Finite Group Databasebeta

← #27438403

Subgroups — #27438403

Order 13568.

1318 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811029C211yesmaximal, characteristic, complemented
2C2×#16810987C211yesmaximal, characteristic, complemented
3#20283025C211yesmaximal, characteristic, complemented
4#20275901C212yesmaximal, complemented
5#20275894C212yesmaximal, complemented
6#20275689C211yesmaximal, characteristic, complemented
7#20275540C211yesmaximal, characteristic, complemented
8#20274976C212yesmaximal, complemented
9#20274962C212yesmaximal, complemented
10#20274477C211yesmaximal, characteristic, complemented
11C2×C53×#532C211yesmaximal, characteristic
12C2×#13133209C2211yescharacteristic
13#16811029—24no—
14#16810987—24no—
15C2×#13133150C2212yes—
16C2×#13133148C2212yes—
17C2×#13133138C2212yes—
18C2×#13133138C2212yes—
19C2×#13133138C2212yes—
20C2×#13133138C2212yes—
21C2×C22⋊#88322C2212yes—
22C2×#13133128C2211yescharacteristic
23C2×#13133128C2211yescharacteristic
24#16810521C2211yescharacteristic
25#16810521C2211yescharacteristic
26#16810507—21no—
27#16810507—21no—
28#16810496—21no—
29#16810496—21no—
30#16810489C2211yescharacteristic
31#16810483C2211yescharacteristic
32#16810483C2211yescharacteristic
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C22×#41×C53—21no—
37C53×#532—24no—
38C53×#532—24no—
39C2×C4⋊Q8×C53C2211yescharacteristic
40C2×#127×C53C2211yescharacteristic
41C2×#124×C53—21no—
42C2×#121×C53—21no—
43C2×#121×C53C2212yes—
44C2×C4×D4×C53—21no—
45C2×C4×D4×C53C2212yes—
46C22×C42×C53C2211yescharacteristic
47C22×#88322C2312yes—
48C22×#88322C2312yes—
49#13133209—22no—
50#13133209—22no—
51C2×C4⋊Dic53C2311yescharacteristic
52C2×C4⋊Dic53C2311yescharacteristic
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2311yescharacteristic
55C2×#11719919—21no—
56C2×#11719919—21no—
57C2×#11719919—21no—
58C2×#11719919—21no—
59C2×C22⋊Dic53—21no—
60C2×C22⋊Dic53—21no—
61#13133150—24no—
62#13133148—24no—
63#13133138—24no—
64#13133138—24no—
65#13133138—24no—
66#13133138—24no—
67C22⋊#88322—24no—
68#13133128—22no—
69#13133128—22no—
70#13133124—21no—
71#13133124—21no—
72#13133124—21no—
73#13133124—21no—
74#13133124—21no—
75#13133124—21no—
76C2×#41×C53—28no—
77C2×#41×C53—44no—
78C2×#41×C53—44no—
79C22×Q8×C53—21no—
80C22×D4×C53—22no—
81C22×D4×C53—21no—
82C22×D4×C53—21no—
83C22×D4×C53C2312yes—
84C23×C4×C53—22no—
85C23×C4×C53D411yescharacteristic
86C23×C4×C53D411yescharacteristic
87C23×C4×C53C2311yescharacteristic
88C4⋊Q8×C53—22no—
89C4⋊Q8×C53D414yes—
90#127×C53—22no—
91#127×C53D414yes—
92#124×C53—28no—
93#121×C53—28no—
94#121×C53—24no—
95#121×C53—24no—
96C4×D4×C53—28no—
97C4×D4×C53—24no—
98C4×D4×C53—24no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53C2312yes—
101C2×C22⋊C4×C53—22no—
102C2×C22⋊C4×C53—22no—
103C2×C42×C53—22no—
104C2×C42×C53—22no—
105C2×C42×C53D411yescharacteristic
106C2×C42×C53D411yescharacteristic
107C2×C42×C53C2311yescharacteristic
108C2×C42×C53C2311yescharacteristic
109C24×C53—22no—
110C24×C53C2×D411yescharacteristic
111#41×C53—432no—
112C2×Q8×C53—24no—
113C2×Q8×C53—42no—
114C2×Q8×C53—42no—
115C2×D4×C53—28no—
116C2×D4×C53—24no—
117C2×D4×C53—24no—
118C2×D4×C53—22no—
119C2×D4×C53—22no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—42no—
124C2×D4×C53—42no—
125C2×D4×C53—42no—
126C2×D4×C53—42no—
127C2×D4×C53—44no—
128C2×D4×C53—44no—
129C2×D4×C53C2×D414yes—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—28no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—41no—
141C22×C4×C53—41no—
142C22×C4×C53—44no—
143C22×C4×C53—44no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2411yescharacteristic
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53C2×D414yes—
157C22⋊C4×C53—28no—
158C22⋊C4×C53—28no—
159C42×C53—22no—
160C42×C53—22no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53C2×D412yes—
164C42×C53C2×D412yes—
165C22⋊Dic53—42no—
166C22⋊Dic53—42no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173#11719919—42no—
174#11719919—42no—
175#11719919—42no—
176#11719919—42no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322#4114yescomplemented
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—24no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53AGL(1,ℤ/8)12yes—
197C23×C53C22×D411yescharacteristic
198Q8×C53—48no—
199D4×C53—24no—
200D4×C53—24no—
201D4×C53—48no—
202D4×C53—48no—
203D4×C53—416no—
204D4×C53—42no—
205D4×C53—42no—
206C2×C4×C53—28no—
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—24no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—44no—
221C2×C4×C53—416no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53#12012yes—
227C2×C4×C53#12012yes—
228C2×C4×C53AGL(1,ℤ/8)12yes—
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C2×#4112yes—
232C2×C4×C532−1+412yes—
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—
245#88322—24no—
246#88322—42no—
247#25676—531nomaximal, Sylow
248C22×C53—22no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—44no—
256C22×C53—44no—
257C22×C53—44no—
258C22×C53—82no—
259C22×C53—82no—
260C22×C53—82no—
261C22×C53—82no—
262C22×C53C2×#12011yescharacteristic
263C22×C53#53211yescharacteristic
264C22×C53#53611yescharacteristic
265C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C53#57511yescharacteristic
268C22×C53#57511yescharacteristic
269C4×C53—22no—
270C4×C53—22no—
271C4×C53—22no—
272C4×C53—41no—
273C4×C53—44no—
274C4×C53—41no—
275C4×C53#54712yes—
276C4×C53#54812yes—
277Dic53—81no—
278Dic53—81no—
279Dic53—81no—
280Dic53—81no—
281C2×#532D5311yescharacteristic
282C2×#501—531no—
283C2×#490—531no—
284#2851—531no—
285#2034—532no—
286#2033—532no—
287#1993—531no—
288#1961—531no—
289#1864—532no—
290#1862—532no—
291#1749—531no—
292C2×C53—42no—
293C2×C53—84no—
294C2×C53#243211yescharacteristic
295C2×C53#300311yescharacteristic
296C2×C53#300511yescharacteristic
297C2×C53#317111yescharacteristic
298C2×C53#318011yescharacteristic
299C2×C53#323411yescharacteristic
300C2×C53#323511yescharacteristic
301C22×#41—21no—
302#532—24no—
303#532—24no—
304C2×C4⋊Q8C2×D5311yescharacteristic
305C2×#127C2×D5311yescharacteristic
306C2×#125—531no—
307C2×#124—21no—
308C2×#121—21no—
309C2×#121C2×D5312yes—
310C2×C4×D4—21no—
311C2×C4×D4C2×D5312yes—
312C22×C42C2×D5311yescharacteristic
313#501—1064no—
314#490—1064no—
315C2×#107—532no—
316C2×#106—532no—
317C2×#102—532no—
318C2×#102—532no—
319C2×#102—532no—
320C2×#102—532no—
321C2×C22⋊C8—532no—
322C2×#97—531no—
323C2×#97—531no—
324#389—531no—
325#389—531no—
326#387—1061no—
327#387—1061no—
328#385—1061no—
329#385—1061no—
330#382—531no—
331#380—531no—
332#380—531no—
333#340—532no—
334#340—532no—
335#336—532no—
336C53#2567611yescharacteristic, complemented, Sylow
337C2×#41—28no—
338C2×#41—44no—
339C2×#41—44no—
340C22×Q8—21no—
341C22×D4—22no—
342C22×D4—21no—
343C22×D4—21no—
344C22×D4C22×D5312yes—
345C23×C4—22no—
346C23×C4C22⋊D5311yescharacteristic
347C23×C4C22⋊D5311yescharacteristic
348C23×C4C22×D5311yescharacteristic
349C22×C8—532no—
350C22×C8—532no—
351C4⋊Q8—22no—
352C4⋊Q8C22⋊D5314yes—
353#127—22no—
354#127C22⋊D5314yes—
355#125—1062no—
356#125—1062no—
357#124—28no—
358#121—28no—
359#121—24no—
360#121—24no—
361C4×D4—28no—
362C4×D4—24no—
363C4×D4—24no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4C22×D5312yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—1061no—
377C2×C22⋊C4—1061no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42C22⋊D5311yescharacteristic
381C2×C42C22⋊D5311yescharacteristic
382C2×C42C22×D5311yescharacteristic
383C2×C42C22×D5311yescharacteristic
384#107—1064no—
385#106—1064no—
386#102—1064no—
387#102—1064no—
388#102—1064no—
389#102—1064no—
390C22⋊C8—1064no—
391#97—1062no—
392#97—1062no—
393#95—1061no—
394#95—1061no—
395#95—1061no—
396#95—1061no—
397#95—1061no—
398#95—1061no—
399C24—22no—
400C24C2×C22⋊D5311yescharacteristic
401#41—432no—
402C2×Q8—24no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×D4—28no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4C2×C22⋊D5314yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4C2×C22⋊D5311yescharacteristic
439C22×C4C2×C22⋊D5311yescharacteristic
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C2×C22⋊D5311yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C23×D5311yescharacteristic
444C2×C8—534no—
445C2×C8—1062no—
446C2×C8—1062no—
447C2×C8—1062no—
448C2×C8—1062no—
449C2×C8—1062no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—1061no—
453C4⋊C4—1061no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4—1062no—
457C4⋊C4—1062no—
458C4⋊C4—2122no—
459C4⋊C4—2122no—
460C4⋊C4—2122no—
461C4⋊C4—2122no—
462C4⋊C4C2×C22⋊D5314yes—
463C22⋊C4—28no—
464C22⋊C4—28no—
465C22⋊C4—2122no—
466C22⋊C4—2122no—
467C42—22no—
468C42—22no—
469C42—21no—
470C42—21no—
471C42C2×C22⋊D5312yes—
472C42C2×C22⋊D5312yes—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—24no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23#1313324312yes—
487C23C22×C22⋊D5311yescharacteristic
488Q8—48no—
489D4—24no—
490D4—24no—
491D4—48no—
492D4—48no—
493D4—416no—
494D4—42no—
495D4—42no—
496C2×C4—28no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—44no—
511C2×C4—416no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4#1313318612yes—
529C2×C4#1313318612yes—
530C2×C4#1313324312yes—
531C2×C4C22×C22⋊D5311yescharacteristic
532C2×C4C22×C22⋊D5311yescharacteristic
533C2×C4#41×D5312yes—
534C2×C4#1313326412yes—
535C8—1064no—
536C8—2122no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22C2×#1313318611yescharacteristic
552C22#1681116911yescharacteristic
553C22#1681119411yescharacteristic
554C22C2×#1313324311yescharacteristic
555C22C2×#1313324311yescharacteristic
556C22#1681142111yescharacteristic
557C22#1681142111yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—44no—
562C4—41no—
563C4—41no—
564C4—4241no—
565C4—4241no—
566C4—4241no—
567C4—4241no—
568C4#1681127212yes—
569C4#1681128012yes—
570C2—42no—
571C2—84no—
572C2#2027833911yescharacteristic
573C2#2028394211yescharacteristic
574C2#2028395711yescharacteristic
575C2#2028530411yescharacteristic
576C2#2028540711yescharacteristic
577C2#2028590111yescharacteristic
578C2#2028590911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#532C2
socleC23×C53C22×D4
hypercenterC2×#532D53
hypocenterC53#25676
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×#16811029C2semidirect
2. C2×#16810987C2semidirect
3. #20283025C2semidirect
4. #20275901C2semidirect
5. #20275894C2semidirect
6. #20275689C2semidirect
7. #20275540C2semidirect
8. #20274976C2semidirect
9. #20274962C2semidirect
10. #20274477C2semidirect
182. C2×#88322#41semidirect
336. C53#25676semidirect

Maximal subgroups

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

subgroupindex
C2×#168110292
C2×#168109872
#202830252
#202759012
#202758942
#202756892
#202755402
#202749762
#202749622
#202744772
C2×C53×#5322
#2567653