Finite Group Databasebeta

← #28319225

Subgroups — #28319225

Order 13568.

1418 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811648C212yesmaximal, complemented
2#20290609C212yesmaximal, complemented
3#20290573C212yesmaximal, complemented
4#20290572C211yesmaximal, characteristic, complemented
5#20290570C211yesmaximal, characteristic, complemented
6#20290538C211yesmaximal, characteristic
7#20290538C211yesmaximal, characteristic
8#20290445C212yesmaximal, complemented
9#20290442C211yesmaximal, characteristic, complemented
10C4×#13133274C211yesmaximal, characteristic, complemented
11C2×#117×D53C211yesmaximal, characteristic
12#16811648—28no—
13C2×#13133284C2212yescomplemented
14C2×C4⋊#88324C2212yescomplemented
15C2×#13133274C2211yescharacteristic
16C2×#13133274C2211yescharacteristic
17C2×#13133274C2211yescharacteristic
18C2×#13133274C2211yescharacteristic
19C2×#13133274C2212yes—
20C2×#13133274C2212yes—
21C2×C22⋊#88324C2212yes—
22C2×C22⋊#88324C2212yes—
23C2×#13133272C2212yescomplemented
24C2×C4×#88324C2212yescomplemented
25#16811489C2212yescomplemented
26#16811489C2212yescomplemented
27#16811489C2212yescomplemented
28#16811489C2212yescomplemented
29C23×C4×D53C2211yescharacteristic
30C2×#13133176C411yescharacteristic
31C2×#13133175C2211yescharacteristic
32C2×#13133175C2211yescharacteristic
33C2×#13133174C411yescharacteristic
34C2×#13133174C411yescharacteristic
35C2×#13133173C2211yescharacteristic
36C2×#13133173C2211yescharacteristic
37#117×D53—28no—
38C2×C4⋊C4×D53C411yescharacteristic
39C2×C4⋊C4×D53C411yescharacteristic
40C2×C22⋊C4×D53C411yescharacteristic
41C2×C22⋊C4×D53C411yescharacteristic
42C2×C42×D53C2211yescharacteristic
43C2×C42×D53C2211yescharacteristic
44C2×#117×C53C411yescharacteristic
45C22×#88324C2312yes—
46C22×#88324C2312yes—
47C22×#88324C2312yes—
48C22×#88324C2312yes—
49#13133284—24no—
50C4⋊#88324—24no—
51#13133274—24no—
52#13133274—24no—
53#13133274—22no—
54#13133274—22no—
55#13133274—22no—
56#13133274—22no—
57C22⋊#88324—24no—
58C22⋊#88324—24no—
59#13133272—24no—
60C4×#88324—24no—
61C24×D53C2×C411yescharacteristic
62C23×Dic53C2311yescharacteristic
63C22×C4×D53—24no—
64C22×C4×D53—24no—
65C22×C4×D53—22no—
66C22×C4×D53—22no—
67C22×C4×D53C2×C411yescharacteristic
68C22×C4×D53C2×C411yescharacteristic
69C22×C4×D53C2×C411yescharacteristic
70C22×C4×D53C2×C411yescharacteristic
71C22×C4×D53C2×C411yescharacteristic
72C22×C4×D53C2×C411yescharacteristic
73C22×C4×D53C2311yescharacteristic
74C22×C4×D53C2311yescharacteristic
75#13133176—24no—
76#13133175—24no—
77#13133175D418yes—
78#13133174—24no—
79#13133174—24no—
80#13133173—24no—
81#13133173—24no—
82C2×C4⋊Dic53C2×C411yescharacteristic
83C2×C4⋊Dic53C2×C411yescharacteristic
84C2×#11719919C2×C411yescharacteristic
85C2×#11719919C2×C411yescharacteristic
86C2×#11719919C2×C411yescharacteristic
87C2×#11719919C2×C411yescharacteristic
88C4⋊C4×D53—24no—
89C4⋊C4×D53—24no—
90C2×C22⋊Dic53C2×C411yescharacteristic
91C2×C22⋊Dic53C2×C411yescharacteristic
92C2×#11719921C2×C411yescharacteristic
93C2×#11719921C2×C411yescharacteristic
94C2×#11719921C2×C411yescharacteristic
95C2×#11719921C2×C411yescharacteristic
96C22⋊C4×D53—24no—
97C22⋊C4×D53—24no—
98C2×C4×Dic53C2×C411yescharacteristic
99C2×C4×Dic53C2×C411yescharacteristic
100C2×C4×Dic53C2311yescharacteristic
101C2×C4×Dic53C2311yescharacteristic
102C2×C4×Dic53C2311yescharacteristic
103C2×C4×Dic53C2311yescharacteristic
104C42×D53—24no—
105C42×D53—24no—
106C23×C4×C53C2×C411yescharacteristic
107#117×C53—24no—
108C2×C4⋊C4×C53C2×C411yescharacteristic
109C2×C4⋊C4×C53C2×C411yescharacteristic
110C2×C22⋊C4×C53C2×C411yescharacteristic
111C2×C22⋊C4×C53C2×C411yescharacteristic
112C2×C42×C53C2×C411yescharacteristic
113C2×C42×C53C2×C411yescharacteristic
114C24×C53C22×C411yescharacteristic
115C23×D53—22no—
116C23×D53—22no—
117C23×D53—22no—
118C23×D53—22no—
119C23×D53—22no—
120C23×D53—41no—
121C23×D53—41no—
122C23×D53C22×C411yescharacteristic
123C23×D53C22×C411yescharacteristic
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53C22×C411yescharacteristic
128C22×C4×C53C22×C411yescharacteristic
129C22×C4×C53C22×C411yescharacteristic
130C22×C4×C53C22×C411yescharacteristic
131C22×C4×C53C22×C411yescharacteristic
132C22×C4×C53C22×C411yescharacteristic
133C22×Dic53—22no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53C22×C411yescharacteristic
137C22×Dic53C22×C411yescharacteristic
138C22×Dic53C22×C411yescharacteristic
139C22×Dic53C22×C411yescharacteristic
140C22×Dic53C22×C411yescharacteristic
141C22×Dic53#4114yes—
142C22×Dic53C2411yescharacteristic
143C2×C4×D53—216no—
144C2×C4×D53—216no—
145C2×C4×D53—22no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×C4×D53—22no—
163C2×C4×D53—22no—
164C2×C4×D53—41no—
165C2×C4×D53—41no—
166C2×C4×D53—41no—
167C2×C4×D53—41no—
168C2×C4×D53C2×D414yes—
169C2×C4×D53#4112yes—
170C2×C4×D53#4112yes—
171C2×#88324—22no—
172C2×#88324—22no—
173C2×#88324—22no—
174C2×#88324—22no—
175C2×#88324—22no—
176C2×#88324—22no—
177C2×#88324—22no—
178C2×#88324—22no—
179C2×#88324—22no—
180C2×#88324—22no—
181C2×#88324—22no—
182C2×#88324—22no—
183C4⋊C4×C53—22no—
184C4⋊C4×C53C22⋊C414yes—
185C22⋊C4×C53—22no—
186C22⋊C4×C53—22no—
187C42×C53—22no—
188C42×C53—22no—
189C22⋊Dic53—22no—
190C22⋊Dic53—22no—
191#11719921—22no—
192#11719921—22no—
193#11719921C22⋊C414yes—
194#11719921C22⋊C414yes—
195C4⋊Dic53—22no—
196C4⋊Dic53C22⋊C414yes—
197#11719919—22no—
198#11719919—22no—
199#11719919—22no—
200#11719919—22no—
201C4×Dic53—22no—
202C4×Dic53—22no—
203C4×Dic53—22no—
204C4×Dic53—22no—
205C4×Dic53C2×D414yes—
206C4×Dic53C2×D414yes—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53#11712yes—
214C23×C53C23×C411yescharacteristic
215C22×D53—21no—
216C22×D53—21no—
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—28no—
220C22×D53—21no—
221C22×D53—21no—
222C22×D53—21no—
223C22×D53—21no—
224C22×D53—21no—
225C22×D53—21no—
226C22×D53—44no—
227C22×D53—44no—
228C22×D53—44no—
229C22×D53—44no—
230C22×D53—44no—
231C22×D53—44no—
232C22×D53C2×C22⋊C412yes—
233C22×D53#11712yes—
234C22×D53#13112yes—
235C22×D53#13112yes—
236C2×C4×C53—28no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53C2×C22⋊C412yes—
252C2×C4×C53C2×C22⋊C412yes—
253C2×C4×C53C2×C22⋊C412yes—
254C2×C4×C53#11712yes—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53C2×C22⋊C412yes—
270C2×Dic53C2×C22⋊C412yes—
271C2×Dic53#11712yes—
272C2×Dic53C22×D412yes—
273C2×Dic53C2×#4112yes—
274C2×Dic53C2×#4111yescharacteristic
275C2×Dic53C2×#4111yescharacteristic
276C4×D53—28no—
277C4×D53—28no—
278C4×D53—24no—
279C4×D53—24no—
280C4×D53—24no—
281C4×D53—24no—
282C4×D53—44no—
283C4×D53—44no—
284C4×D53—44no—
285C4×D53—44no—
286#88324—42no—
287#88324—42no—
288#88324—42no—
289#88324—42no—
290#20716—531nomaximal, Sylow
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53C22×C22⋊C411yescharacteristic
299C22×C53C2×#11711yescharacteristic
300C22×C53C2×#11711yescharacteristic
301C22×C53C2×#13111yescharacteristic
302C22×C53C2×#13111yescharacteristic
303C22×C53#56811yescharacteristic
304C22×C53#56811yescharacteristic
305C2×D53—22no—
306C2×D53—22no—
307C2×D53—22no—
308C2×D53—22no—
309C2×D53—22no—
310C2×D53—22no—
311C2×D53—22no—
312C2×D53—22no—
313C2×D53—22no—
314C2×D53—416no—
315C2×D53—41no—
316C2×D53—41no—
317C2×D53—41no—
318C2×D53—41no—
319C2×D53#43314yescomplemented
320C4×C53—22no—
321C4×C53—22no—
322C4×C53—22no—
323C4×C53—22no—
324C4×C53—41no—
325C4×C53—41no—
326Dic53—22no—
327Dic53—22no—
328Dic53—22no—
329Dic53—41no—
330Dic53—41no—
331Dic53#52512yes—
332Dic53#52512yes—
333C22×#117—531no—
334C2×#433—532no—
335#1818—532no—
336#1798—532no—
337#1797—531no—
338#1797—531no—
339#1779—531no—
340#1779—531no—
341#1750—532no—
342#1749—531no—
343C4×C22⋊C8—531no—
344C2×C53—42no—
345C2×C53#228011yescharacteristic
346C2×C53#288011yescharacteristic
347C2×C53#288011yescharacteristic
348C2×C53C2×#43311yescharacteristic
349C2×C53#292111yescharacteristic
350C2×C53#292411yescharacteristic
351C2×C53#292411yescharacteristic
352D53—24no—
353D53—42no—
354C24×C4—531no—
355C2×#117—531no—
356C2×#117—531no—
357C2×#117—531no—
358C2×#117—531no—
359C2×#117—531no—
360C2×#117—531no—
361C2×#117—531no—
362C2×#117—1068no—
363C2×#117#615711yescharacteristic
364C22×C4⋊C4—531no—
365C22×C4⋊C4—531no—
366C22×C22⋊C4—531no—
367C22×C22⋊C4—531no—
368C22×C42—531no—
369C22×C42—531no—
370#433—1068no—
371C2×C4⋊C8—532no—
372C2×C4⋊C8—532no—
373C2×C22⋊C8—531no—
374C2×C22⋊C8—531no—
375C2×C22⋊C8—531no—
376C2×C22⋊C8—531no—
377C2×C22⋊C8—532no—
378C2×C22⋊C8—532no—
379C2×C22⋊C8—532no—
380C2×C22⋊C8—532no—
381C2×#97—532no—
382C2×C4×C8—532no—
383#336—532no—
384#336—532no—
385#336—532no—
386#336—532no—
387C53#2071611yescharacteristic, complemented, Sylow
388C25—531no—
389C23×C4—531no—
390C23×C4—531no—
391C23×C4—531no—
392C23×C4—531no—
393C23×C4—531no—
394C23×C4—531no—
395C23×C4—531no—
396C23×C4—531no—
397C23×C4—531no—
398C23×C4—1064no—
399C23×C4—1064no—
400C23×C4—1062no—
401C23×C4—1062no—
402C23×C4C2×#615711yescharacteristic
403C22×C8—532no—
404C22×C8—532no—
405C22×C8—532no—
406C22×C8—532no—
407#117—24no—
408#117—538no—
409#117—1064no—
410#117—1064no—
411#117—1064no—
412#117—1064no—
413#117—1064no—
414#117—1064no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—1064no—
422C2×C4⋊C4—1064no—
423C2×C4⋊C4C2×#615711yescharacteristic
424C2×C4⋊C4C2×#615711yescharacteristic
425C2×C22⋊C4—531no—
426C2×C22⋊C4—531no—
427C2×C22⋊C4—531no—
428C2×C22⋊C4—531no—
429C2×C22⋊C4—531no—
430C2×C22⋊C4—531no—
431C2×C22⋊C4—1064no—
432C2×C22⋊C4—1064no—
433C2×C22⋊C4C2×#615711yescharacteristic
434C2×C22⋊C4C2×#615711yescharacteristic
435C2×C42—531no—
436C2×C42—531no—
437C2×C42—531no—
438C2×C42—531no—
439C2×C42—531no—
440C2×C42—531no—
441C2×C42—1064no—
442C2×C42—1064no—
443C2×C42C2×#615711yescharacteristic
444C2×C42C2×#615711yescharacteristic
445C4⋊C8—1064no—
446C4⋊C8—1064no—
447C22⋊C8—1064no—
448C22⋊C8—1064no—
449C22⋊C8—1064no—
450C22⋊C8—1064no—
451C22⋊C8—1062no—
452C22⋊C8—1062no—
453C22⋊C8—1062no—
454C22⋊C8—1062no—
455#97—1064no—
456C4×C8—1064no—
457C24—531no—
458C24—531no—
459C24—1062no—
460C24—1062no—
461C24—1062no—
462C24—1062no—
463C24—1062no—
464C24—2121no—
465C24—2121no—
466C24C22×#615711yescharacteristic
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—531no—
471C22×C4—534no—
472C22×C4—531no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—534no—
476C22×C4—532no—
477C22×C4—532no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—10616no—
492C22×C4—10616no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—2121no—
505C22×C4—2121no—
506C22×C4—2121no—
507C22×C4—2121no—
508C22×C4C22×#615711yescharacteristic
509C22×C4C22×#615711yescharacteristic
510C22×C4C22×#615711yescharacteristic
511C22×C4C22×#615711yescharacteristic
512C22×C4C22×#615711yescharacteristic
513C22×C4C22×#615711yescharacteristic
514C2×C8—1062no—
515C2×C8—1062no—
516C2×C8—1062no—
517C2×C8—1062no—
518C2×C8—1062no—
519C2×C8—1062no—
520C2×C8—1062no—
521C2×C8—1062no—
522C2×C8—1062no—
523C2×C8—1062no—
524C2×C8—1062no—
525C2×C8—1062no—
526C4⋊C4—22no—
527C4⋊C4—534no—
528C4⋊C4—1062no—
529C4⋊C4—1062no—
530C4⋊C4—1062no—
531C4⋊C4—1062no—
532C4⋊C4—1062no—
533C4⋊C4C22⋊#615714yes—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—534no—
537C22⋊C4—534no—
538C22⋊C4—1062no—
539C22⋊C4—1062no—
540C22⋊C4—1062no—
541C22⋊C4—1062no—
542C42—22no—
543C42—22no—
544C42—534no—
545C42—534no—
546C42—1062no—
547C42—1062no—
548C42—1062no—
549C42—1062no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—532no—
557C23—532no—
558C23—532no—
559C23—532no—
560C23—1068no—
561C23—1061no—
562C23—1061no—
563C23—1061no—
564C23—1061no—
565C23—1061no—
566C23—1061no—
567C23—1061no—
568C23—1061no—
569C23—1061no—
570C23—1061no—
571C23—2124no—
572C23—2124no—
573C23—2124no—
574C23—2124no—
575C23—2124no—
576C23—2124no—
577C23#1313329612yes—
578C23C23×#615711yescharacteristic
579C2×C4—28no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—532no—
595C2×C4—531no—
596C2×C4—531no—
597C2×C4—532no—
598C2×C4—532no—
599C2×C4—532no—
600C2×C4—532no—
601C2×C4—1064no—
602C2×C4—1064no—
603C2×C4—1064no—
604C2×C4—1064no—
605C2×C4—1068no—
606C2×C4—1068no—
607C2×C4—1064no—
608C2×C4—1064no—
609C2×C4—1061no—
610C2×C4—1061no—
611C2×C4—1061no—
612C2×C4—1061no—
613C2×C4—1061no—
614C2×C4—1061no—
615C2×C4—1061no—
616C2×C4—1061no—
617C2×C4—1061no—
618C2×C4—1061no—
619C2×C4—1061no—
620C2×C4—1061no—
621C2×C4—2124no—
622C2×C4—2124no—
623C2×C4—2124no—
624C2×C4—2124no—
625C2×C4C2×C22⋊#615712yes—
626C2×C4C2×C22⋊#615712yes—
627C2×C4C2×C22⋊#615712yes—
628C2×C4#1313329612yes—
629C8—2122no—
630C8—2122no—
631C8—2122no—
632C8—2122no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—534no—
641C22—1062no—
642C22—1062no—
643C22—1062no—
644C22—1062no—
645C22—1062no—
646C22—1062no—
647C22—1062no—
648C22—1062no—
649C22—1062no—
650C22—21216no—
651C22—2121no—
652C22—2121no—
653C22—2121no—
654C22—2121no—
655C22C22×C22⋊#615711yescharacteristic
656C22C2×#1313329611yescharacteristic
657C22C2×#1313329611yescharacteristic
658C22D53⋊#13111yescharacteristic
659C22D53⋊#13111yescharacteristic
660C22#1681170211yescharacteristic
661C22#1681170211yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—41no—
667C4—41no—
668C4—532no—
669C4—532no—
670C4—1062no—
671C4—1062no—
672C4—1062no—
673C4—2121no—
674C4—2121no—
675C2—42no—
676C2—1064no—
677C2—2122no—
678C2#2029115911yescharacteristic
679C2#2029130511yescharacteristic
680C2#2029130611yescharacteristic
681C2D53⋊#43311yescharacteristic
682C2#2029139611yescharacteristic
683C2#2029140711yescharacteristic
684C2#2029140711yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC2×#117×C53C4
prime radicalC24×D53C2×C4
cosocleC22×Dic53C24
hypercenterC2×#117#6157
hypocenterC53#20716
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#16811648C2semidirect
2. #20290609C2semidirect
3. #20290573C2semidirect
4. #20290572C2semidirect
5. #20290570C2semidirect
8. #20290445C2semidirect
9. #20290442C2semidirect
10. C4×#13133274C2semidirect
13. C2×#13133284C22semidirect
14. C2×C4⋊#88324C22semidirect
23. C2×#13133272C22semidirect
24. C2×C4×#88324C22semidirect
25. #16811489C22semidirect
26. #16811489C22semidirect
27. #16811489C22semidirect
28. #16811489C22semidirect
319. C2×D53#433semidirect
387. C53#20716semidirect

Maximal subgroups

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

subgroupindex
C2×#168116482
#202906092
#202905732
#202905722
#202905702
#202905382
#202904452
#202904422
C4×#131332742
C2×#117×D532
#2071653