Finite Group Databasebeta

← #27416790

Subgroups — #27416790

Order 13568.

1772 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810944C212yesmaximal, complemented
2#20277098C211yesmaximal, characteristic, complemented
3#20275901C212yesmaximal, complemented
4#20275894C212yesmaximal, complemented
5#20275038C212yesmaximal, complemented
6#20274908C211yesmaximal, characteristic, complemented
7#20274903C211yesmaximal, characteristic, complemented
8#20274646C211yesmaximal, characteristic, complemented
9#20274646C211yesmaximal, characteristic, complemented
10#20274625C211yesmaximal, characteristic, complemented
11C22×#121×C53C211yesmaximal, characteristic
12C2×#13133176C2211yescharacteristic, complemented
13C2×#13133176C2211yescharacteristic, complemented
14#16810944—28no—
15C2×#13133150C2212yescomplemented
16C2×#13133148C2212yescomplemented
17C2×#13133138C2212yescomplemented
18C2×#13133138C2212yescomplemented
19C2×#13133138C2212yescomplemented
20C2×#13133138C2212yescomplemented
21C2×C22⋊#88322C2211yescharacteristic
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2212yescomplemented
24C2×C22⋊#88322C2212yescomplemented
25#16810486—21no—
26#16810483C2211yescharacteristic
27#16810483C2211yescharacteristic
28#16810483C2211yescharacteristic, complemented
29#16810483C2211yescharacteristic, complemented
30#16810479—21no—
31C4×C22⋊Dic53—21no—
32C4×C22⋊Dic53—21no—
33#16810385C2212yescomplemented
34#16810385C2212yescomplemented
35C23×D4×C53—21no—
36C23×D4×C53C2211yescharacteristic
37C24×C4×C53C2211yescharacteristic
38C2×#121×C53—21no—
39C2×#121×C53—21no—
40C2×#121×C53—28no—
41C2×#121×C53C2212yes—
42C2×#121×C53C2212yes—
43C22×C4⋊C4×C53C2211yescharacteristic
44C22×C22⋊C4×C53—21no—
45C22×#88322C2312yes—
46C22×#88322C2312yes—
47#13133176—24no—
48#13133176—42no—
49C2×C4⋊Dic53C2311yescharacteristic
50C2×C4⋊Dic53C2311yescharacteristic
51C2×C4⋊Dic53C2311yescharacteristic
52C2×C4⋊Dic53C2311yescharacteristic
53C2×C22⋊Dic53—21no—
54C2×C22⋊Dic53—21no—
55C2×C22⋊Dic53—21no—
56C2×C22⋊Dic53—21no—
57C2×C22⋊Dic53—21no—
58C2×C22⋊Dic53—21no—
59C2×C4×Dic53—21no—
60C2×C4×Dic53—21no—
61#13133150—24no—
62#13133148—24no—
63#13133138—24no—
64#13133138—24no—
65#13133138—42no—
66#13133138—42no—
67C22⋊#88322—24no—
68C22⋊#88322—22no—
69C22⋊#88322—22no—
70C22⋊#88322—42no—
71#13133124—21no—
72#13133124—21no—
73#13133124—21no—
74#13133124—21no—
75C25×C53—21no—
76C25×C53D411yescharacteristic
77C22×D4×C53—28no—
78C22×D4×C53—22no—
79C22×D4×C53—22no—
80C22×D4×C53—24no—
81C22×D4×C53—21no—
82C22×D4×C53—22no—
83C22×D4×C53—22no—
84C22×D4×C53—42no—
85C22×D4×C53—44no—
86C22×D4×C53—44no—
87C22×D4×C53C2312yes—
88C23×C4×C53—21no—
89C23×C4×C53—24no—
90C23×C4×C53—22no—
91C23×C4×C53—22no—
92C23×C4×C53—42no—
93C23×C4×C53D411yescharacteristic
94C23×C4×C53D411yescharacteristic
95C23×C4×C53D411yescharacteristic
96C23×C4×C53C2311yescharacteristic
97C23×C4×C53C2311yescharacteristic
98C23×C4×C53C2311yescharacteristic
99#121×C53—28no—
100#121×C53—28no—
101#121×C53—24no—
102#121×C53—24no—
103#121×C53—24no—
104#121×C53D418yes—
105C2×C4⋊C4×C53—24no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53C2312yes—
108C2×C22⋊C4×C53—28no—
109C2×C22⋊C4×C53—22no—
110C2×C22⋊C4×C53—22no—
111C24×C53—22no—
112C24×C53—22no—
113C24×C53—22no—
114C24×C53—24no—
115C24×C53—22no—
116C24×C53—22no—
117C24×C53—22no—
118C24×C53—41no—
119C24×C53—41no—
120C24×C53—42no—
121C24×C53—42no—
122C24×C53—42no—
123C24×C53—42no—
124C24×C53—42no—
125C24×C53—42no—
126C24×C53C2×D411yescharacteristic
127C24×C53C2×D411yescharacteristic
128C24×C53C2×D411yescharacteristic
129C2×D4×C53—24no—
130C2×D4×C53—28no—
131C2×D4×C53—28no—
132C2×D4×C53—22no—
133C2×D4×C53—22no—
134C2×D4×C53—22no—
135C2×D4×C53—22no—
136C2×D4×C53—22no—
137C2×D4×C53—48no—
138C2×D4×C53—48no—
139C2×D4×C53—42no—
140C2×D4×C53—432no—
141C2×D4×C53—44no—
142C2×D4×C53—44no—
143C2×D4×C53—44no—
144C2×D4×C53—44no—
145C2×D4×C53—42no—
146C2×D4×C53C2×D414yes—
147C22×C4×C53—22no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—22no—
151C22×C4×C53—22no—
152C22×C4×C53—24no—
153C22×C4×C53—22no—
154C22×C4×C53—22no—
155C22×C4×C53—22no—
156C22×C4×C53—22no—
157C22×C4×C53—22no—
158C22×C4×C53—22no—
159C22×C4×C53—22no—
160C22×C4×C53—22no—
161C22×C4×C53—21no—
162C22×C4×C53—21no—
163C22×C4×C53—22no—
164C22×C4×C53—41no—
165C22×C4×C53—41no—
166C22×C4×C53—41no—
167C22×C4×C53—48no—
168C22×C4×C53—48no—
169C22×C4×C53—42no—
170C22×C4×C53—42no—
171C22×C4×C53C2×D411yescharacteristic
172C22×C4×C53C2×D411yescharacteristic
173C22×C4×C53C2×D414yes—
174C22×C4×C53C2×D411yescharacteristic
175C22×C4×C53#4112yes—
176C22×C4×C53#4112yes—
177C22×C4×C53C2411yescharacteristic
178C22×Dic53—21no—
179C22×Dic53—21no—
180C22×Dic53—21no—
181C22×Dic53—21no—
182C4⋊C4×C53—24no—
183C4⋊C4×C53—22no—
184C4⋊C4×C53C2×D414yes—
185C22⋊C4×C53—28no—
186C22⋊C4×C53—28no—
187C22⋊Dic53—42no—
188C22⋊Dic53—42no—
189C22⋊Dic53—42no—
190C22⋊Dic53—42no—
191C22⋊Dic53—42no—
192C22⋊Dic53—42no—
193C4⋊Dic53—22no—
194C4⋊Dic53—22no—
195C4⋊Dic53—22no—
196C4⋊Dic53—22no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C2×#88322—22no—
204C2×#88322—22no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—28no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—24no—
223C23×C53—44no—
224C23×C53—44no—
225C23×C53—44no—
226C23×C53—44no—
227C23×C53—44no—
228C23×C53—48no—
229C23×C53—48no—
230C23×C53—48no—
231C23×C53—44no—
232C23×C53—42no—
233C23×C53—42no—
234C23×C53—42no—
235C23×C53—42no—
236C23×C53—42no—
237C23×C53—42no—
238C23×C53—84no—
239C23×C53—84no—
240C23×C53—84no—
241C23×C53—84no—
242C23×C53#12012yes—
243C23×C53#12312yes—
244C23×C53#13512yes—
245C23×C53AGL(1,ℤ/8)12yes—
246C23×C53AGL(1,ℤ/8)12yes—
247C23×C53C22×D411yescharacteristic
248D4×C53—24no—
249D4×C53—24no—
250D4×C53—48no—
251D4×C53—416no—
252D4×C53—416no—
253D4×C53—42no—
254D4×C53—42no—
255C2×C4×C53—21no—
256C2×C4×C53—21no—
257C2×C4×C53—21no—
258C2×C4×C53—21no—
259C2×C4×C53—24no—
260C2×C4×C53—24no—
261C2×C4×C53—24no—
262C2×C4×C53—24no—
263C2×C4×C53—21no—
264C2×C4×C53—24no—
265C2×C4×C53—21no—
266C2×C4×C53—21no—
267C2×C4×C53—44no—
268C2×C4×C53—44no—
269C2×C4×C53—44no—
270C2×C4×C53—48no—
271C2×C4×C53—42no—
272C2×C4×C53—42no—
273C2×C4×C53—42no—
274C2×C4×C53—42no—
275C2×C4×C53#12012yes—
276C2×C4×C53#12312yes—
277C2×C4×C53C22×D412yes—
278C2×C4×C53C2×#4111yescharacteristic
279C2×C4×C53C2×#4111yescharacteristic
280C2×C4×C53H(5,𝔽2)12yes—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290C2×Dic53—41no—
291C2×Dic53—41no—
292C2×Dic53—41no—
293#88322—42no—
294#88322—42no—
295#24009—531nomaximal, Sylow
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—22no—
299C22×C53—22no—
300C22×C53—22no—
301C22×C53—22no—
302C22×C53—22no—
303C22×C53—22no—
304C22×C53—22no—
305C22×C53—22no—
306C22×C53—22no—
307C22×C53—22no—
308C22×C53—416no—
309C22×C53—41no—
310C22×C53—41no—
311C22×C53—41no—
312C22×C53—41no—
313C22×C53—41no—
314C22×C53—41no—
315C22×C53—41no—
316C22×C53—41no—
317C22×C53—44no—
318C22×C53—44no—
319C22×C53—44no—
320C22×C53—816no—
321C22×C53—82no—
322C22×C53—82no—
323C22×C53—82no—
324C22×C53—82no—
325C22×C53#48214yescomplemented
326C22×C53C2×#12011yescharacteristic
327C22×C53C2×#12311yescharacteristic
328C22×C53#53411yescharacteristic
329C22×C53C2×#13511yescharacteristic
330C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
331C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
332C22×C53#57511yescharacteristic
333C4×C53—22no—
334C4×C53—22no—
335C4×C53—41no—
336C4×C53—44no—
337C4×C53#54612yes—
338C4×C53#54612yes—
339Dic53—81no—
340Dic53—81no—
341Dic53—81no—
342Dic53—81no—
343C22×#121D5311yescharacteristic
344C2×#482—532no—
345#2280—531no—
346#2034—532no—
347#2033—532no—
348#1875—532no—
349#1853—531no—
350#1852—531no—
351#1786—531no—
352#1786—531no—
353#1779—531no—
354C2×C53—24no—
355C2×C53—42no—
356C2×C53—42no—
357C2×C53—84no—
358C2×C53#240511yescharacteristic
359C2×C53#300211yescharacteristic
360C2×C53#300311yescharacteristic
361C2×C53C2×#48211yescharacteristic
362C2×C53#309411yescharacteristic
363C2×C53#318811yescharacteristic
364C2×C53#319411yescharacteristic
365C23×D4—21no—
366C23×D4C2×D5311yescharacteristic
367C24×C4C2×D5311yescharacteristic
368C2×#121—21no—
369C2×#121—21no—
370C2×#121—28no—
371C2×#121C2×D5312yes—
372C2×#121C2×D5312yes—
373C2×#117—531no—
374C2×#117—531no—
375C22×C4⋊C4C2×D5311yescharacteristic
376C22×C22⋊C4—21no—
377#482—1068no—
378C2×#107—532no—
379C2×#106—532no—
380C2×#102—532no—
381C2×#102—532no—
382C2×#102—532no—
383C2×#102—532no—
384C2×C22⋊C8—532no—
385C2×C22⋊C8—531no—
386C2×C22⋊C8—531no—
387C2×C22⋊C8—532no—
388#381—1061no—
389#380—531no—
390#380—531no—
391#380—531no—
392#380—531no—
393#379—1061no—
394C4×C22⋊C4—1061no—
395C4×C22⋊C4—1061no—
396#340—532no—
397#340—532no—
398C53#2400911yescharacteristic, complemented, Sylow
399C25—21no—
400C25C22⋊D5311yescharacteristic
401C22×D4—28no—
402C22×D4—22no—
403C22×D4—22no—
404C22×D4—24no—
405C22×D4—22no—
406C22×D4—22no—
407C22×D4—21no—
408C22×D4—42no—
409C22×D4—44no—
410C22×D4—44no—
411C22×D4C22×D5312yes—
412C23×C4—21no—
413C23×C4—22no—
414C23×C4—22no—
415C23×C4—24no—
416C23×C4—42no—
417C23×C4C22⋊D5311yescharacteristic
418C23×C4C22⋊D5311yescharacteristic
419C23×C4C22⋊D5311yescharacteristic
420C23×C4C22×D5311yescharacteristic
421C23×C4C22×D5311yescharacteristic
422C23×C4C22×D5311yescharacteristic
423C22×C8—532no—
424C22×C8—532no—
425#121—28no—
426#121—28no—
427#121—24no—
428#121—24no—
429#121—24no—
430#121C22⋊D5318yes—
431#117—1064no—
432#117—2122no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—24no—
435C2×C4⋊C4—531no—
436C2×C4⋊C4—531no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4C22×D5312yes—
440C2×C22⋊C4—28no—
441C2×C22⋊C4—22no—
442C2×C22⋊C4—22no—
443C2×C22⋊C4—1061no—
444C2×C22⋊C4—1061no—
445C2×C22⋊C4—1061no—
446C2×C22⋊C4—1061no—
447C2×C22⋊C4—1061no—
448C2×C22⋊C4—1061no—
449C2×C42—1061no—
450C2×C42—1061no—
451#107—1064no—
452#106—1064no—
453#102—1064no—
454#102—1064no—
455#102—2122no—
456#102—2122no—
457C22⋊C8—1062no—
458C22⋊C8—1062no—
459C22⋊C8—1064no—
460C22⋊C8—2122no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465C24—22no—
466C24—22no—
467C24—22no—
468C24—24no—
469C24—22no—
470C24—22no—
471C24—22no—
472C24—41no—
473C24—41no—
474C24—42no—
475C24—42no—
476C24—42no—
477C24—42no—
478C24—42no—
479C24—42no—
480C24C2×C22⋊D5311yescharacteristic
481C24C2×C22⋊D5311yescharacteristic
482C24C2×C22⋊D5311yescharacteristic
483C2×D4—24no—
484C2×D4—28no—
485C2×D4—28no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—48no—
492C2×D4—48no—
493C2×D4—42no—
494C2×D4—432no—
495C2×D4—44no—
496C2×D4—44no—
497C2×D4—44no—
498C2×D4—44no—
499C2×D4—42no—
500C2×D4C2×C22⋊D5314yes—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—24no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—22no—
518C22×C4—41no—
519C22×C4—41no—
520C22×C4—41no—
521C22×C4—48no—
522C22×C4—48no—
523C22×C4—42no—
524C22×C4—42no—
525C22×C4—1061no—
526C22×C4—1061no—
527C22×C4—1061no—
528C22×C4—1061no—
529C22×C4#1171994712yes—
530C22×C4#1171994712yes—
531C22×C4C2×C22⋊D5311yescharacteristic
532C22×C4C2×C22⋊D5311yescharacteristic
533C22×C4C2×C22⋊D5314yes—
534C22×C4C2×C22⋊D5311yescharacteristic
535C22×C4C23×D5311yescharacteristic
536C2×C8—1062no—
537C2×C8—1062no—
538C2×C8—1062no—
539C2×C8—1062no—
540C2×C8—1062no—
541C2×C8—1062no—
542C4⋊C4—24no—
543C4⋊C4—22no—
544C4⋊C4—1062no—
545C4⋊C4—1062no—
546C4⋊C4—1062no—
547C4⋊C4—1062no—
548C4⋊C4C2×C22⋊D5314yes—
549C22⋊C4—28no—
550C22⋊C4—28no—
551C22⋊C4—2122no—
552C22⋊C4—2122no—
553C22⋊C4—2122no—
554C22⋊C4—2122no—
555C22⋊C4—2122no—
556C22⋊C4—2122no—
557C42—2122no—
558C42—2122no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—28no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—24no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—48no—
583C23—48no—
584C23—48no—
585C23—44no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23#1313318612yes—
597C23#1313319812yes—
598C23#1313323712yes—
599C23#1313324312yes—
600C23#1313324312yes—
601C23C22×C22⋊D5311yescharacteristic
602D4—24no—
603D4—24no—
604D4—48no—
605D4—416no—
606D4—416no—
607D4—42no—
608D4—42no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—21no—
618C2×C4—24no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—2121no—
630C2×C4—2121no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4—2121no—
635C2×C4—2121no—
636C2×C4—2121no—
637C2×C4—2121no—
638C2×C4—2121no—
639C2×C4—2121no—
640C2×C4—2121no—
641C2×C4#1313318612yes—
642C2×C4#1313319812yes—
643C2×C4C22×C22⋊D5312yes—
644C2×C4C2×#1171994711yescharacteristic
645C2×C4C2×#1171994711yescharacteristic
646C2×C4#1313326212yes—
647C8—2122no—
648C8—2122no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—416no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—816no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22#1681094414yescomplemented
679C22C2×#1313318611yescharacteristic
680C22C2×#1313319811yescharacteristic
681C22#1681118211yescharacteristic
682C22C2×#1313323711yescharacteristic
683C22C2×#1313324311yescharacteristic
684C22C2×#1313324311yescharacteristic
685C22#1681142111yescharacteristic
686C4—22no—
687C4—22no—
688C4—41no—
689C4—44no—
690C4—4241no—
691C4—4241no—
692C4—4241no—
693C4—4241no—
694C4#1681126712yes—
695C4#1681126712yes—
696C2—24no—
697C2—42no—
698C2—42no—
699C2—84no—
700C2#2027803811yescharacteristic
701C2#2028393311yescharacteristic
702C2#2028394211yescharacteristic
703C2C2×#1681094411yescharacteristic
704C2#2028465111yescharacteristic
705C2#2028548011yescharacteristic
706C2#2028552911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC22×#121×C53C2
socleC23×C53C22×D4
hypercenterC22×#121D53
hypocenterC53#24009
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16810944C2semidirect
2. #20277098C2semidirect
3. #20275901C2semidirect
4. #20275894C2semidirect
5. #20275038C2semidirect
6. #20274908C2semidirect
7. #20274903C2semidirect
8. #20274646C2semidirect
9. #20274646C2semidirect
10. #20274625C2semidirect
12. C2×#13133176C22semidirect
13. C2×#13133176C22semidirect
15. C2×#13133150C22semidirect
16. C2×#13133148C22semidirect
17. C2×#13133138C22semidirect
18. C2×#13133138C22semidirect
19. C2×#13133138C22semidirect
20. C2×#13133138C22semidirect
23. C2×C22⋊#88322C22semidirect
24. C2×C22⋊#88322C22semidirect
28. #16810483C22semidirect
29. #16810483C22semidirect
33. #16810385C22semidirect
34. #16810385C22semidirect
325. C22×C53#482semidirect
398. C53#24009semidirect
678. C22#16810944semidirect

Maximal subgroups

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

subgroupindex
C2×#168109442
#202770982
#202759012
#202758942
#202750382
#202749082
#202749032
#202746462
#202746252
C22×#121×C532
#2400953