Finite Group Databasebeta

← #27411627

Subgroups — #27411627

Order 13568.

1482 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810663C211yesmaximal, characteristic, complemented
2C2×#16810624C211yesmaximal, characteristic, complemented
3C2×#16810619C211yesmaximal, characteristic, complemented
4C2×#16810590C211yesmaximal, characteristic, complemented
5#20277307C211yesmaximal, characteristic, complemented
6#20275915C212yesmaximal, complemented
7#20275908C212yesmaximal, complemented
8#20275901C212yesmaximal, complemented
9#20275894C212yesmaximal, complemented
10C2×#16810385C211yesmaximal, characteristic, complemented
11C2×C53×#537C211yesmaximal, characteristic
12C22×#11719917C2211yescharacteristic
13C23×#88322C2211yescharacteristic
14C2×#13133176C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16#16810663—24no—
17C2×#13133150C2212yes—
18C2×#13133148C2212yes—
19#16810624—24no—
20#16810619—24no—
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2212yes—
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2212yes—
27#16810590—24no—
28C2×C22⋊#88322C2212yes—
29C2×C22⋊#88322C2212yes—
30#16810530—21no—
31#16810526—21no—
32#16810503—22no—
33#16810483C2212yes—
34#16810483C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37C22×#41×C53C2211yescharacteristic
38C22×#41×C53C2211yescharacteristic
39C53×#537—24no—
40C53×#537—24no—
41C2×C22⋊Q8×C53—21no—
42C2×C22⋊Q8×C53C2212yes—
43C2×#121×C53—21no—
44C2×#121×C53—22no—
45C2×#121×C53C2212yes—
46C22×C4⋊C4×C53C2211yescharacteristic
47C23×Dic53—21no—
48C2×#11719917—22no—
49C2×#11719917—22no—
50C2×#11719917D414yescomplemented
51C22×#88322—22no—
52C22×#88322—22no—
53C22×#88322—22no—
54C22×#88322C2311yescharacteristic
55C22×#88322C2311yescharacteristic
56C22×#88322C2312yes—
57#13133176—22no—
58#13133176—22no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—22no—
61C2×C4⋊Dic53—22no—
62C2×C4⋊Dic53C2312yes—
63C2×C4⋊Dic53C2311yescharacteristic
64C2×C4⋊Dic53C2311yescharacteristic
65C2×C22⋊Dic53—21no—
66C2×C22⋊Dic53—22no—
67C2×C22⋊Dic53—22no—
68C2×C4×Dic53—21no—
69#13133150—24no—
70#13133148—24no—
71#13133141—24no—
72#13133141—22no—
73#13133141—22no—
74#13133138—24no—
75#13133138—22no—
76#13133138—22no—
77C22⋊#88322—24no—
78C22⋊#88322—24no—
79#13133124—22no—
80#13133124—22no—
81C2×#41×C53—22no—
82C2×#41×C53—22no—
83C2×#41×C53—22no—
84C2×#41×C53—22no—
85C2×#41×C53—22no—
86C2×#41×C53—22no—
87C2×#41×C53—22no—
88C2×#41×C53—22no—
89C2×#41×C53—22no—
90C2×#41×C53—22no—
91C2×#41×C53—22no—
92C2×#41×C53D414yes—
93C22×Q8×C53D411yescharacteristic
94C22×Q8×C53C2311yescharacteristic
95C22×D4×C53—21no—
96C22×D4×C53—22no—
97C22×D4×C53D412yes—
98C22×D4×C53D411yescharacteristic
99C22×D4×C53C2311yescharacteristic
100C23×C4×C53—21no—
101C23×C4×C53—21no—
102C23×C4×C53C2312yes—
103C23×C4×C53C2311yescharacteristic
104C22⋊Q8×C53—28no—
105C22⋊Q8×C53—24no—
106C22⋊Q8×C53—24no—
107#121×C53—216no—
108#121×C53—28no—
109#121×C53—24no—
110#121×C53—24no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—22no—
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53D414yes—
115C2×C4⋊C4×C53C2312yes—
116C2×C22⋊C4×C53—22no—
117C2×C22⋊C4×C53—22no—
118C24×C53—21no—
119C24×C53C2×D412yes—
120C24×C53C2×D411yescharacteristic
121#41×C53—28no—
122#41×C53—28no—
123#41×C53—44no—
124#41×C53—44no—
125#41×C53—44no—
126#41×C53—44no—
127#41×C53—44no—
128#41×C53—44no—
129C2×Q8×C53—22no—
130C2×Q8×C53—22no—
131C2×Q8×C53—22no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53C2×D412yes—
138C2×D4×C53—24no—
139C2×D4×C53—24no—
140C2×D4×C53—24no—
141C2×D4×C53—24no—
142C2×D4×C53—28no—
143C2×D4×C53—22no—
144C2×D4×C53—22no—
145C2×D4×C53—22no—
146C2×D4×C53—21no—
147C2×D4×C53—21no—
148C2×D4×C53—21no—
149C2×D4×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—42no—
152C2×D4×C53—42no—
153C2×D4×C53—82no—
154C2×D4×C53—82no—
155C2×D4×C53C2×D412yes—
156C22×C4×C53—22no—
157C22×C4×C53—21no—
158C22×C4×C53—24no—
159C22×C4×C53—24no—
160C22×C4×C53—22no—
161C22×C4×C53—22no—
162C22×C4×C53—22no—
163C22×C4×C53—22no—
164C22×C4×C53—22no—
165C22×C4×C53—22no—
166C22×C4×C53—42no—
167C22×C4×C53—42no—
168C22×C4×C53—81no—
169C22×C4×C53—81no—
170C22×C4×C53C2×D412yes—
171C22×C4×C53C2×D411yescharacteristic
172C22×C4×C53C2×D412yes—
173C22×C4×C53C2×D412yes—
174C22×C4×C53C2×D412yes—
175C22×C4×C53#4112yes—
176C22×C4×C53#4112yes—
177C22×C4×C53#4112yes—
178C22×C4×C53C2411yescharacteristic
179C22×Dic53—21no—
180C22×Dic53—21no—
181C22×Dic53—22no—
182C22×Dic53—42no—
183C22×Dic53—81no—
184C22×Dic53—81no—
185C4⋊C4×C53—28no—
186C4⋊C4×C53—24no—
187C4⋊C4×C53—22no—
188C4⋊C4×C53C2×D414yes—
189C22⋊C4×C53—28no—
190C22⋊C4×C53—28no—
191C22⋊Dic53—42no—
192C22⋊Dic53—82no—
193C22⋊Dic53—82no—
194C4⋊Dic53—22no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—22no—
200C4⋊Dic53—42no—
201C4×Dic53—42no—
202#11719917—28no—
203C2×#88322—22no—
204C2×#88322—22no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—44no—
212C2×#88322C2×D414yes—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—22no—
218C23×C53—22no—
219C23×C53—22no—
220C23×C53—22no—
221C23×C53—22no—
222C23×C53—22no—
223C23×C53—22no—
224C23×C53—22no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53—41no—
231C23×C53#12012yes—
232C23×C53#12112yes—
233C23×C53#12312yes—
234C23×C53C22×D411yescharacteristic
235Q8×C53—22no—
236Q8×C53—22no—
237Q8×C53—41no—
238Q8×C53—41no—
239Q8×C53—44no—
240D4×C53—22no—
241D4×C53—22no—
242D4×C53—48no—
243D4×C53—48no—
244D4×C53—41no—
245D4×C53—41no—
246D4×C53—44no—
247D4×C53—88no—
248C2×C4×C53—22no—
249C2×C4×C53—24no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—42no—
256C2×C4×C53—42no—
257C2×C4×C53—42no—
258C2×C4×C53—42no—
259C2×C4×C53—42no—
260C2×C4×C53—42no—
261C2×C4×C53—48no—
262C2×C4×C53—84no—
263C2×C4×C53#12012yes—
264C2×C4×C53#12114yes—
265C2×C4×C53#12114yes—
266C2×C4×C53#12112yes—
267C2×C4×C53#12314yes—
268C2×C4×C53#12312yes—
269C2×C4×C53#12414yes—
270C2×C4×C53C22×D411yescharacteristic
271C2×C4×C53C22×D411yescharacteristic
272C2×C4×C53C22×D411yescharacteristic
273C2×C4×C53C2×#4111yescharacteristic
274C2×C4×C53C2×#4111yescharacteristic
275C2×C4×C53C2×#4111yescharacteristic
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—42no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—84no—
286#88322—24no—
287#88322—41no—
288#88322—41no—
289#23485—531nomaximal, Sylow
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—42no—
297C22×C53—42no—
298C22×C53—42no—
299C22×C53—42no—
300C22×C53—42no—
301C22×C53—42no—
302C22×C53—42no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53—42no—
306C22×C53—81no—
307C22×C53—81no—
308C22×C53—81no—
309C22×C53—81no—
310C22×C53#39414yes—
311C22×C53C2×#12011yescharacteristic
312C22×C53C2×#12111yescharacteristic
313C22×C53C2×#12311yescharacteristic
314C22×C53#57611yescharacteristic
315C22×C53#57711yescharacteristic
316C22×C53#57711yescharacteristic
317C22×C53#57811yescharacteristic
318C4×C53—42no—
319C4×C53—42no—
320C4×C53—44no—
321C4×C53#39414yes—
322C4×C53C2×#12111yescharacteristic
323C4×C53C2×#12111yescharacteristic
324C4×C53C2×#12311yescharacteristic
325C4×C53C2×#12411yescharacteristic
326Dic53—81no—
327Dic53—81no—
328Dic53—82no—
329C2×#537D5311yescharacteristic
330C2×#428—531no—
331C2×#417—531no—
332C2×#416—531no—
333C2×#408—531no—
334#2311—531no—
335#2036—532no—
336#2035—532no—
337#2034—532no—
338#2033—532no—
339C2×#340—531no—
340C2×C53—24no—
341C2×C53—82no—
342C2×C53—82no—
343C2×C53C2×#39411yescharacteristic
344C2×C53#301211yescharacteristic
345C2×C53#301311yescharacteristic
346C2×C53#306211yescharacteristic
347C2×C53#306311yescharacteristic
348C2×C53#309811yescharacteristic
349C2×C53#309911yescharacteristic
350C22×#41C2×D5311yescharacteristic
351C22×#41C2×D5311yescharacteristic
352C22×M16—531no—
353C23×C8—531no—
354#537—24no—
355#537—24no—
356C2×C22⋊Q8—21no—
357C2×C22⋊Q8C2×D5312yes—
358C2×#121—21no—
359C2×#121—22no—
360C2×#121C2×D5312yes—
361C2×#117—531no—
362C22×C4⋊C4—531no—
363C22×C4⋊C4C2×D5311yescharacteristic
364#428—1064no—
365C2×#107—532no—
366C2×#106—532no—
367#417—1064no—
368#416—1064no—
369C2×#103—531no—
370C2×#103—531no—
371C2×#103—532no—
372C2×#102—531no—
373C2×#102—531no—
374C2×#102—532no—
375#408—1064no—
376C2×C22⋊C8—532no—
377C2×C22⋊C8—532no—
378#391—1061no—
379#390—1061no—
380#386—1062no—
381#380—532no—
382#380—532no—
383#340—532no—
384#340—532no—
385C53#2348511yescharacteristic, complemented, Sylow
386C2×#41—22no—
387C2×#41—22no—
388C2×#41—22no—
389C2×#41—22no—
390C2×#41—22no—
391C2×#41—22no—
392C2×#41—22no—
393C2×#41—22no—
394C2×#41—22no—
395C2×#41—22no—
396C2×#41—22no—
397C2×#41C22⋊D5314yes—
398C22×Q8C22⋊D5311yescharacteristic
399C22×Q8C22×D5311yescharacteristic
400C22×D4—21no—
401C22×D4—22no—
402C22×D4C22⋊D5312yes—
403C22×D4C22⋊D5311yescharacteristic
404C22×D4C22×D5311yescharacteristic
405C23×C4—21no—
406C23×C4—21no—
407C23×C4—1061no—
408C23×C4C22×D5312yes—
409C23×C4C22×D5311yescharacteristic
410C2×M16—534no—
411C2×M16—1062no—
412C2×M16—1062no—
413C22×C8—532no—
414C22×C8—531no—
415C22×C8—531no—
416C22×C8—1062no—
417C22×C8—1062no—
418C22×C8—1062no—
419C22⋊Q8—28no—
420C22⋊Q8—24no—
421C22⋊Q8—24no—
422#121—216no—
423#121—28no—
424#121—24no—
425#121—24no—
426#117—1062no—
427#117—1062no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4—22no—
431C2×C4⋊C4—532no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—531no—
434C2×C4⋊C4—1062no—
435C2×C4⋊C4—1062no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4C22⋊D5314yes—
438C2×C4⋊C4C22×D5312yes—
439C2×C22⋊C4—22no—
440C2×C22⋊C4—22no—
441C2×C22⋊C4—1062no—
442C2×C22⋊C4—1062no—
443C2×C22⋊C4—1061no—
444C2×C42—1061no—
445#107—1064no—
446#106—1064no—
447#103—1064no—
448#103—1062no—
449#103—1062no—
450#102—1064no—
451#102—1062no—
452#102—1062no—
453C22⋊C8—1064no—
454C22⋊C8—1064no—
455#95—1062no—
456#95—1062no—
457C24—21no—
458C24C2×C22⋊D5312yes—
459C24C2×C22⋊D5311yescharacteristic
460#41—28no—
461#41—28no—
462#41—44no—
463#41—44no—
464#41—44no—
465#41—44no—
466#41—44no—
467#41—44no—
468C2×Q8—22no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8C2×C22⋊D5312yes—
477C2×D4—24no—
478C2×D4—24no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—28no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—82no—
493C2×D4—82no—
494C2×D4C2×C22⋊D5312yes—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—81no—
508C22×C4—81no—
509C22×C4—1061no—
510C22×C4—1061no—
511C22×C4—1062no—
512C22×C4—4241no—
513C22×C4—4241no—
514C22×C4—2122no—
515C22×C4D4×D5312yes—
516C22×C4#1171994712yes—
517C22×C4#1171994712yes—
518C22×C4#1171994712yes—
519C22×C4C2×C22⋊D5312yes—
520C22×C4C2×C22⋊D5311yescharacteristic
521C22×C4C2×C22⋊D5312yes—
522C22×C4C2×C22⋊D5312yes—
523C22×C4C23×D5311yescharacteristic
524M16—1068no—
525C2×C8—534no—
526C2×C8—1062no—
527C2×C8—1062no—
528C2×C8—1061no—
529C2×C8—1061no—
530C2×C8—1061no—
531C2×C8—1061no—
532C2×C8—1061no—
533C2×C8—1061no—
534C2×C8—2124no—
535C4⋊C4—28no—
536C4⋊C4—24no—
537C4⋊C4—22no—
538C4⋊C4—1062no—
539C4⋊C4—1061no—
540C4⋊C4—1061no—
541C4⋊C4—1061no—
542C4⋊C4—1061no—
543C4⋊C4—1062no—
544C4⋊C4—2122no—
545C4⋊C4C2×C22⋊D5314yes—
546C22⋊C4—28no—
547C22⋊C4—28no—
548C22⋊C4—4242no—
549C22⋊C4—4242no—
550C22⋊C4—2122no—
551C42—2122no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23#1313318612yes—
571C23#1313318912yes—
572C23#1313319812yes—
573C23C22×C22⋊D5311yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—44no—
577Q8—41no—
578Q8—41no—
579D4—22no—
580D4—22no—
581D4—48no—
582D4—48no—
583D4—44no—
584D4—41no—
585D4—41no—
586D4—88no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—84no—
602C2×C4—4244no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2122no—
606C2×C4—2121no—
607C2×C4—2122no—
608C2×C4—2121no—
609C2×C4—2122no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4#1313318612yes—
613C2×C4#1313318814yes—
614C2×C4#1313318912yes—
615C2×C4#1313318914yes—
616C2×C4#1313319812yes—
617C2×C4#1313319814yes—
618C2×C4#1313320414yes—
619C2×C4C2×D4×D5311yescharacteristic
620C2×C4C22×C22⋊D5311yescharacteristic
621C2×C4C22×C22⋊D5311yescharacteristic
622C2×C4C2×#1171994711yescharacteristic
623C2×C4C2×#1171994711yescharacteristic
624C2×C4C2×#1171994711yescharacteristic
625C8—1064no—
626C8—2121no—
627C8—2121no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22#1681053714yes—
649C22C2×#1313318611yescharacteristic
650C22C2×#1313318911yescharacteristic
651C22C2×#1313319811yescharacteristic
652C22#1681142611yescharacteristic
653C22#1681143111yescharacteristic
654C22#1681143411yescharacteristic
655C22#1681143811yescharacteristic
656C4—44no—
657C4—42no—
658C4—42no—
659C4—4241no—
660C4—4241no—
661C4—4242no—
662C4#1681053714yes—
663C4C2×#1313318811yescharacteristic
664C4C2×#1313318911yescharacteristic
665C4C2×#1313319811yescharacteristic
666C4C2×#1313320411yescharacteristic
667C2—24no—
668C2—82no—
669C2—82no—
670C2C2×#1681053711yescharacteristic
671C2#2028401911yescharacteristic
672C2#2028403011yescharacteristic
673C2#2028439711yescharacteristic
674C2#2028440811yescharacteristic
675C2#2028468911yescharacteristic
676C2#2028470011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#537C2
socleC23×C53C22×D4
hypercenterC2×#537D53
hypocenterC53#23485
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. C2×#16810663C2semidirect
2. C2×#16810624C2semidirect
3. C2×#16810619C2semidirect
4. C2×#16810590C2semidirect
5. #20277307C2semidirect
6. #20275915C2semidirect
7. #20275908C2semidirect
8. #20275901C2semidirect
9. #20275894C2semidirect
10. C2×#16810385C2semidirect
50. C2×#11719917D4semidirect
385. C53#23485semidirect

Maximal subgroups

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

subgroupindex
C2×#168106632
C2×#168106242
C2×#168106192
C2×#168105902
#202773072
#202759152
#202759082
#202759012
#202758942
C2×#168103852
C2×C53×#5372
#2348553