Finite Group Databasebeta

← #27315381

Subgroups — #27315381

Order 13568.

2734 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133193C211yesmaximal, characteristic
2C22×C42×D53C211yesmaximal, characteristic
3C2×#16810530C211yesmaximal, characteristic, complemented
4C2×#16810480C211yesmaximal, characteristic
5#20272946C214yesmaximal, complemented
6#20272945C211yesmaximal, characteristic, complemented
7#20272944C211yesmaximal, characteristic, complemented
8#20272936C214yesmaximal
9C2×C53×#391C211yesmaximal, characteristic, complemented
10C23×C4×D53C2212yes—
11C23×C4×D53C2211yescharacteristic
12C2×#13133193—24no—
13C2×#13133193—24no—
14C2×#13133193C418yescomplemented
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C2212yes—
18C22×#11719921C2212yes—
19C22×#11719921C2212yes—
20C22×C4×Dic53C2212yes—
21C22×C4×Dic53C2211yescharacteristic
22C2×C42×D53—24no—
23C2×C42×D53—24no—
24C2×C42×D53—24no—
25#16810530—24no—
26#16810480—24no—
27C2×#13133124C2214yes—
28C2×#13133124C2214yes—
29C2×#13133124C2214yes—
30C2×#13133124C2212yes—
31C23×Q8×C53C2211yescharacteristic
32C22×C4⋊C4×C53C2211yescharacteristic
33C22×C42×C53C2211yescharacteristic
34C53×#391—24no—
35C2×#95×C53C2214yes—
36C24×D53C2311yescharacteristic
37C23×Dic53C2312yes—
38C23×Dic53C2312yes—
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2312yes—
41C22×C4×D53—24no—
42C22×C4×D53—24no—
43C22×C4×D53—24no—
44C22×C4×D53—22no—
45C22×C4×D53—22no—
46C22×C4×D53—24no—
47C22×C4×D53—22no—
48C22×C4×D53—22no—
49C22×C4×D53—24no—
50C22×C4×D53—24no—
51C22×C4×D53C2×C414yes—
52C22×C4×D53D414yes—
53C22×C4×D53D414yes—
54C22×C4×D53Q818yescomplemented
55#13133193—232no—
56C2×C4⋊Dic53—22no—
57C2×C4⋊Dic53—22no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53—22no—
60C2×C4⋊Dic53—22no—
61C2×C4⋊Dic53C2×C414yescomplemented
62C2×#11719919—24no—
63C2×#11719919—24no—
64C2×#11719919C2×C418yescomplemented
65C2×#11719921—24no—
66C2×#11719921—24no—
67C2×#11719921—24no—
68C2×#11719921—28no—
69C2×#11719921C2×C418yes—
70C2×C4×Dic53—24no—
71C2×C4×Dic53—22no—
72C2×C4×Dic53—22no—
73C2×C4×Dic53—24no—
74C2×C4×Dic53—24no—
75C2×C4×Dic53D414yes—
76C42×D53—232no—
77#13133124—28no—
78#13133124—28no—
79#13133124—24no—
80#13133124—28no—
81C22×Q8×C53—22no—
82C22×Q8×C53—22no—
83C22×Q8×C53—22no—
84C22×Q8×C53—22no—
85C22×Q8×C53—22no—
86C22×Q8×C53—22no—
87C22×Q8×C53C2×C414yescomplemented
88C23×C4×C53C2312yes—
89C23×C4×C53C2312yes—
90C23×C4×C53C2312yes—
91C23×C4×C53C2311yescharacteristic
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53—22no—
95C2×C42×C53—22no—
96C2×C42×C53—22no—
97C2×C42×C53D414yes—
98#95×C53—28no—
99C24×C53C2411yescharacteristic
100C23×D53—21no—
101C23×D53—21no—
102C23×D53—21no—
103C23×D53—21no—
104C23×D53—22no—
105C23×D53—22no—
106C23×D53C22×C412yes—
107C23×D53C2×D412yes—
108C23×D53C2×D412yes—
109C23×D53C2×Q814yes—
110C23×D53#4114yes—
111C2×Q8×C53—28no—
112C2×Q8×C53—28no—
113C2×Q8×C53—28no—
114C2×Q8×C53—44no—
115C2×Q8×C53—44no—
116C2×Q8×C53—44no—
117C2×Q8×C53—44no—
118C22×C4×C53—22no—
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—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—24no—
132C22×C4×C53—24no—
133C22×C4×C53C22×C414yes—
134C22×C4×C53C22×C412yes—
135C22×C4×C53C2×D414yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D412yes—
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2×Q814yes—
141C22×C4×C53#4114yes—
142C22×C4×C53#4112yes—
143C22×C4×C53#4112yes—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—24no—
155C22×Dic53—24no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—24no—
159C22×Dic53—24no—
160C22×Dic53—22no—
161C22×Dic53C22×C414yes—
162C22×Dic53C22×C412yes—
163C22×Dic53C2×D412yes—
164C22×Dic53C2×D412yes—
165C22×Dic53C2×D414yes—
166C22×Dic53C2×Q814yes—
167C22×Dic53#4114yes—
168C2×C4×D53—216no—
169C2×C4×D53—216no—
170C2×C4×D53—216no—
171C2×C4×D53—216no—
172C2×C4×D53—216no—
173C2×C4×D53—28no—
174C2×C4×D53—28no—
175C2×C4×D53—28no—
176C2×C4×D53—28no—
177C2×C4×D53—28no—
178C2×C4×D53—216no—
179C2×C4×D53—28no—
180C2×C4×D53—216no—
181C2×C4×D53C22⋊C4116yes—
182C4⋊C4×C53—44no—
183C42×C53—28no—
184#11719921—216no—
185#11719921—48no—
186C4⋊Dic53—28no—
187C4⋊Dic53—44no—
188#11719919—216no—
189C4×Dic53—216no—
190C4×Dic53—28no—
191C23×C53C23×C411yescharacteristic
192C23×C53C22×D411yescharacteristic
193C23×C53C22×D411yescharacteristic
194C23×C53C22×D411yescharacteristic
195C23×C53C22×D411yescharacteristic
196C23×C53C22×Q812yes—
197C23×C53C2×#4112yes—
198C23×C53C2×#4112yes—
199C23×C53C2×#4111yescharacteristic
200C23×C53C2×#4111yescharacteristic
201C23×C53C2×#4112yes—
202C22×D53—22no—
203C22×D53—22no—
204C22×D53—24no—
205C22×D53—22no—
206C22×D53—22no—
207C22×D53—22no—
208C22×D53—24no—
209C22×D53—24no—
210C22×D53—22no—
211C22×D53—24no—
212C22×D53—24no—
213C22×D53—24no—
214C22×D53—24no—
215C22×D53—24no—
216C22×D53—22no—
217C22×D53—22no—
218C22×D53—24no—
219C22×D53—24no—
220C22×D53C2×C22⋊C414yes—
221C22×D53C4×Q818yes—
222C22×D53C22⋊Q818yes—
223C22×D53#12414yes—
224C22×D53C4⋊Q814yes—
225Q8×C53—416no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
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—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—44no—
241C2×C4×C53—42no—
242C2×C4×C53—44no—
243C2×C4×C53—42no—
244C2×C4×C53C2×C22⋊C414yes—
245C2×C4×C53C4×D414yes—
246C2×C4×C53C4×D414yes—
247C2×C4×C53#12014yes—
248C2×C4×C53#12114yes—
249C2×C4×C53#12114yes—
250C2×C4×C53C22⋊Q818yescomplemented
251C2×C4×C53#12314yes—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—24no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—22no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—24no—
264C2×Dic53—24no—
265C2×Dic53—28no—
266C2×Dic53—44no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—44no—
271C2×Dic53—42no—
272C2×Dic53—44no—
273C2×Dic53—42no—
274C2×Dic53C2×C22⋊C414yes—
275C2×Dic53C22⋊Q818yescomplemented
276C4×D53—264no—
277C4×D53—232no—
278#14702—531nomaximal, Sylow
279C22×C53C22×C22⋊C411yescharacteristic
280C22×C53C2×C4×D411yescharacteristic
281C22×C53C2×C4×D411yescharacteristic
282C22×C53C2×C4×Q812yes—
283C22×C53C4×#4112yes—
284C22×C53C2×#12011yescharacteristic
285C22×C53C2×#12111yescharacteristic
286C22×C53C2×#12111yescharacteristic
287C22×C53C2×C22⋊Q812yes—
288C22×C53C2×C22⋊Q812yes—
289C22×C53C2×C22⋊Q812yes—
290C22×C53C2×#12311yescharacteristic
291C22×C53#52512yes—
292C22×C53#52512yes—
293C22×C53#52512yes—
294C22×C53C2×#12411yescharacteristic
295C22×C53#52912yes—
296C22×C53#52912yes—
297C22×C53C2×C4⋊Q811yescharacteristic
298C22×C53#53211yescharacteristic
299C22×C53#53211yescharacteristic
300C22×C53#53312yes—
301C22×C53#53312yes—
302C2×D53—28no—
303C2×D53—24no—
304C2×D53—24no—
305C2×D53—28no—
306C2×D53—24no—
307C2×D53—24no—
308C2×D53—24no—
309C2×D53—24no—
310C2×D53—28no—
311C2×D53—28no—
312C2×D53#39118yescomplemented
313C4×C53—28no—
314C4×C53—44no—
315C4×C53—82no—
316C4×C53#38618yes—
317Dic53—28no—
318Dic53—24no—
319Dic53—44no—
320Dic53—82no—
321C22×C22⋊Q8—531no—
322C23×C42—531no—
323C2×#391—531no—
324C2×#391D5311yescharacteristic, complemented
325C2×#380—531no—
326#1436—534no—
327C4⋊#95—531no—
328C4⋊#95—531no—
329C22⋊#95—534no—
330C2×C53C2×#38611yescharacteristic
331C2×C53C2×#39111yescharacteristic
332C2×C53#229711yescharacteristic
333C2×C53C4×C22⋊Q812yes—
334C2×C53C4×C22⋊Q812yes—
335C2×C53#284812yes—
336C2×C53#285011yescharacteristic
337C2×C53#285011yescharacteristic
338C2×C53#285512yes—
339C2×C53#285711yescharacteristic
340C2×C53#285711yescharacteristic
341D53—28no—
342C23×Q8C2×D5311yescharacteristic
343C24×C4—531no—
344C24×C4—532no—
345C2×C22⋊Q8—538no—
346C2×C22⋊Q8—1064no—
347C2×C22⋊Q8—1064no—
348C22×C4⋊C4—532no—
349C22×C4⋊C4—531no—
350C22×C4⋊C4—531no—
351C22×C4⋊C4C2×D5311yescharacteristic
352C22×C22⋊C4—532no—
353C22×C22⋊C4—532no—
354C22×C42—531no—
355C22×C42—532no—
356C22×C42—1064no—
357C22×C42—1064no—
358C22×C42—1064no—
359C22×C42C2×D5311yescharacteristic
360#391—24no—
361#391—1064no—
362#380—1064no—
363C2×#95—534no—
364C2×#95—534no—
365C2×#95—534no—
366C2×#95—532no—
367C2×#95C2×D5314yes—
368C53#1470211yescharacteristic, complemented, Sylow
369C25—531no—
370C22×Q8—22no—
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—22no—
374C22×Q8—22no—
375C22×Q8—22no—
376C22×Q8C4×D5314yescomplemented
377C23×C4—531no—
378C23×C4—532no—
379C23×C4—534no—
380C23×C4—534no—
381C23×C4—534no—
382C23×C4—532no—
383C23×C4—538no—
384C23×C4—532no—
385C23×C4—1064no—
386C23×C4—1064no—
387C23×C4—1064no—
388C23×C4—1064no—
389C23×C4—1062no—
390C23×C4—1062no—
391C23×C4—1062no—
392C23×C4—1064no—
393C23×C4—1064no—
394C23×C4—1062no—
395C23×C4C22×D5312yes—
396C23×C4C22×D5312yes—
397C23×C4C22×D5312yes—
398C23×C4C22×D5311yescharacteristic
399C22⋊Q8—10632no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—538no—
404C2×C4⋊C4—534no—
405C2×C4⋊C4—1064no—
406C2×C4⋊C4—1064no—
407C2×C4⋊C4—1062no—
408C2×C4⋊C4—1062no—
409C2×C4⋊C4—1062no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1062no—
412C2×C22⋊C4—538no—
413C2×C22⋊C4—1064no—
414C2×C22⋊C4—1064no—
415C2×C22⋊C4—1068no—
416C2×C22⋊C4—1064no—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—534no—
420C2×C42—10632no—
421C2×C42—1064no—
422C2×C42—1064no—
423C2×C42—1062no—
424C2×C42—1064no—
425C2×C42—1062no—
426C2×C42C22⋊D5314yes—
427#95—28no—
428#95—1068no—
429#95—1064no—
430#95—1068no—
431#95—1068no—
432C24—532no—
433C24—532no—
434C24—532no—
435C24—534no—
436C24—534no—
437C24—1061no—
438C24—1061no—
439C24—1062no—
440C24—1061no—
441C24—1061no—
442C24—1062no—
443C24C23×D5311yescharacteristic
444C2×Q8—28no—
445C2×Q8—28no—
446C2×Q8—28no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8—44no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—532no—
467C22×C4—534no—
468C22×C4—532no—
469C22×C4—532no—
470C22×C4—534no—
471C22×C4—534no—
472C22×C4—5316no—
473C22×C4—534no—
474C22×C4—10616no—
475C22×C4—10616no—
476C22×C4—10616no—
477C22×C4—10616no—
478C22×C4—10616no—
479C22×C4—10616no—
480C22×C4—1061no—
481C22×C4—1062no—
482C22×C4—1061no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1062no—
488C22×C4—1068no—
489C22×C4—1068no—
490C22×C4—1068no—
491C22×C4—1068no—
492C22×C4—1068no—
493C22×C4—10616no—
494C22×C4—1068no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1064no—
498C22×C4—1064no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—1064no—
502C22×C4—1062no—
503C22×C4—1064no—
504C22×C4C2×C4×D5314yes—
505C22×C4C2×C4×D5312yes—
506C22×C4#1171994512yes—
507C22×C4D4×D5312yes—
508C22×C4D4×D5312yes—
509C22×C4D4×D5312yes—
510C22×C4#1171994712yes—
511C22×C4Q8×D5314yes—
512C22×C4#1171994914yes—
513C22×C4C2×C22⋊D5314yes—
514C22×C4C2×C22⋊D5312yes—
515C4⋊C4—44no—
516C4⋊C4—10616no—
517C4⋊C4—1068no—
518C4⋊C4—2124no—
519C22⋊C4—10616no—
520C22⋊C4—2128no—
521C42—28no—
522C42—10616no—
523C42—1068no—
524C23—534no—
525C23—538no—
526C23—538no—
527C23—534no—
528C23—534no—
529C23—1062no—
530C23—1062no—
531C23—1064no—
532C23—1062no—
533C23—1062no—
534C23—1062no—
535C23—1064no—
536C23—1064no—
537C23—1062no—
538C23—1064no—
539C23—1064no—
540C23—1064no—
541C23—1064no—
542C23—1064no—
543C23—1062no—
544C23—1062no—
545C23—1064no—
546C23—1064no—
547C23C22×C4×D5311yescharacteristic
548C23C2×D4×D5311yescharacteristic
549C23C2×D4×D5311yescharacteristic
550C23C2×D4×D5311yescharacteristic
551C23C22×C22⋊D5311yescharacteristic
552C23C2×Q8×D5312yes—
553C23#41×D5312yes—
554C23#41×D5312yes—
555C23C2×#1171994711yescharacteristic
556C23C2×#1171994511yescharacteristic
557C23C2×#1171994912yes—
558Q8—416no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—534no—
578C2×C4—538no—
579C2×C4—10664no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1064no—
583C2×C4—1062no—
584C2×C4—1062no—
585C2×C4—1062no—
586C2×C4—1064no—
587C2×C4—1064no—
588C2×C4—1062no—
589C2×C4—1064no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—10632no—
593C2×C4—1064no—
594C2×C4—1068no—
595C2×C4—2122no—
596C2×C4—2124no—
597C2×C4—2122no—
598C2×C4—2124no—
599C2×C4—2122no—
600C2×C4—2124no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4C22⋊C4×D5314yes—
604C2×C4#1313317714yes—
605C2×C4C4×C22⋊D5314yes—
606C2×C4#1313318514yes—
607C2×C4#1313318814yes—
608C2×C4#1313319014yes—
609C2×C4#1313319318yescomplemented
610C2×C4#1313319914yes—
611C22—538no—
612C22—1068no—
613C22—1068no—
614C22—1064no—
615C22—1064no—
616C22—1064no—
617C22—1068no—
618C22—1064no—
619C22—1064no—
620C22—1064no—
621C22—1068no—
622C22C2×C22⋊C4×D5311yescharacteristic
623C22C2×#1313317711yescharacteristic
624C22C2×C4×C22⋊D5311yescharacteristic
625C22C4×Q8×D5312yes—
626C22C4×#1171994912yes—
627C22C2×#1313318511yescharacteristic
628C22C2×#1313318811yescharacteristic
629C22C2×#1313319011yescharacteristic
630C22C22⋊Q8×D5312yes—
631C22C22⋊Q8×D5312yes—
632C22C2×#1313319312yes—
633C22C2×#1313319911yescharacteristic
634C22#1681113012yes—
635C22#1681113612yes—
636C22#1681113612yes—
637C22#124×D5311yescharacteristic
638C22#1681115212yes—
639C22#1681116012yes—
640C22C4⋊Q8×D5311yescharacteristic
641C22#1681117211yescharacteristic
642C22#1681117411yescharacteristic
643C22#1681117712yes—
644C22#1681117812yes—
645C4—28no—
646C4—44no—
647C4—82no—
648C4—1068no—
649C4—1064no—
650C4—4242no—
651C4—2124no—
652C4#1681049918yes—
653C2—1068no—
654C2C2×#1681049911yescharacteristic
655C2#391×D5311yescharacteristic
656C2#2027720411yescharacteristic
657C2C4×#1313319312yes—
658C2#2027725312yes—
659C2#2028299912yes—
660C2#2028301111yescharacteristic
661C2#2028301911yescharacteristic
662C2#2028305612yes—
663C2#2028306211yescharacteristic
664C2#2028306411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#391C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC2×#391D53
hypocenterC53#14702
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. C2×#16810530C2semidirect
5. #20272946C2semidirect
6. #20272945C2semidirect
7. #20272944C2semidirect
9. C2×C53×#391C2semidirect
14. C2×#13133193C4semidirect
54. C22×C4×D53Q8semidirect
61. C2×C4⋊Dic53C2×C4semidirect
64. C2×#11719919C2×C4semidirect
87. C22×Q8×C53C2×C4semidirect
250. C2×C4×C53C22⋊Q8semidirect
275. C2×Dic53C22⋊Q8semidirect
312. C2×D53#391semidirect
324. C2×#391D53semidirect
368. C53#14702semidirect
376. C22×Q8C4×D53semidirect
609. C2×C4#13133193semidirect

Maximal subgroups

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

subgroupindex
C22×#131331932
C22×C42×D532
C2×#168105302
C2×#168104802
#202729462
#202729452
#202729442
#202729362
C2×C53×#3912
#1470253