Finite Group Databasebeta

← #27305732

Subgroups — #27305732

Order 13568.

1998 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810524C211yesmaximal, characteristic
2C2×#16810501C212yesmaximal
3#20272953C212yesmaximal, complemented
4#20272951C212yesmaximal, complemented
5#20272947C211yesmaximal, characteristic, complemented
6#20272940C212yesmaximal, complemented
7#20272936C212yesmaximal, complemented
8#20272936C212yesmaximal, complemented
9C53×#1436C211yesmaximal, characteristic, complemented
10C23×D212C2211yescharacteristic
11C23×C4×D53C2212yes—
12C22×#11719921C2212yes—
13C22×#11719921C2212yes—
14C22×#11719921C2212yes—
15C22×#11719921C2212yes—
16C22×#11719921C2212yes—
17C22×C4×Dic53C2211yescharacteristic
18C22×C4×Dic53C2211yescharacteristic
19#16810524C418yescomplemented
20#16810501C4116yescomplemented
21C2×#13133124C2212yes—
22C2×#13133124C2211yescharacteristic
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C22×C4⋊C4×C53C2211yescharacteristic
29C22×C42×C53C2211yescharacteristic
30C2×#95×C53C2211yescharacteristic
31C2×#95×C53C2212yes—
32C2×#95×C53C2212yes—
33C24×D53C2312yes—
34C22×D212—41no—
35C22×D212—41no—
36C22×D212—42no—
37C22×D212C2×C418yes—
38C22×D212C2×C414yes—
39C23×Dic53C2312yes—
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C22×C4×D53—42no—
43C22×C4×D53—42no—
44C22×C4×D53—42no—
45C22×C4×D53—42no—
46C22×C4×D53—42no—
47C22×C4×D53—42no—
48C22×C4×D53C2×C418yescomplemented
49C2×#11719921—42no—
50C2×#11719921—42no—
51C2×#11719921—42no—
52C2×#11719921—42no—
53C2×#11719921—42no—
54C2×#11719921—42no—
55C2×#11719921—42no—
56C2×#11719921—42no—
57C2×#11719921—42no—
58C2×#11719921—42no—
59C2×#11719921C2×C418yes—
60C2×#11719921C2×C418yes—
61C2×#11719921C2×C418yescomplemented
62C2×#11719921C2×C418yescomplemented
63C2×#11719921C2×C418yescomplemented
64C2×C4×Dic53—42no—
65C2×C4×Dic53—42no—
66C2×C4×Dic53—41no—
67C2×C4×Dic53C2×C414yescomplemented
68#13133124—42no—
69#13133124—42no—
70#13133124—42no—
71#13133124—42no—
72#13133124—42no—
73#13133124—41no—
74#13133124C2×C418yescomplemented
75C23×C4×C53C2311yescharacteristic
76C23×C4×C53C2312yes—
77C23×C4×C53C2311yescharacteristic
78C23×C4×C53C2312yes—
79C23×C4×C53C2311yescharacteristic
80C2×C4⋊C4×C53—41no—
81C2×C4⋊C4×C53—41no—
82C2×C4⋊C4×C53C2×C414yescomplemented
83C2×C42×C53—42no—
84C2×C42×C53—41no—
85#95×C53—42no—
86#95×C53—41no—
87#95×C53C2×C418yescomplemented
88C24×C53C2411yescharacteristic
89C23×D53—22no—
90C23×D53—22no—
91C23×D53—22no—
92C23×D53—22no—
93C23×D53—22no—
94C23×D53—22no—
95C23×D53—22no—
96C23×D53—22no—
97C23×D53—22no—
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—22no—
101C23×D53C22×C414yes—
102C23×D53C22×C414yes—
103C23×D53C22×C414yes—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
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×C53—22no—
127C22×C4×C53C22×C414yes—
128C22×C4×C53C22×C414yes—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C22×C414yes—
132C22×C4×C53C22×C412yes—
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—22no—
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×Dic53C22×C414yes—
152C22×Dic53C22×C414yes—
153C22×Dic53C22×C414yes—
154C2×D212—48no—
155C2×D212—48no—
156C2×D212—44no—
157C2×D212—44no—
158C2×D212C42116yescomplemented
159C2×C4×D53—48no—
160C2×C4×D53—48no—
161C2×C4×D53—48no—
162C2×C4×D53—48no—
163C2×C4×D53—48no—
164C2×C4×D53—48no—
165C2×C4×D53—48no—
166C4⋊C4×C53—44no—
167C42×C53—44no—
168#11719921—48no—
169#11719921—48no—
170#11719921—48no—
171#11719921—48no—
172#11719921—48no—
173C4×Dic53—44no—
174C4×Dic53—44no—
175C23×C53C23×C412yes—
176C23×C53C23×C411yescharacteristic
177C23×C53H(5,𝔽2)11yescharacteristic
178C23×C53H(5,𝔽2)11yescharacteristic
179C23×C53H(5,𝔽2)12yes—
180C23×C53H(5,𝔽2)12yes—
181C23×C53H(5,𝔽2)12yes—
182C23×C53H(5,𝔽2)12yes—
183C23×C532−1+412yes—
184C22×D53—24no—
185C22×D53—24no—
186C22×D53—24no—
187C22×D53—24no—
188C22×D53—24no—
189C22×D53—24no—
190C22×D53—24no—
191C22×D53—24no—
192C22×D53—24no—
193C22×D53—24no—
194C22×D53—24no—
195C22×D53—24no—
196C22×D53—24no—
197C22×D53—24no—
198C22×D53—24no—
199C22×D53—24no—
200C22×D53—24no—
201C22×D53—24no—
202C22×D53—42no—
203C22×D53—42no—
204C22×D53—42no—
205C22×D53—42no—
206C22×D53—42no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53C2×C4218yes—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—24no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—41no—
239C2×C4×C53—42no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—42no—
243C2×C4×C53—41no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53C2×C4214yes—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—24no—
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—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269D212—416no—
270C4×D53—432no—
271#13832—531nomaximal, Sylow
272C22×C53C22×C4211yescharacteristic
273C22×C53#51812yes—
274C22×C53#51812yes—
275C22×C53#51812yes—
276C22×C53#51812yes—
277C22×C53#51811yescharacteristic
278C22×C53#51812yes—
279C22×C53#51811yescharacteristic
280C22×C53#52012yes—
281C22×C53#52012yes—
282C22×C53#52012yes—
283C22×C53#56012yes—
284C22×C53#56011yescharacteristic
285C22×C53#56012yes—
286C22×C53#56011yescharacteristic
287C22×C53#56012yes—
288C22×C53H(3,𝔽4)11yescharacteristic
289C22×C53#56212yes—
290C22×C53#56212yes—
291C22×C53#56212yes—
292C22×C53#56311yescharacteristic
293C2×D53—28no—
294C2×D53—28no—
295C2×D53—28no—
296C2×D53—44no—
297C2×D53—44no—
298C2×D53—44no—
299C2×D53—44no—
300C2×D53—44no—
301C2×D53—44no—
302C2×D53—44no—
303C2×D53—44no—
304C2×D53—44no—
305C2×D53—44no—
306C2×D53—44no—
307C2×D53—44no—
308C4×C53—24no—
309C4×C53—44no—
310C4×C53—44no—
311C4×C53—42no—
312C4×C53—42no—
313Dic53—44no—
314Dic53—44no—
315Dic53—44no—
316C2×#390—531no—
317C2×#386—532no—
318#1437—532no—
319#1437—532no—
320#1436—531no—
321#1436D5311yescharacteristic, complemented
322#1434—532no—
323C22⋊#95—532no—
324C22⋊#95—532no—
325C2×C53#227311yescharacteristic
326C2×C53#227512yes—
327C2×C53#237512yes—
328C2×C53#237612yes—
329C2×C53#237712yes—
330C2×C53#237712yes—
331C2×C53#237712yes—
332C2×C53#237911yescharacteristic
333C2×C53#238011yescharacteristic
334D53—48no—
335C23×D4—531no—
336C24×C4—532no—
337C22×C4⋊C4C2×D5311yescharacteristic
338C22×C22⋊C4—532no—
339C22×C22⋊C4—532no—
340C22×C22⋊C4—532no—
341C22×C22⋊C4—532no—
342C22×C22⋊C4—532no—
343C22×C42—531no—
344C22×C42—531no—
345C22×C42C2×D5311yescharacteristic
346#390—538no—
347#386—5316no—
348C2×#95—532no—
349C2×#95—532no—
350C2×#95—531no—
351C2×#95—532no—
352C2×#95—532no—
353C2×#95—532no—
354C2×#95—532no—
355C2×#95C2×D5311yescharacteristic
356C2×#95C2×D5312yes—
357C2×#95C2×D5312yes—
358C53#1383211yescharacteristic, complemented, Sylow
359C25—532no—
360C22×D4—538no—
361C22×D4—534no—
362C22×D4—2121no—
363C22×D4—2121no—
364C22×D4—2122no—
365C23×C4—532no—
366C23×C4—532no—
367C23×C4—532no—
368C23×C4—538no—
369C23×C4—2122no—
370C23×C4—2122no—
371C23×C4—2122no—
372C23×C4—2122no—
373C23×C4—2122no—
374C23×C4—2122no—
375C23×C4C22×D5311yescharacteristic
376C23×C4C22×D5311yescharacteristic
377C23×C4C22×D5312yes—
378C23×C4C22×D5312yes—
379C23×C4C22×D5311yescharacteristic
380C2×C4⋊C4—41no—
381C2×C4⋊C4—41no—
382C2×C4⋊C4C4×D5314yescomplemented
383C2×C22⋊C4—538no—
384C2×C22⋊C4—538no—
385C2×C22⋊C4—538no—
386C2×C22⋊C4—538no—
387C2×C22⋊C4—538no—
388C2×C22⋊C4—2122no—
389C2×C22⋊C4—2122no—
390C2×C22⋊C4—2122no—
391C2×C22⋊C4—2122no—
392C2×C22⋊C4—2122no—
393C2×C22⋊C4—2122no—
394C2×C22⋊C4—2122no—
395C2×C22⋊C4—2122no—
396C2×C22⋊C4—2122no—
397C2×C22⋊C4—2122no—
398C2×C42—42no—
399C2×C42—41no—
400C2×C42—534no—
401C2×C42—2122no—
402C2×C42—2121no—
403C2×C42—2122no—
404#95—42no—
405#95—41no—
406#95—538no—
407#95—2121no—
408#95—2122no—
409#95—2122no—
410#95—2122no—
411#95—2122no—
412#95—2122no—
413#95C4×D5318yescomplemented
414C24—534no—
415C24—534no—
416C24—534no—
417C24—1062no—
418C24—1062no—
419C24—1062no—
420C24—1062no—
421C24—1062no—
422C24—1062no—
423C24—1062no—
424C24—1062no—
425C24—1062no—
426C24—1062no—
427C24—1062no—
428C24—1062no—
429C24C23×D5311yescharacteristic
430C2×D4—5316no—
431C2×D4—2128no—
432C2×D4—2128no—
433C2×D4—2124no—
434C2×D4—2124no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—534no—
459C22×C4—534no—
460C22×C4—534no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
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—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—2128no—
480C22×C4—2128no—
481C22×C4—2128no—
482C22×C4—2128no—
483C22×C4—2128no—
484C22×C4—2128no—
485C22×C4—2128no—
486C22×C4C2×C4×D5314yes—
487C22×C4C2×C4×D5314yes—
488C22×C4C2×C4×D5312yes—
489C22×C4C2×C4×D5312yes—
490C22×C4C2×C4×D5314yes—
491C22×C4C2×C4×D5312yes—
492C4⋊C4—44no—
493C22⋊C4—2128no—
494C22⋊C4—2128no—
495C22⋊C4—2128no—
496C22⋊C4—2128no—
497C22⋊C4—2128no—
498C42—44no—
499C42—2124no—
500C42—2124no—
501C23—538no—
502C23—1064no—
503C23—1064no—
504C23—1064no—
505C23—1064no—
506C23—1064no—
507C23—1064no—
508C23—1064no—
509C23—1064no—
510C23—1064no—
511C23—1064no—
512C23—1064no—
513C23—1064no—
514C23—1064no—
515C23—1064no—
516C23—1064no—
517C23—1064no—
518C23—1064no—
519C23—1064no—
520C23—2122no—
521C23—2122no—
522C23—2122no—
523C23—2122no—
524C23—2122no—
525C23—2122no—
526C23—2122no—
527C23—2122no—
528C23—2122no—
529C23—2122no—
530C23—2122no—
531C23—2122no—
532C23—2122no—
533C23—2122no—
534C23—2122no—
535C23—2122no—
536C23C22×C4×D5312yes—
537C23C22×C4×D5311yescharacteristic
538C23#1313326211yescharacteristic
539C23#1313326211yescharacteristic
540C23#1313326212yes—
541C23#1313326312yes—
542C23#1313326312yes—
543C23#1313326312yes—
544C23#1313326512yes—
545D4—21216no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—1064no—
581C2×C4—1064no—
582C2×C4—1064no—
583C2×C4—21232no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4C42×D5314yes—
597C22—1068no—
598C22—1068no—
599C22—1068no—
600C22—2124no—
601C22—2124no—
602C22—2124no—
603C22—2124no—
604C22—2124no—
605C22—2124no—
606C22—2124no—
607C22—2124no—
608C22—2124no—
609C22—2124no—
610C22—2124no—
611C22—2124no—
612C22C2×C42×D5311yescharacteristic
613C22#1681109512yes—
614C22#1681109512yes—
615C22#1681109512yes—
616C22#1681109512yes—
617C22#1681109611yescharacteristic
618C22#1681109711yescharacteristic
619C22#1681109712yes—
620C22#1681110412yes—
621C22#1681110512yes—
622C22#1681110512yes—
623C22#1681136112yes—
624C22#1681136211yescharacteristic
625C22#1681136312yes—
626C22#1681136311yescharacteristic
627C22#1681136412yes—
628C22#1681136511yescharacteristic
629C22#1681136912yes—
630C22#1681136912yes—
631C22#1681137112yes—
632C22#1681137211yescharacteristic
633C4—24no—
634C4—44no—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—2124no—
639C4—2124no—
640C4—2124no—
641C2—2128no—
642C2#2027706711yescharacteristic
643C2#2027707512yes—
644C2#2027785512yes—
645C2#2027786012yes—
646C2#2027787012yes—
647C2#2027787012yes—
648C2#2027787112yes—
649C2#2027788711yescharacteristic
650C2#2027789011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC22×C53C22×C42
FittingC53×#1436C2
socle, cosocleC24×C53C24
prime radicalC23×D212C22
hypercenter#1436D53
hypocenterC53#13832
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
3. #20272953C2semidirect
4. #20272951C2semidirect
5. #20272947C2semidirect
6. #20272940C2semidirect
7. #20272936C2semidirect
8. #20272936C2semidirect
9. C53×#1436C2semidirect
19. #16810524C4semidirect
20. #16810501C4semidirect
48. C22×C4×D53C2×C4semidirect
61. C2×#11719921C2×C4semidirect
62. C2×#11719921C2×C4semidirect
63. C2×#11719921C2×C4semidirect
67. C2×C4×Dic53C2×C4semidirect
74. #13133124C2×C4semidirect
82. C2×C4⋊C4×C53C2×C4semidirect
87. #95×C53C2×C4semidirect
158. C2×D212C42semidirect
321. #1436D53semidirect
358. C53#13832semidirect
382. C2×C4⋊C4C4×D53semidirect
413. #95C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105242
C2×#168105012
#202729532
#202729512
#202729472
#202729402
#202729362
C53×#14362
#1383253