Finite Group Databasebeta

← #27289488

Subgroups — #27289488

Order 13568.

1192 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287916C211yesmaximal, characteristic
2C2×#16810605C211yesmaximal, characteristic
3#20283159C211yesmaximal, characteristic
4C2×#16810395C211yesmaximal, characteristic, complemented
5C2×#16810395C211yesmaximal, characteristic, complemented
6C2×#16810393C211yesmaximal, characteristic, complemented
7C2×C53×#344C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11#16811104—22no—
12#16811103—22no—
13#16811100—21no—
14#16811097—21no—
15#16811096C414yes—
16C4×#11719945—22no—
17#16811091—22no—
18C2×#13133176C2211yescharacteristic
19#117×D53—22no—
20#16810605—22no—
21#16810605C414yescomplemented
22C2×#13133135—21no—
23C2×#13133133—21no—
24#16810591C414yescomplemented
25#16810589—22no—
26#16810586—22no—
27#16810395—22no—
28#16810395—22no—
29#16810393—22no—
30C22×M16×C53C2211yescharacteristic
31C22×M16×C53C2211yescharacteristic
32C2×#117×C53C2211yescharacteristic
33C53×#344—22no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945C2×C412yes—
43C2×#11719945C2×C412yes—
44C2×#11719945D412yes—
45C2×#11719945Q812yes—
46C22×Dic106D411yescharacteristic
47C22×D212D411yescharacteristic
48C22×C22⋊D53C2×C412yes—
49C22×C4×D53D412yes—
50C2×#11719917—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×#11719917C2×C412yescomplemented
62C22×#88322—21no—
63C22×#88322C2×C412yescomplemented
64C4×Dic106—24no—
65#13133182—24no—
66C4×D212—24no—
67#13133179—24no—
68C4×C22⋊D53—28no—
69#13133177—28no—
70#13133176—21no—
71#13133176—21no—
72#13133176D412yes—
73#13133176D412yes—
74#13133175—24no—
75#13133174—24no—
76#13133173—24no—
77C2×C4⋊Dic53—21no—
78C4⋊C4×D53—24no—
79C2×C22⋊Dic53C2×C412yes—
80C22⋊C4×D53—24no—
81C2×C4×Dic53—21no—
82C42×D53—24no—
83#13133135—24no—
84#13133133—24no—
85#13133129—28no—
86C23×C4×C53C2311yescharacteristic
87C2×M16×C53—21no—
88C2×M16×C53—21no—
89C2×M16×C53—21no—
90C2×M16×C53—21no—
91C2×M16×C53—21no—
92C2×M16×C53—21no—
93C2×M16×C53—21no—
94C2×M16×C53—21no—
95C2×M16×C53—21no—
96C2×M16×C53—21no—
97C2×M16×C53C2×C412yescomplemented
98C2×M16×C53C2×C412yescomplemented
99C22×C8×C53—21no—
100C22×C8×C53—21no—
101#117×C53—21no—
102#117×C53—21no—
103#117×C53D412yes—
104#117×C53D412yes—
105C2×C4⋊C4×C53—21no—
106C2×C22⋊C4×C53C2×C412yes—
107C2×C42×C53—21no—
108C24×C53C22×C411yescharacteristic
109C23×D53C22⋊C412yes—
110C22×C4×C53—22no—
111C22×C4×C53C22×C411yescharacteristic
112C22×C4×C53C22×C411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×Q811yescharacteristic
121C22×C4×C53#4111yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C22×C4×C53#4111yescharacteristic
125C2×C22⋊D53—22no—
126C2×C22⋊D53—22no—
127C2×C22⋊D53—22no—
128C2×C22⋊D53—22no—
129C2×C22⋊D53C4214yes—
130C2×C22⋊D53C4⋊C414yes—
131C22×Dic53—22no—
132C22×Dic53C22⋊C412yes—
133#11719945—22no—
134#11719945—22no—
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—22no—
144#11719945—22no—
145#11719945—22no—
146#11719945—22no—
147#11719945—22no—
148#11719945—22no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212C22⋊C412yes—
154C2×D212C4⋊C412yes—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—28no—
160C2×C4×D53C22⋊C414yes—
161C2×C4×D53C4⋊C414yes—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106C22⋊C412yes—
167C2×Dic106C4⋊C412yes—
168M16×C53—22no—
169M16×C53—22no—
170M16×C53—22no—
171M16×C53—22no—
172M16×C53—41no—
173M16×C53—41no—
174M16×C53—41no—
175M16×C53—41no—
176C2×C8×C53—22no—
177C2×C8×C53—22no—
178C2×C8×C53—41no—
179C2×C8×C53—41no—
180C2×C8×C53—41no—
181C2×C8×C53—41no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53—22no—
184C22⋊C4×C53—22no—
185C22⋊C4×C53C22⋊C414yes—
186C42×C53—22no—
187C42×C53—22no—
188C22⋊Dic53—22no—
189C22⋊Dic53C22⋊C414yes—
190#11719921—28no—
191C4⋊Dic53—22no—
192C4⋊Dic53—22no—
193#11719919—28no—
194C4×Dic53—28no—
195C4×Dic53—22no—
196C4×Dic53—22no—
197#11719917—22no—
198#11719917—22no—
199#11719917—41no—
200#11719917—41no—
201#11719917—41no—
202#11719917—41no—
203#11719917—41no—
204#11719917—41no—
205C2×#88322—22no—
206C2×#88322—22no—
207C2×#88322—22no—
208C2×#88322—22no—
209C2×#88322—41no—
210C2×#88322—41no—
211C23×C53—81no—
212C23×C53C2×C4211yescharacteristic
213C23×C53C2×C22⋊C411yescharacteristic
214C23×C53C2×C22⋊C411yescharacteristic
215C23×C53C2×C22⋊C411yescharacteristic
216C23×C53C2×C4⋊C411yescharacteristic
217C23×C53#11711yescharacteristic
218C23×C53#11711yescharacteristic
219C22×D53—22no—
220C22×D53—22no—
221C22×D53—82no—
222C22×D53#9514yes—
223C2×C4×C53—24no—
224C2×C4×C53—82no—
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53C2×C22⋊C411yescharacteristic
227C2×C4×C53C2×C4⋊C411yescharacteristic
228C2×C4×C53C2×C4⋊C411yescharacteristic
229C2×C4×C53C4×D411yescharacteristic
230C2×C4×C53C4×D411yescharacteristic
231C2×C4×C53C4×D411yescharacteristic
232C2×C4×C53C4×D411yescharacteristic
233C2×C4×C53C4×D411yescharacteristic
234C2×C4×C53C4×D411yescharacteristic
235C2×C4×C53C4×D411yescharacteristic
236C2×C4×C53C4×D411yescharacteristic
237C2×C4×C53#12011yescharacteristic
238C2×C4×C53#12011yescharacteristic
239C2×C4×C53#12011yescharacteristic
240C2×C4×C53#12011yescharacteristic
241C2×C4×C53#12111yescharacteristic
242C2×C4×C53#12111yescharacteristic
243C2×C4×C53#12111yescharacteristic
244C2×C4×C53#12111yescharacteristic
245C2×C4×C53C22⋊Q811yescharacteristic
246C2×C4×C53C22⋊Q811yescharacteristic
247C2×C4×C53C22⋊Q811yescharacteristic
248C2×C4×C53C22⋊Q811yescharacteristic
249C2×C4×C53#12311yescharacteristic
250C2×C4×C53#12311yescharacteristic
251C2×C4×C53#12311yescharacteristic
252C2×C4×C53#12311yescharacteristic
253C22⋊D53—24no—
254C22⋊D53—24no—
255C22⋊D53—24no—
256C22⋊D53—24no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—24no—
260C2×Dic53—82no—
261C2×Dic53#9514yes—
262D212—22no—
263D212—22no—
264D212—22no—
265D212—22no—
266C4×D53—24no—
267C4×D53—24no—
268C4×D53—24no—
269C4×D53—24no—
270C4×D53—88no—
271Dic106—22no—
272Dic106—22no—
273Dic106—22no—
274Dic106—22no—
275C8×C53—42no—
276C8×C53—42no—
277#88322—42no—
278#88322—42no—
279#10756—531nomaximal, Sylow
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53C2×#9511yescharacteristic
288C22×C53C4×C22⋊C411yescharacteristic
289C22×C53C4×C22⋊C411yescharacteristic
290C22×C53#37911yescharacteristic
291C22×C53#38011yescharacteristic
292C22×C53#38011yescharacteristic
293C22×C53#38111yescharacteristic
294C2×D53—24no—
295C2×D53—44no—
296C2×D53—44no—
297C2×D53—82no—
298C4×C53—44no—
299C4×C53#38511yescharacteristic
300C4×C53#38511yescharacteristic
301C4×C53#38511yescharacteristic
302C4×C53#38511yescharacteristic
303C4×C53#38611yescharacteristic
304C4×C53#38611yescharacteristic
305C4×C53#38611yescharacteristic
306C4×C53#38611yescharacteristic
307Dic53—24no—
308Dic53—44no—
309#3428—531no—
310C2×#413—531no—
311#2879—531no—
312C2×#344—531no—
313C2×#344—531no—
314C2×#344—531no—
315C2×#344D5311yescharacteristic, complemented
316C2×C53—21no—
317C2×C53—21no—
318C2×C53—21no—
319C2×C53—21no—
320C2×C53—21no—
321C2×C53—21no—
322C2×C53—21no—
323C2×C53C22⋊#9511yescharacteristic
324D53—44no—
325C22×#41—531no—
326C22×M16—531no—
327C22×M16—531no—
328C22×M16C2×D5311yescharacteristic
329C22×M16C2×D5311yescharacteristic
330#520—1062no—
331#520—1062no—
332#519—1061no—
333#518—534no—
334#518—1061no—
335C4×#41—1062no—
336C4×#41—1062no—
337C2×#117—531no—
338C2×#117—1062no—
339C2×#117C2×D5311yescharacteristic
340#413—534no—
341#413—1062no—
342C2×#101—1061no—
343C2×#100—1061no—
344#408—534no—
345#408—1062no—
346#407—1062no—
347#344—22no—
348#344—1062no—
349#344—1062no—
350#344—1062no—
351C53#1075611yescharacteristic, complemented, Sylow
352C2×#41—532no—
353C2×#41—532no—
354C2×#41—532no—
355C2×#41—532no—
356C2×#41—1061no—
357C2×#41—1061no—
358C2×#41—1061no—
359C2×#41—1061no—
360C2×#41—1061no—
361C2×#41—1061no—
362C2×#41—1061no—
363C2×#41—1061no—
364C22×Q8—531no—
365C22×D4—532no—
366C22×D4—531no—
367C23×C4—532no—
368C23×C4C22×D5311yescharacteristic
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—21no—
379C2×M16—532no—
380C2×M16—1061no—
381C2×M16—1061no—
382C2×M16—1061no—
383C2×M16—1061no—
384C2×M16—1061no—
385C2×M16—1061no—
386C2×M16—1061no—
387C2×M16—1061no—
388C2×M16—1061no—
389C2×M16—1061no—
390C2×M16—1061no—
391C2×M16C4×D5312yescomplemented
392C2×M16C4×D5312yescomplemented
393C22×C8—21no—
394C22×C8—21no—
395C22×C8—532no—
396C22×C8—1061no—
397C4×Q8—1064no—
398C4×Q8—1064no—
399C4×D4—1064no—
400C4×D4—1068no—
401C4×D4—1064no—
402C4×D4—1068no—
403#117—21no—
404#117—21no—
405#117—532no—
406#117—532no—
407#117—1064no—
408#117—1064no—
409#117—1064no—
410#117—1061no—
411#117—1061no—
412#117D21212yes—
413#117C22⋊D5312yes—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—1064no—
416C2×C4⋊C4—1061no—
417C2×C22⋊C4—532no—
418C2×C22⋊C4—1064no—
419C2×C22⋊C4C4×D5312yes—
420C2×C42—21no—
421C2×C42—1064no—
422C2×C42—1061no—
423#101—1064no—
424#100—1064no—
425C22⋊C8—1068no—
426C24—532no—
427C24C2×C4×D5311yescharacteristic
428#41—1062no—
429#41—1062no—
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—1062no—
439#41—1062no—
440#41—1062no—
441#41—1062no—
442#41—1062no—
443#41—1062no—
444C2×Q8—532no—
445C2×Q8—532no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×D4—534no—
451C2×D4—534no—
452C2×D4—532no—
453C2×D4—532no—
454C2×D4—1062no—
455C2×D4—1062no—
456C2×D4—1062no—
457C2×D4—1062no—
458C2×D4—1061no—
459C2×D4—1061no—
460C2×D4—1061no—
461C2×D4—1061no—
462C22×C4—22no—
463C22×C4—532no—
464C22×C4—534no—
465C22×C4—534no—
466C22×C4—1068no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4C2×C4×D5311yescharacteristic
473C22×C4C2×C4×D5311yescharacteristic
474C22×C4C2×D21211yescharacteristic
475C22×C4#1171994511yescharacteristic
476C22×C4#1171994511yescharacteristic
477C22×C4D4×D5311yescharacteristic
478C22×C4D4×D5311yescharacteristic
479C22×C4D4×D5311yescharacteristic
480C22×C4D4×D5311yescharacteristic
481C22×C4D4×D5311yescharacteristic
482C22×C4#1171994711yescharacteristic
483C22×C4Q8×D5311yescharacteristic
484C22×C4#1171994911yescharacteristic
485C22×C4C2×C22⋊D5311yescharacteristic
486M16—22no—
487M16—22no—
488M16—22no—
489M16—22no—
490M16—41no—
491M16—41no—
492M16—41no—
493M16—41no—
494M16—1062no—
495M16—1062no—
496M16—2121no—
497M16—2121no—
498M16—2121no—
499M16—2121no—
500M16—2121no—
501M16—2121no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—1062no—
509C2×C8—1062no—
510C2×C8—1062no—
511C2×C8—1062no—
512C2×C8—2121no—
513C2×C8—2121no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—1068no—
517C4⋊C4—1062no—
518C4⋊C4—1062no—
519C22⋊C4—22no—
520C22⋊C4—534no—
521C22⋊C4—1068no—
522C22⋊C4—1062no—
523C22⋊C4#1171992114yes—
524C42—22no—
525C42—22no—
526C42—1068no—
527C42—1062no—
528C42—1062no—
529C23—81no—
530C23—534no—
531C23—1062no—
532C23—1062no—
533C23—4242no—
534C23C42×D5311yescharacteristic
535C23C22⋊C4×D5311yescharacteristic
536C23C22⋊C4×D5311yescharacteristic
537C23C2×#1171992111yescharacteristic
538C23C4⋊C4×D5311yescharacteristic
539C23#1313317411yescharacteristic
540C23#1313317511yescharacteristic
541Q8—1062no—
542Q8—1062no—
543Q8—1062no—
544Q8—1062no—
545D4—1064no—
546D4—1064no—
547D4—1064no—
548D4—1064no—
549D4—1062no—
550D4—1062no—
551D4—1062no—
552D4—1062no—
553C2×C4—24no—
554C2×C4—82no—
555C2×C4—534no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1064no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—1064no—
562C2×C4—1064no—
563C2×C4—4248no—
564C2×C4—4242no—
565C2×C4C22⋊C4×D5311yescharacteristic
566C2×C4C22⋊C4×D5311yescharacteristic
567C2×C4C4⋊C4×D5311yescharacteristic
568C2×C4C4⋊C4×D5311yescharacteristic
569C2×C4#1313317711yescharacteristic
570C2×C4#1313317711yescharacteristic
571C2×C4C4×C22⋊D5311yescharacteristic
572C2×C4C4×C22⋊D5311yescharacteristic
573C2×C4#1313317911yescharacteristic
574C2×C4#1313317911yescharacteristic
575C2×C4C4×D21211yescharacteristic
576C2×C4C4×D21211yescharacteristic
577C2×C4#1313318511yescharacteristic
578C2×C4#1313318511yescharacteristic
579C2×C4#1313318711yescharacteristic
580C2×C4#1313318711yescharacteristic
581C2×C4#1313318811yescharacteristic
582C2×C4#1313319011yescharacteristic
583C2×C4#1313319011yescharacteristic
584C2×C4#1313319211yescharacteristic
585C2×C4#1313319311yescharacteristic
586C2×C4#1313319511yescharacteristic
587C2×C4#1313319511yescharacteristic
588C2×C4#1313319711yescharacteristic
589C2×C4#1313319911yescharacteristic
590C2×C4#1313319911yescharacteristic
591C2×C4#1313320211yescharacteristic
592C2×C4#1313320211yescharacteristic
593C8—42no—
594C8—42no—
595C8—2122no—
596C8—2122no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—1064no—
605C22—4242no—
606C22—2124no—
607C22—2124no—
608C22#95×D5311yescharacteristic
609C22#1681047011yescharacteristic
610C22C4×#1171992111yescharacteristic
611C22#1681047811yescharacteristic
612C22#1681048011yescharacteristic
613C22#1681048211yescharacteristic
614C22#1681048511yescharacteristic
615C4—44no—
616C4—1064no—
617C4—2124no—
618C4#1681049411yescharacteristic
619C4#1681049411yescharacteristic
620C4#1681049711yescharacteristic
621C4#1681049711yescharacteristic
622C4#1681049911yescharacteristic
623C4#1681049911yescharacteristic
624C4#1681050111yescharacteristic
625C4#1681050111yescharacteristic
626C2—21no—
627C2—21no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—2124no—
634C2#2027293611yescharacteristic
subgroupG/H
centerC4#16810501
commutatorC23×C53C2×C42
FrattiniC23×C4C22×D53
FittingC2×C53×#344C2
socleC2×C53C22⋊#95
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenterC2×#344D53
hypocenterC53#10756
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
4. C2×#16810395C2semidirect
5. C2×#16810395C2semidirect
6. C2×#16810393C2semidirect
7. C2×C53×#344C2semidirect
21. #16810605C4semidirect
24. #16810591C4semidirect
61. C2×#11719917C2×C4semidirect
63. C22×#88322C2×C4semidirect
97. C2×M16×C53C2×C4semidirect
98. C2×M16×C53C2×C4semidirect
315. C2×#344D53semidirect
351. C53#10756semidirect
391. C2×M16C4×D53semidirect
392. C2×M16C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202879162
C2×#168106052
#202831592
C2×#168103952
C2×#168103932
C2×C53×#3442
#1075653