Finite Group Databasebeta

← #27309602

Subgroups — #27309602

Order 13568.

1894 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
1C22×#13133182C211yesmaximal, characteristic, complemented
2C2×#16810535C211yesmaximal, characteristic
3C2×#16810516C211yesmaximal, characteristic
4C2×#16810511C212yesmaximal
5#20272985C211yesmaximal, characteristic, complemented
6#20272974C211yesmaximal, characteristic, complemented
7#20272952C212yesmaximal
8#20272949C212yesmaximal, complemented
9#20272946C212yesmaximal, complemented
10#95×Dic53C211yesmaximal, characteristic, complemented
11C53×#1439C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C2×#13133182—28no—
14C2×#13133182—24no—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×#11719919C2212yes—
17C22×C22⋊Dic53C2211yescharacteristic
18C22×C22⋊Dic53C2211yescharacteristic
19C22×C22⋊Dic53C2212yes—
20C22×C4×Dic53C2212yes—
21C22×C4×Dic53C2212yes—
22C22×C4×Dic53C2211yescharacteristic
23#16810535C418yescomplemented
24#16810516—24no—
25#16810511—28no—
26C2×#13133124C2212yes—
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2212yes—
33C2×#13133124C2211yescharacteristic
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2212yes—
37C22×C22⋊C4×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C22×Dic106—22no—
41C22×Dic106—22no—
42C22×Dic106—24no—
43C22×Dic106—24no—
44C22×Dic106C2×C414yes—
45C23×Dic53C2311yescharacteristic
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2312yes—
48C23×Dic53C2312yes—
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2311yescharacteristic
51#13133182—232no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—24no—
54C2×#11719919—24no—
55C2×#11719919—24no—
56C2×#11719919D418yes—
57C2×C22⋊Dic53—24no—
58C2×C22⋊Dic53—24no—
59C2×C22⋊Dic53—22no—
60C2×C22⋊Dic53—22no—
61C2×C22⋊Dic53—24no—
62C2×C22⋊Dic53—24no—
63C2×C22⋊Dic53C2×C418yes—
64C2×C4×Dic53—24no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—24no—
67C2×C4×Dic53—24no—
68C2×C4×Dic53—24no—
69C2×C4×Dic53—24no—
70C2×C4×Dic53—42no—
71C2×C4×Dic53D414yes—
72#13133124—24no—
73#13133124—22no—
74#13133124—24no—
75#13133124—22no—
76#13133124—24no—
77#13133124—24no—
78#13133124—42no—
79#13133124C2×C414yes—
80#13133124C2×C418yes—
81C25×C53C2311yescharacteristic
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2312yes—
84C23×C4×C53C2312yes—
85C23×C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53—24no—
87C2×C4⋊C4×C53—22no—
88C2×C22⋊C4×C53—22no—
89C2×C22⋊C4×C53—24no—
90C2×C22⋊C4×C53—22no—
91C2×C22⋊C4×C53—24no—
92C2×C22⋊C4×C53—42no—
93C2×C22⋊C4×C53C2×C414yes—
94#95×C53—24no—
95#95×C53—22no—
96C24×C53—22no—
97C24×C53—21no—
98C24×C53—21no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53—22no—
102C24×C53—21no—
103C24×C53—21no—
104C24×C53—21no—
105C24×C53—21no—
106C24×C53C22×C412yes—
107C24×C53C2411yescharacteristic
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—21no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
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×C53C22×C412yes—
127C22×C4×C53C22×C412yes—
128C22×C4×C53C2×D414yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53#4114yes—
131C22×C4×C53#4112yes—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—21no—
139C22×Dic53—21no—
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—21no—
150C22×Dic53—21no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53C22×C414yes—
159C22×Dic53C22×C414yes—
160C22×Dic53C2×D414yes—
161C22×Dic53C2×D412yes—
162C22×Dic53C2×D414yes—
163C22×Dic53C2×D412yes—
164C22×Dic53#4114yes—
165C22×Dic53#4114yes—
166C2×Dic106—28no—
167C2×Dic106—216no—
168C2×Dic106—28no—
169C2×Dic106—28no—
170C2×Dic106—216no—
171C4⋊C4×C53—28no—
172C22⋊C4×C53—28no—
173C22⋊C4×C53—48no—
174C22⋊Dic53—216no—
175C22⋊Dic53—44no—
176C22⋊Dic53—44no—
177C4⋊Dic53—44no—
178#11719919—216no—
179C4×Dic53—216no—
180C4×Dic53—28no—
181C4×Dic53—48no—
182C23×C53—24no—
183C23×C53—22no—
184C23×C53—24no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—42no—
190C23×C53—41no—
191C23×C53—42no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53C23×C411yescharacteristic
206C23×C53C22×D411yescharacteristic
207C23×C53C22×D412yes—
208C23×C53C2×#4112yes—
209C23×C53C2×#4112yes—
210C23×C53C2×#4111yescharacteristic
211C23×C53C2×#4112yes—
212C23×C53H(5,𝔽2)11yescharacteristic
213C23×C532−1+411yescharacteristic
214C23×C532−1+411yescharacteristic
215C23×C532−1+411yescharacteristic
216C2×C4×C53—24no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—24no—
220C2×C4×C53—22no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
228C2×C4×C53—42no—
229C2×C4×C53—41no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—41no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53C4×D414yes—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—22no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
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—24no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53C4×D418yes—
272C2×Dic53#12014yes—
273C2×Dic53#12414yes—
274Dic106—232no—
275#14188—531nomaximal, Sylow
276C22×C53—42no—
277C22×C53—44no—
278C22×C53—44no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—81no—
282C22×C53—81no—
283C22×C53—81no—
284C22×C53—81no—
285C22×C53—82no—
286C22×C53—82no—
287C22×C53C2×C4×D412yes—
288C22×C53C2×C4×D411yescharacteristic
289C22×C53C4×#4112yes—
290C22×C53#51911yescharacteristic
291C22×C53#52011yescharacteristic
292C22×C53C2×#12011yescharacteristic
293C22×C53#52511yescharacteristic
294C22×C53#52512yes—
295C22×C53C2×#12411yescharacteristic
296C22×C53#52912yes—
297C22×C53#53211yescharacteristic
298C22×C53#53611yescharacteristic
299C22×C53#53711yescharacteristic
300C22×C53#54111yescharacteristic
301C22×C53#54211yescharacteristic
302C22×C53#54612yes—
303C22×C53#54712yes—
304C22×C53#54812yes—
305C22×C53#54812yes—
306C22×C53#55212yes—
307C22×C53#55612yes—
308C22×C53#55612yes—
309C22×C53#55912yes—
310C4×C53—24no—
311C4×C53—44no—
312C4×C53—44no—
313C4×C53—42no—
314Dic53—24no—
315Dic53—24no—
316Dic53—28no—
317Dic53—44no—
318Dic53—81no—
319Dic53—81no—
320C22×C4×Q8—531no—
321C2×#393—531no—
322C2×#388—532no—
323C2×#388—531no—
324#1446—531no—
325#1442—531no—
326#1439D5311yescharacteristic
327#1437—532no—
328#1436—532no—
329#1436—532no—
330C4×#95—531no—
331C2×C53—82no—
332C2×C53C4×#12011yescharacteristic
333C2×C53C4×#12411yescharacteristic
334C2×C53#232311yescharacteristic
335C2×C53#236212yes—
336C2×C53#236712yes—
337C2×C53#243011yescharacteristic
338C2×C53#243611yescharacteristic
339C2×C53#245812yes—
340C2×C53#246912yes—
341C2×C53#259211yescharacteristic
342C2×C53#260411yescharacteristic
343C23×Q8—531no—
344C2×C4×Q8—1064no—
345C2×C4×Q8—1068no—
346C22×C4⋊C4—532no—
347C22×C4⋊C4—531no—
348C22×C4⋊C4C2×D5311yescharacteristic
349C22×C22⋊C4—532no—
350C22×C22⋊C4—531no—
351C22×C22⋊C4—531no—
352C22×C22⋊C4C2×D5312yes—
353C22×C22⋊C4C2×D5311yescharacteristic
354C22×C42—531no—
355C22×C42—532no—
356C22×C42—532no—
357#393—538no—
358#388—1064no—
359#388—1068no—
360C2×#95—531no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—531no—
364C2×#95—532no—
365C2×#95—532no—
366C2×#95—532no—
367C2×#95—531no—
368C2×#95—532no—
369C2×#95C2×D5311yescharacteristic
370C2×#95C2×D5312yes—
371C53#1418811yescharacteristic, complemented, Sylow
372C25C22×D5311yescharacteristic
373C22×Q8—534no—
374C22×Q8—1064no—
375C22×Q8—1064no—
376C22×Q8—1062no—
377C22×Q8—1062no—
378C23×C4—531no—
379C23×C4—532no—
380C23×C4—531no—
381C23×C4—532no—
382C23×C4—531no—
383C23×C4—531no—
384C23×C4C22×D5311yescharacteristic
385C23×C4C22×D5312yes—
386C23×C4C22×D5312yes—
387C23×C4C22×D5311yescharacteristic
388C4×Q8—10632no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—538no—
392C2×C4⋊C4—1064no—
393C2×C4⋊C4—1064no—
394C2×C4⋊C4—1064no—
395C2×C4⋊C4—1062no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—42no—
401C2×C22⋊C4—538no—
402C2×C22⋊C4—1064no—
403C2×C22⋊C4—1064no—
404C2×C22⋊C4—1064no—
405C2×C22⋊C4—1062no—
406C2×C22⋊C4—1062no—
407C2×C22⋊C4—1064no—
408C2×C22⋊C4C4×D5314yes—
409C2×C42—534no—
410C2×C42—1064no—
411C2×C42—1064no—
412C2×C42—1064no—
413C2×C42—1064no—
414C2×C42—1064no—
415C2×C42—1064no—
416C2×C42—2122no—
417#95—24no—
418#95—22no—
419#95—538no—
420#95—534no—
421#95—1064no—
422#95—1064no—
423#95—1064no—
424#95—1062no—
425#95—1064no—
426#95—1062no—
427#95—2122no—
428C24—22no—
429C24—21no—
430C24—22no—
431C24—22no—
432C24—21no—
433C24—22no—
434C24—21no—
435C24—21no—
436C24—21no—
437C24—21no—
438C24C2×C4×D5312yes—
439C24C23×D5311yescharacteristic
440C2×Q8—10616no—
441C2×Q8—1068no—
442C2×Q8—1068no—
443C2×Q8—10616no—
444C2×Q8—1068no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
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—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—534no—
464C22×C4—532no—
465C22×C4—534no—
466C22×C4—532no—
467C22×C4—534no—
468C22×C4—534no—
469C22×C4—534no—
470C22×C4—534no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1061no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4C2×C4×D5312yes—
498C22×C4C2×C4×D5312yes—
499C22×C4D4×D5314yes—
500C22×C4D4×D5312yes—
501C22×C4#1171994714yes—
502C22×C4#1171994712yes—
503C4⋊C4—28no—
504C4⋊C4—10616no—
505C4⋊C4—2124no—
506C22⋊C4—28no—
507C22⋊C4—48no—
508C22⋊C4—10616no—
509C22⋊C4—2124no—
510C22⋊C4—2124no—
511C42—1068no—
512C42—10616no—
513C42—2128no—
514C23—24no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23C22×C4×D5311yescharacteristic
538C23C2×D4×D5312yes—
539C23C2×D4×D5311yescharacteristic
540C23#41×D5312yes—
541C23#41×D5312yes—
542C23C2×#1171994712yes—
543C23C2×#1171994711yescharacteristic
544C23#1313326211yescharacteristic
545C23#1313326411yescharacteristic
546C23#1313326411yescharacteristic
547C23#1313326511yescharacteristic
548Q8—10632no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—538no—
573C2×C4—534no—
574C2×C4—534no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—1062no—
581C2×C4—1064no—
582C2×C4—1064no—
583C2×C4—1064no—
584C2×C4—1064no—
585C2×C4—1062no—
586C2×C4—1062no—
587C2×C4—1062no—
588C2×C4—1064no—
589C2×C4—1064no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4#1313317714yes—
607C22—42no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—82no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—82no—
617C22—81no—
618C22C4×D4×D5312yes—
619C22C2×#1313317711yescharacteristic
620C22C4×#1171994712yes—
621C22#1681110011yescharacteristic
622C22#1681110411yescharacteristic
623C22#120×D5311yescharacteristic
624C22#1681113111yescharacteristic
625C22#1681113312yes—
626C22#124×D5311yescharacteristic
627C22#1681115412yes—
628C22#1681117111yescharacteristic
629C22#1681119611yescharacteristic
630C22#1681120211yescharacteristic
631C22#1681123311yescharacteristic
632C22#1681124011yescharacteristic
633C22#1681126012yes—
634C22#1681127312yes—
635C22#1681128112yes—
636C22#1681128312yes—
637C22#1681131012yes—
638C22#1681134012yes—
639C22#1681134412yes—
640C22#1681135712yes—
641C4—24no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—1064no—
646C4—1064no—
647C4—1068no—
648C4—4241no—
649C4—4241no—
650C4—2124no—
651C2—82no—
652C2#2027722011yescharacteristic
653C2#2027726211yescharacteristic
654C2#2027739311yescharacteristic
655C2#2027773912yes—
656C2#2027778312yes—
657C2#2027831811yescharacteristic
658C2#2027836811yescharacteristic
659C2#2027863812yes—
660C2#2027879712yes—
661C2#2028019911yescharacteristic
662C2#2028035411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1439C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1439D53
hypocenterC53#14188
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C22×#13133182C2semidirect
5. #20272985C2semidirect
6. #20272974C2semidirect
8. #20272949C2semidirect
9. #20272946C2semidirect
10. #95×Dic53C2semidirect
23. #16810535C4semidirect
371. C53#14188semidirect

Maximal subgroups

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

subgroupindex
C22×#131331822
C2×#168105352
C2×#168105162
C2×#168105112
#202729852
#202729742
#202729522
#202729492
#202729462
#95×Dic532
C53×#14392
#1418853