Finite Group Databasebeta

← #27416752

Subgroups — #27416752

Order 13568.

1844 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133191C211yesmaximal, characteristic
2C2×#16810941C212yesmaximal, complemented
3#20275904C214yesmaximal, complemented
4#20275032C212yesmaximal, complemented
5#20274907C212yesmaximal, complemented
6#20274645C212yesmaximal, complemented
7#20274625C211yesmaximal, characteristic, complemented
8C2×C53×#380C211yesmaximal, characteristic, complemented
9C23×D212C2211yescharacteristic
10C23×C22⋊D53—21no—
11C2×#13133191—28no—
12C2×#13133191—21no—
13C2×#13133191—21no—
14C2×#13133191C2212yes—
15C2×#13133191C2212yes—
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719921—21no—
18#16810941—28no—
19C2×#13133148C2214yescomplemented
20C2×#13133137C2214yescomplemented
21C2×#13133137C2214yescomplemented
22C2×C22⋊#88322C2212yes—
23C2×C22⋊#88322C2212yescomplemented
24C2×C22⋊#88322C2212yescomplemented
25#16810385C2214yescomplemented
26C24×C4×C53C2211yescharacteristic
27C22×C4⋊C4×C53C2211yescharacteristic, complemented
28C22×C4⋊C4×C53C2211yescharacteristic, complemented
29C22×C22⋊C4×C53—21no—
30C53×#380—21no—
31C53×#380—21no—
32C53×#380C2212yes—
33C53×#380C2212yescomplemented
34C2×#95×C53—21no—
35C24×D53—21no—
36C22×D212—21no—
37C22×D212—24no—
38C22×D212—24no—
39C22×D212—42no—
40C22×D212C2312yes—
41C22×C22⋊D53—28no—
42C22×C22⋊D53—22no—
43C22×C22⋊D53—22no—
44C22×C22⋊D53—44no—
45C22×C22⋊D53—44no—
46C23×Dic53—21no—
47C22×#88322C2314yes—
48#13133191—28no—
49#13133191—28no—
50#13133191—24no—
51#13133191—24no—
52#13133191—24no—
53#13133191D418yes—
54C2×C4⋊Dic53—24no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53C2312yes—
57C2×#11719921—28no—
58C2×#11719921—22no—
59C2×#11719921—22no—
60#13133148—28no—
61#13133137—28no—
62#13133137—44no—
63C22⋊#88322—24no—
64C22⋊#88322—24no—
65C22⋊#88322—42no—
66C25×C53D411yescharacteristic
67C23×C4×C53—24no—
68C23×C4×C53—24no—
69C23×C4×C53—21no—
70C23×C4×C53—21no—
71C23×C4×C53—42no—
72C23×C4×C53D411yescharacteristic
73C23×C4×C53D411yescharacteristic
74C23×C4×C53D411yescharacteristic
75C23×C4×C53C2311yescharacteristic
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2311yescharacteristic
78C2×C4⋊C4×C53—21no—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53—24no—
81C2×C4⋊C4×C53—42no—
82C2×C4⋊C4×C53C2312yes—
83C2×C4⋊C4×C53C2312yes—
84C2×C22⋊C4×C53—22no—
85C2×C22⋊C4×C53—22no—
86C2×C22⋊C4×C53—44no—
87#95×C53—22no—
88#95×C53—22no—
89C24×C53—22no—
90C24×C53—22no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53—41no—
94C24×C53—41no—
95C24×C53SD1614yes—
96C24×C53C2×D411yescharacteristic
97C24×C53C2×D411yescharacteristic
98C24×C53C2×D411yescharacteristic
99C23×D53—24no—
100C23×D53—22no—
101C23×D53—42no—
102C23×D53—42no—
103C23×D53—42no—
104C23×D53—42no—
105C23×D53—42no—
106C23×D53—42no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53—416no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D414yes—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53#4114yes—
137C22×C4×C53C2411yescharacteristic
138C2×C22⋊D53—28no—
139C2×C22⋊D53—28no—
140C2×C22⋊D53—432no—
141C2×C22⋊D53—44no—
142C2×C22⋊D53—44no—
143C2×C22⋊D53—44no—
144C2×C22⋊D53—44no—
145C22×Dic53—24no—
146C22×Dic53—22no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C2×D212—24no—
150C2×D212—24no—
151C2×D212—24no—
152C2×D212—22no—
153C2×D212—416no—
154C2×D212—42no—
155C2×D212—42no—
156C2×D212C2×D414yes—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—24no—
159C4⋊C4×C53—42no—
160C4⋊C4×C53—42no—
161C22⋊C4×C53—44no—
162C22⋊C4×C53—44no—
163#11719921—28no—
164#11719921—28no—
165C4⋊Dic53—24no—
166C4⋊Dic53—22no—
167C4⋊Dic53C2×D414yes—
168C2×#88322—24no—
169C2×#88322—24no—
170C2×#88322—24no—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—28no—
186C23×C53—28no—
187C23×C53—44no—
188C23×C53—44no—
189C23×C53—44no—
190C23×C53—44no—
191C23×C53—44no—
192C23×C53#12012yes—
193C23×C53#12312yes—
194C23×C53C2×SD1612yes—
195C23×C53C2×SD1612yes—
196C23×C53C2×SD1612yes—
197C23×C53AGL(1,ℤ/8)12yes—
198C23×C53AGL(1,ℤ/8)12yes—
199C23×C53C22×D411yescharacteristic
200C22×D53—24no—
201C22×D53—48no—
202C22×D53—48no—
203C22×D53—48no—
204C22×D53—42no—
205C22×D53—42no—
206C22×D53—42no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—84no—
211C22×D53—84no—
212C22×D53—84no—
213C22×D53—84no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—28no—
219C2×C4×C53—28no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—44no—
224C2×C4×C53—44no—
225C2×C4×C53—44no—
226C2×C4×C53—44no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—84no—
234C2×C4×C53—84no—
235C2×C4×C53#12012yes—
236C2×C4×C53#12312yes—
237C2×C4×C53C22×D412yes—
238C2×C4×C53C2×#4112yes—
239C2×C4×C53H(5,𝔽2)12yes—
240C22⋊D53—416no—
241C22⋊D53—416no—
242C2×Dic53—24no—
243C2×Dic53—48no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246D212—28no—
247D212—48no—
248D212—44no—
249#88322—44no—
250#24005—531nomaximal, Sylow
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—416no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53#45014yes—
272C22×C53#48114yescomplemented
273C22×C53C2×#12011yescharacteristic
274C22×C53C2×#12311yescharacteristic
275C22×C53#53411yescharacteristic
276C22×C53C22×SD1611yescharacteristic
277C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C53#57511yescharacteristic
280C2×D53—44no—
281C2×D53—44no—
282C2×D53—44no—
283C2×D53—816no—
284C2×D53—82no—
285C2×D53—82no—
286C2×D53—82no—
287C2×D53—82no—
288C4×C53—22no—
289C4×C53—22no—
290C4×C53—41no—
291C4×C53—82no—
292C4×C53—82no—
293C4×C53#54614yes—
294Dic53—44no—
295C22×#121—531no—
296C2×#481—532no—
297C2×#380D5311yescharacteristic, complemented
298#2034—534no—
299#1874—532no—
300#1853—532no—
301#1786—532no—
302#1779—531no—
303C2×C53—24no—
304C2×C53—24no—
305C2×C53—42no—
306C2×C53#240511yescharacteristic
307C2×C53C2×#45011yescharacteristic
308C2×C53#300611yescharacteristic
309C2×C53C2×#48111yescharacteristic
310C2×C53#309411yescharacteristic
311C2×C53#318711yescharacteristic
312C2×C53#319211yescharacteristic
313D53—84no—
314C23×D4—531no—
315C23×D4—1061no—
316C24×C4C2×D5311yescharacteristic
317C2×#121—532no—
318C2×#121—532no—
319C2×#121—1068no—
320C2×#121—1061no—
321C2×#121—1061no—
322C22×C4⋊C4—531no—
323C22×C4⋊C4C2×D5311yescharacteristic, complemented
324C22×C4⋊C4C2×D5311yescharacteristic, complemented
325C22×C22⋊C4—21no—
326C22×C22⋊C4—1061no—
327#481—1068no—
328C2×#106—534no—
329C2×#102—534no—
330C2×#102—534no—
331C2×C22⋊C8—532no—
332C2×C22⋊C8—532no—
333C2×C22⋊C8—532no—
334#380—21no—
335#380—21no—
336#380C2×D5312yes—
337#380C2×D5312yescomplemented
338C2×#95—21no—
339#340—534no—
340C53#2400511yescharacteristic, complemented, Sylow
341C25—1061no—
342C25C22⋊D5311yescharacteristic
343C22×D4—532no—
344C22×D4—1064no—
345C22×D4—1068no—
346C22×D4—1062no—
347C22×D4—1062no—
348C22×D4—1064no—
349C22×D4—1061no—
350C22×D4—2122no—
351C22×D4—2124no—
352C22×D4—2124no—
353C23×C4—24no—
354C23×C4—24no—
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—42no—
358C23×C4—1061no—
359C23×C4C22⋊D5311yescharacteristic
360C23×C4C22⋊D5311yescharacteristic
361C23×C4C22⋊D5311yescharacteristic
362C23×C4C22×D5311yescharacteristic
363C23×C4C22×D5311yescharacteristic
364C23×C4C22×D5311yescharacteristic
365C22×C8—534no—
366#121—538no—
367#121—1064no—
368#121—1064no—
369#121—1068no—
370#121—1068no—
371#121—1064no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—42no—
376C2×C4⋊C4—532no—
377C2×C4⋊C4—1064no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4C22×D5312yes—
380C2×C4⋊C4C22×D5312yes—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—44no—
384C2×C22⋊C4—1068no—
385C2×C22⋊C4—1062no—
386C2×C22⋊C4—1062no—
387#106—1068no—
388#102—1068no—
389#102—2124no—
390C22⋊C8—1064no—
391C22⋊C8—1064no—
392C22⋊C8—2122no—
393#95—22no—
394#95—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—41no—
400C24—41no—
401C24—1064no—
402C24—1062no—
403C24—2122no—
404C24—2122no—
405C24—2122no—
406C24—2122no—
407C24—2122no—
408C24—2122no—
409C24#1171992414yes—
410C24C2×C22⋊D5311yescharacteristic
411C24C2×C22⋊D5311yescharacteristic
412C24C2×C22⋊D5311yescharacteristic
413C2×D4—534no—
414C2×D4—1064no—
415C2×D4—1068no—
416C2×D4—1068no—
417C2×D4—1062no—
418C2×D4—1064no—
419C2×D4—1064no—
420C2×D4—21216no—
421C2×D4—2122no—
422C2×D4—21232no—
423C2×D4—2124no—
424C2×D4—2124no—
425C2×D4—2124no—
426C2×D4—2124no—
427C2×D4—2122no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—416no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—1064no—
454C22×C4—1062no—
455C22×C4—2122no—
456C22×C4—2122no—
457C22×C4#1171994514yes—
458C22×C4D4×D5314yes—
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C23×D5311yescharacteristic
463C2×C8—1064no—
464C2×C8—1064no—
465C2×C8—1064no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—534no—
471C4⋊C4—1064no—
472C4⋊C4—1062no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—1068no—
476C22⋊C4—1068no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—28no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—28no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—1064no—
499C23—4244no—
500C23—4244no—
501C23—4244no—
502C23—4244no—
503C23—2128no—
504C23—2128no—
505C23—2128no—
506C23—2122no—
507C23—2122no—
508C23—2122no—
509C23—2122no—
510C23—2122no—
511C23—2122no—
512C23#1313318512yes—
513C23#1313320012yes—
514C23C2×#1171992412yes—
515C23C2×#1171992412yes—
516C23C2×#1171992412yes—
517C23#1313324212yes—
518C23#1313324312yes—
519C23C22×C22⋊D5311yescharacteristic
520D4—1068no—
521D4—2128no—
522D4—21216no—
523D4—21216no—
524D4—2124no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—1064no—
547C2×C4—2128no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4#1313318512yes—
551C2×C4#1313320012yes—
552C2×C4C2×D4×D5312yes—
553C2×C4C2×#1171994512yes—
554C2×C4#1313326212yes—
555C8—2124no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—416no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—42416no—
577C22—4242no—
578C22—4242no—
579C22—4242no—
580C22—4242no—
581C22—2124no—
582C22—2124no—
583C22—2124no—
584C22#1681077314yes—
585C22#1681094114yescomplemented
586C22C2×#1313318511yescharacteristic
587C22C2×#1313320011yescharacteristic
588C22#1681118211yescharacteristic
589C22C22×#1171992411yescharacteristic
590C22C2×#1313324211yescharacteristic
591C22C2×#1313324311yescharacteristic
592C22#1681142111yescharacteristic
593C4—22no—
594C4—22no—
595C4—41no—
596C4—82no—
597C4—82no—
598C4—2124no—
599C4#1681126314yes—
600C2—24no—
601C2—24no—
602C2—42no—
603C2—4244no—
604C2#2027804211yescharacteristic
605C2C2×#1681077311yescharacteristic
606C2#2028396511yescharacteristic
607C2C2×#1681094111yescharacteristic
608C2#2028465511yescharacteristic
609C2#2028546811yescharacteristic
610C2#2028551211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#380C2
socleC23×C53C22×D4
prime radicalC22×#13133191C2
hypercenterC2×#380D53
hypocenterC53#24005
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#16810941C2semidirect
3. #20275904C2semidirect
4. #20275032C2semidirect
5. #20274907C2semidirect
6. #20274645C2semidirect
7. #20274625C2semidirect
8. C2×C53×#380C2semidirect
19. C2×#13133148C22semidirect
20. C2×#13133137C22semidirect
21. C2×#13133137C22semidirect
23. C2×C22⋊#88322C22semidirect
24. C2×C22⋊#88322C22semidirect
25. #16810385C22semidirect
27. C22×C4⋊C4×C53C22semidirect
28. C22×C4⋊C4×C53C22semidirect
33. C53×#380C22semidirect
272. C22×C53#481semidirect
297. C2×#380D53semidirect
323. C22×C4⋊C4C2×D53semidirect
324. C22×C4⋊C4C2×D53semidirect
337. #380C2×D53semidirect
340. C53#24005semidirect
585. C22#16810941semidirect

Maximal subgroups

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

subgroupindex
C22×#131331912
C2×#168109412
#202759042
#202750322
#202749072
#202746452
#202746252
C2×C53×#3802
#2400553