Finite Group Databasebeta

← #27289496

Subgroups — #27289496

Order 13568.

1168 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811389C211yesmaximal, characteristic
2C2×#16810596C211yesmaximal, characteristic
3#20283159C211yesmaximal, characteristic
4C2×#16810394C211yesmaximal, characteristic, complemented
5#20274454C211yesmaximal, characteristic, complemented
6#20274453C211yesmaximal, characteristic, complemented
7C2×C53×#344C211yesmaximal, 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#16810596—22no—
20#16810596—22no—
21C2×#13133131C414yes—
22#16810591—22no—
23#16810589C414yescomplemented
24#16810586—22no—
25#16810394—22no—
26#16810392—22no—
27#16810391—22no—
28C22×M16×C53C2211yescharacteristic
29C22×M16×C53C2211yescharacteristic
30C2×#117×C53C2211yescharacteristic
31C53×#344—22no—
32C2×#11719945—21no—
33C2×#11719945—21no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945C2×C412yes—
41C2×#11719945C2×C412yes—
42C2×#11719945D412yes—
43C2×#11719945D412yes—
44C22×Dic106Q811yescharacteristic
45C22×D212Q811yescharacteristic
46C22×C22⋊D53C2×C412yes—
47C22×C4×D53D412yes—
48#13133225—28no—
49#13133225—28no—
50#13133223—28no—
51#13133223—28no—
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×#11719917C2×C412yes—
61C2×#11719917C2×C412yes—
62C2×#11719917D412yes—
63C2×#11719917D412yes—
64C2×#11719912—24no—
65M16×D53—28no—
66M16×D53—28no—
67C22×#88322—21no—
68C22×#88322—21no—
69C2×C8×D53—24no—
70#13133176—21no—
71#13133176—21no—
72#13133176—21no—
73#13133176—21no—
74C2×C4⋊Dic53—21no—
75C2×C22⋊Dic53C2×C412yes—
76C2×C4×Dic53—21no—
77#13133131—28no—
78#13133129—28no—
79C23×C4×C53C2311yescharacteristic
80C2×M16×C53—21no—
81C2×M16×C53—21no—
82C2×M16×C53—21no—
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×C53C2×C412yes—
90C2×M16×C53D412yes—
91C2×M16×C53D412yes—
92C22×C8×C53—21no—
93C22×C8×C53C2×C412yes—
94#117×C53—21no—
95#117×C53—21no—
96#117×C53—21no—
97#117×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C22⋊C4×C53C2×C412yes—
100C2×C42×C53—21no—
101C24×C53C22×C411yescharacteristic
102C23×D53C4⋊C412yes—
103C22×C4×C53—22no—
104C22×C4×C53C22×C411yescharacteristic
105C22×C4×C53C22×C411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×Q811yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C22×C4×C53#4111yescharacteristic
118C2×C22⋊D53—22no—
119C2×C22⋊D53—22no—
120C2×C22⋊D53—22no—
121C2×C22⋊D53—22no—
122C2×C22⋊D53C4214yes—
123C2×C22⋊D53C22⋊C414yes—
124C22×Dic53—22no—
125C22×Dic53C4⋊C412yes—
126#11719945—22no—
127#11719945—22no—
128#11719945—22no—
129#11719945—22no—
130#11719945—22no—
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—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212C4⋊C412yes—
147C2×D212C4⋊C412yes—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53C22⋊C414yes—
153C2×C4×D53C22⋊C414yes—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106C4⋊C412yes—
159C2×Dic106C4⋊C412yes—
160M16×C53—22no—
161M16×C53—22no—
162M16×C53—22no—
163M16×C53—41no—
164M16×C53—41no—
165M16×C53—41no—
166M16×C53—41no—
167M16×C53C22⋊C414yes—
168C2×C8×C53—22no—
169C2×C8×C53—22no—
170C2×C8×C53—22no—
171C2×C8×C53—22no—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C4⋊C4×C53—41no—
175C4⋊C4×C53—41no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C42×C53—41no—
179C42×C53—41no—
180C22⋊Dic53—22no—
181C22⋊Dic53—22no—
182C4⋊Dic53—41no—
183C4⋊Dic53—41no—
184C4×Dic53—41no—
185C4×Dic53—41no—
186#11719917—22no—
187#11719917—22no—
188#11719917—22no—
189#11719917—22no—
190#11719917—22no—
191#11719917—41no—
192#11719917—41no—
193#11719917C22⋊C414yes—
194C2×#88322—22no—
195C2×#88322—22no—
196C2×#88322—22no—
197C2×#88322—22no—
198C2×#88322—41no—
199C2×#88322—41no—
200#11719912—216no—
201C8×D53—216no—
202C23×C53—81no—
203C23×C53C2×C4211yescharacteristic
204C23×C53C2×C22⋊C411yescharacteristic
205C23×C53C2×C4⋊C411yescharacteristic
206C23×C53C4×D411yescharacteristic
207C23×C53C4×D411yescharacteristic
208C23×C53C4×D411yescharacteristic
209C23×C53C4×D411yescharacteristic
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—82no—
213C22×D53#9514yes—
214C2×C4×C53—42no—
215C2×C4×C53—82no—
216C2×C4×C53C2×C22⋊C411yescharacteristic
217C2×C4×C53C2×C22⋊C411yescharacteristic
218C2×C4×C53C2×C22⋊C411yescharacteristic
219C2×C4×C53C2×C22⋊C411yescharacteristic
220C2×C4×C53C2×C4⋊C411yescharacteristic
221C2×C4×C53C2×C4⋊C411yescharacteristic
222C2×C4×C53#11711yescharacteristic
223C2×C4×C53#11711yescharacteristic
224C2×C4×C53C4×D411yescharacteristic
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53#12011yescharacteristic
229C2×C4×C53#12011yescharacteristic
230C2×C4×C53#12011yescharacteristic
231C2×C4×C53#12011yescharacteristic
232C2×C4×C53#12111yescharacteristic
233C2×C4×C53#12111yescharacteristic
234C2×C4×C53#12111yescharacteristic
235C2×C4×C53#12111yescharacteristic
236C2×C4×C53C22⋊Q811yescharacteristic
237C2×C4×C53C22⋊Q811yescharacteristic
238C2×C4×C53C22⋊Q811yescharacteristic
239C2×C4×C53C22⋊Q811yescharacteristic
240C2×C4×C53#12311yescharacteristic
241C2×C4×C53#12311yescharacteristic
242C2×C4×C53#12311yescharacteristic
243C2×C4×C53#12311yescharacteristic
244C22⋊D53—24no—
245C22⋊D53—24no—
246C22⋊D53—24no—
247C22⋊D53—24no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—42no—
251C2×Dic53—82no—
252C2×Dic53#9514yes—
253D212—22no—
254D212—22no—
255D212—22no—
256D212—22no—
257C4×D53—24no—
258C4×D53—24no—
259C4×D53—24no—
260C4×D53—24no—
261Dic106—22no—
262Dic106—22no—
263Dic106—22no—
264Dic106—22no—
265C8×C53—24no—
266C8×C53—42no—
267#88322—24no—
268#88322—42no—
269#10757—531nomaximal, Sylow
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53C2×#9511yescharacteristic
278C22×C53C4×C22⋊C411yescharacteristic
279C22×C53C4×C22⋊C411yescharacteristic
280C22×C53#38511yescharacteristic
281C22×C53#38511yescharacteristic
282C22×C53#38611yescharacteristic
283C22×C53#38611yescharacteristic
284C2×D53—24no—
285C2×D53—44no—
286C2×D53—44no—
287C2×D53—82no—
288C4×C53—82no—
289C4×C53#37911yescharacteristic
290C4×C53#38011yescharacteristic
291C4×C53#38011yescharacteristic
292C4×C53#38111yescharacteristic
293C4×C53#38511yescharacteristic
294C4×C53#38511yescharacteristic
295C4×C53#38611yescharacteristic
296C4×C53#38611yescharacteristic
297Dic53—24no—
298Dic53—82no—
299C2×#568—531no—
300C2×#410—531no—
301#2879—531no—
302C2×#344—531no—
303C2×#344D5311yescharacteristic, complemented
304#1742—531no—
305#1742—531no—
306C2×C53—21no—
307C2×C53—21no—
308C2×C53—21no—
309C2×C53—21no—
310C2×C53—21no—
311C2×C53—21no—
312C2×C53—21no—
313C2×C53C22⋊#9511yescharacteristic
314D53—44no—
315C22×#41—531no—
316#568—534no—
317#568—1062no—
318#568—1062no—
319#568—1062no—
320C2×#131—1062no—
321C2×#131—1062no—
322C22×M16—531no—
323C22×M16—531no—
324C22×M16—1062no—
325C22×M16C2×D5311yescharacteristic
326C22×M16C2×D5311yescharacteristic
327C2×#117—531no—
328C2×#117C2×D5311yescharacteristic
329#410—1062no—
330#410—1062no—
331C2×#99—534no—
332#408—534no—
333#408—1062no—
334#407—1062no—
335#344—22no—
336#344—1062no—
337#343—1062no—
338#343—1062no—
339C53#1075711yescharacteristic, complemented, Sylow
340C2×#41—532no—
341C2×#41—532no—
342C2×#41—532no—
343C2×#41—532no—
344C2×#41—1061no—
345C2×#41—1061no—
346C2×#41—1061no—
347C2×#41—1061no—
348C2×#41—1061no—
349C2×#41—1061no—
350C2×#41—1061no—
351C2×#41—1061no—
352C22×Q8—531no—
353C22×D4—532no—
354C22×D4—531no—
355C23×C4—532no—
356C23×C4C22×D5311yescharacteristic
357#131—1068no—
358#131—1068no—
359#131—1068no—
360#131—1068no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—532no—
371C2×M16—532no—
372C2×M16—532no—
373C2×M16—532no—
374C2×M16—1064no—
375C2×M16—1068no—
376C2×M16—1068no—
377C2×M16—1061no—
378C2×M16—1061no—
379C2×M16—1061no—
380C2×M16—1061no—
381C2×M16—1061no—
382C2×M16—1061no—
383C2×M16—1061no—
384C2×M16—1061no—
385C2×M16C4×D5312yes—
386C2×M16D21212yes—
387C2×M16C22⋊D5312yes—
388C22×C8—21no—
389C22×C8—1064no—
390C22×C8—1061no—
391C22×C8—1061no—
392C22×C8C4×D5312yes—
393#117—21no—
394#117—21no—
395#117—21no—
396#117—21no—
397#117—1061no—
398#117—1061no—
399#117—1061no—
400#117—1061no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—1061no—
403C2×C22⋊C4—532no—
404C2×C22⋊C4C4×D5312yes—
405C2×C42—21no—
406C2×C42—1061no—
407#99—1068no—
408C22⋊C8—1068no—
409C24—532no—
410C24C2×C4×D5311yescharacteristic
411#41—1062no—
412#41—1062no—
413#41—1062no—
414#41—1062no—
415#41—1062no—
416#41—1062no—
417#41—1062no—
418#41—1062no—
419#41—1062no—
420#41—1062no—
421#41—1062no—
422#41—1062no—
423#41—1062no—
424#41—1062no—
425#41—1062no—
426#41—1062no—
427C2×Q8—532no—
428C2×Q8—532no—
429C2×Q8—1061no—
430C2×Q8—1061no—
431C2×Q8—1061no—
432C2×Q8—1061no—
433C2×D4—534no—
434C2×D4—534no—
435C2×D4—532no—
436C2×D4—532no—
437C2×D4—1062no—
438C2×D4—1062no—
439C2×D4—1062no—
440C2×D4—1062no—
441C2×D4—1061no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C22×C4—22no—
446C22×C4—532no—
447C22×C4—534no—
448C22×C4—534no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4C2×C4×D5311yescharacteristic
455C22×C4C2×C4×D5311yescharacteristic
456C22×C4C2×D21211yescharacteristic
457C22×C4#1171994511yescharacteristic
458C22×C4#1171994511yescharacteristic
459C22×C4D4×D5311yescharacteristic
460C22×C4D4×D5311yescharacteristic
461C22×C4D4×D5311yescharacteristic
462C22×C4D4×D5311yescharacteristic
463C22×C4D4×D5311yescharacteristic
464C22×C4#1171994711yescharacteristic
465C22×C4Q8×D5311yescharacteristic
466C22×C4#1171994911yescharacteristic
467C22×C4C2×C22⋊D5311yescharacteristic
468M16—22no—
469M16—22no—
470M16—22no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16—41no—
475M16—534no—
476M16—10616no—
477M16—1062no—
478M16—1062no—
479M16—1062no—
480M16—1062no—
481M16—1062no—
482M16—2121no—
483M16—2121no—
484M16#1171992114yes—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—10616no—
492C2×C8—1062no—
493C2×C8—1062no—
494C2×C8—1062no—
495C2×C8—1062no—
496C2×C8—2121no—
497C2×C8—2121no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—2121no—
501C4⋊C4—2121no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—1062no—
505C22⋊C4—1062no—
506C42—41no—
507C42—41no—
508C42—2121no—
509C42—2121no—
510C23—81no—
511C23—534no—
512C23—1062no—
513C23—1062no—
514C23—4242no—
515C23C42×D5311yescharacteristic
516C23C22⋊C4×D5311yescharacteristic
517C23C4⋊C4×D5311yescharacteristic
518C23#1313317711yescharacteristic
519C23C4×C22⋊D5311yescharacteristic
520C23#1313317911yescharacteristic
521C23C4×D21211yescharacteristic
522Q8—1062no—
523Q8—1062no—
524Q8—1062no—
525Q8—1062no—
526D4—1064no—
527D4—1064no—
528D4—1064no—
529D4—1064no—
530D4—1062no—
531D4—1062no—
532D4—1062no—
533D4—1062no—
534C2×C4—42no—
535C2×C4—82no—
536C2×C4—534no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—1064no—
540C2×C4—1064no—
541C2×C4—1064no—
542C2×C4—1064no—
543C2×C4—4242no—
544C2×C4—2122no—
545C2×C4C22⋊C4×D5311yescharacteristic
546C2×C4C22⋊C4×D5311yescharacteristic
547C2×C4C22⋊C4×D5311yescharacteristic
548C2×C4C2×#1171992111yescharacteristic
549C2×C4C4⋊C4×D5311yescharacteristic
550C2×C4C4⋊C4×D5311yescharacteristic
551C2×C4#1313317411yescharacteristic
552C2×C4#1313317511yescharacteristic
553C2×C4#1313317711yescharacteristic
554C2×C4C4×C22⋊D5311yescharacteristic
555C2×C4#1313317911yescharacteristic
556C2×C4C4×D21211yescharacteristic
557C2×C4#1313318511yescharacteristic
558C2×C4#1313318511yescharacteristic
559C2×C4#1313318711yescharacteristic
560C2×C4#1313318711yescharacteristic
561C2×C4#1313318811yescharacteristic
562C2×C4#1313319011yescharacteristic
563C2×C4#1313319011yescharacteristic
564C2×C4#1313319211yescharacteristic
565C2×C4#1313319311yescharacteristic
566C2×C4#1313319511yescharacteristic
567C2×C4#1313319511yescharacteristic
568C2×C4#1313319711yescharacteristic
569C2×C4#1313319911yescharacteristic
570C2×C4#1313319911yescharacteristic
571C2×C4#1313320211yescharacteristic
572C2×C4#1313320211yescharacteristic
573C8—24no—
574C8—42no—
575C8—1064no—
576C8—2122no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—1064no—
585C22—4242no—
586C22—2124no—
587C22—2124no—
588C22#95×D5311yescharacteristic
589C22#1681047011yescharacteristic
590C22C4×#1171992111yescharacteristic
591C22#1681049411yescharacteristic
592C22#1681049711yescharacteristic
593C22#1681049911yescharacteristic
594C22#1681050111yescharacteristic
595C4—82no—
596C4—1064no—
597C4—4242no—
598C4#1681047811yescharacteristic
599C4#1681048011yescharacteristic
600C4#1681048211yescharacteristic
601C4#1681048511yescharacteristic
602C4#1681049411yescharacteristic
603C4#1681049711yescharacteristic
604C4#1681049911yescharacteristic
605C4#1681050111yescharacteristic
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—2124no—
614C2#2027293611yescharacteristic
subgroupG/H
centerC4#16810499
commutatorC23×C53C2×C42
FrattiniC23×C4C22×D53
FittingC2×C53×#344C2
socleC2×C53C22⋊#95
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenterC2×#344D53
hypocenterC53#10757
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×#16810394C2semidirect
5. #20274454C2semidirect
6. #20274453C2semidirect
7. C2×C53×#344C2semidirect
12. #16811389C4semidirect
23. #16810589C4semidirect
303. C2×#344D53semidirect
339. C53#10757semidirect

Maximal subgroups

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

subgroupindex
C2×#168113892
C2×#168105962
#202831592
C2×#168103942
#202744542
#202744532
C2×C53×#3442
#1075753