Finite Group Databasebeta

← #27305966

Subgroups — #27305966

Order 13568.

3686 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133173C211yesmaximal, characteristic, complemented
2C23×C22⋊Dic53C211yesmaximal, characteristic
3C2×#16810484C211yesmaximal, characteristic, complemented
4C2×#16810479C211yesmaximal, characteristic
5#20272958C214yesmaximal, complemented
6#20272952C214yesmaximal, complemented
7#20272937C211yesmaximal, characteristic, complemented
8#20272937C211yesmaximal, characteristic, complemented
9C2×C53×#379C211yesmaximal, characteristic
10C24×Dic53C2211yescharacteristic
11C24×Dic53C2211yescharacteristic
12C2×#13133173—24no—
13C2×#13133173—24no—
14C2×#13133173—24no—
15C22×#11719919C2212yescomplemented
16C22×C22⋊Dic53—24no—
17C22×C22⋊Dic53—24no—
18C22×C22⋊Dic53C418yescomplemented
19C22×C22⋊Dic53C2211yescharacteristic
20C22×C22⋊Dic53C2214yes—
21C22×C22⋊Dic53C2212yes—
22C22×C22⋊Dic53C2212yes—
23C22×C22⋊Dic53C2211yescharacteristic
24C22×C4×Dic53C2212yescomplemented
25#16810484—24no—
26#16810479—24no—
27C2×#13133124C2212yescomplemented
28C2×#13133124C2212yescomplemented
29C2×#13133124C2214yescomplemented
30C2×#13133124C2214yescomplemented
31C26×C53C2211yescharacteristic
32C22×C22⋊C4×C53C2211yescharacteristic
33C22×C22⋊C4×C53C2214yes—
34C22×C22⋊C4×C53C2211yescharacteristic
35C53×#379—24no—
36C23×Dic53—22no—
37C23×Dic53—22no—
38C23×Dic53—22no—
39C23×Dic53—22no—
40C23×Dic53—22no—
41C23×Dic53—22no—
42C23×Dic53—44no—
43C23×Dic53C2×C414yes—
44C23×Dic53C2×C414yes—
45C23×Dic53D414yes—
46C23×Dic53D414yes—
47C23×Dic53C2312yes—
48C23×Dic53C2312yes—
49C23×Dic53C2312yes—
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2311yescharacteristic
52#13133173—232no—
53C2×#11719919—24no—
54C2×#11719919—24no—
55C2×#11719919—24no—
56C2×C22⋊Dic53—22no—
57C2×C22⋊Dic53—22no—
58C2×C22⋊Dic53—22no—
59C2×C22⋊Dic53—22no—
60C2×C22⋊Dic53—28no—
61C2×C22⋊Dic53—22no—
62C2×C22⋊Dic53—232no—
63C2×C22⋊Dic53—24no—
64C2×C22⋊Dic53—24no—
65C2×C22⋊Dic53—24no—
66C2×C22⋊Dic53—24no—
67C2×C22⋊Dic53—28no—
68C2×C22⋊Dic53—44no—
69C2×C22⋊Dic53C2×C418yes—
70C2×C22⋊Dic53C2×C418yes—
71C2×C22⋊Dic53D414yes—
72C2×C4×Dic53—24no—
73C2×C4×Dic53—24no—
74C2×C4×Dic53—24no—
75#13133124—24no—
76#13133124—24no—
77#13133124—44no—
78#13133124—44no—
79C25×C53—22no—
80C25×C53—22no—
81C25×C53—28no—
82C25×C53—22no—
83C25×C53—22no—
84C25×C53—22no—
85C25×C53—22no—
86C25×C53—44no—
87C25×C53C2×C414yes—
88C25×C53C2312yes—
89C25×C53C2311yescharacteristic
90C23×C4×C53C2312yes—
91C23×C4×C53C2312yes—
92C2×C22⋊C4×C53—22no—
93C2×C22⋊C4×C53—22no—
94C2×C22⋊C4×C53—22no—
95C2×C22⋊C4×C53—22no—
96C2×C22⋊C4×C53—28no—
97C2×C22⋊C4×C53—22no—
98C2×C22⋊C4×C53—28no—
99C2×C22⋊C4×C53—44no—
100C2×C22⋊C4×C53D414yes—
101C24×C53—24no—
102C24×C53—24no—
103C24×C53—216no—
104C24×C53—216no—
105C24×C53—28no—
106C24×C53—216no—
107C24×C53—22no—
108C24×C53—22no—
109C24×C53—22no—
110C24×C53—22no—
111C24×C53—24no—
112C24×C53—22no—
113C24×C53—22no—
114C24×C53—24no—
115C24×C53—24no—
116C24×C53—28no—
117C24×C53—28no—
118C24×C53—48no—
119C24×C53—416no—
120C24×C53—416no—
121C24×C53—416no—
122C24×C53—416no—
123C24×C53—48no—
124C24×C53—48no—
125C24×C53—44no—
126C24×C53—44no—
127C24×C53—44no—
128C24×C53—44no—
129C24×C53C22×C414yes—
130C24×C53C22×C412yes—
131C24×C53C2×D412yes—
132C24×C53C2×D412yes—
133C24×C53C2×D412yes—
134C24×C53C2×D412yes—
135C24×C53#4118yes—
136C24×C53#4112yes—
137C24×C53#4112yes—
138C24×C53C2411yescharacteristic
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—24no—
142C22×C4×C53—24no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—24no—
147C22×C4×C53—24no—
148C22×C4×C53C2×D414yes—
149C22×Dic53—28no—
150C22×Dic53—28no—
151C22×Dic53—28no—
152C22×Dic53—28no—
153C22×Dic53—28no—
154C22×Dic53—28no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—28no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—24no—
169C22×Dic53—24no—
170C22×Dic53—24no—
171C22×Dic53—24no—
172C22×Dic53—22no—
173C22×Dic53—22no—
174C22×Dic53—22no—
175C22×Dic53—24no—
176C22×Dic53—24no—
177C22×Dic53—48no—
178C22×Dic53—48no—
179C22×Dic53—48no—
180C22×Dic53C22⋊C4116yescomplemented
181C22×Dic53C22×C414yes—
182C22×Dic53C22×C412yes—
183C22×Dic53C22×C412yes—
184C22×Dic53C2×D414yes—
185C22×Dic53C2×D412yes—
186C22×Dic53C2×D412yes—
187C22×Dic53C2×D412yes—
188C22×Dic53C2×D412yes—
189C22⋊C4×C53—28no—
190C22⋊C4×C53—416no—
191C22⋊C4×C53—44no—
192C22⋊Dic53—28no—
193C22⋊Dic53—216no—
194C22⋊Dic53—216no—
195C22⋊Dic53—416no—
196C22⋊Dic53—44no—
197#11719919—216no—
198C4×Dic53—216no—
199C23×C53—24no—
200C23×C53—24no—
201C23×C53—24no—
202C23×C53—24no—
203C23×C53—24no—
204C23×C53—24no—
205C23×C53—28no—
206C23×C53—28no—
207C23×C53—28no—
208C23×C53—28no—
209C23×C53—232no—
210C23×C53—24no—
211C23×C53—24no—
212C23×C53—24no—
213C23×C53—24no—
214C23×C53—24no—
215C23×C53—24no—
216C23×C53—28no—
217C23×C53—28no—
218C23×C53—28no—
219C23×C53—28no—
220C23×C53—28no—
221C23×C53—28no—
222C23×C53—432no—
223C23×C53—432no—
224C23×C53—416no—
225C23×C53—432no—
226C23×C53—432no—
227C23×C53—432no—
228C23×C53—432no—
229C23×C53—42no—
230C23×C53—42no—
231C23×C53—42no—
232C23×C53—42no—
233C23×C53—44no—
234C23×C53—44no—
235C23×C53—44no—
236C23×C53—44no—
237C23×C53—44no—
238C23×C53—44no—
239C23×C53—416no—
240C23×C53C2×C22⋊C414yes—
241C23×C53#11718yescomplemented
242C23×C53C4×D414yes—
243C23×C53C4×D414yes—
244C23×C53#12014yes—
245C23×C53#12114yes—
246C23×C53#12114yes—
247C23×C53#12314yes—
248C23×C53C23×C411yescharacteristic
249C23×C53C22×D411yescharacteristic
250C23×C53C22×D411yescharacteristic
251C23×C53C22×D411yescharacteristic
252C23×C53C22×D411yescharacteristic
253C23×C53C2×#4114yes—
254C23×C53C2×#4111yescharacteristic
255C23×C53C2×#4111yescharacteristic
256C23×C53H(5,𝔽2)14yes—
257C2×C4×C53—24no—
258C2×C4×C53—28no—
259C2×C4×C53—42no—
260C2×C4×C53—42no—
261C2×C4×C53—44no—
262C2×C4×C53—42no—
263C2×C4×C53—42no—
264C2×C4×C53—42no—
265C2×C4×C53—44no—
266C2×C4×C53—44no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—22no—
273C2×Dic53—22no—
274C2×Dic53—22no—
275C2×Dic53—22no—
276C2×Dic53—22no—
277C2×Dic53—22no—
278C2×Dic53—22no—
279C2×Dic53—232no—
280C2×Dic53—24no—
281C2×Dic53—24no—
282C2×Dic53—24no—
283C2×Dic53—24no—
284C2×Dic53—28no—
285C2×Dic53—416no—
286C2×Dic53—42no—
287C2×Dic53—42no—
288C2×Dic53—44no—
289C2×Dic53—44no—
290C2×Dic53—44no—
291C2×Dic53—42no—
292C2×Dic53—42no—
293C2×Dic53—42no—
294C2×Dic53—44no—
295C2×Dic53—44no—
296C2×Dic53C2×C22⋊C414yes—
297C2×Dic53C2×C22⋊C414yes—
298#13868—531nomaximal, Sylow
299C22×C53—24no—
300C22×C53—28no—
301C22×C53—28no—
302C22×C53—28no—
303C22×C53—28no—
304C22×C53—28no—
305C22×C53—28no—
306C22×C53—24no—
307C22×C53—28no—
308C22×C53—28no—
309C22×C53—28no—
310C22×C53—464no—
311C22×C53—44no—
312C22×C53—44no—
313C22×C53—44no—
314C22×C53—48no—
315C22×C53—48no—
316C22×C53—48no—
317C22×C53—48no—
318C22×C53—44no—
319C22×C53#38618yes—
320C22×C53C22×C22⋊C411yescharacteristic
321C22×C53C2×#11712yes—
322C22×C53C2×C4×D411yescharacteristic
323C22×C53C2×C4×D411yescharacteristic
324C22×C53#51812yes—
325C22×C53C2×#12011yescharacteristic
326C22×C53C2×#12111yescharacteristic
327C22×C53C2×#12111yescharacteristic
328C22×C53C2×#12311yescharacteristic
329C22×C53#52512yes—
330C22×C53#52512yes—
331C22×C53#53512yes—
332C22×C53#53512yes—
333C22×C53#54614yes—
334C22×C53#54614yes—
335C22×C53#55114yes—
336C22×C53#55114yes—
337C4×C53—44no—
338C4×C53—82no—
339Dic53—24no—
340Dic53—24no—
341Dic53—44no—
342Dic53—44no—
343Dic53—82no—
344C22×#117—531no—
345C23×C22⋊C4—531no—
346C2×#381—531no—
347C2×#379—531no—
348C2×#379D5311yescharacteristic
349#1439—534no—
350#1437—534no—
351C22⋊#95—531no—
352C22⋊#95—531no—
353C2×C53—28no—
354C2×C53—48no—
355C2×C53#228011yescharacteristic
356C2×C53C2×#38611yescharacteristic
357C2×C53#230911yescharacteristic
358C2×C53#234212yes—
359C2×C53#234212yes—
360C2×C53#240412yes—
361C2×C53#241112yes—
362C2×C53#241112yes—
363C2×C53#248212yes—
364C26C2×D5311yescharacteristic
365C24×C4—531no—
366C24×C4—531no—
367C2×#117—1064no—
368C2×#117—1064no—
369C2×#117—1064no—
370C22×C4⋊C4—532no—
371C22×C22⋊C4—538no—
372C22×C22⋊C4—532no—
373C22×C22⋊C4—532no—
374C22×C22⋊C4—531no—
375C22×C22⋊C4—531no—
376C22×C22⋊C4—534no—
377C22×C22⋊C4—1064no—
378C22×C22⋊C4—1064no—
379C22×C22⋊C4C2×D5311yescharacteristic
380C22×C22⋊C4C2×D5314yes—
381C22×C22⋊C4C2×D5311yescharacteristic
382C22×C42—532no—
383#381—1064no—
384#379—24no—
385#379—1064no—
386C2×#95—532no—
387C2×#95—532no—
388C2×#95—534no—
389C2×#95—534no—
390C53#1386811yescharacteristic, complemented, Sylow
391C25—22no—
392C25—22no—
393C25—28no—
394C25—22no—
395C25—22no—
396C25—22no—
397C25—22no—
398C25—44no—
399C25C4×D5314yes—
400C25C22×D5312yes—
401C25C22×D5311yescharacteristic
402C23×C4—534no—
403C23×C4—534no—
404C23×C4—534no—
405C23×C4—531no—
406C23×C4—531no—
407C23×C4—532no—
408C23×C4—532no—
409C23×C4—534no—
410C23×C4—532no—
411C23×C4—1062no—
412C23×C4—1062no—
413C23×C4—1062no—
414C23×C4—1062no—
415C23×C4—1062no—
416C23×C4—1062no—
417C23×C4—2124no—
418C23×C4C22×D5312yes—
419C23×C4C22×D5312yes—
420#117—10632no—
421C2×C4⋊C4—1064no—
422C2×C4⋊C4—1064no—
423C2×C4⋊C4—1064no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—28no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—28no—
431C2×C22⋊C4—44no—
432C2×C22⋊C4—538no—
433C2×C22⋊C4—538no—
434C2×C22⋊C4—534no—
435C2×C22⋊C4—10632no—
436C2×C22⋊C4—1064no—
437C2×C22⋊C4—1064no—
438C2×C22⋊C4—1064no—
439C2×C22⋊C4—1064no—
440C2×C22⋊C4—1062no—
441C2×C22⋊C4—1062no—
442C2×C22⋊C4—1068no—
443C2×C22⋊C4—1062no—
444C2×C22⋊C4—1062no—
445C2×C22⋊C4—1062no—
446C2×C22⋊C4—1068no—
447C2×C22⋊C4—2124no—
448C2×C22⋊C4C22⋊D5314yes—
449C2×C42—1064no—
450C2×C42—1064no—
451C2×C42—1064no—
452#95—1064no—
453#95—1064no—
454#95—2124no—
455#95—2124no—
456C24—24no—
457C24—24no—
458C24—216no—
459C24—216no—
460C24—28no—
461C24—216no—
462C24—22no—
463C24—22no—
464C24—22no—
465C24—22no—
466C24—22no—
467C24—24no—
468C24—22no—
469C24—24no—
470C24—24no—
471C24—28no—
472C24—28no—
473C24—48no—
474C24—416no—
475C24—416no—
476C24—416no—
477C24—416no—
478C24—48no—
479C24—48no—
480C24—44no—
481C24—44no—
482C24—44no—
483C24—44no—
484C24C2×C4×D5314yes—
485C24C2×C4×D5312yes—
486C24#1171994512yes—
487C24D4×D5312yes—
488C24D4×D5312yes—
489C24D4×D5312yes—
490C24#1171994718yes—
491C24#1171994712yes—
492C24C2×C22⋊D5312yes—
493C24C23×D5311yescharacteristic
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—24no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—5316no—
504C22×C4—532no—
505C22×C4—532no—
506C22×C4—532no—
507C22×C4—532no—
508C22×C4—532no—
509C22×C4—532no—
510C22×C4—534no—
511C22×C4—534no—
512C22×C4—1068no—
513C22×C4—1068no—
514C22×C4—1068no—
515C22×C4—1068no—
516C22×C4—1068no—
517C22×C4—1068no—
518C22×C4—1061no—
519C22×C4—1061no—
520C22×C4—1061no—
521C22×C4—1061no—
522C22×C4—1061no—
523C22×C4—1061no—
524C22×C4—1061no—
525C22×C4—1061no—
526C22×C4—1068no—
527C22×C4—1062no—
528C22×C4—1062no—
529C22×C4—1062no—
530C22×C4—1062no—
531C22×C4—1064no—
532C22×C4—1064no—
533C22×C4—1064no—
534C22×C4—1064no—
535C22×C4—1062no—
536C22×C4—1062no—
537C22×C4—1064no—
538C22×C4—1062no—
539C22×C4—1064no—
540C22×C4—2128no—
541C22×C4—2128no—
542C22×C4—2128no—
543C22×C4C2×C22⋊D5314yes—
544C4⋊C4—10616no—
545C22⋊C4—28no—
546C22⋊C4—416no—
547C22⋊C4—44no—
548C22⋊C4—10616no—
549C22⋊C4—10616no—
550C22⋊C4—1068no—
551C22⋊C4—21216no—
552C22⋊C4—2124no—
553C42—10616no—
554C23—28no—
555C23—28no—
556C23—28no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—28no—
564C23—232no—
565C23—28no—
566C23—28no—
567C23—28no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—28no—
575C23—28no—
576C23—28no—
577C23—432no—
578C23—432no—
579C23—432no—
580C23—416no—
581C23—432no—
582C23—432no—
583C23—432no—
584C23—416no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23C22⋊C4×D5314yes—
596C23#1313317318yescomplemented
597C23#1313317714yes—
598C23C4×C22⋊D5314yes—
599C23#1313318514yes—
600C23#1313318814yes—
601C23#1313319014yes—
602C23#1313319914yes—
603C23C22×C4×D5311yescharacteristic
604C23C2×D4×D5311yescharacteristic
605C23C2×D4×D5311yescharacteristic
606C23C2×D4×D5311yescharacteristic
607C23C22×C22⋊D5311yescharacteristic
608C23C2×#1171994714yes—
609C23C2×#1171994711yescharacteristic
610C23C2×#1171994511yescharacteristic
611C23#1313326214yes—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—534no—
623C2×C4—534no—
624C2×C4—10632no—
625C2×C4—1062no—
626C2×C4—1062no—
627C2×C4—1062no—
628C2×C4—1062no—
629C2×C4—1062no—
630C2×C4—1062no—
631C2×C4—1062no—
632C2×C4—1062no—
633C2×C4—1062no—
634C2×C4—1062no—
635C2×C4—1062no—
636C2×C4—1062no—
637C2×C4—1064no—
638C2×C4—1064no—
639C2×C4—1064no—
640C2×C4—1064no—
641C2×C4—1068no—
642C2×C4—2124no—
643C2×C4—2122no—
644C2×C4—2122no—
645C2×C4—2122no—
646C2×C4—2124no—
647C2×C4—21216no—
648C2×C4—2124no—
649C2×C4—2122no—
650C2×C4—2122no—
651C2×C4—2124no—
652C2×C4—2124no—
653C22—28no—
654C22—24no—
655C22—28no—
656C22—28no—
657C22—28no—
658C22—28no—
659C22—24no—
660C22—28no—
661C22—28no—
662C22—28no—
663C22—28no—
664C22—464no—
665C22—48no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—48no—
670C22—48no—
671C22—44no—
672C22—48no—
673C22#1681049918yes—
674C22C2×C22⋊C4×D5311yescharacteristic
675C22C2×#1313317312yes—
676C22C2×#1313317711yescharacteristic
677C22C2×C4×C22⋊D5311yescharacteristic
678C22#1681109612yes—
679C22C2×#1313318511yescharacteristic
680C22C2×#1313318811yescharacteristic
681C22C2×#1313319011yescharacteristic
682C22C2×#1313319911yescharacteristic
683C22#1681112912yes—
684C22#1681112912yes—
685C22#1681118812yes—
686C22#1681118812yes—
687C22#1681126014yes—
688C22#1681126714yes—
689C22#1681129914yes—
690C22#1681130414yes—
691C4—44no—
692C4—82no—
693C4—1064no—
694C4—1064no—
695C4—4242no—
696C4—2124no—
697C4—2124no—
698C2—28no—
699C2—48no—
700C2#2027709311yescharacteristic
701C2C2×#1681049911yescharacteristic
702C2#2027729611yescharacteristic
703C2#2027756012yes—
704C2#2027756612yes—
705C2#2027803112yes—
706C2#2027809612yes—
707C2#2027810212yes—
708C2#2027898612yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#379C2
socle, cosocleC24×C53C24
prime radicalC26×C53C22
hypercenterC2×#379D53
hypocenterC53#13868
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
1. C22×#13133173C2semidirect
3. C2×#16810484C2semidirect
5. #20272958C2semidirect
6. #20272952C2semidirect
7. #20272937C2semidirect
8. #20272937C2semidirect
15. C22×#11719919C22semidirect
18. C22×C22⋊Dic53C4semidirect
24. C22×C4×Dic53C22semidirect
27. C2×#13133124C22semidirect
28. C2×#13133124C22semidirect
29. C2×#13133124C22semidirect
30. C2×#13133124C22semidirect
180. C22×Dic53C22⋊C4semidirect
241. C23×C53#117semidirect
390. C53#13868semidirect
596. C23#13133173semidirect

Maximal subgroups

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

subgroupindex
C22×#131331732
C23×C22⋊Dic532
C2×#168104842
C2×#168104792
#202729582
#202729522
#202729372
C2×C53×#3792
#1386853