Finite Group Databasebeta

← #27352667

Subgroups — #27352667

Order 13568.

950 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811178C211yesmaximal, characteristic
2#20283345C211yesmaximal, characteristic
3#20283345C211yesmaximal, characteristic
4#20273451C214yesmaximal, complemented
5#20273433C211yesmaximal, characteristic, complemented
6#20273432C211yesmaximal, characteristic, complemented
7#20273431C211yesmaximal, characteristic, complemented
8#20273378C214yesmaximal, complemented
9C53×#1556C211yesmaximal, characteristic, complemented
10#16811178—21no—
11#16811178—21no—
12#16811178—21no—
13#16811178—21no—
14#16811178C412yes—
15#16811178C412yes—
16#16811178C2212yes—
17#16811178C2212yes—
18C2×#13133217C2211yescharacteristic
19C2×#13133208C412yes—
20C2×#13133197—21no—
21C2×#13133193—21no—
22C2×Q8×Dic53—21no—
23C2×#13133182—21no—
24C2×#13133175C412yes—
25C2×C42×D53C2211yescharacteristic
26#16810680C2211yescharacteristic
27#16810680C2211yescharacteristic
28#16810678—21no—
29#16810678—21no—
30#16810675C2212yes—
31#16810675C2212yes—
32#16810673—21no—
33#16810673—21no—
34#16810645C2212yes—
35#16810645C2212yes—
36#16810585—21no—
37#16810585—21no—
38#16810575—21no—
39#16810575—21no—
40#16810370C2214yescomplemented
41#16810369C2214yescomplemented
42#16810368C2214yescomplemented
43C2×C4⋊Q8×C53C2211yescharacteristic
44C53×#432C2211yescharacteristic
45C53×#432C2211yescharacteristic
46C53×#332C2214yescomplemented
47C22×Dic106—21no—
48C22×C4×D53C2×C411yescharacteristic
49C22×C4×D53D411yescharacteristic
50C22×C4×D53D411yescharacteristic
51C2×#11719912—22no—
52C2×#11719912—22no—
53#13133217—21no—
54#13133217—21no—
55#13133217C2×C412yes—
56#13133217C2312yes—
57#13133208—22no—
58#13133208—22no—
59#13133208C2×C414yes—
60#13133208C2×C414yes—
61#13133197—22no—
62#13133197—22no—
63#13133197—22no—
64#13133197—22no—
65#13133193—22no—
66#13133193—22no—
67#13133193—22no—
68#13133193—22no—
69Q8×Dic53—22no—
70Q8×Dic53—22no—
71Q8×Dic53—22no—
72Q8×Dic53—22no—
73#13133182—22no—
74#13133182—22no—
75#13133182—22no—
76#13133182—22no—
77#13133175—22no—
78#13133175—22no—
79#13133175C2×C414yes—
80#13133175C2×C414yes—
81C2×C4⋊Dic53—21no—
82C2×C4⋊Dic53—22no—
83C2×C4⋊Dic53C2×C412yes—
84C2×#11719919—22no—
85C2×#11719921—22no—
86C2×C4×Dic53—22no—
87C2×C4×Dic53C2×C411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C42×D53C2×C411yescharacteristic
91C42×D53C2×C411yescharacteristic
92C42×D53D411yescharacteristic
93C42×D53D411yescharacteristic
94C42×D53D411yescharacteristic
95C42×D53D411yescharacteristic
96C42×D53C2311yescharacteristic
97C42×D53C2311yescharacteristic
98C4⋊#88322C2312yes—
99C4⋊#88322C2312yes—
100#13133146—22no—
101#13133146—22no—
102C22⋊#88322—21no—
103C22⋊#88322—21no—
104#13133129—22no—
105#13133129—22no—
106#13133128—21no—
107#13133128—21no—
108C8×Dic53—22no—
109C8×Dic53—22no—
110C22×Q8×C53—21no—
111C4⋊Q8×C53—21no—
112C4⋊Q8×C53—21no—
113C4⋊Q8×C53C2×C412yes—
114C4⋊Q8×C53C2312yes—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2×C412yes—
117C2×C42×C53C2311yescharacteristic
118C4⋊C8×C53C2312yes—
119C4⋊C8×C53C2312yes—
120C22⋊C8×C53—21no—
121C22⋊C8×C53—21no—
122#97×C53—21no—
123#97×C53—21no—
124C23×D53C22⋊C411yescharacteristic
125C2×Q8×C53—22no—
126C2×Q8×C53—22no—
127C2×Q8×C53—44no—
128C22×C4×C53—22no—
129C22×C4×C53C22×C411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×Dic53—22no—
133C22×Dic53C22⋊C411yescharacteristic
134C22×Dic53C22⋊C411yescharacteristic
135C22×Dic53C22⋊C411yescharacteristic
136C2×C4×D53—24no—
137C2×C4×D53—42no—
138C2×C4×D53—42no—
139C2×C4×D53C22⋊C411yescharacteristic
140C2×C4×D53C22⋊C411yescharacteristic
141C2×C4×D53C22⋊C411yescharacteristic
142C2×C4×D53C22⋊C411yescharacteristic
143C2×C4×D53C22⋊C411yescharacteristic
144C2×C4×D53C22⋊C411yescharacteristic
145C2×C4×D53C22×C411yescharacteristic
146C2×C4×D53C22×C411yescharacteristic
147C2×C4×D53C2×D411yescharacteristic
148C2×C4×D53C2×D411yescharacteristic
149C2×C4×D53C2×D411yescharacteristic
150C2×C4×D53C2×D411yescharacteristic
151C2×Dic106—22no—
152C2×Dic106—22no—
153C2×Dic106—44no—
154C2×C8×C53—22no—
155C2×C8×C53—22no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53C22×C414yes—
160C42×C53C22×C411yescharacteristic
161C42×C53C2×D411yescharacteristic
162C42×C53C2×D411yescharacteristic
163C42×C53C2411yescharacteristic
164#11719921—24no—
165#11719921—24no—
166C4⋊Dic53—22no—
167C4⋊Dic53—22no—
168C4⋊Dic53—22no—
169C4⋊Dic53—24no—
170C4⋊Dic53—24no—
171C4⋊Dic53C22×C414yes—
172#11719919—24no—
173#11719919—24no—
174C4×Dic53—24no—
175C4×Dic53—24no—
176C4×Dic53C22⋊C411yescharacteristic
177C4×Dic53C22⋊C411yescharacteristic
178C4×Dic53C22⋊C411yescharacteristic
179C4×Dic53C22⋊C411yescharacteristic
180C4×Dic53C22⋊C411yescharacteristic
181C4×Dic53C22⋊C411yescharacteristic
182C4×Dic53C22×C411yescharacteristic
183C4×Dic53C22×C411yescharacteristic
184C4×Dic53C2×D411yescharacteristic
185C4×Dic53C2×D411yescharacteristic
186C4×Dic53C2×D411yescharacteristic
187C4×Dic53C2×D411yescharacteristic
188C2×#88322—22no—
189C2×#88322—22no—
190#11719912—44no—
191#11719912—44no—
192C23×C53C2×C22⋊C411yescharacteristic
193C22×D53—22no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53C2×C22⋊C411yescharacteristic
197C22×D53C2×C22⋊C411yescharacteristic
198Q8×C53—44no—
199Q8×C53—44no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—24no—
204C2×C4×C53—44no—
205C2×C4×C53C2×C22⋊C411yescharacteristic
206C2×C4×C53C2×C22⋊C411yescharacteristic
207C2×C4×C53C2×C22⋊C411yescharacteristic
208C2×C4×C53C23×C411yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C2×Dic53—24no—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—44no—
215C2×Dic53#10114yes—
216C2×Dic53C2×C22⋊C411yescharacteristic
217C2×Dic53C2×C22⋊C411yescharacteristic
218C2×Dic53C2×C22⋊C411yescharacteristic
219C2×Dic53C2×C22⋊C411yescharacteristic
220C2×Dic53C2×C22⋊C411yescharacteristic
221C2×Dic53C2×C22⋊C411yescharacteristic
222C4×D53—22no—
223C4×D53—22no—
224C4×D53—22no—
225C4×D53—22no—
226C4×D53—22no—
227C4×D53—22no—
228C4×D53—22no—
229C4×D53—22no—
230C4×D53—41no—
231C4×D53—41no—
232C4×D53—41no—
233C4×D53—41no—
234Dic106—44no—
235Dic106—44no—
236C8×C53—42no—
237C8×C53—42no—
238#88322—42no—
239#88322—42no—
240#17739—531nomaximal, Sylow
241C22×C53—21no—
242C22×C53—21no—
243C22×C53C2×#10112yes—
244C22×C53C22×C22⋊C411yescharacteristic
245C2×D53—21no—
246C2×D53—21no—
247C2×D53—21no—
248C2×D53—21no—
249C2×D53—21no—
250C2×D53—21no—
251C2×D53—44no—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—44no—
257C4×C53#57512yes—
258C4×C53#57512yes—
259Dic53—21no—
260Dic53—21no—
261Dic53—21no—
262Dic53—21no—
263Dic53—44no—
264Dic53C2×#10112yes—
265Dic53C2×#10112yes—
266C2×#533—531no—
267#2920—531no—
268#2920—531no—
269#1561—534no—
270#1556—531no—
271#1556—531no—
272#1556—531no—
273#1556D5311yescharacteristic, complemented
274#1544—534no—
275C2×C53—22no—
276C2×C53C22×#10111yescharacteristic
277C2×C53#289611yescharacteristic
278C2×C53#289611yescharacteristic
279D53—41no—
280D53—41no—
281#533—532no—
282#533—532no—
283#533—532no—
284#533—532no—
285#533—1061no—
286#533—1061no—
287#533—1061no—
288#533—1061no—
289C2×C4⋊Q8—531no—
290C2×C4⋊Q8C2×D5311yescharacteristic
291C2×#125—532no—
292C2×C22⋊Q8—1061no—
293C2×C22⋊Q8—1061no—
294C2×C4×Q8—1061no—
295C2×C4×Q8—1061no—
296C2×#117—532no—
297C22×C42—531no—
298#432—531no—
299#432—531no—
300#432—1061no—
301#432—1061no—
302#432C2×D5311yescharacteristic
303#432C2×D5311yescharacteristic
304#431—532no—
305#431—532no—
306#431—1061no—
307#431—1061no—
308#423—532no—
309#423—532no—
310#407—1061no—
311#407—1061no—
312C4×M16—1061no—
313C4×M16—1061no—
314#332—534no—
315#332—534no—
316#332—534no—
317#332C2×D5314yescomplemented
318C53#1773911yescharacteristic, complemented, Sylow
319C22×Q8—21no—
320C22×Q8—1061no—
321C23×C4—531no—
322C23×C4—531no—
323C23×C4—531no—
324C2×M16—1062no—
325C2×M16—1062no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8—532no—
329C4⋊Q8—532no—
330C4⋊Q8—1061no—
331C4⋊Q8—1061no—
332C4⋊Q8C4×D5312yes—
333C4⋊Q8C22×D5312yes—
334#125—534no—
335#125—534no—
336#125—1062no—
337#125—1062no—
338C22⋊Q8—1062no—
339C22⋊Q8—1062no—
340C22⋊Q8—1062no—
341C22⋊Q8—1062no—
342C22⋊Q8—1062no—
343C22⋊Q8—1062no—
344C22⋊Q8—1062no—
345C22⋊Q8—1062no—
346C4×Q8—1062no—
347C4×Q8—1062no—
348C4×Q8—1062no—
349C4×Q8—1062no—
350C4×Q8—1062no—
351C4×Q8—1062no—
352C4×Q8—1062no—
353C4×Q8—1062no—
354#117—534no—
355#117—534no—
356#117—1062no—
357#117—1062no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—532no—
360C2×C4⋊C4—1062no—
361C2×C4⋊C4—1062no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4C4×D5312yes—
364C2×C22⋊C4—1062no—
365C2×C42—531no—
366C2×C42—531no—
367C2×C42—531no—
368C2×C42—531no—
369C2×C42—531no—
370C2×C42—531no—
371C2×C42—531no—
372C2×C42—531no—
373C2×C42—531no—
374C2×C42—531no—
375C2×C42—531no—
376C2×C42—1062no—
377C2×C42C22×D5311yescharacteristic
378C4⋊C8—532no—
379C4⋊C8—532no—
380C4⋊C8—1062no—
381C4⋊C8—1062no—
382C4⋊C8C22×D5312yes—
383C4⋊C8C22×D5312yes—
384C22⋊C8—21no—
385C22⋊C8—21no—
386C22⋊C8—1062no—
387C22⋊C8—1062no—
388C22⋊C8—1061no—
389C22⋊C8—1061no—
390#97—21no—
391#97—21no—
392#97—1061no—
393#97—1061no—
394C4×C8—1062no—
395C4×C8—1062no—
396C24—531no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—44no—
400C2×Q8—1062no—
401C2×Q8—1062no—
402C2×Q8—2124no—
403C22×C4—22no—
404C22×C4—531no—
405C22×C4—531no—
406C22×C4—531no—
407C22×C4—531no—
408C22×C4—531no—
409C22×C4—531no—
410C22×C4—531no—
411C22×C4—531no—
412C22×C4—531no—
413C22×C4—531no—
414C22×C4—531no—
415C22×C4—531no—
416C22×C4—531no—
417C22×C4—531no—
418C22×C4—531no—
419C22×C4—1064no—
420C22×C4—1062no—
421C22×C4—2122no—
422C22×C4—2122no—
423C22×C4C2×C4×D5311yescharacteristic
424C22×C4D4×D5311yescharacteristic
425C22×C4D4×D5311yescharacteristic
426M16—2124no—
427M16—2124no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—1062no—
431C2×C8—1062no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—534no—
436C4⋊C4—1064no—
437C4⋊C4—1064no—
438C4⋊C4—1064no—
439C4⋊C4—1064no—
440C4⋊C4—1062no—
441C4⋊C4—1062no—
442C4⋊C4—1062no—
443C4⋊C4C2×C4×D5314yes—
444C22⋊C4—1064no—
445C22⋊C4—1064no—
446C42—531no—
447C42—531no—
448C42—531no—
449C42—531no—
450C42—531no—
451C42—531no—
452C42—531no—
453C42—531no—
454C42—531no—
455C42—531no—
456C42—531no—
457C42—531no—
458C42—1064no—
459C42—1064no—
460C42C2×C4×D5311yescharacteristic
461C42D4×D5311yescharacteristic
462C42D4×D5311yescharacteristic
463C42C23×D5311yescharacteristic
464C23—531no—
465C23—531no—
466C23—1062no—
467C23—2121no—
468C23—2121no—
469C23C22⋊C4×D5311yescharacteristic
470Q8—44no—
471Q8—44no—
472Q8—2124no—
473Q8—2124no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—44no—
479C2×C4—531no—
480C2×C4—531no—
481C2×C4—534no—
482C2×C4—531no—
483C2×C4—531no—
484C2×C4—531no—
485C2×C4—531no—
486C2×C4—1062no—
487C2×C4—1062no—
488C2×C4—1062no—
489C2×C4—1062no—
490C2×C4—1062no—
491C2×C4—1062no—
492C2×C4—1062no—
493C2×C4—1062no—
494C2×C4—1064no—
495C2×C4—2121no—
496C2×C4—2121no—
497C2×C4—2121no—
498C2×C4—2121no—
499C2×C4—2121no—
500C2×C4—2121no—
501C2×C4—2124no—
502C2×C4C22⋊C4×D5311yescharacteristic
503C2×C4C22⋊C4×D5311yescharacteristic
504C2×C4C22⋊C4×D5311yescharacteristic
505C2×C4C22×C4×D5311yescharacteristic
506C2×C4C2×D4×D5311yescharacteristic
507C2×C4C2×D4×D5311yescharacteristic
508C8—42no—
509C8—42no—
510C8—2122no—
511C8—2122no—
512C22—21no—
513C22—21no—
514C22—1061no—
515C22—1061no—
516C22—1061no—
517C22—1061no—
518C22—1061no—
519C22—1061no—
520C22—2124no—
521C22#101×D5312yes—
522C22C2×C22⋊C4×D5311yescharacteristic
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—44no—
528C4—532no—
529C4—532no—
530C4—1061no—
531C4—1061no—
532C4—1061no—
533C4—1061no—
534C4—2124no—
535C4#1681141912yes—
536C4#1681142212yes—
537C2—22no—
538C2—2121no—
539C2—2121no—
540C2C2×#101×D5311yescharacteristic
541C2#2028322911yescharacteristic
542C2#2028323311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1556C2
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC42×C53C24
hypercenter#1556D53
hypocenterC53#17739
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
4. #20273451C2semidirect
5. #20273433C2semidirect
6. #20273432C2semidirect
7. #20273431C2semidirect
8. #20273378C2semidirect
9. C53×#1556C2semidirect
40. #16810370C22semidirect
41. #16810369C22semidirect
42. #16810368C22semidirect
46. C53×#332C22semidirect
273. #1556D53semidirect
317. #332C2×D53semidirect
318. C53#17739semidirect

Maximal subgroups

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

subgroupindex
C2×#168111782
#202833452
#202734512
#202734332
#202734322
#202734312
#202733782
C53×#15562
#1773953