Finite Group Databasebeta

← #27393917

Subgroups — #27393917

Order 13568.

1098 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811218C211yesmaximal, characteristic
2#20277332C211yesmaximal, characteristic, complemented
3#20277315C211yesmaximal, characteristic, complemented
4#20275700C211yesmaximal, characteristic, complemented
5#20275699C211yesmaximal, characteristic, complemented
6#20275552C211yesmaximal, characteristic
7#20275170C214yesmaximal, complemented
8#20274701C214yesmaximal, complemented
9C53×#1964C211yesmaximal, characteristic
10#16811218—22no—
11#16811218—22no—
12#16811218—22no—
13#16811218—22no—
14C2×#13133210C412yescomplemented
15C2×#13133207C412yescomplemented
16C2×#13133198C412yescomplemented
17C2×#13133198C2212yes—
18C2×#13133196C2211yescharacteristic
19C2×#13133194C412yescomplemented
20C2×D4×Dic53C2212yes—
21C2×#13133173C2212yes—
22C22×#11719919C2211yescharacteristic
23C2×#13133132C2214yes—
24#16810520—22no—
25#16810518—22no—
26#16810507—22no—
27#16810505—22no—
28#16810484C2212yes—
29#16810483C2211yescharacteristic
30#16810481C2212yes—
31#16810481C2212yes—
32C4×C22⋊Dic53C2211yescharacteristic
33#16810389C2214yes—
34#16810389C2214yes—
35C2×#121×C53C2211yescharacteristic
36C2×#99×C53C2214yes—
37C53×#380C2211yescharacteristic
38C4×C22⋊C4×C53C2211yescharacteristic
39C22×Dic106C2×C411yescharacteristic
40C23×Dic53C2×C412yes—
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43#13133210—24no—
44#13133210—24no—
45#13133207—24no—
46#13133207—24no—
47#13133198—24no—
48#13133198—24no—
49#13133198—24no—
50#13133198—24no—
51#13133196—21no—
52#13133196—21no—
53#13133196—21no—
54#13133196—21no—
55#13133194—24no—
56#13133194—24no—
57D4×Dic53—24no—
58D4×Dic53—24no—
59#13133173—44no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53C2×C412yes—
62C2×C4⋊Dic53C2×C411yescharacteristic
63C2×#11719919—22no—
64C2×#11719919—22no—
65C2×#11719919—22no—
66C2×#11719919C2×C412yes—
67C2×#11719919C2×C412yes—
68C2×#11719919C2×C412yes—
69C2×#11719919C2×C412yes—
70C2×#11719919C2×C412yes—
71C2×#11719919D412yes—
72C2×#11719919D412yes—
73C2×#11719919D412yes—
74C2×#11719919D412yes—
75C2×C22⋊Dic53C2×C412yes—
76C2×C22⋊Dic53C2×C412yes—
77C2×C22⋊Dic53C2×C412yes—
78C2×C22⋊Dic53C2312yes—
79C2×C22⋊Dic53C2312yes—
80C2×C22⋊Dic53C2312yes—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53C2×C412yes—
84#13133132—44no—
85#13133124—21no—
86#13133124—21no—
87#13133124—22no—
88#13133124—24no—
89#13133124—22no—
90C22×D4×C53C2×C411yescharacteristic
91C22×D4×C53C2312yes—
92C23×C4×C53C2311yescharacteristic
93#121×C53—21no—
94#121×C53—21no—
95#121×C53—21no—
96#121×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53C2×C411yescharacteristic
99C2×C22⋊C4×C53C2×C412yes—
100C2×C22⋊C4×C53C2312yes—
101C2×C22⋊C4×C53C2312yes—
102C2×C42×C53—21no—
103#99×C53—44no—
104#95×C53—21no—
105#95×C53—21no—
106C24×C53C22×C412yes—
107C24×C53C2411yescharacteristic
108C2×D4×C53—22no—
109C2×D4×C53—22no—
110C2×D4×C53—24no—
111C2×D4×C53—24no—
112C2×D4×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53—81no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—81no—
118C22×C4×C53C22×C412yes—
119C22×C4×C53C22×C411yescharacteristic
120C22×C4×C53C22×C411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×Dic53—22no—
126C22×Dic53—22no—
127C22×Dic53—24no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—81no—
131C22×Dic53—82no—
132C22×Dic53C22×C412yes—
133C22×Dic53C22×C412yes—
134C22×Dic53C22×C412yes—
135C22×Dic53C22×C412yes—
136C22×Dic53C2×D412yes—
137C22×Dic53C2×D412yes—
138C22×Dic53C2×D412yes—
139C22×Dic53C2×D412yes—
140C2×Dic106—21no—
141C2×Dic106—21no—
142C2×Dic106—81no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—81no—
146C22⋊C4×C53—22no—
147C22⋊C4×C53—22no—
148C22⋊C4×C53—22no—
149C22⋊C4×C53—24no—
150C22⋊C4×C53—22no—
151C42×C53—81no—
152C22⋊Dic53—22no—
153C22⋊Dic53—22no—
154C22⋊Dic53—24no—
155C22⋊Dic53—24no—
156C22⋊Dic53—24no—
157C22⋊Dic53—22no—
158C22⋊Dic53—24no—
159C22⋊Dic53—22no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—24no—
163C4⋊Dic53—81no—
164#11719919—22no—
165#11719919—22no—
166#11719919—24no—
167#11719919—24no—
168#11719919—44no—
169#11719919—82no—
170#11719919—82no—
171#11719919C22⋊C414yes—
172#11719919C22⋊C414yes—
173#11719919C22⋊C414yes—
174#11719919C22⋊C414yes—
175C4×Dic53—24no—
176C4×Dic53—44no—
177C4×Dic53—81no—
178C23×C53—22no—
179C23×C53—24no—
180C23×C53—41no—
181C23×C53—41no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53C23×C411yescharacteristic
185C23×C53C22×D411yescharacteristic
186C23×C53C22×D411yescharacteristic
187C23×C53H(5,𝔽2)12yes—
188C23×C532−1+412yes—
189D4×C53—42no—
190D4×C53—42no—
191D4×C53—44no—
192D4×C53—81no—
193D4×C53—81no—
194C2×C4×C53—22no—
195C2×C4×C53—24no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53C2×C22⋊C411yescharacteristic
209C2×C4×C53C2×C22⋊C411yescharacteristic
210C2×C4×C53C2×C22⋊C411yescharacteristic
211C2×C4×C53C2×C22⋊C411yescharacteristic
212C2×Dic53—24no—
213C2×Dic53—22no—
214C2×Dic53—22no—
215C2×Dic53—24no—
216C2×Dic53—22no—
217C2×Dic53—22no—
218C2×Dic53—42no—
219C2×Dic53—42no—
220C2×Dic53—42no—
221C2×Dic53—42no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—84no—
230C2×Dic53C2×C22⋊C414yes—
231C2×Dic53C2×C22⋊C414yes—
232Dic106—81no—
233Dic106—81no—
234#21877—531nomaximal, Sylow
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53—84no—
247C22×C53C22×C22⋊C411yescharacteristic
248C22×C53#52012yes—
249C22×C53#53511yescharacteristic
250C22×C53#53611yescharacteristic
251C22×C53#53712yes—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—42no—
257C4×C53—82no—
258C4×C53—82no—
259Dic53—24no—
260Dic53—24no—
261Dic53—42no—
262Dic53—42no—
263Dic53—82no—
264Dic53—82no—
265C2×#539—531no—
266#2316—531no—
267#2313—531no—
268#1996—531no—
269#1996—531no—
270#1964—531no—
271#1964D5311yescharacteristic
272#1899—534no—
273#1799—534no—
274C2×C53—21no—
275C2×C53—22no—
276C2×C53—22no—
277C2×C53—21no—
278C2×C53—82no—
279C2×C53#231111yescharacteristic
280C2×C53#288311yescharacteristic
281C2×C53#288411yescharacteristic
282#539—1062no—
283#539—1062no—
284#539—1062no—
285#539—1062no—
286C2×#126—532no—
287C2×#125—532no—
288C2×#123—532no—
289C2×#123—532no—
290C2×C22⋊Q8—531no—
291C2×C22⋊Q8—532no—
292C2×#121C2×D5311yescharacteristic
293C2×C4×D4—532no—
294C2×#117—532no—
295C22×C4⋊C4—531no—
296C2×#99—534no—
297C2×#99C2×D5314yes—
298#389—1062no—
299#389—1062no—
300#387—1062no—
301#387—1062no—
302#381—532no—
303#380—531no—
304#380—532no—
305#380—532no—
306#380C2×D5311yescharacteristic
307C4×C22⋊C4—531no—
308C4×C22⋊C4C2×D5311yescharacteristic
309#342—534no—
310#342—534no—
311C53#2187711yescharacteristic, complemented, Sylow
312C22×Q8—531no—
313C22×D4C4×D5311yescharacteristic
314C22×D4C22×D5312yes—
315C23×C4—531no—
316C23×C4—531no—
317C23×C4—532no—
318C23×C4C22×D5311yescharacteristic
319#126—1064no—
320#126—1064no—
321#125—1064no—
322#125—1064no—
323#123—1064no—
324#123—1064no—
325#123—1064no—
326#123—1064no—
327C22⋊Q8—1064no—
328C22⋊Q8—1064no—
329C22⋊Q8—1061no—
330C22⋊Q8—1061no—
331C22⋊Q8—1061no—
332C22⋊Q8—1061no—
333#121—21no—
334#121—21no—
335#121—21no—
336#121—21no—
337C4×D4—1064no—
338C4×D4—1064no—
339#117—2124no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—532no—
342C2×C4⋊C4—532no—
343C2×C4⋊C4—532no—
344C2×C4⋊C4—532no—
345C2×C4⋊C4—532no—
346C2×C4⋊C4—532no—
347C2×C4⋊C4—532no—
348C2×C4⋊C4—532no—
349C2×C4⋊C4—532no—
350C2×C4⋊C4—531no—
351C2×C4⋊C4—532no—
352C2×C4⋊C4—1062no—
353C2×C4⋊C4—1062no—
354C2×C4⋊C4—1061no—
355C2×C4⋊C4—1062no—
356C2×C4⋊C4C4×D5311yescharacteristic
357C2×C22⋊C4—532no—
358C2×C22⋊C4—532no—
359C2×C22⋊C4—532no—
360C2×C22⋊C4—532no—
361C2×C22⋊C4—532no—
362C2×C22⋊C4—532no—
363C2×C22⋊C4C4×D5312yes—
364C2×C22⋊C4C22×D5312yes—
365C2×C22⋊C4C22×D5312yes—
366C2×C42—21no—
367C2×C42—532no—
368C2×C42—1061no—
369C2×C42—1062no—
370#99—44no—
371#99—2124no—
372#95—21no—
373#95—21no—
374#95—1062no—
375#95—1062no—
376#95—1064no—
377#95—1061no—
378#95—1061no—
379C24C2×C4×D5312yes—
380C24C23×D5311yescharacteristic
381C2×Q8—1061no—
382C2×Q8—1061no—
383C2×Q8—4241no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—81no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—81no—
394C22×C4—532no—
395C22×C4—532no—
396C22×C4—532no—
397C22×C4—532no—
398C22×C4—532no—
399C22×C4—532no—
400C22×C4—532no—
401C22×C4—532no—
402C22×C4—1064no—
403C22×C4—1062no—
404C22×C4—1062no—
405C22×C4—1062no—
406C22×C4—1062no—
407C22×C4—4242no—
408C22×C4—4241no—
409C22×C4C2×C4×D5312yes—
410C22×C4C2×C4×D5311yescharacteristic
411C22×C4C2×C4×D5311yescharacteristic
412C22×C4D4×D5311yescharacteristic
413C22×C4D4×D5311yescharacteristic
414C22×C4D4×D5311yescharacteristic
415C22×C4D4×D5311yescharacteristic
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—81no—
419C4⋊C4—534no—
420C4⋊C4—534no—
421C4⋊C4—534no—
422C4⋊C4—534no—
423C4⋊C4—1064no—
424C4⋊C4—1064no—
425C4⋊C4—1064no—
426C4⋊C4—1062no—
427C4⋊C4—1062no—
428C4⋊C4—1061no—
429C4⋊C4—1061no—
430C4⋊C4—4242no—
431C4⋊C4—4242no—
432C4⋊C4—4241no—
433C4⋊C4—2124no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—1064no—
440C22⋊C4—1064no—
441C22⋊C4—1064no—
442C22⋊C4—1062no—
443C22⋊C4—1062no—
444C22⋊C4—1062no—
445C22⋊C4—1064no—
446C22⋊C4—1062no—
447C42—81no—
448C42—1064no—
449C42—4241no—
450C42—2124no—
451C23—22no—
452C23—24no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23C22×C4×D5311yescharacteristic
458C23C2×D4×D5311yescharacteristic
459C23C2×D4×D5311yescharacteristic
460C23#1313326212yes—
461C23#1313326412yes—
462Q8—4241no—
463Q8—4241no—
464D4—42no—
465D4—42no—
466D4—44no—
467D4—81no—
468D4—81no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—534no—
484C2×C4—534no—
485C2×C4—1064no—
486C2×C4—1062no—
487C2×C4—1062no—
488C2×C4—1064no—
489C2×C4—1062no—
490C2×C4—1062no—
491C2×C4—4244no—
492C2×C4—2122no—
493C2×C4—2122no—
494C2×C4—2122no—
495C2×C4—2122no—
496C2×C4—2122no—
497C2×C4—2122no—
498C2×C4—2122no—
499C2×C4—2122no—
500C2×C4—2122no—
501C2×C4—2122no—
502C2×C4—2122no—
503C2×C4C22⋊C4×D5311yescharacteristic
504C2×C4C22⋊C4×D5311yescharacteristic
505C2×C4C22⋊C4×D5311yescharacteristic
506C2×C4C22⋊C4×D5311yescharacteristic
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—84no—
519C22C2×C22⋊C4×D5311yescharacteristic
520C22#1681110212yes—
521C22#1681118611yescharacteristic
522C22#1681119311yescharacteristic
523C22#1681120212yes—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—42no—
529C4—82no—
530C4—82no—
531C4—1064no—
532C4—1064no—
533C4—4242no—
534C4—4242no—
535C4—2122no—
536C4—2122no—
537C2—21no—
538C2—22no—
539C2—22no—
540C2—21no—
541C2—82no—
542C2#2027730511yescharacteristic
543C2#2028317311yescharacteristic
544C2#2028317711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1964C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#121×C53C22
cosocleC24×C53C24
hypercenter#1964D53
hypocenterC53#21877
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. #20277332C2semidirect
3. #20277315C2semidirect
4. #20275700C2semidirect
5. #20275699C2semidirect
7. #20275170C2semidirect
8. #20274701C2semidirect
14. C2×#13133210C4semidirect
15. C2×#13133207C4semidirect
16. C2×#13133198C4semidirect
19. C2×#13133194C4semidirect
311. C53#21877semidirect

Maximal subgroups

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

subgroupindex
C2×#168112182
#202773322
#202773152
#202757002
#202756992
#202755522
#202751702
#202747012
C53×#19642
#2187753