Finite Group Databasebeta

← #27384983

Subgroups — #27384983

Order 13568.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287918C211yesmaximal, characteristic
2C2×#16810665C211yesmaximal, characteristic, complemented
3C2×#16810624C211yesmaximal, characteristic
4#20277338C211yesmaximal, characteristic, complemented
5#20274950C214yesmaximal, complemented
6#20274943C214yesmaximal, complemented
7#20274453C211yesmaximal, characteristic, complemented
8#20274425C211yesmaximal, characteristic, complemented
9C2×C53×#421C211yesmaximal, characteristic
10C22×#11719917C2211yescharacteristic
11#16811106—21no—
12#16811106—21no—
13#16811106C2212yes—
14#16811106C2212yes—
15#16811101—21no—
16#16811098—22no—
17#16811098—21no—
18#41×Dic53—22no—
19#41×Dic53—22no—
20C2×#13133176—22no—
21C2×#13133176C2211yescharacteristic
22C2×#13133176C2211yescharacteristic
23C22×C4⋊Dic53C2211yescharacteristic
24#16810665—22no—
25#16810665C2214yescomplemented
26C2×#13133151—21no—
27C2×#13133149—21no—
28#16810624—22no—
29#16810624C2214yes—
30C2×#13133141—21no—
31C2×#13133138—21no—
32#16810522—22no—
33#16810498—22no—
34#16810493C2212yescomplemented
35#16810489C2212yescomplemented
36#16810391C2212yescomplemented
37#16810391C2212yescomplemented
38#16810383C2214yescomplemented
39C22×#41×C53C2211yescharacteristic
40C22×M16×C53C2211yescharacteristic
41C2×#117×C53C2211yescharacteristic
42C53×#421C418yescomplemented
43C2×#104×C53C2214yes—
44C23×Dic53—21no—
45C2×#11719917—21no—
46C2×#11719917—22no—
47C2×#11719917—22no—
48C2×#11719917C2312yes—
49C22×#88322—21no—
50Q8×Dic53—22no—
51Q8×Dic53—22no—
52Q8×Dic53—22no—
53Q8×Dic53—22no—
54D4×Dic53—22no—
55D4×Dic53—22no—
56D4×Dic53—24no—
57D4×Dic53—24no—
58D4×Dic53—22no—
59D4×Dic53—22no—
60D4×Dic53—24no—
61D4×Dic53—24no—
62#13133176—24no—
63#13133176—24no—
64#13133176—22no—
65#13133176—22no—
66#13133176—21no—
67#13133176—21no—
68#13133176—21no—
69#13133176—21no—
70#13133176—21no—
71#13133176—21no—
72#13133176D412yes—
73#13133176C2312yes—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53—22no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—24no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—21no—
81C2×C4⋊Dic53—21no—
82C2×C4⋊Dic53D411yescharacteristic
83C2×C4⋊Dic53D411yescharacteristic
84C2×C4⋊Dic53C2312yes—
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C22⋊Dic53—24no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53D412yes—
90C2×C4×Dic53—24no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53D412yes—
93#13133151—24no—
94#13133149—24no—
95#13133141—24no—
96#13133138—24no—
97#13133124—24no—
98C2×#41×C53—21no—
99C2×#41×C53—21no—
100C2×#41×C53—22no—
101C2×#41×C53—22no—
102C2×#41×C53C2×C414yes—
103C2×#41×C53D412yes—
104C2×#41×C53Q814yes—
105C2×#41×C53C2312yes—
106C22×Q8×C53D411yescharacteristic
107C22×D4×C53—21no—
108C22×D4×C53D411yescharacteristic
109C23×C4×C53—21no—
110C23×C4×C53C2311yescharacteristic
111C2×M16×C53—21no—
112C2×M16×C53—22no—
113C2×M16×C53C2×C414yescomplemented
114C2×M16×C53C2312yes—
115C22×C8×C53—21no—
116#117×C53—22no—
117#117×C53C2×C414yescomplemented
118C2×C4⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C42×C53C2312yes—
121#104×C53C2×C4116yescomplemented
122C24×C53—21no—
123C24×C53C2×D411yescharacteristic
124#41×C53—24no—
125#41×C53—24no—
126#41×C53—24no—
127#41×C53—24no—
128#41×C53—24no—
129#41×C53—22no—
130#41×C53—22no—
131#41×C53C4⋊C418yescomplemented
132#41×C53C22×C414yes—
133#41×C53C2×Q814yes—
134C2×Q8×C53—21no—
135C2×Q8×C53—22no—
136C2×Q8×C53C4⋊C414yescomplemented
137C2×Q8×C53C2×D412yes—
138C2×D4×C53—24no—
139C2×D4×C53—22no—
140C2×D4×C53—24no—
141C2×D4×C53—22no—
142C2×D4×C53—21no—
143C2×D4×C53—22no—
144C2×D4×C53C4⋊C414yescomplemented
145C2×D4×C53C2×D412yes—
146C22×C4×C53—21no—
147C22×C4×C53—24no—
148C22×C4×C53—22no—
149C22×C4×C53—24no—
150C22×C4×C53—22no—
151C22×C4×C53—22no—
152C22×C4×C53C22×C412yes—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×Q812yes—
159C22×C4×C53#4112yes—
160C22×C4×C53#4112yes—
161C22×C4×C53C2411yescharacteristic
162C22×Dic53—24no—
163C22×Dic53—24no—
164C22×Dic53—21no—
165C22×Dic53—21no—
166C22×Dic53—22no—
167C22×Dic53—41no—
168C22×Dic53—41no—
169C22×Dic53—82no—
170C22×Dic53C2×D412yes—
171C22×Dic53C2×D412yes—
172M16×C53—24no—
173M16×C53—22no—
174M16×C53C22×C414yes—
175C2×C8×C53—24no—
176C2×C8×C53—22no—
177C4⋊C4×C53—24no—
178C22⋊C4×C53—24no—
179C42×C53—22no—
180C42×C53C22×C414yes—
181C22⋊Dic53—24no—
182C22⋊Dic53—24no—
183C22⋊Dic53—22no—
184C22⋊Dic53—22no—
185C22⋊Dic53—22no—
186C22⋊Dic53—22no—
187C4⋊Dic53—24no—
188C4⋊Dic53—24no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—22no—
196C4⋊Dic53—22no—
197C4⋊Dic53—41no—
198C4⋊Dic53—41no—
199C4⋊Dic53—84no—
200C4⋊Dic53C2×D412yes—
201C4⋊Dic53C2×D412yes—
202C4×Dic53—24no—
203C4×Dic53—24no—
204C4×Dic53—22no—
205C4×Dic53—22no—
206C4×Dic53—22no—
207C4×Dic53—22no—
208#11719917—22no—
209#11719917—22no—
210#11719917—42no—
211C2×#88322—22no—
212C2×#88322—42no—
213C23×C53—22no—
214C23×C53—24no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—81no—
218C23×C53—81no—
219C23×C53C4×D412yes—
220C23×C53#12011yescharacteristic
221C23×C53#12011yescharacteristic
222C23×C53C22⋊Q812yes—
223C23×C53C22×D411yescharacteristic
224Q8×C53—22no—
225Q8×C53—24no—
226Q8×C53C2×C4⋊C414yes—
227D4×C53—24no—
228D4×C53—24no—
229D4×C53—28no—
230D4×C53—22no—
231D4×C53—24no—
232D4×C53C2×C4⋊C414yes—
233C2×C4×C53—22no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—28no—
238C2×C4×C53—24no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53C2×C4⋊C412yes—
242C2×C4×C53C2×C4⋊C412yes—
243C2×C4×C53C4×D412yes—
244C2×C4×C53C4×D412yes—
245C2×C4×C53C4×D412yes—
246C2×C4×C53#12011yescharacteristic
247C2×C4×C53#12011yescharacteristic
248C2×C4×C53C22⋊Q812yes—
249C2×C4×C53C22⋊Q812yes—
250C2×C4×C53C22⋊Q812yes—
251C2×C4×C53#12312yes—
252C2×C4×C53#12312yes—
253C2×C4×C53C23×C411yescharacteristic
254C2×C4×C53C22×D411yescharacteristic
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C22×Q811yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—44no—
266C2×Dic53—42no—
267C2×Dic53—81no—
268C2×Dic53—81no—
269C2×Dic53—84no—
270C2×Dic53—84no—
271C2×Dic53—82no—
272C2×Dic53—82no—
273C8×C53—24no—
274#88322—42no—
275#20865—531nomaximal, Sylow
276C22×C53—21no—
277C22×C53—21no—
278C22×C53—21no—
279C22×C53—21no—
280C22×C53—21no—
281C22×C53—21no—
282C22×C53—24no—
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—42no—
287C22×C53—84no—
288C22×C53#38512yes—
289C22×C53#38512yes—
290C22×C53C2×C4×D411yescharacteristic
291C22×C53C2×#12011yescharacteristic
292C22×C53C2×C22⋊Q811yescharacteristic
293C4×C53—24no—
294C4×C53—44no—
295C4×C53#38512yes—
296C4×C53#38512yes—
297C4×C53C22×C4⋊C411yescharacteristic
298C4×C53C2×C4×D411yescharacteristic
299C4×C53C2×C22⋊Q811yescharacteristic
300C4×C53C2×#12311yescharacteristic
301Dic53—42no—
302Dic53—42no—
303Dic53—42no—
304Dic53—42no—
305Dic53—82no—
306#3428—531no—
307C2×#428—531no—
308C2×#421D5311yescharacteristic
309C2×#417—531no—
310#2317—531no—
311#1860—534no—
312#1859—534no—
313#1742—531no—
314#1733—531no—
315C2×C53—21no—
316C2×C53—22no—
317C2×C53—21no—
318C2×C53—22no—
319C2×C53—44no—
320C2×C53C2×#38511yescharacteristic
321C2×C53#302111yescharacteristic
322C2×C53#302211yescharacteristic
323C22×#41C2×D5311yescharacteristic
324C22×M16—531no—
325C22×M16C2×D5311yescharacteristic
326#520—532no—
327#520—532no—
328#520—1061no—
329#520—1061no—
330#519—1061no—
331#518—1062no—
332#518—1061no—
333C4×#41—1062no—
334C4×#41—1062no—
335C2×#117—531no—
336C2×#117—531no—
337C2×#117—1062no—
338C2×#117C2×D5311yescharacteristic
339C22×C4⋊C4—531no—
340#428—534no—
341#428—1062no—
342C2×#107—1061no—
343C2×#106—1061no—
344#421Dic5318yescomplemented
345C2×#104C2×D5314yes—
346#417—534no—
347#417—1062no—
348C2×#103—1061no—
349C2×#102—1061no—
350#389—1062no—
351#385—1062no—
352#384—532no—
353#382—532no—
354#343—532no—
355#343—532no—
356#339—534no—
357C53#2086511yescharacteristic, complemented, Sylow
358C2×#41—21no—
359C2×#41—21no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41Dic10614yes—
363C2×#41D21212yes—
364C2×#41C2×Dic5314yes—
365C2×#41C22×D5312yes—
366C22×Q8D21211yescharacteristic
367C22×D4—21no—
368C22×D4D21211yescharacteristic
369C23×C4—21no—
370C23×C4—1061no—
371C23×C4C22×D5311yescharacteristic
372C2×M16—21no—
373C2×M16—22no—
374C2×M16—532no—
375C2×M16—1061no—
376C2×M16—1062no—
377C2×M16—1062no—
378C2×M16C2×Dic5314yescomplemented
379C2×M16C22×D5312yes—
380C22×C8—21no—
381C22×C8—1061no—
382C4×Q8—1062no—
383C4×Q8—1062no—
384C4×Q8—1062no—
385C4×Q8—1062no—
386C4×D4—1064no—
387C4×D4—1064no—
388C4×D4—1064no—
389C4×D4—1064no—
390C4×D4—1062no—
391C4×D4—1062no—
392C4×D4—1062no—
393C4×D4—1062no—
394#117—22no—
395#117—532no—
396#117—532no—
397#117—1062no—
398#117—1062no—
399#117—1064no—
400#117—1064no—
401#117—1061no—
402#117—1061no—
403#117—1061no—
404#117—1061no—
405#117—1061no—
406#117—1061no—
407#117C2×Dic5314yescomplemented
408C2×C4⋊C4—21no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—532no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1062no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1064no—
421C2×C4⋊C4—1061no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—532no—
424C2×C22⋊C4—1064no—
425C2×C22⋊C4—1061no—
426C2×C42—532no—
427C2×C42—1064no—
428C2×C42—1061no—
429C2×C42C22×D5312yes—
430#107—1064no—
431#106—1064no—
432#104C2×Dic53116yescomplemented
433#103—1064no—
434#102—1064no—
435#95—1064no—
436C24—21no—
437C24D4×D5311yescharacteristic
438#41—24no—
439#41—24no—
440#41—24no—
441#41—24no—
442#41—24no—
443#41—22no—
444#41—22no—
445#41C4⋊Dic5318yescomplemented
446#41C2×Dic10614yes—
447#41C22×Dic5314yes—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8C4⋊Dic5314yescomplemented
451C2×Q8C2×D21212yes—
452C2×D4—24no—
453C2×D4—22no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4C4⋊Dic5314yescomplemented
459C2×D4C2×D21212yes—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—532no—
467C22×C4—532no—
468C22×C4—1064no—
469C22×C4—1064no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1062no—
473C22×C4—4242no—
474C22×C4—2121no—
475C22×C4—2121no—
476C22×C4C2×Dic10612yes—
477C22×C4C2×D21211yescharacteristic
478C22×C4C2×D21211yescharacteristic
479C22×C4D4×D5311yescharacteristic
480C22×C4D4×D5311yescharacteristic
481C22×C4D4×D5311yescharacteristic
482C22×C4#1171994712yes—
483C22×C4#1171994712yes—
484C22×C4C22×Dic5312yes—
485C22×C4C23×D5311yescharacteristic
486M16—24no—
487M16—22no—
488M16—1062no—
489M16—1062no—
490M16—2122no—
491M16C22×Dic5314yes—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—1062no—
495C2×C8—2122no—
496C4⋊C4—24no—
497C4⋊C4—532no—
498C4⋊C4—532no—
499C4⋊C4—1064no—
500C4⋊C4—1064no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1062no—
508C4⋊C4—1062no—
509C4⋊C4—4244no—
510C4⋊C4—2121no—
511C4⋊C4—2121no—
512C22⋊C4—24no—
513C22⋊C4—1064no—
514C22⋊C4—1064no—
515C22⋊C4—1062no—
516C22⋊C4—1062no—
517C22⋊C4—1062no—
518C22⋊C4—1062no—
519C42—22no—
520C42—1064no—
521C42—1064no—
522C42—1062no—
523C42—1062no—
524C42—1062no—
525C42—1062no—
526C42C22×Dic5314yes—
527C23—22no—
528C23—24no—
529C23—41no—
530C23—41no—
531C23—81no—
532C23—81no—
533C23D4×Dic5312yes—
534C23#1313318711yescharacteristic
535C23#1313318711yescharacteristic
536C23#1313319412yes—
537C23C2×D4×D5311yescharacteristic
538Q8—24no—
539Q8—22no—
540Q8C2×C4⋊Dic5314yes—
541D4—28no—
542D4—24no—
543D4—24no—
544D4—24no—
545D4—22no—
546D4C2×C4⋊Dic5314yes—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—4242no—
560C2×C4—4242no—
561C2×C4—4241no—
562C2×C4—4241no—
563C2×C4—4244no—
564C2×C4—4244no—
565C2×C4—2124no—
566C2×C4—2122no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4C2×C4⋊Dic5312yes—
570C2×C4C2×C4⋊Dic5312yes—
571C2×C4D4×Dic5312yes—
572C2×C4D4×Dic5312yes—
573C2×C4D4×Dic5312yes—
574C2×C4#1313318711yescharacteristic
575C2×C4#1313318711yescharacteristic
576C2×C4#1313319412yes—
577C2×C4#1313319412yes—
578C2×C4#1313319412yes—
579C2×C4#1313320112yes—
580C2×C4#1313320112yes—
581C2×C4C23×Dic5311yescharacteristic
582C2×C4C2×D4×D5311yescharacteristic
583C2×C4C22×D21211yescharacteristic
584C2×C4C22×Dic10611yescharacteristic
585C2×C4C2×#1171994711yescharacteristic
586C2×C4C2×#1171994711yescharacteristic
587C8—24no—
588C8—2122no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—84no—
601C22#1681049812yes—
602C22#1681049812yes—
603C22C2×D4×Dic5311yescharacteristic
604C22C2×#1313318711yescharacteristic
605C22C2×#1313319411yescharacteristic
606C4—24no—
607C4—44no—
608C4—4242no—
609C4—2122no—
610C4—2122no—
611C4—2122no—
612C4—2122no—
613C4#1681049812yes—
614C4#1681049812yes—
615C4C22×C4⋊Dic5311yescharacteristic
616C4C2×D4×Dic5311yescharacteristic
617C4C2×#1313319411yescharacteristic
618C4C2×#1313320111yescharacteristic
619C2—21no—
620C2—22no—
621C2—22no—
622C2—21no—
623C2—44no—
624C2C2×#1681049811yescharacteristic
625C2#2028408811yescharacteristic
626C2#2028409511yescharacteristic
subgroupG/H
centerC22C2×#13133187
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#421C2
socleC22×C53C2×#120
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenterC2×#421D53
hypocenterC53#20865
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
2. C2×#16810665C2semidirect
4. #20277338C2semidirect
5. #20274950C2semidirect
6. #20274943C2semidirect
7. #20274453C2semidirect
8. #20274425C2semidirect
25. #16810665C22semidirect
34. #16810493C22semidirect
35. #16810489C22semidirect
36. #16810391C22semidirect
37. #16810391C22semidirect
38. #16810383C22semidirect
42. C53×#421C4semidirect
113. C2×M16×C53C2×C4semidirect
117. #117×C53C2×C4semidirect
121. #104×C53C2×C4semidirect
131. #41×C53C4⋊C4semidirect
136. C2×Q8×C53C4⋊C4semidirect
144. C2×D4×C53C4⋊C4semidirect
344. #421Dic53semidirect
357. C53#20865semidirect
378. C2×M16C2×Dic53semidirect
407. #117C2×Dic53semidirect
432. #104C2×Dic53semidirect
445. #41C4⋊Dic53semidirect
450. C2×Q8C4⋊Dic53semidirect
458. C2×D4C4⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
#202879182
C2×#168106652
C2×#168106242
#202773382
#202749502
#202749432
#202744532
#202744252
C2×C53×#4212
#2086553