Finite Group Databasebeta

← #27306783

Subgroups — #27306783

Order 13568.

1862 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133173C211yesmaximal, characteristic
2C2×#16810488C211yesmaximal, characteristic, complemented
3C2×#16810468C211yesmaximal, characteristic, complemented
4#20272969C211yesmaximal, characteristic, complemented
5#20272965C211yesmaximal, characteristic, complemented
6#20272960C211yesmaximal, characteristic
7#20272952C212yesmaximal
8#20272946C212yesmaximal, complemented
9#20272940C212yesmaximal, complemented
10#20272937C212yesmaximal
11C53×#1439C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C2×#13133173—24no—
14C2×#13133173—24no—
15C2×#13133173C418yescomplemented
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2211yescharacteristic
19C22×C22⋊Dic53C2212yes—
20C22×C22⋊Dic53C2211yescharacteristic
21C22×C4×Dic53C2212yes—
22C22×C4×Dic53C2211yescharacteristic
23C22×C4×Dic53C2211yescharacteristic
24#16810488—24no—
25#16810468—24no—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2212yes—
37C22×C22⋊C4×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2211yescharacteristic
40C23×Dic53—22no—
41C23×Dic53—22no—
42C23×Dic53—22no—
43C23×Dic53—22no—
44C23×Dic53—22no—
45C23×Dic53—22no—
46C23×Dic53C2×C414yes—
47C23×Dic53C2312yes—
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2312yes—
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2311yescharacteristic
52C23×Dic53C2311yescharacteristic
53#13133173—232no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—42no—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919C2×C414yes—
61C2×#11719919C2×C414yes—
62C2×C22⋊Dic53—24no—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53—24no—
66C2×C22⋊Dic53—42no—
67C2×C22⋊Dic53C2×C414yes—
68C2×C4×Dic53—24no—
69C2×C4×Dic53—22no—
70C2×C4×Dic53—22no—
71C2×C4×Dic53—24no—
72C2×C4×Dic53—24no—
73C2×C4×Dic53C2×C414yes—
74C2×C4×Dic53C2×C414yes—
75C2×C4×Dic53D414yes—
76C2×C4×Dic53D414yes—
77#13133124—22no—
78#13133124—24no—
79#13133124—24no—
80#13133124—24no—
81#13133124—24no—
82#13133124—24no—
83#13133124—24no—
84#13133124—42no—
85#13133124—42no—
86C25×C53C2311yescharacteristic
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
89C23×C4×C53C2312yes—
90C23×C4×C53C2311yescharacteristic
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—42no—
93C2×C22⋊C4×C53—24no—
94C2×C22⋊C4×C53—22no—
95C2×C22⋊C4×C53—22no—
96C2×C22⋊C4×C53—24no—
97C2×C22⋊C4×C53—42no—
98C2×C22⋊C4×C53C2×C414yes—
99#95×C53—22no—
100#95×C53—24no—
101C24×C53—21no—
102C24×C53—21no—
103C24×C53—22no—
104C24×C53—22no—
105C24×C53—21no—
106C24×C53—21no—
107C24×C53—22no—
108C24×C53—22no—
109C24×C53C22×C412yes—
110C24×C53#4112yes—
111C24×C53#4112yes—
112C24×C53C2411yescharacteristic
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
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—22no—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C22×C412yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53#4114yes—
137C22×Dic53—28no—
138C22×Dic53—28no—
139C22×Dic53—28no—
140C22×Dic53—28no—
141C22×Dic53—28no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—28no—
153C22×Dic53—28no—
154C22×Dic53—22no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—22no—
171C22×Dic53C22×C412yes—
172C22×Dic53C22×C412yes—
173C22×Dic53C22×C412yes—
174C22×Dic53C22×C412yes—
175C22×Dic53C2×D412yes—
176C22×Dic53C2×D412yes—
177C22×Dic53C2×D412yes—
178C22×Dic53C2×D412yes—
179C22×Dic53#4114yes—
180C22×Dic53#4112yes—
181C22×Dic53#4112yes—
182C4⋊C4×C53—44no—
183C22⋊C4×C53—28no—
184C22⋊C4×C53—48no—
185C22⋊Dic53—28no—
186C22⋊Dic53—48no—
187C4⋊Dic53—44no—
188#11719919—28no—
189#11719919—28no—
190C4×Dic53—216no—
191C4×Dic53—28no—
192C4×Dic53C22⋊C4116yescomplemented
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—24no—
196C23×C53—24no—
197C23×C53—22no—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53—24no—
202C23×C53—24no—
203C23×C53—24no—
204C23×C53—24no—
205C23×C53—42no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—42no—
211C23×C53—42no—
212C23×C53—42no—
213C23×C53—42no—
214C23×C53—42no—
215C23×C53#11714yes—
216C23×C53C23×C411yescharacteristic
217C23×C53C22×D411yescharacteristic
218C23×C53C22×D411yescharacteristic
219C23×C53C2×#4111yescharacteristic
220C23×C53C2×#4111yescharacteristic
221C23×C53C2×#4112yes—
222C23×C53C2×#4111yescharacteristic
223C23×C53C2×#4112yes—
224C23×C53C2×#4111yescharacteristic
225C23×C53H(5,𝔽2)12yes—
226C23×C532−1+412yes—
227C2×C4×C53—24no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—24no—
231C2×C4×C53—22no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—42no—
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×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53C2×C22⋊C414yes—
251C2×Dic53—232no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—24no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—24no—
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—24no—
270C2×Dic53—24no—
271C2×Dic53—24no—
272C2×Dic53—24no—
273C2×Dic53—42no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53—42no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287C2×Dic53C2×C22⋊C414yes—
288C2×Dic53C2×C22⋊C414yes—
289C2×Dic53#11714yes—
290#13957—531nomaximal, Sylow
291C22×C53—28no—
292C22×C53—24no—
293C22×C53—42no—
294C22×C53—44no—
295C22×C53—44no—
296C22×C53—44no—
297C22×C53—42no—
298C22×C53—42no—
299C22×C53—42no—
300C22×C53—42no—
301C22×C53—42no—
302C22×C53C22×C22⋊C411yescharacteristic
303C22×C53C2×#11711yescharacteristic
304C22×C53C2×#11711yescharacteristic
305C22×C53C4×#4112yes—
306C22×C53#52012yes—
307C22×C53#52511yescharacteristic
308C22×C53#52512yes—
309C22×C53#52511yescharacteristic
310C22×C53#52511yescharacteristic
311C22×C53#52511yescharacteristic
312C22×C53#53211yescharacteristic
313C22×C53#53211yescharacteristic
314C22×C53#53511yescharacteristic
315C22×C53#53611yescharacteristic
316C22×C53#53712yes—
317C22×C53#55112yes—
318C22×C53#55112yes—
319C22×C53#55212yes—
320C22×C53#55212yes—
321C22×C53#55212yes—
322C22×C53#55212yes—
323C22×C53#55512yes—
324C22×C53#55512yes—
325C4×C53—24no—
326C4×C53—44no—
327C4×C53—42no—
328C4×C53—82no—
329Dic53—24no—
330Dic53—24no—
331Dic53—24no—
332Dic53—42no—
333Dic53—44no—
334Dic53—82no—
335C22×#117—531no—
336C2×#382—531no—
337C2×#376—531no—
338#1441—531no—
339#1441—531no—
340#1439—531no—
341#1439D5311yescharacteristic
342#1437—532no—
343#1436—532no—
344#1434—532no—
345C22⋊#95—532no—
346C2×C53—44no—
347C2×C53#228011yescharacteristic
348C2×C53#229711yescharacteristic
349C2×C53#231111yescharacteristic
350C2×C53#234312yes—
351C2×C53#234612yes—
352C2×C53#260912yes—
353C2×C53#260912yes—
354C2×C53#261011yescharacteristic
355C2×C53#261011yescharacteristic
356C2×C53#261111yescharacteristic
357C2×C53#261111yescharacteristic
358C24×C4—531no—
359C2×#117—538no—
360C2×#117—1064no—
361C2×#117—1064no—
362C22×C4⋊C4—531no—
363C22×C4⋊C4—531no—
364C22×C4⋊C4—531no—
365C22×C4⋊C4C2×D5311yescharacteristic
366C22×C22⋊C4—531no—
367C22×C22⋊C4—532no—
368C22×C22⋊C4C2×D5312yes—
369C22×C22⋊C4C2×D5311yescharacteristic
370C22×C42—531no—
371C22×C42—531no—
372C22×C42—532no—
373#382—1064no—
374#376—1064no—
375C2×#95—532no—
376C2×#95—532no—
377C2×#95—532no—
378C2×#95—532no—
379C2×#95—532no—
380C2×#95—532no—
381C2×#95—531no—
382C2×#95—532no—
383C2×#95—532no—
384C2×#95C2×D5312yes—
385C2×#95C2×D5311yescharacteristic
386C53#1395711yescharacteristic, complemented, Sylow
387C25C22×D5311yescharacteristic
388C23×C4—531no—
389C23×C4—531no—
390C23×C4—531no—
391C23×C4—534no—
392C23×C4—531no—
393C23×C4—532no—
394C23×C4—532no—
395C23×C4—1062no—
396C23×C4—1062no—
397C23×C4—1062no—
398C23×C4—1062no—
399C23×C4—1062no—
400C23×C4—1062no—
401C23×C4C22×D5312yes—
402C23×C4C22×D5311yescharacteristic
403C23×C4C22×D5312yes—
404C23×C4C22×D5311yescharacteristic
405#117—10632no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—42no—
408C2×C4⋊C4—534no—
409C2×C4⋊C4—534no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1062no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—2122no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—24no—
419C2×C22⋊C4—24no—
420C2×C22⋊C4—42no—
421C2×C22⋊C4—534no—
422C2×C22⋊C4—1062no—
423C2×C22⋊C4—1062no—
424C2×C22⋊C4—1064no—
425C2×C22⋊C4—1064no—
426C2×C22⋊C4—2122no—
427C2×C22⋊C4C4×D5314yes—
428C2×C42—534no—
429C2×C42—534no—
430C2×C42—534no—
431C2×C42—534no—
432C2×C42—1062no—
433C2×C42—1062no—
434C2×C42—1064no—
435C2×C42—1064no—
436C2×C42—1064no—
437#95—22no—
438#95—24no—
439#95—1064no—
440#95—1064no—
441#95—1064no—
442#95—1064no—
443#95—1064no—
444#95—1064no—
445#95—1062no—
446#95—2122no—
447#95—2122no—
448C24—21no—
449C24—21no—
450C24—22no—
451C24—21no—
452C24—21no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24C2×C4×D5312yes—
457C24#1171994712yes—
458C24#1171994912yes—
459C24C23×D5311yescharacteristic
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—532no—
480C22×C4—532no—
481C22×C4—532no—
482C22×C4—532no—
483C22×C4—532no—
484C22×C4—532no—
485C22×C4—532no—
486C22×C4—532no—
487C22×C4—532no—
488C22×C4—532no—
489C22×C4—534no—
490C22×C4—1068no—
491C22×C4—1068no—
492C22×C4—1068no—
493C22×C4—1068no—
494C22×C4—1068no—
495C22×C4—1068no—
496C22×C4—1068no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1062no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1062no—
503C22×C4—1061no—
504C22×C4—1061no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1062no—
508C22×C4—1061no—
509C22×C4—1061no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4—1062no—
515C22×C4—1062no—
516C22×C4—1062no—
517C22×C4—1062no—
518C22×C4—1062no—
519C22×C4—1062no—
520C22×C4—1062no—
521C22×C4—1062no—
522C22×C4—1062no—
523C22×C4—1062no—
524C22×C4C2×C4×D5312yes—
525C22×C4C2×C4×D5312yes—
526C22×C4D4×D5312yes—
527C22×C4D4×D5312yes—
528C22×C4#1171994714yes—
529C4⋊C4—44no—
530C4⋊C4—1068no—
531C4⋊C4—1068no—
532C4⋊C4—2124no—
533C22⋊C4—28no—
534C22⋊C4—48no—
535C22⋊C4—1068no—
536C22⋊C4—2128no—
537C42—5316no—
538C42—1068no—
539C42—10616no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—24no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23#1313317514yes—
563C23C22×C4×D5311yescharacteristic
564C23C2×D4×D5311yescharacteristic
565C23C2×D4×D5311yescharacteristic
566C23#41×D5312yes—
567C23#41×D5311yescharacteristic
568C23#41×D5311yescharacteristic
569C23C2×#1171994711yescharacteristic
570C23C2×#1171994712yes—
571C23C2×#1171994911yescharacteristic
572C23#1313326212yes—
573C23#1313326412yes—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—534no—
598C2×C4—534no—
599C2×C4—534no—
600C2×C4—10632no—
601C2×C4—1062no—
602C2×C4—1062no—
603C2×C4—1062no—
604C2×C4—1062no—
605C2×C4—1064no—
606C2×C4—1064no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—1062no—
615C2×C4—1062no—
616C2×C4—1062no—
617C2×C4—1064no—
618C2×C4—1064no—
619C2×C4—1062no—
620C2×C4—1064no—
621C2×C4—1064no—
622C2×C4—2122no—
623C2×C4—2122no—
624C2×C4—2122no—
625C2×C4—2122no—
626C2×C4—2122no—
627C2×C4—2122no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4—2122no—
635C2×C4—2122no—
636C2×C4C22⋊C4×D5314yes—
637C22—28no—
638C22—24no—
639C22—42no—
640C22—44no—
641C22—44no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22C2×C22⋊C4×D5311yescharacteristic
649C22#117×D5311yescharacteristic
650C22C2×#1313317511yescharacteristic
651C22C4×#1171994712yes—
652C22#1681110212yes—
653C22#1681113311yescharacteristic
654C22#1681113312yes—
655C22#1681113311yescharacteristic
656C22#1681113611yescharacteristic
657C22#1681113611yescharacteristic
658C22#1681117111yescharacteristic
659C22#1681117211yescharacteristic
660C22#1681118811yescharacteristic
661C22#1681119511yescharacteristic
662C22#1681120012yes—
663C22#1681130012yes—
664C22#1681130412yes—
665C22#1681130912yes—
666C22#1681131012yes—
667C22#1681131012yes—
668C22#1681131512yes—
669C22#1681133512yes—
670C22#1681133712yes—
671C4—24no—
672C4—44no—
673C4—42no—
674C4—82no—
675C4—1064no—
676C4—1064no—
677C4—1064no—
678C4—4242no—
679C4—2122no—
680C4—2124no—
681C2—44no—
682C2#2027709711yescharacteristic
683C2#2027720211yescharacteristic
684C2#2027730211yescharacteristic
685C2#2027757312yes—
686C2#2027759712yes—
687C2#2028038612yes—
688C2#2028038812yes—
689C2#2028039511yescharacteristic
690C2#2028040311yescharacteristic
691C2#2028040611yescharacteristic
692C2#2028041411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1439C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1439D53
hypocenterC53#13957
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#16810488C2semidirect
3. C2×#16810468C2semidirect
4. #20272969C2semidirect
5. #20272965C2semidirect
8. #20272946C2semidirect
9. #20272940C2semidirect
15. C2×#13133173C4semidirect
192. C4×Dic53C22⋊C4semidirect
386. C53#13957semidirect

Maximal subgroups

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

subgroupindex
C22×#131331732
C2×#168104882
C2×#168104682
#202729692
#202729652
#202729602
#202729522
#202729462
#202729402
#202729372
C53×#14392
#1395753