Finite Group Databasebeta

← #27389667

Subgroups — #27389667

Order 13568.

1880 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic53C211yesmaximal, characteristic
2C2×#16810774C211yesmaximal, characteristic
3C2×#16810714C212yesmaximal, complemented
4#20277243C211yesmaximal, characteristic, complemented
5#20275047C212yesmaximal, complemented
6#20275020C211yesmaximal, characteristic, complemented
7#20274974C212yesmaximal, complemented
8#20274907C211yesmaximal, characteristic, complemented
9#102×Dic53C212yesmaximal, complemented
10C22⋊C8×Dic53C211yesmaximal, characteristic, complemented
11C53×#1786C211yesmaximal, characteristic
12C24×Dic53—21no—
13C22×#11719923C2212yes—
14C2×#13133196C2211yescharacteristic
15C2×#13133182C2211yescharacteristic, complemented
16C2×D4×Dic53—28no—
17C2×D4×Dic53—21no—
18C2×D4×Dic53—21no—
19C2×D4×Dic53C2212yes—
20C2×D4×Dic53C2212yes—
21C22×C4⋊Dic53C2211yescharacteristic
22C22×C22⋊Dic53—21no—
23C22×C4×Dic53C2211yescharacteristic
24#16810774C418yescomplemented
25#16810714—28no—
26C2×#13133148C2212yescomplemented
27C2×#13133142C2212yescomplemented
28C2×#13133138C2212yes—
29C2×C22⋊#88322C2211yescharacteristic
30C2×C8×Dic53C2212yescomplemented
31#16810528C2211yescharacteristic, complemented
32#16810519C2211yescharacteristic, complemented
33#16810505—21no—
34#16810495—21no—
35C4⋊C4×Dic53C2211yescharacteristic, complemented
36#16810474—21no—
37C22⋊C4×Dic53—21no—
38#16810386C2212yescomplemented
39#16810385C2212yescomplemented
40#16810377C2212yescomplemented
41C23×D4×C53C2211yescharacteristic
42C2×#102×C53C2212yes—
43C2×#102×C53C2212yes—
44C2×C22⋊C8×C53C2211yescharacteristic
45C53×#380C2211yescharacteristic
46C22×Dic106C2311yescharacteristic
47C23×Dic53—28no—
48C23×Dic53—21no—
49C23×Dic53—22no—
50C23×Dic53—22no—
51C23×Dic53—44no—
52C23×Dic53—44no—
53C23×Dic53D411yescharacteristic
54C23×Dic53D411yescharacteristic
55C2×#11719923—24no—
56C2×#11719923—24no—
57C2×#11719923C2×C418yescomplemented
58C22×#88322C2312yes—
59#13133196—22no—
60#13133196C2×C414yescomplemented
61#13133182—24no—
62D4×Dic53—28no—
63D4×Dic53—28no—
64D4×Dic53—24no—
65D4×Dic53—24no—
66D4×Dic53—24no—
67D4×Dic53D418yes—
68C2×C4⋊Dic53—24no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2312yes—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×C22⋊Dic53—28no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—22no—
78C2×C22⋊Dic53—22no—
79C2×C4×Dic53—24no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86#13133148—24no—
87#13133142—24no—
88#13133138C2×C418yescomplemented
89C22⋊#88322C2×C414yescomplemented
90C8×Dic53—24no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95C25×C53—21no—
96C22×D4×C53—24no—
97C22×D4×C53—24no—
98C22×D4×C53—21no—
99C22×D4×C53—22no—
100C22×D4×C53C2×C414yes—
101C22×D4×C53C2312yes—
102C23×C4×C53C2311yescharacteristic
103C22×C8×C53C2312yes—
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C22⋊C4×C53—21no—
107#102×C53—24no—
108#102×C53—24no—
109C22⋊C8×C53—22no—
110#95×C53—21no—
111C24×C53—24no—
112C24×C53—24no—
113C24×C53—24no—
114C24×C53—22no—
115C24×C53—42no—
116C24×C53—42no—
117C24×C53—42no—
118C24×C53—42no—
119C24×C53C2×D411yescharacteristic
120C2×D4×C53—24no—
121C2×D4×C53—216no—
122C2×D4×C53—24no—
123C2×D4×C53—216no—
124C2×D4×C53—24no—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—22no—
128C2×D4×C53C22×C414yes—
129C2×D4×C53C2×D414yes—
130C2×D4×C53#4114yes—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53C22×C412yes—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53#4112yes—
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—24no—
142C22×Dic53—28no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—28no—
149C22×Dic53—41no—
150C22×Dic53—44no—
151C22×Dic53—432no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—44no—
155C22×Dic53—44no—
156C22×Dic53—44no—
157C22×Dic53SD1614yescomplemented
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53C2×D411yescharacteristic
162C2×Dic106—22no—
163C2×Dic106—22no—
164C2×Dic106—21no—
165C2×Dic106C22×C412yescomplemented
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—22no—
171C22⋊C4×C53—42no—
172C22⋊Dic53—28no—
173C22⋊Dic53—24no—
174C22⋊Dic53—28no—
175#11719923—216no—
176C4⋊Dic53—24no—
177C4⋊Dic53—22no—
178C4⋊Dic53—21no—
179C4⋊Dic53C22×C412yescomplemented
180C4⋊Dic53C2×D414yes—
181#11719919—24no—
182#11719919—42no—
183#11719919—42no—
184C4×Dic53—22no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—42no—
188C4×Dic53—42no—
189C4×Dic53C2×D414yes—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322C22×C414yescomplemented
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—216no—
198C23×C53—24no—
199C23×C53—24no—
200C23×C53—24no—
201C23×C53—48no—
202C23×C53—48no—
203C23×C53—48no—
204C23×C53—48no—
205C23×C53—48no—
206C23×C53—42no—
207C23×C53—42no—
208C23×C53—42no—
209C23×C53—42no—
210C23×C53—48no—
211C23×C53C4×D412yes—
212C23×C53C2×SD1612yes—
213C23×C53#13512yes—
214C23×C53C22×D411yescharacteristic
215D4×C53—216no—
216D4×C53—24no—
217D4×C53—24no—
218D4×C53—24no—
219D4×C53C4×D418yes—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53C4×D412yes—
231C2×C4×C53C23×C411yescharacteristic
232C2×C4×C53C22×D412yes—
233C2×C4×C53C2×#4112yes—
234C2×C4×C53C2×#4111yescharacteristic
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—28no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—24no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—44no—
247C2×Dic53—416no—
248C2×Dic53—48no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—416no—
256C2×Dic53#12012yes—
257C2×Dic53#12012yes—
258C2×Dic53C2×SD1612yes—
259C2×Dic53C2×SD1612yes—
260C2×Dic53AGL(1,ℤ/8)12yes—
261Dic106—24no—
262Dic106—42no—
263C8×C53—42no—
264#88322—24no—
265C4×#450—531nomaximal, Sylow
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—28no—
273C22×C53—432no—
274C22×C53—44no—
275C22×C53—44no—
276C22×C53—44no—
277C22×C53—44no—
278C22×C53—44no—
279C22×C53—44no—
280C22×C53C4×SD1614yescomplemented
281C22×C53C2×C4×D411yescharacteristic
282C22×C53C2×#12011yescharacteristic
283C22×C53#52511yescharacteristic
284C22×C53C22×SD1611yescharacteristic
285C22×C53C2×#13511yescharacteristic
286C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C53#57511yescharacteristic
288C4×C53—22no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53C2×C4×D412yes—
292C4×C53#52512yes—
293Dic53—22no—
294Dic53—22no—
295Dic53—42no—
296Dic53—44no—
297Dic53—41no—
298Dic53—42no—
299Dic53#45014yescomplemented
300C22×C4×D4—531no—
301C2×#450—531no—
302C2×C4×SD16—532no—
303C4×C22⋊Q8—531no—
304#1877—532no—
305#1872—531no—
306#1864—532no—
307#1853—531no—
308#1786D5311yescharacteristic
309C4×#102—532no—
310C4×C22⋊C8—531no—
311C2×C53—24no—
312C2×C53—48no—
313C2×C53C4×#12011yescharacteristic
314C2×C53C2×C4×SD1611yescharacteristic
315C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C53C2×#45011yescharacteristic
317C2×C53#300211yescharacteristic
318C2×C53#310111yescharacteristic
319C2×C53#310511yescharacteristic
320C23×D4C2×D5311yescharacteristic
321C24×C4—1061no—
322C22×SD16—532no—
323C2×C22⋊Q8—531no—
324C2×C4×Q8—531no—
325C2×C4×D4—532no—
326C2×C4×D4—532no—
327C2×C4×D4—1061no—
328C2×C4×D4—1061no—
329C2×C4×D4—1068no—
330C22×C4⋊C4—531no—
331C22×C22⋊C4—1061no—
332C22×C42—531no—
333#450—538no—
334C4×SD16—1068no—
335C2×#106—532no—
336C2×#103—532no—
337C2×#102—532no—
338C2×#102C2×D5312yes—
339C2×#102C2×D5312yes—
340C2×C22⋊C8—531no—
341C2×C22⋊C8C2×D5311yescharacteristic
342C2×C4×C8—532no—
343#391—531no—
344#389—531no—
345#387—1061no—
346#385—1061no—
347#380C2×D5311yescharacteristic
348C4×C4⋊C4—531no—
349C4×C4⋊C4—1061no—
350C4×C22⋊C4—1061no—
351#340—532no—
352#340—532no—
353#336—532no—
354C53C4×#45011yescharacteristic, complemented, Sylow
355C25—21no—
356C22×Q8—531no—
357C22×D4—24no—
358C22×D4—24no—
359C22×D4—21no—
360C22×D4—22no—
361C22×D4C4×D5314yes—
362C22×D4C22×D5312yes—
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—1062no—
366C23×C4—1062no—
367C23×C4—1068no—
368C23×C4—1061no—
369C23×C4—2124no—
370C23×C4—2124no—
371C23×C4C22×D5311yescharacteristic
372C2×SD16—538no—
373C2×SD16—1064no—
374C2×SD16—1064no—
375C22×C8—532no—
376C22×C8C22×D5312yes—
377C22⋊Q8—534no—
378C22⋊Q8—1062no—
379C4×Q8—1064no—
380C4×D4—538no—
381C4×D4—1064no—
382C4×D4—1064no—
383C4×D4—1068no—
384C4×D4—1068no—
385C4×D4—1064no—
386C2×C4⋊C4—532no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—1064no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1062no—
397C2×C22⋊C4—1062no—
398C2×C22⋊C4—1068no—
399C2×C22⋊C4—1061no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—1064no—
405C2×C42—1061no—
406C2×C42—1061no—
407#106—1064no—
408#103—1064no—
409#102—24no—
410#102—24no—
411#102—538no—
412C22⋊C8—22no—
413C22⋊C8—534no—
414C4×C8—1064no—
415#95—21no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420C24—24no—
421C24—24no—
422C24—24no—
423C24—22no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24D4×D5311yescharacteristic
429C2×Q8—532no—
430C2×Q8—1061no—
431C2×Q8—1062no—
432C2×Q8—1062no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—216no—
436C2×D4—216no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4C2×C4×D5314yes—
442C2×D4D4×D5314yes—
443C2×D4#1171994714yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—531no—
449C22×C4—534no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1068no—
457C22×C4—1064no—
458C22×C4—1068no—
459C22×C4—1062no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1062no—
463C22×C4—2124no—
464C22×C4—2124no—
465C22×C4—2124no—
466C22×C4—2121no—
467C22×C4—21232no—
468C22×C4—2124no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4C2×C4×D5312yes—
472C22×C4D4×D5311yescharacteristic
473C22×C4#1171994712yes—
474C22×C4C23×D5311yescharacteristic
475SD16—10616no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—534no—
480C2×C8—1062no—
481C2×C8—1062no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—532no—
485C4⋊C4—534no—
486C4⋊C4—1064no—
487C4⋊C4—1064no—
488C4⋊C4—1062no—
489C4⋊C4—1061no—
490C4⋊C4—2122no—
491C4⋊C4—2122no—
492C22⋊C4—42no—
493C22⋊C4—1068no—
494C22⋊C4—1068no—
495C22⋊C4—1064no—
496C42—534no—
497C42—1062no—
498C42—1062no—
499C42—1062no—
500C42—2122no—
501C42—2122no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—216no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—48no—
520C23#1313317712yes—
521C23SD16×D5312yes—
522C23#1313323612yes—
523C23C2×D4×D5311yescharacteristic
524Q8—1064no—
525Q8—2122no—
526D4—216no—
527D4—24no—
528D4—24no—
529D4—24no—
530D4#1313317718yes—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—532no—
542C2×C4—532no—
543C2×C4—532no—
544C2×C4—532no—
545C2×C4—532no—
546C2×C4—1068no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—1061no—
554C2×C4—1064no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—21216no—
558C2×C4—2124no—
559C2×C4—21216no—
560C2×C4—2128no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2122no—
564C2×C4—2121no—
565C2×C4—2122no—
566C2×C4—2121no—
567C2×C4#1313317712yes—
568C2×C4C22×C4×D5311yescharacteristic
569C2×C4C2×D4×D5312yes—
570C2×C4C2×#1171994712yes—
571C2×C4C2×#1171994711yescharacteristic
572C8—42no—
573C8—1064no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—28no—
581C22—432no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22#1681071414yescomplemented
589C22C2×#1313317711yescharacteristic
590C22#120×D5311yescharacteristic
591C22#1681112911yescharacteristic
592C22C2×SD16×D5311yescharacteristic
593C22C2×#1313323611yescharacteristic
594C22AGL(1,ℤ/8)×D5311yescharacteristic
595C22#1681141911yescharacteristic
596C4—22no—
597C4—81no—
598C4—81no—
599C4—534no—
600C4—1062no—
601C4—1062no—
602C4—2121no—
603C4—2122no—
604C4—2122no—
605C4—2124no—
606C4C2×#1313317712yes—
607C4#1681112912yes—
608C2—24no—
609C2—48no—
610C2#2027721411yescharacteristic
611C2C2×#1681071411yescharacteristic
612C2#2028351911yescharacteristic
613C2#450×D5311yescharacteristic
614C2#2028392811yescharacteristic
615C2#2028471811yescharacteristic
616C2#2028475711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1786C2
socleC23×C53C22×D4
prime radicalC23×D4×C53C22
cosocleC22×C4×C53C24
hypercenter#1786D53
hypocenterC53C4×#450
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
3. C2×#16810714C2semidirect
4. #20277243C2semidirect
5. #20275047C2semidirect
6. #20275020C2semidirect
7. #20274974C2semidirect
8. #20274907C2semidirect
9. #102×Dic53C2semidirect
10. C22⋊C8×Dic53C2semidirect
15. C2×#13133182C22semidirect
24. #16810774C4semidirect
26. C2×#13133148C22semidirect
27. C2×#13133142C22semidirect
30. C2×C8×Dic53C22semidirect
31. #16810528C22semidirect
32. #16810519C22semidirect
35. C4⋊C4×Dic53C22semidirect
38. #16810386C22semidirect
39. #16810385C22semidirect
40. #16810377C22semidirect
57. C2×#11719923C2×C4semidirect
60. #13133196C2×C4semidirect
88. #13133138C2×C4semidirect
89. C22⋊#88322C2×C4semidirect
157. C22×Dic53SD16semidirect
165. C2×Dic106C22×C4semidirect
179. C4⋊Dic53C22×C4semidirect
192. C2×#88322C22×C4semidirect
280. C22×C53C4×SD16semidirect
299. Dic53#450semidirect
354. C53C4×#450semidirect
588. C22#16810714semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic532
C2×#168107742
C2×#168107142
#202772432
#202750472
#202750202
#202749742
#202749072
#102×Dic532
C22⋊C8×Dic532
C53×#17862
C4×#45053