Finite Group Databasebeta

← #27447176

Subgroups — #27447176

Order 13568.

856 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810916C211yesmaximal, characteristic, complemented
2C2×#16810708C211yesmaximal, characteristic
3#20278563C211yesmaximal, characteristic
4#20277636C211yesmaximal, characteristic, complemented
5#20276235C211yesmaximal, characteristic
6#20276196C211yesmaximal, characteristic, complemented
7#20276098C211yesmaximal, characteristic
8#20276047C211yesmaximal, characteristic, complemented
9#20275449C211yesmaximal, characteristic, complemented
10#20275273C211yesmaximal, characteristic, complemented
11#20275070C211yesmaximal, characteristic
12#20275068C211yesmaximal, characteristic
13#20274985C211yesmaximal, characteristic, complemented
14#20274963C211yesmaximal, characteristic
15C53×#1863C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133197—21no—
19C2×Q8×Dic53C2211yescharacteristic
20C2×Q8×Dic53C2211yescharacteristic
21C2×#13133179C2211yescharacteristic
22#16810916—24no—
23#16810708—24no—
24C2×#13133151C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133146C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133139C2211yescharacteristic
31C2×#13133139C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×C8×Dic53C2211yescharacteristic
34#16810545—21no—
35#16810536—21no—
36#16810525C2211yescharacteristic
37#16810524C2211yescharacteristic
38#16810522—21no—
39#16810522—21no—
40#16810522C2211yescharacteristic
41#16810522C2211yescharacteristic
42#16810522C2211yescharacteristic
43#16810497—21no—
44#16810489—21no—
45#16810489C2211yescharacteristic
46C4×C4⋊Dic53—21no—
47C4×C4⋊Dic53C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C2×C4×Q8×C53C2211yescharacteristic
55C2×#107×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C53×#389C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C22×Dic106—21no—
62C22×D212C2311yescharacteristic
63C22×C4×D53—21no—
64C2×#11719924—22no—
65C2×#11719924—22no—
66C2×#11719924Q814yescomplemented
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133216—22no—
70#13133216—41no—
71#13133197—44no—
72Q8×Dic53—22no—
73Q8×Dic53—22no—
74Q8×Dic53—41no—
75Q8×Dic53—41no—
76#13133179—22no—
77#13133179—41no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—21no—
81C2×C4⋊Dic53—21no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53D411yescharacteristic
84C2×C4⋊Dic53D411yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×#11719919—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×#11719921—21no—
93C2×#11719921—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53—21no—
96C2×C4×Dic53C2311yescharacteristic
97C2×C4×Dic53C2311yescharacteristic
98#13133151—22no—
99#13133150—22no—
100#13133149—41no—
101#13133148—22no—
102#13133146—22no—
103#13133141—22no—
104#13133139—22no—
105#13133139—41no—
106#13133137—22no—
107C8×Dic53—22no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124D411yescharacteristic
114#13133124D411yescharacteristic
115C22×Q8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C22×C8×C53C2311yescharacteristic
118C4×Q8×C53—22no—
119C4×Q8×C53—22no—
120C2×C4⋊C4×C53—21no—
121C2×C4⋊C4×C53—21no—
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C42×C53—21no—
126C2×C42×C53C2311yescharacteristic
127#107×C53—22no—
128#103×C53—22no—
129#103×C53D414yes—
130#95×C53—21no—
131C23×D53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53C2×D412yes—
136C2×Q8×C53C2×Q812yes—
137C2×Q8×C53#4112yes—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212C2×Q812yes—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—81no—
161C2×Dic106—42no—
162C2×Dic106—42no—
163C2×Dic106—81no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53C2×D412yes—
170C4⋊C4×C53—24no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—42no—
176C4⋊C4×C53C2×D412yes—
177C4⋊C4×C53#4112yes—
178C42×C53—24no—
179C42×C53—21no—
180C42×C53—21no—
181#11719924—28no—
182#11719921—42no—
183#11719921—42no—
184#11719921—42no—
185#11719921—42no—
186#11719921—81no—
187C4⋊Dic53—22no—
188C4⋊Dic53—22no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—42no—
196C4⋊Dic53—42no—
197C4⋊Dic53—42no—
198C4⋊Dic53—42no—
199C4⋊Dic53—42no—
200#11719919—42no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—42no—
206C4×Dic53—42no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322C2×Q812yes—
213C23×C53C22×D411yescharacteristic
214C22×D53—22no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—41no—
219C22×D53—41no—
220C22×D53—41no—
221Q8×C53—22no—
222Q8×C53—22no—
223Q8×C53—22no—
224Q8×C53C22⋊Q814yescomplemented
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—22no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
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—41no—
243C2×C4×C53C22×D411yescharacteristic
244C2×C4×C53C22×Q811yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53H(5,𝔽2)11yescharacteristic
248C2×C4×C532−1+411yescharacteristic
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53#13512yes—
269C2×Dic53AGL(1,ℤ/8)12yes—
270D212—22no—
271D212—22no—
272D212—41no—
273D212—41no—
274C4×D53—84no—
275Dic106—84no—
276C8×C53—22no—
277C8×C53—41no—
278#88322—22no—
279#88322—41no—
280#26377—531nomaximal, Sylow
281C22×C53#52511yescharacteristic
282C22×C53D4211yescharacteristic
283C22×C53D4×Q811yescharacteristic
284C22×C53C2×#13511yescharacteristic
285C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C53#57711yescharacteristic
287C22×C53#57811yescharacteristic
288C2×D53—42no—
289C2×D53—42no—
290C2×D53—42no—
291C2×D53—81no—
292C2×D53—81no—
293C2×D53—81no—
294C2×D53—81no—
295C4×C53—24no—
296C4×C53—41no—
297C4×C53—42no—
298C4×C53—41no—
299C4×C53—42no—
300C4×C53—81no—
301C4×C53C2×C22⋊Q811yescharacteristic
302C4×C53#53911yescharacteristic
303C4×C53#54711yescharacteristic
304C4×C53#55411yescharacteristic
305Dic53—22no—
306Dic53—22no—
307Dic53—41no—
308Dic53—41no—
309Dic53—81no—
310Dic53—81no—
311Dic53—81no—
312C2×#477—531no—
313C2×C4×SD16—531no—
314#2452—531no—
315#2351—531no—
316#2092—531no—
317#2085—531no—
318#2067—531no—
319#2058—531no—
320#1942—531no—
321#1916—531no—
322#1880—531no—
323#1880—531no—
324#1865—531no—
325#1863D5311yescharacteristic, complemented
326#1862—531no—
327C2×C53#244911yescharacteristic
328C2×C53#310611yescharacteristic
329C2×C53#312111yescharacteristic
330C2×C53#326511yescharacteristic
331C2×C53#328511yescharacteristic
332C2×C53#338111yescharacteristic
333C2×C53#338611yescharacteristic
334D53—82no—
335C22×SD16—531no—
336C2×C4⋊Q8—531no—
337C2×C22⋊Q8—1061no—
338C2×C4×Q8—531no—
339C2×C4×Q8—531no—
340C2×C4×Q8C2×D5311yescharacteristic
341C2×C4×D4—531no—
342#477—1064no—
343C4×SD16—1064no—
344C2×#107—531no—
345C2×#107—531no—
346C2×#107C2×D5311yescharacteristic
347C2×#106—531no—
348C2×#106—531no—
349C2×C4⋊C8—531no—
350C2×#103—531no—
351C2×#103C2×D5311yescharacteristic
352C2×#103C2×D5311yescharacteristic
353C2×#102—531no—
354C2×#102—531no—
355C2×#102—531no—
356C2×C4×C8—531no—
357#396—1061no—
358#393—1061no—
359#390—531no—
360#390—531no—
361#389—531no—
362#389—531no—
363#389—531no—
364#389—1061no—
365#389—1061no—
366#389C2×D5311yescharacteristic
367#385—1061no—
368#382—531no—
369#382—1061no—
370C4×C4⋊C4—531no—
371C4×C4⋊C4—1061no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340C2×D5311yescharacteristic
376#336—531no—
377#336—531no—
378#336—531no—
379#336C2×D5311yescharacteristic
380C53#2637711yescharacteristic, complemented, Sylow
381C22×Q8—1061no—
382C22×Q8C22×D5311yescharacteristic
383C22×D4—531no—
384C23×C4—1061no—
385C2×SD16—534no—
386C2×SD16—1062no—
387C2×SD16—1062no—
388C22×C8—531no—
389C22×C8—531no—
390C22×C8C22×D5311yescharacteristic
391C22×C8C22×D5311yescharacteristic
392C4⋊Q8—1062no—
393C4⋊Q8—2121no—
394C22⋊Q8—2124no—
395C4×Q8—22no—
396C4×Q8—22no—
397C4×Q8—1062no—
398C4×Q8—1062no—
399C4×Q8—2121no—
400C4×Q8—2121no—
401C4×D4—1062no—
402C4×D4—2121no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C42—21no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—1061no—
428C2×C42—1061no—
429C2×C42C22×D5311yescharacteristic
430#107—22no—
431#107—1062no—
432#107—1062no—
433#106—1062no—
434#106—2121no—
435C4⋊C8—1062no—
436#103—22no—
437#103—1062no—
438#103D21214yes—
439#102—1062no—
440#102—1062no—
441#102—2121no—
442C4×C8—1062no—
443#95—21no—
444#95—531no—
445#95—531no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451C24—1061no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—4241no—
456C2×Q8—2122no—
457C2×Q8—2122no—
458C2×Q8C2×D21212yes—
459C2×Q8#1171994712yes—
460C2×Q8Q8×D5312yes—
461C2×D4—532no—
462C2×D4—1061no—
463C2×D4—1061no—
464C2×D4—1061no—
465C2×D4—1061no—
466C2×D4—1061no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—531no—
472C22×C4—531no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—4241no—
479C22×C4—2122no—
480C22×C4—2122no—
481C22×C4D4×D5311yescharacteristic
482C22×C4D4×D5311yescharacteristic
483C22×C4C23×D5311yescharacteristic
484SD16—1068no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—532no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8C2×D21212yes—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—1062no—
504C4⋊C4—1062no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C4⋊C4—2122no—
516C4⋊C4—2122no—
517C4⋊C4C2×D21212yes—
518C4⋊C4#1171994712yes—
519C22⋊C4—4241no—
520C22⋊C4—2122no—
521C22⋊C4—2122no—
522C22⋊C4—2122no—
523C22⋊C4—2122no—
524C42—24no—
525C42—21no—
526C42—21no—
527C42—1061no—
528C42—1061no—
529C42—1061no—
530C42—1061no—
531C42—2122no—
532C42—2122no—
533C23—1062no—
534C23—2121no—
535C23—2121no—
536C23—2121no—
537C23—2121no—
538C23—2121no—
539C23—2121no—
540C23C2×D4×D5311yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—22no—
544Q8—4244no—
545Q8#1313319714yescomplemented
546D4—1062no—
547D4—1062no—
548D4—2121no—
549D4—2121no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—532no—
569C2×C4—532no—
570C2×C4—1061no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—4244no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4C22×D21211yescharacteristic
591C2×C4C2×Q8×D5311yescharacteristic
592C2×C4C2×#1171994711yescharacteristic
593C2×C4C2×#1171994711yescharacteristic
594C2×C4#1313326311yescharacteristic
595C2×C4#1313326411yescharacteristic
596C8—22no—
597C8—41no—
598C8—1062no—
599C8—2121no—
600C22—4241no—
601C22—4241no—
602C22—4241no—
603C22—4241no—
604C22—2122no—
605C22—2122no—
606C22—2122no—
607C22#1681112911yescharacteristic
608C22D4×D21211yescharacteristic
609C22#1681128611yescharacteristic
610C22#135×D5311yescharacteristic
611C22AGL(1,ℤ/8)×D5311yescharacteristic
612C22#1681143211yescharacteristic
613C22#1681143611yescharacteristic
614C4—24no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—42no—
619C4—81no—
620C4—1062no—
621C4—1062no—
622C4—4241no—
623C4—4241no—
624C4—4241no—
625C4—2121no—
626C4—2121no—
627C4C2×#1313319711yescharacteristic
628C4#1681121711yescharacteristic
629C4#1681127711yescharacteristic
630C4#1681133111yescharacteristic
631C2—4242no—
632C2#2027852211yescharacteristic
633C2#2028476811yescharacteristic
634C2#2028494511yescharacteristic
635C2#2028621211yescharacteristic
636C2#2028645711yescharacteristic
637C2#2028759311yescharacteristic
638C2#2028763911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1863C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#1863D53
hypocenterC53#26377
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
1. C2×#16810916C2semidirect
4. #20277636C2semidirect
6. #20276196C2semidirect
8. #20276047C2semidirect
9. #20275449C2semidirect
10. #20275273C2semidirect
13. #20274985C2semidirect
15. C53×#1863C2semidirect
66. C2×#11719924Q8semidirect
224. Q8×C53C22⋊Q8semidirect
325. #1863D53semidirect
380. C53#26377semidirect
545. Q8#13133197semidirect

Maximal subgroups

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

subgroupindex
C2×#168109162
C2×#168107082
#202785632
#202776362
#202762352
#202761962
#202760982
#202760472
#202754492
#202752732
#202750702
#202750682
#202749852
#202749632
C53×#18632
#2637753