Finite Group Databasebeta

← #27411521

Subgroups — #27411521

Order 13568.

1148 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810641C211yesmaximal, characteristic
2C2×#16810641C211yesmaximal, characteristic
3C2×#16810596C211yesmaximal, characteristic
4#20275884C218yesmaximal, complemented
5C2×#16810393C211yesmaximal, characteristic, complemented
6#20274425C211yesmaximal, characteristic, complemented
7C2×#16810379C211yesmaximal, characteristic, complemented
8C53×#3458C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13#16810641—22no—
14#16810641—22no—
15#16810641—22no—
16#16810641C2214yes—
17C2×#13133145—22no—
18C2×#13133145C2214yes—
19#16810596—22no—
20#16810596C2214yes—
21C2×#13133131—22no—
22#16810393C2214yescomplemented
23#16810383C2214yescomplemented
24#16810379C2214yescomplemented
25C53×#544—22no—
26C53×#544—22no—
27C53×#544—22no—
28C53×#543—22no—
29C53×#533—22no—
30C53×#533—22no—
31C53×#533C2214yescomplemented
32C2×#117×C53C2211yescharacteristic
33C2×#117×C53C2211yescharacteristic
34C2×#117×C53C2211yescharacteristic
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945—21no—
44C2×#11719945—21no—
45C2×#11719945Q812yescomplemented
46C2×#11719945C2312yes—
47C22×Dic106Q811yescharacteristic, complemented
48C22×D212Q811yescharacteristic, complemented
49C22×C22⋊D53—21no—
50C22×C4×D53—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719917—21no—
59C2×#11719917—21no—
60C2×#11719917—21no—
61C2×#11719917C2312yes—
62C2×#11719917C2312yes—
63C22×#88322—21no—
64C22×#88322—21no—
65#13133176—21no—
66#13133176—21no—
67#13133176—21no—
68#13133176C2312yes—
69C2×C4⋊Dic53—21no—
70C2×C22⋊Dic53—21no—
71C2×C4×Dic53—21no—
72#13133145—28no—
73#13133145—28no—
74#13133131—28no—
75C23×C4×C53C2311yescharacteristic
76C4⋊Q8×C53—24no—
77C4⋊Q8×C53—24no—
78C4⋊Q8×C53—24no—
79#125×C53—24no—
80#125×C53—24no—
81#125×C53—24no—
82C22⋊Q8×C53—28no—
83C22⋊Q8×C53—28no—
84C22⋊Q8×C53—28no—
85C4×Q8×C53—28no—
86#117×C53—21no—
87#117×C53—21no—
88#117×C53—21no—
89#117×C53—21no—
90#117×C53—21no—
91#117×C53—21no—
92#117×C53D412yes—
93#117×C53D412yes—
94#117×C53D412yes—
95#117×C53D412yes—
96#117×C53C2312yes—
97#117×C53C2312yes—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53D412yes—
104C2×C42×C53—21no—
105C2×C42×C53D412yes—
106C2×C42×C53C2312yes—
107C24×C53C2×D411yescharacteristic
108C23×D53—21no—
109C2×Q8×C53—28no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×Q811yescharacteristic
119C22×C4×C53C2×Q811yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C22×C4×C53#4111yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C22×C4×C53#4111yescharacteristic
125C22×C4×C53#4111yescharacteristic
126C22×C4×C53C2411yescharacteristic
127C2×C22⋊D53—22no—
128C2×C22⋊D53—22no—
129C2×C22⋊D53—41no—
130C2×C22⋊D53—41no—
131C2×C22⋊D53—41no—
132C2×C22⋊D53—41no—
133C22×Dic53—21no—
134C22×Dic53—41no—
135#11719945—22no—
136#11719945—22no—
137#11719945—22no—
138#11719945—22no—
139#11719945—22no—
140#11719945—22no—
141#11719945—22no—
142#11719945—22no—
143#11719945—41no—
144#11719945—41no—
145#11719945—41no—
146#11719945—41no—
147#11719945—41no—
148#11719945—41no—
149#11719945—41no—
150#11719945—41no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212C2×Q812yes—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—41no—
160C2×C4×D53—41no—
161C2×C4×D53—41no—
162C2×C4×D53—41no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106C2×Q812yes—
169C4⋊C4×C53—28no—
170C4⋊C4×C53—28no—
171C4⋊C4×C53—28no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—22no—
178C22⋊C4×C53—22no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53C2×D414yes—
184C42×C53—22no—
185C42×C53—22no—
186C42×C53—22no—
187C42×C53—22no—
188C42×C53C2×D414yes—
189C42×C53C2×D414yes—
190C22⋊Dic53—22no—
191C22⋊Dic53—41no—
192C4⋊Dic53—22no—
193C4⋊Dic53—41no—
194C4×Dic53—22no—
195C4×Dic53—41no—
196#11719917—22no—
197#11719917—22no—
198#11719917—22no—
199#11719917—22no—
200#11719917—41no—
201#11719917—41no—
202#11719917—41no—
203#11719917—41no—
204C2×#88322—22no—
205C2×#88322—22no—
206C2×#88322—41no—
207C2×#88322—41no—
208C2×#88322—41no—
209C2×#88322—41no—
210C23×C53—81no—
211C23×C53#12011yescharacteristic
212C23×C53#12011yescharacteristic
213C23×C53C22⋊Q811yescharacteristic
214C23×C53C22⋊Q811yescharacteristic
215C23×C53#12411yescharacteristic
216C23×C53#12411yescharacteristic
217C23×C53C22×D411yescharacteristic
218C22×D53—22no—
219C22×D53—41no—
220C22×D53—41no—
221C22×D53—82no—
222Q8×C53—88no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—82no—
227C2×C4×C53—82no—
228C2×C4×C53—82no—
229C2×C4×C53#12011yescharacteristic
230C2×C4×C53#12011yescharacteristic
231C2×C4×C53C22⋊Q811yescharacteristic
232C2×C4×C53C22⋊Q811yescharacteristic
233C2×C4×C53C22⋊Q811yescharacteristic
234C2×C4×C53C22⋊Q811yescharacteristic
235C2×C4×C53C22⋊Q811yescharacteristic
236C2×C4×C53C22⋊Q811yescharacteristic
237C2×C4×C53C22⋊Q811yescharacteristic
238C2×C4×C53C22⋊Q811yescharacteristic
239C2×C4×C53C22⋊Q811yescharacteristic
240C2×C4×C53C22⋊Q811yescharacteristic
241C2×C4×C53#12411yescharacteristic
242C2×C4×C53#12411yescharacteristic
243C2×C4×C53#12411yescharacteristic
244C2×C4×C53#12411yescharacteristic
245C2×C4×C53#12411yescharacteristic
246C2×C4×C53#12411yescharacteristic
247C2×C4×C53#12411yescharacteristic
248C2×C4×C53#12411yescharacteristic
249C2×C4×C53#12411yescharacteristic
250C2×C4×C53#12411yescharacteristic
251C2×C4×C53C22×D411yescharacteristic
252C2×C4×C53C22×D411yescharacteristic
253C2×C4×C53C22×Q811yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C22⋊D53—42no—
258C22⋊D53—42no—
259C22⋊D53—42no—
260C22⋊D53—42no—
261C2×Dic53—22no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—42no—
265C2×Dic53—82no—
266D212—22no—
267D212—22no—
268D212—41no—
269D212—41no—
270C4×D53—42no—
271C4×D53—42no—
272C4×D53—42no—
273C4×D53—42no—
274Dic106—22no—
275Dic106—22no—
276Dic106—41no—
277Dic106—41no—
278#88322—42no—
279#88322—42no—
280#23472—531nomaximal, Sylow
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53#39311yescharacteristic
289C22×C53#39311yescharacteristic
290C22×C53#39311yescharacteristic
291C22×C53#39311yescharacteristic
292C22×C53C2×#12011yescharacteristic
293C22×C53C2×C22⋊Q811yescharacteristic
294C22×C53C2×#12411yescharacteristic
295C2×D53—42no—
296C2×D53—82no—
297C2×D53—82no—
298C2×D53—82no—
299C4×C53—44no—
300C4×C53—44no—
301C4×C53—44no—
302C4×C53#39311yescharacteristic
303C4×C53#39311yescharacteristic
304C4×C53#39311yescharacteristic
305C4×C53#39311yescharacteristic
306C4×C53C2×C22⋊Q811yescharacteristic
307C4×C53C2×C22⋊Q811yescharacteristic
308C4×C53C2×#12411yescharacteristic
309C4×C53C2×#12411yescharacteristic
310Dic53—42no—
311Dic53—82no—
312#3458D5311yescharacteristic, complemented
313C2×#421—531no—
314C2×#421—531no—
315C2×#410—531no—
316#2031—538no—
317C2×#344—531no—
318#1733—531no—
319C2×H(3,ℤ/4)—531no—
320C2×C53—21no—
321C2×C53—21no—
322C2×C53—21no—
323C2×C53—21no—
324C2×C53—21no—
325C2×C53—21no—
326C2×C53—21no—
327C2×C53C2×#39311yescharacteristic
328D53—82no—
329C22×#41—531no—
330C22×M16—531no—
331C22×M16—531no—
332#544—22no—
333#544—22no—
334#544—22no—
335#543—22no—
336#533—22no—
337#533—22no—
338#533C2×D5314yescomplemented
339C2×#117—531no—
340C2×#117C2×D5311yescharacteristic
341C2×#117C2×D5311yescharacteristic
342C2×#117C2×D5311yescharacteristic
343#421—534no—
344#421—1062no—
345#421—1062no—
346#421—1062no—
347C2×#104—534no—
348C2×#104—1062no—
349#410—534no—
350#410—1062no—
351C2×#99—1062no—
352#344—534no—
353#339—534no—
354H(3,ℤ/4)—534no—
355C53#2347211yescharacteristic, complemented, Sylow
356C2×#41—532no—
357C2×#41—532no—
358C2×#41—1061no—
359C2×#41—1061no—
360C2×#41—1061no—
361C2×#41—1061no—
362C2×#41—1061no—
363C2×#41—1061no—
364C2×#41—1061no—
365C2×#41—1061no—
366C2×#41—1061no—
367C2×#41—1061no—
368C22×Q8—531no—
369C22×D4—531no—
370C22×D4—1061no—
371C23×C4—1061no—
372C23×C4C22×D5311yescharacteristic
373C2×M16—532no—
374C2×M16—532no—
375C2×M16—1061no—
376C2×M16—1061no—
377C2×M16—1061no—
378C2×M16—1061no—
379C2×M16—1061no—
380C2×M16—1061no—
381C2×M16—1061no—
382C2×M16—1061no—
383C2×M16—1061no—
384C2×M16—1061no—
385C22×C8—1061no—
386C22×C8—1061no—
387C4⋊Q8—24no—
388C4⋊Q8—24no—
389C4⋊Q8—24no—
390#125—24no—
391#125—24no—
392#125—24no—
393C22⋊Q8—28no—
394C22⋊Q8—28no—
395C22⋊Q8—28no—
396C4×Q8—28no—
397#117—21no—
398#117—21no—
399#117—21no—
400#117—21no—
401#117—21no—
402#117—21no—
403#117—532no—
404#117—1061no—
405#117—1061no—
406#117—1061no—
407#117C22⋊D5312yes—
408#117C22⋊D5312yes—
409#117C22⋊D5312yes—
410#117C22⋊D5312yes—
411#117C22×D5312yes—
412#117C22×D5312yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—1061no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4C22⋊D5312yes—
421C2×C42—21no—
422C2×C42—1061no—
423C2×C42C22⋊D5312yes—
424C2×C42C22×D5312yes—
425#104—1068no—
426#104—1068no—
427#99—1068no—
428C24—1061no—
429C24C2×C22⋊D5311yescharacteristic
430#41—1062no—
431#41—1062no—
432#41—1062no—
433#41—1062no—
434#41—1062no—
435#41—1062no—
436#41—1062no—
437#41—1062no—
438#41—2121no—
439#41—2121no—
440#41—2121no—
441#41—2121no—
442#41—2121no—
443#41—2121no—
444#41—2121no—
445#41—2121no—
446C2×Q8—28no—
447C2×Q8—532no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C2×D4—532no—
454C2×D4—1062no—
455C2×D4—1062no—
456C2×D4—1061no—
457C2×D4—1061no—
458C2×D4—1061no—
459C2×D4—1061no—
460C2×D4—1061no—
461C2×D4—2121no—
462C2×D4—2121no—
463C2×D4—2121no—
464C2×D4—2121no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—1061no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—2121no—
472C22×C4—2121no—
473C22×C4—2121no—
474C22×C4—2121no—
475C22×C4—2121no—
476C22×C4Q8×D5311yescharacteristic
477C22×C4Q8×D5311yescharacteristic
478C22×C4#1171994911yescharacteristic
479C22×C4#1171994911yescharacteristic
480C22×C4#1171994911yescharacteristic
481C22×C4#1171994911yescharacteristic
482C22×C4#1171994911yescharacteristic
483C22×C4#1171994911yescharacteristic
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C2×C22⋊D5311yescharacteristic
486C22×C4C2×C22⋊D5311yescharacteristic
487C22×C4C2×C22⋊D5311yescharacteristic
488C22×C4C2×C22⋊D5311yescharacteristic
489C22×C4C23×D5311yescharacteristic
490M16—1062no—
491M16—1062no—
492M16—1062no—
493M16—1062no—
494M16—2121no—
495M16—2121no—
496M16—2121no—
497M16—2121no—
498C2×C8—1062no—
499C2×C8—1062no—
500C2×C8—2121no—
501C2×C8—2121no—
502C2×C8—2121no—
503C2×C8—2121no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—1062no—
514C4⋊C4—2121no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—1062no—
521C22⋊C4—2121no—
522C22⋊C4C2×C22⋊D5314yes—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—1062no—
528C42—2121no—
529C42C2×C22⋊D5314yes—
530C42C2×C22⋊D5314yes—
531C23—81no—
532C23—1062no—
533C23—4242no—
534C23—2121no—
535C23—2121no—
536C23#1313318611yescharacteristic
537C23#1313318611yescharacteristic
538C23#1313319311yescharacteristic
539C23#1313319311yescharacteristic
540C23#1313320611yescharacteristic
541C23#1313320611yescharacteristic
542C23C22×C22⋊D5311yescharacteristic
543Q8—88no—
544Q8—1062no—
545Q8—1062no—
546Q8—2121no—
547Q8—2121no—
548D4—1062no—
549D4—1062no—
550D4—2122no—
551D4—2122no—
552D4—2122no—
553D4—2122no—
554D4—2121no—
555D4—2121no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—1062no—
563C2×C4—4242no—
564C2×C4—2122no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4#1313318611yescharacteristic
572C2×C4#1313318611yescharacteristic
573C2×C4#1313319311yescharacteristic
574C2×C4#1313319311yescharacteristic
575C2×C4#1313319311yescharacteristic
576C2×C4#1313319311yescharacteristic
577C2×C4#1313319311yescharacteristic
578C2×C4#1313319311yescharacteristic
579C2×C4#1313319311yescharacteristic
580C2×C4#1313319311yescharacteristic
581C2×C4#1313319311yescharacteristic
582C2×C4#1313319311yescharacteristic
583C2×C4#1313320611yescharacteristic
584C2×C4#1313320611yescharacteristic
585C2×C4#1313320611yescharacteristic
586C2×C4#1313320611yescharacteristic
587C2×C4#1313320611yescharacteristic
588C2×C4#1313320611yescharacteristic
589C2×C4#1313320611yescharacteristic
590C2×C4#1313320611yescharacteristic
591C2×C4#1313320611yescharacteristic
592C2×C4#1313320611yescharacteristic
593C2×C4C22×C22⋊D5311yescharacteristic
594C2×C4C22×C22⋊D5311yescharacteristic
595C2×C4C2×Q8×D5311yescharacteristic
596C2×C4C2×#1171994911yescharacteristic
597C2×C4C2×#1171994911yescharacteristic
598C2×C4C2×#1171994911yescharacteristic
599C8—2122no—
600C8—2122no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—4242no—
609C22—4242no—
610C22—4242no—
611C22—2122no—
612C22#1681053411yescharacteristic
613C22#1681053411yescharacteristic
614C22#1681053411yescharacteristic
615C22#1681053411yescharacteristic
616C22C2×#1313318611yescharacteristic
617C22C2×#1313319311yescharacteristic
618C22C2×#1313320611yescharacteristic
619C4—44no—
620C4—44no—
621C4—44no—
622C4—4242no—
623C4—2122no—
624C4#1681053411yescharacteristic
625C4#1681053411yescharacteristic
626C4#1681053411yescharacteristic
627C4#1681053411yescharacteristic
628C4C2×#1313319311yescharacteristic
629C4C2×#1313319311yescharacteristic
630C4C2×#1313320611yescharacteristic
631C4C2×#1313320611yescharacteristic
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—4242no—
640C2C2×#1681053411yescharacteristic
subgroupG/H
centerC4C2×#13133206
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3458C2
socleC2×C53C2×#393
prime radicalC22×#11719945C22
hypercenter#3458D53
hypocenterC53#23472
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
4. #20275884C2semidirect
5. C2×#16810393C2semidirect
6. #20274425C2semidirect
7. C2×#16810379C2semidirect
8. C53×#3458C2semidirect
22. #16810393C22semidirect
23. #16810383C22semidirect
24. #16810379C22semidirect
31. C53×#533C22semidirect
45. C2×#11719945Q8semidirect
47. C22×Dic106Q8semidirect
48. C22×D212Q8semidirect
312. #3458D53semidirect
338. #533C2×D53semidirect
355. C53#23472semidirect

Maximal subgroups

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

subgroupindex
C2×#168106412
C2×#168105962
#202758842
C2×#168103932
#202744252
C2×#168103792
C53×#34582
#2347253