Finite Group Databasebeta

← #27383981

Subgroups — #27383981

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
1C22×#13133174C211yesmaximal, characteristic
2C2×#16810683C212yesmaximal, complemented
3#20274785C212yesmaximal, complemented
4#20274696C212yesmaximal, complemented
5#20274695C211yesmaximal, characteristic, complemented
6#20274624C211yesmaximal, characteristic
7#20274624C211yesmaximal, characteristic
8#20274480C212yesmaximal, complemented
9#20274479C211yesmaximal, characteristic, complemented
10C22⋊C8×Dic53C211yesmaximal, characteristic, complemented
11C53×#1797C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C2×#13133174—28no—
14C2×#13133174C411yescharacteristic
15C2×#13133174C411yescharacteristic
16C2×#13133174C411yescharacteristic
17C2×#13133174C411yescharacteristic
18C2×#13133174C2211yescharacteristic
19C2×#13133174C2211yescharacteristic
20C2×#13133174C2211yescharacteristic
21C2×#13133174C2211yescharacteristic
22C22×#11719919C411yescharacteristic
23C22×#11719919C411yescharacteristic
24C22×#11719921C411yescharacteristic
25C22×#11719921C411yescharacteristic
26C22×C4×Dic53C2211yescharacteristic
27#16810683—28no—
28C2×C4⋊#88322C2212yescomplemented
29C2×C22⋊#88322C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31C2×#13133129C2212yes—
32C2×#13133129C2212yes—
33C2×#13133129C2212yes—
34C2×#13133129C2212yes—
35C2×#13133127C2212yescomplemented
36C2×C8×Dic53C2212yescomplemented
37#16810378C2212yescomplemented
38#16810377C2212yescomplemented
39#16810377C2212yescomplemented
40C22×C42×C53C2211yescharacteristic
41C2×C4⋊C8×C53C2212yescomplemented
42C2×C22⋊C8×C53C2211yescharacteristic
43C2×C22⋊C8×C53C2211yescharacteristic
44C53×#336C2212yescomplemented
45C24×D53C2×C411yescharacteristic
46C23×Dic53C2×C411yescharacteristic
47C23×Dic53C2×C411yescharacteristic
48C23×Dic53C2×C411yescharacteristic
49C22×C4×D53—22no—
50C22×C4×D53—22no—
51C22×C4×D53—24no—
52C22×C4×D53—24no—
53C22×C4×D53C2×C411yescharacteristic
54C22×C4×D53C2×C411yescharacteristic
55C22×C4×D53C2311yescharacteristic
56C22×C4×D53C2311yescharacteristic
57C22×#88322C2312yes—
58C22×#88322C2312yes—
59#13133174—24no—
60#13133174—24no—
61#13133174—24no—
62#13133174—24no—
63#13133174—24no—
64#13133174—24no—
65#13133174—24no—
66#13133174D418yes—
67C2×#11719919—24no—
68C2×#11719919—24no—
69C2×#11719919C2×C411yescharacteristic
70C2×#11719919C2×C411yescharacteristic
71C2×#11719919C2×C411yescharacteristic
72C2×#11719919C2×C411yescharacteristic
73C2×#11719919C2×C411yescharacteristic
74C2×#11719919C2×C411yescharacteristic
75C2×#11719919C2×C411yescharacteristic
76C2×#11719919C2×C411yescharacteristic
77C2×#11719921—24no—
78C2×#11719921—24no—
79C2×#11719921C2×C411yescharacteristic
80C2×#11719921C2×C411yescharacteristic
81C2×#11719921C2×C411yescharacteristic
82C2×#11719921C2×C411yescharacteristic
83C2×#11719921C2×C411yescharacteristic
84C2×#11719921C2×C411yescharacteristic
85C2×#11719921C2×C411yescharacteristic
86C2×#11719921C2×C411yescharacteristic
87C2×C4×Dic53—24no—
88C2×C4×Dic53C2×C411yescharacteristic
89C2×C4×Dic53C2×C411yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92C4⋊#88322—24no—
93C22⋊#88322—22no—
94C22⋊#88322—22no—
95#13133129—24no—
96#13133129—24no—
97#13133129—24no—
98#13133129—24no—
99#13133127—24no—
100C8×Dic53—24no—
101C23×C4×C53C2×C411yescharacteristic
102C23×C4×C53C2×C411yescharacteristic
103C23×C4×C53C2311yescharacteristic
104C22×C8×C53C2312yes—
105C22×C8×C53C2312yes—
106C2×C42×C53—24no—
107C2×C42×C53C2×C411yescharacteristic
108C2×C42×C53C2×C411yescharacteristic
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C4⋊C8×C53—24no—
112C22⋊C8×C53—22no—
113C22⋊C8×C53—22no—
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×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—41no—
131C22×C4×C53—41no—
132C22×C4×C53C22×C411yescharacteristic
133C22×C4×C53C22×C411yescharacteristic
134C22×C4×C53C22×C411yescharacteristic
135C22×C4×C53C22×C411yescharacteristic
136C22×C4×C53C22×C411yescharacteristic
137C22×C4×C53#4112yes—
138C22×C4×C53#4112yes—
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—41no—
148C22×Dic53—41no—
149C22×Dic53C22×C411yescharacteristic
150C22×Dic53C22×C411yescharacteristic
151C22×Dic53C22×C411yescharacteristic
152C22×Dic53C22×C411yescharacteristic
153C22×Dic53C22×C411yescharacteristic
154C22×Dic53C22×C411yescharacteristic
155C2×C4×D53—216no—
156C2×C4×D53—216no—
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—21no—
164C2×C4×D53—21no—
165C2×C4×D53—21no—
166C2×C4×D53—21no—
167C2×C4×D53—22no—
168C2×C4×D53—22no—
169C2×C4×D53C2×D414yes—
170C2×C4×D53#4114yes—
171C2×C8×C53—22no—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C2×C8×C53—22no—
177C42×C53—22no—
178C42×C53—22no—
179C42×C53—22no—
180C42×C53C2×D414yes—
181#11719921—22no—
182#11719921—22no—
183#11719921—22no—
184#11719921—22no—
185#11719921—22no—
186#11719921—22no—
187#11719921C22⋊C414yes—
188#11719921C22⋊C414yes—
189#11719919—22no—
190#11719919—22no—
191#11719919—22no—
192#11719919—22no—
193#11719919—22no—
194#11719919—22no—
195#11719919C22⋊C414yes—
196#11719919C22⋊C414yes—
197C4×Dic53—22no—
198C4×Dic53—22no—
199C4×Dic53—22no—
200C4×Dic53C2×D414yes—
201C2×#88322—22no—
202C2×#88322—22no—
203C2×#88322—22no—
204C2×#88322—22no—
205C2×#88322—22no—
206C2×#88322—22no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53#11712yes—
212C23×C53#13112yes—
213C23×C53#13112yes—
214C23×C53C23×C411yescharacteristic
215C22×D53—28no—
216C22×D53—21no—
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—21no—
220C22×D53—21no—
221C22×D53—21no—
222C22×D53—21no—
223C22×D53—21no—
224C22×D53—21no—
225C22×D53—21no—
226C22×D53—21no—
227C22×D53—21no—
228C22×D53—44no—
229C22×D53—44no—
230C22×D53—44no—
231C22×D53—44no—
232C22×D53—44no—
233C22×D53—44no—
234C22×D53C2×C22⋊C412yes—
235C22×D53#11712yes—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
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—44no—
251C2×C4×C53—44no—
252C2×C4×C53C2×C22⋊C412yes—
253C2×C4×C53C2×C22⋊C412yes—
254C2×C4×C53#11712yes—
255C2×C4×C53C22×D412yes—
256C2×C4×C53C2×#4112yes—
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×Dic53—28no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—44no—
275C2×Dic53—44no—
276C2×Dic53C2×C22⋊C412yes—
277C2×Dic53C2×C22⋊C412yes—
278C2×Dic53C2×C22⋊C412yes—
279C2×Dic53#11712yes—
280C4×D53—28no—
281C4×D53—28no—
282C4×D53—24no—
283C4×D53—24no—
284C4×D53—24no—
285C4×D53—24no—
286C8×C53—42no—
287C8×C53—42no—
288#88322—42no—
289#88322—42no—
290#20716—531nomaximal, Sylow
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—22no—
297C22×C53#43314yescomplemented
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—416no—
312C2×D53—41no—
313C2×D53—41no—
314C2×D53—41no—
315C2×D53—41no—
316C2×D53—41no—
317C2×D53—41no—
318C2×D53—41no—
319C2×D53—41no—
320C4×C53—22no—
321C4×C53—22no—
322C4×C53—22no—
323C4×C53—41no—
324C4×C53—41no—
325C4×C53#52512yes—
326C4×C53#52512yes—
327Dic53—22no—
328Dic53—22no—
329Dic53—22no—
330Dic53—22no—
331Dic53—41no—
332Dic53—41no—
333C22×#117—531no—
334C2×#433—532no—
335#1818—532no—
336#1798—532no—
337#1797—531no—
338#1797D5311yescharacteristic, complemented
339#1779—531no—
340#1779—531no—
341#1750—532no—
342#1749—531no—
343C4×C22⋊C8—531no—
344C2×C53—24no—
345C2×C53#228011yescharacteristic
346C2×C53#288011yescharacteristic
347C2×C53#288011yescharacteristic
348C2×C53C2×#43311yescharacteristic
349C2×C53#292111yescharacteristic
350C2×C53#292411yescharacteristic
351C2×C53#292411yescharacteristic
352D53—42no—
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—531no—
363C2×#117—1068no—
364C22×C4⋊C4—531no—
365C22×C4⋊C4—531no—
366C22×C22⋊C4—531no—
367C22×C22⋊C4—531no—
368C22×C42—531no—
369C22×C42C2×D5311yescharacteristic
370#433—1068no—
371C2×C4⋊C8—532no—
372C2×C4⋊C8C2×D5312yescomplemented
373C2×C22⋊C8—531no—
374C2×C22⋊C8—531no—
375C2×C22⋊C8—532no—
376C2×C22⋊C8—532no—
377C2×C22⋊C8—532no—
378C2×C22⋊C8—532no—
379C2×C22⋊C8C2×D5311yescharacteristic
380C2×C22⋊C8C2×D5311yescharacteristic
381C2×#97—532no—
382C2×C4×C8—532no—
383#336—532no—
384#336—532no—
385#336—532no—
386#336C2×D5312yescomplemented
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—1064no—
397C23×C4—1064no—
398C23×C4—1062no—
399C23×C4—1062no—
400C23×C4C4×D5311yescharacteristic
401C23×C4C4×D5311yescharacteristic
402C23×C4C22×D5311yescharacteristic
403C22×C8—532no—
404C22×C8—532no—
405C22×C8C22×D5312yes—
406C22×C8C22×D5312yes—
407#117—538no—
408#117—1064no—
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—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—1064no—
424C2×C4⋊C4—1064no—
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—531no—
432C2×C22⋊C4—531no—
433C2×C22⋊C4—1064no—
434C2×C22⋊C4—1064no—
435C2×C42—24no—
436C2×C42—531no—
437C2×C42—531no—
438C2×C42—531no—
439C2×C42—531no—
440C2×C42—1064no—
441C2×C42C4×D5311yescharacteristic
442C2×C42C4×D5311yescharacteristic
443C2×C42C22×D5311yescharacteristic
444C2×C42C22×D5311yescharacteristic
445C4⋊C8—24no—
446C4⋊C8—1064no—
447C22⋊C8—22no—
448C22⋊C8—22no—
449C22⋊C8—1064no—
450C22⋊C8—1064no—
451C22⋊C8—1064no—
452C22⋊C8—1064no—
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—
466C24C2×C4×D5311yescharacteristic
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—534no—
480C22×C4—534no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—10616no—
488C22×C4—10616no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—2121no—
505C22×C4—2121no—
506C22×C4C2×C4×D5311yescharacteristic
507C22×C4C2×C4×D5311yescharacteristic
508C22×C4C2×C4×D5311yescharacteristic
509C22×C4C2×C4×D5311yescharacteristic
510C22×C4C2×C4×D5311yescharacteristic
511C22×C4#1171994712yes—
512C22×C4#1171994912yes—
513C22×C4C23×D5311yescharacteristic
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—1062no—
521C2×C8—1062no—
522C2×C8—1062no—
523C2×C8—1062no—
524C2×C8—1062no—
525C2×C8—1062no—
526C4⋊C4—534no—
527C4⋊C4—534no—
528C4⋊C4—1062no—
529C4⋊C4—1062no—
530C4⋊C4—1062no—
531C4⋊C4—1062no—
532C4⋊C4—1062no—
533C4⋊C4—1062no—
534C22⋊C4—534no—
535C22⋊C4—534no—
536C22⋊C4—1062no—
537C22⋊C4—1062no—
538C22⋊C4—1062no—
539C22⋊C4—1062no—
540C22⋊C4—1062no—
541C22⋊C4—1062no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—534no—
546C42—1062no—
547C42—1062no—
548C42—1062no—
549C42D4×D5314yes—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—532no—
555C23—532no—
556C23—1068no—
557C23—1061no—
558C23—1061no—
559C23—1061no—
560C23—1061no—
561C23—1061no—
562C23—1061no—
563C23—1061no—
564C23—1061no—
565C23—1061no—
566C23—1061no—
567C23—1061no—
568C23—1061no—
569C23—2124no—
570C23—2124no—
571C23—2124no—
572C23—2124no—
573C23—2124no—
574C23—2124no—
575C23#1313317512yes—
576C23#1313322312yes—
577C23#1313322512yes—
578C23C22×C4×D5311yescharacteristic
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—532no—
596C2×C4—532no—
597C2×C4—532no—
598C2×C4—532no—
599C2×C4—1068no—
600C2×C4—1068no—
601C2×C4—1068no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1061no—
606C2×C4—1061no—
607C2×C4—1061no—
608C2×C4—1061no—
609C2×C4—1061no—
610C2×C4—1064no—
611C2×C4—1064no—
612C2×C4—1064no—
613C2×C4—1064no—
614C2×C4—1061no—
615C2×C4—1061no—
616C2×C4—1061no—
617C2×C4—1061no—
618C2×C4—1061no—
619C2×C4—1061no—
620C2×C4—2124no—
621C2×C4—2124no—
622C2×C4C22⋊C4×D5312yes—
623C2×C4C22⋊C4×D5312yes—
624C2×C4#1313317512yes—
625C2×C4C2×D4×D5312yes—
626C2×C4#41×D5312yes—
627C2×C4C2×#1171994711yescharacteristic
628C2×C4C2×#1171994911yescharacteristic
629C8—42no—
630C8—42no—
631C8—2122no—
632C8—2122no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—1062no—
640C22—1062no—
641C22—1062no—
642C22—1062no—
643C22—1062no—
644C22—1062no—
645C22—21216no—
646C22—2121no—
647C22—2121no—
648C22—2121no—
649C22—2121no—
650C22—2121no—
651C22—2121no—
652C22—2121no—
653C22—2121no—
654C22#1681068314yescomplemented
655C22C2×C22⋊C4×D5311yescharacteristic
656C22#117×D5311yescharacteristic
657C22C2×#1313317511yescharacteristic
658C22C2×#1313322311yescharacteristic
659C22C2×#1313322511yescharacteristic
660C22#1681138911yescharacteristic
661C22#1681138911yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—41no—
666C4—41no—
667C4—1062no—
668C4—1062no—
669C4—1062no—
670C4—1062no—
671C4—2121no—
672C4—2121no—
673C4#1681113312yes—
674C4#1681113612yes—
675C2—24no—
676C2—2122no—
677C2—2122no—
678C2#2027709711yescharacteristic
679C2#2028316111yescharacteristic
680C2#2028316511yescharacteristic
681C2C2×#1681068311yescharacteristic
682C2#2028335211yescharacteristic
683C2#2028336811yescharacteristic
684C2#2028337411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D53
commutator, socleC23×C53C23×C4
FittingC53×#1797C2
prime radicalC24×D53C2×C4
cosocleC22×C4×C53C24
hypercenter#1797D53
hypocenterC53#20716
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. C2×#16810683C2semidirect
3. #20274785C2semidirect
4. #20274696C2semidirect
5. #20274695C2semidirect
8. #20274480C2semidirect
9. #20274479C2semidirect
10. C22⋊C8×Dic53C2semidirect
11. C53×#1797C2semidirect
28. C2×C4⋊#88322C22semidirect
35. C2×#13133127C22semidirect
36. C2×C8×Dic53C22semidirect
37. #16810378C22semidirect
38. #16810377C22semidirect
39. #16810377C22semidirect
41. C2×C4⋊C8×C53C22semidirect
44. C53×#336C22semidirect
297. C22×C53#433semidirect
338. #1797D53semidirect
372. C2×C4⋊C8C2×D53semidirect
386. #336C2×D53semidirect
387. C53#20716semidirect
654. C22#16810683semidirect

Maximal subgroups

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

subgroupindex
C22×#131331742
C2×#168106832
#202747852
#202746962
#202746952
#202746242
#202744802
#202744792
C22⋊C8×Dic532
C53×#17972
#2071653