Finite Group Databasebeta

← #27315446

Subgroups — #27315446

Order 13568.

1770 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133210C211yesmaximal, characteristic
2C2×#16810490C211yesmaximal, characteristic, complemented
3C2×C42×Dic53C211yesmaximal, characteristic, complemented
4#20272952C212yesmaximal
5#20272952C212yesmaximal
6#20272952C211yesmaximal, characteristic
7#20272950C211yesmaximal, characteristic, complemented
8#20272949C212yesmaximal, complemented
9#20272948C211yesmaximal, characteristic, complemented
10#20272946C212yesmaximal, complemented
11C53×#1437C211yesmaximal, characteristic
12C2×#13133210—24no—
13C2×#13133210—24no—
14C2×#13133210C418yescomplemented
15C22×C4⋊Dic53C2211yescharacteristic
16C22×#11719919C2211yescharacteristic
17C22×#11719919C2211yescharacteristic
18C22×C22⋊Dic53C2212yes—
19C22×C22⋊Dic53C2211yescharacteristic
20C22×C22⋊Dic53C2211yescharacteristic
21C22×C4×Dic53C2212yes—
22C22×C4×Dic53C2212yes—
23C22×C4×Dic53C2211yescharacteristic
24C22×C4×Dic53C2211yescharacteristic
25#16810490—24no—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C42×Dic53—24no—
35C22×C22⋊C4×C53C2212yes—
36C22×C22⋊C4×C53C2211yescharacteristic
37C22×C42×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2211yescharacteristic
40C23×Dic53C2312yes—
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2312yes—
46#13133210—232no—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53C2×C414yes—
50C2×#11719919—22no—
51C2×#11719919—22no—
52C2×#11719919—22no—
53C2×#11719919—22no—
54C2×#11719919C2×C414yes—
55C2×#11719919C2×C414yes—
56C2×C22⋊Dic53—22no—
57C2×C22⋊Dic53—22no—
58C2×C22⋊Dic53—22no—
59C2×C22⋊Dic53—22no—
60C2×C22⋊Dic53—24no—
61C2×C22⋊Dic53—24no—
62C2×C22⋊Dic53—24no—
63C2×C22⋊Dic53C2×C414yes—
64C2×C22⋊Dic53C2×C414yes—
65C2×C4×Dic53—22no—
66C2×C4×Dic53—24no—
67C2×C4×Dic53—24no—
68C2×C4×Dic53—24no—
69C2×C4×Dic53—24no—
70C2×C4×Dic53—24no—
71C2×C4×Dic53—24no—
72C2×C4×Dic53—24no—
73C2×C4×Dic53C2×C414yes—
74C2×C4×Dic53D414yes—
75C2×C4×Dic53D414yes—
76#13133124—22no—
77#13133124—24no—
78#13133124—24no—
79#13133124—24no—
80#13133124—24no—
81#13133124—24no—
82#13133124—24no—
83#13133124—24no—
84C25×C53C2311yescharacteristic
85C23×C4×C53C2312yes—
86C23×C4×C53C2311yescharacteristic
87C23×C4×C53C2311yescharacteristic
88C23×C4×C53C2312yes—
89C2×C22⋊C4×C53—22no—
90C2×C22⋊C4×C53—22no—
91C2×C22⋊C4×C53—24no—
92C2×C22⋊C4×C53—24no—
93C2×C22⋊C4×C53—24no—
94C2×C22⋊C4×C53C2×C414yes—
95C2×C42×C53—22no—
96C2×C42×C53—24no—
97#95×C53—22no—
98#95×C53—24no—
99C24×C53—21no—
100C24×C53—21no—
101C24×C53—22no—
102C24×C53—22no—
103C24×C53—22no—
104C24×C53—22no—
105C24×C53—22no—
106C24×C53—22no—
107C24×C53C22×C412yes—
108C24×C53C2411yescharacteristic
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
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×C53C22×C412yes—
127C22×C4×C53C22×C412yes—
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53#4114yes—
131C22×C4×C53#4114yes—
132C22×C4×C53#4114yes—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53C22×C412yes—
157C22×Dic53C22×C412yes—
158C22×Dic53C22×C412yes—
159C22×Dic53C22×C412yes—
160C22×Dic53C2×D412yes—
161C22×Dic53C2×D412yes—
162C22×Dic53C2×D412yes—
163C22×Dic53C2×D412yes—
164C22×Dic53#4114yes—
165C22×Dic53#4114yes—
166C22×Dic53#4114yes—
167C22⋊C4×C53—28no—
168C22⋊C4×C53—48no—
169C42×C53—28no—
170C22⋊Dic53—28no—
171C22⋊Dic53—28no—
172C22⋊Dic53—48no—
173C4⋊Dic53—28no—
174#11719919—28no—
175#11719919—28no—
176C4×Dic53—216no—
177C4×Dic53—216no—
178C4×Dic53—28no—
179C4×Dic53C22⋊C4116yescomplemented
180C23×C53—22no—
181C23×C53—24no—
182C23×C53—24no—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53C23×C411yescharacteristic
202C23×C53C22×D411yescharacteristic
203C23×C53C22×D411yescharacteristic
204C23×C53C2×#4112yes—
205C23×C53C2×#4112yes—
206C23×C53C2×#4112yes—
207C23×C53C2×#4112yes—
208C23×C53C2×#4112yes—
209C23×C53C2×#4112yes—
210C2×C4×C53—24no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—22no—
221C2×C4×C53—24no—
222C2×C4×C53—42no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53C2×C22⋊C414yes—
233C2×C4×C53#12618yescomplemented
234C2×Dic53—24no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—24no—
238C2×Dic53—24no—
239C2×Dic53—24no—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—24no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53C2×C22⋊C414yes—
266C2×Dic53C2×C22⋊C414yes—
267C2×Dic53#12618yescomplemented
268#14711—531nomaximal, Sylow
269C22×C53—44no—
270C22×C53—44no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—42no—
274C22×C53—82no—
275C22×C53—82no—
276C22×C53—82no—
277C22×C53—82no—
278C22×C53C22×C22⋊C411yescharacteristic
279C22×C53C4×#4112yes—
280C22×C53C4×#4112yes—
281C22×C53C4×#4111yescharacteristic
282C22×C53C4×#4111yescharacteristic
283C22×C53#52511yescharacteristic
284C22×C53#52511yescharacteristic
285C22×C53#52512yes—
286C22×C53#52512yes—
287C22×C53C2×#12612yes—
288C22×C53C2×#12612yes—
289C22×C53#52912yes—
290C22×C53#52912yes—
291C22×C53#52912yes—
292C22×C53#52912yes—
293C22×C53#52912yes—
294C22×C53#52912yes—
295C22×C53#53212yes—
296C22×C53#53212yes—
297C22×C53#53211yescharacteristic
298C22×C53#53211yescharacteristic
299C4×C53—28no—
300C4×C53—24no—
301C4×C53—42no—
302C4×C53—82no—
303Dic53—28no—
304Dic53—24no—
305Dic53—24no—
306Dic53—42no—
307Dic53—42no—
308Dic53—82no—
309C22×#126—531no—
310C2×#383—531no—
311C2×C43—531no—
312#1437—532no—
313#1437—532no—
314#1437—531no—
315#1437D5311yescharacteristic
316#1436—531no—
317#1436—532no—
318#1436—532no—
319#1436—531no—
320C2×C53—82no—
321C2×C53#229711yescharacteristic
322C2×C53#229711yescharacteristic
323C2×C53#229711yescharacteristic
324C2×C53C4×#12612yes—
325C2×C53C4×#12612yes—
326C2×C53#285212yes—
327C2×C53#285212yes—
328C2×C53#286012yes—
329C2×C53#286012yes—
330C2×#126—538no—
331C2×#126—1064no—
332C2×#126—1064no—
333C22×C4⋊C4—531no—
334C22×C4⋊C4—531no—
335C22×C4⋊C4—531no—
336C22×C22⋊C4—531no—
337C22×C22⋊C4—531no—
338C22×C22⋊C4—532no—
339C22×C22⋊C4C2×D5312yes—
340C22×C22⋊C4C2×D5311yescharacteristic
341C22×C42—531no—
342C22×C42—532no—
343C22×C42—532no—
344C22×C42—531no—
345C22×C42C2×D5311yescharacteristic
346#383—1064no—
347C2×#95—532no—
348C2×#95—532no—
349C2×#95—532no—
350C2×#95—532no—
351C2×#95—532no—
352C2×#95—532no—
353C2×#95—532no—
354C2×#95—531no—
355C2×#95C2×D5312yes—
356C2×#95C2×D5311yescharacteristic
357C43—1064no—
358C53#1471111yescharacteristic, complemented, Sylow
359C25C22×D5311yescharacteristic
360C23×C4—531no—
361C23×C4—531no—
362C23×C4—532no—
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—532no—
366C23×C4C22×D5312yes—
367C23×C4C22×D5311yescharacteristic
368C23×C4C22×D5311yescharacteristic
369C23×C4C22×D5312yes—
370#126—10632no—
371C2×C4⋊C4—534no—
372C2×C4⋊C4—534no—
373C2×C4⋊C4—534no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4—1062no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4—1062no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—534no—
386C2×C22⋊C4—534no—
387C2×C22⋊C4—1062no—
388C2×C22⋊C4—1062no—
389C2×C22⋊C4—1062no—
390C2×C22⋊C4—1062no—
391C2×C22⋊C4—1064no—
392C2×C22⋊C4—1064no—
393C2×C22⋊C4—1064no—
394C2×C22⋊C4C4×D5314yes—
395C2×C42—22no—
396C2×C42—24no—
397C2×C42—534no—
398C2×C42—534no—
399C2×C42—534no—
400C2×C42—1064no—
401C2×C42—1064no—
402C2×C42—1064no—
403C2×C42—1064no—
404C2×C42—1064no—
405C2×C42—1064no—
406C2×C42—1062no—
407C2×C42—1064no—
408#95—22no—
409#95—24no—
410#95—1064no—
411#95—1064no—
412#95—1064no—
413#95—1064no—
414#95—1064no—
415#95—1064no—
416#95—1062no—
417#95—1064no—
418C24—21no—
419C24—21no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24C2×C4×D5312yes—
427C24C23×D5311yescharacteristic
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—532no—
446C22×C4—532no—
447C22×C4—532no—
448C22×C4—532no—
449C22×C4—532no—
450C22×C4—532no—
451C22×C4—534no—
452C22×C4—534no—
453C22×C4—534no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4C2×C4×D5312yes—
480C22×C4C2×C4×D5312yes—
481C22×C4#1171994514yes—
482C22×C4D4×D5312yes—
483C22×C4D4×D5312yes—
484C22×C4#1171994714yes—
485C22×C4#1171994714yes—
486C4⋊C4—1068no—
487C4⋊C4—1068no—
488C4⋊C4—1068no—
489C22⋊C4—28no—
490C22⋊C4—48no—
491C22⋊C4—1068no—
492C22⋊C4—1068no—
493C22⋊C4—2128no—
494C42—28no—
495C42—5316no—
496C42—1068no—
497C42—10616no—
498C42—10616no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23C22×C4×D5311yescharacteristic
521C23C2×D4×D5311yescharacteristic
522C23C2×D4×D5311yescharacteristic
523C23#41×D5312yes—
524C23#41×D5312yes—
525C23#41×D5312yes—
526C23C2×#1171994712yes—
527C23C2×#1171994712yes—
528C23C2×#1171994512yes—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—534no—
552C2×C4—534no—
553C2×C4—538no—
554C2×C4—1064no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1064no—
558C2×C4—1064no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—1064no—
562C2×C4—1064no—
563C2×C4—1064no—
564C2×C4—1064no—
565C2×C4—1064no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1064no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4C22⋊C4×D5314yes—
586C2×C4#1313321018yescomplemented
587C22—44no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22C2×C22⋊C4×D5311yescharacteristic
597C22C4×#1171994712yes—
598C22C4×#1171994712yes—
599C22#1681109111yescharacteristic
600C22C4×#1171994511yescharacteristic
601C22#1681112911yescharacteristic
602C22#1681113111yescharacteristic
603C22#1681113212yes—
604C22#1681113312yes—
605C22#126×D5312yes—
606C22C2×#1313321012yes—
607C22#1681115012yes—
608C22#1681115112yes—
609C22#1681115212yes—
610C22#1681115312yes—
611C22#1681115412yes—
612C22#1681115512yes—
613C22#1681116812yes—
614C22#1681117012yes—
615C22#1681117111yescharacteristic
616C22#1681117211yescharacteristic
617C4—28no—
618C4—24no—
619C4—42no—
620C4—82no—
621C4—1068no—
622C4—1064no—
623C4—1064no—
624C4—4242no—
625C4—2122no—
626C4—2122no—
627C2—82no—
628C2#2027720011yescharacteristic
629C2#2027720111yescharacteristic
630C2#2027720211yescharacteristic
631C2C4×#1313321012yes—
632C2#2027726912yes—
633C2#2028302812yes—
634C2#2028302912yes—
635C2#2028307912yes—
636C2#2028308112yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1437C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1437D53
hypocenterC53#14711
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×#16810490C2semidirect
3. C2×C42×Dic53C2semidirect
7. #20272950C2semidirect
8. #20272949C2semidirect
9. #20272948C2semidirect
10. #20272946C2semidirect
14. C2×#13133210C4semidirect
179. C4×Dic53C22⋊C4semidirect
233. C2×C4×C53#126semidirect
267. C2×Dic53#126semidirect
358. C53#14711semidirect
586. C2×C4#13133210semidirect

Maximal subgroups

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

subgroupindex
C22×#131332102
C2×#168104902
C2×C42×Dic532
#202729522
#202729502
#202729492
#202729482
#202729462
C53×#14372
#1471153