Finite Group Databasebeta

← #27289631

Subgroups — #27289631

Order 13568.

1186 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×#16811389C211yesmaximal, characteristic
2C2×#16810639C211yesmaximal, characteristic
3C2×#16810622C211yesmaximal, characteristic
4C2×#16810395C211yesmaximal, characteristic, complemented
5#20274434C211yesmaximal, characteristic, complemented
6C2×#16810381C211yesmaximal, characteristic, complemented
7C2×C53×#343C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9#16811389—22no—
10#16811389—22no—
11#16811389—22no—
12#16811389C414yescomplemented
13C2×#13133225—22no—
14C2×#13133223—22no—
15C22×#11719917C2211yescharacteristic
16C22×#11719917C2211yescharacteristic
17C2×M16×D53—22no—
18C2×#13133176C2211yescharacteristic
19#16810639—22no—
20#16810639C414yescomplemented
21C2×#13133143—22no—
22#16810622—22no—
23#16810622—22no—
24C2×#13133140C412yescomplemented
25C2×#13133137C412yescomplemented
26#16810395—22no—
27#16810385—22no—
28#16810381—22no—
29C22×M16×C53C2211yescharacteristic
30C22×M16×C53C2211yescharacteristic
31C22×C4⋊C4×C53C2211yescharacteristic
32C53×#343—22no—
33C2×#11719945—21no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945C2×C412yes—
40C2×#11719945C2×C412yes—
41C2×#11719945D412yes—
42C2×#11719945D412yes—
43C2×#11719945D412yes—
44C2×#11719945Q812yes—
45C22×Dic106C2×C411yescharacteristic
46C22×D212C2×C411yescharacteristic
47C22×C22⋊D53—21no—
48C22×C4×D53—21no—
49#13133225—28no—
50#13133225—28no—
51#13133223—28no—
52#13133223—28no—
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×#11719917—21no—
62C2×#11719917—21no—
63C2×#11719917C2×C412yes—
64C2×#11719917D412yes—
65C2×#11719912—24no—
66M16×D53—28no—
67M16×D53—28no—
68C22×#88322C2×C412yes—
69C22×#88322D412yes—
70C2×C8×D53—24no—
71#13133176—21no—
72#13133176—21no—
73#13133176—21no—
74#13133176C2×C412yes—
75C2×C4⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C4×Dic53—21no—
78#13133143—28no—
79#13133140—24no—
80#13133137—24no—
81C23×C4×C53—21no—
82C23×C4×C53C2311yescharacteristic
83C2×M16×C53—21no—
84C2×M16×C53—21no—
85C2×M16×C53—21no—
86C2×M16×C53—21no—
87C2×M16×C53—21no—
88C2×M16×C53—21no—
89C2×M16×C53—21no—
90C2×M16×C53—21no—
91C2×M16×C53C2×C412yes—
92C2×M16×C53C2×C412yes—
93C2×M16×C53D412yes—
94C2×M16×C53D412yes—
95C22×C8×C53—21no—
96C22×C8×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53C2×C412yes—
103C24×C53#4111yescharacteristic
104C23×D53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—41no—
107C22×C4×C53—41no—
108C22×C4×C53—82no—
109C22×C4×C53C22×C411yescharacteristic
110C22×C4×C53C22×C411yescharacteristic
111C22×C4×C53C22×C411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
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×Q811yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C22×C4×C53#4111yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C2×C22⋊D53—22no—
124C2×C22⋊D53—22no—
125C2×C22⋊D53—22no—
126C2×C22⋊D53—22no—
127C2×C22⋊D53—22no—
128C2×C22⋊D53—22no—
129C22×Dic53—21no—
130C22×Dic53—41no—
131#11719945—22no—
132#11719945—22no—
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#11719945C22⋊C414yes—
144#11719945C22⋊C414yes—
145#11719945C4⋊C414yes—
146#11719945C4⋊C414yes—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212C4212yes—
151C2×D212C22⋊C412yes—
152C2×D212C4⋊C412yes—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106C4212yes—
163C2×Dic106C22⋊C412yes—
164C2×Dic106C4⋊C412yes—
165M16×C53—22no—
166M16×C53—22no—
167M16×C53—22no—
168M16×C53—22no—
169M16×C53—22no—
170M16×C53—41no—
171M16×C53—41no—
172M16×C53C22⋊C414yes—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C2×C8×C53—22no—
177C2×C8×C53—41no—
178C2×C8×C53—41no—
179C4⋊C4×C53—22no—
180C4⋊C4×C53—22no—
181C4⋊C4×C53—41no—
182C4⋊C4×C53—41no—
183C22⋊Dic53—22no—
184C22⋊Dic53—41no—
185C4⋊Dic53—22no—
186C4⋊Dic53—41no—
187C4×Dic53—22no—
188C4×Dic53—41no—
189#11719917—22no—
190#11719917—22no—
191#11719917—22no—
192#11719917—22no—
193#11719917—41no—
194#11719917—41no—
195#11719917—41no—
196#11719917—41no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322C22⋊C414yes—
203#11719912—216no—
204C8×D53—216no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53#11711yescharacteristic
208C23×C53C4×D411yescharacteristic
209C23×C53C4×D411yescharacteristic
210C23×C53#12111yescharacteristic
211C23×C53C22⋊Q811yescharacteristic
212C23×C53#12311yescharacteristic
213C23×C53#12311yescharacteristic
214C22×D53—22no—
215C22×D53—22no—
216C22×D53—22no—
217C22×D53—81no—
218C22×D53—81no—
219C2×C4×C53—42no—
220C2×C4×C53—82no—
221C2×C4×C53—82no—
222C2×C4×C53—82no—
223C2×C4×C53C2×C4211yescharacteristic
224C2×C4×C53C2×C22⋊C411yescharacteristic
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53C2×C22⋊C411yescharacteristic
227C2×C4×C53C2×C22⋊C411yescharacteristic
228C2×C4×C53C2×C22⋊C411yescharacteristic
229C2×C4×C53C2×C4⋊C411yescharacteristic
230C2×C4×C53C2×C4⋊C411yescharacteristic
231C2×C4×C53C2×C4⋊C411yescharacteristic
232C2×C4×C53#11711yescharacteristic
233C2×C4×C53C4×D411yescharacteristic
234C2×C4×C53C4×D411yescharacteristic
235C2×C4×C53C4×D411yescharacteristic
236C2×C4×C53C4×D411yescharacteristic
237C2×C4×C53C4×D411yescharacteristic
238C2×C4×C53C4×D411yescharacteristic
239C2×C4×C53#12011yescharacteristic
240C2×C4×C53#12011yescharacteristic
241C2×C4×C53#12011yescharacteristic
242C2×C4×C53#12011yescharacteristic
243C2×C4×C53#12111yescharacteristic
244C2×C4×C53#12111yescharacteristic
245C2×C4×C53#12111yescharacteristic
246C2×C4×C53C22⋊Q811yescharacteristic
247C2×C4×C53C22⋊Q811yescharacteristic
248C2×C4×C53C22⋊Q811yescharacteristic
249C2×C4×C53#12311yescharacteristic
250C2×C4×C53#12311yescharacteristic
251C22⋊D53—24no—
252C22⋊D53—24no—
253C22⋊D53—24no—
254C22⋊D53—24no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—42no—
259C2×Dic53—81no—
260C2×Dic53—81no—
261D212—22no—
262D212—22no—
263D212—22no—
264D212#9514yes—
265C4×D53—24no—
266C4×D53—24no—
267C4×D53—24no—
268C4×D53—24no—
269Dic106—22no—
270Dic106—22no—
271Dic106—22no—
272Dic106#9514yes—
273C8×C53—24no—
274C8×C53—42no—
275#88322—24no—
276#88322—42no—
277#10784—531nomaximal, Sylow
278C22×C53—21no—
279C22×C53—21no—
280C22×C53—21no—
281C22×C53—21no—
282C22×C53—21no—
283C22×C53—21no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53C4×C22⋊C411yescharacteristic
289C22×C53#38011yescharacteristic
290C22×C53#38111yescharacteristic
291C22×C53#38511yescharacteristic
292C22×C53#38511yescharacteristic
293C22×C53#38611yescharacteristic
294C22×C53#38611yescharacteristic
295C2×D53—24no—
296C2×D53—42no—
297C2×D53—42no—
298C2×D53—82no—
299C2×D53—82no—
300C4×C53—82no—
301C4×C53C2×#9511yescharacteristic
302C4×C53C4×C22⋊C411yescharacteristic
303C4×C53#37911yescharacteristic
304C4×C53#38011yescharacteristic
305C4×C53#38511yescharacteristic
306C4×C53#38511yescharacteristic
307C4×C53#38611yescharacteristic
308C4×C53#38611yescharacteristic
309Dic53—24no—
310Dic53—82no—
311C2×#568—531no—
312C2×#421—531no—
313C2×#417—531no—
314C2×#344—531no—
315C2×#343D5311yescharacteristic, complemented
316#1736—531no—
317C2×#338—531no—
318C2×C53—21no—
319C2×C53—21no—
320C2×C53—21no—
321C2×C53—21no—
322C2×C53—21no—
323C2×C53—21no—
324C2×C53C22⋊#9511yescharacteristic
325C2×C53#190211yescharacteristic
326C2×C53#190311yescharacteristic
327D53—44no—
328C22×#41—531no—
329#568—534no—
330#568—1062no—
331#568—1062no—
332#568—1062no—
333C2×#131—1062no—
334C2×#131—1062no—
335C22×M16—531no—
336C22×M16—531no—
337C22×M16—1062no—
338C22×M16C2×D5311yescharacteristic
339C22×M16C2×D5311yescharacteristic
340C2×#117—531no—
341C22×C4⋊C4C2×D5311yescharacteristic
342#421—534no—
343#421—1062no—
344C2×#104—1062no—
345#417—1062no—
346#417—1062no—
347C2×#103—532no—
348C2×#102—532no—
349#344—1062no—
350#343—22no—
351#340—1062no—
352#338—1062no—
353C53#1078411yescharacteristic, complemented, Sylow
354C2×#41—532no—
355C2×#41—532no—
356C2×#41—532no—
357C2×#41—532no—
358C2×#41—532no—
359C2×#41—532no—
360C2×#41—1061no—
361C2×#41—1061no—
362C2×#41—1061no—
363C2×#41—1061no—
364C2×#41—1061no—
365C2×#41—1061no—
366C22×Q8—531no—
367C22×D4—531no—
368C22×D4—1061no—
369C23×C4—21no—
370C23×C4—1061no—
371C23×C4C22×D5311yescharacteristic
372#131—1068no—
373#131—1068no—
374#131—1068no—
375#131—1068no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—21no—
379C2×M16—21no—
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—21no—
384C2×M16—532no—
385C2×M16—532no—
386C2×M16—1064no—
387C2×M16—1068no—
388C2×M16—1061no—
389C2×M16—1061no—
390C2×M16—1061no—
391C2×M16—1061no—
392C2×M16—1068no—
393C2×M16—1061no—
394C2×M16—1061no—
395C2×M16—1061no—
396C2×M16—1061no—
397C2×M16—1061no—
398C2×M16—1061no—
399C2×M16C4×D5312yes—
400C2×M16C4×D5312yes—
401C2×M16D21212yes—
402C2×M16C22⋊D5312yes—
403C22×C8—21no—
404C22×C8—21no—
405C22×C8—532no—
406C22×C8—532no—
407C22×C8—1064no—
408#117—532no—
409#117—1061no—
410#117—1061no—
411#117—1061no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4C4×D5312yes—
419C2×C22⋊C4—1061no—
420C2×C42—1061no—
421#104—1068no—
422#103—1064no—
423#102—1064no—
424C24—1061no—
425C24#1171994511yescharacteristic
426#41—534no—
427#41—534no—
428#41—534no—
429#41—534no—
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—
442C2×Q8—532no—
443C2×Q8—532no—
444C2×Q8—532no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×D4—532no—
449C2×D4—532no—
450C2×D4—532no—
451C2×D4—1062no—
452C2×D4—1062no—
453C2×D4—1062no—
454C2×D4—1062no—
455C2×D4—1062no—
456C2×D4—1062no—
457C2×D4—1061no—
458C2×D4—1061no—
459C2×D4—1061no—
460C22×C4—22no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—82no—
464C22×C4—1061no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—2121no—
472C22×C4C2×C4×D5311yescharacteristic
473C22×C4C2×C4×D5311yescharacteristic
474C22×C4C2×C4×D5311yescharacteristic
475C22×C4C2×D21211yescharacteristic
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—22no—
491M16—41no—
492M16—41no—
493M16—10616no—
494M16—1062no—
495M16—1062no—
496M16—1062no—
497M16—1062no—
498M16—2121no—
499M16—2121no—
500M16—2121no—
501M16—2121no—
502M16#1171992114yes—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—534no—
510C2×C8—10616no—
511C2×C8—1062no—
512C2×C8—1062no—
513C2×C8—1062no—
514C2×C8—1062no—
515C2×C8—1062no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—1062no—
521C4⋊C4—2121no—
522C22⋊C4—1062no—
523C22⋊C4—2121no—
524C42—1062no—
525C42—2121no—
526C23—41no—
527C23—41no—
528C23—1062no—
529C23—1062no—
530C23—1062no—
531C23—4241no—
532C23—4241no—
533C23#1313317411yescharacteristic
534C23C4×C22⋊D5311yescharacteristic
535C23C4×D21211yescharacteristic
536C23#1313319011yescharacteristic
537C23#1313319511yescharacteristic
538C23#1313319911yescharacteristic
539C23#1313320211yescharacteristic
540Q8—534no—
541Q8—1062no—
542Q8—1062no—
543Q8—1062no—
544D4—534no—
545D4—1064no—
546D4—1064no—
547D4—1064no—
548D4—1064no—
549D4—1062no—
550D4—1062no—
551D4—1062no—
552C2×C4—42no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—1064no—
562C2×C4—1064no—
563C2×C4—4241no—
564C2×C4—4241no—
565C2×C4—2122no—
566C2×C4C42×D5311yescharacteristic
567C2×C4C22⋊C4×D5311yescharacteristic
568C2×C4C22⋊C4×D5311yescharacteristic
569C2×C4C22⋊C4×D5311yescharacteristic
570C2×C4C22⋊C4×D5311yescharacteristic
571C2×C4C2×#1171992111yescharacteristic
572C2×C4C4⋊C4×D5311yescharacteristic
573C2×C4C4⋊C4×D5311yescharacteristic
574C2×C4C4⋊C4×D5311yescharacteristic
575C2×C4#1313317511yescharacteristic
576C2×C4#1313317711yescharacteristic
577C2×C4#1313317711yescharacteristic
578C2×C4C4×C22⋊D5311yescharacteristic
579C2×C4#1313317911yescharacteristic
580C2×C4#1313317911yescharacteristic
581C2×C4C4×D21211yescharacteristic
582C2×C4#1313318511yescharacteristic
583C2×C4#1313318511yescharacteristic
584C2×C4#1313318711yescharacteristic
585C2×C4#1313318711yescharacteristic
586C2×C4#1313318811yescharacteristic
587C2×C4#1313319011yescharacteristic
588C2×C4#1313319211yescharacteristic
589C2×C4#1313319311yescharacteristic
590C2×C4#1313319511yescharacteristic
591C2×C4#1313319711yescharacteristic
592C2×C4#1313319911yescharacteristic
593C2×C4#1313320211yescharacteristic
594C8—24no—
595C8—42no—
596C8—1064no—
597C8—2122no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—1064no—
609C22—4242no—
610C22—4242no—
611C22—2122no—
612C22—2122no—
613C22C4×#1171992111yescharacteristic
614C22#1681048211yescharacteristic
615C22#1681048511yescharacteristic
616C22#1681049411yescharacteristic
617C22#1681049711yescharacteristic
618C22#1681049911yescharacteristic
619C22#1681050111yescharacteristic
620C4—82no—
621C4—1064no—
622C4—4242no—
623C4#95×D5311yescharacteristic
624C4#1681047011yescharacteristic
625C4#1681047811yescharacteristic
626C4#1681048011yescharacteristic
627C4#1681049411yescharacteristic
628C4#1681049711yescharacteristic
629C4#1681049911yescharacteristic
630C4#1681050111yescharacteristic
631C2—21no—
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—2124no—
638C2#2027293611yescharacteristic
639C2#2027519111yescharacteristic
640C2#2027519811yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC2×C53×#343C2
socleC22×C53#386
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenterC2×#343D53
hypocenterC53#10784
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
4. C2×#16810395C2semidirect
5. #20274434C2semidirect
6. C2×#16810381C2semidirect
7. C2×C53×#343C2semidirect
12. #16811389C4semidirect
20. #16810639C4semidirect
24. C2×#13133140C4semidirect
25. C2×#13133137C4semidirect
315. C2×#343D53semidirect
353. C53#10784semidirect

Maximal subgroups

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

subgroupindex
C2×#168113892
C2×#168106392
C2×#168106222
C2×#168103952
#202744342
C2×#168103812
C2×C53×#3432
#1078453