Finite Group Databasebeta

← #27383883

Subgroups — #27383883

Order 13568.

1502 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D53C211yesmaximal, characteristic
2C22×#13133129C211yesmaximal, characteristic
3#20277094C211yesmaximal, characteristic
4#20274809C212yesmaximal, complemented
5#20274808C212yesmaximal, complemented
6#20274804C212yesmaximal, complemented
7#20274800C212yesmaximal, complemented
8#20274432C211yesmaximal, characteristic, complemented
9#20274431C211yesmaximal, characteristic, complemented
10#20274431C211yesmaximal, characteristic, complemented
11C53×#1735C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C23×#88322C2211yescharacteristic
14C2×#13133176C2211yescharacteristic
15C2×#13133176C2211yescharacteristic
16C2×#13133175C2211yescharacteristic
17C2×#13133175C2211yescharacteristic
18C2×#13133174—21no—
19C2×#13133173—21no—
20#117×D53—24no—
21#117×D53C418yescomplemented
22C2×C4⋊C4×D53C2211yescharacteristic
23C2×C4⋊C4×D53C2211yescharacteristic
24C2×C22⋊C4×D53—21no—
25C2×C42×D53—21no—
26C2×#13133129—24no—
27C2×#13133129C2212yes—
28C2×#13133129C2212yes—
29#16810484—21no—
30#16810480C2212yes—
31#16810480C2212yes—
32#16810478—21no—
33C4×#11719921—21no—
34C4×#11719921—21no—
35#16810386C2212yescomplemented
36#16810386C2212yescomplemented
37#16810385C2212yescomplemented
38#16810385C2212yescomplemented
39#16810385C2212yescomplemented
40#16810385C2212yescomplemented
41C23×C8×C53C2211yescharacteristic
42C2×#117×C53C2211yescharacteristic
43C2×#117×C53C2211yescharacteristic
44C53×#340C2212yescomplemented
45C53×#340C2212yescomplemented
46C24×D53D411yescharacteristic
47C23×Dic53D411yescharacteristic
48C22×C4×D53—22no—
49C22×C4×D53—22no—
50C22×C4×D53—22no—
51C22×C4×D53—22no—
52C22×C4×D53—21no—
53C22×C4×D53—21no—
54C22×C4×D53C2×C414yes—
55C22×C4×D53D411yescharacteristic
56C22×C4×D53D411yescharacteristic
57C22×C4×D53D414yes—
58C22×C4×D53C2311yescharacteristic
59C22×C4×D53C2311yescharacteristic
60C22×#88322—22no—
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322C2312yes—
64#13133176—22no—
65#13133176—22no—
66#13133176—22no—
67#13133176C2×C414yes—
68#13133175—22no—
69#13133175—22no—
70#13133175C2×C414yes—
71#13133175C2×C414yes—
72#13133174—28no—
73#13133173—28no—
74C2×C4⋊Dic53C2312yes—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C4⋊C4×D53—22no—
80C4⋊C4×D53—22no—
81C4⋊C4×D53C2×C414yes—
82C4⋊C4×D53C2×C414yes—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921—22no—
88C2×#11719921—22no—
89C22⋊C4×D53—28no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C42×D53—28no—
95#13133129—24no—
96#13133129—24no—
97#13133124—22no—
98#13133124—22no—
99C23×C4×C53C2311yescharacteristic
100C22×C8×C53—22no—
101C22×C8×C53—22no—
102C22×C8×C53—22no—
103C22×C8×C53C2312yes—
104#117×C53—22no—
105#117×C53—22no—
106#117×C53—22no—
107#117×C53C2×C414yes—
108C2×C4⋊C4×C53C2312yes—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C42×C53—21no—
114C2×C42×C53—21no—
115C24×C53C2×D411yescharacteristic
116C23×D53—22no—
117C23×D53—22no—
118C23×D53—24no—
119C23×D53—42no—
120C23×D53C22⋊C414yes—
121C23×D53C2×D411yescharacteristic
122C23×D53C2×D411yescharacteristic
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D412yes—
129C22×C4×C53#4112yes—
130C22×C4×C53#4112yes—
131C22×C4×C53#4112yes—
132C22×C4×C53#4112yes—
133C22×C4×C53C2411yescharacteristic
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53C22⋊C414yes—
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C2×C4×D53—28no—
143C2×C4×D53—28no—
144C2×C4×D53—28no—
145C2×C4×D53—28no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
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—21no—
157C2×C4×D53—21no—
158C2×C4×D53—24no—
159C2×C4×D53—24no—
160C2×C4×D53—44no—
161C2×C4×D53—44no—
162C2×C4×D53C22⋊C414yes—
163C2×C4×D53C22⋊C414yes—
164C2×C4×D53C22×C412yes—
165C2×C4×D53C22×C412yes—
166C2×C4×D53C2×D412yes—
167C2×C4×D53C2×D412yes—
168C2×C8×C53—22no—
169C2×C8×C53—22no—
170C2×C8×C53—22no—
171C2×C8×C53—44no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53C22×C412yes—
177C4⋊C4×C53C22×C412yes—
178C22⋊C4×C53—24no—
179C22⋊C4×C53—42no—
180C42×C53—24no—
181C42×C53—42no—
182C22⋊Dic53—24no—
183C22⋊Dic53—42no—
184#11719921—24no—
185#11719921—24no—
186#11719921—44no—
187#11719921—44no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—22no—
191C4⋊Dic53—22no—
192C4⋊Dic53C22×C412yes—
193C4⋊Dic53C22×C412yes—
194#11719919—24no—
195#11719919—24no—
196C4×Dic53—24no—
197C4×Dic53—24no—
198C4×Dic53—24no—
199C4×Dic53—42no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C2×#88322—44no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53C2×C22⋊C412yes—
207C23×C53#12312yes—
208C23×C53#12312yes—
209C23×C53C22×D411yescharacteristic
210C22×D53—21no—
211C22×D53—21no—
212C22×D53—22no—
213C22×D53—21no—
214C22×D53—21no—
215C22×D53—22no—
216C22×D53—22no—
217C22×D53—28no—
218C22×D53—28no—
219C22×D53—44no—
220C22×D53—48no—
221C22×D53—48no—
222C22×D53C2×C22⋊C412yes—
223C22×D53C2×C22⋊C412yes—
224C22×D53#13514yes—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53C2×C22⋊C412yes—
233C2×C4×C53#11714yes—
234C2×C4×C53#11714yes—
235C2×C4×C53#12314yes—
236C2×C4×C53#12312yes—
237C2×C4×C53#12314yes—
238C2×C4×C53#12312yes—
239C2×C4×C53C23×C411yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—28no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53C2×C22⋊C412yes—
258C2×Dic53C2×C22⋊C412yes—
259C4×D53—22no—
260C4×D53—22no—
261C4×D53—22no—
262C4×D53—22no—
263C4×D53—22no—
264C4×D53—22no—
265C4×D53—22no—
266C4×D53—22no—
267C4×D53—28no—
268C4×D53—28no—
269C4×D53—48no—
270C4×D53—48no—
271C8×C53—42no—
272#88322—42no—
273#20699—531nomaximal, Sylow
274C22×C53—24no—
275C22×C53—24no—
276C22×C53—22no—
277C22×C53—22no—
278C22×C53#38114yes—
279C22×C53C22×C22⋊C411yescharacteristic
280C22×C53C2×#12311yescharacteristic
281C22×C53C2×#12311yescharacteristic
282C22×C53C2×#13512yes—
283C22×C53#57512yes—
284C2×D53—24no—
285C2×D53—24no—
286C2×D53—22no—
287C2×D53—22no—
288C2×D53—22no—
289C2×D53—22no—
290C2×D53—22no—
291C2×D53—416no—
292C2×D53—42no—
293C2×D53—42no—
294C2×D53#41614yes—
295C4×C53—42no—
296C4×C53—42no—
297C4×C53—82no—
298C4×C53#38114yes—
299C4×C53C2×#11711yescharacteristic
300C4×C53C2×#11711yescharacteristic
301C4×C53C2×#12311yescharacteristic
302C4×C53C2×#12311yescharacteristic
303Dic53—22no—
304Dic53—22no—
305Dic53—42no—
306Dic53—42no—
307Dic53—82no—
308C22×#117—531no—
309C22×C22⋊C8—531no—
310#2280—531no—
311#1825—532no—
312#1825—532no—
313#1824—532no—
314#1823—532no—
315#1735—531no—
316#1735—531no—
317#1735—531no—
318#1735D5311yescharacteristic, complemented
319C2×C53—24no—
320C2×C53C2×#38111yescharacteristic
321C2×C53C2×#41611yescharacteristic
322C2×C53#289611yescharacteristic
323C2×C53#309112yes—
324C2×C53#309112yes—
325D53—24no—
326D53—42no—
327C24×C4—531no—
328C23×C8—531no—
329C23×C8C2×D5311yescharacteristic
330C2×#117—531no—
331C2×#117—538no—
332C2×#117—531no—
333C2×#117—531no—
334C2×#117—531no—
335C2×#117—1061no—
336C2×#117—1061no—
337C2×#117—1064no—
338C2×#117C2×D5311yescharacteristic
339C2×#117C2×D5311yescharacteristic
340C22×C4⋊C4—531no—
341C22×C4⋊C4—531no—
342C22×C22⋊C4—1061no—
343C22×C42—1061no—
344C2×C22⋊C8—532no—
345C2×C22⋊C8—532no—
346C2×C22⋊C8—1064no—
347#381—1061no—
348#380—532no—
349#380—532no—
350#379—1061no—
351C4×C22⋊C4—1061no—
352C4×C22⋊C4—1061no—
353#340—532no—
354#340—532no—
355#340—532no—
356#340—532no—
357#340—532no—
358#340—532no—
359#340C2×D5312yescomplemented
360#340C2×D5312yescomplemented
361C53#2069911yescharacteristic, complemented, Sylow
362C25—531no—
363C23×C4—531no—
364C23×C4—534no—
365C23×C4—534no—
366C23×C4—531no—
367C23×C4—531no—
368C23×C4—531no—
369C23×C4—531no—
370C23×C4—1061no—
371C23×C4—1061no—
372C23×C4—1062no—
373C23×C4—1062no—
374C23×C4—1062no—
375C23×C4—1062no—
376C23×C4C22×D5311yescharacteristic
377C22×C8—22no—
378C22×C8—22no—
379C22×C8—22no—
380C22×C8—532no—
381C22×C8—1062no—
382C22×C8—1062no—
383C22×C8—1062no—
384C22×C8C22×D5312yes—
385#117—22no—
386#117—22no—
387#117—22no—
388#117—534no—
389#117—534no—
390#117—534no—
391#117—1068no—
392#117—1068no—
393#117—1062no—
394#117—1062no—
395#117—1062no—
396#117—1062no—
397#117—1062no—
398#117C4×D5314yes—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—534no—
402C2×C4⋊C4—534no—
403C2×C4⋊C4—532no—
404C2×C4⋊C4—1062no—
405C2×C4⋊C4—1062no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4C22×D5312yes—
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—1068no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1062no—
418C2×C22⋊C4—1062no—
419C2×C22⋊C4—1061no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42—1068no—
423C2×C42—1061no—
424C2×C42—1061no—
425C2×C42—1061no—
426C2×C42—1061no—
427C22⋊C8—1064no—
428C22⋊C8—1064no—
429#95—1062no—
430#95—1062no—
431C24—531no—
432C24—531no—
433C24—534no—
434C24—1064no—
435C24—1062no—
436C24—1062no—
437C24—2122no—
438C24D4×D5311yescharacteristic
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—534no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—534no—
446C22×C4—534no—
447C22×C4—532no—
448C22×C4—532no—
449C22×C4—532no—
450C22×C4—532no—
451C22×C4—1064no—
452C22×C4—1064no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1068no—
456C22×C4—1068no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1068no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1068no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1062no—
474C22×C4—2124no—
475C22×C4—2124no—
476C22×C4C2×C4×D5312yes—
477C22×C4#1171994512yes—
478C22×C4#1171994512yes—
479C22×C4D4×D5312yes—
480C22×C4D4×D5311yescharacteristic
481C22×C4#1171994712yes—
482C22×C4#1171994912yes—
483C22×C4C23×D5311yescharacteristic
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—44no—
488C2×C8—1062no—
489C2×C8—1062no—
490C2×C8—1062no—
491C2×C8—2124no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—532no—
497C4⋊C4—532no—
498C4⋊C4—1064no—
499C4⋊C4—1064no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1062no—
503C4⋊C4—1062no—
504C4⋊C4C2×C4×D5312yes—
505C4⋊C4C2×C4×D5312yes—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—1064no—
509C22⋊C4—1064no—
510C22⋊C4—1064no—
511C22⋊C4—2124no—
512C22⋊C4—2124no—
513C22⋊C4—2122no—
514C42—24no—
515C42—42no—
516C42—1064no—
517C42—1064no—
518C42—1064no—
519C42—2122no—
520C23—22no—
521C23—22no—
522C23—532no—
523C23—534no—
524C23—532no—
525C23—1068no—
526C23—1061no—
527C23—1061no—
528C23—1062no—
529C23—1061no—
530C23—1061no—
531C23—1062no—
532C23—1062no—
533C23—1068no—
534C23—2124no—
535C23—2128no—
536C23—2128no—
537C23C22⋊C4×D5312yes—
538C23#1313319912yes—
539C23#1313320212yes—
540C23C2×D4×D5311yescharacteristic
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—532no—
549C2×C4—532no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1068no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1068no—
564C2×C4—1068no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—2128no—
568C2×C4—2128no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4C22⋊C4×D5312yes—
575C2×C4#1313317414yes—
576C2×C4#1313317514yes—
577C2×C4#1313319914yes—
578C2×C4#1313319912yes—
579C2×C4#1313320214yes—
580C2×C4#1313320212yes—
581C2×C4C22×C4×D5311yescharacteristic
582C2×C4C2×D4×D5311yescharacteristic
583C2×C4C2×#1171994711yescharacteristic
584C2×C4C2×#1171994511yescharacteristic
585C2×C4C2×#1171994511yescharacteristic
586C2×C4C2×#1171994911yescharacteristic
587C8—42no—
588C8—2122no—
589C22—22no—
590C22—22no—
591C22—24no—
592C22—24no—
593C22—534no—
594C22—1062no—
595C22—1062no—
596C22—1064no—
597C22—1064no—
598C22—1062no—
599C22—1062no—
600C22—1062no—
601C22—21216no—
602C22—2122no—
603C22—2122no—
604C22#1681048514yes—
605C22C2×C22⋊C4×D5311yescharacteristic
606C22C2×#1313319911yescharacteristic
607C22C2×#1313320211yescharacteristic
608C22#135×D5312yes—
609C22#1681142012yes—
610C4—42no—
611C4—42no—
612C4—82no—
613C4—1062no—
614C4—1062no—
615C4—4242no—
616C4—2122no—
617C4—2122no—
618C4#1681048514yes—
619C4C2×#1313317411yescharacteristic
620C4C2×#1313317511yescharacteristic
621C4C2×#1313319911yescharacteristic
622C4C2×#1313320211yescharacteristic
623C2—24no—
624C2—1064no—
625C2—2122no—
626C2C2×#1681048511yescharacteristic
627C2#416×D5311yescharacteristic
628C2#2028323011yescharacteristic
629C2#2028462612yes—
630C2#2028463212yes—
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1735C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenter#1735D53
hypocenterC53#20699
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
4. #20274809C2semidirect
5. #20274808C2semidirect
6. #20274804C2semidirect
7. #20274800C2semidirect
8. #20274432C2semidirect
9. #20274431C2semidirect
10. #20274431C2semidirect
11. C53×#1735C2semidirect
21. #117×D53C4semidirect
35. #16810386C22semidirect
36. #16810386C22semidirect
37. #16810385C22semidirect
38. #16810385C22semidirect
39. #16810385C22semidirect
40. #16810385C22semidirect
44. C53×#340C22semidirect
45. C53×#340C22semidirect
318. #1735D53semidirect
359. #340C2×D53semidirect
360. #340C2×D53semidirect
361. C53#20699semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D532
C22×#131331292
#202770942
#202748092
#202748082
#202748042
#202748002
#202744322
#202744312
C53×#17352
#2069953