Finite Group Databasebeta

← #27417990

Subgroups — #27417990

Order 13568.

1238 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×#16810910C211yesmaximal, characteristic, complemented
2C2×C4×#11719922C211yesmaximal, characteristic, complemented
3#20279628C211yesmaximal, characteristic, complemented
4#20275901C212yesmaximal, complemented
5#20275711C212yesmaximal, complemented
6#20275677C211yesmaximal, characteristic, complemented
7#20275510C211yesmaximal, characteristic, complemented
8#20275037C212yesmaximal, complemented
9#20274914C212yesmaximal, complemented
10#20274715C211yesmaximal, characteristic, complemented
11C2×C53×#555C211yesmaximal, characteristic, complemented
12C22×#11719922C2211yescharacteristic
13C2×#13133191C2212yes—
14C2×#13133188—21no—
15C2×C4×D212C2211yescharacteristic
16#16810910—24no—
17C4×#11719922—24no—
18C2×#13133150C2211yescharacteristic
19C2×#13133148C2212yes—
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2212yes—
23C2×#13133137C2212yes—
24C2×#13133137C2211yescharacteristic
25C2×C22⋊#88322C2212yes—
26C2×C22⋊#88322C2212yes—
27C2×C4×#88322C2211yescharacteristic
28#16810539—22no—
29#16810493C2211yescharacteristic
30#16810483C2212yes—
31#16810480—21no—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35C53×#555—24no—
36C53×#555—24no—
37C2×C4⋊Q8×C53C2211yescharacteristic
38C2×#124×C53—22no—
39C2×#121×C53—21no—
40C2×#121×C53C2212yes—
41C2×C4×D4×C53C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C2×#117×C53—21no—
44C2×#117×C53C2212yes—
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—22no—
47C22×C4×D53—21no—
48C2×#11719922—22no—
49C2×#11719922—22no—
50C2×#11719922—22no—
51C22×#88322C2311yescharacteristic
52C22×#88322C2312yes—
53C22×#88322C2311yescharacteristic
54#13133191—24no—
55#13133191—42no—
56#13133188—44no—
57C4×D212—22no—
58C4×D212—22no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53C2311yescharacteristic
62C2×C4⋊Dic53C2312yes—
63C2×C22⋊Dic53—22no—
64C2×#11719921—21no—
65C2×#11719921—22no—
66#13133150—22no—
67#13133148—24no—
68C4⋊#88322—22no—
69#13133138—22no—
70#13133138—42no—
71#13133137—24no—
72#13133137—22no—
73C22⋊#88322—24no—
74C22⋊#88322—42no—
75C4×#88322—22no—
76#13133124—22no—
77C22×Q8×C53—21no—
78C22×D4×C53—22no—
79C22×D4×C53D411yescharacteristic
80C22×D4×C53C2311yescharacteristic
81C23×C4×C53—21no—
82C23×C4×C53C2312yes—
83C4⋊Q8×C53—22no—
84C4⋊Q8×C53D414yescomplemented
85#124×C53—216no—
86#121×C53—28no—
87#121×C53—24no—
88#121×C53—24no—
89C4×D4×C53—22no—
90C4×D4×C53—22no—
91C4×D4×C53—22no—
92C4×D4×C53D414yescomplemented
93#117×C53—28no—
94#117×C53—24no—
95#117×C53—24no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2312yes—
100C2×C22⋊C4×C53—22no—
101C2×C22⋊C4×C53—22no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53D412yes—
104C2×C42×C53—22no—
105C2×C42×C53C2311yescharacteristic
106C24×C53—21no—
107C24×C53C2×D412yes—
108C23×D53—21no—
109C2×Q8×C53—24no—
110C2×Q8×C53—81no—
111C2×Q8×C53—81no—
112C2×D4×C53—28no—
113C2×D4×C53—22no—
114C2×D4×C53—22no—
115C2×D4×C53—22no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—82no—
122C2×D4×C53—82no—
123C2×D4×C53C2×D412yes—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53—24no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C2×C22⋊D53—44no—
140C2×C22⋊D53—44no—
141C2×C22⋊D53—82no—
142C22×Dic53—21no—
143C22×Dic53—22no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212#4112yes—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53C2×D412yes—
158C4⋊C4×C53C2×D414yes—
159C22⋊C4×C53—28no—
160C22⋊C4×C53—28no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—24no—
163C42×C53—28no—
164C42×C53C2×D412yes—
165C42×C53C2×D412yes—
166C22⋊Dic53—44no—
167#11719922—28no—
168#11719921—44no—
169#11719921—42no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—22no—
173C4⋊Dic53—22no—
174C4⋊Dic53—42no—
175C4⋊Dic53—81no—
176C2×#88322—21no—
177C2×#88322—21no—
178C2×#88322—21no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—21no—
182C2×#88322—22no—
183C2×#88322—21no—
184C2×#88322#4112yes—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53C22×D411yescharacteristic
200C22×D53—22no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—41no—
207Q8×C53—84no—
208D4×C53—22no—
209D4×C53—22no—
210D4×C53—41no—
211D4×C53—41no—
212D4×C53—44no—
213D4×C53—88no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—24no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—48no—
230C2×C4×C53#12012yes—
231C2×C4×C53#12012yes—
232C2×C4×C53#13512yes—
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
240C22⋊D53—88no—
241C2×Dic53—41no—
242C2×Dic53—42no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247D212—22no—
248D212—22no—
249D212—41no—
250D212—41no—
251C4×D53—84no—
252#88322—22no—
253#88322—41no—
254#88322—42no—
255#24117—531nomaximal, Sylow
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53C2×#12011yescharacteristic
271C22×C53#54611yescharacteristic
272C22×C53#54711yescharacteristic
273C22×C53C2×#13511yescharacteristic
274C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C53#57611yescharacteristic
276C22×C53#57711yescharacteristic
277C2×D53—42no—
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—81no—
281C2×D53—81no—
282C2×D53—81no—
283C2×D53—81no—
284C4×C53—22no—
285C4×C53—42no—
286C4×C53—42no—
287C4×C53—44no—
288C4×C53#44914yes—
289C4×C53#52511yescharacteristic
290C4×C53#53711yescharacteristic
291C4×C53#54811yescharacteristic
292C4×C53#55011yescharacteristic
293Dic53—81no—
294Dic53—82no—
295C2×#555D5311yescharacteristic, complemented
296C2×#476—531no—
297C2×C4×D8—531no—
298#2537—531no—
299#2034—532no—
300#1999—532no—
301#1990—531no—
302#1954—531no—
303#1875—532no—
304#1854—532no—
305#1802—531no—
306C2×C53—82no—
307C2×C53—82no—
308C2×C53#240711yescharacteristic
309C2×C53C2×#44911yescharacteristic
310C2×C53#301211yescharacteristic
311C2×C53#330311yescharacteristic
312C2×C53#332511yescharacteristic
313C2×C53#334111yescharacteristic
314C2×C53#335411yescharacteristic
315D53—82no—
316C22×D8—531no—
317#555—24no—
318#555—24no—
319C2×C4⋊Q8C2×D5311yescharacteristic
320C2×#124—22no—
321C2×#121—21no—
322C2×#121—532no—
323C2×#121—1061no—
324C2×#121C2×D5312yes—
325C2×C4×D4—531no—
326C2×C4×D4C2×D5311yescharacteristic
327C2×C4×D4C2×D5311yescharacteristic
328C2×#117—21no—
329C2×#117C2×D5312yes—
330#476—1064no—
331C4×D8—1064no—
332C2×#107—531no—
333C2×#106—532no—
334C2×C4⋊C8—531no—
335C2×#102—531no—
336C2×#102—531no—
337C2×#102—532no—
338C2×#102—532no—
339C2×C22⋊C8—532no—
340C2×C22⋊C8—532no—
341C2×C4×C8—531no—
342#394—1062no—
343#384—531no—
344#380—532no—
345#380—1061no—
346#340—532no—
347#340—532no—
348#340—532no—
349C53#2411711yescharacteristic, complemented, Sylow
350C22×Q8—21no—
351C22×D4—22no—
352C22×D4—531no—
353C22×D4—1062no—
354C22×D4C22⋊D5311yescharacteristic
355C22×D4C22×D5311yescharacteristic
356C23×C4—21no—
357C23×C4—1061no—
358C23×C4C22×D5312yes—
359C2×D8—1062no—
360C2×D8—1062no—
361C2×D8—1062no—
362C22×C8—531no—
363C22×C8—531no—
364C22×C8—532no—
365C4⋊Q8—22no—
366C4⋊Q8C22⋊D5314yescomplemented
367#124—216no—
368#121—28no—
369#121—24no—
370#121—24no—
371#121—1064no—
372#121—2122no—
373#121—2124no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—1062no—
378C4×D4—1062no—
379C4×D4C22⋊D5314yescomplemented
380#117—28no—
381#117—24no—
382#117—24no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—532no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4C22⋊D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5312yes—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1062no—
395C2×C22⋊C4—1061no—
396C2×C22⋊C4—1062no—
397C2×C22⋊C4C22⋊D5312yes—
398C2×C42—22no—
399C2×C42C22×D5311yescharacteristic
400#107—1062no—
401#106—1064no—
402C4⋊C8—1062no—
403#102—1062no—
404#102—1062no—
405#102—1064no—
406#102—2122no—
407C22⋊C8—1064no—
408C22⋊C8—2122no—
409C4×C8—1062no—
410#95—1062no—
411C24—21no—
412C24—1061no—
413C24C2×C22⋊D5312yes—
414C2×Q8—24no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×D4—28no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4—532no—
429C2×D4—1061no—
430C2×D4—1061no—
431C2×D4—1061no—
432C2×D4—1061no—
433C2×D4—1061no—
434C2×D4—4242no—
435C2×D4—2124no—
436C2×D4—2124no—
437C2×D4C2×C22⋊D5312yes—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—1061no—
450C22×C4—1062no—
451C22×C4—2122no—
452C22×C4—2122no—
453C22×C4—2122no—
454C22×C4C2×C22⋊D5312yes—
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C23×D5311yescharacteristic
458D8—1068no—
459C2×C8—532no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1062no—
464C2×C8—1062no—
465C2×C8—1061no—
466C2×C8—1062no—
467C2×C8—1061no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1062no—
475C4⋊C4—1062no—
476C4⋊C4—4241no—
477C4⋊C4—2122no—
478C4⋊C4C2×C22⋊D5314yes—
479C4⋊C4C2×C22⋊D5312yes—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—2124no—
485C22⋊C4—2124no—
486C22⋊C4—2122no—
487C42—28no—
488C42C2×C22⋊D5312yes—
489C42C2×C22⋊D5312yes—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—1062no—
505C23—2121no—
506C23—2121no—
507C23—2121no—
508C23—2121no—
509C23—2121no—
510C23—2121no—
511C23C22×C22⋊D5311yescharacteristic
512Q8—84no—
513D4—22no—
514D4—22no—
515D4—44no—
516D4—41no—
517D4—41no—
518D4—88no—
519D4—1062no—
520D4—1062no—
521D4—4248no—
522D4—2121no—
523D4—2121no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—4244no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4#1313318612yes—
548C2×C4#1313318612yes—
549C2×C4#1313323712yes—
550C2×C4#1313324212yes—
551C2×C4C22×C22⋊D5311yescharacteristic
552C2×C4C22×C22⋊D5311yescharacteristic
553C2×C4#41×D5311yescharacteristic
554C2×C4#41×D5311yescharacteristic
555C2×C4#1313326311yescharacteristic
556C2×C4#1313326411yescharacteristic
557C8—1062no—
558C8—2122no—
559C8—2121no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—4241no—
575C22—4241no—
576C22—4241no—
577C22—4241no—
578C22—2122no—
579C22—2122no—
580C22—2122no—
581C22C2×#1313318611yescharacteristic
582C22#1681126111yescharacteristic
583C22#1681127211yescharacteristic
584C22C2×#1313323711yescharacteristic
585C22C2×#1313324211yescharacteristic
586C22#1681142611yescharacteristic
587C22#1681143411yescharacteristic
588C4—22no—
589C4—42no—
590C4—42no—
591C4—44no—
592C4—4241no—
593C4—4242no—
594C4#1681076814yes—
595C4#1681113011yescharacteristic
596C4#1681120111yescharacteristic
597C4#1681128011yescharacteristic
598C4#1681129211yescharacteristic
599C2—82no—
600C2—82no—
601C2—4242no—
602C2#2027805411yescharacteristic
603C2C2×#1681076811yescharacteristic
604C2#2028401911yescharacteristic
605C2#2028663711yescharacteristic
606C2#2028695111yescharacteristic
607C2#2028713311yescharacteristic
608C2#2028730411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#555C2
socleC23×C53C22×D4
hypercenterC2×#555D53
hypocenterC53#24117
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×#16810910C2semidirect
2. C2×C4×#11719922C2semidirect
3. #20279628C2semidirect
4. #20275901C2semidirect
5. #20275711C2semidirect
6. #20275677C2semidirect
7. #20275510C2semidirect
8. #20275037C2semidirect
9. #20274914C2semidirect
10. #20274715C2semidirect
11. C2×C53×#555C2semidirect
84. C4⋊Q8×C53D4semidirect
92. C4×D4×C53D4semidirect
295. C2×#555D53semidirect
349. C53#24117semidirect
366. C4⋊Q8C22⋊D53semidirect
379. C4×D4C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
C2×C4×#117199222
#202796282
#202759012
#202757112
#202756772
#202755102
#202750372
#202749142
#202747152
C2×C53×#5552
#2411753