Finite Group Databasebeta

← #27384815

Subgroups — #27384815

Order 13568.

1590 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D53C211yesmaximal, characteristic
2C2×#16810659C211yesmaximal, characteristic, complemented
3C22×#13133129C211yesmaximal, characteristic
4#20277094C211yesmaximal, characteristic
5#20274913C212yesmaximal, complemented
6#20274911C212yesmaximal, complemented
7#20274906C212yesmaximal, complemented
8#20274901C212yesmaximal, complemented
9#20274432C211yesmaximal, characteristic, complemented
10#20274431C211yesmaximal, characteristic, complemented
11C2×C53×#427C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C23×#88322C2211yescharacteristic
14C2×#13133176C2211yescharacteristic
15C2×#13133176C2211yescharacteristic
16C2×#13133175C2212yes—
17C2×#13133174—21no—
18C2×#13133173—21no—
19#117×D53—24no—
20#117×D53—24no—
21C2×C4⋊C4×D53C2212yes—
22C2×C22⋊C4×D53—21no—
23C2×C42×D53—21no—
24#16810659—24no—
25C2×#13133150C2212yescomplemented
26C2×#13133148C2212yescomplemented
27C2×#13133129C418yescomplemented
28C2×#13133129C2211yescharacteristic
29C2×#13133129C2211yescharacteristic
30C2×#13133129C2211yescharacteristic
31C2×#13133129C2211yescharacteristic
32#16810484—21no—
33#16810480C2212yes—
34#16810480C2212yes—
35#16810478—21no—
36C4×#11719921—21no—
37C4×#11719921—21no—
38#16810386C2212yescomplemented
39#16810386C2212yescomplemented
40#16810385C2212yescomplemented
41#16810385C2212yescomplemented
42C23×C8×C53C2211yescharacteristic
43C2×#117×C53C2211yescharacteristic
44C2×#117×C53C2211yescharacteristic
45C53×#427—24no—
46C2×#107×C53C2212yescomplemented
47C2×#106×C53C2212yescomplemented
48C24×D53D411yescharacteristic
49C23×Dic53D411yescharacteristic
50C22×C4×D53—22no—
51C22×C4×D53—22no—
52C22×C4×D53—22no—
53C22×C4×D53—22no—
54C22×C4×D53—22no—
55C22×C4×D53C2×C414yes—
56C22×C4×D53D411yescharacteristic
57C22×C4×D53D411yescharacteristic
58C22×C4×D53Q814yes—
59C22×C4×D53C2311yescharacteristic
60C22×C4×D53C2311yescharacteristic
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322C2×C414yescomplemented
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133176—22no—
67#13133176—22no—
68#13133176—22no—
69#13133176D414yes—
70#13133175—24no—
71#13133175—24no—
72#13133174—28no—
73#13133173—28no—
74C2×C4⋊Dic53C2312yes—
75C2×C4⋊Dic53C2312yes—
76C2×#11719919—22no—
77C4⋊C4×D53—24no—
78C4⋊C4×D53—24no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×#11719921—22no—
82C2×#11719921—22no—
83C2×#11719921—22no—
84C22⋊C4×D53—28no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—22no—
88C42×D53—28no—
89#13133150—24no—
90#13133148—24no—
91#13133129C2×C414yes—
92#13133129C2×C414yes—
93#13133129C2×C414yescomplemented
94#13133129C2×C414yescomplemented
95#13133124—22no—
96#13133124—22no—
97C23×C4×C53C2311yescharacteristic
98C22×C8×C53—22no—
99C22×C8×C53—22no—
100C22×C8×C53C2×C414yescomplemented
101C22×C8×C53C2311yescharacteristic
102C22×C8×C53C2311yescharacteristic
103#117×C53—22no—
104#117×C53—22no—
105#117×C53—22no—
106#117×C53D414yes—
107C2×C4⋊C4×C53C2312yes—
108C2×C4⋊C4×C53C2312yes—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C42×C53—21no—
112C2×C42×C53—21no—
113#107×C53—24no—
114#106×C53—24no—
115C24×C53C2×D411yescharacteristic
116C23×D53—22no—
117C23×D53—22no—
118C23×D53—24no—
119C23×D53—42no—
120C23×D53C4⋊C414yes—
121C23×D53C2×D411yescharacteristic
122C23×D53C2×D411yescharacteristic
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53C22×C412yes—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D412yes—
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2×Q812yes—
130C22×C4×C53#4112yes—
131C22×C4×C53#4112yes—
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53C4⋊C414yes—
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C2×C4×D53—28no—
141C2×C4×D53—28no—
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—28no—
145C2×C4×D53—28no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—21no—
149C2×C4×D53—21no—
150C2×C4×D53—21no—
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C4×D53—28no—
157C2×C4×D53—44no—
158C2×C4×D53—44no—
159C2×C4×D53C4⋊C414yes—
160C2×C4×D53C4⋊C414yes—
161C2×C4×D53C22×C412yes—
162C2×C4×D53C22×C412yes—
163C2×C4×D53C2×Q812yes—
164C2×C4×D53C2×Q812yes—
165C2×C8×C53—28no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53C22×C412yes—
171C2×C8×C53C22×C412yes—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53C2×D414yes—
176C22⋊C4×C53—24no—
177C22⋊C4×C53—42no—
178C42×C53—24no—
179C42×C53—42no—
180C22⋊Dic53—24no—
181C22⋊Dic53—42no—
182#11719921—28no—
183#11719921—44no—
184#11719921—44no—
185C4⋊Dic53—22no—
186C4⋊Dic53—22no—
187C4⋊Dic53—22no—
188C4⋊Dic53C2×D414yes—
189#11719919—28no—
190C4×Dic53—28no—
191C4×Dic53—24no—
192C4×Dic53—42no—
193C2×#88322—28no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322C22×C412yes—
199C2×#88322C22×C412yes—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53C2×C4⋊C412yes—
203C23×C53#12012yes—
204C23×C53#12312yes—
205C23×C53C22×D411yescharacteristic
206C22×D53—21no—
207C22×D53—21no—
208C22×D53—22no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—22no—
212C22×D53—22no—
213C22×D53—28no—
214C22×D53—28no—
215C22×D53—44no—
216C22×D53—48no—
217C22×D53—48no—
218C22×D53C2×C4⋊C412yes—
219C22×D53C2×C4⋊C412yes—
220C22×D53#13514yes—
221C2×C4×C53—24no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53C2×C4⋊C412yes—
228C2×C4×C53C4×D414yes—
229C2×C4×C53C4×D414yes—
230C2×C4×C53#12012yes—
231C2×C4×C53C22⋊Q814yes—
232C2×C4×C53C22⋊Q814yes—
233C2×C4×C53#12312yes—
234C2×C4×C53C23×C411yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×Q811yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—28no—
245C2×Dic53—24no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53C2×C4⋊C412yes—
252C2×Dic53C2×C4⋊C412yes—
253C4×D53—22no—
254C4×D53—22no—
255C4×D53—22no—
256C4×D53—22no—
257C4×D53—22no—
258C4×D53—22no—
259C4×D53—22no—
260C4×D53—22no—
261C4×D53—28no—
262C4×D53—28no—
263C4×D53—416no—
264C8×C53—22no—
265C8×C53—22no—
266#88322—22no—
267#88322—22no—
268#20843—531nomaximal, Sylow
269C22×C53—24no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—24no—
273C22×C53#38514yes—
274C22×C53C22×C4⋊C411yescharacteristic
275C22×C53C2×#12011yescharacteristic
276C22×C53C2×#12311yescharacteristic
277C22×C53C2×#13512yes—
278C22×C53#57512yes—
279C2×D53—24no—
280C2×D53—22no—
281C2×D53—24no—
282C2×D53—22no—
283C2×D53—22no—
284C2×D53—22no—
285C2×D53—22no—
286C2×D53—416no—
287C2×D53—42no—
288C2×D53—42no—
289C2×D53#42714yescomplemented
290C4×C53—44no—
291C4×C53—82no—
292C4×C53#38514yes—
293C4×C53C2×C4×D411yescharacteristic
294C4×C53C2×C4×D411yescharacteristic
295C4×C53C2×C22⋊Q811yescharacteristic
296C4×C53C2×C22⋊Q811yescharacteristic
297Dic53—22no—
298Dic53—22no—
299Dic53—44no—
300Dic53—82no—
301C22×#117—531no—
302C2×#427—531no—
303C2×#427D5311yescharacteristic, complemented
304C22×C22⋊C8—531no—
305#2280—531no—
306#1854—532no—
307#1854—532no—
308#1853—532no—
309#1852—532no—
310#1735—531no—
311#1735—531no—
312C2×C53—24no—
313C2×C53C2×#38511yescharacteristic
314C2×C53C2×#42711yescharacteristic
315C2×C53#291111yescharacteristic
316C2×C53#300212yes—
317C2×C53#309112yes—
318D53—24no—
319D53—42no—
320C24×C4—531no—
321C23×C8—531no—
322C23×C8C2×D5311yescharacteristic
323C2×#117—531no—
324C2×#117—531no—
325C2×#117—532no—
326C2×#117—1061no—
327C2×#117—1061no—
328C2×#117—1064no—
329C2×#117—1064no—
330C2×#117C2×D5311yescharacteristic
331C2×#117C2×D5311yescharacteristic
332C22×C4⋊C4—532no—
333C22×C22⋊C4—1061no—
334C22×C42—1061no—
335#427—24no—
336#427—1064no—
337C2×#107—532no—
338C2×#107C2×D5312yescomplemented
339C2×#106—532no—
340C2×#106C2×D5312yescomplemented
341C2×C22⋊C8—538no—
342C2×C22⋊C8—531no—
343C2×C22⋊C8—531no—
344C2×C22⋊C8—531no—
345C2×C22⋊C8—531no—
346#381—1061no—
347#380—532no—
348#380—532no—
349#379—1061no—
350C4×C22⋊C4—1061no—
351C4×C22⋊C4—1061no—
352#340—532no—
353#340—532no—
354#340—532no—
355#340—532no—
356C53#2084311yescharacteristic, complemented, Sylow
357C25—531no—
358C23×C4—531no—
359C23×C4—534no—
360C23×C4—531no—
361C23×C4—531no—
362C23×C4—534no—
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—1062no—
366C23×C4—1062no—
367C23×C4—1062no—
368C23×C4—1062no—
369C23×C4—1062no—
370C23×C4C22×D5311yescharacteristic
371C22×C8—22no—
372C22×C8—22no—
373C22×C8—534no—
374C22×C8—531no—
375C22×C8—531no—
376C22×C8—1062no—
377C22×C8—1062no—
378C22×C8C4×D5314yescomplemented
379C22×C8C22×D5311yescharacteristic
380C22×C8C22×D5311yescharacteristic
381#117—22no—
382#117—22no—
383#117—22no—
384#117—534no—
385#117—1068no—
386#117—1068no—
387#117—1064no—
388#117—1062no—
389#117—1064no—
390#117—1062no—
391#117—1062no—
392#117C22⋊D5314yes—
393C2×C4⋊C4—532no—
394C2×C4⋊C4—532no—
395C2×C4⋊C4—1064no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4—1064no—
398C2×C4⋊C4C22×D5312yes—
399C2×C4⋊C4C22×D5312yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1068no—
403C2×C22⋊C4—1062no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1062no—
406C2×C22⋊C4—1062no—
407C2×C22⋊C4—1061no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—1068no—
411C2×C42—1062no—
412C2×C42—1061no—
413C2×C42—1061no—
414#107—24no—
415#107—1064no—
416#106—24no—
417#106—1064no—
418C22⋊C8—534no—
419C22⋊C8—534no—
420C22⋊C8—534no—
421C22⋊C8—534no—
422#95—1062no—
423#95—1062no—
424C24—531no—
425C24—531no—
426C24—534no—
427C24—1064no—
428C24—1062no—
429C24—1062no—
430C24—2122no—
431C24D4×D5311yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—534no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—534no—
438C22×C4—534no—
439C22×C4—532no—
440C22×C4—532no—
441C22×C4—532no—
442C22×C4—532no—
443C22×C4—1068no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1068no—
447C22×C4—1068no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1068no—
451C22×C4—1068no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—2124no—
465C22×C4—2124no—
466C22×C4C2×C4×D5312yes—
467C22×C4#1171994512yes—
468C22×C4D4×D5312yes—
469C22×C4D4×D5311yescharacteristic
470C22×C4Q8×D5312yes—
471C22×C4#1171994912yes—
472C22×C4C2×C22⋊D5312yes—
473C22×C4C23×D5311yescharacteristic
474C2×C8—28no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—532no—
480C2×C8—532no—
481C2×C8—1068no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8C2×C4×D5312yes—
487C2×C8C2×C4×D5312yes—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—534no—
492C4⋊C4—1068no—
493C4⋊C4—1062no—
494C4⋊C4—1062no—
495C4⋊C4—1062no—
496C4⋊C4C2×C22⋊D5314yes—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—1068no—
500C22⋊C4—1064no—
501C22⋊C4—2124no—
502C22⋊C4—2124no—
503C22⋊C4—2122no—
504C42—24no—
505C42—42no—
506C42—1068no—
507C42—1064no—
508C42—2122no—
509C23—22no—
510C23—22no—
511C23—532no—
512C23—534no—
513C23—532no—
514C23—1068no—
515C23—1061no—
516C23—1061no—
517C23—1062no—
518C23—1061no—
519C23—1061no—
520C23—1062no—
521C23—1062no—
522C23—1068no—
523C23—2124no—
524C23—2128no—
525C23—2128no—
526C23C4⋊C4×D5312yes—
527C23#1313318512yes—
528C23#1313320212yes—
529C23C2×D4×D5311yescharacteristic
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—532no—
537C2×C4—532no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1068no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—1062no—
551C2×C4—1068no—
552C2×C4—1068no—
553C2×C4—1064no—
554C2×C4—21216no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4C4⋊C4×D5312yes—
561C2×C4C4×C22⋊D5314yes—
562C2×C4#1313317914yes—
563C2×C4#1313318512yes—
564C2×C4#1313319314yes—
565C2×C4#1313319514yes—
566C2×C4#1313320212yes—
567C2×C4C22×C4×D5311yescharacteristic
568C2×C4C2×D4×D5311yescharacteristic
569C2×C4C22×C22⋊D5311yescharacteristic
570C2×C4C2×Q8×D5311yescharacteristic
571C2×C4C2×#1171994511yescharacteristic
572C2×C4C2×#1171994911yescharacteristic
573C8—22no—
574C8—22no—
575C8—1062no—
576C8—1062no—
577C22—24no—
578C22—22no—
579C22—22no—
580C22—24no—
581C22—534no—
582C22—1064no—
583C22—1062no—
584C22—1062no—
585C22—1064no—
586C22—1062no—
587C22—1062no—
588C22—1062no—
589C22—21216no—
590C22—2122no—
591C22—2122no—
592C22#1681049414yes—
593C22C2×C4⋊C4×D5311yescharacteristic
594C22C2×#1313318511yescharacteristic
595C22C2×#1313320211yescharacteristic
596C22#135×D5312yes—
597C22#1681142012yes—
598C4—44no—
599C4—82no—
600C4—1062no—
601C4—1062no—
602C4—4242no—
603C4—2124no—
604C4#1681049414yes—
605C4C2×C4×C22⋊D5311yescharacteristic
606C4C2×#1313317911yescharacteristic
607C4C2×#1313319311yescharacteristic
608C4C2×#1313319511yescharacteristic
609C2—24no—
610C2—1064no—
611C2—2122no—
612C2C2×#1681049411yescharacteristic
613C2#427×D5311yescharacteristic
614C2#2028330311yescharacteristic
615C2#2028392912yes—
616C2#2028463212yes—
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#427C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenterC2×#427D53
hypocenterC53#20843
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. C2×#16810659C2semidirect
5. #20274913C2semidirect
6. #20274911C2semidirect
7. #20274906C2semidirect
8. #20274901C2semidirect
9. #20274432C2semidirect
10. #20274431C2semidirect
11. C2×C53×#427C2semidirect
25. C2×#13133150C22semidirect
26. C2×#13133148C22semidirect
27. C2×#13133129C4semidirect
38. #16810386C22semidirect
39. #16810386C22semidirect
40. #16810385C22semidirect
41. #16810385C22semidirect
46. C2×#107×C53C22semidirect
47. C2×#106×C53C22semidirect
63. C22×#88322C2×C4semidirect
93. #13133129C2×C4semidirect
94. #13133129C2×C4semidirect
100. C22×C8×C53C2×C4semidirect
289. C2×D53#427semidirect
303. C2×#427D53semidirect
338. C2×#107C2×D53semidirect
340. C2×#106C2×D53semidirect
356. C53#20843semidirect
378. C22×C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D532
C2×#168106592
C22×#131331292
#202770942
#202749132
#202749112
#202749062
#202749012
#202744322
#202744312
C2×C53×#4272
#2084353