Finite Group Databasebeta

← #27305420

Subgroups — #27305420

Order 13568.

1710 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810515C212yesmaximal
2C2×C4×#11719921C211yesmaximal, characteristic
3#20272951C212yesmaximal
4#20272949C212yesmaximal, complemented
5#20272945C211yesmaximal, characteristic, complemented
6#20272941C212yesmaximal, complemented
7#20272936C211yesmaximal, characteristic
8#20272936C211yesmaximal, characteristic
9#95×Dic53C211yesmaximal, characteristic, complemented
10#95×Dic53C211yesmaximal, characteristic, complemented
11C53×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×C4⋊Dic53C2212yes—
14C22×#11719921C2211yescharacteristic
15C22×#11719921C2212yes—
16C22×#11719921C2212yes—
17C22×#11719921C2211yescharacteristic
18C22×C4×Dic53C2212yes—
19C22×C4×Dic53C2211yescharacteristic
20C22×C4×Dic53C2212yes—
21#16810515C4116yescomplemented
22C4×#11719921C418yescomplemented
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2212yes—
32C22×C42×C53C2211yescharacteristic
33C2×#95×C53C2211yescharacteristic
34C2×#95×C53C2211yescharacteristic
35C2×#95×C53C2212yes—
36C2×#95×C53C2212yes—
37C24×D53C2311yescharacteristic
38C23×Dic53C2312yes—
39C23×Dic53C2312yes—
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43C22×C4×D53—22no—
44C22×C4×D53—22no—
45C22×C4×D53—42no—
46C22×C4×D53—42no—
47C22×C4×D53C2×C414yes—
48C2×C4⋊Dic53—42no—
49C2×C4⋊Dic53—42no—
50C2×C4⋊Dic53C2×C418yescomplemented
51C2×#11719921—24no—
52C2×#11719921—42no—
53C2×#11719921—42no—
54C2×#11719921—42no—
55C2×#11719921—42no—
56C2×#11719921C2×C414yes—
57C2×#11719921C2×C418yes—
58C2×#11719921C2×C418yes—
59C2×#11719921C2×C418yes—
60C2×#11719921C2×C414yes—
61C2×C4×Dic53—24no—
62C2×C4×Dic53—42no—
63C2×C4×Dic53—42no—
64C2×C4×Dic53—42no—
65C2×C4×Dic53—42no—
66C2×C4×Dic53—42no—
67C2×C4×Dic53C2×C414yes—
68C2×C4×Dic53C2×C418yescomplemented
69#13133124—22no—
70#13133124—22no—
71#13133124—24no—
72#13133124—42no—
73#13133124—42no—
74#13133124—42no—
75#13133124C2×C414yes—
76#13133124C2×C414yes—
77#13133124C2×C418yescomplemented
78C23×C4×C53C2311yescharacteristic
79C23×C4×C53C2312yes—
80C23×C4×C53C2312yes—
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2311yescharacteristic
83C2×C42×C53—42no—
84C2×C42×C53C2×C414yes—
85#95×C53—22no—
86#95×C53—22no—
87#95×C53—42no—
88#95×C53C2×C418yescomplemented
89C24×C53C2411yescharacteristic
90C23×D53—22no—
91C23×D53—21no—
92C23×D53—21no—
93C23×D53—21no—
94C23×D53—21no—
95C23×D53—22no—
96C23×D53—22no—
97C23×D53—21no—
98C23×D53—21no—
99C23×D53C22×C414yes—
100C23×D53C22×C412yes—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53C22×C414yes—
123C22×C4×C53C22×C414yes—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C22×C414yes—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C22×C412yes—
128C22×C4×C53#4112yes—
129C22×C4×C53#4112yes—
130C22×Dic53—22no—
131C22×Dic53—21no—
132C22×Dic53—22no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53C22×C412yes—
153C22×Dic53C22×C414yes—
154C22×Dic53C22×C414yes—
155C22×Dic53C22×C412yes—
156C22×Dic53C22×C414yes—
157C22×Dic53C22×C412yes—
158C22×Dic53#4114yes—
159C2×C4×D53—28no—
160C2×C4×D53—44no—
161C2×C4×D53—44no—
162C2×C4×D53—48no—
163C2×C4×D53—48no—
164C42×C53—44no—
165#11719921—28no—
166#11719921—48no—
167#11719921—48no—
168#11719921C42116yescomplemented
169C4⋊Dic53—48no—
170C4×Dic53—44no—
171C4×Dic53—48no—
172C4×Dic53—48no—
173C23×C53C23×C411yescharacteristic
174C23×C53C23×C412yes—
175C23×C53C2×#4112yes—
176C23×C53C2×#4111yescharacteristic
177C23×C53C2×#4111yescharacteristic
178C23×C53H(5,𝔽2)12yes—
179C23×C53H(5,𝔽2)12yes—
180C23×C53H(5,𝔽2)11yescharacteristic
181C23×C53H(5,𝔽2)11yescharacteristic
182C23×C532−1+411yescharacteristic
183C23×C532−1+411yescharacteristic
184C22×D53—22no—
185C22×D53—22no—
186C22×D53—24no—
187C22×D53—24no—
188C22×D53—22no—
189C22×D53—22no—
190C22×D53—24no—
191C22×D53—24no—
192C22×D53—24no—
193C22×D53—22no—
194C22×D53—22no—
195C22×D53—24no—
196C22×D53—42no—
197C22×D53—42no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—42no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—42no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53C2×C4214yes—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—24no—
210C2×C4×C53—22no—
211C2×C4×C53—24no—
212C2×C4×C53—22no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—22no—
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—22no—
223C2×C4×C53—24no—
224C2×C4×C53—42no—
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×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53C2×C4214yes—
235C2×C4×C53#11714yes—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—24no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—22no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—42no—
254C2×Dic53—42no—
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×Dic53C2×C4214yes—
264C2×Dic53#11714yes—
265C4×D53—416no—
266#13788—531nomaximal, Sylow
267C22×C53C22×C4211yescharacteristic
268C22×C53C2×#11711yescharacteristic
269C22×C53C2×#11711yescharacteristic
270C22×C53C4×#4112yes—
271C22×C53C4×#4112yes—
272C22×C53#51812yes—
273C22×C53#51811yescharacteristic
274C22×C53#51811yescharacteristic
275C22×C53#51812yes—
276C22×C53#52012yes—
277C22×C53#52011yescharacteristic
278C22×C53#52011yescharacteristic
279C22×C53#52012yes—
280C22×C53#53812yes—
281C22×C53#53812yes—
282C22×C53#53812yes—
283C22×C53#53912yes—
284C22×C53#54012yes—
285C22×C53#54011yescharacteristic
286C22×C53#54111yescharacteristic
287C22×C53#54212yes—
288C22×C53#54211yescharacteristic
289C22×C53#54211yescharacteristic
290C2×D53—24no—
291C2×D53—24no—
292C2×D53—24no—
293C2×D53—42no—
294C2×D53—42no—
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—44no—
298C2×D53—44no—
299C2×D53—44no—
300C2×D53—44no—
301C4×C53—24no—
302C4×C53—24no—
303C4×C53—42no—
304C4×C53—44no—
305C4×C53—44no—
306Dic53—24no—
307Dic53—24no—
308Dic53—42no—
309Dic53—44no—
310Dic53—44no—
311C2×#388—532no—
312C2×C4×C22⋊C4—531no—
313#1437—532no—
314#1436—532no—
315C4⋊#95—531no—
316#1434—532no—
317#1434D5311yescharacteristic, complemented
318C22⋊#95—531no—
319C22⋊#95—531no—
320C4×#95—531no—
321C4×#95—531no—
322C2×C53C4×#11711yescharacteristic
323C2×C53#227511yescharacteristic
324C2×C53#227511yescharacteristic
325C2×C53#231211yescharacteristic
326C2×C53#231411yescharacteristic
327C2×C53#231811yescharacteristic
328C2×C53#231911yescharacteristic
329C2×C53#233312yes—
330C2×C53#233412yes—
331C2×C53#233512yes—
332C2×C53#233712yes—
333D53—44no—
334C24×C4—531no—
335C22×C4⋊C4—532no—
336C22×C22⋊C4—531no—
337C22×C22⋊C4—531no—
338C22×C22⋊C4—532no—
339C22×C22⋊C4—532no—
340C22×C42—531no—
341C22×C42—532no—
342C22×C42—532no—
343C22×C42C2×D5311yescharacteristic
344#388—5316no—
345C4×C22⋊C4—538no—
346C2×#95—532no—
347C2×#95—531no—
348C2×#95—531no—
349C2×#95—531no—
350C2×#95—531no—
351C2×#95—532no—
352C2×#95—532no—
353C2×#95—532no—
354C2×#95—532no—
355C2×#95C2×D5311yescharacteristic
356C2×#95C2×D5311yescharacteristic
357C2×#95C2×D5312yes—
358C2×#95C2×D5312yes—
359C53#1378811yescharacteristic, complemented, Sylow
360C25—531no—
361C23×C4—531no—
362C23×C4—531no—
363C23×C4—532no—
364C23×C4—532no—
365C23×C4—531no—
366C23×C4—534no—
367C23×C4—1062no—
368C23×C4—1062no—
369C23×C4—2122no—
370C23×C4—2122no—
371C23×C4C22×D5311yescharacteristic
372C23×C4C22×D5312yes—
373C23×C4C22×D5312yes—
374C23×C4C22×D5311yescharacteristic
375C23×C4C22×D5311yescharacteristic
376C2×C4⋊C4—538no—
377C2×C4⋊C4—2122no—
378C2×C4⋊C4—2122no—
379C2×C22⋊C4—534no—
380C2×C22⋊C4—538no—
381C2×C22⋊C4—534no—
382C2×C22⋊C4—538no—
383C2×C22⋊C4—538no—
384C2×C22⋊C4—1064no—
385C2×C22⋊C4—2122no—
386C2×C22⋊C4—2122no—
387C2×C22⋊C4—2122no—
388C2×C22⋊C4—2122no—
389C2×C42—42no—
390C2×C42—534no—
391C2×C42—538no—
392C2×C42—1064no—
393C2×C42—2122no—
394C2×C42—2122no—
395C2×C42—2122no—
396C2×C42—2122no—
397C2×C42—2122no—
398C2×C42C4×D5314yes—
399#95—22no—
400#95—22no—
401#95—42no—
402#95—534no—
403#95—534no—
404#95—538no—
405#95—1064no—
406#95—1062no—
407#95—1062no—
408#95—2122no—
409#95—2122no—
410#95—2122no—
411#95C4×D5318yescomplemented
412C24—534no—
413C24—532no—
414C24—1062no—
415C24—1061no—
416C24—1061no—
417C24—1062no—
418C24—1061no—
419C24—1061no—
420C24—1062no—
421C24—1061no—
422C24—1061no—
423C24C23×D5311yescharacteristic
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—532no—
446C22×C4—534no—
447C22×C4—532no—
448C22×C4—534no—
449C22×C4—532no—
450C22×C4—534no—
451C22×C4—534no—
452C22×C4—1062no—
453C22×C4—1061no—
454C22×C4—1062no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1061no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1061no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1068no—
475C22×C4—2124no—
476C22×C4—2124no—
477C22×C4—2128no—
478C22×C4—2128no—
479C22×C4C2×C4×D5314yes—
480C22×C4C2×C4×D5312yes—
481C22×C4C2×C4×D5314yes—
482C22×C4C2×C4×D5312yes—
483C22×C4C2×C4×D5314yes—
484C22×C4C2×C4×D5312yes—
485C22×C4#1171994712yes—
486C22×C4#1171994912yes—
487C4⋊C4—2128no—
488C22⋊C4—5316no—
489C22⋊C4—1068no—
490C22⋊C4—2128no—
491C22⋊C4—2128no—
492C42—44no—
493C42—2124no—
494C42—2128no—
495C42—2128no—
496C23—534no—
497C23—1062no—
498C23—1062no—
499C23—1064no—
500C23—1064no—
501C23—1064no—
502C23—1062no—
503C23—1064no—
504C23—1062no—
505C23—1064no—
506C23—1062no—
507C23—1064no—
508C23—1062no—
509C23—2122no—
510C23—2122no—
511C23—2122no—
512C23—2122no—
513C23—2122no—
514C23—2121no—
515C23—2121no—
516C23—2122no—
517C23—2121no—
518C23—2121no—
519C23C22×C4×D5311yescharacteristic
520C23C22×C4×D5312yes—
521C23#41×D5312yes—
522C23C2×#1171994711yescharacteristic
523C23C2×#1171994911yescharacteristic
524C23#1313326212yes—
525C23#1313326311yescharacteristic
526C23#1313326312yes—
527C23#1313326311yescharacteristic
528C23#1313326411yescharacteristic
529C23#1313326411yescharacteristic
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—534no—
558C2×C4—534no—
559C2×C4—1062no—
560C2×C4—1064no—
561C2×C4—1062no—
562C2×C4—1064no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1064no—
566C2×C4—1064no—
567C2×C4—1062no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—1064no—
571C2×C4—1064no—
572C2×C4—1062no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—1062no—
576C2×C4—21216no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4C42×D5314yes—
588C2×C4#1313317514yes—
589C22—1064no—
590C22—1064no—
591C22—1064no—
592C22—2124no—
593C22—2122no—
594C22—2122no—
595C22—2122no—
596C22—2122no—
597C22—2124no—
598C22—2124no—
599C22—2124no—
600C22C2×C42×D5311yescharacteristic
601C22#117×D5311yescharacteristic
602C22C2×#1313317511yescharacteristic
603C22C4×#1171994712yes—
604C22C4×#1171994912yes—
605C22#1681109512yes—
606C22#1681109611yescharacteristic
607C22#1681109711yescharacteristic
608C22#1681109712yes—
609C22#1681110212yes—
610C22#1681110312yes—
611C22#1681110311yescharacteristic
612C22#1681110311yescharacteristic
613C22#1681121012yes—
614C22#1681121212yes—
615C22#1681121312yes—
616C22#1681121412yes—
617C22#1681122512yes—
618C22#1681122911yescharacteristic
619C22#1681123611yescharacteristic
620C22#1681124112yes—
621C22#1681124211yescharacteristic
622C22#1681124211yescharacteristic
623C4—24no—
624C4—24no—
625C4—44no—
626C4—42no—
627C4—44no—
628C4—1064no—
629C4—1064no—
630C4—2122no—
631C4—2124no—
632C4—2124no—
633C2—2124no—
634C2C4×#1313317511yescharacteristic
635C2#2027707411yescharacteristic
636C2#2027707711yescharacteristic
637C2#2027731211yescharacteristic
638C2#2027732511yescharacteristic
639C2#2027734611yescharacteristic
640C2#2027735511yescharacteristic
641C2#2027747912yes—
642C2#2027749312yes—
643C2#2027749812yes—
644C2#2027752212yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC22×C53C22×C42
FittingC53×#1434C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1434D53
hypocenterC53#13788
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #20272949C2semidirect
5. #20272945C2semidirect
6. #20272941C2semidirect
9. #95×Dic53C2semidirect
10. #95×Dic53C2semidirect
11. C53×#1434C2semidirect
21. #16810515C4semidirect
22. C4×#11719921C4semidirect
50. C2×C4⋊Dic53C2×C4semidirect
68. C2×C4×Dic53C2×C4semidirect
77. #13133124C2×C4semidirect
88. #95×C53C2×C4semidirect
168. #11719921C42semidirect
317. #1434D53semidirect
359. C53#13788semidirect
411. #95C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105152
C2×C4×#117199212
#202729512
#202729492
#202729452
#202729412
#202729362
#95×Dic532
C53×#14342
#1378853