Finite Group Databasebeta

← #27315127

Subgroups — #27315127

Order 13568.

1702 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×#13133195C211yesmaximal, characteristic
2C2×#16810485C211yesmaximal, characteristic
3C2×#16810480C211yesmaximal, characteristic
4#20272985C212yesmaximal, complemented
5#20272981C212yesmaximal, complemented
6#20272969C211yesmaximal, characteristic, complemented
7#20272968C211yesmaximal, characteristic, complemented
8#20272965C211yesmaximal, characteristic, complemented
9#20272963C212yesmaximal
10#20272963C212yesmaximal
11C53×#1441C211yesmaximal, characteristic, complemented
12C23×Dic106C2211yescharacteristic
13C23×C4×D53C2211yescharacteristic
14C2×#13133195—24no—
15C2×#13133195—24no—
16C2×#13133195C418yescomplemented
17C22×C4⋊Dic53C2211yescharacteristic
18C22×#11719919C2211yescharacteristic
19C22×#11719919C2211yescharacteristic
20C22×#11719921C2211yescharacteristic
21C22×#11719921C2211yescharacteristic
22C22×#11719921C2212yes—
23C22×#11719921C2212yes—
24#16810485—24no—
25#16810480—24no—
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2212yes—
40C24×D53C2311yescharacteristic
41C22×Dic106—22no—
42C22×Dic106—22no—
43C22×Dic106—42no—
44C22×Dic106—42no—
45C22×Dic106C2×C414yes—
46C23×Dic53C2312yes—
47C23×Dic53C2312yes—
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2311yescharacteristic
51C22×C4×D53—42no—
52C22×C4×D53—42no—
53C22×C4×D53C2×C414yes—
54C22×C4×D53D414yes—
55C22×C4×D53D414yes—
56#13133195—232no—
57C2×C4⋊Dic53—22no—
58C2×C4⋊Dic53—42no—
59C2×#11719919—22no—
60C2×#11719919—22no—
61C2×#11719919—22no—
62C2×#11719919—22no—
63C2×#11719919C2×C414yes—
64C2×#11719919C2×C414yes—
65C2×#11719921—22no—
66C2×#11719921—22no—
67C2×#11719921—22no—
68C2×#11719921—22no—
69C2×#11719921—24no—
70C2×#11719921—24no—
71C2×#11719921—42no—
72C2×#11719921—42no—
73C2×#11719921—42no—
74C2×#11719921—42no—
75C2×#11719921C2×C414yes—
76C2×#11719921C2×C414yes—
77#13133124—22no—
78#13133124—24no—
79#13133124—24no—
80#13133124—42no—
81#13133124—42no—
82#13133124—42no—
83#13133124—42no—
84#13133124—42no—
85#13133124—42no—
86C23×C4×C53C2312yes—
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2311yescharacteristic
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53—42no—
95C2×C4⋊C4×C53C2×C414yes—
96#95×C53—22no—
97#95×C53—42no—
98#95×C53—42no—
99C24×C53C2411yescharacteristic
100C23×D53—21no—
101C23×D53—21no—
102C23×D53—22no—
103C23×D53—22no—
104C23×D53—22no—
105C23×D53—22no—
106C23×D53—21no—
107C23×D53—21no—
108C23×D53C22×C412yes—
109C23×D53C2×D412yes—
110C23×D53C2×D412yes—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
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×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53C22×C412yes—
138C22×C4×C53C22×C412yes—
139C22×C4×C53C22×C412yes—
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2×D412yes—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
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×Dic53C22×C412yes—
169C22×Dic53C22×C412yes—
170C22×Dic53C22×C412yes—
171C22×Dic53C2×D412yes—
172C22×Dic53C2×D412yes—
173C2×C4×D53—48no—
174C2×C4×D53—48no—
175C2×C4×D53—44no—
176C2×C4×D53—44no—
177C2×C4×D53C22⋊C4116yes—
178C2×Dic106—28no—
179C2×Dic106—48no—
180C2×Dic106—48no—
181C2×Dic106—44no—
182C2×Dic106—44no—
183C4⋊C4×C53—28no—
184C4⋊C4×C53—44no—
185#11719921—28no—
186#11719921—28no—
187#11719921—48no—
188#11719921—48no—
189C4⋊Dic53—44no—
190#11719919—28no—
191#11719919—28no—
192C23×C53C23×C411yescharacteristic
193C23×C53C22×D411yescharacteristic
194C23×C53C22×D411yescharacteristic
195C23×C53H(5,𝔽2)12yes—
196C23×C53H(5,𝔽2)11yescharacteristic
197C23×C53H(5,𝔽2)12yes—
198C23×C53H(5,𝔽2)11yescharacteristic
199C23×C532−1+412yes—
200C23×C532−1+411yescharacteristic
201C23×C532−1+412yes—
202C23×C532−1+411yescharacteristic
203C22×D53—24no—
204C22×D53—24no—
205C22×D53—22no—
206C22×D53—22no—
207C22×D53—22no—
208C22×D53—22no—
209C22×D53—24no—
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—24no—
213C22×D53—22no—
214C22×D53—22no—
215C22×D53—22no—
216C22×D53—22no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—42no—
222C22×D53—42no—
223C22×D53—42no—
224C22×D53—42no—
225C22×D53C2×C22⋊C414yes—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—24no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—42no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
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×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53C2×C22⋊C414yes—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—24no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—42no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
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—
288C4×D53—416no—
289Dic106—416no—
290#14660—531nomaximal, Sylow
291C22×C53C22×C22⋊C411yescharacteristic
292C22×C53#51812yes—
293C22×C53#51912yes—
294C22×C53#52011yescharacteristic
295C22×C53#52011yescharacteristic
296C22×C53#53412yes—
297C22×C53#53411yescharacteristic
298C22×C53#53411yescharacteristic
299C22×C53#53511yescharacteristic
300C22×C53#53611yescharacteristic
301C22×C53#53611yescharacteristic
302C22×C53#53612yes—
303C22×C53#53611yescharacteristic
304C22×C53#53711yescharacteristic
305C22×C53#53711yescharacteristic
306C22×C53#56212yes—
307C22×C53#56212yes—
308C22×C53#56212yes—
309C22×C53#56212yes—
310C22×C53#56312yes—
311C22×C53#56312yes—
312C22×C53#56312yes—
313C22×C53#56312yes—
314C2×D53—24no—
315C2×D53—24no—
316C2×D53—24no—
317C2×D53—42no—
318C2×D53—42no—
319C2×D53—42no—
320C2×D53—42no—
321C2×D53—44no—
322C2×D53—44no—
323C2×D53—44no—
324C2×D53—44no—
325C4×C53—24no—
326C4×C53—42no—
327C4×C53—42no—
328C4×C53—82no—
329C4×C53—82no—
330Dic53—24no—
331Dic53—42no—
332Dic53—42no—
333Dic53—82no—
334Dic53—82no—
335C22×C22⋊Q8—531no—
336C2×#381—531no—
337C2×#380—531no—
338#1446—532no—
339#1445—532no—
340#1441—531no—
341#1441—531no—
342#1441—531no—
343#1441D5311yescharacteristic, complemented
344#1440—532no—
345#1440—532no—
346C2×C53#230811yescharacteristic
347C2×C53#231011yescharacteristic
348C2×C53#231111yescharacteristic
349C2×C53#237812yes—
350C2×C53#237812yes—
351C2×C53#281412yes—
352C2×C53#281412yes—
353C2×C53#281611yescharacteristic
354C2×C53#281611yescharacteristic
355C2×C53#281711yescharacteristic
356C2×C53#281711yescharacteristic
357D53—44no—
358C23×Q8—531no—
359C24×C4—531no—
360C2×C22⋊Q8—538no—
361C2×C22⋊Q8—1064no—
362C2×C22⋊Q8—1064no—
363C22×C4⋊C4—531no—
364C22×C4⋊C4—531no—
365C22×C4⋊C4—531no—
366C22×C4⋊C4C2×D5311yescharacteristic
367C22×C4⋊C4C2×D5311yescharacteristic
368C22×C22⋊C4—531no—
369C22×C22⋊C4—531no—
370C22×C22⋊C4—532no—
371C22×C22⋊C4—532no—
372#381—1064no—
373#380—1064no—
374C2×#95—531no—
375C2×#95—532no—
376C2×#95—532no—
377C2×#95—532no—
378C2×#95—532no—
379C2×#95—532no—
380C2×#95—532no—
381C2×#95—532no—
382C2×#95—532no—
383C2×#95C2×D5311yescharacteristic
384C2×#95C2×D5312yes—
385C2×#95C2×D5312yes—
386C53#1466011yescharacteristic, complemented, Sylow
387C25—531no—
388C22×Q8—534no—
389C22×Q8—1062no—
390C22×Q8—1062no—
391C22×Q8—2122no—
392C22×Q8—2122no—
393C23×C4—531no—
394C23×C4—534no—
395C23×C4—534no—
396C23×C4—534no—
397C23×C4—531no—
398C23×C4—531no—
399C23×C4—532no—
400C23×C4—532no—
401C23×C4—2122no—
402C23×C4—2122no—
403C23×C4C22×D5312yes—
404C23×C4C22×D5312yes—
405C23×C4C22×D5311yescharacteristic
406C23×C4C22×D5311yescharacteristic
407C23×C4C22×D5311yescharacteristic
408C22⋊Q8—10632no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—42no—
413C2×C4⋊C4—534no—
414C2×C4⋊C4—534no—
415C2×C4⋊C4—1062no—
416C2×C4⋊C4—1062no—
417C2×C4⋊C4—1062no—
418C2×C4⋊C4—1062no—
419C2×C4⋊C4—1062no—
420C2×C4⋊C4—2122no—
421C2×C4⋊C4C4×D5314yes—
422C2×C22⋊C4—534no—
423C2×C22⋊C4—534no—
424C2×C22⋊C4—1062no—
425C2×C22⋊C4—1062no—
426C2×C22⋊C4—1062no—
427C2×C22⋊C4—1062no—
428C2×C22⋊C4—1064no—
429C2×C22⋊C4—1064no—
430C2×C22⋊C4—2122no—
431C2×C22⋊C4—2122no—
432C2×C22⋊C4—2122no—
433C2×C22⋊C4—2122no—
434#95—22no—
435#95—42no—
436#95—42no—
437#95—1064no—
438#95—1064no—
439#95—1062no—
440#95—2122no—
441#95—2122no—
442#95—2122no—
443#95—2122no—
444#95—2122no—
445#95—2122no—
446C24—532no—
447C24—532no—
448C24—532no—
449C24—1061no—
450C24—1062no—
451C24—1061no—
452C24—1062no—
453C24—1062no—
454C24—1061no—
455C24—1062no—
456C24—1061no—
457C24C23×D5311yescharacteristic
458C2×Q8—1068no—
459C2×Q8—2128no—
460C2×Q8—2128no—
461C2×Q8—2124no—
462C2×Q8—2124no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—532no—
490C22×C4—532no—
491C22×C4—532no—
492C22×C4—5316no—
493C22×C4—532no—
494C22×C4—532no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1062no—
503C22×C4—1061no—
504C22×C4—1062no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1062no—
508C22×C4—1062no—
509C22×C4—1062no—
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—2128no—
522C22×C4—2128no—
523C22×C4—2124no—
524C22×C4—2124no—
525C22×C4C2×C4×D5312yes—
526C22×C4C2×C4×D5312yes—
527C22×C4C2×C4×D5312yes—
528C22×C4D4×D5312yes—
529C22×C4D4×D5312yes—
530C4⋊C4—28no—
531C4⋊C4—44no—
532C4⋊C4—1068no—
533C4⋊C4—1068no—
534C4⋊C4—2124no—
535C22⋊C4—1068no—
536C22⋊C4—1068no—
537C22⋊C4—2128no—
538C22⋊C4—2128no—
539C23—534no—
540C23—1064no—
541C23—1064no—
542C23—1062no—
543C23—1062no—
544C23—1064no—
545C23—1062no—
546C23—1062no—
547C23—1062no—
548C23—1062no—
549C23—1064no—
550C23—1062no—
551C23—1062no—
552C23—1062no—
553C23—1062no—
554C23—2122no—
555C23—2122no—
556C23—2122no—
557C23—2122no—
558C23—2122no—
559C23—2122no—
560C23—2122no—
561C23—2122no—
562C23C22×C4×D5311yescharacteristic
563C23C2×D4×D5311yescharacteristic
564C23C2×D4×D5311yescharacteristic
565C23#1313326212yes—
566C23#1313326211yescharacteristic
567C23#1313326312yes—
568C23#1313326311yescharacteristic
569C23#1313326412yes—
570C23#1313326411yescharacteristic
571C23#1313326512yes—
572C23#1313326511yescharacteristic
573Q8—21216no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—534no—
605C2×C4—1064no—
606C2×C4—1064no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—1062no—
611C2×C4—1064no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—21216no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4—2122no—
621C2×C4—2122no—
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—2121no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4—2121no—
635C2×C4—2121no—
636C2×C4C22⋊C4×D5314yes—
637C22—1064no—
638C22—1064no—
639C22—1064no—
640C22—2124no—
641C22—2124no—
642C22—2124no—
643C22—2124no—
644C22—2122no—
645C22—2122no—
646C22—2122no—
647C22—2122no—
648C22C2×C22⋊C4×D5311yescharacteristic
649C22#1681109512yes—
650C22#1681109912yes—
651C22#1681110311yescharacteristic
652C22#1681110411yescharacteristic
653C22#1681118112yes—
654C22#1681118311yescharacteristic
655C22#1681118411yescharacteristic
656C22#1681118911yescharacteristic
657C22#1681119311yescharacteristic
658C22#1681119612yes—
659C22#1681119611yescharacteristic
660C22#1681119811yescharacteristic
661C22#1681120211yescharacteristic
662C22#1681120311yescharacteristic
663C22#1681136812yes—
664C22#1681136912yes—
665C22#1681137012yes—
666C22#1681137112yes—
667C22#1681137212yes—
668C22#1681137412yes—
669C22#1681137512yes—
670C22#1681137712yes—
671C4—24no—
672C4—42no—
673C4—42no—
674C4—82no—
675C4—82no—
676C4—1064no—
677C4—4242no—
678C4—4242no—
679C4—2122no—
680C4—2122no—
681C2—2124no—
682C2#2027728911yescharacteristic
683C2#2027729811yescharacteristic
684C2#2027730411yescharacteristic
685C2#2027787712yes—
686C2#2027788012yes—
687C2#2028273312yes—
688C2#2028273512yes—
689C2#2028274611yescharacteristic
690C2#2028274811yescharacteristic
691C2#2028274911yescharacteristic
692C2#2028275111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1441C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1441D53
hypocenterC53#14660
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
4. #20272985C2semidirect
5. #20272981C2semidirect
6. #20272969C2semidirect
7. #20272968C2semidirect
8. #20272965C2semidirect
11. C53×#1441C2semidirect
16. C2×#13133195C4semidirect
343. #1441D53semidirect
386. C53#14660semidirect

Maximal subgroups

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

subgroupindex
C22×#131331952
C2×#168104852
C2×#168104802
#202729852
#202729812
#202729692
#202729682
#202729652
#202729632
C53×#14412
#1466053