Finite Group Databasebeta

← #27402513

Subgroups — #27402513

Order 13568.

986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719913C211yesmaximal, characteristic
2C2×#16810710C211yesmaximal, characteristic
3C2×#16810710C211yesmaximal, characteristic
4#20277634C211yesmaximal, characteristic, complemented
5#20277624C211yesmaximal, characteristic
6#20275444C211yesmaximal, characteristic, complemented
7#20274970C211yesmaximal, characteristic, complemented
8#20274968C211yesmaximal, characteristic, complemented
9#20274963C211yesmaximal, characteristic
10#20274961C211yesmaximal, characteristic
11#20274876C211yesmaximal, characteristic, complemented
12#20274729C211yesmaximal, characteristic, complemented
13#20274724C211yesmaximal, characteristic
14C8×#11719919C211yesmaximal, characteristic, complemented
15C53×#1941C211yesmaximal, characteristic, complemented
16C22×#11719913C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×#13133182C2211yescharacteristic
19C2×#13133182C2211yescharacteristic
20C2×C4×D212C2211yescharacteristic
21C2×#13133179C2211yescharacteristic
22C2×#13133179C2211yescharacteristic
23C4×#11719913—24no—
24#16810710—24no—
25#16810710C418yescomplemented
26C2×#13133150C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133149C2211yescharacteristic
29C2×#13133146C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133140C2211yescharacteristic
33C2×#13133139C2211yescharacteristic
34C2×#13133137C2211yescharacteristic
35C2×#13133137C2211yescharacteristic
36C2×C8×Dic53C2211yescharacteristic
37C2×C8×Dic53C2211yescharacteristic
38#16810522C2211yescharacteristic
39#16810522C2211yescharacteristic
40#16810520—21no—
41#16810518—21no—
42#16810497—21no—
43#16810494—21no—
44#16810488C2211yescharacteristic
45#16810488C2211yescharacteristic
46#16810487—21no—
47#16810480—21no—
48C4⋊C4×Dic53—21no—
49C4×#11719919C2211yescharacteristic
50#95×D53—21no—
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810378C2211yescharacteristic
54#16810377C2211yescharacteristic
55C2×#106×C53C2211yescharacteristic
56C2×#106×C53C2211yescharacteristic
57C2×C4×C8×C53C2211yescharacteristic
58C53×#389C2211yescharacteristic
59C53×#389C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×Dic106C2311yescharacteristic
63C22×D212C2311yescharacteristic
64C22×C4×D53—21no—
65C22×C4×D53—21no—
66C22×C4×D53—21no—
67C2×#11719913C2×C414yes—
68C2×#11719913D414yes—
69C2×#11719913Q814yes—
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C4×Dic106—22no—
73C4×Dic106—41no—
74#13133182—22no—
75#13133182—22no—
76#13133182—41no—
77#13133182C2×C414yescomplemented
78C4×D212—22no—
79C4×D212—41no—
80#13133179—22no—
81#13133179—22no—
82#13133179—41no—
83#13133179C2×C414yes—
84C2×C4⋊Dic53—21no—
85C2×C4⋊Dic53—21no—
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×#11719919D411yescharacteristic
92C2×#11719919D411yescharacteristic
93C2×#11719921—21no—
94C2×#11719921—21no—
95C2×#11719921—21no—
96C2×C4×Dic53—21no—
97C2×C4×Dic53—21no—
98C2×C4×Dic53—21no—
99C2×C4×Dic53C2311yescharacteristic
100C2×C4×Dic53C2311yescharacteristic
101#13133150—41no—
102#13133150—41no—
103#13133149—22no—
104#13133146—22no—
105#13133142—22no—
106#13133140—22no—
107#13133140C2×C414yescomplemented
108#13133139—22no—
109#13133137—22no—
110#13133137C2×C414yes—
111C8×Dic53—22no—
112C8×Dic53C2×C414yescomplemented
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118#13133124—21no—
119#13133124D411yescharacteristic
120#13133124D411yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C22×C8×C53C2311yescharacteristic
123C2×C4⋊C4×C53—21no—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C42×C53C2311yescharacteristic
130#106×C53—22no—
131#106×C53C2×C414yescomplemented
132C4×C8×C53—22no—
133#95×C53—21no—
134#95×C53—21no—
135C23×D53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212C22×C412yes—
154C2×D212C2×D412yes—
155C2×D212C2×Q812yes—
156C2×C4×D53—24no—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—42no—
161C2×C4×D53—81no—
162C2×C4×D53—81no—
163C2×C4×D53—81no—
164C2×C4×D53—81no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106C22×C412yes—
169C2×Dic106C2×D412yes—
170C2×Dic106C2×Q812yes—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53C22×C412yes—
175C2×C8×C53C2×D412yes—
176C2×C8×C53C2×Q812yes—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—42no—
184C4⋊C4×C53—81no—
185C4⋊C4×C53C22×C412yes—
186C4⋊C4×C53C22×C412yes—
187C42×C53—21no—
188C42×C53—21no—
189#11719921—24no—
190#11719921—42no—
191#11719921—42no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—42no—
197C4⋊Dic53—81no—
198#11719919—22no—
199#11719919—24no—
200#11719919—22no—
201#11719919—42no—
202#11719919—42no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—24no—
206C4×Dic53—42no—
207C4×Dic53—42no—
208C4×Dic53C22×C412yes—
209C4×Dic53#4112yes—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322C22×C412yes—
216#11719913C4⋊C4116yescomplemented
217C23×C53C22×D411yescharacteristic
218C22×D53—22no—
219C22×D53—22no—
220C22×D53—22no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53#13512yes—
243C2×C4×C53C23×C411yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×Q811yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53H(5,𝔽2)11yescharacteristic
248C2×C4×C532−1+411yescharacteristic
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—21no—
252C2×Dic53—22no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53C4×D412yes—
268C2×Dic53C4×D412yes—
269C2×Dic53#13512yes—
270D212—22no—
271D212—22no—
272D212—22no—
273D212C2×C4⋊C414yes—
274C4×D53—84no—
275C4×D53—84no—
276C4×D53—84no—
277Dic106—22no—
278Dic106—22no—
279Dic106—22no—
280Dic106C2×C4⋊C414yes—
281C8×C53—41no—
282C8×C53C2×C4⋊C414yes—
283#88322—22no—
284#88322—41no—
285#22683—531nomaximal, Sylow
286C22×C53C2×C4×D411yescharacteristic
287C22×C53D4211yescharacteristic
288C22×C53D4×Q811yescharacteristic
289C22×C53C2×#13511yescharacteristic
290C22×C53C2×#13511yescharacteristic
291C22×C53#57711yescharacteristic
292C22×C53#57711yescharacteristic
293C2×D53—24no—
294C2×D53—42no—
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C4×C53—22no—
301C4×C53—42no—
302C4×C53—42no—
303C4×C53—41no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53C22×C4⋊C411yescharacteristic
307C4×C53#52011yescharacteristic
308C4×C53#54711yescharacteristic
309C4×C53#55411yescharacteristic
310Dic53—24no—
311Dic53—22no—
312Dic53—22no—
313Dic53—22no—
314Dic53—41no—
315Dic53—81no—
316Dic53—81no—
317C2×C4×SD16—531no—
318C2×C4×SD16—531no—
319C2×C4×SD16—531no—
320#2351—531no—
321#2349—531no—
322#1941—531no—
323#1941D5311yescharacteristic, complemented
324#1863—531no—
325#1863—531no—
326#1862—531no—
327#1862—531no—
328#1846—531no—
329#1805—531no—
330#1804—531no—
331C8×C4⋊C4—531no—
332C2×C53D4×C4⋊C411yescharacteristic
333C2×C53C4×#13511yescharacteristic
334C2×C53#294911yescharacteristic
335C2×C53#328211yescharacteristic
336C2×C53#328211yescharacteristic
337C2×C53#338111yescharacteristic
338C2×C53#338111yescharacteristic
339D53—44no—
340C22×SD16—531no—
341C2×C4×Q8—531no—
342C2×C4×Q8—531no—
343C2×C4×Q8—531no—
344C2×C4×D4—531no—
345C2×C4×D4—531no—
346C2×C4×D4—531no—
347C4×SD16—538no—
348C4×SD16—1064no—
349C4×SD16—1064no—
350C2×#107—531no—
351C2×#107—531no—
352C2×#106—531no—
353C2×#106C2×D5311yescharacteristic
354C2×#106C2×D5311yescharacteristic
355C2×C4⋊C8—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×#102—531no—
360C2×#102—531no—
361C2×#102—531no—
362C2×C4×C8—531no—
363C2×C4×C8—531no—
364C2×C4×C8C2×D5311yescharacteristic
365#389—531no—
366#389—531no—
367#389—1061no—
368#389—1061no—
369#389C2×D5311yescharacteristic
370#389C2×D5311yescharacteristic
371#385—1061no—
372#385—1061no—
373#382—531no—
374#382—531no—
375#382—1061no—
376#380—1061no—
377C4×C4⋊C4—531no—
378C4×C4⋊C4—1061no—
379C2×#95—1061no—
380#340—531no—
381#340—531no—
382#340C2×D5311yescharacteristic
383#340C2×D5311yescharacteristic
384#336—531no—
385#336—531no—
386C53#2268311yescharacteristic, complemented, Sylow
387C22×Q8—531no—
388C22×D4—531no—
389C23×C4—1061no—
390C23×C4—1061no—
391C23×C4—1061no—
392C2×SD16—534no—
393C2×SD16—534no—
394C2×SD16—534no—
395C22×C8—531no—
396C22×C8—531no—
397C22×C8C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399C4×Q8—534no—
400C4×Q8—1062no—
401C4×Q8—1062no—
402C4×Q8—1062no—
403C4×Q8—2121no—
404C4×Q8—2121no—
405C4×D4—534no—
406C4×D4—1062no—
407C4×D4—1062no—
408C4×D4—1062no—
409C4×D4—2121no—
410C4×D4—2121no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—1061no—
432C2×C42—1061no—
433C2×C42—1061no—
434C2×C42C22×D5311yescharacteristic
435#107—2121no—
436#107—2121no—
437#106—22no—
438#106—1062no—
439#106C4×D5314yescomplemented
440C4⋊C8—1062no—
441#103—534no—
442#103—1062no—
443#103—1062no—
444#102—534no—
445#102—1062no—
446#102—1062no—
447C4×C8—22no—
448C4×C8—534no—
449C4×C8—1062no—
450#95—21no—
451#95—21no—
452#95—531no—
453#95—531no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459#95—1061no—
460C24—1061no—
461C2×Q8—532no—
462C2×Q8—532no—
463C2×Q8—532no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8—1061no—
467C2×D4—532no—
468C2×D4—532no—
469C2×D4—532no—
470C2×D4—1061no—
471C2×D4—1061no—
472C2×D4—1061no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—1064no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—4241no—
486C22×C4—4241no—
487C22×C4—4241no—
488C22×C4—4241no—
489C22×C4—2122no—
490C22×C4—2122no—
491C22×C4—2122no—
492C22×C4—2122no—
493C22×C4D4×D5311yescharacteristic
494C22×C4D4×D5311yescharacteristic
495C22×C4C23×D5311yescharacteristic
496SD16—5316no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—532no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8C2×C4×D5312yes—
507C2×C8D4×D5312yes—
508C2×C8Q8×D5312yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—1062no—
518C4⋊C4—1064no—
519C4⋊C4—1062no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—4241no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4—2122no—
528C4⋊C4C2×C4×D5312yes—
529C4⋊C4C2×C4×D5312yes—
530C22⋊C4—1064no—
531C22⋊C4—2122no—
532C22⋊C4—2122no—
533C42—21no—
534C42—21no—
535C42—532no—
536C42—532no—
537C42—1064no—
538C42—1061no—
539C42—1061no—
540C42—2122no—
541C42—2122no—
542C23—1062no—
543C23—1062no—
544C23—1062no—
545C23—2121no—
546C23—2121no—
547C23—2121no—
548C23—2121no—
549C23C2×D4×D5311yescharacteristic
550Q8—534no—
551Q8—1062no—
552Q8—1062no—
553Q8—1062no—
554D4—534no—
555D4—1062no—
556D4—1062no—
557D4—1062no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—532no—
576C2×C4—532no—
577C2×C4—532no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1062no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—4244no—
591C2×C4—4244no—
592C2×C4—4244no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4#1313323812yes—
600C2×C4C22×C4×D5311yescharacteristic
601C2×C4C2×D4×D5311yescharacteristic
602C2×C4C2×Q8×D5311yescharacteristic
603C2×C4#41×D5311yescharacteristic
604C2×C4#1313326311yescharacteristic
605C2×C4#1313326511yescharacteristic
606C8—41no—
607C8—1062no—
608C8—2121no—
609C8C4⋊C4×D5314yes—
610C22—1064no—
611C22—2122no—
612C22—2122no—
613C22—2122no—
614C22—2122no—
615C22—2122no—
616C22—2122no—
617C22C4×D4×D5311yescharacteristic
618C22#1681125711yescharacteristic
619C22#1681128811yescharacteristic
620C22#135×D5311yescharacteristic
621C22C2×#1313323811yescharacteristic
622C22#1681142911yescharacteristic
623C22#1681143011yescharacteristic
624C4—22no—
625C4—42no—
626C4—42no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—1062no—
631C4—1064no—
632C4—1062no—
633C4—1062no—
634C4—4241no—
635C4—4241no—
636C4—2121no—
637C4C2×C4⋊C4×D5311yescharacteristic
638C4#1681110511yescharacteristic
639C4#1681127511yescharacteristic
640C4#1681132911yescharacteristic
641C2—2124no—
642C2#2027761611yescharacteristic
643C2C4×#1313323811yescharacteristic
644C2#2028356911yescharacteristic
645C2#2028641911yescharacteristic
646C2#2028642911yescharacteristic
647C2#2028758711yescharacteristic
648C2#2028759511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1941C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1941D53
hypocenterC53#22683
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
4. #20277634C2semidirect
6. #20275444C2semidirect
7. #20274970C2semidirect
8. #20274968C2semidirect
11. #20274876C2semidirect
12. #20274729C2semidirect
14. C8×#11719919C2semidirect
15. C53×#1941C2semidirect
25. #16810710C4semidirect
77. #13133182C2×C4semidirect
107. #13133140C2×C4semidirect
112. C8×Dic53C2×C4semidirect
131. #106×C53C2×C4semidirect
216. #11719913C4⋊C4semidirect
323. #1941D53semidirect
386. C53#22683semidirect
439. #106C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199132
C2×#168107102
#202776342
#202776242
#202754442
#202749702
#202749682
#202749632
#202749612
#202748762
#202747292
#202747242
C8×#117199192
C53×#19412
#2268353