Finite Group Databasebeta

← #27306015

Subgroups — #27306015

Order 13568.

1918 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810484C211yesmaximal, characteristic
2C2×#16810484C211yesmaximal, characteristic
3C2×#16810481C211yesmaximal, characteristic
4C22×#13133124C211yesmaximal, characteristic
5#20272975C212yesmaximal, complemented
6#20272970C212yesmaximal, complemented
7#20272948C212yesmaximal, complemented
8#20272942C212yesmaximal, complemented
9#20272938C211yesmaximal, characteristic
10#20272937C211yesmaximal, characteristic
11C2×C4×C22⋊C4×C53C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C24×Dic53C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×C22⋊Dic53C2211yescharacteristic
16C22×C22⋊Dic53C2211yescharacteristic
17#16810484—24no—
18#16810484—24no—
19#16810481—24no—
20C2×#13133124C418yescomplemented
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2211yescharacteristic
34C24×C4×C53C2211yescharacteristic
35C22×C22⋊C4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2211yescharacteristic
37C22×C42×C53C2212yes—
38C4×C22⋊C4×C53—24no—
39C2×#95×C53C2212yes—
40C23×Dic53—22no—
41C23×Dic53—22no—
42C23×Dic53—22no—
43C23×Dic53—22no—
44C23×Dic53—22no—
45C23×Dic53—22no—
46C23×Dic53—22no—
47C23×Dic53—22no—
48C23×Dic53—24no—
49C23×Dic53—42no—
50C23×Dic53C2×C414yes—
51C23×Dic53C2×C414yes—
52C23×Dic53C2312yes—
53C23×Dic53C2312yes—
54C23×Dic53C2311yescharacteristic
55C23×Dic53C2311yescharacteristic
56C23×Dic53C2311yescharacteristic
57C23×Dic53C2311yescharacteristic
58C2×#11719919—24no—
59C2×#11719919—24no—
60C2×#11719919—42no—
61C2×C22⋊Dic53—22no—
62C2×C22⋊Dic53—22no—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53—22no—
66C2×C22⋊Dic53—22no—
67#13133124—24no—
68#13133124—24no—
69#13133124—24no—
70#13133124—24no—
71#13133124—24no—
72#13133124—24no—
73#13133124—24no—
74#13133124—42no—
75#13133124—42no—
76#13133124C2×C414yes—
77#13133124C2×C414yes—
78#13133124C2×C414yes—
79#13133124C2×C414yes—
80C25×C53C2311yescharacteristic
81C23×C4×C53—22no—
82C23×C4×C53—22no—
83C23×C4×C53—24no—
84C23×C4×C53—42no—
85C23×C4×C53C2×C414yes—
86C23×C4×C53C2311yescharacteristic
87C23×C4×C53C2312yes—
88C23×C4×C53C2312yes—
89C23×C4×C53C2311yescharacteristic
90C2×C22⋊C4×C53—22no—
91C2×C22⋊C4×C53—22no—
92C2×C22⋊C4×C53—22no—
93C2×C22⋊C4×C53—22no—
94C2×C22⋊C4×C53D414yes—
95C2×C22⋊C4×C53D414yes—
96C2×C42×C53—24no—
97C2×C42×C53—24no—
98C2×C42×C53—42no—
99#95×C53—24no—
100C24×C53—22no—
101C24×C53—22no—
102C24×C53—22no—
103C24×C53—22no—
104C24×C53C22×C412yes—
105C24×C53C2×D412yes—
106C24×C53C2×D412yes—
107C24×C53#4112yes—
108C24×C53#4112yes—
109C24×C53#4112yes—
110C24×C53#4112yes—
111C24×C53C2411yescharacteristic
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—28no—
116C22×C4×C53—21no—
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—21no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—28no—
133C22×C4×C53—44no—
134C22×C4×C53—48no—
135C22×C4×C53—48no—
136C22×C4×C53C22×C412yes—
137C22×C4×C53C22×C412yes—
138C22×C4×C53C2×D414yes—
139C22×C4×C53C2×D414yes—
140C22×C4×C53#4114yes—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—22no—
144C22×Dic53—28no—
145C22×Dic53—28no—
146C22×Dic53—28no—
147C22×Dic53—28no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—22no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—22no—
164C22×Dic53—28no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—22no—
171C22×Dic53—22no—
172C22×Dic53—22no—
173C22×Dic53—22no—
174C22×Dic53—22no—
175C22×Dic53—22no—
176C22×Dic53—22no—
177C22×Dic53—22no—
178C22×Dic53—22no—
179C22×Dic53—44no—
180C22×Dic53—48no—
181C22×Dic53—48no—
182C22×Dic53—44no—
183C22×Dic53—44no—
184C22×Dic53—44no—
185C22×Dic53—44no—
186C22×Dic53C22×C412yes—
187C22×Dic53C22×C412yes—
188C22×Dic53C22×C412yes—
189C22×Dic53C22×C412yes—
190C22×Dic53#4114yes—
191C22⋊C4×C53—28no—
192C22⋊C4×C53—28no—
193C42×C53—48no—
194C22⋊Dic53—44no—
195C22⋊Dic53—44no—
196#11719919—48no—
197C23×C53—22no—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53—24no—
202C23×C53—24no—
203C23×C53—24no—
204C23×C53—24no—
205C23×C53—24no—
206C23×C53—24no—
207C23×C53—24no—
208C23×C53—24no—
209C23×C53—24no—
210C23×C53—24no—
211C23×C53—24no—
212C23×C53—24no—
213C23×C53#11714yes—
214C23×C53C4×D414yes—
215C23×C53C4×D414yes—
216C23×C53#12114yes—
217C23×C53#12314yes—
218C23×C53#12414yes—
219C23×C53#12614yes—
220C23×C53C23×C411yescharacteristic
221C23×C53C22×D411yescharacteristic
222C23×C53C22×D411yescharacteristic
223C23×C53C2×#4112yes—
224C23×C53C2×#4112yes—
225C23×C53C2×#4111yescharacteristic
226C23×C53C2×#4111yescharacteristic
227C23×C53C2×#4111yescharacteristic
228C23×C53C2×#4111yescharacteristic
229C23×C53H(5,𝔽2)12yes—
230C23×C532−1+412yes—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—24no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—24no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—416no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53#11714yes—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—416no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—416no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53—42no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287C2×Dic53—42no—
288C2×Dic53—42no—
289C2×Dic53—42no—
290C2×Dic53—42no—
291C2×Dic53—42no—
292C2×Dic53—42no—
293C2×Dic53—42no—
294C2×Dic53—42no—
295C2×Dic53#11714yes—
296#13874—531nomaximal, Sylow
297C22×C53—24no—
298C22×C53—24no—
299C22×C53—24no—
300C22×C53—24no—
301C22×C53—24no—
302C22×C53—28no—
303C22×C53—28no—
304C22×C53—28no—
305C22×C53—28no—
306C22×C53—24no—
307C22×C53#38818yes—
308C22×C53C2×#11711yescharacteristic
309C22×C53C2×#11711yescharacteristic
310C22×C53C2×#11711yescharacteristic
311C22×C53C2×C4×D411yescharacteristic
312C22×C53C2×C4×D411yescharacteristic
313C22×C53#51811yescharacteristic
314C22×C53#51911yescharacteristic
315C22×C53C2×#12111yescharacteristic
316C22×C53C2×#12311yescharacteristic
317C22×C53C2×#12411yescharacteristic
318C22×C53C2×#12611yescharacteristic
319C22×C53#52912yes—
320C22×C53#52912yes—
321C22×C53#53912yes—
322C22×C53#53912yes—
323C22×C53#54612yes—
324C22×C53#54612yes—
325C22×C53#54712yes—
326C22×C53#54712yes—
327C22×C53#55112yes—
328C22×C53#55112yes—
329C22×C53#55212yes—
330C22×C53#55212yes—
331C4×C53—24no—
332C4×C53—42no—
333C4×C53—44no—
334C4×C53—44no—
335Dic53—24no—
336Dic53—42no—
337Dic53—42no—
338Dic53—42no—
339Dic53—82no—
340Dic53—82no—
341C2×#381—531no—
342C2×#381—531no—
343C2×#380—531no—
344C2×C4×C22⋊C4D5311yescharacteristic
345C22×#95—531no—
346#1443—532no—
347#1442—532no—
348#1436—532no—
349#1434—532no—
350C22⋊#95—531no—
351C22⋊#95—531no—
352C2×C53—28no—
353C2×C53#228311yescharacteristic
354C2×C53C2×#38811yescharacteristic
355C2×C53#231311yescharacteristic
356C2×C53#234211yescharacteristic
357C2×C53#234211yescharacteristic
358C2×C53#234411yescharacteristic
359C2×C53#234411yescharacteristic
360C2×C53#241912yes—
361C2×C53#242512yes—
362C2×C53#248812yes—
363C2×C53#249312yes—
364C24×C4—531no—
365C24×C4—531no—
366C24×C4C2×D5311yescharacteristic
367C22×C4⋊C4—532no—
368C22×C22⋊C4—531no—
369C22×C22⋊C4—531no—
370C22×C22⋊C4C2×D5311yescharacteristic
371C22×C22⋊C4C2×D5311yescharacteristic
372C22×C42C2×D5312yes—
373#381—1064no—
374#381—1064no—
375#380—1064no—
376C4×C22⋊C4—24no—
377C2×#95—538no—
378C2×#95—531no—
379C2×#95—531no—
380C2×#95—531no—
381C2×#95—531no—
382C2×#95—532no—
383C2×#95—532no—
384C2×#95—532no—
385C2×#95—532no—
386C2×#95—532no—
387C2×#95—532no—
388C2×#95—532no—
389C2×#95—532no—
390C2×#95—532no—
391C2×#95C2×D5312yes—
392C53#1387411yescharacteristic, complemented, Sylow
393C25C22×D5311yescharacteristic
394C23×C4—22no—
395C23×C4—22no—
396C23×C4—24no—
397C23×C4—42no—
398C23×C4—531no—
399C23×C4—534no—
400C23×C4—531no—
401C23×C4—531no—
402C23×C4—531no—
403C23×C4—534no—
404C23×C4—532no—
405C23×C4—532no—
406C23×C4—1064no—
407C23×C4—1062no—
408C23×C4—1062no—
409C23×C4—1062no—
410C23×C4—1062no—
411C23×C4—1062no—
412C23×C4—1062no—
413C23×C4—1062no—
414C23×C4—1062no—
415C23×C4—2122no—
416C23×C4C4×D5314yes—
417C23×C4C22×D5311yescharacteristic
418C23×C4C22×D5312yes—
419C23×C4C22×D5312yes—
420C23×C4C22×D5311yescharacteristic
421C2×C4⋊C4—1064no—
422C2×C4⋊C4—1064no—
423C2×C4⋊C4—2122no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—1062no—
429C2×C22⋊C4—1062no—
430C2×C22⋊C4—1062no—
431C2×C22⋊C4—1062no—
432C2×C22⋊C4—1062no—
433C2×C22⋊C4—1062no—
434C2×C22⋊C4D21214yes—
435C2×C22⋊C4C22⋊D5314yes—
436C2×C42—24no—
437C2×C42—24no—
438C2×C42—42no—
439#95—24no—
440#95—534no—
441#95—534no—
442#95—534no—
443#95—534no—
444#95—1064no—
445#95—1064no—
446#95—1064no—
447#95—1064no—
448#95—1064no—
449#95—1064no—
450#95—1064no—
451#95—2122no—
452#95—2122no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24—22no—
457C24C2×C4×D5312yes—
458C24C2×D21212yes—
459C24#1171994512yes—
460C24#1171994512yes—
461C24#1171994512yes—
462C24#1171994512yes—
463C24C2×C22⋊D5312yes—
464C24C23×D5311yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—28no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—28no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—532no—
490C22×C4—534no—
491C22×C4—532no—
492C22×C4—532no—
493C22×C4—532no—
494C22×C4—1068no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1062no—
498C22×C4—1068no—
499C22×C4—1068no—
500C22×C4—1068no—
501C22×C4—1068no—
502C22×C4—1061no—
503C22×C4—1061no—
504C22×C4—1061no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1062no—
508C22×C4—1061no—
509C22×C4—1061no—
510C22×C4—1061no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—1061no—
514C22×C4—1061no—
515C22×C4—1061no—
516C22×C4—1061no—
517C22×C4—1062no—
518C22×C4—1062no—
519C22×C4—1062no—
520C22×C4—1062no—
521C22×C4—1062no—
522C22×C4—1062no—
523C22×C4—1062no—
524C22×C4—1062no—
525C22×C4—1062no—
526C22×C4—1062no—
527C22×C4—1062no—
528C22×C4—1062no—
529C22×C4—1062no—
530C22×C4—1062no—
531C22×C4—1062no—
532C22×C4—2124no—
533C22×C4—2128no—
534C22×C4—2128no—
535C22×C4—2124no—
536C22×C4—2124no—
537C22×C4—2124no—
538C22×C4—2124no—
539C22×C4C2×C4×D5312yes—
540C22×C4C2×C4×D5312yes—
541C22×C4C2×D21214yes—
542C22×C4#1171994714yes—
543C22×C4C2×C22⋊D5314yes—
544C4⋊C4—2128no—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—2124no—
548C22⋊C4—2124no—
549C42—48no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23#1313317414yes—
567C23C4×C22⋊D5314yes—
568C23C4×D21214yes—
569C23#1313319114yes—
570C23#1313320014yes—
571C23#1313320514yes—
572C23#1313321114yes—
573C23C22×C4×D5311yescharacteristic
574C23C22×C22⋊D5311yescharacteristic
575C23C22×D21211yescharacteristic
576C23#41×D5312yes—
577C23C2×#1171994712yes—
578C23C2×#1171994511yescharacteristic
579C23C2×#1171994511yescharacteristic
580C23C2×#1171994511yescharacteristic
581C23C2×#1171994511yescharacteristic
582C23#1313326212yes—
583C23#1313326412yes—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—416no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—534no—
609C2×C4—1064no—
610C2×C4—1064no—
611C2×C4—1062no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—1062no—
615C2×C4—1062no—
616C2×C4—1062no—
617C2×C4—1062no—
618C2×C4—1062no—
619C2×C4—1062no—
620C2×C4—1062no—
621C2×C4—1062no—
622C2×C4—21216no—
623C2×C4—2122no—
624C2×C4—2122no—
625C2×C4—2122no—
626C2×C4—2122no—
627C2×C4—2122no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4—2122no—
633C2×C4—2122no—
634C2×C4—2122no—
635C2×C4—2122no—
636C2×C4—2122no—
637C2×C4—2122no—
638C2×C4—21216no—
639C2×C4—2122no—
640C2×C4—2121no—
641C2×C4—2121no—
642C2×C4—2121no—
643C2×C4—2121no—
644C2×C4—2121no—
645C2×C4—2121no—
646C2×C4—2121no—
647C2×C4—2121no—
648C2×C4#1313317314yes—
649C22—28no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—28no—
657C22—28no—
658C22—28no—
659C22#1681051418yes—
660C22#117×D5311yescharacteristic
661C22C2×#1313317311yescharacteristic
662C22C2×#1313317411yescharacteristic
663C22C2×C4×C22⋊D5311yescharacteristic
664C22C2×C4×D21211yescharacteristic
665C22#1681109611yescharacteristic
666C22#1681110011yescharacteristic
667C22C2×#1313319111yescharacteristic
668C22C2×#1313320011yescharacteristic
669C22C2×#1313320511yescharacteristic
670C22C2×#1313321111yescharacteristic
671C22#1681115312yes—
672C22#1681115312yes—
673C22#1681121512yes—
674C22#1681121512yes—
675C22#1681126712yes—
676C22#1681126812yes—
677C22#1681127212yes—
678C22#1681127712yes—
679C22#1681129912yes—
680C22#1681130212yes—
681C22#1681130612yes—
682C22#1681131312yes—
683C4—24no—
684C4—42no—
685C4—44no—
686C4—44no—
687C4—1064no—
688C4—4242no—
689C4—4242no—
690C4—2122no—
691C4—2122no—
692C4—2122no—
693C2—28no—
694C2#2027710911yescharacteristic
695C2C2×#1681051411yescharacteristic
696C2#2027731511yescharacteristic
697C2#2027756011yescharacteristic
698C2#2027756711yescharacteristic
699C2#2027758211yescharacteristic
700C2#2027758511yescharacteristic
701C2#2027818612yes—
702C2#2027826012yes—
703C2#2027903512yes—
704C2#2027909412yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C4×C22⋊C4×C53C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×C4×C22⋊C4D53
hypocenterC53#13874
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. #20272975C2semidirect
6. #20272970C2semidirect
7. #20272948C2semidirect
8. #20272942C2semidirect
20. C2×#13133124C4semidirect
392. C53#13874semidirect

Maximal subgroups

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

subgroupindex
C2×#168104842
C2×#168104812
C22×#131331242
#202729752
#202729702
#202729482
#202729422
#202729382
#202729372
C2×C4×C22⋊C4×C532
#1387453