Finite Group Databasebeta

← #27411631

Subgroups — #27411631

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×#16811201C211yesmaximal, characteristic
2C2×#16810666C211yesmaximal, characteristic, complemented
3C2×#16810622C211yesmaximal, characteristic, complemented
4C2×#16810617C211yesmaximal, characteristic, complemented
5C2×#16810590C211yesmaximal, characteristic, complemented
6#20275918C212yesmaximal, complemented
7#20275911C212yesmaximal, complemented
8#20275904C212yesmaximal, complemented
9#20275897C212yesmaximal, complemented
10C2×#16810385C211yesmaximal, characteristic, complemented
11C53×#2311C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14C23×#88322C2211yescharacteristic
15#16811201—24no—
16#16811201—24no—
17C2×#13133196C2212yes—
18C2×#13133193—21no—
19C2×#13133191C2212yes—
20C2×#13133189—22no—
21C2×#13133188—21no—
22C22×C4⋊Dic53C2211yescharacteristic
23#16810666—24no—
24C2×#13133150C2212yes—
25C2×#13133148C2212yes—
26#16810622—24no—
27#16810617—24no—
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133140C2212yes—
31C2×#13133137C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×#13133137C2212yes—
34#16810590—24no—
35C2×C22⋊#88322C2212yes—
36C2×C22⋊#88322C2212yes—
37#16810385C2212yes—
38#16810385C2212yes—
39C22×#41×C53C2211yescharacteristic
40C2×#117×C53C2211yescharacteristic
41C22×C4⋊C4×C53C2211yescharacteristic
42C53×#391—21no—
43C53×#390—21no—
44C53×#386—22no—
45C53×#380C2212yes—
46C53×#380C2212yes—
47C2×#11719945—22no—
48C2×#11719945—22no—
49C2×#11719945—22no—
50C2×#11719945—22no—
51C2×#11719945—22no—
52C2×#11719945D414yes—
53C22×Dic106C2311yescharacteristic
54C22×D212C2311yescharacteristic
55C22×C22⋊D53—21no—
56C22×C22⋊D53—22no—
57C23×Dic53—21no—
58C22×C4×D53—21no—
59C2×#11719917—22no—
60C2×#11719917—22no—
61C2×#11719917D414yescomplemented
62C22×#88322—22no—
63C22×#88322—22no—
64C22×#88322—22no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2312yes—
68#13133196—24no—
69#13133196—24no—
70#13133193—28no—
71#13133191—24no—
72#13133191—24no—
73#13133189—216no—
74#13133188—28no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53D414yes—
78C2×C4⋊Dic53C2312yes—
79C2×#11719919—22no—
80C2×C22⋊Dic53—22no—
81C2×#11719921—22no—
82#13133150—24no—
83#13133148—24no—
84#13133140—24no—
85#13133140—22no—
86#13133140—22no—
87#13133137—24no—
88#13133137—22no—
89#13133137—22no—
90C22⋊#88322—24no—
91C22⋊#88322—24no—
92C2×#41×C53—22no—
93C2×#41×C53—22no—
94C2×#41×C53—22no—
95C2×#41×C53—22no—
96C2×#41×C53—22no—
97C2×#41×C53—22no—
98C22×Q8×C53D411yescharacteristic
99C22×D4×C53D411yescharacteristic
100C22×D4×C53D412yes—
101C23×C4×C53—21no—
102C23×C4×C53C2312yes—
103C23×C4×C53C2311yescharacteristic
104#117×C53—22no—
105#117×C53—22no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—22no—
108C2×C4⋊C4×C53—22no—
109C2×C4⋊C4×C53C2312yes—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—22no—
114C2×C22⋊C4×C53—22no—
115C2×C42×C53—21no—
116#95×C53—22no—
117#95×C53—22no—
118C24×C53C2×D412yes—
119C24×C53C2×D411yescharacteristic
120C23×D53—21no—
121#41×C53—44no—
122#41×C53—44no—
123#41×C53—44no—
124#41×C53—44no—
125C2×Q8×C53—22no—
126C2×Q8×C53—22no—
127C2×Q8×C53—22no—
128C2×D4×C53—24no—
129C2×D4×C53—24no—
130C2×D4×C53—24no—
131C2×D4×C53—22no—
132C2×D4×C53—22no—
133C2×D4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—22no—
143C22×C4×C53—42no—
144C22×C4×C53—81no—
145C22×C4×C53—81no—
146C22×C4×C53C2×D412yes—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D412yes—
149C22×C4×C53C2×D412yes—
150C22×C4×C53C2×D412yes—
151C22×C4×C53#4112yes—
152C22×C4×C53#4112yes—
153C22×C4×C53#4112yes—
154C22×C4×C53C2411yescharacteristic
155C2×C22⋊D53—24no—
156C2×C22⋊D53—28no—
157C2×C22⋊D53—42no—
158C2×C22⋊D53—42no—
159C2×C22⋊D53—82no—
160C2×C22⋊D53—82no—
161C22×Dic53—22no—
162C22×Dic53—21no—
163C22×Dic53—24no—
164C22×Dic53—81no—
165C22×Dic53—81no—
166#11719945—28no—
167#11719945—28no—
168#11719945—44no—
169#11719945—44no—
170C2×D212—21no—
171C2×D212—21no—
172C2×D212—21no—
173C2×D212—21no—
174C2×D212—21no—
175C2×D212C2×D412yes—
176C2×C4×D53—24no—
177C2×C4×D53—42no—
178C2×C4×D53—42no—
179C2×Dic106—21no—
180C2×Dic106—21no—
181C2×Dic106—21no—
182C2×Dic106—21no—
183C2×Dic106—21no—
184C2×Dic106C2×D412yes—
185C4⋊C4×C53—22no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53—22no—
191C4⋊C4×C53—42no—
192C22⋊C4×C53—42no—
193C22⋊C4×C53—82no—
194C22⋊C4×C53—82no—
195C42×C53—42no—
196C22⋊Dic53—28no—
197#11719921—28no—
198C4⋊Dic53—24no—
199C4⋊Dic53—22no—
200C4⋊Dic53C2×D414yes—
201#11719919—28no—
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#12012yes—
225C23×C53#12112yes—
226C23×C53#12312yes—
227C23×C53C22×D411yescharacteristic
228C22×D53—22no—
229C22×D53—41no—
230C22×D53—41no—
231C22×D53—41no—
232C22×D53—41no—
233C22×D53—41no—
234C22×D53—41no—
235Q8×C53—44no—
236D4×C53—48no—
237D4×C53—44no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—42no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—84no—
255C2×C4×C53#12012yes—
256C2×C4×C53#12112yes—
257C2×C4×C53#12114yes—
258C2×C4×C53#12114yes—
259C2×C4×C53#12314yes—
260C2×C4×C53#12312yes—
261C2×C4×C53#12414yes—
262C2×C4×C53C22×D411yescharacteristic
263C2×C4×C53C22×D411yescharacteristic
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C22⋊D53—48no—
269C22⋊D53—88no—
270C2×Dic53—22no—
271C2×Dic53—24no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—84no—
277D212—22no—
278D212—22no—
279D212—41no—
280D212—41no—
281C4×D53—48no—
282Dic106—22no—
283Dic106—22no—
284Dic106—41no—
285Dic106—41no—
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#39414yes—
304C22×C53C2×#12011yescharacteristic
305C22×C53C2×#12111yescharacteristic
306C22×C53C2×#12311yescharacteristic
307C22×C53#57611yescharacteristic
308C22×C53#57711yescharacteristic
309C22×C53#57711yescharacteristic
310C22×C53#57811yescharacteristic
311C2×D53—42no—
312C2×D53—42no—
313C2×D53—42no—
314C2×D53—81no—
315C2×D53—81no—
316C2×D53—81no—
317C2×D53—81no—
318C4×C53—42no—
319C4×C53—81no—
320C4×C53—81no—
321C4×C53—82no—
322C4×C53#39414yes—
323C4×C53C2×#12111yescharacteristic
324C4×C53C2×#12111yescharacteristic
325C4×C53C2×#12311yescharacteristic
326C4×C53C2×#12411yescharacteristic
327Dic53—42no—
328Dic53—44no—
329C2×#537—531no—
330C2×#428—531no—
331C2×#417—531no—
332C2×#416—531no—
333C2×#408—531no—
334#2311D5311yescharacteristic, complemented
335#2036—532no—
336#2035—532no—
337#2034—532no—
338#2033—532no—
339C2×#340—531no—
340C2×C53—24no—
341C2×C53—82no—
342C2×C53C2×#39411yescharacteristic
343C2×C53#301211yescharacteristic
344C2×C53#301311yescharacteristic
345C2×C53#306211yescharacteristic
346C2×C53#306311yescharacteristic
347C2×C53#309811yescharacteristic
348C2×C53#309911yescharacteristic
349D53—82no—
350C22×#41—531no—
351C22×#41C2×D5311yescharacteristic
352C22×M16—531no—
353C23×C8—531no—
354#537—1064no—
355#537—1064no—
356C2×C22⋊Q8—532no—
357C2×C22⋊Q8—1061no—
358C2×#121—532no—
359C2×#121—1061no—
360C2×#121—1062no—
361C2×#117C2×D5311yescharacteristic
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—21no—
379#390—21no—
380#386—22no—
381#380C2×D5312yes—
382#380C2×D5312yes—
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—534no—
393C2×#41—1062no—
394C2×#41—1062no—
395C2×#41—1062no—
396C2×#41—1062no—
397C2×#41—1062no—
398C22×Q8—531no—
399C22×Q8C22⋊D5311yescharacteristic
400C22×D4—531no—
401C22×D4—1062no—
402C22×D4—1061no—
403C22×D4C22⋊D5312yes—
404C22×D4C22⋊D5311yescharacteristic
405C23×C4—21no—
406C23×C4—1061no—
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—1064no—
420C22⋊Q8—1064no—
421C22⋊Q8—1068no—
422#121—1064no—
423#121—1064no—
424#121—1068no—
425#121—10616no—
426#117—22no—
427#117—22no—
428C2×C4⋊C4—22no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—534no—
432C2×C4⋊C4—532no—
433C2×C4⋊C4—1062no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1062no—
436C2×C4⋊C4C22×D5312yes—
437C2×C4⋊C4C22×D5311yescharacteristic
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—22no—
441C2×C22⋊C4—22no—
442C2×C22⋊C4—1062no—
443C2×C22⋊C4—1062no—
444C2×C42—21no—
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—22no—
456#95—22no—
457C24—1061no—
458C24C2×C22⋊D5312yes—
459C24C2×C22⋊D5311yescharacteristic
460#41—44no—
461#41—44no—
462#41—44no—
463#41—44no—
464#41—1068no—
465#41—1068no—
466#41—2124no—
467#41—2124no—
468C2×Q8—22no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—532no—
472C2×Q8—1061no—
473C2×Q8—1061no—
474C2×Q8—1061no—
475C2×Q8—1061no—
476C2×Q8—1061no—
477C2×D4—24no—
478C2×D4—24no—
479C2×D4—24no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—532no—
484C2×D4—1068no—
485C2×D4—1064no—
486C2×D4—1061no—
487C2×D4—1061no—
488C2×D4—1061no—
489C2×D4—1061no—
490C2×D4—1061no—
491C2×D4—4242no—
492C2×D4—4242no—
493C2×D4—2122no—
494C2×D4—2122no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—42no—
505C22×C4—81no—
506C22×C4—81no—
507C22×C4—1064no—
508C22×C4—1062no—
509C22×C4—1061no—
510C22×C4—1064no—
511C22×C4—4241no—
512C22×C4—4241no—
513C22×C4—2122no—
514C22×C4—2122no—
515C22×C4#1171994512yes—
516C22×C4#1171994512yes—
517C22×C4D4×D5312yes—
518C22×C4D4×D5312yes—
519C22×C4D4×D5312yes—
520C22×C4#1171994712yes—
521C22×C4C2×C22⋊D5312yes—
522C22×C4C2×C22⋊D5311yescharacteristic
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—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—22no—
541C4⋊C4—42no—
542C4⋊C4—534no—
543C4⋊C4—1068no—
544C4⋊C4—1064no—
545C4⋊C4—1062no—
546C22⋊C4—42no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—1068no—
550C22⋊C4—1068no—
551C42—42no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—1062no—
564C23—2121no—
565C23—2121no—
566C23—2121no—
567C23—2121no—
568C23—2121no—
569C23—2121no—
570C23#1313318512yes—
571C23#1313318912yes—
572C23#1313320012yes—
573C23C22×C22⋊D5311yescharacteristic
574Q8—44no—
575Q8—1062no—
576Q8—1062no—
577Q8—2121no—
578Q8—2121no—
579D4—48no—
580D4—44no—
581D4—1062no—
582D4—1062no—
583D4—4248no—
584D4—2128no—
585D4—2121no—
586D4—2121no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—84no—
604C2×C4—1064no—
605C2×C4—1062no—
606C2×C4—4244no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2128no—
612C2×C4#1313318512yes—
613C2×C4#1313318912yes—
614C2×C4#1313319014yes—
615C2×C4#1313319014yes—
616C2×C4#1313319914yes—
617C2×C4#1313320012yes—
618C2×C4#1313320314yes—
619C2×C4C2×D4×D5311yescharacteristic
620C2×C4C2×D4×D5311yescharacteristic
621C2×C4C2×D4×D5311yescharacteristic
622C2×C4C2×#1171994711yescharacteristic
623C2×C4C2×#1171994511yescharacteristic
624C2×C4C2×#1171994511yescharacteristic
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—4241no—
642C22—4241no—
643C22—4241no—
644C22—4241no—
645C22—2122no—
646C22—2122no—
647C22—2122no—
648C22#1681053814yes—
649C22C2×#1313318511yescharacteristic
650C22C2×#1313318911yescharacteristic
651C22C2×#1313320011yescharacteristic
652C22#1681142611yescharacteristic
653C22#1681143111yescharacteristic
654C22#1681143411yescharacteristic
655C22#1681143811yescharacteristic
656C4—42no—
657C4—81no—
658C4—81no—
659C4—82no—
660C4—2124no—
661C4—2122no—
662C4#1681053814yes—
663C4C2×#1313319011yescharacteristic
664C4C2×#1313319011yescharacteristic
665C4C2×#1313319911yescharacteristic
666C4C2×#1313320311yescharacteristic
667C2—24no—
668C2—82no—
669C2—4242no—
670C2C2×#1681053811yescharacteristic
671C2#2028401611yescharacteristic
672C2#2028402711yescharacteristic
673C2#2028439111yescharacteristic
674C2#2028440211yescharacteristic
675C2#2028469311yescharacteristic
676C2#2028470411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2311C2
socleC23×C53C22×D4
prime radicalC2×#16811201C2
hypercenter#2311D53
hypocenterC53#23485
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#16810666C2semidirect
3. C2×#16810622C2semidirect
4. C2×#16810617C2semidirect
5. C2×#16810590C2semidirect
6. #20275918C2semidirect
7. #20275911C2semidirect
8. #20275904C2semidirect
9. #20275897C2semidirect
10. C2×#16810385C2semidirect
11. C53×#2311C2semidirect
61. C2×#11719917D4semidirect
334. #2311D53semidirect
385. C53#23485semidirect

Maximal subgroups

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

subgroupindex
C2×#168112012
C2×#168106662
C2×#168106222
C2×#168106172
C2×#168105902
#202759182
#202759112
#202759042
#202758972
C2×#168103852
C53×#23112
#2348553