Finite Group Databasebeta

← #28262494

Subgroups — #28262494

Order 13568.

2020 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288635C211yesmaximal, characteristic, complemented
2#20286886C214yesmaximal, complemented
3#20286811C214yesmaximal, complemented
4#20286766C214yesmaximal, complemented
5#20286691C214yesmaximal, complemented
6C2×#16810991C211yesmaximal, characteristic, complemented
7#20284678C212yesmaximal, complemented
8#20284537C211yesmaximal, characteristic, complemented
9#20284402C212yesmaximal, complemented
10#20284198C211yesmaximal, characteristic, complemented
11#20284030C212yesmaximal, complemented
12#20283513C211yesmaximal, characteristic, complemented
13#20283433C211yesmaximal, characteristic, complemented
14#20283248C212yesmaximal, complemented
15C53×#3499C211yesmaximal, characteristic, complemented
16C2×#13133248C2212yes—
17C2×#13133237C2211yescharacteristic
18C2×#13133224C2212yes—
19#16811319C2214yes—
20#16811316—24no—
21#16811313C2214yes—
22#16811234—21no—
23#16811227—21no—
24#16811201C2212yes—
25#16811152—22no—
26C2×#13133209C2211yescharacteristic
27#16811106C2212yes—
28C4×#11719945C2211yescharacteristic
29#16810991C2214yes—
30#16810991C2214yes—
31#16810961C2214yes—
32#16810961C2214yes—
33#16810956C2214yes—
34#16810951C2214yes—
35#16810930C2214yes—
36#16810923C2214yes—
37#16810879C2214yes—
38#16810871C2214yes—
39#16810862C2214yes—
40#16810854C2214yes—
41#16810830C2214yes—
42#16810825C2214yes—
43#16810785C2214yes—
44#16810779C2214yes—
45#16810768C2214yes—
46#16810762C2214yes—
47#16810733C2212yes—
48#16810728C2212yes—
49#16810723C2212yes—
50#16810721C2212yes—
51#16810695C2214yes—
52#16810695C2214yes—
53#16810666C2212yes—
54#16810659C2211yescharacteristic
55#16810647C2211yescharacteristic
56#16810632C2212yes—
57#16810632C2214yes—
58#16810624C2214yes—
59#16810624C2212yes—
60#16810619C2214yes—
61#16810619C2211yescharacteristic
62#16810617C2211yescharacteristic
63C2×#13133141C2212yes—
64C2×#13133141C2214yes—
65C2×#13133141C2212yes—
66#16810590C2212yes—
67C2×#13133128C2211yescharacteristic
68C2×2−1+4×C53C2212yes—
69D4×Q8×C53C2214yes—
70C53×#547—24no—
71C53×#547C2214yes—
72C53×#536—22no—
73C53×#536C2212yes—
74C53×#532—21no—
75C53×#532C2211yescharacteristic
76C2×C4⋊Q8×C53—21no—
77C2×C4⋊Q8×C53C2211yescharacteristic
78C4×#41×C53C2211yescharacteristic
79C2×#11719945C2311yescharacteristic
80#13133248—44no—
81#13133237—24no—
82C2×#11719925C2311yescharacteristic
83C2×#11719925C2314yes—
84C2×#11719923C2311yescharacteristic
85C2×#11719924C2311yescharacteristic
86C2×#11719924C2314yes—
87C2×#11719922C2311yescharacteristic
88#13133224—44no—
89C2×#11719917C2312yes—
90C2×#11719917C2314yes—
91C22×#88322C2311yescharacteristic
92C22×#88322C2311yescharacteristic
93C22×#88322C2314yes—
94#13133212—24no—
95#13133210—24no—
96#13133209—21no—
97#13133209—21no—
98#13133209C2314yes—
99#13133207—24no—
100#13133199—24no—
101#13133198—24no—
102#13133196C2314yes—
103#13133193—22no—
104#13133191C2314yes—
105#13133189—24no—
106#13133189—24no—
107#13133188—22no—
108Q8×Dic53—22no—
109C4×Dic106C2312yes—
110C4×D212C2312yes—
111D4×Dic53—24no—
112D4×Dic53—22no—
113C4×C22⋊D53—22no—
114#13133177—24no—
115#13133176C2314yes—
116#13133176C2311yescharacteristic
117#13133174—21no—
118#13133173—24no—
119C2×C4⋊Dic53C2314yes—
120C2×C4⋊Dic53C2312yes—
121C2×#11719919—22no—
122C42×D53—21no—
123#13133150C2314yes—
124#13133150C2312yes—
125#13133148C2314yes—
126#13133148C2312yes—
127C4⋊#88322C2314yes—
128#13133141C2314yes—
129#13133141C2314yes—
130#13133141C2314yes—
131#13133141C2314yes—
132#13133141C2314yes—
133#13133141C2314yes—
134#13133141C2314yes—
135#13133141C2312yes—
136#13133140C2312yes—
137#13133138C2314yes—
138#13133138C2312yes—
139#13133137C2312yes—
140C22⋊#88322C2314yes—
141C22⋊#88322C2314yes—
142#13133128C2312yes—
143#13133128C2312yes—
1442−1+4×C53—28no—
1452−1+4×C53—44no—
146C2×#41×C53—22no—
147C2×#41×C53—24no—
148C2×#41×C53C2314yes—
149C2×#41×C53C2311yescharacteristic
150C2×#41×C53C2312yes—
151C22×Q8×C53—22no—
152C22×Q8×C53C2312yes—
153C22×Q8×C53C2314yes—
154C4⋊Q8×C53—24no—
155C4⋊Q8×C53—21no—
156C4⋊Q8×C53—21no—
157C4⋊Q8×C53—22no—
158C4⋊Q8×C53D411yescharacteristic
159C4⋊Q8×C53C2314yes—
160#127×C53D411yescharacteristic
161#124×C53—22no—
162#124×C53D412yes—
163#123×C53—24no—
164#123×C53—24no—
165#123×C53—24no—
166C22⋊Q8×C53—24no—
167C22⋊Q8×C53—24no—
168C22⋊Q8×C53C2314yes—
169C22⋊Q8×C53C2314yes—
170#121×C53—24no—
171#121×C53D414yes—
172C4×Q8×C53C2312yes—
173C4×D4×C53—24no—
174C4×D4×C53—22no—
175C4×D4×C53C2312yes—
176C4×D4×C53C2314yes—
177#117×C53—21no—
178#117×C53C2311yescharacteristic
179C2×C4⋊C4×C53—24no—
180C2×C4⋊C4×C53—22no—
181C2×C42×C53—21no—
182C2×C42×C53C2311yescharacteristic
183#41×C53—24no—
184#41×C53—22no—
185#41×C53—24no—
186#41×C53—28no—
187#41×C53—44no—
188#41×C53—48no—
189#41×C53—416no—
190#41×C53#4114yes—
191C2×Q8×C53—24no—
192C2×Q8×C53—22no—
193C2×Q8×C53—28no—
194C2×Q8×C53—24no—
195C2×Q8×C53—24no—
196C2×Q8×C53—44no—
197C2×Q8×C53—48no—
198C2×Q8×C53C2×D412yes—
199C2×Q8×C53C2414yes—
200C2×Q8×C53C2411yescharacteristic
201C2×D4×C53—21no—
202C2×D4×C53—24no—
203C2×D4×C53C2×D414yes—
204C2×D4×C53C2×D412yes—
205C2×D4×C53C2×D414yes—
206C2×D4×C53C2411yescharacteristic
207C22×C4×C53—21no—
208C22×C4×C53—22no—
209C22×C4×C53—24no—
210C22×C4×C53—22no—
211C22×C4×C53—24no—
212C22×C4×C53C2×D412yes—
213C22×C4×C53C2414yes—
214C22×C4×C53C2411yescharacteristic
215C2×C22⋊D53—21no—
216C2×C22⋊D53—24no—
217C22×Dic53—22no—
218C22×Dic53—24no—
219#11719945—22no—
220#11719945—22no—
221C2×D212C2411yescharacteristic
222C2×C4×D53—21no—
223C2×C4×D53—22no—
224C2×Dic106C2411yescharacteristic
225C4⋊C4×C53—24no—
226C4⋊C4×C53—24no—
227C4⋊C4×C53—24no—
228C4⋊C4×C53—24no—
229C4⋊C4×C53—22no—
230C4⋊C4×C53C2×D414yes—
231C4⋊C4×C53C2412yes—
232C22⋊C4×C53—21no—
233C22⋊C4×C53—24no—
234C22⋊C4×C53—24no—
235C22⋊C4×C53—22no—
236C22⋊C4×C53C2×D414yes—
237C22⋊C4×C53C2×D414yes—
238C42×C53—21no—
239C42×C53—22no—
240C42×C53C2×D411yescharacteristic
241C42×C53C2×D411yescharacteristic
242C42×C53C2412yes—
243C22⋊Dic53—24no—
244C22⋊Dic53—21no—
245C22⋊Dic53—24no—
246#11719925—22no—
247#11719925—44no—
248#11719924—22no—
249#11719924—44no—
250#11719923—22no—
251#11719922—22no—
252#11719921—22no—
253#11719921—24no—
254C4⋊Dic53—22no—
255C4⋊Dic53—24no—
256C4⋊Dic53C2412yes—
257C4⋊Dic53C2414yes—
258#11719919—24no—
259#11719919—24no—
260#11719919—24no—
261#11719919—22no—
262C4×Dic53—22no—
263C4×Dic53—21no—
264C4×Dic53—24no—
265#11719917—24no—
266#11719917—44no—
267C2×#88322—22no—
268C2×#88322—22no—
269C2×#88322—44no—
270C2×#88322C2411yescharacteristic
271C2×#88322C2411yescharacteristic
272C2×#88322C2412yes—
273C2×#88322C2414yes—
274C23×C53—21no—
275C23×C53C22×D414yes—
276C23×C53C22×D411yescharacteristic
277C22×D53—21no—
278Q8×C53—24no—
279Q8×C53—21no—
280Q8×C53—24no—
281Q8×C53—24no—
282Q8×C53—48no—
283Q8×C53—44no—
284Q8×C53C2×#4112yes—
285D4×C53—24no—
286D4×C53—28no—
287D4×C53—24no—
288D4×C53—21no—
289D4×C53—44no—
290D4×C53—48no—
291D4×C53C2×#4112yes—
292C2×C4×C53—21no—
293C2×C4×C53—24no—
294C2×C4×C53—24no—
295C2×C4×C53—24no—
296C2×C4×C53—24no—
297C2×C4×C53—22no—
298C2×C4×C53—42no—
299C2×C4×C53—48no—
300C2×C4×C53—48no—
301C2×C4×C53—44no—
302C2×C4×C53—44no—
303C2×C4×C53C22×D412yes—
304C2×C4×C53C22×D412yes—
305C2×C4×C53C22×D414yes—
306C2×C4×C53C22×D411yescharacteristic
307C2×C4×C53C2×#4112yes—
308C2×C4×C53H(5,𝔽2)12yes—
309C2×C4×C53C2511yescharacteristic
310C22⋊D53—42no—
311C22⋊D53—84no—
312C2×Dic53—21no—
313C2×Dic53—22no—
314C2×Dic53—24no—
315C2×Dic53—84no—
316C2×Dic53—82no—
317D212—21no—
318D212—21no—
319C4×D53—42no—
320C4×D53—82no—
321Dic106—21no—
322Dic106—21no—
323#88322—21no—
324#88322—21no—
325#88322—22no—
326#88322—24no—
327#61224—531nomaximal, Sylow
328C22×C53—22no—
329C22×C53—24no—
330C22×C53—24no—
331C22×C53—22no—
332C22×C53—24no—
333C22×C53—42no—
334C22×C53#54612yes—
335C22×C53C23×D411yescharacteristic
336C2×D53—42no—
337C2×D53—41no—
338C4×C53—22no—
339C4×C53—22no—
340C4×C53—22no—
341C4×C53—24no—
342C4×C53—42no—
343C4×C53—44no—
344C4×C53#54612yes—
345C4×C53C22×#4111yescharacteristic
346C4×C53C2×H(5,𝔽2)11yescharacteristic
347Dic53—42no—
348Dic53—44no—
349Dic53—41no—
350#3512—531no—
351#3499D5311yescharacteristic, complemented
352#3320—534no—
353#3315—534no—
354#3312—534no—
355#3307—534no—
356C2×#491—531no—
357#3097—532no—
358#3080—531no—
359#3063—532no—
360#3039—531no—
361#3013—532no—
362#2943—531no—
363#2932—531no—
364#2898—532no—
365C2×C53—22no—
366C2×C53—44no—
367C2×C53—42no—
368C2×C53C2×#54611yescharacteristic
369C2×C53#358612yes—
370D53—81no—
371C2×2−1+4C2×D5312yes—
372C2×#137—532no—
373C2×#135—531no—
374C2×#131—532no—
375#553—534no—
376#553—1064no—
377#552—534no—
378D4×Q8C2×D5314yes—
379#547—24no—
380#547C2×D5314yes—
381#541—1061no—
382#540—1061no—
383#537—532no—
384#536—22no—
385#536C2×D5312yes—
386#532—21no—
387#532C2×D5311yescharacteristic
388C2×C4⋊Q8—21no—
389C2×C4⋊Q8C2×D5311yescharacteristic
390#529—1062no—
391C2×#125—531no—
392#520—532no—
393C4×#41—531no—
394C4×#41C2×D5311yescharacteristic
395#491—534no—
396#491—534no—
397#485—534no—
398#485—534no—
399#484—534no—
400#483—534no—
401#479—534no—
402#478—534no—
403#470—534no—
404#468—534no—
405#467—534no—
406#465—534no—
407#462—534no—
408#461—534no—
409#452—534no—
410#451—534no—
411#449—534no—
412#448—534no—
413#442—532no—
414#441—532no—
415#440—532no—
416#440—532no—
417#435—534no—
418#435—534no—
419#428—532no—
420#427—531no—
421#423—531no—
422#419—532no—
423#419—534no—
424#417—532no—
425#417—534no—
426#416—531no—
427#416—534no—
428#416—531no—
429C2×#103—532no—
430C2×#103—532no—
431C2×#103—534no—
432#408—532no—
433C2×#97—531no—
434C53#6122411yescharacteristic, complemented, Sylow
4352−1+4—28no—
4362−1+4—44no—
437C2×#41—22no—
438C2×#41—24no—
439C2×#41—531no—
440C2×#41C22×D5314yes—
441C2×#41C22×D5311yescharacteristic
442C2×#41C22×D5312yes—
443C22×Q8—22no—
444C22×Q8C22×D5312yes—
445C22×Q8C22×D5314yes—
446#137—2124no—
447#135—1064no—
448C2×Q16—531no—
449C2×Q16—534no—
450C2×SD16—531no—
451C2×SD16—531no—
452C2×SD16—534no—
453C2×D8—531no—
454#131—2124no—
455C2×M16—532no—
456C2×M16—534no—
457C22×C8—531no—
458C22×C8—531no—
459C22×C8—534no—
460C4⋊Q8—22no—
461C4⋊Q8—24no—
462C4⋊Q8—21no—
463C4⋊Q8—21no—
464C4⋊Q8C22⋊D5311yescharacteristic
465C4⋊Q8C22×D5314yes—
466#127C22⋊D5311yescharacteristic
467#126—1064no—
468#126—1064no—
469#125—534no—
470#125—1064no—
471#125—1061no—
472#125—1061no—
473#124—22no—
474#124C22⋊D5312yes—
475#123—24no—
476#123—24no—
477#123—24no—
478#123—1064no—
479#123—1064no—
480C22⋊Q8—24no—
481C22⋊Q8—24no—
482C22⋊Q8—534no—
483C22⋊Q8—1062no—
484C22⋊Q8C22×D5314yes—
485C22⋊Q8C22×D5314yes—
486#121—24no—
487#121—534no—
488#121—1062no—
489#121—1064no—
490#121—1064no—
491#121C22⋊D5314yes—
492C4×Q8—532no—
493C4×Q8—1062no—
494C4×Q8C22×D5312yes—
495C4×D4—24no—
496C4×D4—22no—
497C4×D4—532no—
498C4×D4—1062no—
499C4×D4—1064no—
500C4×D4—1062no—
501C4×D4—1064no—
502C4×D4C22×D5312yes—
503C4×D4C22×D5314yes—
504#117—21no—
505#117—531no—
506#117—534no—
507#117—1064no—
508#117—1061no—
509#117C22×D5311yescharacteristic
510C2×C4⋊C4—22no—
511C2×C4⋊C4—24no—
512C2×C4⋊C4—532no—
513C2×C4⋊C4—534no—
514C2×C4⋊C4—1062no—
515C2×C42—21no—
516C2×C42—1061no—
517C2×C42C22×D5311yescharacteristic
518#107—532no—
519#107—534no—
520#106—532no—
521#106—534no—
522C4⋊C8—534no—
523#103—532no—
524#103—534no—
525#103—534no—
526#103—534no—
527#103—534no—
528#103—534no—
529#103—534no—
530#103—534no—
531#103—532no—
532#102—532no—
533#102—534no—
534#102—532no—
535C22⋊C8—534no—
536C22⋊C8—534no—
537#97—532no—
538#97—532no—
539#41—24no—
540#41—24no—
541#41—22no—
542#41—28no—
543#41—44no—
544#41—48no—
545#41—416no—
546#41—1062no—
547#41—1062no—
548#41#1171994714yes—
549C2×Q8—24no—
550C2×Q8—22no—
551C2×Q8—28no—
552C2×Q8—24no—
553C2×Q8—24no—
554C2×Q8—44no—
555C2×Q8—48no—
556C2×Q8—531no—
557C2×Q8C2×C22⋊D5312yes—
558C2×Q8C23×D5314yes—
559C2×Q8C23×D5311yescharacteristic
560C2×D4—21no—
561C2×D4—24no—
562C2×D4—531no—
563C2×D4—1064no—
564C2×D4—1061no—
565C2×D4C2×C22⋊D5314yes—
566C2×D4C2×C22⋊D5312yes—
567C2×D4C2×C22⋊D5314yes—
568C2×D4C23×D5311yescharacteristic
569C22×C4—21no—
570C22×C4—22no—
571C22×C4—24no—
572C22×C4—22no—
573C22×C4—24no—
574C22×C4—1062no—
575C22×C4—1064no—
576C22×C4—1062no—
577C22×C4—1061no—
578C22×C4C2×C22⋊D5312yes—
579C22×C4C23×D5314yes—
580C22×C4C23×D5311yescharacteristic
581Q16—1062no—
582Q16—2124no—
583SD16—1062no—
584SD16—1062no—
585SD16—2124no—
586D8—1062no—
587M16—1064no—
588M16—2124no—
589C2×C8—531no—
590C2×C8—531no—
591C2×C8—532no—
592C2×C8—534no—
593C2×C8—1062no—
594C2×C8—1062no—
595C2×C8—2124no—
596C4⋊C4—24no—
597C4⋊C4—24no—
598C4⋊C4—24no—
599C4⋊C4—24no—
600C4⋊C4—22no—
601C4⋊C4—532no—
602C4⋊C4—534no—
603C4⋊C4—1062no—
604C4⋊C4—1064no—
605C4⋊C4—1064no—
606C4⋊C4—1064no—
607C4⋊C4—1062no—
608C4⋊C4—1064no—
609C4⋊C4C2×C22⋊D5314yes—
610C4⋊C4C23×D5312yes—
611C22⋊C4—21no—
612C22⋊C4—24no—
613C22⋊C4—24no—
614C22⋊C4—22no—
615C22⋊C4—1061no—
616C22⋊C4—1064no—
617C22⋊C4—1064no—
618C22⋊C4—1062no—
619C22⋊C4—1064no—
620C22⋊C4C2×C22⋊D5314yes—
621C22⋊C4C2×C22⋊D5314yes—
622C42—21no—
623C42—22no—
624C42—1061no—
625C42—1062no—
626C42—1064no—
627C42C2×C22⋊D5311yescharacteristic
628C42C2×C22⋊D5311yescharacteristic
629C42C23×D5312yes—
630C23—21no—
631C23—1061no—
632C23C22×C22⋊D5314yes—
633C23C22×C22⋊D5311yescharacteristic
634Q8—24no—
635Q8—21no—
636Q8—24no—
637Q8—24no—
638Q8—48no—
639Q8—44no—
640Q8—1061no—
641Q8—1061no—
642Q8C2×#1171994712yes—
643D4—24no—
644D4—28no—
645D4—24no—
646D4—21no—
647D4—44no—
648D4—48no—
649D4—1061no—
650D4—1061no—
651D4—4244no—
652D4—2122no—
653D4C2×#1171994712yes—
654C2×C4—21no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—22no—
659C2×C4—24no—
660C2×C4—42no—
661C2×C4—48no—
662C2×C4—48no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—1062no—
666C2×C4—1061no—
667C2×C4—1064no—
668C2×C4—4242no—
669C2×C4—4244no—
670C2×C4—4242no—
671C2×C4—2122no—
672C2×C4C22×C22⋊D5312yes—
673C2×C4C22×C22⋊D5312yes—
674C2×C4C22×C22⋊D5314yes—
675C2×C4C22×C22⋊D5311yescharacteristic
676C2×C4C2×#1171994712yes—
677C2×C4#1313326212yes—
678C2×C4C24×D5311yescharacteristic
679C8—1061no—
680C8—1061no—
681C8—1062no—
682C8—1064no—
683C22—22no—
684C22—22no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—42no—
689C22—2121no—
690C22—2122no—
691C22#1681126712yes—
692C22C23×C22⋊D5311yescharacteristic
693C4—22no—
694C4—24no—
695C4—22no—
696C4—22no—
697C4—44no—
698C4—42no—
699C4—2121no—
700C4—2122no—
701C4—2124no—
702C4#1681126712yes—
703C4C22×#1171994711yescharacteristic
704C4C2×#1313326211yescharacteristic
705C2—22no—
706C2—44no—
707C2—42no—
708C2—4241no—
709C2C2×#1681126711yescharacteristic
710C2#2028950212yes—
subgroupG/H
centerC22C23×C22⋊D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC53×#3499C2
socleC22×C53C23×D4
hypercenter#3499D53
hypocenterC53#61224
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #532 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #20288635C2semidirect
2. #20286886C2semidirect
3. #20286811C2semidirect
4. #20286766C2semidirect
5. #20286691C2semidirect
6. C2×#16810991C2semidirect
7. #20284678C2semidirect
8. #20284537C2semidirect
9. #20284402C2semidirect
10. #20284198C2semidirect
11. #20284030C2semidirect
12. #20283513C2semidirect
13. #20283433C2semidirect
14. #20283248C2semidirect
15. C53×#3499C2semidirect
351. #3499D53semidirect
434. C53#61224semidirect

Maximal subgroups

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

subgroupindex
#202886352
#202868862
#202868112
#202867662
#202866912
C2×#168109912
#202846782
#202845372
#202844022
#202841982
#202840302
#202835132
#202834332
#202832482
C53×#34992
#6122453