Finite Group Databasebeta

← #27385316

Subgroups — #27385316

Order 13568.

2258 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811130C211yesmaximal, characteristic
2C22×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3C2×#16810624C211yesmaximal, characteristic, complemented
4C2×#16810586C211yesmaximal, characteristic, complemented
5#20275029C214yesmaximal, complemented
6#20275019C214yesmaximal, complemented
7#20274435C211yesmaximal, characteristic, complemented
8#20274434C211yesmaximal, characteristic, complemented
9C2×C53×#417C211yesmaximal, characteristic, complemented
10C23×C4×D53—21no—
11C23×C4×D53C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13#16811130—24no—
14#16811130C418yescomplemented
15C2×#13133198—21no—
16C2×#13133193C2212yes—
17C2×#13133188C2212yes—
18C2×#13133185—21no—
19C2×C4×C22⋊D53—22no—
20C2×#13133176C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C22×#11719919—21no—
23C2×C4⋊C4×D53—22no—
24C2×C4⋊C4×D53—24no—
25C2×C4⋊C4×D53—24no—
26C2×C4⋊C4×D53C2214yes—
27#16810624—24no—
28C2×#13133141C2212yescomplemented
29C2×#13133138C2212yescomplemented
30#16810586—24no—
31C2×#13133129C2214yes—
32#16810386C2214yescomplemented
33#16810385C2214yescomplemented
34C22×#41×C53C2211yescharacteristic
35C22×M16×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C53×#417—24no—
38C2×#103×C53C2212yescomplemented
39C2×#102×C53C2212yescomplemented
40C24×D53D411yescharacteristic
41C22×C22⋊D53—22no—
42C23×Dic53—21no—
43C23×Dic53D411yescharacteristic
44C22×C4×D53—22no—
45C22×C4×D53—22no—
46C22×C4×D53—22no—
47C22×C4×D53—22no—
48C22×C4×D53—28no—
49C22×C4×D53—24no—
50C22×C4×D53—48no—
51C22×C4×D53C2×C414yes—
52C22×C4×D53D412yes—
53C22×C4×D53D414yes—
54C22×C4×D53C2312yes—
55C2×#11719917—22no—
56C2×#11719917—22no—
57C2×#11719917—22no—
58C22×#88322C2312yes—
59#13133198—28no—
60#13133193—24no—
61#13133193C2×C418yescomplemented
62#13133188—24no—
63#13133188C2×C418yescomplemented
64#13133185—28no—
65C4×C22⋊D53—216no—
66#13133176—22no—
67#13133176C2×C414yescomplemented
68C2×C4⋊Dic53—22no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53D414yescomplemented
71C2×C4⋊Dic53C2312yes—
72C2×C4⋊Dic53C2312yes—
73C2×#11719919—28no—
74C2×#11719919—24no—
75C2×#11719919—22no—
76C4⋊C4×D53—216no—
77C4⋊C4×D53—28no—
78C4⋊C4×D53—28no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—22no—
81C2×#11719921—22no—
82C2×C4×Dic53—21no—
83#13133141—24no—
84#13133138—24no—
85#13133129—28no—
86C2×#41×C53—22no—
87C2×#41×C53—22no—
88C2×#41×C53—22no—
89C2×#41×C53—22no—
90C2×#41×C53—22no—
91C2×#41×C53C2×C414yescomplemented
92C22×Q8×C53C2311yescharacteristic
93C22×D4×C53—21no—
94C22×D4×C53C2311yescharacteristic
95C23×C4×C53—21no—
96C23×C4×C53—21no—
97C23×C4×C53C2311yescharacteristic
98C2×M16×C53—22no—
99C2×M16×C53—22no—
100C2×M16×C53—22no—
101C22×C8×C53C2312yes—
102C2×C4⋊C4×C53—22no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53D414yescomplemented
105C2×C4⋊C4×C53C2312yes—
106#103×C53—24no—
107#102×C53—24no—
108C24×C53—21no—
109C24×C53C2×D411yescharacteristic
110C23×D53—22no—
111C23×D53—22no—
112C23×D53—24no—
113C23×D53—24no—
114C23×D53C22⋊C414yes—
115C23×D53C2×D412yes—
116#41×C53—28no—
117#41×C53—28no—
118#41×C53—44no—
119#41×C53—44no—
120C2×Q8×C53—21no—
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53C22×C412yescomplemented
126C2×D4×C53—24no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—42no—
133C2×D4×C53—42no—
134C2×D4×C53C22×C412yescomplemented
135C22×C4×C53—24no—
136C22×C4×C53—22no—
137C22×C4×C53—21no—
138C22×C4×C53—24no—
139C22×C4×C53—44no—
140C22×C4×C53—42no—
141C22×C4×C53—42no—
142C22×C4×C53C22×C412yes—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D412yes—
147C22×C4×C53#4112yes—
148C22×C4×C53#4112yes—
149C22×C4×C53C2411yescharacteristic
150C2×C22⋊D53—28no—
151C2×C22⋊D53—44no—
152C2×C22⋊D53—44no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—24no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—44no—
159C22×Dic53C22⋊C414yes—
160C22×Dic53C2×D412yes—
161C2×C4×D53—28no—
162C2×C4×D53—28no—
163C2×C4×D53—24no—
164C2×C4×D53—28no—
165C2×C4×D53—28no—
166C2×C4×D53—24no—
167C2×C4×D53—22no—
168C2×C4×D53—22no—
169C2×C4×D53—22no—
170C2×C4×D53—22no—
171C2×C4×D53—216no—
172C2×C4×D53—432no—
173C2×C4×D53—48no—
174C2×C4×D53—48no—
175C2×C4×D53C22⋊C418yes—
176C2×C4×D53C22×C414yes—
177C2×C4×D53C2×D414yes—
178M16×C53—44no—
179C2×C8×C53—22no—
180C2×C8×C53—22no—
181C2×C8×C53—22no—
182C4⋊C4×C53—24no—
183C4⋊C4×C53—22no—
184C4⋊C4×C53C2×D414yes—
185C22⋊Dic53—24no—
186C22⋊Dic53—28no—
187#11719921—28no—
188C4⋊Dic53—24no—
189C4⋊Dic53—22no—
190C4⋊Dic53—22no—
191C4⋊Dic53C22×C414yescomplemented
192C4⋊Dic53C2×D414yes—
193#11719919—216no—
194#11719919—28no—
195C4×Dic53—24no—
196#11719917—44no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—42no—
206C23×C53—42no—
207C23×C53C2×C22⋊C412yes—
208C23×C53#12012yes—
209C23×C53#12312yes—
210C23×C53C22×D411yescharacteristic
211C22×D53—28no—
212C22×D53—216no—
213C22×D53—28no—
214C22×D53—22no—
215C22×D53—22no—
216C22×D53—22no—
217C22×D53—22no—
218C22×D53—28no—
219C22×D53—216no—
220C22×D53C2×C22⋊C414yes—
221C22×D53AGL(1,ℤ/8)14yes—
222C22×D53#13714yes—
223Q8×C53—22no—
224Q8×C53—22no—
225Q8×C53—41no—
226Q8×C53—41no—
227D4×C53—22no—
228D4×C53—22no—
229D4×C53—48no—
230D4×C53—41no—
231D4×C53—41no—
232C2×C4×C53—22no—
233C2×C4×C53—24no—
234C2×C4×C53—48no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—44no—
238C2×C4×C53—48no—
239C2×C4×C53C2×C22⋊C412yes—
240C2×C4×C53C4×D414yes—
241C2×C4×C53C4×D414yes—
242C2×C4×C53#12012yes—
243C2×C4×C53#12114yes—
244C2×C4×C53#12114yes—
245C2×C4×C53#12312yes—
246C2×C4×C53C23×C411yescharacteristic
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C22⋊D53—416no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—28no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—48no—
259C2×Dic53—44no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53C2×C22⋊C414yes—
263C4×D53—24no—
264C4×D53—24no—
265C4×D53—24no—
266C4×D53—24no—
267C4×D53—216no—
268C4×D53—432no—
269C8×C53—42no—
270#88322—42no—
271#20909—531nomaximal, Sylow
272C22×C53—24no—
273C22×C53—24no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—44no—
277C22×C53—42no—
278C22×C53—82no—
279C22×C53—81no—
280C22×C53—81no—
281C22×C53#38614yes—
282C22×C53C22×C22⋊C411yescharacteristic
283C22×C53C2×#12011yescharacteristic
284C22×C53C2×#12311yescharacteristic
285C22×C53C2×AGL(1,ℤ/8)12yes—
286C22×C53C2×#13712yes—
287C2×D53—232no—
288C2×D53—24no—
289C2×D53—24no—
290C2×D53—24no—
291C2×D53—24no—
292C2×D53—24no—
293C2×D53—24no—
294C2×D53#41718yescomplemented
295C4×C53—44no—
296C4×C53—42no—
297C4×C53#38614yes—
298C4×C53C2×C4×D411yescharacteristic
299C4×C53C2×C4×D411yescharacteristic
300C4×C53C2×#12111yescharacteristic
301C4×C53C2×#12111yescharacteristic
302Dic53—24no—
303Dic53—44no—
304Dic53—44no—
305C2×#525—531no—
306C23×C4⋊C4—531no—
307C2×#417—531no—
308C2×#417D5311yescharacteristic, complemented
309C2×#407—531no—
310#1874—534no—
311#1872—534no—
312#1736—531no—
313#1736—531no—
314C2×C53—24no—
315C2×C53—82no—
316C2×C53C2×#38611yescharacteristic
317C2×C53C2×#41712yes—
318C2×C53#300311yescharacteristic
319C2×C53#300411yescharacteristic
320C2×C53#309212yes—
321D53—28no—
322C22×#41C2×D5311yescharacteristic
323C24×C4—531no—
324C24×C4—1061no—
325C22×M16—531no—
326C22×M16C2×D5311yescharacteristic
327#525—538no—
328#525—1064no—
329C2×#123—1061no—
330C2×C22⋊Q8—532no—
331C2×#121—532no—
332C2×#120—1061no—
333C2×C4×D4—1062no—
334C2×#117—531no—
335C22×C4⋊C4—531no—
336C22×C4⋊C4—534no—
337C22×C4⋊C4—1064no—
338C22×C4⋊C4—1062no—
339C22×C4⋊C4—1061no—
340C22×C4⋊C4—1064no—
341C22×C4⋊C4C2×D5311yescharacteristic
342#417—24no—
343#417—1064no—
344C2×#103—532no—
345C2×#103C2×D5312yescomplemented
346C2×#102—532no—
347C2×#102C2×D5312yescomplemented
348#407—1064no—
349C2×C22⋊C8—534no—
350#340—534no—
351#340—534no—
352C53#2090911yescharacteristic, complemented, Sylow
353C25—531no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41C4×D5314yescomplemented
360C22×Q8C22×D5311yescharacteristic
361C22×D4—21no—
362C22×D4—1062no—
363C22×D4C22×D5311yescharacteristic
364C23×C4—21no—
365C23×C4—21no—
366C23×C4—531no—
367C23×C4—534no—
368C23×C4—532no—
369C23×C4—534no—
370C23×C4—532no—
371C23×C4—1064no—
372C23×C4—1068no—
373C23×C4—1062no—
374C23×C4—1062no—
375C23×C4—1062no—
376C23×C4—1062no—
377C23×C4—1061no—
378C23×C4—2128no—
379C23×C4C22×D5311yescharacteristic
380C2×M16—22no—
381C2×M16—22no—
382C2×M16—22no—
383C2×M16—1062no—
384C2×M16—1062no—
385C2×M16—1062no—
386C22×C8—532no—
387C22×C8C22×D5312yes—
388#123—1068no—
389C22⋊Q8—538no—
390C22⋊Q8—1064no—
391#121—538no—
392#121—1064no—
393#120—1068no—
394C4×D4—10616no—
395#117—534no—
396#117—1062no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—534no—
400C2×C4⋊C4—532no—
401C2×C4⋊C4—532no—
402C2×C4⋊C4—1068no—
403C2×C4⋊C4—10616no—
404C2×C4⋊C4—1064no—
405C2×C4⋊C4—1062no—
406C2×C4⋊C4—1068no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1062no—
409C2×C4⋊C4—1068no—
410C2×C4⋊C4C22⋊D5314yescomplemented
411C2×C4⋊C4C22×D5312yes—
412C2×C22⋊C4—1062no—
413C2×C22⋊C4—1062no—
414C2×C22⋊C4—1061no—
415C2×C42—1061no—
416#103—24no—
417#103—1064no—
418#102—24no—
419#102—1064no—
420C22⋊C8—1068no—
421C24—21no—
422C24—532no—
423C24—534no—
424C24—1064no—
425C24—1064no—
426C24—1062no—
427C24—1062no—
428C24D4×D5311yescharacteristic
429#41—28no—
430#41—28no—
431#41—44no—
432#41—44no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×C4×D5312yescomplemented
439C2×D4—24no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—1068no—
448C2×D4—2124no—
449C2×D4—2124no—
450C2×D4C2×C4×D5312yescomplemented
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—534no—
459C22×C4—532no—
460C22×C4—538no—
461C22×C4—534no—
462C22×C4—534no—
463C22×C4—10616no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1068no—
467C22×C4—1068no—
468C22×C4—1064no—
469C22×C4—1068no—
470C22×C4—1068no—
471C22×C4—1064no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1064no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—2128no—
480C22×C4—21232no—
481C22×C4—2128no—
482C22×C4—2124no—
483C22×C4C2×C4×D5312yes—
484C22×C4#1171994512yes—
485C22×C4D4×D5311yescharacteristic
486C22×C4D4×D5312yes—
487C22×C4D4×D5312yes—
488C22×C4#1171994712yes—
489C22×C4C2×C22⋊D5312yes—
490C22×C4C23×D5311yescharacteristic
491M16—44no—
492M16—2124no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—1062no—
497C2×C8—1062no—
498C2×C8—1062no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—534no—
502C4⋊C4—534no—
503C4⋊C4—1068no—
504C4⋊C4—10616no—
505C4⋊C4—1064no—
506C4⋊C4—1062no—
507C4⋊C4—1062no—
508C4⋊C4C2×C22⋊D5314yes—
509C22⋊C4—1068no—
510C22⋊C4—1068no—
511C22⋊C4—1064no—
512C42—1064no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—534no—
521C23—534no—
522C23—534no—
523C23—10616no—
524C23—1068no—
525C23—1068no—
526C23—10616no—
527C23—1068no—
528C23—1062no—
529C23—1062no—
530C23—1062no—
531C23—1062no—
532C23C22⋊C4×D5312yes—
533C23#1313318512yes—
534C23#1313319912yes—
535C23C2×D4×D5311yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540D4—22no—
541D4—22no—
542D4—48no—
543D4—41no—
544D4—41no—
545D4—21216no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—48no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—534no—
554C2×C4—1068no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—10616no—
558C2×C4—1064no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—1064no—
562C2×C4—1064no—
563C2×C4—1064no—
564C2×C4—21232no—
565C2×C4—2128no—
566C2×C4—2124no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4C22⋊C4×D5312yes—
570C2×C4#1313317714yes—
571C2×C4C4×C22⋊D5314yes—
572C2×C4#1313318512yes—
573C2×C4#1313318814yes—
574C2×C4#1313319014yes—
575C2×C4#1313319912yes—
576C2×C4C22×C4×D5311yescharacteristic
577C2×C4C2×D4×D5311yescharacteristic
578C2×C4C2×D4×D5311yescharacteristic
579C2×C4C22×C22⋊D5311yescharacteristic
580C2×C4C2×#1171994711yescharacteristic
581C2×C4C2×#1171994511yescharacteristic
582C8—42no—
583C8—2122no—
584C22—22no—
585C22—22no—
586C22—24no—
587C22—24no—
588C22—44no—
589C22—42no—
590C22—81no—
591C22—81no—
592C22—82no—
593C22—538no—
594C22—10632no—
595C22—1064no—
596C22—1064no—
597C22—1064no—
598C22—1064no—
599C22—1064no—
600C22—1064no—
601C22#1681049914yes—
602C22C2×C22⋊C4×D5311yescharacteristic
603C22C2×#1313318511yescharacteristic
604C22C2×#1313319911yescharacteristic
605C22AGL(1,ℤ/8)×D5312yes—
606C22#137×D5312yes—
607C4—44no—
608C4—42no—
609C4—1064no—
610C4—2124no—
611C4—2124no—
612C4#1681049914yes—
613C4C2×#1313317711yescharacteristic
614C4C2×C4×C22⋊D5311yescharacteristic
615C4C2×#1313318811yescharacteristic
616C4C2×#1313319011yescharacteristic
617C2—24no—
618C2—82no—
619C2—1068no—
620C2C2×#1681049911yescharacteristic
621C2#417×D5312yes—
622C2#2028394011yescharacteristic
623C2#2028394711yescharacteristic
624C2#2028463612yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#417C2
socleC23×C53C22×D4
prime radicalC2×#16811130C2
cosocleC22×C4×C53C24
hypercenterC2×#417D53
hypocenterC53#20909
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D53C2semidirect
3. C2×#16810624C2semidirect
4. C2×#16810586C2semidirect
5. #20275029C2semidirect
6. #20275019C2semidirect
7. #20274435C2semidirect
8. #20274434C2semidirect
9. C2×C53×#417C2semidirect
14. #16811130C4semidirect
28. C2×#13133141C22semidirect
29. C2×#13133138C22semidirect
32. #16810386C22semidirect
33. #16810385C22semidirect
38. C2×#103×C53C22semidirect
39. C2×#102×C53C22semidirect
61. #13133193C2×C4semidirect
63. #13133188C2×C4semidirect
67. #13133176C2×C4semidirect
70. C2×C4⋊Dic53D4semidirect
91. C2×#41×C53C2×C4semidirect
104. C2×C4⋊C4×C53D4semidirect
125. C2×Q8×C53C22×C4semidirect
134. C2×D4×C53C22×C4semidirect
191. C4⋊Dic53C22×C4semidirect
294. C2×D53#417semidirect
308. C2×#417D53semidirect
345. C2×#103C2×D53semidirect
347. C2×#102C2×D53semidirect
352. C53#20909semidirect
359. C2×#41C4×D53semidirect
410. C2×C4⋊C4C22⋊D53semidirect
438. C2×Q8C2×C4×D53semidirect
450. C2×D4C2×C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111302
C22×C4⋊C4×D532
C2×#168106242
C2×#168105862
#202750292
#202750192
#202744352
#202744342
C2×C53×#4172
#2090953