Finite Group Databasebeta

← #27418001

Subgroups — #27418001

Order 13568.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810910C211yesmaximal, characteristic, complemented
2C2×C4×#11719924C211yesmaximal, characteristic, complemented
3#20279628C211yesmaximal, characteristic
4#20275908C212yesmaximal, complemented
5#20275716C212yesmaximal, complemented
6#20275677C211yesmaximal, characteristic, complemented
7#20275510C211yesmaximal, characteristic, complemented
8#20275032C212yesmaximal, complemented
9#20274908C212yesmaximal, complemented
10#20274720C211yesmaximal, characteristic, complemented
11C2×C53×#554C211yesmaximal, characteristic, complemented
12C22×#11719924C2211yescharacteristic
13C2×#13133191C2212yes—
14C2×#13133188—21no—
15C2×C4×D212C2211yescharacteristic
16#16810910—24no—
17C4×#11719924—24no—
18C2×#13133148C2212yes—
19C2×#13133148C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2212yes—
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×#554—24no—
36C53×#554—24no—
37C2×C4⋊Q8×C53C2211yescharacteristic
38C2×#125×C53—21no—
39C2×C22⋊Q8×C53—22no—
40C2×C22⋊Q8×C53C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C2×C4×D4×C53—21no—
43C2×C4×D4×C53C2211yescharacteristic
44C22×C4⋊C4×C53C2212yes—
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—22no—
47C22×C4×D53—21no—
48C2×#11719924—22no—
49C2×#11719924—22no—
50C2×#11719924—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#13133148—24no—
67#13133148—22no—
68C4⋊#88322—22no—
69#13133141—22no—
70#13133141—42no—
71#13133137—24no—
72#13133137—22no—
73C22⋊#88322—24no—
74C22⋊#88322—42no—
75C4×#88322—22no—
76#13133124—22no—
77C22×Q8×C53—21no—
78C22×Q8×C53C2311yescharacteristic
79C22×D4×C53D411yescharacteristic
80C23×C4×C53—22no—
81C23×C4×C53C2312yes—
82C4⋊Q8×C53—22no—
83C4⋊Q8×C53D414yescomplemented
84#125×C53—28no—
85C22⋊Q8×C53—216no—
86C22⋊Q8×C53—24no—
87C22⋊Q8×C53—24no—
88C4×Q8×C53—22no—
89C4×Q8×C53D414yescomplemented
90C4×D4×C53—28no—
91C4×D4×C53—22no—
92C4×D4×C53—22no—
93C2×C4⋊C4×C53—24no—
94C2×C4⋊C4×C53—24no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2312yes—
102C2×C22⋊C4×C53—22no—
103C2×C22⋊C4×C53D412yes—
104C2×C42×C53—21no—
105C2×C42×C53C2311yescharacteristic
106C24×C53C2×D412yes—
107C23×D53—21no—
108C2×Q8×C53—24no—
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—81no—
115C2×Q8×C53—81no—
116C2×Q8×C53C2×D412yes—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53SD1614yescomplemented
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53—28no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—82no—
131C22×C4×C53—82no—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2411yescharacteristic
136C2×C22⋊D53—44no—
137C2×C22⋊D53—44no—
138C2×C22⋊D53—82no—
139C22×Dic53—21no—
140C22×Dic53—22no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212#4112yes—
147C2×C4×D53—42no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C4⋊C4×C53—28no—
151C4⋊C4×C53—28no—
152C4⋊C4×C53—24no—
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—24no—
161C42×C53—24no—
162C42×C53C2×D412yes—
163C42×C53C2×D412yes—
164C22⋊Dic53—44no—
165#11719924—28no—
166#11719921—44no—
167#11719921—42no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53—42no—
173C4⋊Dic53—81no—
174C2×#88322—21no—
175C2×#88322—21no—
176C2×#88322—21no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—21no—
180C2×#88322—22no—
181C2×#88322—21no—
182C2×#88322#4112yes—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53C2×SD1614yes—
190C23×C53C22×D411yescharacteristic
191C22×D53—22no—
192C22×D53—41no—
193C22×D53—41no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—41no—
198Q8×C53—22no—
199Q8×C53—22no—
200Q8×C53—41no—
201Q8×C53—41no—
202Q8×C53—84no—
203D4×C53—28no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—22no—
207C2×C4×C53—24no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—42no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—88no—
223C2×C4×C53#12012yes—
224C2×C4×C53#12012yes—
225C2×C4×C53C2×SD1612yes—
226C2×C4×C53#13712yes—
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53H(5,𝔽2)11yescharacteristic
232C2×C4×C532−1+411yescharacteristic
233C22⋊D53—88no—
234C2×Dic53—41no—
235C2×Dic53—42no—
236C2×Dic53—41no—
237C2×Dic53—42no—
238C2×Dic53—41no—
239C2×Dic53—42no—
240D212—22no—
241D212—22no—
242D212—41no—
243D212—41no—
244C4×D53—84no—
245#88322—22no—
246#88322—41no—
247#88322—42no—
248#24118—531nomaximal, Sylow
249C22×C53—24no—
250C22×C53—24no—
251C22×C53—24no—
252C22×C53—42no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53C2×#12011yescharacteristic
257C22×C53#54611yescharacteristic
258C22×C53#54711yescharacteristic
259C22×C53C22×SD1611yescharacteristic
260C22×C53C2×#13711yescharacteristic
261C22×C53#57611yescharacteristic
262C22×C53#57711yescharacteristic
263C2×D53—42no—
264C2×D53—42no—
265C2×D53—42no—
266C2×D53—81no—
267C2×D53—81no—
268C2×D53—81no—
269C2×D53—81no—
270C4×C53—22no—
271C4×C53—42no—
272C4×C53—42no—
273C4×C53—42no—
274C4×C53—44no—
275C4×C53#44814yes—
276C4×C53#52511yescharacteristic
277C4×C53#53711yescharacteristic
278C4×C53#54811yescharacteristic
279C4×C53#55011yescharacteristic
280Dic53—81no—
281Dic53—82no—
282C2×#554D5311yescharacteristic, complemented
283C2×#476—531no—
284C2×C4×SD16—531no—
285#2537—531no—
286#2035—532no—
287#2000—532no—
288#1990—531no—
289#1954—531no—
290#1874—532no—
291#1853—532no—
292#1803—531no—
293C2×C53—44no—
294C2×C53#240711yescharacteristic
295C2×C53C2×#44811yescharacteristic
296C2×C53#301211yescharacteristic
297C2×C53#329511yescharacteristic
298C2×C53#333111yescharacteristic
299C2×C53#333511yescharacteristic
300C2×C53#335711yescharacteristic
301D53—82no—
302C22×SD16—531no—
303#554—24no—
304#554—24no—
305C2×C4⋊Q8C2×D5311yescharacteristic
306C2×#125—21no—
307C2×C22⋊Q8—22no—
308C2×C22⋊Q8C2×D5312yes—
309C2×#121—532no—
310C2×#121—1061no—
311C2×C4×Q8C2×D5311yescharacteristic
312C2×C4×D4—21no—
313C2×C4×D4—531no—
314C2×C4×D4C2×D5311yescharacteristic
315C22×C4⋊C4C2×D5312yes—
316#476—1064no—
317C4×SD16—1064no—
318C2×#106—531no—
319C2×#106—532no—
320C2×C4⋊C8—531no—
321C2×#103—531no—
322C2×#103—532no—
323C2×#102—531no—
324C2×#102—532no—
325C2×C22⋊C8—532no—
326C2×C22⋊C8—532no—
327C2×C4×C8—531no—
328#394—1062no—
329#384—531no—
330#380—532no—
331#380—1061no—
332#340—532no—
333#340—532no—
334#340—532no—
335C53#2411811yescharacteristic, complemented, Sylow
336C22×Q8—21no—
337C22×Q8C22×D5311yescharacteristic
338C22×D4—531no—
339C22×D4—1062no—
340C22×D4C22⋊D5311yescharacteristic
341C23×C4—22no—
342C23×C4—1061no—
343C23×C4C22×D5312yes—
344C2×SD16—1062no—
345C2×SD16—1062no—
346C2×SD16—1062no—
347C22×C8—531no—
348C22×C8—531no—
349C22×C8—532no—
350C4⋊Q8—22no—
351C4⋊Q8C22⋊D5314yescomplemented
352#125—28no—
353C22⋊Q8—216no—
354C22⋊Q8—24no—
355C22⋊Q8—24no—
356#121—1064no—
357#121—2122no—
358#121—2124no—
359C4×Q8—22no—
360C4×Q8C22⋊D5314yescomplemented
361C4×D4—28no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—1062no—
365C4×D4—1062no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—532no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4C22⋊D5311yescharacteristic
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5312yes—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—1062no—
381C2×C22⋊C4—1061no—
382C2×C22⋊C4—1062no—
383C2×C22⋊C4C22⋊D5312yes—
384C2×C42—21no—
385C2×C42C22×D5311yescharacteristic
386#106—1062no—
387#106—1064no—
388C4⋊C8—1062no—
389#103—1062no—
390#103—2122no—
391#102—1062no—
392#102—1064no—
393C22⋊C8—1064no—
394C22⋊C8—2122no—
395C4×C8—1062no—
396#95—1062no—
397C24—1061no—
398C24C2×C22⋊D5312yes—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—81no—
406C2×Q8—81no—
407C2×Q8C2×C22⋊D5312yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—532no—
411C2×D4—1061no—
412C2×D4—1061no—
413C2×D4—1061no—
414C2×D4—1061no—
415C2×D4—1061no—
416C2×D4—4242no—
417C2×D4—2124no—
418C2×D4—2124no—
419C2×D4#1171992414yescomplemented
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—1061no—
433C22×C4—1062no—
434C22×C4—2122no—
435C22×C4—2122no—
436C22×C4—2122no—
437C22×C4C2×C22⋊D5312yes—
438C22×C4C2×C22⋊D5311yescharacteristic
439C22×C4C2×C22⋊D5311yescharacteristic
440C22×C4C23×D5311yescharacteristic
441SD16—1068no—
442C2×C8—532no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1062no—
447C2×C8—1062no—
448C2×C8—1061no—
449C2×C8—1062no—
450C2×C8—1061no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1062no—
461C4⋊C4—1062no—
462C4⋊C4—4241no—
463C4⋊C4—2122no—
464C4⋊C4C2×C22⋊D5312yes—
465C4⋊C4C2×C22⋊D5314yes—
466C22⋊C4—28no—
467C22⋊C4—24no—
468C22⋊C4—2124no—
469C22⋊C4—2124no—
470C22⋊C4—2122no—
471C42—24no—
472C42C2×C22⋊D5312yes—
473C42C2×C22⋊D5312yes—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—1062no—
481C23—2121no—
482C23—2121no—
483C23—2121no—
484C23—2121no—
485C23—2121no—
486C23—2121no—
487C23C2×#1171992414yes—
488C23C22×C22⋊D5311yescharacteristic
489Q8—22no—
490Q8—22no—
491Q8—41no—
492Q8—41no—
493Q8—84no—
494D4—28no—
495D4—1062no—
496D4—1062no—
497D4—4248no—
498D4—2121no—
499D4—2121no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—88no—
519C2×C4—4244no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2122no—
523C2×C4—2121no—
524C2×C4—2122no—
525C2×C4—2122no—
526C2×C4#1313318612yes—
527C2×C4#1313318612yes—
528C2×C4C2×#1171992412yes—
529C2×C4#1313324812yes—
530C2×C4C22×C22⋊D5311yescharacteristic
531C2×C4C22×C22⋊D5311yescharacteristic
532C2×C4#41×D5311yescharacteristic
533C2×C4#41×D5311yescharacteristic
534C2×C4#1313326311yescharacteristic
535C2×C4#1313326411yescharacteristic
536C8—1062no—
537C8—2122no—
538C8—2121no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—4241no—
547C22—4241no—
548C22—4241no—
549C22—4241no—
550C22—2122no—
551C22—2122no—
552C22—2122no—
553C22C2×#1313318611yescharacteristic
554C22#1681126111yescharacteristic
555C22#1681127211yescharacteristic
556C22C22×#1171992411yescharacteristic
557C22C2×#1313324811yescharacteristic
558C22#1681142611yescharacteristic
559C22#1681143411yescharacteristic
560C4—22no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—44no—
565C4—4242no—
566C4—4241no—
567C4#1681076214yes—
568C4#1681113011yescharacteristic
569C4#1681120111yescharacteristic
570C4#1681128011yescharacteristic
571C4#1681129211yescharacteristic
572C2—44no—
573C2—4242no—
574C2#2027805411yescharacteristic
575C2C2×#1681076211yescharacteristic
576C2#2028401911yescharacteristic
577C2#2028653311yescharacteristic
578C2#2028702511yescharacteristic
579C2#2028706611yescharacteristic
580C2#2028734111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#554C2
socleC23×C53C22×D4
prime radical#20279628C2
hypercenterC2×#554D53
hypocenterC53#24118
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16810910C2semidirect
2. C2×C4×#11719924C2semidirect
4. #20275908C2semidirect
5. #20275716C2semidirect
6. #20275677C2semidirect
7. #20275510C2semidirect
8. #20275032C2semidirect
9. #20274908C2semidirect
10. #20274720C2semidirect
11. C2×C53×#554C2semidirect
83. C4⋊Q8×C53D4semidirect
89. C4×Q8×C53D4semidirect
119. C2×D4×C53SD16semidirect
282. C2×#554D53semidirect
335. C53#24118semidirect
351. C4⋊Q8C22⋊D53semidirect
360. C4×Q8C22⋊D53semidirect
419. C2×D4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
C2×C4×#117199242
#202796282
#202759082
#202757162
#202756772
#202755102
#202750322
#202749082
#202747202
C2×C53×#5542
#2411853