Finite Group Databasebeta

← #27385177

Subgroups — #27385177

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#20277205C211yesmaximal, characteristic
4#20277198C211yesmaximal, characteristic, complemented
5#20274994C218yesmaximal, complemented
6#20274448C211yesmaximal, characteristic, complemented
7#20274447C211yesmaximal, characteristic, complemented
8C53×#2296C211yesmaximal, 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
19C2×#13133176C2211yescharacteristic
20#117×D53—22no—
21#16810596C418yescomplemented
22C2×#13133131C2214yes—
23#16810529—21no—
24#16810525—21no—
25#16810521—22no—
26#16810502C2214yes—
27#16810496C2214yes—
28C4×#11719919—22no—
29C4×#11719921—22no—
30#16810390C2214yes—
31#16810389C2214yes—
32C2×#117×C53C2211yescharacteristic
33C2×#117×C53C2211yescharacteristic
34C22×C42×C53C2211yescharacteristic
35C53×#389—22no—
36C53×#385C2214yes—
37C4×C4⋊C4×C53—22no—
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—
49C22×C4×D53D412yes—
50C4×Dic106—24no—
51#13133182—24no—
52C4×D212—24no—
53#13133179—24no—
54C4×C22⋊D53—28no—
55#13133177—28no—
56#13133176—21no—
57#13133176—21no—
58#13133176—21no—
59#13133176—21no—
60#13133176—22no—
61#13133176—22no—
62#13133176—22no—
63#13133176C2×C414yescomplemented
64#13133175—24no—
65#13133174—24no—
66#13133173—24no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53D412yes—
70C2×#11719919—24no—
71C4⋊C4×D53—24no—
72C2×C22⋊Dic53C2312yes—
73C2×C22⋊Dic53C2312yes—
74C2×C22⋊Dic53C2312yes—
75C2×#11719921—24no—
76C22⋊C4×D53—24no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53D412yes—
80C42×D53—24no—
81#13133131C2×C4116yescomplemented
82#13133124—24no—
83#13133124—24no—
84C23×C4×C53C2312yes—
85C23×C4×C53C2311yescharacteristic
86#117×C53—21no—
87#117×C53—21no—
88#117×C53—21no—
89#117×C53—21no—
90#117×C53—22no—
91#117×C53C2×C414yescomplemented
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—24no—
94C2×C4⋊C4×C53D412yes—
95C2×C22⋊C4×C53C2312yes—
96C2×C22⋊C4×C53C2312yes—
97C2×C42×C53—21no—
98C2×C42×C53—21no—
99C2×C42×C53—21no—
100C2×C42×C53—21no—
101C2×C42×C53—21no—
102C2×C42×C53—21no—
103C2×C42×C53—21no—
104C2×C42×C53D412yes—
105#95×C53—24no—
106C24×C53C2411yescharacteristic
107C23×D53C2×D412yes—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—44no—
113C22×C4×C53C22×C412yes—
114C22×C4×C53C2×D414yes—
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×Q812yes—
122C22×C4×C53#4112yes—
123C22×C4×C53#4112yes—
124C2×C22⋊D53—22no—
125C2×C22⋊D53—22no—
126C2×C22⋊D53—22no—
127C2×C22⋊D53—22no—
128C2×C22⋊D53C22×C414yes—
129C2×C22⋊D53C2×Q814yes—
130C22×Dic53—22no—
131C22×Dic53—22no—
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—48no—
159C22⋊C4×C53—22no—
160C22⋊C4×C53—22no—
161C22⋊C4×C53—22no—
162C22⋊C4×C53C22×C414yes—
163C42×C53—24no—
164C42×C53—22no—
165C42×C53—22no—
166C42×C53—41no—
167C42×C53—41no—
168C42×C53—41no—
169C42×C53—41no—
170C22⋊Dic53—22no—
171C22⋊Dic53—22no—
172C22⋊Dic53—22no—
173C22⋊Dic53—22no—
174C22⋊Dic53—22no—
175C22⋊Dic53C22×C414yes—
176#11719921—28no—
177#11719921—48no—
178C4⋊Dic53—24no—
179C4⋊Dic53—22no—
180C4⋊Dic53—22no—
181C4⋊Dic53—42no—
182#11719919—28no—
183#11719919—48no—
184C4×Dic53—28no—
185C4×Dic53—24no—
186C4×Dic53—22no—
187C4×Dic53—22no—
188C4×Dic53—42no—
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—24no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—44no—
210C2×C4×C53—42no—
211C2×C4×C53—44no—
212C2×C4×C53C2×C4⋊C412yes—
213C2×C4×C53C2×C4⋊C412yes—
214C2×C4×C53C4×D412yes—
215C2×C4×C53C4×D412yes—
216C2×C4×C53C4×D412yes—
217C2×C4×C53C4×D412yes—
218C2×C4×C53#12011yescharacteristic
219C2×C4×C53#12011yescharacteristic
220C2×C4×C53#12011yescharacteristic
221C2×C4×C53#12011yescharacteristic
222C2×C4×C53C22⋊Q812yes—
223C2×C4×C53C22⋊Q812yes—
224C2×C4×C53C22⋊Q812yes—
225C2×C4×C53C22⋊Q812yes—
226C2×C4×C53#12312yes—
227C2×C4×C53#12312yes—
228C22⋊D53—24no—
229C22⋊D53—24no—
230C22⋊D53—24no—
231C22⋊D53—24no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—24no—
235C2×Dic53—24no—
236C2×Dic53—44no—
237C2×Dic53—44no—
238C2×Dic53—44no—
239C2×Dic53—42no—
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—44no—
270C4×C53#38512yes—
271C4×C53#38512yes—
272C4×C53#38512yes—
273C4×C53#38512yes—
274Dic53—24no—
275Dic53—44no—
276Dic53—44no—
277Dic53—82no—
278#3428—531no—
279C2×#410—531no—
280#2297—531no—
281#2296—531no—
282#2296D5311yescharacteristic, complemented
283#1867—538no—
284#1740—531no—
285#1740—531no—
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—531no—
306C2×#117—1062no—
307C2×#117C2×D5311yescharacteristic
308C2×#117C2×D5311yescharacteristic
309C22×C42C2×D5311yescharacteristic
310#410—538no—
311C2×#99—534no—
312#391—1061no—
313#390—1061no—
314#389—22no—
315#389—1062no—
316#386—534no—
317#385—534no—
318#385C2×D5314yes—
319C4×C4⋊C4—22no—
320C4×C4⋊C4—1062no—
321C4×C22⋊C4—1062no—
322#342—534no—
323#342—534no—
324C53#2088811yescharacteristic, complemented, Sylow
325C2×#41—534no—
326C2×#41—534no—
327C2×#41—1062no—
328C2×#41—1061no—
329C2×#41—1061no—
330C2×#41—1062no—
331C2×#41—1061no—
332C2×#41—1061no—
333C22×Q8—531no—
334C22×D4—532no—
335C22×D4—531no—
336C23×C4—532no—
337C23×C4C22×D5312yes—
338C23×C4C22×D5311yescharacteristic
339C4×Q8—1064no—
340C4×Q8—1064no—
341C4×D4—1064no—
342C4×D4—1068no—
343C4×D4—1064no—
344C4×D4—1068no—
345#117—22no—
346#117—21no—
347#117—21no—
348#117—21no—
349#117—21no—
350#117—534no—
351#117—1064no—
352#117—1064no—
353#117—1064no—
354#117—1061no—
355#117—1061no—
356#117—1062no—
357#117—1061no—
358#117—1061no—
359#117—1062no—
360#117—1062no—
361#117C4×D5314yescomplemented
362C2×C4⋊C4—21no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—532no—
365C2×C4⋊C4—1064no—
366C2×C4⋊C4—1061no—
367C2×C4⋊C4—1064no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4C22⋊D5312yes—
370C2×C22⋊C4—532no—
371C2×C22⋊C4—532no—
372C2×C22⋊C4—532no—
373C2×C22⋊C4—1064no—
374C2×C22⋊C4—1064no—
375C2×C22⋊C4C22×D5312yes—
376C2×C22⋊C4C22×D5312yes—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—532no—
385C2×C42—1064no—
386C2×C42—1061no—
387C2×C42—1061no—
388C2×C42C22⋊D5312yes—
389#99—5316no—
390#95—24no—
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—22no—
420C22×C4—22no—
421C22×C4—44no—
422C22×C4—532no—
423C22×C4—538no—
424C22×C4—534no—
425C22×C4—1068no—
426C22×C4—1062no—
427C22×C4—1062no—
428C22×C4—1064no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4C2×C4×D5312yes—
432C22×C4#1171994512yes—
433C22×C4D4×D5311yescharacteristic
434C22×C4D4×D5311yescharacteristic
435C22×C4D4×D5311yescharacteristic
436C22×C4D4×D5311yescharacteristic
437C22×C4Q8×D5312yes—
438C22×C4#1171994912yes—
439C22×C4C2×C22⋊D5314yes—
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C2×C22⋊D5311yescharacteristic
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—48no—
446C4⋊C4—1068no—
447C4⋊C4—1064no—
448C4⋊C4—1062no—
449C4⋊C4—1062no—
450C4⋊C4—2128no—
451C4⋊C4—2122no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—534no—
456C22⋊C4—1068no—
457C22⋊C4—1062no—
458C22⋊C4—1062no—
459C22⋊C4—1062no—
460C22⋊C4—1062no—
461C22⋊C4—1062no—
462C22⋊C4—2128no—
463C22⋊C4C2×C4×D5314yes—
464C42—24no—
465C42—22no—
466C42—22no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—1068no—
472C42—1064no—
473C42—1062no—
474C42—1062no—
475C42—2122no—
476C23—42no—
477C23—534no—
478C23—1062no—
479C23—1062no—
480C23—2124no—
481C23C22×C4×D5311yescharacteristic
482C23C2×D4×D5311yescharacteristic
483C23C2×D4×D5311yescharacteristic
484C23C22×C22⋊D5311yescharacteristic
485C23C2×Q8×D5311yescharacteristic
486C23C2×#1171994511yescharacteristic
487C23C2×#1171994911yescharacteristic
488Q8—1064no—
489Q8—1064no—
490D4—1064no—
491D4—1064no—
492D4—1064no—
493D4—1064no—
494D4—1064no—
495D4—1064no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—534no—
508C2×C4—1062no—
509C2×C4—1062no—
510C2×C4—1068no—
511C2×C4—1068no—
512C2×C4—1064no—
513C2×C4—1064no—
514C2×C4—4248no—
515C2×C4—2124no—
516C2×C4—2122no—
517C2×C4—2124no—
518C2×C4—2124no—
519C2×C4C4⋊C4×D5312yes—
520C2×C4C4⋊C4×D5312yes—
521C2×C4C4×C22⋊D5312yes—
522C2×C4C4×C22⋊D5312yes—
523C2×C4#1313317912yes—
524C2×C4#1313317912yes—
525C2×C4#1313318511yescharacteristic
526C2×C4#1313318511yescharacteristic
527C2×C4#1313318511yescharacteristic
528C2×C4#1313318511yescharacteristic
529C2×C4#1313319312yes—
530C2×C4#1313319312yes—
531C2×C4#1313319512yes—
532C2×C4#1313319512yes—
533C2×C4#1313320212yes—
534C2×C4#1313320212yes—
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×C4×C22⋊D5311yescharacteristic
548C22C2×#1313317911yescharacteristic
549C22C2×#1313318511yescharacteristic
550C22C2×#1313319311yescharacteristic
551C22C2×#1313319511yescharacteristic
552C22C2×#1313320211yescharacteristic
553C4—44no—
554C4—44no—
555C4—44no—
556C4—1064no—
557C4—4242no—
558C4—2124no—
559C4—2124no—
560C4#1681049412yes—
561C4#1681049412yes—
562C4#1681049412yes—
563C4#1681049412yes—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—2124no—
571C2C2×#1681049411yescharacteristic
572C2#2028410612yes—
subgroupG/H
centerC2×C4#13133185
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#2296C2
socleC22×C53C2×#120
prime radicalC22×#11719945C22
cosocleC24×C53C24
hypercenter#2296D53
hypocenterC53#20888
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2296

Complements

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

NH ≅ G/Ntype
4. #20277198C2semidirect
5. #20274994C2semidirect
6. #20274448C2semidirect
7. #20274447C2semidirect
8. C53×#2296C2semidirect
21. #16810596C4semidirect
63. #13133176C2×C4semidirect
81. #13133131C2×C4semidirect
91. #117×C53C2×C4semidirect
145. C2×D212C4⋊C4semidirect
150. C2×C4×D53C4⋊C4semidirect
154. C2×Dic106C4⋊C4semidirect
282. #2296D53semidirect
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
#202772052
#202771982
#202749942
#202744482
#202744472
C53×#22962
#2088853