Finite Group Databasebeta

← #27289654

Subgroups — #27289654

Order 13568.

1164 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811389C211yesmaximal, characteristic
2C2×#16810641C211yesmaximal, characteristic
3#20283232C211yesmaximal, characteristic
4C2×#16810393C211yesmaximal, characteristic, complemented
5#20274454C211yesmaximal, characteristic, complemented
6#20274442C211yesmaximal, characteristic, complemented
7C2×C53×H(3,ℤ/4)C211yesmaximal, 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
17C22×#11719917C2211yescharacteristic
18C2×M16×D53—22no—
19#16810641—22no—
20#16810641—22no—
21C2×#13133145C414yescomplemented
22#16810631C412yescomplemented
23#16810627C412yescomplemented
24#16810622—22no—
25#16810617—22no—
26#16810393—22no—
27#16810392—22no—
28#16810388—22no—
29C22×M16×C53C2211yescharacteristic
30C2×#117×C53C2211yescharacteristic
31C2×#117×C53C2211yescharacteristic
32C53×H(3,ℤ/4)—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×#11719917—21no—
64C2×#11719917—21no—
65C2×#11719917—21no—
66C2×#11719917C2×C412yes—
67C2×#11719917C2×C412yes—
68C2×#11719917C2×C412yes—
69C2×#11719917D412yes—
70C2×#11719917D412yes—
71C2×#11719912—24no—
72M16×D53—28no—
73M16×D53—28no—
74C22×#88322—21no—
75C22×#88322—21no—
76C22×#88322—21no—
77C2×C8×D53—24no—
78#13133145—28no—
79#13133140—24no—
80#13133137—24no—
81C23×C4×C53C2311yescharacteristic
82C2×M16×C53—21no—
83C2×M16×C53—21no—
84C2×M16×C53—21no—
85C2×M16×C53—21no—
86C2×M16×C53C2×C412yes—
87C2×M16×C53D412yes—
88C22×C8×C53D412yes—
89#117×C53—21no—
90#117×C53—21no—
91#117×C53—21no—
92#117×C53—21no—
93#117×C53—21no—
94#117×C53—21no—
95#117×C53—21no—
96#117×C53C2×C412yes—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C22⋊C4×C53—21no—
100C2×C22⋊C4×C53—21no—
101C2×C42×C53—21no—
102C2×C42×C53C2×C412yes—
103C24×C53#4111yescharacteristic
104C23×D53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—41no—
107C22×C4×C53C22×C411yescharacteristic
108C22×C4×C53C22×C411yescharacteristic
109C22×C4×C53C22×C411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×Q811yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C22×C4×C53#4111yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C2×C22⋊D53—22no—
122C2×C22⋊D53—22no—
123C2×C22⋊D53—22no—
124C2×C22⋊D53—22no—
125C2×C22⋊D53—22no—
126C2×C22⋊D53—22no—
127C22×Dic53—21no—
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#11719945C22⋊C414yes—
141#11719945C22⋊C414yes—
142#11719945C4⋊C414yes—
143#11719945C4⋊C414yes—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212C4212yes—
148C2×D212C22⋊C412yes—
149C2×D212C4⋊C412yes—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106C4212yes—
160C2×Dic106C22⋊C412yes—
161C2×Dic106C4⋊C412yes—
162M16×C53—22no—
163M16×C53—22no—
164M16×C53—22no—
165M16×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53C22⋊C414yes—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—41no—
171C4⋊C4×C53—41no—
172C4⋊C4×C53—41no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—41no—
175C22⋊C4×C53—41no—
176C22⋊C4×C53—41no—
177C42×C53—22no—
178C42×C53—22no—
179C42×C53—22no—
180C42×C53—41no—
181#11719917—22no—
182#11719917—22no—
183#11719917—22no—
184#11719917—22no—
185#11719917—22no—
186#11719917—22no—
187#11719917—22no—
188#11719917—41no—
189#11719917—41no—
190#11719917—41no—
191#11719917—41no—
192#11719917C22⋊C414yes—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322—22no—
196C2×#88322—22no—
197C2×#88322—22no—
198C2×#88322—41no—
199C2×#88322—41no—
200C2×#88322—41no—
201C2×#88322—41no—
202#11719912—216no—
203C8×D53—216no—
204C23×C53—81no—
205C23×C53#11711yescharacteristic
206C23×C53C4×D411yescharacteristic
207C23×C53C4×D411yescharacteristic
208C23×C53#12111yescharacteristic
209C23×C53C22⋊Q811yescharacteristic
210C23×C53#12311yescharacteristic
211C23×C53#12311yescharacteristic
212C22×D53—22no—
213C22×D53—22no—
214C22×D53—22no—
215C22×D53—82no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—82no—
219C2×C4×C53—82no—
220C2×C4×C53C2×C4211yescharacteristic
221C2×C4×C53C2×C22⋊C411yescharacteristic
222C2×C4×C53C2×C22⋊C411yescharacteristic
223C2×C4×C53C2×C22⋊C411yescharacteristic
224C2×C4×C53C2×C22⋊C411yescharacteristic
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53C2×C4⋊C411yescharacteristic
227C2×C4×C53C2×C4⋊C411yescharacteristic
228C2×C4×C53C2×C4⋊C411yescharacteristic
229C2×C4×C53#11711yescharacteristic
230C2×C4×C53C4×D411yescharacteristic
231C2×C4×C53C4×D411yescharacteristic
232C2×C4×C53C4×D411yescharacteristic
233C2×C4×C53C4×D411yescharacteristic
234C2×C4×C53C4×D411yescharacteristic
235C2×C4×C53C4×D411yescharacteristic
236C2×C4×C53#12011yescharacteristic
237C2×C4×C53#12011yescharacteristic
238C2×C4×C53#12011yescharacteristic
239C2×C4×C53#12011yescharacteristic
240C2×C4×C53#12111yescharacteristic
241C2×C4×C53#12111yescharacteristic
242C2×C4×C53#12111yescharacteristic
243C2×C4×C53C22⋊Q811yescharacteristic
244C2×C4×C53C22⋊Q811yescharacteristic
245C2×C4×C53C22⋊Q811yescharacteristic
246C2×C4×C53#12311yescharacteristic
247C2×C4×C53#12311yescharacteristic
248C22⋊D53—24no—
249C22⋊D53—24no—
250C22⋊D53—24no—
251C22⋊D53—24no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255D212—22no—
256D212—22no—
257D212—22no—
258D212#9514yes—
259C4×D53—24no—
260C4×D53—24no—
261C4×D53—24no—
262C4×D53—24no—
263Dic106—22no—
264Dic106—22no—
265Dic106—22no—
266Dic106#9514yes—
267C8×C53—24no—
268#88322—24no—
269#88322—42no—
270#88322—42no—
271#10787—531nomaximal, Sylow
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53C4×C22⋊C411yescharacteristic
280C22×C53#38011yescharacteristic
281C22×C53#38111yescharacteristic
282C22×C53#38511yescharacteristic
283C22×C53#38511yescharacteristic
284C22×C53#38611yescharacteristic
285C22×C53#38611yescharacteristic
286C2×D53—24no—
287C2×D53—44no—
288C2×D53—82no—
289C2×D53—82no—
290C4×C53—82no—
291C4×C53—82no—
292C4×C53C2×#9511yescharacteristic
293C4×C53C4×C22⋊C411yescharacteristic
294C4×C53#37911yescharacteristic
295C4×C53#38011yescharacteristic
296C4×C53#38511yescharacteristic
297C4×C53#38511yescharacteristic
298C4×C53#38611yescharacteristic
299C4×C53#38611yescharacteristic
300Dic53—24no—
301C2×#568—531no—
302C2×#421—531no—
303#2896—531no—
304C2×#344—531no—
305#1742—531no—
306#1738—531no—
307C2×H(3,ℤ/4)D5311yescharacteristic, complemented
308C2×C53—21no—
309C2×C53—21no—
310C2×C53—21no—
311C2×C53—21no—
312C2×C53—21no—
313C2×C53—21no—
314C2×C53—21no—
315C2×C53C22⋊#9511yescharacteristic
316D53—44no—
317C22×#41—531no—
318#568—534no—
319#568—1062no—
320#568—1062no—
321#568—1062no—
322C2×#131—1062no—
323C2×#131—1062no—
324C22×M16—531no—
325C22×M16—531no—
326C22×M16—531no—
327C22×M16—1062no—
328C22×M16C2×D5311yescharacteristic
329C2×#117C2×D5311yescharacteristic
330C2×#117C2×D5311yescharacteristic
331#421—1062no—
332#421—1062no—
333C2×#104—534no—
334#419—532no—
335#418—532no—
336#417—1062no—
337#416—1062no—
338#344—1062no—
339#343—1062no—
340#341—1062no—
341H(3,ℤ/4)—22no—
342C53#1078711yescharacteristic, complemented, Sylow
343C2×#41—532no—
344C2×#41—532no—
345C2×#41—532no—
346C2×#41—532no—
347C2×#41—532no—
348C2×#41—532no—
349C2×#41—1061no—
350C2×#41—1061no—
351C2×#41—1061no—
352C2×#41—1061no—
353C2×#41—1061no—
354C2×#41—1061no—
355C22×Q8—531no—
356C22×D4—531no—
357C22×D4—1061no—
358C23×C4—1061no—
359C23×C4C22×D5311yescharacteristic
360#131—1068no—
361#131—1068no—
362#131—1068no—
363#131—1068no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—21no—
368C2×M16—532no—
369C2×M16—532no—
370C2×M16—532no—
371C2×M16—532no—
372C2×M16—532no—
373C2×M16—1064no—
374C2×M16—1068no—
375C2×M16—1068no—
376C2×M16—1061no—
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×M16—1061no—
386C2×M16—1061no—
387C2×M16—1061no—
388C2×M16—1061no—
389C2×M16C4×D5312yes—
390C2×M16C22⋊D5312yes—
391C22×C8—1064no—
392C22×C8—1061no—
393C22×C8—1061no—
394C22×C8—1061no—
395C22×C8D21212yes—
396#117—21no—
397#117—21no—
398#117—21no—
399#117—21no—
400#117—21no—
401#117—21no—
402#117—21no—
403#117C4×D5312yes—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C42—21no—
409C2×C42C4×D5312yes—
410#104—1068no—
411#103—1064no—
412#102—1064no—
413C24—1061no—
414C24#1171994711yescharacteristic
415#41—534no—
416#41—534no—
417#41—534no—
418#41—534no—
419#41—1062no—
420#41—1062no—
421#41—1062no—
422#41—1062no—
423#41—1062no—
424#41—1062no—
425#41—1062no—
426#41—1062no—
427#41—1062no—
428#41—1062no—
429#41—1062no—
430#41—1062no—
431C2×Q8—532no—
432C2×Q8—532no—
433C2×Q8—532no—
434C2×Q8—1061no—
435C2×Q8—1061no—
436C2×Q8—1061no—
437C2×D4—532no—
438C2×D4—532no—
439C2×D4—532no—
440C2×D4—1062no—
441C2×D4—1062no—
442C2×D4—1062no—
443C2×D4—1062no—
444C2×D4—1062no—
445C2×D4—1062no—
446C2×D4—1061no—
447C2×D4—1061no—
448C2×D4—1061no—
449C22×C4—22no—
450C22×C4—41no—
451C22×C4—1061no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4C2×C4×D5311yescharacteristic
459C22×C4C2×C4×D5311yescharacteristic
460C22×C4C2×C4×D5311yescharacteristic
461C22×C4C2×D21211yescharacteristic
462C22×C4#1171994511yescharacteristic
463C22×C4#1171994511yescharacteristic
464C22×C4D4×D5311yescharacteristic
465C22×C4D4×D5311yescharacteristic
466C22×C4D4×D5311yescharacteristic
467C22×C4D4×D5311yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4Q8×D5311yescharacteristic
470C22×C4#1171994911yescharacteristic
471C22×C4C2×C22⋊D5311yescharacteristic
472M16—22no—
473M16—22no—
474M16—22no—
475M16—22no—
476M16—534no—
477M16—10616no—
478M16—1062no—
479M16—1062no—
480M16—1062no—
481M16—1062no—
482M16—1062no—
483M16—1062no—
484M16—1062no—
485M16—2121no—
486M16—2121no—
487M16—2121no—
488M16—2121no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—10616no—
492C2×C8—1062no—
493C2×C8—1062no—
494C2×C8—1062no—
495C2×C8—1062no—
496C2×C8—1062no—
497C2×C8—2121no—
498C2×C8—2121no—
499C2×C8—2121no—
500C2×C8—2121no—
501C2×C8#1171992114yes—
502C4⋊C4—22no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C22⋊C4—22no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—41no—
514C23—81no—
515C23—1062no—
516C23—1062no—
517C23—1062no—
518C23—4242no—
519C23#1313317511yescharacteristic
520C23#1313317711yescharacteristic
521C23#1313317711yescharacteristic
522C23#1313318811yescharacteristic
523C23#1313319711yescharacteristic
524C23#1313319911yescharacteristic
525C23#1313319911yescharacteristic
526Q8—534no—
527Q8—1062no—
528Q8—1062no—
529Q8—1062no—
530D4—534no—
531D4—1064no—
532D4—1064no—
533D4—1064no—
534D4—1064no—
535D4—1062no—
536D4—1062no—
537D4—1062no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—1062no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1064no—
546C2×C4—1064no—
547C2×C4—1064no—
548C2×C4—1064no—
549C2×C4C42×D5311yescharacteristic
550C2×C4C22⋊C4×D5311yescharacteristic
551C2×C4C22⋊C4×D5311yescharacteristic
552C2×C4C22⋊C4×D5311yescharacteristic
553C2×C4C22⋊C4×D5311yescharacteristic
554C2×C4C2×#1171992111yescharacteristic
555C2×C4C4⋊C4×D5311yescharacteristic
556C2×C4C4⋊C4×D5311yescharacteristic
557C2×C4C4⋊C4×D5311yescharacteristic
558C2×C4#1313317411yescharacteristic
559C2×C4C4×C22⋊D5311yescharacteristic
560C2×C4C4×C22⋊D5311yescharacteristic
561C2×C4#1313317911yescharacteristic
562C2×C4#1313317911yescharacteristic
563C2×C4C4×D21211yescharacteristic
564C2×C4C4×D21211yescharacteristic
565C2×C4#1313318511yescharacteristic
566C2×C4#1313318511yescharacteristic
567C2×C4#1313318711yescharacteristic
568C2×C4#1313318711yescharacteristic
569C2×C4#1313319011yescharacteristic
570C2×C4#1313319011yescharacteristic
571C2×C4#1313319211yescharacteristic
572C2×C4#1313319311yescharacteristic
573C2×C4#1313319511yescharacteristic
574C2×C4#1313319511yescharacteristic
575C2×C4#1313320211yescharacteristic
576C2×C4#1313320211yescharacteristic
577C8—24no—
578C8—1064no—
579C8—2122no—
580C8—2122no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—1064no—
589C22—4242no—
590C22—4242no—
591C22—2124no—
592C22#1681047011yescharacteristic
593C22#1681048011yescharacteristic
594C22#1681048511yescharacteristic
595C22#1681049711yescharacteristic
596C22#1681049711yescharacteristic
597C22#1681049911yescharacteristic
598C22#1681049911yescharacteristic
599C4—82no—
600C4—82no—
601C4—1064no—
602C4#95×D5311yescharacteristic
603C4C4×#1171992111yescharacteristic
604C4#1681047811yescharacteristic
605C4#1681048211yescharacteristic
606C4#1681049411yescharacteristic
607C4#1681049411yescharacteristic
608C4#1681050111yescharacteristic
609C4#1681050111yescharacteristic
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—21no—
615C2—21no—
616C2—21no—
617C2—2124no—
618C2#2027293611yescharacteristic
subgroupG/H
centerC4#16810501
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC2×C53×H(3,ℤ/4)C2
socleC2×C53C22⋊#95
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenterC2×H(3,ℤ/4)D53
hypocenterC53#10787
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
4. C2×#16810393C2semidirect
5. #20274454C2semidirect
6. #20274442C2semidirect
7. C2×C53×H(3,ℤ/4)C2semidirect
12. #16811389C4semidirect
21. C2×#13133145C4semidirect
22. #16810631C4semidirect
23. #16810627C4semidirect
307. C2×H(3,ℤ/4)D53semidirect
342. C53#10787semidirect

Maximal subgroups

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

subgroupindex
C2×#168113892
C2×#168106412
#202832322
C2×#168103932
#202744542
#202744422
C2×C53×H(3,ℤ/4)2
#1078753