Finite Group Databasebeta

← #27306064

Subgroups — #27306064

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
1C2×#16810492C211yesmaximal, characteristic, complemented
2C2×#16810488C211yesmaximal, characteristic, complemented
3C2×#16810483C211yesmaximal, characteristic
4C2×C4×C22⋊Dic53C211yesmaximal, characteristic
5#20272958C214yesmaximal
6#20272948C214yesmaximal, complemented
7#20272944C211yesmaximal, characteristic, complemented
8#20272944C211yesmaximal, characteristic, complemented
9C22×#117×C53C211yesmaximal, characteristic
10C22×C4⋊Dic53C2211yescharacteristic
11C22×C4⋊Dic53C2211yescharacteristic
12C22×#11719919C2214yes—
13C22×C22⋊Dic53C2212yes—
14C22×C22⋊Dic53C2212yes—
15C22×C4×Dic53C2211yescharacteristic
16C22×C4×Dic53C2211yescharacteristic
17#16810492—24no—
18#16810488—24no—
19#16810483—24no—
20C4×C22⋊Dic53C418yescomplemented
21C2×#13133124C2214yes—
22C2×#13133124C2212yes—
23C2×#13133124C2214yes—
24C2×#13133124C2214yes—
25C2×#13133124C2212yes—
26C24×C4×C53C2211yescharacteristic
27C2×#117×C53—24no—
28C2×#117×C53—24no—
29C2×#117×C53—24no—
30C22×C4⋊C4×C53C2211yescharacteristic
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×C22⋊C4×C53C2212yes—
33C22×C42×C53C2212yes—
34C23×Dic53C2312yes—
35C23×Dic53C2312yes—
36C23×Dic53C2312yes—
37C23×Dic53C2312yes—
38C2×C4⋊Dic53—22no—
39C2×C4⋊Dic53—22no—
40C2×C4⋊Dic53—22no—
41C2×C4⋊Dic53—22no—
42C2×C4⋊Dic53—22no—
43C2×C4⋊Dic53—22no—
44C2×#11719919—28no—
45C2×#11719919—28no—
46C2×#11719919—44no—
47C2×C22⋊Dic53—24no—
48C2×C22⋊Dic53—24no—
49C2×C22⋊Dic53—24no—
50C2×C22⋊Dic53—44no—
51C2×C22⋊Dic53C2×C418yes—
52C2×C4×Dic53—22no—
53C2×C4×Dic53—22no—
54C2×C4×Dic53—22no—
55C2×C4×Dic53—22no—
56C2×C4×Dic53C2×C414yes—
57C2×C4×Dic53C2×C414yes—
58#13133124—24no—
59#13133124—28no—
60#13133124—28no—
61#13133124—44no—
62#13133124C2×C418yes—
63C25×C53C2311yescharacteristic
64C23×C4×C53—22no—
65C23×C4×C53—22no—
66C23×C4×C53—22no—
67C23×C4×C53—22no—
68C23×C4×C53C2×C414yes—
69C23×C4×C53D414yes—
70C23×C4×C53Q814yes—
71C23×C4×C53C2312yes—
72C23×C4×C53C2312yes—
73C23×C4×C53C2311yescharacteristic
74C23×C4×C53C2311yescharacteristic
75#117×C53—232no—
76C2×C4⋊C4×C53—22no—
77C2×C4⋊C4×C53—22no—
78C2×C4⋊C4×C53—22no—
79C2×C4⋊C4×C53—22no—
80C2×C4⋊C4×C53D414yes—
81C2×C4⋊C4×C53D414yes—
82C2×C22⋊C4×C53—24no—
83C2×C22⋊C4×C53—24no—
84C2×C22⋊C4×C53—24no—
85C2×C42×C53—24no—
86C2×C42×C53—24no—
87C2×C42×C53—24no—
88C24×C53—21no—
89C24×C53—21no—
90C24×C53—21no—
91C24×C53—21no—
92C24×C53—22no—
93C24×C53—22no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53C22×C412yes—
97C24×C53C2×D412yes—
98C24×C53C2×Q812yes—
99C24×C53C2411yescharacteristic
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—28no—
105C22×C4×C53—28no—
106C22×C4×C53—28no—
107C22×C4×C53—28no—
108C22×C4×C53—28no—
109C22×C4×C53—28no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—24no—
115C22×C4×C53—24no—
116C22×C4×C53—24no—
117C22×C4×C53—24no—
118C22×C4×C53C4⋊C4116yescomplemented
119C22×C4×C53C22×C412yes—
120C22×C4×C53C22×C412yes—
121C22×C4×C53C2×D414yes—
122C22×C4×C53C2×D414yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53C2×D412yes—
125C22×C4×C53C2×D412yes—
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2×Q812yes—
128C22×C4×C53C2×Q812yes—
129C22×C4×C53#4112yes—
130C22×C4×C53#4112yes—
131C22×Dic53—22no—
132C22×Dic53—22no—
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—24no—
146C22×Dic53—24no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—24no—
150C22×Dic53—24no—
151C22×Dic53C22×C414yes—
152C22×Dic53C22×C414yes—
153C22×Dic53#4114yes—
154C22×Dic53#4114yes—
155C4⋊C4×C53—28no—
156C4⋊C4×C53—28no—
157C22⋊C4×C53—216no—
158C42×C53—216no—
159C22⋊Dic53—216no—
160C22⋊Dic53—48no—
161C4⋊Dic53—44no—
162C4⋊Dic53—44no—
163#11719919—416no—
164C4×Dic53—28no—
165C4×Dic53—28no—
166C23×C53—24no—
167C23×C53—22no—
168C23×C53—22no—
169C23×C53—24no—
170C23×C53—22no—
171C23×C53—22no—
172C23×C53—22no—
173C23×C53—22no—
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—24no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53—24no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53C2×C4⋊C414yes—
189C23×C53C23×C411yescharacteristic
190C23×C53C22×D411yescharacteristic
191C23×C53C22×D411yescharacteristic
192C23×C53C22×D411yescharacteristic
193C23×C53C22×Q811yescharacteristic
194C23×C53C2×#4112yes—
195C23×C53C2×#4112yes—
196C23×C53C2×#4111yescharacteristic
197C23×C53C2×#4111yescharacteristic
198C23×C53H(5,𝔽2)12yes—
199C23×C532−1+412yes—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—232no—
207C2×C4×C53—24no—
208C2×C4×C53—24no—
209C2×C4×C53—28no—
210C2×C4×C53—28no—
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×C53C2×C4⋊C414yes—
219C2×C4×C53C4×D414yes—
220C2×C4×C53C4×D414yes—
221C2×C4×C53#12014yes—
222C2×C4×C53C22⋊Q814yes—
223C2×C4×C53C22⋊Q814yes—
224C2×C4×C53#12314yes—
225C2×Dic53—24no—
226C2×Dic53—24no—
227C2×Dic53—24no—
228C2×Dic53—24no—
229C2×Dic53—24no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—24no—
233C2×Dic53—24no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—44no—
245C2×Dic53—44no—
246C2×Dic53—44no—
247C2×Dic53—44no—
248C2×Dic53#11718yescomplemented
249#13879—531nomaximal, Sylow
250C22×C53—24no—
251C22×C53—24no—
252C22×C53—24no—
253C22×C53—44no—
254C22×C53—44no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—44no—
258C22×C53—44no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53C22×C4⋊C411yescharacteristic
262C22×C53C2×#11712yes—
263C22×C53C2×C4×D411yescharacteristic
264C22×C53C2×C4×D411yescharacteristic
265C22×C53#52012yes—
266C22×C53C2×#12011yescharacteristic
267C22×C53C2×C22⋊Q811yescharacteristic
268C22×C53C2×C22⋊Q811yescharacteristic
269C22×C53C2×#12311yescharacteristic
270C22×C53#53211yescharacteristic
271C22×C53#53211yescharacteristic
272C22×C53#53312yes—
273C22×C53#53511yescharacteristic
274C22×C53#53611yescharacteristic
275C22×C53#54412yes—
276C22×C53#54612yes—
277C22×C53#54712yes—
278C22×C53#54812yes—
279C22×C53#55012yes—
280C22×C53#55212yes—
281C22×C53#55312yes—
282C22×C53#55512yes—
283C22×C53#55612yes—
284C4×C53—24no—
285C4×C53—44no—
286C4×C53—44no—
287C4×C53#38518yes—
288Dic53—28no—
289Dic53—44no—
290Dic53—82no—
291Dic53—82no—
292C22×#117D5311yescharacteristic
293C2×#384—531no—
294C2×#382—531no—
295C2×#380—531no—
296C2×C4×C22⋊C4—531no—
297#1439—534no—
298#1436—534no—
299C4⋊#95—531no—
300C4⋊#95—531no—
301C2×C53—44no—
302C2×C53#228511yescharacteristic
303C2×C53C2×#38511yescharacteristic
304C2×C53#231711yescharacteristic
305C2×C53#235212yes—
306C2×C53#235412yes—
307C2×C53#243111yescharacteristic
308C2×C53#243211yescharacteristic
309C2×C53#243912yes—
310C2×C53#244112yes—
311C2×C53#249911yescharacteristic
312C2×C53#250011yescharacteristic
313C24×C4C2×D5311yescharacteristic
314C2×#117—24no—
315C2×#117—24no—
316C2×#117—24no—
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⋊C4C2×D5312yes—
325C22×C42—531no—
326C22×C42—531no—
327C22×C42C2×D5312yes—
328#384—1064no—
329#382—1064no—
330#380—1064no—
331C4×C22⋊C4—538no—
332C2×#95—532no—
333C2×#95—534no—
334C2×#95—532no—
335C2×#95—534no—
336C2×#95—534no—
337C53#1387911yescharacteristic, complemented, Sylow
338C25C22×D5311yescharacteristic
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—22no—
342C23×C4—22no—
343C23×C4—532no—
344C23×C4—532no—
345C23×C4—532no—
346C23×C4—532no—
347C23×C4Dic10614yes—
348C23×C4C4×D5314yes—
349C23×C4C22⋊D5314yes—
350C23×C4C22×D5312yes—
351C23×C4C22×D5312yes—
352C23×C4C22×D5311yescharacteristic
353C23×C4C22×D5311yescharacteristic
354#117—232no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—1068no—
360C2×C4⋊C4—1062no—
361C2×C4⋊C4—1062no—
362C2×C4⋊C4—1068no—
363C2×C4⋊C4—1062no—
364C2×C4⋊C4—1062no—
365C2×C4⋊C4—1062no—
366C2×C4⋊C4—1062no—
367C2×C4⋊C4—2124no—
368C2×C4⋊C4C22⋊D5314yes—
369C2×C4⋊C4C22⋊D5314yes—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—538no—
374C2×C22⋊C4—1064no—
375C2×C22⋊C4—1064no—
376C2×C22⋊C4—1064no—
377C2×C22⋊C4—2124no—
378C2×C42—24no—
379C2×C42—24no—
380C2×C42—24no—
381C2×C42—534no—
382C2×C42—534no—
383C2×C42—1062no—
384C2×C42—1062no—
385C2×C42—1062no—
386C2×C42—1062no—
387#95—538no—
388#95—1068no—
389#95—1064no—
390#95—1068no—
391#95—2124no—
392C24—21no—
393C24—21no—
394C24—22no—
395C24—22no—
396C24—21no—
397C24—21no—
398C24—22no—
399C24—22no—
400C24C2×Dic10612yes—
401C24C2×C4×D5312yes—
402C24C2×C22⋊D5312yes—
403C24C23×D5311yescharacteristic
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—534no—
423C22×C4—534no—
424C22×C4—534no—
425C22×C4—534no—
426C22×C4—1062no—
427C22×C4—1062no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1064no—
441C22×C4—1064no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1064no—
445C22×C4—1064no—
446C22×C4#11719919116yescomplemented
447C22×C4C2×Dic10612yes—
448C22×C4C2×Dic10612yes—
449C22×C4C2×C4×D5312yes—
450C22×C4C2×C4×D5312yes—
451C22×C4#1171994512yes—
452C22×C4#1171994512yes—
453C22×C4C2×C22⋊D5314yes—
454C22×C4C2×C22⋊D5314yes—
455C22×C4C2×C22⋊D5312yes—
456C22×C4C2×C22⋊D5312yes—
457C22×C4C2×C22⋊D5312yes—
458C22×C4C2×C22⋊D5312yes—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—21216no—
462C4⋊C4—2124no—
463C4⋊C4—2124no—
464C22⋊C4—216no—
465C22⋊C4—10616no—
466C22⋊C4—2128no—
467C42—216no—
468C42—1068no—
469C42—1068no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C2×#1171991914yes—
493C23C22×C4×D5311yescharacteristic
494C23C22×C22⋊D5311yescharacteristic
495C23C22×C22⋊D5311yescharacteristic
496C23C22×C22⋊D5311yescharacteristic
497C23C22×Dic10611yescharacteristic
498C23#41×D5312yes—
499C23#41×D5312yes—
500C23C2×#1171994511yescharacteristic
501C23C2×#1171994511yescharacteristic
502C23#1313326312yes—
503C23#1313326412yes—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—232no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—538no—
522C2×C4—1064no—
523C2×C4—1064no—
524C2×C4—1064no—
525C2×C4—1064no—
526C2×C4—1064no—
527C2×C4—1064no—
528C2×C4—1064no—
529C2×C4—1064no—
530C2×C4—1064no—
531C2×C4—2124no—
532C2×C4—2124no—
533C2×C4—2124no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4—2124no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4C2×#1171991914yes—
546C2×C4C2×#1171991914yes—
547C2×C4C4×C22⋊D5314yes—
548C2×C4C4×C22⋊D5314yes—
549C2×C4#1313318614yes—
550C2×C4#1313319614yes—
551C2×C4#1313319614yes—
552C2×C4#1313320014yes—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22C22×#1171991911yescharacteristic
565C22#117×D5312yes—
566C22C2×C4×C22⋊D5311yescharacteristic
567C22C2×C4×C22⋊D5311yescharacteristic
568C22#1681110312yes—
569C22C2×#1313318611yescharacteristic
570C22C2×#1313319611yescharacteristic
571C22C2×#1313319611yescharacteristic
572C22C2×#1313320011yescharacteristic
573C22#1681116911yescharacteristic
574C22#1681116911yescharacteristic
575C22#1681117612yes—
576C22#1681118711yescharacteristic
577C22#1681119411yescharacteristic
578C22#1681125112yes—
579C22#1681126112yes—
580C22#1681127212yes—
581C22#1681128012yes—
582C22#1681129212yes—
583C22#1681131212yes—
584C22#1681131712yes—
585C22#1681133612yes—
586C22#1681134112yes—
587C4—24no—
588C4—44no—
589C4—44no—
590C4—1068no—
591C4—4242no—
592C4—4242no—
593C4—2124no—
594C4#1681049618yes—
595C2—44no—
596C2#2027711811yescharacteristic
597C2C2×#1681049611yescharacteristic
598C2#2027733511yescharacteristic
599C2#2027764512yes—
600C2#2027766512yes—
601C2#2027833011yescharacteristic
602C2#2027833911yescharacteristic
603C2#2027840012yes—
604C2#2027842212yes—
605C2#2027916411yescharacteristic
606C2#2027917311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC22×#117×C53C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC22×#117D53
hypocenterC53#13879
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. C2×#16810492C2semidirect
2. C2×#16810488C2semidirect
6. #20272948C2semidirect
7. #20272944C2semidirect
8. #20272944C2semidirect
20. C4×C22⋊Dic53C4semidirect
118. C22×C4×C53C4⋊C4semidirect
248. C2×Dic53#117semidirect
337. C53#13879semidirect
446. C22×C4#11719919semidirect

Maximal subgroups

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

subgroupindex
C2×#168104922
C2×#168104882
C2×#168104832
C2×C4×C22⋊Dic532
#202729582
#202729482
#202729442
C22×#117×C532
#1387953