Finite Group Databasebeta

← #27447174

Subgroups — #27447174

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×#16810708C211yesmaximal, characteristic
2#20278561C211yesmaximal, characteristic
3#20277633C211yesmaximal, characteristic, complemented
4#20276233C211yesmaximal, characteristic
5#20276199C211yesmaximal, characteristic, complemented
6#20276102C211yesmaximal, characteristic
7#20276045C211yesmaximal, characteristic, complemented
8#20275446C211yesmaximal, characteristic, complemented
9#20275269C211yesmaximal, characteristic, complemented
10#20275070C211yesmaximal, characteristic
11#20275068C211yesmaximal, characteristic
12#20274981C211yesmaximal, characteristic, complemented
13#20274971C211yesmaximal, characteristic, complemented
14#20274959C211yesmaximal, characteristic
15C2×C53×#477C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×#13133193—21no—
18C2×Q8×Dic53C2211yescharacteristic
19C2×Q8×Dic53C2211yescharacteristic
20C2×#13133179C2211yescharacteristic
21#16810708—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30#16810545—21no—
31#16810534—21no—
32#16810525C2211yescharacteristic
33#16810524C2211yescharacteristic
34#16810522C2211yescharacteristic
35#16810521—21no—
36#16810521C2211yescharacteristic
37#16810519—21no—
38#16810519C2211yescharacteristic
39#16810519C2211yescharacteristic
40#16810494—21no—
41#16810488—21no—
42#16810488C2211yescharacteristic
43C4×#11719919—21no—
44C4×#11719919C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×C4⋊Q8×C53C2211yescharacteristic
54C2×C4×Q8×C53C2211yescharacteristic
55C53×#477—24no—
56C2×#107×C53C2211yescharacteristic
57C2×#107×C53C2211yescharacteristic
58C2×C4⋊C8×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#103×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C2×#11719924—22no—
64C2×#11719924—22no—
65C2×#11719924Q814yescomplemented
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133193—44no—
69Q8×Dic53—22no—
70Q8×Dic53—22no—
71Q8×Dic53—22no—
72Q8×Dic53—41no—
73#13133179—22no—
74#13133179—41no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919D411yescharacteristic
85C2×#11719919D411yescharacteristic
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53C2311yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95#13133151—22no—
96#13133148—22no—
97#13133148—41no—
98#13133141—22no—
99#13133139—22no—
100#13133139—22no—
101#13133137—41no—
102C8×Dic53—22no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124D411yescharacteristic
110#13133124D411yescharacteristic
111C22×Q8×C53—21no—
112C22×Q8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C4⋊Q8×C53—22no—
116C4⋊Q8×C53—41no—
117C4×Q8×C53—22no—
118C4×Q8×C53—41no—
119C2×C4⋊C4×C53—21no—
120C2×C4⋊C4×C53—21no—
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C42×C53—21no—
125C2×C42×C53C2311yescharacteristic
126#107×C53—22no—
127#107×C53—22no—
128C4⋊C8×C53—22no—
129#103×C53—22no—
130#103×C53D414yes—
131C23×D53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—42no—
136C2×Q8×C53—42no—
137C2×Q8×C53—81no—
138C2×Q8×C53C2×D412yes—
139C2×Q8×C53C2×Q812yes—
140C2×Q8×C53#4112yes—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212C2×Q812yes—
161C2×C4×D53—42no—
162C2×C4×D53—42no—
163C2×C4×D53—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—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—42no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53C2×D412yes—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—42no—
181#11719924—28no—
182#11719921—42no—
183#11719921—42no—
184#11719921—42no—
185#11719921—42no—
186#11719921—81no—
187C4⋊Dic53—24no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—42no—
194C4⋊Dic53—42no—
195C4⋊Dic53#4112yes—
196#11719919—22no—
197#11719919—22no—
198#11719919—42no—
199#11719919—42no—
200#11719919—42no—
201C4×Dic53—24no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—21no—
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×C53—84no—
225Q8×C53C22⋊Q814yescomplemented
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—21no—
231C2×C4×C53—22no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
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—41no—
243C2×C4×C53#13512yes—
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×Q811yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53H(5,𝔽2)11yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C2×Dic53—21no—
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—21no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53AGL(1,ℤ/8)12yes—
271D212—22no—
272D212—22no—
273D212—41no—
274D212—41no—
275C4×D53—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—22no—
297C4×C53—42no—
298C4×C53—41no—
299C4×C53—81no—
300C4×C53—81no—
301C4×C53C2×C22⋊Q811yescharacteristic
302C4×C53#53911yescharacteristic
303C4×C53#54711yescharacteristic
304C4×C53#55411yescharacteristic
305Dic53—22no—
306Dic53—41no—
307Dic53—42no—
308Dic53—41no—
309Dic53—41no—
310Dic53—81no—
311Dic53—81no—
312C2×#477D5311yescharacteristic, complemented
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#1863—531no—
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⋊Q8C2×D5311yescharacteristic
337C2×C22⋊Q8—1061no—
338C2×C4×Q8—531no—
339C2×C4×Q8—531no—
340C2×C4×Q8C2×D5311yescharacteristic
341C2×C4×D4—531no—
342#477—24no—
343C4×SD16—1064no—
344C2×#107—531no—
345C2×#107C2×D5311yescharacteristic
346C2×#107C2×D5311yescharacteristic
347C2×#106—531no—
348C2×#106—531no—
349C2×C4⋊C8C2×D5311yescharacteristic
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—531no—
365#389—1061no—
366#389—1061no—
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#340—531no—
376#336—531no—
377#336—531no—
378#336—531no—
379#336—531no—
380C53#2637711yescharacteristic, complemented, Sylow
381C22×Q8—21no—
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—22no—
393C4⋊Q8—41no—
394C22⋊Q8—2124no—
395C4×Q8—22no—
396C4×Q8—41no—
397C4×Q8—1062no—
398C4×Q8—1062no—
399C4×Q8—1062no—
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—22no—
432#107—1062no—
433#106—1062no—
434#106—2121no—
435C4⋊C8—22no—
436#103—22no—
437#103—1062no—
438#103C22⋊D5314yes—
439#102—1062no—
440#102—1062no—
441#102—2121no—
442C4×C8—1062no—
443#95—531no—
444#95—531no—
445#95—1061no—
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—42no—
456C2×Q8—42no—
457C2×Q8—81no—
458C2×Q8Q8×D5312yes—
459C2×Q8#1171994912yes—
460C2×Q8C2×C22⋊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×C22⋊D5312yes—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—532no—
505C4⋊C4—1062no—
506C4⋊C4—1064no—
507C4⋊C4—1062no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C4⋊C4—2122no—
516C4⋊C4—2122no—
517C4⋊C4—2122no—
518C4⋊C4C2×C22⋊D5312yes—
519C22⋊C4—4241no—
520C22⋊C4—2122no—
521C22⋊C4—2122no—
522C22⋊C4—2122no—
523C22⋊C4—2122no—
524C42—21no—
525C42—21no—
526C42—42no—
527C42—1064no—
528C42—1061no—
529C42—1061no—
530C42—1061no—
531C42—1061no—
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—84no—
545Q8#1313319314yescomplemented
546D4—1062no—
547D4—1062no—
548D4—2121no—
549D4—2121no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—532no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1062no—
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#1313324012yes—
590C2×C4C22×C22⋊D5311yescharacteristic
591C2×C4C2×Q8×D5311yescharacteristic
592C2×C4#41×D5311yescharacteristic
593C2×C4C2×#1171994911yescharacteristic
594C2×C4#1313326211yescharacteristic
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#1681113611yescharacteristic
608C22D4×C22⋊D5311yescharacteristic
609C22#1681128611yescharacteristic
610C22C2×#1313324011yescharacteristic
611C22AGL(1,ℤ/8)×D5311yescharacteristic
612C22#1681143311yescharacteristic
613C22#1681143611yescharacteristic
614C4—24no—
615C4—22no—
616C4—41no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—1062no—
621C4—4241no—
622C4—4241no—
623C4—2121no—
624C4—2121no—
625C4—2122no—
626C4—2121no—
627C4C2×#1313319311yescharacteristic
628C4#1681122211yescharacteristic
629C4#1681127211yescharacteristic
630C4#1681132711yescharacteristic
631C2—4242no—
632C2#2027852011yescharacteristic
633C2#2028477811yescharacteristic
634C2#2028494911yescharacteristic
635C2#2028620411yescharacteristic
636C2#2028645611yescharacteristic
637C2#2028758911yescharacteristic
638C2#2028763911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#477C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenterC2×#477D53
hypocenterC53#26377
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #20277633C2semidirect
5. #20276199C2semidirect
7. #20276045C2semidirect
8. #20275446C2semidirect
9. #20275269C2semidirect
12. #20274981C2semidirect
13. #20274971C2semidirect
15. C2×C53×#477C2semidirect
65. C2×#11719924Q8semidirect
225. Q8×C53C22⋊Q8semidirect
312. C2×#477D53semidirect
380. C53#26377semidirect
545. Q8#13133193semidirect

Maximal subgroups

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

subgroupindex
C2×#168107082
#202785612
#202776332
#202762332
#202761992
#202761022
#202760452
#202754462
#202752692
#202750702
#202750682
#202749812
#202749712
#202749592
C2×C53×#4772
#2637753