
328C4×D8—1064no—
329C2×#107—531no—
330C2×C4⋊C8—531no—
331C2×C4⋊C8—532no—
332C2×#102—532no—
333C2×#102—532no—
334C2×#102—532no—
335C2×#102—531no—
336C2×#102—531no—
337C2×#102—532no—
338C2×C4×C8—531no—
339#392—22no—
340#390C2×D5312yes—
341#389—532no—
342#385—1062no—
343#384—531no—
344#380—1061no—
345C4×C4⋊C4C2×D5311yescharacteristic
346#340—532no—
347#336—532no—
348C53#2472611yescharacteristic, complemented, Sylow
349C22×D4—21no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4—21no—
353C22×D4—531no—
354C22×D4C22×D5312yes—
355C22×D4C22×D5311yescharacteristic
356C23×C4—21no—
357C23×C4—1061no—
358C23×C4—1062no—
359C2×D8—534no—
360C2×D8—1062no—
361C2×D8—1062no—
362C2×D8—1064no—
363C2×D8—1064no—
364C2×D8—1064no—
365C22×C8—531no—
366C22×C8—532no—
367C22×C8—531no—
368#127—24no—
369#127—22no—
370#127—22no—
371#127—42no—
372#121—44no—
373#121—44no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—534no—
377C4×D4—1064no—
378C4×D4—1062no—
379C4×D4—2122no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—532no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1062no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—2124no—
386C2×C4⋊C4C22⋊D5311yescharacteristic
387C2×C4⋊C4C22⋊D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1062no—
394C2×C42C22×D5312yes—
395C2×C42C22×D5311yescharacteristic
396#107—1062no—
397C4⋊C8—1064no—
398C4⋊C8—1062no—
399#102—1062no—
400#102—1062no—
401#102—1064no—
402#102—1064no—
403#102—1064no—
404#102—2122no—
405C4×C8—1062no—
406#95—1062no—
407#95C22⋊D5312yes—
408C24—21no—
409C24—22no—
410C24—1061no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—81no—
431C2×D4—82no—
432C2×D4—81no—
433C2×D4—82no—
434C2×D4—532no—
435C2×D4—532no—
436C2×D4—532no—
437C2×D4—1061no—
438C2×D4—1061no—
439C2×D4—1061no—
440C2×D4D4×D5312yes—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—1061no—
446C22×C4—1064no—
447C22×C4—1062no—
448C22×C4—4242no—
449C22×C4—2124no—
450C22×C4—2124no—
451C22×C4—2122no—
452C22×C4—2122no—
453C22×C4C2×C22⋊D5312yes—
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5312yes—
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C23×D5311yescharacteristic
458D8—1068no—
459D8—2128no—
460C2×C8—532no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1062no—
465C2×C8—1062no—
466C2×C8—1062no—
467C2×C8—1061no—
468C2×C8—1061no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—532no—
473C4⋊C4—1061no—
474C4⋊C4—1062no—
475C4⋊C4—1062no—
476C4⋊C4—4241no—
477C4⋊C4—2124no—
478C4⋊C4—2122no—
479C4⋊C4#1171992214yescomplemented
480C22⋊C4—44no—
481C22⋊C4—42no—
482C22⋊C4—44no—
483C22⋊C4—1064no—
484C22⋊C4—2124no—
485C42—22no—
486C42—22no—
487C42D4×D5312yes—
488C42#1171994712yes—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—1062no—
504C23—1062no—
505C23—1062no—
506C23—2121no—
507C23—2121no—
508C23—2121no—
509C23—2121no—
510C23C22×C22⋊D5311yescharacteristic
511D4—22no—
512D4—22no—
513D4—42no—
514D4—41no—
515D4—42no—
516D4—42no—
517D4—41no—
518D4—42no—
519D4—88no—
520D4—84no—
521D4—88no—
522D4—84no—
523D4—534no—
524D4—1062no—
525D4—1062no—
526D4—1062no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—1062no—
539C2×C4—4248no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2128no—
546C2×C4#1313318812yes—
547C2×C4#1313318812yes—
548C2×C4C2×#1171992214yes—
549C2×C4C2×#1171992212yes—
550C2×C4#1313324312yes—
551C2×C4C2×D4×D5311yescharacteristic
552C2×C4C2×D4×D5311yescharacteristic
553C2×C4#41×D5311yescharacteristic
554C2×C4C2×#1171994711yescharacteristic
555C2×C4#1313326211yescharacteristic
556C2×C4#1313326411yescharacteristic
557C8—1062no—
558C8—2122no—
559C8—2121no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—1064no—
575C22—2122no—
576C22—2122no—
577C22—2122no—
578C22—2122no—
579C22—2122no—
580C22—2122no—
581C22C2×#1313318811yescharacteristic
582C22D4×C22⋊D5311yescharacteristic
583C22#1681128011yescharacteristic
584C22C22×#1171992211yescharacteristic
585C22C2×#1313324311yescharacteristic
586C22#1681142611yescharacteristic
587C22#1681143411yescharacteristic
588C4—22no—
589C4—24no—
590C4—42no—
591C4—81no—
592C4—4242no—
593C4—2122no—
594C4#1681081514yescomplemented
595C4#121×D5311yescharacteristic
596C4#1681120211yescharacteristic
597C4#1681127311yescharacteristic
598C4#1681132011yescharacteristic
599C2—82no—
600C2—162no—
601C2—2124no—
602C2#2027815111yescharacteristic
603C2C2×#1681081511yescharacteristic
604C2#2028425011yescharacteristic
605C2#2028631111yescharacteristic
606C2D4×#1171992211yescharacteristic
607C2#2028734611yescharacteristic
608C2#2028741811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2541C2
socleC23×C53C22×D4
hypercenter#2541D53
hypocenterC53#24726
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C2×#16810815C2semidirect
2. C2×#16810815C2semidirect
3. C2×C4×#11719922C2semidirect
4. C4⋊C4×D212C2semidirect
5. #20276103C2semidirect
6. #20275946C2semidirect
7. #20275522C2semidirect
8. #20275434C2semidirect
9. #20275066C2semidirect
10. #20274715C2semidirect
11. C53×#2541C2semidirect
52. C2×#11719922D4semidirect
59. C4×D212D4semidirect
156. C4⋊C4×C53D8semidirect
249. D212#121semidirect
288. C4×C53#459semidirect
298. #2541D53semidirect
348. C53#24726semidirect
479. C4⋊C4#11719922semidirect
594. C4#16810815semidirect

Maximal subgroups

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

subgroupindex
C2×#168108152
C2×C4×#117199222
C4⋊C4×D2122
#202761032
#202759462
#202755222
#202754342
#202750662
#202747152
C53×#25412
#2472653