Finite Group Databasebeta

← #27306071

Subgroups — #27306071

Order 13568.

2006 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133175C211yesmaximal, characteristic
2C2×#16810492C211yesmaximal, characteristic, complemented
3C2×#16810480C211yesmaximal, characteristic
4C2×C4×#11719921C211yesmaximal, characteristic
5#20272959C214yesmaximal
6#20272946C214yesmaximal, complemented
7#20272945C211yesmaximal, characteristic, complemented
8#20272944C211yesmaximal, characteristic, complemented
9C2×C53×#382C211yesmaximal, characteristic, complemented
10C23×C4×D53C2211yescharacteristic
11C2×#13133175—24no—
12C2×#13133175—24no—
13C2×#13133175—24no—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16C22×#11719919C2214yes—
17C22×#11719921C2212yes—
18C22×#11719921C2212yes—
19C22×#11719921C2212yes—
20C22×C4×Dic53C2212yes—
21C22×C4×Dic53C2211yescharacteristic
22#16810492—24no—
23#16810480—24no—
24C4×#11719921C418yescomplemented
25C2×#13133124C2214yes—
26C2×#13133124C2212yes—
27C2×#13133124C2214yes—
28C2×#13133124C2212yes—
29C22×C4⋊C4×C53C2211yescharacteristic
30C22×C4⋊C4×C53C2211yescharacteristic
31C22×C42×C53C2211yescharacteristic
32C53×#382—24no—
33C2×#95×C53C2214yes—
34C24×D53C2311yescharacteristic
35C23×Dic53C2312yes—
36C23×Dic53C2312yes—
37C23×Dic53C2311yescharacteristic
38C23×Dic53C2312yes—
39C22×C4×D53—22no—
40C22×C4×D53—22no—
41C22×C4×D53—22no—
42C22×C4×D53—22no—
43C22×C4×D53C2×C414yes—
44C22×C4×D53D414yes—
45C22×C4×D53Q814yes—
46#13133175—232no—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—22no—
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53D414yes—
53C2×#11719919—28no—
54C2×#11719919—28no—
55C2×#11719919—44no—
56C2×#11719921—24no—
57C2×#11719921—24no—
58C2×#11719921—24no—
59C2×#11719921—24no—
60C2×#11719921—24no—
61C2×#11719921—24no—
62C2×#11719921—44no—
63C2×#11719921C2×C418yes—
64C2×C4×Dic53—24no—
65C2×C4×Dic53—22no—
66C2×C4×Dic53—22no—
67C2×C4×Dic53—24no—
68C2×C4×Dic53—24no—
69C2×C4×Dic53C2×C414yes—
70#13133124—24no—
71#13133124—28no—
72#13133124—44no—
73#13133124C2×C418yes—
74C23×C4×C53C2312yes—
75C23×C4×C53C2312yes—
76C23×C4×C53C2312yes—
77C23×C4×C53C2311yescharacteristic
78C2×C4⋊C4×C53—22no—
79C2×C4⋊C4×C53—22no—
80C2×C4⋊C4×C53—22no—
81C2×C4⋊C4×C53—22no—
82C2×C4⋊C4×C53—22no—
83C2×C4⋊C4×C53D414yes—
84C2×C42×C53—22no—
85C2×C42×C53—22no—
86C2×C42×C53C2×C414yes—
87#95×C53—28no—
88C24×C53C2411yescharacteristic
89C23×D53—21no—
90C23×D53—21no—
91C23×D53—22no—
92C23×D53—22no—
93C23×D53—21no—
94C23×D53—21no—
95C23×D53—22no—
96C23×D53—22no—
97C23×D53C22×C412yes—
98C23×D53C2×D412yes—
99C23×D53C2×Q812yes—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—24no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—24no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—24no—
114C22×C4×C53—24no—
115C22×C4×C53C22×C414yes—
116C22×C4×C53C22×C412yes—
117C22×C4×C53C2×D414yes—
118C22×C4×C53C2×D412yes—
119C22×C4×C53C2×D412yes—
120C22×C4×C53C2×D412yes—
121C22×C4×C53C2×Q812yes—
122C22×C4×C53#4112yes—
123C22×C4×C53#4112yes—
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53—22no—
127C22×Dic53—22no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—24no—
135C22×Dic53—24no—
136C22×Dic53—22no—
137C22×Dic53—24no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—24no—
141C22×Dic53C22×C412yes—
142C22×Dic53C22×C414yes—
143C22×Dic53C2×D414yes—
144C22×Dic53C2×D412yes—
145C22×Dic53C2×Q812yes—
146C22×Dic53#4114yes—
147C22×Dic53#4114yes—
148C2×C4×D53—28no—
149C2×C4×D53—28no—
150C2×C4×D53—28no—
151C2×C4×D53—28no—
152C2×C4×D53—28no—
153C2×C4×D53—28no—
154C2×C4×D53C4⋊C4116yescomplemented
155C4⋊C4×C53—28no—
156C4⋊C4×C53—44no—
157C42×C53—28no—
158#11719921—216no—
159#11719921—216no—
160#11719921—48no—
161C4⋊Dic53—28no—
162C4⋊Dic53—44no—
163#11719919—416no—
164C4×Dic53—216no—
165C4×Dic53—28no—
166C23×C53C23×C411yescharacteristic
167C23×C53C22×D411yescharacteristic
168C23×C53C22×D411yescharacteristic
169C23×C53C22×D411yescharacteristic
170C23×C53C22×Q811yescharacteristic
171C23×C53C2×#4112yes—
172C23×C53C2×#4112yes—
173C23×C53C2×#4111yescharacteristic
174C23×C53C2×#4111yescharacteristic
175C23×C53H(5,𝔽2)12yes—
176C23×C532−1+412yes—
177C22×D53—24no—
178C22×D53—22no—
179C22×D53—22no—
180C22×D53—24no—
181C22×D53—22no—
182C22×D53—22no—
183C22×D53—22no—
184C22×D53—22no—
185C22×D53—22no—
186C22×D53—22no—
187C22×D53—24no—
188C22×D53—22no—
189C22×D53—22no—
190C22×D53—24no—
191C22×D53—42no—
192C22×D53—42no—
193C22×D53—42no—
194C22×D53—42no—
195C22×D53—42no—
196C22×D53—42no—
197C22×D53—42no—
198C22×D53—42no—
199C22×D53C2×C4⋊C414yes—
200C2×C4×C53—24no—
201C2×C4×C53—24no—
202C2×C4×C53—24no—
203C2×C4×C53—24no—
204C2×C4×C53—28no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—44no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—44no—
216C2×C4×C53—44no—
217C2×C4×C53C2×C4⋊C414yes—
218C2×C4×C53C4×D414yes—
219C2×C4×C53C4×D414yes—
220C2×C4×C53#12014yes—
221C2×C4×C53C22⋊Q814yes—
222C2×C4×C53C22⋊Q814yes—
223C2×C4×C53#12314yes—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—24no—
227C2×Dic53—24no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—24no—
233C2×Dic53—24no—
234C2×Dic53—24no—
235C2×Dic53—24no—
236C2×Dic53—24no—
237C2×Dic53—28no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—44no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—44no—
245C2×Dic53—44no—
246C2×Dic53C2×C4⋊C414yes—
247C2×Dic53#11718yescomplemented
248C4×D53—232no—
249#13879—531nomaximal, Sylow
250C22×C53C22×C4⋊C411yescharacteristic
251C22×C53C2×#11712yes—
252C22×C53C2×C4×D411yescharacteristic
253C22×C53C2×C4×D411yescharacteristic
254C22×C53#52012yes—
255C22×C53C2×#12011yescharacteristic
256C22×C53C2×C22⋊Q811yescharacteristic
257C22×C53C2×C22⋊Q811yescharacteristic
258C22×C53C2×#12311yescharacteristic
259C22×C53#53211yescharacteristic
260C22×C53#53211yescharacteristic
261C22×C53#53312yes—
262C22×C53#53511yescharacteristic
263C22×C53#53611yescharacteristic
264C22×C53#54412yes—
265C22×C53#54612yes—
266C22×C53#54712yes—
267C22×C53#54812yes—
268C22×C53#55012yes—
269C22×C53#55212yes—
270C22×C53#55312yes—
271C22×C53#55512yes—
272C22×C53#55612yes—
273C2×D53—24no—
274C2×D53—24no—
275C2×D53—24no—
276C2×D53—44no—
277C2×D53—44no—
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—44no—
281C2×D53—44no—
282C2×D53—42no—
283C2×D53—42no—
284C4×C53—44no—
285C4×C53—44no—
286C4×C53—82no—
287C4×C53#38518yes—
288Dic53—28no—
289Dic53—24no—
290Dic53—44no—
291Dic53—82no—
292C22×#117—531no—
293C2×#384—531no—
294C2×#382D5311yescharacteristic, complemented
295C2×#380—531no—
296C2×C4×C22⋊C4—531no—
297#1439—534no—
298#1436—534no—
299C4⋊#95—531no—
300C4⋊#95—531no—
301C2×C53#228511yescharacteristic
302C2×C53C2×#38511yescharacteristic
303C2×C53#231711yescharacteristic
304C2×C53#235212yes—
305C2×C53#235412yes—
306C2×C53#243111yescharacteristic
307C2×C53#243211yescharacteristic
308C2×C53#243912yes—
309C2×C53#244112yes—
310C2×C53#249911yescharacteristic
311C2×C53#250011yescharacteristic
312D53—44no—
313C24×C4—531no—
314C2×#117—1064no—
315C2×#117—1064no—
316C2×#117—1064no—
317C22×C4⋊C4—531no—
318C22×C4⋊C4—531no—
319C22×C4⋊C4—534no—
320C22×C4⋊C4C2×D5311yescharacteristic
321C22×C4⋊C4C2×D5311yescharacteristic
322C22×C22⋊C4—532no—
323C22×C22⋊C4—532no—
324C22×C22⋊C4—532no—
325C22×C42—531no—
326C22×C42—532no—
327C22×C42C2×D5311yescharacteristic
328#384—1064no—
329#382—24no—
330#380—1064no—
331C4×C22⋊C4—538no—
332C2×#95—532no—
333C2×#95—534no—
334C2×#95—532no—
335C2×#95—534no—
336C2×#95C2×D5314yes—
337C53#1387911yescharacteristic, complemented, Sylow
338C25—531no—
339C23×C4—532no—
340C23×C4—531no—
341C23×C4—534no—
342C23×C4—534no—
343C23×C4—534no—
344C23×C4—532no—
345C23×C4—532no—
346C23×C4—1062no—
347C23×C4—1062no—
348C23×C4—1062no—
349C23×C4—1062no—
350C23×C4C22×D5312yes—
351C23×C4C22×D5312yes—
352C23×C4C22×D5312yes—
353C23×C4C22×D5311yescharacteristic
354#117—10632no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—534no—
361C2×C4⋊C4—1062no—
362C2×C4⋊C4—1062no—
363C2×C4⋊C4—1068no—
364C2×C4⋊C4—1068no—
365C2×C4⋊C4—1062no—
366C2×C4⋊C4—1062no—
367C2×C4⋊C4—1062no—
368C2×C4⋊C4—2124no—
369C2×C4⋊C4C22⋊D5314yes—
370C2×C22⋊C4—538no—
371C2×C22⋊C4—1064no—
372C2×C22⋊C4—1064no—
373C2×C22⋊C4—1064no—
374C2×C22⋊C4—1064no—
375C2×C22⋊C4—1064no—
376C2×C22⋊C4—1064no—
377C2×C22⋊C4—2124no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—534no—
381C2×C42—1064no—
382C2×C42—1064no—
383C2×C42—1062no—
384C2×C42—1062no—
385C2×C42—1064no—
386C2×C42C4×D5314yes—
387#95—28no—
388#95—538no—
389#95—1068no—
390#95—1064no—
391#95—2124no—
392C24—532no—
393C24—532no—
394C24—532no—
395C24—1061no—
396C24—1061no—
397C24—1062no—
398C24—1062no—
399C24—1061no—
400C24—1061no—
401C24—1062no—
402C24—1062no—
403C24C23×D5311yescharacteristic
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—532no—
420C22×C4—534no—
421C22×C4—532no—
422C22×C4—532no—
423C22×C4—534no—
424C22×C4—534no—
425C22×C4—5316no—
426C22×C4—534no—
427C22×C4—1061no—
428C22×C4—1062no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1062no—
432C22×C4—1061no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1068no—
436C22×C4—1068no—
437C22×C4—1068no—
438C22×C4—1068no—
439C22×C4—1068no—
440C22×C4—1068no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1064no—
444C22×C4—1064no—
445C22×C4—1062no—
446C22×C4—1064no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1064no—
450C22×C4C2×C4×D5314yes—
451C22×C4C2×C4×D5312yes—
452C22×C4#1171994512yes—
453C22×C4D4×D5312yes—
454C22×C4D4×D5312yes—
455C22×C4Q8×D5312yes—
456C22×C4#1171994912yes—
457C22×C4C2×C22⋊D5314yes—
458C22×C4C2×C22⋊D5312yes—
459C4⋊C4—28no—
460C4⋊C4—44no—
461C4⋊C4—1068no—
462C4⋊C4—21216no—
463C4⋊C4—2124no—
464C22⋊C4—10616no—
465C22⋊C4—10616no—
466C22⋊C4—2128no—
467C42—28no—
468C42—10616no—
469C42—1068no—
470C23—534no—
471C23—1064no—
472C23—1062no—
473C23—1062no—
474C23—1064no—
475C23—1062no—
476C23—1062no—
477C23—1062no—
478C23—1062no—
479C23—1062no—
480C23—1062no—
481C23—1064no—
482C23—1062no—
483C23—1062no—
484C23—1064no—
485C23—2122no—
486C23—2122no—
487C23—2122no—
488C23—2122no—
489C23—2122no—
490C23—2122no—
491C23—2122no—
492C23—2122no—
493C23C22×C4×D5311yescharacteristic
494C23C2×D4×D5311yescharacteristic
495C23C2×D4×D5311yescharacteristic
496C23C22×C22⋊D5311yescharacteristic
497C23C2×Q8×D5311yescharacteristic
498C23#41×D5312yes—
499C23#41×D5312yes—
500C23C2×#1171994511yescharacteristic
501C23C2×#1171994911yescharacteristic
502C23#1313326312yes—
503C23#1313326412yes—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—534no—
522C2×C4—538no—
523C2×C4—1064no—
524C2×C4—1064no—
525C2×C4—1064no—
526C2×C4—1064no—
527C2×C4—1062no—
528C2×C4—1064no—
529C2×C4—1062no—
530C2×C4—1064no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—10632no—
536C2×C4—1068no—
537C2×C4—1064no—
538C2×C4—2124no—
539C2×C4—2124no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2124no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4C4⋊C4×D5314yes—
547C2×C4C4×C22⋊D5314yes—
548C2×C4#1313317914yes—
549C2×C4#1313318514yes—
550C2×C4#1313319314yes—
551C2×C4#1313319514yes—
552C2×C4#1313320214yes—
553C22—1064no—
554C22—1064no—
555C22—1064no—
556C22—2122no—
557C22—2122no—
558C22—2122no—
559C22—2124no—
560C22—2124no—
561C22—2122no—
562C22—2124no—
563C22—2124no—
564C22C2×C4⋊C4×D5311yescharacteristic
565C22#117×D5312yes—
566C22C2×C4×C22⋊D5311yescharacteristic
567C22C2×#1313317911yescharacteristic
568C22#1681110312yes—
569C22C2×#1313318511yescharacteristic
570C22C2×#1313319311yescharacteristic
571C22C2×#1313319511yescharacteristic
572C22C2×#1313320211yescharacteristic
573C22#1681117211yescharacteristic
574C22#1681117211yescharacteristic
575C22#1681117912yes—
576C22#1681119111yescharacteristic
577C22#1681119311yescharacteristic
578C22#1681125312yes—
579C22#1681126612yes—
580C22#1681127212yes—
581C22#1681128112yes—
582C22#1681129212yes—
583C22#1681131112yes—
584C22#1681132612yes—
585C22#1681133612yes—
586C22#1681134112yes—
587C4—44no—
588C4—44no—
589C4—82no—
590C4—1064no—
591C4—1068no—
592C4—4242no—
593C4—2124no—
594C4#1681049418yes—
595C2—2124no—
596C2#2027712111yescharacteristic
597C2C2×#1681049411yescharacteristic
598C2#2027734011yescharacteristic
599C2#2027764112yes—
600C2#2027766512yes—
601C2#2027833711yescharacteristic
602C2#2027834611yescharacteristic
603C2#2027840912yes—
604C2#2027842412yes—
605C2#2027917111yescharacteristic
606C2#2027918011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#382C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC2×#382D53
hypocenterC53#13879
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. C2×#16810492C2semidirect
6. #20272946C2semidirect
7. #20272945C2semidirect
8. #20272944C2semidirect
9. C2×C53×#382C2semidirect
24. C4×#11719921C4semidirect
154. C2×C4×D53C4⋊C4semidirect
247. C2×Dic53#117semidirect
294. C2×#382D53semidirect
337. C53#13879semidirect

Maximal subgroups

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

subgroupindex
C22×#131331752
C2×#168104922
C2×#168104802
C2×C4×#117199212
#202729592
#202729462
#202729452
#202729442
C2×C53×#3822
#1387953