Finite Group Databasebeta

← #27385180

Subgroups — #27385180

Order 13568.

1502 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287916C211yesmaximal, characteristic
2C2×#16810596C211yesmaximal, characteristic
3#20277203C211yesmaximal, characteristic
4#20277199C211yesmaximal, characteristic, complemented
5#20277197C211yesmaximal, characteristic, complemented
6#20274997C218yesmaximal, complemented
7#20274447C211yesmaximal, characteristic, complemented
8C53×#1740C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10#16811104—22no—
11#16811103—22no—
12#16811100—21no—
13#16811097—21no—
14#16811096C2214yes—
15C4×#11719945—22no—
16#16811091—22no—
17C2×#13133176C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19#117×D53—22no—
20C22×C4×Dic53C2211yescharacteristic
21#16810596C418yescomplemented
22C2×#13133131C2214yes—
23#16810528—21no—
24#16810524—21no—
25#16810522—22no—
26#16810519—22no—
27#16810500C2214yes—
28#16810498C2214yes—
29#16810495C2214yes—
30C4×#11719919—22no—
31C4×C4⋊Dic53—22no—
32C4×#11719921—22no—
33#16810389C2214yes—
34C2×#117×C53C2211yescharacteristic
35C2×#117×C53C2211yescharacteristic
36C2×#117×C53C2211yescharacteristic
37C53×#342C2214yes—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—22no—
43C2×#11719945—22no—
44C2×#11719945C2×C414yes—
45C2×#11719945Q814yes—
46C22×Dic106D411yescharacteristic
47C22×D212D411yescharacteristic
48C22×C22⋊D53C2312yes—
49C23×Dic53C2312yes—
50C22×C4×D53D412yes—
51C4×Dic106—24no—
52#13133182—24no—
53C4×D212—24no—
54#13133179—24no—
55C4×C22⋊D53—28no—
56#13133177—28no—
57#13133176—21no—
58#13133176—21no—
59#13133176—21no—
60#13133176—21no—
61#13133176—22no—
62#13133176C2×C414yescomplemented
63#13133175—24no—
64#13133174—24no—
65#13133173—24no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—24no—
68C2×C4⋊Dic53D412yes—
69C2×#11719919—24no—
70C4⋊C4×D53—24no—
71C2×C22⋊Dic53C2312yes—
72C2×C22⋊Dic53C2312yes—
73C2×#11719921—24no—
74C22⋊C4×D53—24no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53D412yes—
83C42×D53—24no—
84#13133131C2×C4116yescomplemented
85#13133124—24no—
86#13133124—24no—
87#13133124—24no—
88C23×C4×C53C2311yescharacteristic
89#117×C53—21no—
90#117×C53—21no—
91#117×C53—21no—
92#117×C53—21no—
93#117×C53—22no—
94#117×C53—22no—
95#117×C53—22no—
96#117×C53C2×C414yescomplemented
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53D412yes—
100C2×C22⋊C4×C53C2312yes—
101C2×C22⋊C4×C53C2312yes—
102C2×C22⋊C4×C53C2312yes—
103C2×C42×C53—21no—
104C2×C42×C53—21no—
105C2×C42×C53D412yes—
106C24×C53C2411yescharacteristic
107C23×D53C2×D412yes—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53C22×C412yes—
111C22×C4×C53C2×D414yes—
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×Q812yes—
119C22×C4×C53#4112yes—
120C22×C4×C53#4112yes—
121C2×C22⋊D53—22no—
122C2×C22⋊D53—22no—
123C2×C22⋊D53—22no—
124C2×C22⋊D53—22no—
125C2×C22⋊D53C22×C414yes—
126C2×C22⋊D53C2×Q814yes—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—44no—
132C22×Dic53C2×D412yes—
133C22×Dic53C2×D414yes—
134#11719945—24no—
135#11719945—24no—
136#11719945—24no—
137#11719945—24no—
138#11719945—24no—
139#11719945—24no—
140#11719945—24no—
141#11719945—24no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—22no—
145C2×D212C4⋊C414yescomplemented
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—24no—
149C2×C4×D53—28no—
150C2×C4×D53C4⋊C418yescomplemented
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—22no—
154C2×Dic106C4⋊C414yescomplemented
155C4⋊C4×C53—24no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—42no—
159C22⋊C4×C53—22no—
160C22⋊C4×C53—22no—
161C22⋊C4×C53—22no—
162C22⋊C4×C53—22no—
163C22⋊C4×C53—22no—
164C22⋊C4×C53C22×C414yes—
165C42×C53—24no—
166C42×C53—22no—
167C42×C53—22no—
168C42×C53—42no—
169C22⋊Dic53—22no—
170C22⋊Dic53—22no—
171C22⋊Dic53—22no—
172C22⋊Dic53C22×C414yes—
173#11719921—28no—
174#11719921—48no—
175C4⋊Dic53—24no—
176C4⋊Dic53—22no—
177C4⋊Dic53—22no—
178C4⋊Dic53—48no—
179#11719919—28no—
180#11719919—48no—
181C4×Dic53—28no—
182C4×Dic53—24no—
183C4×Dic53—22no—
184C4×Dic53—22no—
185C4×Dic53—41no—
186C4×Dic53—41no—
187C4×Dic53—41no—
188C4×Dic53—41no—
189C23×C53—42no—
190C23×C53C23×C411yescharacteristic
191C23×C53C22×D411yescharacteristic
192C23×C53C22×D411yescharacteristic
193C23×C53C22×D411yescharacteristic
194C23×C53C22×Q811yescharacteristic
195C23×C53C2×#4111yescharacteristic
196C23×C53C2×#4111yescharacteristic
197C22×D53—22no—
198C22×D53—22no—
199C22×D53—44no—
200C22×D53C2×C4⋊C414yes—
201C2×C4×C53—24no—
202C2×C4×C53—24no—
203C2×C4×C53—44no—
204C2×C4×C53—44no—
205C2×C4×C53—44no—
206C2×C4×C53—42no—
207C2×C4×C53C2×C4⋊C412yes—
208C2×C4×C53C2×C4⋊C412yes—
209C2×C4×C53C4×D412yes—
210C2×C4×C53C4×D412yes—
211C2×C4×C53C4×D412yes—
212C2×C4×C53C4×D412yes—
213C2×C4×C53#12011yescharacteristic
214C2×C4×C53#12011yescharacteristic
215C2×C4×C53#12011yescharacteristic
216C2×C4×C53#12011yescharacteristic
217C2×C4×C53C22⋊Q812yes—
218C2×C4×C53C22⋊Q812yes—
219C2×C4×C53C22⋊Q812yes—
220C2×C4×C53C22⋊Q812yes—
221C2×C4×C53#12312yes—
222C2×C4×C53#12312yes—
223C22⋊D53—24no—
224C22⋊D53—24no—
225C22⋊D53—24no—
226C22⋊D53—24no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—24no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—24no—
233C2×Dic53—24no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—44no—
238C2×Dic53—42no—
239C2×Dic53—44no—
240C2×Dic53C2×C4⋊C414yes—
241D212—24no—
242D212—24no—
243C4×D53—28no—
244C4×D53—28no—
245C4×D53—88no—
246Dic106—24no—
247Dic106—24no—
248#20888—531nomaximal, Sylow
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53C22×C4⋊C411yescharacteristic
257C22×C53C2×C4×D411yescharacteristic
258C22×C53C2×C4×D411yescharacteristic
259C22×C53C2×#12011yescharacteristic
260C22×C53C2×C22⋊Q811yescharacteristic
261C22×C53C2×C22⋊Q811yescharacteristic
262C22×C53C2×#12311yescharacteristic
263C2×D53—24no—
264C2×D53—44no—
265C2×D53—44no—
266C2×D53—82no—
267C4×C53—44no—
268C4×C53—44no—
269C4×C53—82no—
270C4×C53#38512yes—
271C4×C53#38512yes—
272C4×C53#38512yes—
273C4×C53#38512yes—
274Dic53—24no—
275Dic53—44no—
276Dic53—44no—
277Dic53—44no—
278#3428—531no—
279C2×#410—531no—
280#2297—531no—
281#2296—531no—
282#2296—531no—
283#1867—538no—
284#1740—531no—
285#1740D5311yescharacteristic, complemented
286C2×C53—21no—
287C2×C53—21no—
288C2×C53—21no—
289C2×C53—21no—
290C2×C53—21no—
291C2×C53—21no—
292C2×C53C2×#38511yescharacteristic
293C2×C53#302512yes—
294D53—44no—
295C22×#41—531no—
296#520—1062no—
297#520—1062no—
298#519—1061no—
299#518—534no—
300#518—1061no—
301C4×#41—1062no—
302C4×#41—1062no—
303C2×#117—531no—
304C2×#117—531no—
305C2×#117—1062no—
306C2×#117C2×D5311yescharacteristic
307C2×#117C2×D5311yescharacteristic
308C2×#117C2×D5311yescharacteristic
309C22×C42—531no—
310#410—538no—
311C2×#99—534no—
312#391—1061no—
313#390—1061no—
314#389—1062no—
315#389—1062no—
316#386—534no—
317#385—534no—
318#385—534no—
319C4×C4⋊C4—1062no—
320C4×C4⋊C4—1062no—
321C4×C22⋊C4—1062no—
322#342—534no—
323#342C2×D5314yes—
324C53#2088811yescharacteristic, complemented, Sylow
325C2×#41—534no—
326C2×#41—534no—
327C2×#41—1061no—
328C2×#41—1061no—
329C2×#41—1062no—
330C2×#41—1061no—
331C2×#41—1061no—
332C2×#41—1062no—
333C22×Q8—531no—
334C22×D4—532no—
335C22×D4—531no—
336C23×C4—532no—
337C23×C4—532no—
338C23×C4C22×D5311yescharacteristic
339C4×Q8—1064no—
340C4×Q8—1064no—
341C4×D4—1064no—
342C4×D4—1068no—
343C4×D4—1064no—
344C4×D4—1068no—
345#117—21no—
346#117—21no—
347#117—21no—
348#117—21no—
349#117—22no—
350#117—22no—
351#117—22no—
352#117—534no—
353#117—1064no—
354#117—1064no—
355#117—1064no—
356#117—1061no—
357#117—1061no—
358#117—1062no—
359#117—1061no—
360#117—1061no—
361#117C4×D5314yescomplemented
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—532no—
365C2×C4⋊C4—1064no—
366C2×C4⋊C4—1061no—
367C2×C4⋊C4—1064no—
368C2×C4⋊C4—1064no—
369C2×C4⋊C4D21212yes—
370C2×C22⋊C4—532no—
371C2×C22⋊C4—532no—
372C2×C22⋊C4—1064no—
373C2×C22⋊C4—1064no—
374C2×C22⋊C4C22×D5312yes—
375C2×C22⋊C4C22×D5312yes—
376C2×C22⋊C4C22×D5312yes—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—532no—
380C2×C42—1064no—
381C2×C42—1061no—
382C2×C42—1061no—
383C2×C42—1061no—
384C2×C42—1061no—
385C2×C42—1061no—
386C2×C42—1061no—
387C2×C42—1061no—
388C2×C42D21212yes—
389#99—5316no—
390#95—1064no—
391#95—1064no—
392#95—1064no—
393C24—532no—
394C24C23×D5311yescharacteristic
395#41—1064no—
396#41—1064no—
397#41—1064no—
398#41—1064no—
399#41—1064no—
400#41—1064no—
401#41—1064no—
402#41—1064no—
403C2×Q8—534no—
404C2×Q8—1061no—
405C2×Q8—1061no—
406C2×Q8—1062no—
407C2×D4—534no—
408C2×D4—534no—
409C2×D4—534no—
410C2×D4—1062no—
411C2×D4—1062no—
412C2×D4—1062no—
413C2×D4—1062no—
414C2×D4—1061no—
415C2×D4—1061no—
416C2×D4—1062no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—532no—
420C22×C4—538no—
421C22×C4—534no—
422C22×C4—1068no—
423C22×C4—1062no—
424C22×C4—1062no—
425C22×C4—1064no—
426C22×C4—1062no—
427C22×C4—1062no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—2124no—
431C22×C4C2×C4×D5312yes—
432C22×C4C2×D21214yes—
433C22×C4C2×D21211yescharacteristic
434C22×C4C2×D21211yescharacteristic
435C22×C4#1171994512yes—
436C22×C4D4×D5311yescharacteristic
437C22×C4D4×D5311yescharacteristic
438C22×C4D4×D5311yescharacteristic
439C22×C4D4×D5311yescharacteristic
440C22×C4#1171994712yes—
441C22×C4Q8×D5312yes—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—42no—
446C4⋊C4—1068no—
447C4⋊C4—1064no—
448C4⋊C4—1062no—
449C4⋊C4—1062no—
450C4⋊C4—2128no—
451C4⋊C4—2128no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—534no—
458C22⋊C4—1068no—
459C22⋊C4—1062no—
460C22⋊C4—1062no—
461C22⋊C4—1062no—
462C22⋊C4—2128no—
463C22⋊C4C2×C4×D5314yes—
464C42—24no—
465C42—22no—
466C42—22no—
467C42—42no—
468C42—1068no—
469C42—1064no—
470C42—1062no—
471C42—1062no—
472C42—2121no—
473C42—2121no—
474C42—2121no—
475C42—2121no—
476C23—42no—
477C23—534no—
478C23—1062no—
479C23—1062no—
480C23—2124no—
481C23C22×C4×D5311yescharacteristic
482C23C2×D4×D5311yescharacteristic
483C23C2×D4×D5311yescharacteristic
484C23C22×D21211yescharacteristic
485C23C2×Q8×D5311yescharacteristic
486C23C2×#1171994711yescharacteristic
487C23C2×#1171994511yescharacteristic
488Q8—1064no—
489Q8—1064no—
490D4—1064no—
491D4—1064no—
492D4—1064no—
493D4—1064no—
494D4—1064no—
495D4—1064no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—534no—
503C2×C4—1062no—
504C2×C4—1062no—
505C2×C4—1068no—
506C2×C4—1068no—
507C2×C4—1064no—
508C2×C4—1064no—
509C2×C4—1064no—
510C2×C4—1064no—
511C2×C4—1064no—
512C2×C4—4248no—
513C2×C4—2124no—
514C2×C4—2122no—
515C2×C4—2122no—
516C2×C4—2122no—
517C2×C4—2122no—
518C2×C4—2124no—
519C2×C4C4⋊C4×D5312yes—
520C2×C4C4⋊C4×D5312yes—
521C2×C4#1313317712yes—
522C2×C4#1313317712yes—
523C2×C4C4×D21212yes—
524C2×C4C4×D21212yes—
525C2×C4#1313318711yescharacteristic
526C2×C4#1313318711yescharacteristic
527C2×C4#1313318711yescharacteristic
528C2×C4#1313318711yescharacteristic
529C2×C4#1313319512yes—
530C2×C4#1313319512yes—
531C2×C4#1313319712yes—
532C2×C4#1313319712yes—
533C2×C4#1313319912yes—
534C2×C4#1313319912yes—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—1064no—
543C22—4242no—
544C22—2124no—
545C22—2124no—
546C22C2×C4⋊C4×D5311yescharacteristic
547C22C2×#1313317711yescharacteristic
548C22C2×C4×D21211yescharacteristic
549C22C2×#1313318711yescharacteristic
550C22C2×#1313319511yescharacteristic
551C22C2×#1313319711yescharacteristic
552C22C2×#1313319911yescharacteristic
553C4—44no—
554C4—44no—
555C4—82no—
556C4—1064no—
557C4—2124no—
558C4—2124no—
559C4—2124no—
560C4#1681049712yes—
561C4#1681049712yes—
562C4#1681049712yes—
563C4#1681049712yes—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—2124no—
571C2C2×#1681049711yescharacteristic
572C2#2028410912yes—
subgroupG/H
centerC2×C4#13133187
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1740C2
socleC22×C53C2×#120
prime radicalC22×#11719945C22
cosocleC24×C53C24
hypercenter#1740D53
hypocenterC53#20888
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
4. #20277199C2semidirect
5. #20277197C2semidirect
6. #20274997C2semidirect
7. #20274447C2semidirect
8. C53×#1740C2semidirect
21. #16810596C4semidirect
62. #13133176C2×C4semidirect
84. #13133131C2×C4semidirect
96. #117×C53C2×C4semidirect
145. C2×D212C4⋊C4semidirect
150. C2×C4×D53C4⋊C4semidirect
154. C2×Dic106C4⋊C4semidirect
285. #1740D53semidirect
324. C53#20888semidirect
361. #117C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202879162
C2×#168105962
#202772032
#202771992
#202771972
#202749972
#202744472
C53×#17402
#2088853