Finite Group Databasebeta

← #27385475

Subgroups — #27385475

Order 13568.

2046 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×#13133243C211yesmaximal, characteristic
2C2×#16811091C211yesmaximal, characteristic, complemented
3C2×#16810625C211yesmaximal, characteristic, complemented
4C2×#16810526C211yesmaximal, characteristic, complemented
5#20275067C214yesmaximal, complemented
6#20275060C214yesmaximal, complemented
7#20274436C211yesmaximal, characteristic, complemented
8#20274408C211yesmaximal, characteristic, complemented
9C2×C53×#418C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C2×#13133243—24no—
12C2×#13133243—24no—
13C2×#13133243C418yescomplemented
14C22×#11719923C2212yes—
15C22×#11719922C2212yes—
16C22×#11719917C2211yescharacteristic
17#16811091—24no—
18#16811091—24no—
19C2×#13133182C2212yescomplemented
20C2×#13133179C2212yescomplemented
21C2×#13133177—22no—
22C2×#13133174—21no—
23C22×C4⋊Dic53C2211yescharacteristic
24C22×C22⋊Dic53—22no—
25C22×C4×Dic53C2211yescharacteristic
26C2×C42×D53—21no—
27#16810625—24no—
28C2×#13133142C2212yescomplemented
29C2×#13133139C2212yescomplemented
30#16810526—21no—
31#16810526—21no—
32#16810526C2214yes—
33#16810385C2214yescomplemented
34#16810377C2214yescomplemented
35C23×D4×C53C2211yescharacteristic
36C22×M16×C53C2211yescharacteristic
37C2×#117×C53C2211yescharacteristic
38C53×#418—24no—
39C2×#102×C53C2214yes—
40C2×#11719945—22no—
41C2×#11719945—22no—
42C2×#11719945—22no—
43C2×#11719945C2×C414yes—
44C2×#11719945D414yes—
45C2×#11719945D414yes—
46C22×Dic106C2311yescharacteristic
47C22×D212C2311yescharacteristic
48C22×C22⋊D53—21no—
49C23×Dic53—21no—
50C23×Dic53D412yes—
51C22×C4×D53—21no—
52C22×C4×D53—22no—
53#13133243—232no—
54C2×#11719923—24no—
55C2×#11719923—24no—
56C2×#11719923C2×C418yescomplemented
57C2×#11719922—24no—
58C2×#11719922—24no—
59C2×#11719922C2×C418yescomplemented
60C2×#11719917—22no—
61C2×#11719917—22no—
62C2×#11719917C2×C414yescomplemented
63C22×#88322C2312yes—
64#13133182—24no—
65#13133182—24no—
66#13133179—24no—
67#13133179—24no—
68#13133177—216no—
69#13133174—28no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53C2312yes—
74C2×#11719919—22no—
75C2×C22⋊Dic53—24no—
76C2×C22⋊Dic53—24no—
77C2×C22⋊Dic53—48no—
78C2×#11719921—22no—
79C2×C4×Dic53—22no—
80C2×C4×Dic53—22no—
81C2×C4×Dic53—22no—
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53C2312yes—
85C42×D53—28no—
86#13133142—24no—
87#13133139—24no—
88C25×C53—21no—
89C22×D4×C53—22no—
90C22×D4×C53—22no—
91C22×D4×C53—22no—
92C22×D4×C53—21no—
93C22×D4×C53—22no—
94C22×D4×C53—22no—
95C22×D4×C53C2×C414yescomplemented
96C22×D4×C53C2312yes—
97C23×C4×C53C2311yescharacteristic
98C2×M16×C53—22no—
99C2×M16×C53—22no—
100C2×M16×C53—22no—
101C22×C8×C53C2312yes—
102#117×C53—22no—
103#117×C53D414yes—
104C2×C4⋊C4×C53C2312yes—
105C2×C22⋊C4×C53—21no—
106C2×C42×C53—21no—
107#102×C53—28no—
108C24×C53—24no—
109C24×C53—22no—
110C24×C53—44no—
111C24×C53—44no—
112C24×C53—42no—
113C24×C53—42no—
114C24×C53C2×D411yescharacteristic
115C23×D53—21no—
116C2×D4×C53—28no—
117C2×D4×C53—28no—
118C2×D4×C53—24no—
119C2×D4×C53—22no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—22no—
124C2×D4×C53—42no—
125C2×D4×C53—44no—
126C2×D4×C53—42no—
127C2×D4×C53—44no—
128C2×D4×C53C22×C414yes—
129C22×C4×C53—22no—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D412yes—
135C22×C4×C53#4112yes—
136C22×C4×C53#4112yes—
137C22×C4×C53C2411yescharacteristic
138C2×C22⋊D53—24no—
139C2×C22⋊D53—24no—
140C2×C22⋊D53—24no—
141C22×Dic53—21no—
142C22×Dic53—24no—
143C22×Dic53—24no—
144C22×Dic53—22no—
145C22×Dic53—42no—
146C22×Dic53—44no—
147C22×Dic53—42no—
148C22×Dic53C2×D412yes—
149C22×Dic53C2×D412yes—
150#11719945—28no—
151#11719945—28no—
152#11719945—28no—
153#11719945C22⋊C4116yescomplemented
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212C22×C412yes—
158C2×D212C2×D412yes—
159C2×D212C2×D412yes—
160C2×C4×D53—24no—
161C2×C4×D53—24no—
162C2×C4×D53—24no—
163C2×C4×D53—28no—
164C2×C4×D53—44no—
165C2×C4×D53—44no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106C22×C412yes—
170C2×Dic106C2×D412yes—
171C2×Dic106C2×D412yes—
172M16×C53—44no—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53C2×D414yes—
178C22⋊C4×C53—24no—
179C42×C53—24no—
180C22⋊Dic53—48no—
181C22⋊Dic53—48no—
182#11719923—216no—
183#11719922—216no—
184#11719921—28no—
185C4⋊Dic53—22no—
186C4⋊Dic53—22no—
187C4⋊Dic53—42no—
188#11719919—28no—
189C4×Dic53—28no—
190C4×Dic53—22no—
191C4×Dic53—22no—
192C4×Dic53—22no—
193C4×Dic53—22no—
194#11719917—28no—
195C2×#88322—22no—
196C2×#88322—22no—
197C2×#88322C22×C414yes—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53—24no—
201C23×C53—416no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—44no—
205C23×C53—44no—
206C23×C53—48no—
207C23×C53—88no—
208C23×C53—84no—
209C23×C53—84no—
210C23×C53C4×D412yes—
211C23×C53#12012yes—
212C23×C53#12112yes—
213C23×C53C22×D411yescharacteristic
214C22×D53—22no—
215C22×D53—22no—
216C22×D53—22no—
217C22×D53—42no—
218C22×D53—42no—
219D4×C53—24no—
220D4×C53—24no—
221D4×C53—42no—
222D4×C53—42no—
223D4×C53—48no—
224C2×C4×C53—24no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53C2×C22⋊C414yes—
228C2×C4×C53C4×D412yes—
229C2×C4×C53C4×D414yes—
230C2×C4×C53#12012yes—
231C2×C4×C53#12114yes—
232C2×C4×C53#12112yes—
233C2×C4×C53#12314yes—
234C2×C4×C53C23×C411yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C22⋊D53—216no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—48no—
250C2×Dic53—42no—
251C2×Dic53—44no—
252C2×Dic53—84no—
253C2×Dic53AGL(1,ℤ/8)14yescomplemented
254D212—22no—
255D212—22no—
256D212—22no—
257D212C2×C22⋊C414yescomplemented
258C4×D53—216no—
259C4×D53—416no—
260Dic106—22no—
261Dic106—22no—
262Dic106—22no—
263Dic106C2×C22⋊C414yescomplemented
264C8×C53—42no—
265#88322—24no—
266#20928—531nomaximal, Sylow
267C22×C53—24no—
268C22×C53—24no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—48no—
272C22×C53—44no—
273C22×C53—816no—
274C22×C53—84no—
275C22×C53—82no—
276C22×C53—82no—
277C22×C53#38614yes—
278C22×C53C2×C4×D411yescharacteristic
279C22×C53C2×#12011yescharacteristic
280C22×C53C2×#12111yescharacteristic
281C22×C53C2×AGL(1,ℤ/8)12yes—
282C22×C53#57512yes—
283C2×D53—24no—
284C2×D53—44no—
285C2×D53—44no—
286C2×D53—42no—
287C2×D53—42no—
288C4×C53—44no—
289C4×C53#38614yes—
290C4×C53C22×C22⋊C411yescharacteristic
291C4×C53C2×C4×D411yescharacteristic
292C4×C53C2×#12111yescharacteristic
293C4×C53C2×#12311yescharacteristic
294Dic53—24no—
295Dic53—24no—
296Dic53—42no—
297Dic53—82no—
298C22×AGL(1,ℤ/8)—531no—
299C2×C4×#41—531no—
300C2×#417—531no—
301C2×#418D5311yescharacteristic, complemented
302C2×#390—531no—
303#1880—534no—
304#1879—534no—
305#1736—531no—
306#1728—531no—
307C2×C53—24no—
308C2×C53—84no—
309C2×C53C2×#38611yescharacteristic
310C2×C53C4×AGL(1,ℤ/8)12yes—
311C2×C53#300612yes—
312C2×C53#303811yescharacteristic
313C2×C53#304011yescharacteristic
314D53—44no—
315C22×#41—531no—
316C23×D4C2×D5311yescharacteristic
317C2×AGL(1,ℤ/8)—538no—
318C2×AGL(1,ℤ/8)—1064no—
319C2×AGL(1,ℤ/8)—1064no—
320C22×SD16—532no—
321C22×D8—532no—
322C22×M16—531no—
323C22×M16C2×D5311yescharacteristic
324C4×#41—1064no—
325C4×#41—1064no—
326C2×C4×Q8—532no—
327C2×C4×D4—532no—
328C2×C4×D4—1062no—
329C2×#117—1061no—
330C2×#117C2×D5311yescharacteristic
331C22×C4⋊C4—531no—
332C22×C22⋊C4—1062no—
333C22×C42—531no—
334C22×C42—1061no—
335#418—24no—
336#417—1064no—
337C2×#103—532no—
338C2×#102—532no—
339C2×#102C2×D5314yes—
340#390—534no—
341#390—1061no—
342#390—1061no—
343#340—534no—
344#336—534no—
345C53#2092811yescharacteristic, complemented, Sylow
346C25—21no—
347C2×#41—534no—
348C2×#41—534no—
349C2×#41—534no—
350C2×#41—1062no—
351C2×#41—1062no—
352C2×#41—1062no—
353C22×Q8—531no—
354C22×D4—22no—
355C22×D4—22no—
356C22×D4—22no—
357C22×D4—21no—
358C22×D4—22no—
359C22×D4—22no—
360C22×D4—531no—
361C22×D4—1061no—
362C22×D4C4×D5314yescomplemented
363C22×D4C22×D5312yes—
364C23×C4—532no—
365C23×C4—1062no—
366C23×C4—1061no—
367C23×C4—1061no—
368C23×C4C22×D5311yescharacteristic
369AGL(1,ℤ/8)—10632no—
370C2×SD16—538no—
371C2×SD16—1064no—
372C2×SD16—1064no—
373C2×D8—538no—
374C2×D8—1064no—
375C2×D8—1064no—
376C2×M16—22no—
377C2×M16—22no—
378C2×M16—22no—
379C2×M16—534no—
380C2×M16—1062no—
381C2×M16—1062no—
382C22×C8—532no—
383C22×C8C22×D5312yes—
384C4×Q8—1064no—
385C4×Q8—1064no—
386C4×D4—10616no—
387C4×D4—1064no—
388C4×D4—1064no—
389#117—22no—
390#117—1068no—
391#117C22⋊D5314yes—
392C2×C4⋊C4—532no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4C22×D5312yes—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—1062no—
400C2×C22⋊C4—1064no—
401C2×C22⋊C4—1064no—
402C2×C22⋊C4—2128no—
403C2×C42—21no—
404C2×C42—531no—
405C2×C42—531no—
406C2×C42—532no—
407C2×C42—1068no—
408C2×C42—1062no—
409C2×C42—1062no—
410C2×C42—1062no—
411#103—1064no—
412#102—28no—
413#102—1064no—
414C24—24no—
415C24—22no—
416C24—44no—
417C24—44no—
418C24—42no—
419C24—42no—
420C24—1061no—
421C24D4×D5311yescharacteristic
422#41—5316no—
423#41—1068no—
424#41—1068no—
425#41—1068no—
426C2×Q8—532no—
427C2×Q8—532no—
428C2×Q8—532no—
429C2×Q8—1061no—
430C2×Q8—1061no—
431C2×Q8—1061no—
432C2×D4—28no—
433C2×D4—24no—
434C2×D4—28no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—42no—
443C2×D4—44no—
444C2×D4—532no—
445C2×D4—532no—
446C2×D4—532no—
447C2×D4—1064no—
448C2×D4—1064no—
449C2×D4—1064no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4—1061no—
453C2×D4C2×C4×D5314yes—
454C22×C4—22no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—1068no—
458C22×C4—1064no—
459C22×C4—1061no—
460C22×C4—1064no—
461C22×C4—1064no—
462C22×C4—1064no—
463C22×C4—1064no—
464C22×C4—1062no—
465C22×C4—2124no—
466C22×C4—2124no—
467C22×C4—2122no—
468C22×C4—2124no—
469C22×C4—2122no—
470C22×C4C2×C4×D5312yes—
471C22×C4#1171994512yes—
472C22×C4D4×D5312yes—
473C22×C4D4×D5312yes—
474C22×C4D4×D5311yescharacteristic
475C22×C4#1171994712yes—
476C22×C4C2×C22⋊D5312yes—
477C22×C4C23×D5311yescharacteristic
478SD16—10616no—
479D8—10616no—
480M16—44no—
481M16—1068no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—534no—
486C2×C8—1062no—
487C2×C8—1062no—
488C4⋊C4—22no—
489C4⋊C4—1068no—
490C4⋊C4—1062no—
491C4⋊C4—1062no—
492C4⋊C4—2122no—
493C4⋊C4C2×C22⋊D5314yes—
494C22⋊C4—24no—
495C22⋊C4—1068no—
496C22⋊C4—2128no—
497C22⋊C4—2128no—
498C42—24no—
499C42—1068no—
500C42—1062no—
501C42—1062no—
502C42—1062no—
503C42—1062no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—416no—
508C23—48no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—84no—
514C23—84no—
515C23—88no—
516C23—1062no—
517C23—1062no—
518C23—1062no—
519C23—2122no—
520C23—2122no—
521C23#1313317712yes—
522C23#1313318512yes—
523C23#1313319012yes—
524C23C2×D4×D5311yescharacteristic
525Q8—534no—
526Q8—1062no—
527Q8—1062no—
528Q8—1062no—
529D4—24no—
530D4—24no—
531D4—48no—
532D4—42no—
533D4—42no—
534D4—534no—
535D4—10616no—
536D4—1062no—
537D4—1062no—
538D4—1062no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—534no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—10616no—
549C2×C4—1062no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—4244no—
553C2×C4—21216no—
554C2×C4—2128no—
555C2×C4—2122no—
556C2×C4—2124no—
557C2×C4C22⋊C4×D5314yes—
558C2×C4#1313317712yes—
559C2×C4C4×C22⋊D5314yes—
560C2×C4#1313318512yes—
561C2×C4#1313318814yes—
562C2×C4#1313319012yes—
563C2×C4#1313319914yes—
564C2×C4C22×C4×D5311yescharacteristic
565C2×C4C2×D4×D5311yescharacteristic
566C2×C4C2×D4×D5311yescharacteristic
567C2×C4C22×C22⋊D5311yescharacteristic
568C2×C4C2×#1171994711yescharacteristic
569C2×C4C2×#1171994511yescharacteristic
570C8—42no—
571C8—1064no—
572C22—22no—
573C22—22no—
574C22—24no—
575C22—24no—
576C22—48no—
577C22—44no—
578C22—816no—
579C22—82no—
580C22—82no—
581C22—84no—
582C22—1064no—
583C22—2122no—
584C22—2122no—
585C22—2124no—
586C22—2124no—
587C22#1681049914yes—
588C22C2×#1313317711yescharacteristic
589C22C2×#1313318511yescharacteristic
590C22C2×#1313319011yescharacteristic
591C22AGL(1,ℤ/8)×D5312yes—
592C22#1681141912yes—
593C4—44no—
594C4—1064no—
595C4—1064no—
596C4—4242no—
597C4—2122no—
598C4#1681049914yes—
599C4C2×C22⋊C4×D5311yescharacteristic
600C4C2×C4×C22⋊D5311yescharacteristic
601C4C2×#1313318811yescharacteristic
602C4C2×#1313319911yescharacteristic
603C2—24no—
604C2—84no—
605C2—2124no—
606C2C2×#1681049911yescharacteristic
607C2#2028351912yes—
608C2#2028396312yes—
609C2#2028419411yescharacteristic
610C2#2028420811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#418C2
socleC23×C53C22×D4
prime radicalC22×#13133243C2
cosocleC22×C4×C53C24
hypercenterC2×#418D53
hypocenterC53#20928
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×#16811091C2semidirect
3. C2×#16810625C2semidirect
4. C2×#16810526C2semidirect
5. #20275067C2semidirect
6. #20275060C2semidirect
7. #20274436C2semidirect
8. #20274408C2semidirect
9. C2×C53×#418C2semidirect
13. C2×#13133243C4semidirect
19. C2×#13133182C22semidirect
20. C2×#13133179C22semidirect
28. C2×#13133142C22semidirect
29. C2×#13133139C22semidirect
33. #16810385C22semidirect
34. #16810377C22semidirect
56. C2×#11719923C2×C4semidirect
59. C2×#11719922C2×C4semidirect
62. C2×#11719917C2×C4semidirect
95. C22×D4×C53C2×C4semidirect
153. #11719945C22⋊C4semidirect
253. C2×Dic53AGL(1,ℤ/8)semidirect
257. D212C2×C22⋊C4semidirect
263. Dic106C2×C22⋊C4semidirect
301. C2×#418D53semidirect
345. C53#20928semidirect
362. C22×D4C4×D53semidirect

Maximal subgroups

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

subgroupindex
C22×#131332432
C2×#168110912
C2×#168106252
C2×#168105262
#202750672
#202750602
#202744362
#202744082
C2×C53×#4182
#2092853