Finite Group Databasebeta

← #27393771

Subgroups — #27393771

Order 13568.

1222 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811175C211yesmaximal, characteristic
2C2×#16810673C211yesmaximal, characteristic
3C2×#16810492C211yesmaximal, characteristic
4#20275674C212yesmaximal, complemented
5#20275515C212yesmaximal, complemented
6#20275023C211yesmaximal, characteristic
7#20275022C211yesmaximal, characteristic
8#20275020C211yesmaximal, characteristic
9#20274719C212yesmaximal, complemented
10#20274718C212yesmaximal, complemented
11C53×#1872C211yesmaximal, characteristic
12#16811175—24no—
13#16811175C418yescomplemented
14C2×#13133216C2211yescharacteristic
15C2×#13133215C2211yescharacteristic
16C2×#13133207—21no—
17C2×#13133196C2211yescharacteristic
18C2×#13133194—21no—
19C2×Q8×Dic53C2211yescharacteristic
20C2×#13133182—21no—
21C2×#13133182C2211yescharacteristic
22C2×#13133173—21no—
23C22×C4⋊Dic53C2211yescharacteristic
24C22×#11719919—21no—
25C22×#11719919—21no—
26C22×C4×Dic53C2211yescharacteristic
27#16810673—24no—
28C2×#13133146C2212yes—
29C2×#13133142C2212yes—
30C2×#13133141C2212yes—
31C2×#13133140C2212yes—
32C2×C22⋊#88322C2211yescharacteristic
33C2×C8×Dic53C2212yes—
34#16810492—21no—
35#16810492—21no—
36#16810492C2212yes—
37#16810492C2212yes—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×C22⋊Q8×C53C2211yescharacteristic
42C22×C4⋊C4×C53C2211yescharacteristic
43C2×#103×C53C2212yes—
44C2×C22⋊C8×C53C2211yescharacteristic
45C53×#340C2212yes—
46C22×Dic106—21no—
47C22×Dic106C2311yescharacteristic
48C23×Dic53—21no—
49C23×Dic53D411yescharacteristic
50C23×Dic53D411yescharacteristic
51C22×#88322C2312yes—
52#13133216—22no—
53#13133216C2×C414yescomplemented
54#13133215—22no—
55#13133215C2×C414yescomplemented
56#13133207—28no—
57#13133196—22no—
58#13133196C2×C414yes—
59#13133194—28no—
60Q8×Dic53—22no—
61Q8×Dic53C2×C414yescomplemented
62#13133182—28no—
63#13133182—22no—
64#13133182C2×C414yescomplemented
65#13133173—28no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—42no—
69C2×C4⋊Dic53C2312yes—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—22no—
77C2×#11719919—22no—
78C2×#11719919—22no—
79C2×#11719919—22no—
80C2×#11719919—44no—
81C2×#11719919—44no—
82C2×C22⋊Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2×C414yes—
86C2×C4×Dic53D414yes—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133146—24no—
92#13133142—42no—
93#13133141—42no—
94#13133140—42no—
95C22⋊#88322—22no—
96C8×Dic53—24no—
97C22×Q8×C53C2311yescharacteristic
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100C22×C8×C53C2312yes—
101C22⋊Q8×C53—22no—
102C22⋊Q8×C53C2×C414yes—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—42no—
106C2×C4⋊C4×C53C2312yes—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—21no—
109#103×C53—42no—
110C22⋊C8×C53—22no—
111C24×C53C2×D411yescharacteristic
112C2×Q8×C53—21no—
113C2×Q8×C53—22no—
114C2×Q8×C53—22no—
115C2×Q8×C53C22×C412yes—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2411yescharacteristic
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—22no—
133C22×Dic53—42no—
134C22×Dic53—42no—
135C22×Dic53—42no—
136C22×Dic53C22⋊C414yes—
137C22×Dic53C22⋊C414yes—
138C22×Dic53SD1614yes—
139C22×Dic53Q1614yes—
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C2×Dic106—22no—
145C2×Dic106—22no—
146C2×Dic106—21no—
147C2×Dic106—24no—
148C2×Dic106—42no—
149C2×Dic106—42no—
150C2×Dic106C22×C412yes—
151C2×C8×C53—22no—
152C2×C8×C53—22no—
153C2×C8×C53—22no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—42no—
159C4⋊C4×C53C22×C412yes—
160C22⋊C4×C53—24no—
161C22⋊Dic53—24no—
162C4⋊Dic53—21no—
163C4⋊Dic53—24no—
164C4⋊Dic53—22no—
165C4⋊Dic53—22no—
166C4⋊Dic53—42no—
167C4⋊Dic53C22×C412yes—
168#11719919—24no—
169#11719919—24no—
170#11719919—24no—
171#11719919—24no—
172#11719919—24no—
173#11719919—44no—
174#11719919—44no—
175#11719919—44no—
176#11719919—44no—
177C4×Dic53—24no—
178C4×Dic53—24no—
179C4×Dic53C22⋊C414yes—
180C4×Dic53C22⋊C414yes—
181C4×Dic53C22×C412yes—
182C4×Dic53C22×C412yes—
183C4×Dic53C2×D412yes—
184C4×Dic53C2×D412yes—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53C2×C22⋊C412yes—
193C23×C53C2×SD1612yes—
194C23×C53C2×Q1612yes—
195C23×C53C22×D411yescharacteristic
196Q8×C53—24no—
197Q8×C53—42no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—24no—
201C2×C4×C53—24no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—84no—
212C2×C4×C53C2×C22⋊C412yes—
213C2×C4×C53C23×C411yescharacteristic
214C2×C4×C53C22×D411yescharacteristic
215C2×C4×C53H(5,𝔽2)12yes—
216C2×C4×C532−1+412yes—
217C2×Dic53—21no—
218C2×Dic53—21no—
219C2×Dic53—28no—
220C2×Dic53—22no—
221C2×Dic53—22no—
222C2×Dic53—41no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—84no—
230C2×Dic53#103116yescomplemented
231C2×Dic53C2×C22⋊C412yes—
232C2×Dic53C2×C22⋊C412yes—
233C2×Dic53C2×C22⋊C412yes—
234C2×Dic53C2×C22⋊C412yes—
235C2×Dic53C2×SD1612yes—
236C2×Dic53C2×SD1612yes—
237C2×Dic53C2×Q1612yes—
238C2×Dic53C2×Q1612yes—
239C2×Dic53AGL(1,ℤ/8)12yes—
240C2×Dic53#13712yes—
241Dic106—24no—
242Dic106—42no—
243Dic106—48no—
244C8×C53—42no—
245#88322—42no—
246#21860—531nomaximal, Sylow
247C22×C53—22no—
248C22×C53—22no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—22no—
253C22×C53C2×#10314yes—
254C22×C53C22×C22⋊C411yescharacteristic
255C22×C53#53511yescharacteristic
256C22×C53#53611yescharacteristic
257C22×C53C22×SD1611yescharacteristic
258C22×C53C22×Q1611yescharacteristic
259C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C53C2×#13711yescharacteristic
261C4×C53—22no—
262C4×C53—42no—
263C4×C53—42no—
264C4×C53—82no—
265C4×C53#52012yes—
266C4×C53#53712yes—
267Dic53—22no—
268Dic53—42no—
269Dic53—42no—
270Dic53—82no—
271Dic53C2×#10314yes—
272Dic53C2×#10314yes—
273Dic53#41714yes—
274C2×#533—531no—
275C2×#431—531no—
276C2×#384—531no—
277#1990—532no—
278#1956—532no—
279#1872—531no—
280#1872—531no—
281#1872—531no—
282#1872D5311yescharacteristic
283#1803—532no—
284#1803—532no—
285C2×C53—24no—
286C2×C53#231111yescharacteristic
287C2×C53C22×#10311yescharacteristic
288C2×C53C2×#41711yescharacteristic
289C2×C53#320811yescharacteristic
290C2×C53#320911yescharacteristic
291C2×C53#322911yescharacteristic
292C2×C53#323011yescharacteristic
293#533—538no—
294#533—1064no—
295C2×C4⋊Q8—531no—
296C2×C4⋊Q8—531no—
297C2×#125—1061no—
298C2×C22⋊Q8—531no—
299C2×C22⋊Q8—1061no—
300C2×C22⋊Q8C2×D5311yescharacteristic
301C2×C4×Q8—531no—
302C2×C4×Q8—531no—
303C2×C4×Q8—1061no—
304C2×#117—1061no—
305C22×C4⋊C4—531no—
306C22×C4⋊C4—1061no—
307C22×C4⋊C4—1061no—
308C22×C4⋊C4C2×D5311yescharacteristic
309C22×C42—531no—
310#431—1064no—
311C2×C4⋊C8—532no—
312C2×#103—532no—
313C2×#103—532no—
314C2×#103—532no—
315C2×#103C2×D5312yes—
316C2×C22⋊C8—531no—
317C2×C22⋊C8C2×D5311yescharacteristic
318C2×C4×C8—532no—
319#384—532no—
320#384—532no—
321#384—1061no—
322#384—1061no—
323#340—532no—
324#340—532no—
325#340—532no—
326#340C2×D5312yes—
327C53#2186011yescharacteristic, complemented, Sylow
328C22×Q8—531no—
329C22×Q8—1061no—
330C22×Q8C22×D5311yescharacteristic
331C23×C4—21no—
332C23×C4—531no—
333C23×C4—531no—
334C23×C4—1061no—
335C23×C4C22×D5311yescharacteristic
336C22×C8—532no—
337C22×C8C22×D5312yes—
338C4⋊Q8—534no—
339C4⋊Q8—534no—
340C4⋊Q8—1062no—
341C4⋊Q8—1062no—
342#125—1068no—
343C22⋊Q8—22no—
344C22⋊Q8—534no—
345C22⋊Q8—1068no—
346C22⋊Q8—1062no—
347C22⋊Q8C4×D5314yes—
348C4×Q8—534no—
349C4×Q8—534no—
350C4×Q8—1062no—
351C4×Q8—1068no—
352C4×Q8—1062no—
353#117—1068no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4—531no—
358C2×C4⋊C4—532no—
359C2×C4⋊C4—1061no—
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4—1061no—
365C2×C4⋊C4—1062no—
366C2×C4⋊C4—1062no—
367C2×C4⋊C4—1062no—
368C2×C4⋊C4—1062no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4—2122no—
371C2×C4⋊C4—2124no—
372C2×C4⋊C4—2124no—
373C2×C4⋊C4C22×D5312yes—
374C2×C4⋊C4C22×D5311yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—1061no—
377C2×C42—534no—
378C2×C42—531no—
379C2×C42—531no—
380C2×C42—534no—
381C2×C42—531no—
382C2×C42—531no—
383C2×C42—1061no—
384C2×C42—1061no—
385C4⋊C8—1064no—
386#103—42no—
387#103—2122no—
388#103—2122no—
389#103—2122no—
390C22⋊C8—22no—
391C22⋊C8—1062no—
392C4×C8—1064no—
393C24D4×D5311yescharacteristic
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—532no—
398C2×Q8—1064no—
399C2×Q8—1061no—
400C2×Q8—1062no—
401C2×Q8—1062no—
402C2×Q8—2122no—
403C2×Q8—2122no—
404C2×Q8C2×C4×D5312yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—534no—
414C22×C4—534no—
415C22×C4—534no—
416C22×C4—531no—
417C22×C4—531no—
418C22×C4—531no—
419C22×C4—534no—
420C22×C4—531no—
421C22×C4—1062no—
422C22×C4—1062no—
423C22×C4—1061no—
424C22×C4—1061no—
425C22×C4—1062no—
426C22×C4—2122no—
427C22×C4—2122no—
428C22×C4—2122no—
429C22×C4C2×C4×D5312yes—
430C22×C4D4×D5311yescharacteristic
431C22×C4D4×D5312yes—
432C22×C4C23×D5311yescharacteristic
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—1062no—
437C2×C8—1062no—
438C2×C8—1062no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—532no—
445C4⋊C4—1064no—
446C4⋊C4—1064no—
447C4⋊C4—1064no—
448C4⋊C4—1064no—
449C4⋊C4—1064no—
450C4⋊C4—1064no—
451C4⋊C4—1061no—
452C4⋊C4—1062no—
453C4⋊C4—1062no—
454C4⋊C4—2122no—
455C4⋊C4—2124no—
456C4⋊C4—2124no—
457C4⋊C4—2124no—
458C4⋊C4—2124no—
459C4⋊C4C2×C4×D5312yes—
460C22⋊C4—24no—
461C22⋊C4—1064no—
462C42—534no—
463C42—534no—
464C42—532no—
465C42—532no—
466C42—532no—
467C42—532no—
468C42—1064no—
469C42—1064no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23C22⋊C4×D5312yes—
475C23SD16×D5312yes—
476C23Q16×D5312yes—
477C23C2×D4×D5311yescharacteristic
478Q8—24no—
479Q8—42no—
480Q8—1064no—
481Q8—2128no—
482Q8—2122no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—84no—
497C2×C4—532no—
498C2×C4—532no—
499C2×C4—532no—
500C2×C4—5316no—
501C2×C4—532no—
502C2×C4—532no—
503C2×C4—532no—
504C2×C4—532no—
505C2×C4—532no—
506C2×C4—532no—
507C2×C4—532no—
508C2×C4—1068no—
509C2×C4—1061no—
510C2×C4—1061no—
511C2×C4—1062no—
512C2×C4—1062no—
513C2×C4—4244no—
514C2×C4—2122no—
515C2×C4—2122no—
516C2×C4—2122no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4C22⋊C4×D5312yes—
522C2×C4C22×C4×D5311yescharacteristic
523C2×C4C2×D4×D5311yescharacteristic
524C2×C4#1313326212yes—
525C2×C4#1313326412yes—
526C8—42no—
527C8—2122no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22#103×D5314yes—
535C22C2×C22⋊C4×D5311yescharacteristic
536C22#1681118811yescharacteristic
537C22#1681119511yescharacteristic
538C22C2×SD16×D5311yescharacteristic
539C22C2×Q16×D5311yescharacteristic
540C22AGL(1,ℤ/8)×D5311yescharacteristic
541C22#137×D5311yescharacteristic
542C4—22no—
543C4—42no—
544C4—42no—
545C4—82no—
546C4—534no—
547C4—534no—
548C4—534no—
549C4—1062no—
550C4—4242no—
551C4—2122no—
552C4—2122no—
553C4#1681110212yes—
554C4#1681120012yes—
555C2—24no—
556C2#2027730211yescharacteristic
557C2C2×#103×D5311yescharacteristic
558C2#417×D5311yescharacteristic
559C2#2028565411yescharacteristic
560C2#2028566111yescharacteristic
561C2#2028585711yescharacteristic
562C2#2028586411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1872C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenter#1872D53
hypocenterC53#21860
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #20275674C2semidirect
5. #20275515C2semidirect
9. #20274719C2semidirect
10. #20274718C2semidirect
13. #16811175C4semidirect
53. #13133216C2×C4semidirect
55. #13133215C2×C4semidirect
61. Q8×Dic53C2×C4semidirect
64. #13133182C2×C4semidirect
230. C2×Dic53#103semidirect
327. C53#21860semidirect

Maximal subgroups

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

subgroupindex
C2×#168111752
C2×#168106732
C2×#168104922
#202756742
#202755152
#202750232
#202750222
#202750202
#202747192
#202747182
C53×#18722
#2186053