Finite Group Databasebeta

← #27385832

Subgroups — #27385832

Order 13568.

1818 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133224C211yesmaximal, characteristic, complemented
2C2×#16811201C211yesmaximal, characteristic
3C2×#16810622C211yesmaximal, characteristic, complemented
4C2×#16810617C211yesmaximal, characteristic, complemented
5#20275151C212yesmaximal, complemented
6#20275146C212yesmaximal, complemented
7#20275141C212yesmaximal, complemented
8#20275136C212yesmaximal, complemented
9#20274436C211yesmaximal, characteristic, complemented
10C2×#16810385C211yesmaximal, characteristic, complemented
11C53×#2311C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C2×#13133224—24no—
14C2×#13133224—24no—
15C2×#13133224—24no—
16C22×#11719917C2212yes—
17C22×#11719917C2211yescharacteristic
18C23×#88322C2212yes—
19C23×#88322C2211yescharacteristic
20#16811201—24no—
21#16811201C418yescomplemented
22C2×#13133196C2212yes—
23C2×#13133193—21no—
24C2×#13133191C2212yes—
25C2×#13133189—22no—
26C2×#13133188—21no—
27C22×C4⋊Dic53C2211yescharacteristic
28#16810622—24no—
29#16810617—24no—
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133140C2212yes—
33C2×#13133137C2211yescharacteristic
34C2×#13133137C2211yescharacteristic
35C2×#13133137C2212yes—
36#16810385C2212yes—
37#16810385C2212yes—
38#16810385C2212yes—
39#16810385C2212yes—
40C22×#41×C53C2211yescharacteristic
41C2×#117×C53C2211yescharacteristic
42C22×C4⋊C4×C53C2211yescharacteristic
43C53×#391—21no—
44C53×#390—21no—
45C53×#386—22no—
46C53×#380C2212yes—
47C53×#380C2212yes—
48C2×#11719945—22no—
49C2×#11719945—22no—
50C2×#11719945—22no—
51C2×#11719945—22no—
52C2×#11719945—22no—
53C2×#11719945C2×C414yes—
54C22×Dic106C2311yescharacteristic
55C22×D212C2311yescharacteristic
56C22×C22⋊D53—21no—
57C22×C22⋊D53—22no—
58C23×Dic53—21no—
59C22×C4×D53—21no—
60#13133224—232no—
61C2×#11719917—24no—
62C2×#11719917—24no—
63C2×#11719917—24no—
64C2×#11719917—22no—
65C2×#11719917—22no—
66C2×#11719917D414yescomplemented
67C22×#88322—24no—
68C22×#88322—24no—
69C22×#88322—24no—
70C22×#88322—22no—
71C22×#88322—22no—
72C22×#88322D414yescomplemented
73C22×#88322C2311yescharacteristic
74C22×#88322C2312yes—
75C22×#88322C2311yescharacteristic
76#13133196—24no—
77#13133196C2×C418yescomplemented
78#13133193—28no—
79#13133191—24no—
80#13133191C2×C418yescomplemented
81#13133189—216no—
82#13133188—28no—
83C2×C4⋊Dic53—21no—
84C2×C4⋊Dic53—22no—
85C2×C4⋊Dic53C2×C414yes—
86C2×C4⋊Dic53C2312yes—
87C2×#11719919—22no—
88C2×C22⋊Dic53—22no—
89C2×#11719921—22no—
90#13133140—22no—
91#13133140—22no—
92#13133140—24no—
93#13133137—22no—
94#13133137—22no—
95#13133137—24no—
96C2×#41×C53—22no—
97C2×#41×C53—22no—
98C2×#41×C53—22no—
99C2×#41×C53—22no—
100C2×#41×C53C2×C414yes—
101C2×#41×C53D414yes—
102C22×Q8×C53D411yescharacteristic
103C22×D4×C53D411yescharacteristic
104C22×D4×C53D412yes—
105C23×C4×C53—21no—
106C23×C4×C53C2312yes—
107C23×C4×C53C2311yescharacteristic
108#117×C53—22no—
109#117×C53—22no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—22no—
112C2×C4⋊C4×C53—22no—
113C2×C4⋊C4×C53C2312yes—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—22no—
118C2×C22⋊C4×C53—22no—
119C2×C42×C53—21no—
120#95×C53—22no—
121#95×C53—22no—
122C24×C53C2×D412yes—
123C24×C53C2×D411yescharacteristic
124C23×D53—21no—
125#41×C53—28no—
126#41×C53—28no—
127#41×C53—28no—
128#41×C53—28no—
129C2×Q8×C53—22no—
130C2×Q8×C53—22no—
131C2×Q8×C53C22⋊C414yes—
132C2×D4×C53—24no—
133C2×D4×C53—24no—
134C2×D4×C53—22no—
135C2×D4×C53—22no—
136C2×D4×C53C22⋊C414yes—
137C2×D4×C53C22⋊C418yes—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—22no—
145C22×C4×C53—42no—
146C22×C4×C53—81no—
147C22×C4×C53—81no—
148C22×C4×C53C22×C414yes—
149C22×C4×C53C22×C412yes—
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D414yes—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2×D412yes—
155C22×C4×C53C2×D412yes—
156C22×C4×C53#4112yes—
157C22×C4×C53#4112yes—
158C22×C4×C53C2411yescharacteristic
159C2×C22⋊D53—24no—
160C2×C22⋊D53—28no—
161C2×C22⋊D53—42no—
162C2×C22⋊D53—42no—
163C2×C22⋊D53—82no—
164C2×C22⋊D53—82no—
165C22×Dic53—22no—
166C22×Dic53—21no—
167C22×Dic53—24no—
168C22×Dic53—81no—
169C22×Dic53—81no—
170#11719945—28no—
171#11719945—28no—
172#11719945—44no—
173#11719945—44no—
174C2×D212—21no—
175C2×D212—21no—
176C2×D212—21no—
177C2×D212—21no—
178C2×D212—21no—
179C2×D212C22×C412yes—
180C2×C4×D53—24no—
181C2×C4×D53—42no—
182C2×C4×D53—42no—
183C2×Dic106—21no—
184C2×Dic106—21no—
185C2×Dic106—21no—
186C2×Dic106—21no—
187C2×Dic106—21no—
188C2×Dic106C22×C412yes—
189C4⋊C4×C53—22no—
190C4⋊C4×C53—21no—
191C4⋊C4×C53—21no—
192C4⋊C4×C53—22no—
193C4⋊C4×C53—21no—
194C4⋊C4×C53—21no—
195C4⋊C4×C53—42no—
196C22⋊C4×C53—42no—
197C22⋊C4×C53—82no—
198C22⋊C4×C53—82no—
199C42×C53—42no—
200C22⋊Dic53—28no—
201#11719921—28no—
202C4⋊Dic53—24no—
203C4⋊Dic53—22no—
204C4⋊Dic53C22×C414yes—
205#11719919—28no—
206#11719917—216no—
207#11719917—28no—
208C2×#88322—216no—
209C2×#88322—28no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—22no—
213C2×#88322—21no—
214C2×#88322—22no—
215C2×#88322—21no—
216C2×#88322C2×D412yes—
217C2×#88322C2×D414yes—
218C2×#88322C2×D412yes—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—22no—
224C23×C53—22no—
225C23×C53—22no—
226C23×C53—22no—
227C23×C53—22no—
228C23×C53—22no—
229C23×C53C2×C22⋊C414yes—
230C23×C53C2×C22⋊C412yes—
231C23×C53#12112yes—
232C23×C53#12112yes—
233C23×C53C22×D411yescharacteristic
234C22×D53—22no—
235C22×D53—41no—
236C22×D53—41no—
237C22×D53—41no—
238C22×D53—41no—
239C22×D53—41no—
240C22×D53—41no—
241Q8×C53—28no—
242D4×C53—216no—
243D4×C53—28no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—24no—
247C2×C4×C53—24no—
248C2×C4×C53—24no—
249C2×C4×C53—24no—
250C2×C4×C53—42no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—42no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—42no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—84no—
260C2×C4×C53C2×C22⋊C414yes—
261C2×C4×C53C2×C22⋊C412yes—
262C2×C4×C53C4×D414yes—
263C2×C4×C53C4×D414yes—
264C2×C4×C53#12014yes—
265C2×C4×C53#12112yes—
266C2×C4×C53#12112yes—
267C2×C4×C53#12314yes—
268C2×C4×C53C23×C411yescharacteristic
269C2×C4×C53C22×D411yescharacteristic
270C2×C4×C53C22×D411yescharacteristic
271C2×C4×C53C22×D411yescharacteristic
272C2×C4×C53C2×#4111yescharacteristic
273C2×C4×C53C2×#4111yescharacteristic
274C22⋊D53—48no—
275C22⋊D53—88no—
276C2×Dic53—22no—
277C2×Dic53—24no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—84no—
283D212—22no—
284D212—22no—
285D212—41no—
286D212—41no—
287C4×D53—48no—
288Dic106—22no—
289Dic106—22no—
290Dic106—41no—
291Dic106—41no—
292#88322—22no—
293#88322—24no—
294#88322—22no—
295#20976—531nomaximal, Sylow
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—22no—
299C22×C53—22no—
300C22×C53—22no—
301C22×C53—22no—
302C22×C53—24no—
303C22×C53—24no—
304C22×C53—24no—
305C22×C53—42no—
306C22×C53—42no—
307C22×C53—42no—
308C22×C53—42no—
309C22×C53#38614yes—
310C22×C53C22×C22⋊C411yescharacteristic
311C22×C53C2×#12111yescharacteristic
312C22×C53C2×#12111yescharacteristic
313C22×C53#57611yescharacteristic
314C22×C53#57711yescharacteristic
315C22×C53#57711yescharacteristic
316C22×C53#57811yescharacteristic
317C2×D53—42no—
318C2×D53—42no—
319C2×D53—42no—
320C2×D53—81no—
321C2×D53—81no—
322C2×D53—81no—
323C2×D53—81no—
324C4×C53—24no—
325C4×C53—81no—
326C4×C53—81no—
327C4×C53—82no—
328C4×C53#38614yes—
329C4×C53C2×C4×D411yescharacteristic
330C4×C53C2×C4×D411yescharacteristic
331C4×C53C2×#12011yescharacteristic
332C4×C53C2×#12311yescharacteristic
333Dic53—44no—
334Dic53—42no—
335C22×#131—531no—
336C2×#537—531no—
337C2×#417—531no—
338C2×#416—531no—
339#2311D5311yescharacteristic, complemented
340#1895—532no—
341#1894—532no—
342#1893—532no—
343#1892—532no—
344#1736—531no—
345C2×#340—531no—
346C2×C53—24no—
347C2×C53—44no—
348C2×C53C2×#38611yescharacteristic
349C2×C53#289711yescharacteristic
350C2×C53#289811yescharacteristic
351C2×C53#306011yescharacteristic
352C2×C53#306111yescharacteristic
353C2×C53#306211yescharacteristic
354C2×C53#306311yescharacteristic
355D53—82no—
356C22×#41—531no—
357C22×#41C2×D5311yescharacteristic
358C2×#131—1064no—
359C2×#131—1064no—
360C2×#131—1064no—
361C22×M16—531no—
362C22×M16—532no—
363C23×C8—531no—
364C23×C8—532no—
365#537—538no—
366#537—1064no—
367C2×C22⋊Q8—532no—
368C2×C22⋊Q8—1061no—
369C2×#121—532no—
370C2×#121—1061no—
371C2×#121—1062no—
372C2×#117C2×D5311yescharacteristic
373C22×C4⋊C4—531no—
374C22×C4⋊C4C2×D5311yescharacteristic
375#417—1064no—
376#416—1064no—
377C2×#103—531no—
378C2×#103—531no—
379C2×#103—532no—
380C2×#102—531no—
381C2×#102—531no—
382C2×#102—532no—
383#391—21no—
384#390—21no—
385#386—22no—
386#380C2×D5312yes—
387#380C2×D5312yes—
388#340—532no—
389#340—532no—
390#340—532no—
391#340—532no—
392C53#2097611yescharacteristic, complemented, Sylow
393C2×#41—22no—
394C2×#41—22no—
395C2×#41—22no—
396C2×#41—22no—
397C2×#41—534no—
398C2×#41—1062no—
399C2×#41—1062no—
400C2×#41—1062no—
401C2×#41—1062no—
402C2×#41—1062no—
403C2×#41C4×D5314yes—
404C2×#41D21214yes—
405C22×Q8—531no—
406C22×Q8C22⋊D5311yescharacteristic
407C22×D4—531no—
408C22×D4—1062no—
409C22×D4—1061no—
410C22×D4C22⋊D5312yes—
411C22×D4C22⋊D5311yescharacteristic
412C23×C4—21no—
413C23×C4—1061no—
414C23×C4—1061no—
415C23×C4C22×D5312yes—
416C23×C4C22×D5311yescharacteristic
417#131—10632no—
418C2×M16—534no—
419C2×M16—1064no—
420C2×M16—1064no—
421C2×M16—1064no—
422C2×M16—1062no—
423C2×M16—1062no—
424C22×C8—534no—
425C22×C8—531no—
426C22×C8—531no—
427C22×C8—532no—
428C22×C8—1064no—
429C22×C8—1064no—
430C22×C8—1064no—
431C22×C8—1062no—
432C22×C8—1062no—
433C22⋊Q8—538no—
434C22⋊Q8—1068no—
435C22⋊Q8—1064no—
436#121—538no—
437#121—1068no—
438#121—10616no—
439#121—1064no—
440#117—22no—
441#117—22no—
442C2×C4⋊C4—22no—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—534no—
446C2×C4⋊C4—532no—
447C2×C4⋊C4—1062no—
448C2×C4⋊C4—1061no—
449C2×C4⋊C4—1062no—
450C2×C4⋊C4C22×D5312yes—
451C2×C4⋊C4C22×D5311yescharacteristic
452C2×C4⋊C4C22×D5311yescharacteristic
453C2×C22⋊C4—21no—
454C2×C22⋊C4—22no—
455C2×C22⋊C4—22no—
456C2×C22⋊C4—1062no—
457C2×C22⋊C4—1062no—
458C2×C42—21no—
459#103—1062no—
460#103—1062no—
461#103—1064no—
462#102—1062no—
463#102—1062no—
464#102—1064no—
465#95—22no—
466#95—22no—
467C24—1061no—
468C24C2×C22⋊D5312yes—
469C24C2×C22⋊D5311yescharacteristic
470#41—28no—
471#41—28no—
472#41—28no—
473#41—28no—
474#41—1068no—
475#41—1068no—
476#41—2124no—
477#41—2124no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—532no—
481C2×Q8—1061no—
482C2×Q8—1061no—
483C2×Q8—1061no—
484C2×Q8—1061no—
485C2×Q8—1061no—
486C2×Q8#1171992114yes—
487C2×D4—24no—
488C2×D4—24no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—532no—
492C2×D4—1068no—
493C2×D4—1064no—
494C2×D4—1061no—
495C2×D4—1061no—
496C2×D4—1061no—
497C2×D4—1061no—
498C2×D4—1061no—
499C2×D4—4242no—
500C2×D4—4242no—
501C2×D4—2122no—
502C2×D4—2122no—
503C2×D4#1171992118yes—
504C2×D4#1171992114yes—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—42no—
513C22×C4—81no—
514C22×C4—81no—
515C22×C4—1064no—
516C22×C4—1062no—
517C22×C4—1061no—
518C22×C4—1064no—
519C22×C4—4241no—
520C22×C4—4241no—
521C22×C4—2122no—
522C22×C4—2122no—
523C22×C4C2×C4×D5314yes—
524C22×C4C2×C4×D5312yes—
525C22×C4C2×D21214yes—
526C22×C4C2×D21212yes—
527C22×C4D4×D5312yes—
528C22×C4D4×D5312yes—
529C22×C4#1171994712yes—
530C22×C4#1171994712yes—
531C22×C4C2×C22⋊D5312yes—
532C22×C4C2×C22⋊D5311yescharacteristic
533C22×C4C23×D5311yescharacteristic
534M16—10616no—
535M16—1068no—
536C2×C8—532no—
537C2×C8—532no—
538C2×C8—534no—
539C2×C8—10616no—
540C2×C8—1068no—
541C2×C8—1061no—
542C2×C8—1062no—
543C2×C8—1061no—
544C2×C8—1061no—
545C2×C8—1061no—
546C2×C8—1062no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—22no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—42no—
554C4⋊C4—534no—
555C4⋊C4—1064no—
556C4⋊C4—1068no—
557C4⋊C4—1062no—
558C22⋊C4—42no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—1068no—
562C22⋊C4—1068no—
563C42—42no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—1062no—
575C23—2121no—
576C23—2121no—
577C23—2121no—
578C23—2121no—
579C23—2121no—
580C23—2121no—
581C23C2×#1171992114yes—
582C23C2×#1171992112yes—
583C23#1313318912yes—
584C23#1313318912yes—
585C23C22×C22⋊D5311yescharacteristic
586Q8—28no—
587Q8—1062no—
588Q8—1062no—
589Q8—2121no—
590Q8—2121no—
591D4—216no—
592D4—28no—
593D4—1062no—
594D4—1062no—
595D4—4248no—
596D4—2128no—
597D4—2121no—
598D4—2121no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—84no—
615C2×C4—1064no—
616C2×C4—1062no—
617C2×C4—4244no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2122no—
621C2×C4—2122no—
622C2×C4—2128no—
623C2×C4C2×#1171992114yes—
624C2×C4C2×#1171992112yes—
625C2×C4#1313317714yes—
626C2×C4#1313317714yes—
627C2×C4#1313318714yes—
628C2×C4#1313318912yes—
629C2×C4#1313318912yes—
630C2×C4#1313320114yes—
631C2×C4C22×C4×D5311yescharacteristic
632C2×C4C2×D4×D5311yescharacteristic
633C2×C4C2×D4×D5311yescharacteristic
634C2×C4C22×D21211yescharacteristic
635C2×C4C2×#1171994711yescharacteristic
636C2×C4C2×#1171994711yescharacteristic
637C8—1062no—
638C8—1062no—
639C8—1064no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—4241no—
654C22—4241no—
655C22—4241no—
656C22—4241no—
657C22—2122no—
658C22—2122no—
659C22—2122no—
660C22#1681050014yes—
661C22C22×#1171992111yescharacteristic
662C22C2×#1313318911yescharacteristic
663C22C2×#1313318911yescharacteristic
664C22#1681142611yescharacteristic
665C22#1681143111yescharacteristic
666C22#1681143411yescharacteristic
667C22#1681143811yescharacteristic
668C4—24no—
669C4—81no—
670C4—81no—
671C4—82no—
672C4—2124no—
673C4—2122no—
674C4#1681050014yes—
675C4C2×#1313317711yescharacteristic
676C4C2×#1313317711yescharacteristic
677C4C2×#1313318711yescharacteristic
678C4C2×#1313320111yescharacteristic
679C2—24no—
680C2—44no—
681C2—4242no—
682C2C2×#1681050011yescharacteristic
683C2#2028323811yescharacteristic
684C2#2028324511yescharacteristic
685C2#2028437411yescharacteristic
686C2#2028438211yescharacteristic
687C2#2028439111yescharacteristic
688C2#2028440211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2311C2
socleC23×C53C22×D4
prime radicalC2×#16811201C2
cosocleC22×C4×C53C24
hypercenter#2311D53
hypocenterC53#20976
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
1. C22×#13133224C2semidirect
3. C2×#16810622C2semidirect
4. C2×#16810617C2semidirect
5. #20275151C2semidirect
6. #20275146C2semidirect
7. #20275141C2semidirect
8. #20275136C2semidirect
9. #20274436C2semidirect
10. C2×#16810385C2semidirect
11. C53×#2311C2semidirect
21. #16811201C4semidirect
66. C2×#11719917D4semidirect
72. C22×#88322D4semidirect
77. #13133196C2×C4semidirect
80. #13133191C2×C4semidirect
339. #2311D53semidirect
392. C53#20976semidirect

Maximal subgroups

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

subgroupindex
C22×#131332242
C2×#168112012
C2×#168106222
C2×#168106172
#202751512
#202751462
#202751412
#202751362
#202744362
C2×#168103852
C53×#23112
#2097653