Finite Group Databasebeta

← #27306913

Subgroups — #27306913

Order 13568.

1918 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810496C211yesmaximal, characteristic
2C2×#16810484C211yesmaximal, characteristic
3C2×#16810484C211yesmaximal, characteristic
4C22×#13133124C211yesmaximal, characteristic
5#20272975C212yesmaximal, complemented
6#20272970C212yesmaximal, complemented
7#20272948C212yesmaximal, complemented
8#20272942C212yesmaximal, complemented
9#20272937C211yesmaximal, characteristic
10#20272937C211yesmaximal, characteristic
11C2×C4×C22⋊C4×C53C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C24×Dic53C2211yescharacteristic
14C22×#11719919C2211yescharacteristic
15C22×#11719919C2211yescharacteristic
16C22×C22⋊Dic53C2211yescharacteristic
17C22×C22⋊Dic53C2211yescharacteristic
18#16810496—24no—
19#16810484—24no—
20#16810484C418yescomplemented
21C2×#13133124—24no—
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
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—24no—
48C23×Dic53—22no—
49C23×Dic53—22no—
50C23×Dic53—42no—
51C23×Dic53C2×C414yes—
52C23×Dic53C2311yescharacteristic
53C23×Dic53C2311yescharacteristic
54C23×Dic53C2311yescharacteristic
55C23×Dic53C2311yescharacteristic
56C23×Dic53C2312yes—
57C23×Dic53C2312yes—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919—24no—
61C2×#11719919—42no—
62C2×C22⋊Dic53—22no—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53—22no—
66C2×C22⋊Dic53—22no—
67C2×C22⋊Dic53C2×C414yes—
68#13133124—22no—
69#13133124—22no—
70#13133124—24no—
71#13133124—24no—
72#13133124—24no—
73#13133124—24no—
74#13133124—24no—
75#13133124—24no—
76#13133124—42no—
77#13133124—42no—
78#13133124C2×C418yescomplemented
79#13133124C2×C418yescomplemented
80C25×C53C2311yescharacteristic
81C23×C4×C53—22no—
82C23×C4×C53—22no—
83C23×C4×C53—24no—
84C23×C4×C53—42no—
85C23×C4×C53D414yes—
86C23×C4×C53C2312yes—
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
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×C53C2×C414yes—
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—21no—
124C22×C4×C53—22no—
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×C414yes—
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D414yes—
139C22×C4×C53C2×D412yes—
140C22×C4×C53#4114yes—
141C22×Dic53—28no—
142C22×Dic53—28no—
143C22×Dic53—28no—
144C22×Dic53—28no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—22no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—21no—
166C22×Dic53—21no—
167C22×Dic53—21no—
168C22×Dic53—21no—
169C22×Dic53—22no—
170C22×Dic53—21no—
171C22×Dic53—21no—
172C22×Dic53—22no—
173C22×Dic53—22no—
174C22×Dic53—22no—
175C22×Dic53—48no—
176C22×Dic53—48no—
177C22×Dic53—44no—
178C22×Dic53—44no—
179C22×Dic53—44no—
180C22×Dic53—44no—
181C22×Dic53—44no—
182C22×Dic53—44no—
183C22×Dic53C22×C412yes—
184C22×Dic53C22×C412yes—
185C22×Dic53C22×C414yes—
186C22×Dic53#4114yes—
187C22⋊C4×C53—28no—
188C22⋊C4×C53—28no—
189C42×C53—48no—
190C22⋊Dic53—28no—
191C22⋊Dic53—44no—
192#11719919—44no—
193#11719919—44no—
194C23×C53—24no—
195C23×C53—24no—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53—24no—
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#11714yes—
211C23×C53C4×D414yes—
212C23×C53C4×D414yes—
213C23×C53#12114yes—
214C23×C53#12314yes—
215C23×C53#12414yes—
216C23×C53#12614yes—
217C23×C53C23×C411yescharacteristic
218C23×C53C22×D411yescharacteristic
219C23×C53C22×D411yescharacteristic
220C23×C53C2×#4112yes—
221C23×C53C2×#4112yes—
222C23×C53C2×#4111yescharacteristic
223C23×C53C2×#4111yescharacteristic
224C23×C53C2×#4111yescharacteristic
225C23×C53C2×#4111yescharacteristic
226C23×C53H(5,𝔽2)12yes—
227C23×C532−1+412yes—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—24no—
238C2×C4×C53—22no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—416no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
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#12314yes—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—416no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—416no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287C2×Dic53—42no—
288C2×Dic53—42no—
289#13972—531nomaximal, Sylow
290C22×C53—24no—
291C22×C53—24no—
292C22×C53—28no—
293C22×C53—28no—
294C22×C53—24no—
295C22×C53—24no—
296C22×C53—28no—
297C22×C53—28no—
298C22×C53—24no—
299C22×C53—24no—
300C22×C53#38818yes—
301C22×C53C2×#11711yescharacteristic
302C22×C53C2×C4×D411yescharacteristic
303C22×C53C2×C4×D411yescharacteristic
304C22×C53C4×#4112yes—
305C22×C53#52012yes—
306C22×C53C2×#12111yescharacteristic
307C22×C53C2×#12311yescharacteristic
308C22×C53C2×#12311yescharacteristic
309C22×C53#52511yescharacteristic
310C22×C53C2×#12411yescharacteristic
311C22×C53C2×#12611yescharacteristic
312C22×C53#52912yes—
313C22×C53#53411yescharacteristic
314C22×C53#53611yescharacteristic
315C22×C53#53912yes—
316C22×C53#54612yes—
317C22×C53#54712yes—
318C22×C53#55112yes—
319C22×C53#55112yes—
320C22×C53#55112yes—
321C22×C53#55212yes—
322C22×C53#55212yes—
323C22×C53#55212yes—
324C4×C53—24no—
325C4×C53—44no—
326C4×C53—42no—
327C4×C53—44no—
328Dic53—44no—
329Dic53—42no—
330Dic53—44no—
331Dic53—42no—
332Dic53—81no—
333Dic53—81no—
334C2×#385—531no—
335C2×#381—531no—
336C2×#381—531no—
337C2×C4×C22⋊C4D5311yescharacteristic
338C22×#95—531no—
339#1443—532no—
340#1442—532no—
341#1436—532no—
342#1434—532no—
343C22⋊#95—531no—
344C22⋊#95—531no—
345C2×C53—28no—
346C2×C53C2×#38811yescharacteristic
347C2×C53C4×#12311yescharacteristic
348C2×C53#232211yescharacteristic
349C2×C53#234312yes—
350C2×C53#235912yes—
351C2×C53#241211yescharacteristic
352C2×C53#241311yescharacteristic
353C2×C53#241912yes—
354C2×C53#249312yes—
355C2×C53#250311yescharacteristic
356C2×C53#256111yescharacteristic
357C24×C4—531no—
358C24×C4—531no—
359C24×C4C2×D5311yescharacteristic
360C22×C4⋊C4—531no—
361C22×C4⋊C4—531no—
362C22×C22⋊C4—531no—
363C22×C22⋊C4—531no—
364C22×C22⋊C4C2×D5311yescharacteristic
365C22×C22⋊C4C2×D5311yescharacteristic
366C22×C42C2×D5312yes—
367#385—1064no—
368#381—538no—
369#381—1064no—
370C4×C22⋊C4—24no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95—532no—
374C2×#95—532no—
375C2×#95—531no—
376C2×#95—532no—
377C2×#95—532no—
378C2×#95—532no—
379C2×#95—532no—
380C2×#95—531no—
381C2×#95—532no—
382C2×#95—532no—
383C2×#95—1064no—
384C2×#95C2×D5312yes—
385C53#1397211yescharacteristic, complemented, Sylow
386C25C22×D5311yescharacteristic
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—24no—
390C23×C4—42no—
391C23×C4—534no—
392C23×C4—531no—
393C23×C4—531no—
394C23×C4—532no—
395C23×C4—532no—
396C23×C4—531no—
397C23×C4—531no—
398C23×C4—1064no—
399C23×C4—1062no—
400C23×C4—1062no—
401C23×C4—1062no—
402C23×C4—1062no—
403C23×C4—1062no—
404C23×C4—1062no—
405C23×C4—1062no—
406C23×C4—1062no—
407C23×C4—1062no—
408C23×C4—2122no—
409C23×C4D21214yes—
410C23×C4C22×D5312yes—
411C23×C4C22×D5311yescharacteristic
412C23×C4C22×D5312yes—
413C23×C4C22×D5311yescharacteristic
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1062no—
416C2×C4⋊C4—1064no—
417C2×C4⋊C4—2122no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—534no—
423C2×C22⋊C4—1062no—
424C2×C22⋊C4—1062no—
425C2×C22⋊C4—1062no—
426C2×C22⋊C4—1062no—
427C2×C22⋊C4—1062no—
428C2×C22⋊C4C4×D5314yes—
429C2×C22⋊C4C22⋊D5314yes—
430C2×C42—24no—
431C2×C42—24no—
432C2×C42—42no—
433#95—24no—
434#95—538no—
435#95—538no—
436#95—1064no—
437#95—1064no—
438#95—1064no—
439#95—1064no—
440#95—1062no—
441#95—1062no—
442#95—1064no—
443#95—1064no—
444#95—2122no—
445#95—2122no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24C2×C4×D5312yes—
451C24C2×D21212yes—
452C24#1171994512yes—
453C24#1171994512yes—
454C24#1171994512yes—
455C24#1171994512yes—
456C24C2×C22⋊D5312yes—
457C24C23×D5311yescharacteristic
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—28no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—28no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—532no—
483C22×C4—532no—
484C22×C4—534no—
485C22×C4—534no—
486C22×C4—1068no—
487C22×C4—1068no—
488C22×C4—1068no—
489C22×C4—1068no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1061no—
494C22×C4—1061no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1062no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1061no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—1061no—
514C22×C4—1062no—
515C22×C4—1062no—
516C22×C4—1061no—
517C22×C4—1061no—
518C22×C4—1062no—
519C22×C4—1062no—
520C22×C4—2128no—
521C22×C4—2128no—
522C22×C4—2124no—
523C22×C4—2124no—
524C22×C4—2124no—
525C22×C4—2124no—
526C22×C4—2124no—
527C22×C4—2124no—
528C22×C4C2×C4×D5314yes—
529C22×C4C2×D21212yes—
530C22×C4C2×D21212yes—
531C22×C4#1171994714yes—
532C22×C4C2×C22⋊D5314yes—
533C4⋊C4—2124no—
534C4⋊C4—2124no—
535C22⋊C4—28no—
536C22⋊C4—28no—
537C22⋊C4—1068no—
538C22⋊C4—2124no—
539C42—48no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23#1313317414yes—
557C23C4×C22⋊D5314yes—
558C23C4×D21214yes—
559C23#1313319114yes—
560C23#1313320014yes—
561C23#1313320514yes—
562C23#1313321114yes—
563C23C22×C4×D5311yescharacteristic
564C23C22×C22⋊D5311yescharacteristic
565C23C22×D21211yescharacteristic
566C23#41×D5312yes—
567C23C2×#1171994712yes—
568C23C2×#1171994511yescharacteristic
569C23C2×#1171994511yescharacteristic
570C23C2×#1171994511yescharacteristic
571C23C2×#1171994511yescharacteristic
572C23#1313326212yes—
573C23#1313326412yes—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—416no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—1064no—
599C2×C4—1064no—
600C2×C4—1064no—
601C2×C4—1064no—
602C2×C4—1062no—
603C2×C4—1062no—
604C2×C4—1064no—
605C2×C4—1064no—
606C2×C4—1064no—
607C2×C4—1064no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2122no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—21216no—
619C2×C4—21216no—
620C2×C4—2122no—
621C2×C4—2122no—
622C2×C4—2122no—
623C2×C4—2122no—
624C2×C4—2121no—
625C2×C4—2121no—
626C2×C4—2122no—
627C2×C4—2122no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4—2121no—
631C2×C4—2122no—
632C2×C4—2121no—
633C2×C4—2122no—
634C2×C4#1313320114yes—
635C22—28no—
636C22—24no—
637C22—28no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—24no—
645C22#1681051418yes—
646C22C2×#1313317411yescharacteristic
647C22C2×C4×C22⋊D5311yescharacteristic
648C22C2×C4×D21211yescharacteristic
649C22C4×#1171994712yes—
650C22#1681110212yes—
651C22C2×#1313319111yescharacteristic
652C22C2×#1313320011yescharacteristic
653C22C2×#1313320111yescharacteristic
654C22#1681113511yescharacteristic
655C22C2×#1313320511yescharacteristic
656C22C2×#1313321111yescharacteristic
657C22#1681115312yes—
658C22#1681118511yescharacteristic
659C22#1681119911yescharacteristic
660C22#1681121512yes—
661C22#1681126712yes—
662C22#1681127212yes—
663C22#1681129912yes—
664C22#1681130212yes—
665C22#1681130212yes—
666C22#1681130612yes—
667C22#1681131312yes—
668C22#1681131312yes—
669C4—24no—
670C4—42no—
671C4—44no—
672C4—44no—
673C4—4241no—
674C4—4241no—
675C4—2124no—
676C4—2122no—
677C4—2124no—
678C4—2122no—
679C2—28no—
680C2C2×#1681051411yescharacteristic
681C2C4×#1313320111yescharacteristic
682C2#2027738611yescharacteristic
683C2#2027756912yes—
684C2#2027770112yes—
685C2#2027810511yescharacteristic
686C2#2027811611yescharacteristic
687C2#2027818612yes—
688C2#2027909412yes—
689C2#2027920011yescharacteristic
690C2#2027986811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C4×C22⋊C4×C53C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×C4×C22⋊C4D53
hypocenterC53#13972
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. #16810484C4semidirect
78. #13133124C2×C4semidirect
79. #13133124C2×C4semidirect
385. C53#13972semidirect

Maximal subgroups

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

subgroupindex
C2×#168104962
C2×#168104842
C22×#131331242
#202729752
#202729702
#202729482
#202729422
#202729372
C2×C4×C22⋊C4×C532
#1397253