Finite Group Databasebeta

← #27313272

Subgroups — #27313272

Order 13568.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810516C212yesmaximal, complemented
2#20272983C212yesmaximal, complemented
3#20272960C212yesmaximal, complemented
4#20272956C211yesmaximal, characteristic
5#20272956C211yesmaximal, characteristic
6#20272952C212yesmaximal, complemented
7#20272944C212yesmaximal, complemented
8#20272938C212yesmaximal, complemented
9C22×#123×C53C211yesmaximal, characteristic
10C22×C4⋊Dic53C2212yes—
11C22×C22⋊Dic53C2212yes—
12C22×C22⋊Dic53C2212yes—
13C22×C22⋊Dic53C2211yescharacteristic
14C22×C22⋊Dic53C2212yes—
15C22×C22⋊Dic53C2211yescharacteristic
16C22×C4×Dic53C2212yes—
17#16810516—28no—
18C2×#13133124C2212yes—
19C2×#13133124C2212yes—
20C2×#13133124C2212yes—
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2212yes—
26C2×#13133124C2211yescharacteristic
27C23×D4×C53C2211yescharacteristic
28C24×C4×C53C2211yescharacteristic
29C2×#123×C53—28no—
30C2×#123×C53C418yescomplemented
31C22×C4⋊C4×C53C2212yes—
32C22×C22⋊C4×C53C2212yes—
33C22×C22⋊C4×C53C2211yescharacteristic
34C23×Dic53C2312yes—
35C23×Dic53C2312yes—
36C23×Dic53C2312yes—
37C23×Dic53C2312yes—
38C2×C4⋊Dic53—24no—
39C2×C4⋊Dic53—24no—
40C2×C4⋊Dic53—42no—
41C2×C22⋊Dic53—24no—
42C2×C22⋊Dic53—24no—
43C2×C22⋊Dic53—24no—
44C2×C22⋊Dic53—24no—
45C2×C22⋊Dic53—24no—
46C2×C22⋊Dic53—24no—
47C2×C22⋊Dic53—41no—
48C2×C22⋊Dic53—42no—
49C2×C22⋊Dic53—42no—
50C2×C22⋊Dic53—42no—
51C2×C22⋊Dic53—42no—
52C2×C22⋊Dic53—41no—
53C2×C22⋊Dic53—42no—
54C2×C4×Dic53—24no—
55C2×C4×Dic53—24no—
56C2×C4×Dic53—42no—
57#13133124—24no—
58#13133124—24no—
59#13133124—41no—
60#13133124—42no—
61#13133124—42no—
62#13133124—42no—
63#13133124—42no—
64#13133124—41no—
65#13133124—42no—
66C25×C53C2311yescharacteristic
67C25×C53C2311yescharacteristic
68C22×D4×C53—24no—
69C22×D4×C53—42no—
70C22×D4×C53—41no—
71C22×D4×C53—41no—
72C22×D4×C53C2×C414yes—
73C23×C4×C53—24no—
74C23×C4×C53—42no—
75C23×C4×C53—41no—
76C23×C4×C53—41no—
77C23×C4×C53C2×C414yes—
78C23×C4×C53C2312yes—
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2312yes—
81#123×C53—232no—
82C2×C4⋊C4×C53—24no—
83C2×C4⋊C4×C53—24no—
84C2×C4⋊C4×C53C2×C418yes—
85C2×C22⋊C4×C53—24no—
86C2×C22⋊C4×C53—24no—
87C2×C22⋊C4×C53C2×C418yes—
88C2×C22⋊C4×C53C2×C414yes—
89C2×C22⋊C4×C53D418yes—
90C24×C53—22no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53—21no—
94C24×C53—21no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53—21no—
102C24×C53—21no—
103C24×C53—22no—
104C24×C53—22no—
105C24×C53C22×C412yes—
106C24×C53C22×C412yes—
107C24×C53C2×D414yes—
108C24×C53C2411yescharacteristic
109C2×D4×C53—28no—
110C2×D4×C53—44no—
111C2×D4×C53—44no—
112C2×D4×C53—48no—
113C2×D4×C53—48no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—28no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—44no—
130C22×C4×C53—44no—
131C22×C4×C53—48no—
132C22×C4×C53—48no—
133C22×C4×C53C22×C414yes—
134C22×C4×C53C22×C412yes—
135C22×C4×C53C22×C414yes—
136C22×C4×C53C2×D414yes—
137C22×C4×C53C2×D414yes—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53#4114yes—
165C22×Dic53#4114yes—
166C4⋊C4×C53—216no—
167C22⋊C4×C53—216no—
168C22⋊C4×C53C22⋊C4116yescomplemented
169C22⋊Dic53—44no—
170C22⋊Dic53—48no—
171C22⋊Dic53—48no—
172C22⋊Dic53—48no—
173C22⋊Dic53—44no—
174C4⋊Dic53—48no—
175C4×Dic53—48no—
176C23×C53—24no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53—24no—
180C23×C53—24no—
181C23×C53—24no—
182C23×C53—24no—
183C23×C53—24no—
184C23×C53—24no—
185C23×C53—24no—
186C23×C53—22no—
187C23×C53—24no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—24no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—41no—
196C23×C53—42no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—42no—
200C23×C53—41no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53—42no—
206C23×C53—42no—
207C23×C53—42no—
208C23×C53—42no—
209C23×C53—42no—
210C23×C53—42no—
211C23×C53—41no—
212C23×C53—42no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—42no—
216C23×C53—41no—
217C23×C53C2×C22⋊C414yes—
218C23×C53C23×C411yescharacteristic
219C23×C53C22×D412yes—
220C23×C53C2×#4112yes—
221C23×C53C2×#4112yes—
222C23×C53H(5,𝔽2)12yes—
223C23×C53H(5,𝔽2)11yescharacteristic
224C23×C53H(5,𝔽2)11yescharacteristic
225C23×C53H(5,𝔽2)12yes—
226C23×C532−1+412yes—
227D4×C53—416no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—22no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—416no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53C2×C22⋊C418yes—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276#14506—531nomaximal, Sylow
277C22×C53—24no—
278C22×C53—28no—
279C22×C53—44no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—44no—
283C22×C53—44no—
284C22×C53—44no—
285C22×C53—44no—
286C22×C53—42no—
287C22×C53—44no—
288C22×C53—44no—
289C22×C53—42no—
290C22×C53—42no—
291C22×C53—82no—
292C22×C53—82no—
293C22×C53—82no—
294C22×C53—82no—
295C22×C53C22×C22⋊C411yescharacteristic
296C22×C53C4×#4112yes—
297C22×C53#51811yescharacteristic
298C22×C53#51811yescharacteristic
299C22×C53#52012yes—
300C22×C53#53512yes—
301C22×C53#53512yes—
302C22×C53#53812yes—
303C22×C53#53912yes—
304C22×C53#54012yes—
305C22×C53#54212yes—
306C22×C53#54612yes—
307C22×C53#54612yes—
308C22×C53#54712yes—
309C22×C53#54712yes—
310C22×C53#55111yescharacteristic
311C22×C53#55312yes—
312C22×C53#55911yescharacteristic
313C22×C53#56011yescharacteristic
314C22×C53#56212yes—
315C22×C53#56311yescharacteristic
316C4×C53—28no—
317C4×C53—44no—
318C4×C53—42no—
319Dic53—44no—
320Dic53—44no—
321Dic53—82no—
322Dic53—82no—
323C22×#123D5311yescharacteristic
324C2×#388—532no—
325#1445—532no—
326#1439—532no—
327#1438—531no—
328#1438—531no—
329#1437—532no—
330C4⋊#95—532no—
331C22⋊#95—532no—
332C2×C53—44no—
333C2×C53—82no—
334C2×C53#230911yescharacteristic
335C2×C53#233512yes—
336C2×C53#233912yes—
337C2×C53#237411yescharacteristic
338C2×C53#237611yescharacteristic
339C2×C53#270312yes—
340C2×C53#270612yes—
341C2×C53#271012yes—
342C2×C53#271812yes—
343C23×D4C2×D5311yescharacteristic
344C24×C4C2×D5311yescharacteristic
345C2×#123—28no—
346C2×#123Dic5318yescomplemented
347C22×C4⋊C4—532no—
348C22×C4⋊C4C2×D5312yes—
349C22×C22⋊C4—532no—
350C22×C22⋊C4—532no—
351C22×C22⋊C4—531no—
352C22×C22⋊C4—532no—
353C22×C22⋊C4—531no—
354C22×C22⋊C4C2×D5312yes—
355C22×C22⋊C4C2×D5311yescharacteristic
356C22×C42—532no—
357#388—1068no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—532no—
364C2×#95—531no—
365C2×#95—532no—
366C2×#95—531no—
367C53#1450611yescharacteristic, complemented, Sylow
368C25C22×D5311yescharacteristic
369C25C22×D5311yescharacteristic
370C22×D4—24no—
371C22×D4—41no—
372C22×D4—41no—
373C22×D4—42no—
374C22×D4C2×Dic5314yes—
375C23×C4—24no—
376C23×C4—41no—
377C23×C4—41no—
378C23×C4—42no—
379C23×C4—532no—
380C23×C4—532no—
381C23×C4—532no—
382C23×C4—532no—
383C23×C4C2×Dic5314yes—
384C23×C4C22×D5312yes—
385C23×C4C22×D5311yescharacteristic
386C23×C4C22×D5312yes—
387#123—232no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—1064no—
391C2×C4⋊C4—1064no—
392C2×C4⋊C4—2122no—
393C2×C4⋊C4C2×Dic5318yes—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—1064no—
397C2×C22⋊C4—1064no—
398C2×C22⋊C4—1064no—
399C2×C22⋊C4—1064no—
400C2×C22⋊C4—1064no—
401C2×C22⋊C4—1064no—
402C2×C22⋊C4—2121no—
403C2×C22⋊C4—2122no—
404C2×C22⋊C4—2122no—
405C2×C22⋊C4—2122no—
406C2×C22⋊C4—2122no—
407C2×C22⋊C4—2121no—
408C2×C22⋊C4—2122no—
409C2×C22⋊C4C2×Dic5318yes—
410C2×C22⋊C4C2×Dic5314yes—
411C2×C22⋊C4C22⋊D5318yes—
412C2×C42—1064no—
413C2×C42—1064no—
414C2×C42—2122no—
415#95—1064no—
416#95—1064no—
417#95—2122no—
418#95—2122no—
419#95—2122no—
420#95—2122no—
421#95—2121no—
422#95—2122no—
423#95—2121no—
424C24—22no—
425C24—22no—
426C24—21no—
427C24—21no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—21no—
435C24—21no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24C22×Dic5312yes—
440C24C22×Dic5312yes—
441C24C2×C22⋊D5314yes—
442C24C23×D5311yescharacteristic
443C2×D4—28no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—48no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—534no—
468C22×C4—534no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4C22×Dic5314yes—
496C22×C4C22×Dic5312yes—
497C22×C4C22×Dic5314yes—
498C22×C4C2×C22⋊D5314yes—
499C22×C4C2×C22⋊D5314yes—
500C4⋊C4—216no—
501C4⋊C4—2128no—
502C22⋊C4—216no—
503C22⋊C4—2124no—
504C22⋊C4—2128no—
505C22⋊C4—2128no—
506C22⋊C4—2128no—
507C22⋊C4—2124no—
508C22⋊C4C22⋊Dic53116yescomplemented
509C42—2128no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23C2×C22⋊Dic5314yes—
552C23C23×Dic5311yescharacteristic
553C23C22×C22⋊D5312yes—
554C23#41×D5312yes—
555C23#41×D5312yes—
556C23#1313326211yescharacteristic
557C23#1313326211yescharacteristic
558C23#1313326212yes—
559C23#1313326312yes—
560C23#1313326412yes—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—416no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—1064no—
582C2×C4—1064no—
583C2×C4—1064no—
584C2×C4—1064no—
585C2×C4—1064no—
586C2×C4—1064no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4C2×C22⋊Dic5318yes—
610C22—28no—
611C22—24no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×C22⋊Dic5311yescharacteristic
629C22#41×Dic5312yes—
630C22#1681109811yescharacteristic
631C22#1681109811yescharacteristic
632C22#1681110612yes—
633C22#1681119012yes—
634C22#1681119012yes—
635C22#1681120912yes—
636C22#1681121612yes—
637C22#1681122512yes—
638C22#1681124112yes—
639C22#1681126112yes—
640C22#1681126112yes—
641C22#1681127212yes—
642C22#1681127212yes—
643C22#1681130111yescharacteristic
644C22#1681131812yes—
645C22#1681135711yescharacteristic
646C22#1681136311yescharacteristic
647C22#1681137012yes—
648C22#1681137611yescharacteristic
649C4—28no—
650C4—44no—
651C4—42no—
652C4—4242no—
653C4—4242no—
654C4—2124no—
655C4—2124no—
656C2—44no—
657C2—82no—
658C2#2027729711yescharacteristic
659C2#2027750912yes—
660C2#2027754512yes—
661C2#2027785111yescharacteristic
662C2#2027786311yescharacteristic
663C2#2028135212yes—
664C2#2028139712yes—
665C2#2028145312yes—
666C2#2028156112yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC22×#123×C53C2
socle, cosocleC24×C53C24
prime radicalC23×D4×C53C22
hypercenterC22×#123D53
hypocenterC53#14506
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C2×#16810516C2semidirect
2. #20272983C2semidirect
3. #20272960C2semidirect
6. #20272952C2semidirect
7. #20272944C2semidirect
8. #20272938C2semidirect
30. C2×#123×C53C4semidirect
168. C22⋊C4×C53C22⋊C4semidirect
346. C2×#123Dic53semidirect
367. C53#14506semidirect
508. C22⋊C4C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105162
#202729832
#202729602
#202729562
#202729522
#202729442
#202729382
C22×#123×C532
#1450653