Finite Group Databasebeta

← #27306303

Subgroups — #27306303

Order 13568.

1818 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810490C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Dic53C211yesmaximal, characteristic
3C22×#13133124C211yesmaximal, characteristic
4#20272976C212yesmaximal, complemented
5#20272975C212yesmaximal, complemented
6#20272941C211yesmaximal, characteristic, complemented
7#20272940C211yesmaximal, characteristic, complemented
8#20272939C211yesmaximal, characteristic, complemented
9#20272937C212yesmaximal
10#20272937C212yesmaximal
11C53×C22⋊#95C211yesmaximal, characteristic
12C24×Dic53C2212yes—
13C22×C22⋊Dic53C2211yescharacteristic
14C22×C22⋊Dic53C2211yescharacteristic
15C22×C4×Dic53C2211yescharacteristic
16C22×C4×Dic53C2211yescharacteristic
17#16810490—24no—
18C4×C22⋊Dic53C418yescomplemented
19C2×#13133124—24no—
20C2×#13133124C2212yes—
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2211yescharacteristic
33C24×C4×C53C2211yescharacteristic
34C22×C22⋊C4×C53C2212yes—
35C2×#95×C53C2211yescharacteristic
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C23×Dic53—24no—
39C23×Dic53—24no—
40C23×Dic53—24no—
41C23×Dic53—24no—
42C23×Dic53—24no—
43C23×Dic53—42no—
44C23×Dic53—42no—
45C23×Dic53C2311yescharacteristic
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2312yes—
50C23×Dic53C2312yes—
51C2×C22⋊Dic53—24no—
52C2×C22⋊Dic53—22no—
53C2×C22⋊Dic53—22no—
54C2×C22⋊Dic53C2×C414yes—
55C2×C22⋊Dic53C2×C414yes—
56C2×C4×Dic53—22no—
57C2×C4×Dic53—22no—
58C2×C4×Dic53—42no—
59C2×C4×Dic53C2×C414yescomplemented
60C2×C4×Dic53C2×C414yescomplemented
61#13133124—22no—
62#13133124—24no—
63#13133124—24no—
64#13133124—24no—
65#13133124—24no—
66#13133124—24no—
67#13133124—42no—
68#13133124—42no—
69#13133124—42no—
70#13133124—42no—
71#13133124C2×C414yescomplemented
72#13133124C2×C414yescomplemented
73#13133124D414yes—
74C25×C53C2311yescharacteristic
75C23×C4×C53—24no—
76C23×C4×C53—22no—
77C23×C4×C53—22no—
78C23×C4×C53—42no—
79C23×C4×C53C2×C414yes—
80C23×C4×C53C2312yes—
81C23×C4×C53C2312yes—
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2311yescharacteristic
84C2×C22⋊C4×C53—24no—
85C2×C22⋊C4×C53—24no—
86C2×C22⋊C4×C53—24no—
87#95×C53—22no—
88#95×C53—22no—
89#95×C53—42no—
90C24×C53—21no—
91C24×C53—21no—
92C24×C53—22no—
93C24×C53—22no—
94C24×C53—21no—
95C24×C53—21no—
96C24×C53C22×C412yes—
97C24×C53#4114yes—
98C24×C53#4112yes—
99C24×C53#4114yes—
100C24×C53#4112yes—
101C24×C53C2411yescharacteristic
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—28no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—28no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
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—44no—
125C22×C4×C53—48no—
126C22×C4×C53—48no—
127C22×C4×C53C22×C412yes—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53#4112yes—
131C22×Dic53—216no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—21no—
138C22×Dic53—21no—
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—21no—
153C22×Dic53—21no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—48no—
163C22×Dic53—48no—
164C22×Dic53—48no—
165C22×Dic53—48no—
166C22×Dic53—48no—
167C22×Dic53—48no—
168C22×Dic53C22×C412yes—
169C22×Dic53C22×C412yes—
170C22×Dic53C22×C412yes—
171C22×Dic53C22×C412yes—
172C22×Dic53C2×D414yes—
173C22×Dic53#4114yes—
174C22⋊C4×C53—48no—
175C22⋊Dic53—28no—
176C22⋊Dic53—28no—
177C4×Dic53—28no—
178C4×Dic53—44no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—22no—
183C23×C53—24no—
184C23×C53—24no—
185C23×C53—24no—
186C23×C53—24no—
187C23×C53—22no—
188C23×C53—24no—
189C23×C53—22no—
190C23×C53—24no—
191C23×C53—24no—
192C23×C53—24no—
193C23×C53—24no—
194C23×C53—22no—
195C23×C53—24no—
196C23×C53—22no—
197C23×C53#11714yes—
198C23×C53#11714yes—
199C23×C53#11714yes—
200C23×C53#12618yes—
201C23×C53#12618yes—
202C23×C53C23×C411yescharacteristic
203C23×C53C22×D411yescharacteristic
204C23×C53C2×#4112yes—
205C23×C53C2×#4112yes—
206C23×C53C2×#4111yescharacteristic
207C23×C53C2×#4111yescharacteristic
208C23×C53C2×#4112yes—
209C23×C53C2×#4111yescharacteristic
210C23×C53H(5,𝔽2)12yes—
211C23×C53H(5,𝔽2)11yescharacteristic
212C23×C532−1+411yescharacteristic
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—416no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
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×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53C4×D414yes—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—432no—
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×Dic53—41no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—41no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—41no—
273C2×Dic53—42no—
274C2×Dic53—41no—
275#13901—531nomaximal, Sylow
276C22×C53—24no—
277C22×C53—24no—
278C22×C53—24no—
279C22×C53—24no—
280C22×C53—24no—
281C22×C53—28no—
282C22×C53—28no—
283C22×C53—28no—
284C22×C53—28no—
285C22×C53—24no—
286C22×C53#38318yescomplemented
287C22×C53C2×#11711yescharacteristic
288C22×C53C2×#11711yescharacteristic
289C22×C53C2×#11711yescharacteristic
290C22×C53C2×C4×D411yescharacteristic
291C22×C53C4×#4111yescharacteristic
292C22×C53#51811yescharacteristic
293C22×C53#52011yescharacteristic
294C22×C53#52512yes—
295C22×C53C2×#12612yes—
296C22×C53C2×#12612yes—
297C22×C53#52912yes—
298C22×C53#53812yes—
299C22×C53#53912yes—
300C22×C53#54611yescharacteristic
301C22×C53#54612yes—
302C22×C53#54811yescharacteristic
303C22×C53#55112yes—
304C22×C53#55112yes—
305C22×C53#55112yes—
306C22×C53#55212yes—
307C22×C53#55211yescharacteristic
308C22×C53#55312yes—
309C22×C53#55311yescharacteristic
310C4×C53—24no—
311C4×C53—42no—
312C4×C53—82no—
313C4×C53—82no—
314Dic53—42no—
315Dic53—42no—
316Dic53—42no—
317Dic53—44no—
318Dic53—44no—
319Dic53—42no—
320C2×#383—531no—
321C2×C4×C22⋊C4—531no—
322C22×#95—531no—
323#1443—532no—
324#1443—532no—
325#1434—531no—
326#1434—531no—
327#1434—531no—
328C22⋊#95—532no—
329C22⋊#95—532no—
330C22⋊#95D5311yescharacteristic
331C2×C53—28no—
332C2×C53C2×#38311yescharacteristic
333C2×C53#229011yescharacteristic
334C2×C53#231811yescharacteristic
335C2×C53#234211yescharacteristic
336C2×C53#234311yescharacteristic
337C2×C53#235211yescharacteristic
338C2×C53#235311yescharacteristic
339C2×C53#248012yes—
340C2×C53#248112yes—
341C2×C53#249312yes—
342C2×C53#251012yes—
343C24×C4—532no—
344C24×C4C2×D5311yescharacteristic
345C22×C22⋊C4—531no—
346C22×C22⋊C4—531no—
347C22×C22⋊C4C2×D5312yes—
348C22×C42—531no—
349C22×C42—531no—
350#383—1064no—
351C4×C22⋊C4—538no—
352C2×#95—531no—
353C2×#95—531no—
354C2×#95—531no—
355C2×#95—532no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—531no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—532no—
364C2×#95—532no—
365C2×#95—1064no—
366C2×#95C2×D5311yescharacteristic
367C2×#95C2×D5311yescharacteristic
368C2×#95C2×D5312yes—
369C53#1390111yescharacteristic, complemented, Sylow
370C25C22×D5311yescharacteristic
371C23×C4—24no—
372C23×C4—22no—
373C23×C4—22no—
374C23×C4—42no—
375C23×C4—531no—
376C23×C4—531no—
377C23×C4—531no—
378C23×C4—531no—
379C23×C4—532no—
380C23×C4—532no—
381C23×C4—1064no—
382C23×C4—1064no—
383C23×C4—1064no—
384C23×C4—1064no—
385C23×C4—1064no—
386C23×C4—2122no—
387C23×C4—2122no—
388C23×C4C4×D5314yes—
389C23×C4C22×D5312yes—
390C23×C4C22×D5312yes—
391C23×C4C22×D5311yescharacteristic
392C23×C4C22×D5311yescharacteristic
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—534no—
397C2×C22⋊C4—534no—
398C2×C22⋊C4—1064no—
399C2×C22⋊C4—1062no—
400C2×C22⋊C4—1062no—
401C2×C42—534no—
402C2×C42—534no—
403C2×C42—1062no—
404C2×C42—1062no—
405C2×C42—2122no—
406#95—22no—
407#95—22no—
408#95—42no—
409#95—534no—
410#95—534no—
411#95—534no—
412#95—1064no—
413#95—1064no—
414#95—1064no—
415#95—1064no—
416#95—1064no—
417#95—1062no—
418#95—2122no—
419#95—2122no—
420#95—2122no—
421#95—2122no—
422C24—21no—
423C24—22no—
424C24—21no—
425C24—22no—
426C24—21no—
427C24—21no—
428C24C2×C4×D5312yes—
429C24#1171994514yes—
430C24#1171994714yes—
431C24#1171994712yes—
432C24#1171994912yes—
433C24C23×D5311yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—28no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—532no—
460C22×C4—532no—
461C22×C4—532no—
462C22×C4—532no—
463C22×C4—534no—
464C22×C4—534no—
465C22×C4—10616no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—2128no—
497C22×C4—2128no—
498C22×C4—2128no—
499C22×C4—2128no—
500C22×C4—2128no—
501C22×C4—2128no—
502C22×C4C2×C4×D5312yes—
503C22×C4C2×C4×D5312yes—
504C22×C4D4×D5312yes—
505C22×C4#1171994712yes—
506C22⋊C4—48no—
507C22⋊C4—1068no—
508C22⋊C4—1068no—
509C42—1068no—
510C42—2124no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—24no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23#1313317314yes—
530C23#1313317414yes—
531C23#1313317514yes—
532C23#1313321018yes—
533C23#1313321218yes—
534C23C22×C4×D5311yescharacteristic
535C23C2×D4×D5311yescharacteristic
536C23#41×D5312yes—
537C23C2×#1171994711yescharacteristic
538C23C2×#1171994712yes—
539C23C2×#1171994711yescharacteristic
540C23C2×#1171994512yes—
541C23C2×#1171994911yescharacteristic
542C23#1313326212yes—
543C23#1313326311yescharacteristic
544C23#1313326411yescharacteristic
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—416no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—1064no—
570C2×C4—1064no—
571C2×C4—1064no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1062no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—1064no—
585C2×C4—1064no—
586C2×C4—1064no—
587C2×C4—21232no—
588C2×C4—2121no—
589C2×C4—2122no—
590C2×C4—2121no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2121no—
595C2×C4—2122no—
596C2×C4—2121no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4#1313317714yes—
607C22—24no—
608C22—28no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—28no—
615C22—28no—
616C22—28no—
617C22#1681049018yescomplemented
618C22C2×#1313317311yescharacteristic
619C22C2×#1313317411yescharacteristic
620C22C2×#1313317511yescharacteristic
621C22C2×#1313317711yescharacteristic
622C22#1681109111yescharacteristic
623C22#1681109611yescharacteristic
624C22#1681110311yescharacteristic
625C22#1681113212yes—
626C22C2×#1313321012yes—
627C22C2×#1313321212yes—
628C22#1681115312yes—
629C22#1681120612yes—
630C22#1681121512yes—
631C22#1681126012yes—
632C22#1681126411yescharacteristic
633C22#1681127911yescharacteristic
634C22#1681129612yes—
635C22#1681130012yes—
636C22#1681130412yes—
637C22#1681131012yes—
638C22#1681131511yescharacteristic
639C22#1681131612yes—
640C22#1681132511yescharacteristic
641C4—24no—
642C4—42no—
643C4—82no—
644C4—82no—
645C4—2122no—
646C4—2122no—
647C4—2122no—
648C4—2122no—
649C4—2124no—
650C4—2124no—
651C2—28no—
652C2C2×#1681049011yescharacteristic
653C2#2027715411yescharacteristic
654C2#2027734211yescharacteristic
655C2#2027756611yescharacteristic
656C2#2027757711yescharacteristic
657C2#2027764611yescharacteristic
658C2#2027765711yescharacteristic
659C2#2027896312yes—
660C2#2027897812yes—
661C2#2027910212yes—
662C2#2027927712yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C22⋊#95C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC22⋊#95D53
hypocenterC53#13901
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
1. C2×#16810490C2semidirect
4. #20272976C2semidirect
5. #20272975C2semidirect
6. #20272941C2semidirect
7. #20272940C2semidirect
8. #20272939C2semidirect
18. C4×C22⋊Dic53C4semidirect
59. C2×C4×Dic53C2×C4semidirect
60. C2×C4×Dic53C2×C4semidirect
71. #13133124C2×C4semidirect
72. #13133124C2×C4semidirect
286. C22×C53#383semidirect
369. C53#13901semidirect
617. C22#16810490semidirect

Maximal subgroups

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

subgroupindex
C2×#168104902
C2×C4×C22⋊Dic532
C22×#131331242
#202729762
#202729752
#202729412
#202729402
#202729392
#202729372
C53×C22⋊#952
#1390153