Finite Group Databasebeta

← #27306335

Subgroups — #27306335

Order 13568.

2574 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic53C211yesmaximal, characteristic
2C2×#16810526C211yesmaximal, characteristic
3C2×#16810492C211yesmaximal, characteristic, complemented
4C2×#16810481C214yesmaximal, complemented
5C2×C4×#11719919C211yesmaximal, characteristic, complemented
6#20272958C214yesmaximal, complemented
7#20272945C211yesmaximal, characteristic, complemented
8#20272944C211yesmaximal, characteristic, complemented
9C2×C53×#390C211yesmaximal, characteristic
10C24×Dic53C2212yes—
11C2×D4×Dic53—24no—
12C2×D4×Dic53—24no—
13C2×D4×Dic53C418yescomplemented
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16C22×#11719919C2212yes—
17C22×#11719919C2212yes—
18C22×#11719919C2212yes—
19C22×C22⋊Dic53C2214yes—
20C22×C22⋊Dic53C2212yes—
21C22×C4×Dic53C2211yescharacteristic
22C22×C4×Dic53C2211yescharacteristic
23#16810526—24no—
24#16810492—24no—
25#16810481—216no—
26C4×#11719919—24no—
27C2×#13133124C2214yes—
28C2×#13133124C2214yes—
29C2×#13133124C2212yes—
30C23×D4×C53C2211yescharacteristic
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×C22⋊C4×C53C2214yes—
33C22×C42×C53C2211yescharacteristic
34C53×#390—24no—
35C23×Dic53—24no—
36C23×Dic53—24no—
37C23×Dic53—28no—
38C23×Dic53—28no—
39C23×Dic53C2×C418yes—
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2311yescharacteristic
45D4×Dic53—232no—
46C2×C4⋊Dic53—22no—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—22no—
51C2×C4⋊Dic53C2×C414yes—
52C2×#11719919—24no—
53C2×#11719919—28no—
54C2×#11719919—24no—
55C2×#11719919—24no—
56C2×#11719919—24no—
57C2×#11719919—44no—
58C2×#11719919D418yescomplemented
59C2×C22⋊Dic53—28no—
60C2×C22⋊Dic53—24no—
61C2×C22⋊Dic53—24no—
62C2×C22⋊Dic53—28no—
63C2×C22⋊Dic53—44no—
64C2×C22⋊Dic53C2×C418yes—
65C2×C4×Dic53—22no—
66C2×C4×Dic53—22no—
67C2×C4×Dic53—22no—
68C2×C4×Dic53C2×C414yes—
69C2×C4×Dic53D414yes—
70C2×C4×Dic53D414yes—
71#13133124—28no—
72#13133124—24no—
73#13133124—28no—
74C25×C53C2312yes—
75C22×D4×C53—22no—
76C22×D4×C53—22no—
77C22×D4×C53—22no—
78C22×D4×C53—22no—
79C22×D4×C53C2×C414yes—
80C22×D4×C53D414yes—
81C22×D4×C53Q814yes—
82C23×C4×C53C2312yes—
83C23×C4×C53C2312yes—
84C23×C4×C53C2311yescharacteristic
85C2×C4⋊C4×C53—22no—
86C2×C4⋊C4×C53—22no—
87C2×C4⋊C4×C53—22no—
88C2×C22⋊C4×C53—28no—
89C2×C22⋊C4×C53—28no—
90C2×C22⋊C4×C53—44no—
91C2×C42×C53—22no—
92C2×C42×C53—22no—
93C2×C42×C53—22no—
94C24×C53—24no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—24no—
100C24×C53—24no—
101C24×C53—24no—
102C24×C53C22×C414yes—
103C24×C53C2×D414yes—
104C24×C53C2×Q814yes—
105C24×C53C2411yescharacteristic
106C2×D4×C53—28no—
107C2×D4×C53—28no—
108C2×D4×C53—28no—
109C2×D4×C53—28no—
110C2×D4×C53—28no—
111C2×D4×C53—28no—
112C2×D4×C53C4⋊C4116yescomplemented
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—24no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—24no—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C2×D414yes—
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2×D412yes—
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2×Q812yes—
130C22×C4×C53#4112yes—
131C22×C4×C53#4112yes—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—24no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—24no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—216no—
145C22×Dic53—232no—
146C22×Dic53—24no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—24no—
151C22×Dic53—416no—
152C22×Dic53—48no—
153C22×Dic53—48no—
154C22×Dic53C22×C414yes—
155C22×Dic53C22×C412yes—
156C22×Dic53C22×C412yes—
157C22×Dic53C2×D414yes—
158C22×Dic53C2×D412yes—
159C22×Dic53C2×D412yes—
160C22×Dic53C2×D412yes—
161C22×Dic53C2×D412yes—
162C22×Dic53C2×D414yes—
163C22×Dic53#4114yes—
164C4⋊C4×C53—44no—
165C22⋊C4×C53—416no—
166C42×C53—44no—
167C22⋊Dic53—216no—
168C22⋊Dic53—416no—
169C4⋊Dic53—28no—
170C4⋊Dic53—44no—
171#11719919—216no—
172#11719919—48no—
173#11719919—48no—
174C4×Dic53—44no—
175C4×Dic53C22⋊C4116yescomplemented
176C23×C53—24no—
177C23×C53—24no—
178C23×C53—28no—
179C23×C53—28no—
180C23×C53—24no—
181C23×C53—24no—
182C23×C53—24no—
183C23×C53—24no—
184C23×C53—28no—
185C23×C53—28no—
186C23×C53—28no—
187C23×C53—28no—
188C23×C53—44no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—44no—
194C23×C53—44no—
195C23×C53—44no—
196C23×C53—44no—
197C23×C53—44no—
198C23×C53C2×C4⋊C418yes—
199C23×C53C23×C411yescharacteristic
200C23×C53C22×D411yescharacteristic
201C23×C53C22×D411yescharacteristic
202C23×C53C22×D412yes—
203C23×C53C22×D411yescharacteristic
204C23×C53C22×Q811yescharacteristic
205C23×C53C2×#4112yes—
206C23×C53C2×#4111yescharacteristic
207C23×C53C2×#4111yescharacteristic
208C23×C53H(5,𝔽2)12yes—
209C23×C532−1+412yes—
210D4×C53—232no—
211C2×C4×C53—24no—
212C2×C4×C53—24no—
213C2×C4×C53—24no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—44no—
222C2×C4×C53—44no—
223C2×C4×C53C2×C22⋊C414yes—
224C2×C4×C53C2×C4⋊C414yes—
225C2×C4×C53#11714yes—
226C2×C4×C53#12114yes—
227C2×C4×C53#12114yes—
228C2×C4×C53C22⋊Q814yes—
229C2×C4×C53C22⋊Q814yes—
230C2×Dic53—24no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—24no—
234C2×Dic53—24no—
235C2×Dic53—28no—
236C2×Dic53—24no—
237C2×Dic53—24no—
238C2×Dic53—24no—
239C2×Dic53—44no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—432no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—44no—
251C2×Dic53—44no—
252C2×Dic53C2×C22⋊C414yes—
253C2×Dic53C2×C22⋊C414yes—
254C2×Dic53C4×D418yes—
255C2×Dic53#12118yes—
256C2×Dic53#12414yes—
257C2×Dic53#12714yes—
258#13906—531nomaximal, Sylow
259C22×C53—216no—
260C22×C53—28no—
261C22×C53—44no—
262C22×C53—44no—
263C22×C53—48no—
264C22×C53—48no—
265C22×C53—48no—
266C22×C53—44no—
267C22×C53—44no—
268C22×C53—44no—
269C22×C53—44no—
270C22×C53C22×C22⋊C411yescharacteristic
271C22×C53C22×C4⋊C411yescharacteristic
272C22×C53C2×#11711yescharacteristic
273C22×C53C2×C4×D412yes—
274C22×C53#52012yes—
275C22×C53C2×#12112yes—
276C22×C53C2×#12111yescharacteristic
277C22×C53C2×#12111yescharacteristic
278C22×C53C2×C22⋊Q811yescharacteristic
279C22×C53C2×C22⋊Q811yescharacteristic
280C22×C53C2×#12411yescharacteristic
281C22×C53C2×#12711yescharacteristic
282C22×C53#53511yescharacteristic
283C22×C53#53611yescharacteristic
284C22×C53#53712yes—
285C22×C53D4212yes—
286C22×C53#54712yes—
287C22×C53#54712yes—
288C22×C53D4×Q812yes—
289C22×C53#55012yes—
290C22×C53#55412yes—
291C22×C53#55512yes—
292C22×C53#55612yes—
293C4×C53—44no—
294C4×C53—82no—
295C4×C53#38018yescomplemented
296Dic53—24no—
297Dic53—44no—
298Dic53—44no—
299Dic53—82no—
300Dic53#39018yescomplemented
301C22×C4×D4—531no—
302C2×#390—531no—
303C2×#390D5311yescharacteristic
304C2×#384—531no—
305C2×#380—534no—
306C2×C4×C4⋊C4—531no—
307#1439—534no—
308C4⋊#95—531no—
309C4⋊#95—531no—
310C2×C53—48no—
311C2×C53C2×#38011yescharacteristic
312C2×C53C2×#39011yescharacteristic
313C2×C53#231111yescharacteristic
314C2×C53D4×C4⋊C412yes—
315C2×C53#235412yes—
316C2×C53#253712yes—
317C2×C53#253812yes—
318C2×C53#254111yescharacteristic
319C2×C53#254211yescharacteristic
320C2×C53#254711yescharacteristic
321C2×C53#256311yescharacteristic
322C23×D4C2×D5311yescharacteristic
323C24×C4—532no—
324C2×C4×D4—538no—
325C2×C4×D4—1064no—
326C2×C4×D4—1064no—
327C22×C4⋊C4—532no—
328C22×C4⋊C4—531no—
329C22×C4⋊C4—532no—
330C22×C4⋊C4—531no—
331C22×C4⋊C4—532no—
332C22×C4⋊C4C2×D5311yescharacteristic
333C22×C22⋊C4—532no—
334C22×C22⋊C4—534no—
335C22×C22⋊C4C2×D5314yes—
336C22×C42—531no—
337C22×C42—531no—
338C22×C42C2×D5311yescharacteristic
339#390—24no—
340#390—1064no—
341#384—1064no—
342#380—10616no—
343C4×C4⋊C4—1064no—
344C2×#95—534no—
345C2×#95—534no—
346C2×#95—532no—
347C53#1390611yescharacteristic, complemented, Sylow
348C25C22×D5312yes—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4—22no—
353C22×D4Dic10614yes—
354C22×D4C4×D5314yes—
355C22×D4C22⋊D5314yes—
356C23×C4—531no—
357C23×C4—531no—
358C23×C4—538no—
359C23×C4—532no—
360C23×C4—532no—
361C23×C4—532no—
362C23×C4—1064no—
363C23×C4—1064no—
364C23×C4—1068no—
365C23×C4—1068no—
366C23×C4C22×D5312yes—
367C23×C4C22×D5312yes—
368C23×C4C22×D5311yescharacteristic
369C4×D4—10632no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—534no—
374C2×C4⋊C4—538no—
375C2×C4⋊C4—1064no—
376C2×C4⋊C4—1064no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4—1068no—
380C2×C4⋊C4—1064no—
381C2×C4⋊C4—1064no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4—1062no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4—2124no—
386C2×C22⋊C4—28no—
387C2×C22⋊C4—28no—
388C2×C22⋊C4—44no—
389C2×C22⋊C4—538no—
390C2×C22⋊C4—1064no—
391C2×C22⋊C4—1064no—
392C2×C22⋊C4—1068no—
393C2×C22⋊C4—1068no—
394C2×C22⋊C4—2124no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—534no—
399C2×C42—534no—
400C2×C42—534no—
401C2×C42—1062no—
402C2×C42—1062no—
403C2×C42—1062no—
404#95—1064no—
405#95—1068no—
406#95—1068no—
407C24—24no—
408C24—22no—
409C24—22no—
410C24—24no—
411C24—22no—
412C24—22no—
413C24—24no—
414C24—24no—
415C24C2×Dic10614yes—
416C24C2×C4×D5314yes—
417C24C2×C22⋊D5314yes—
418C24C23×D5311yescharacteristic
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4—28no—
423C2×D4—28no—
424C2×D4—28no—
425C2×D4#11719919116yescomplemented
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—532no—
438C22×C4—532no—
439C22×C4—532no—
440C22×C4—534no—
441C22×C4—534no—
442C22×C4—532no—
443C22×C4—532no—
444C22×C4—532no—
445C22×C4—534no—
446C22×C4—534no—
447C22×C4—10616no—
448C22×C4—10632no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1064no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1064no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1064no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1064no—
466C22×C4—21216no—
467C22×C4—2128no—
468C22×C4—2128no—
469C22×C4C2×Dic10612yes—
470C22×C4C2×C4×D5312yes—
471C22×C4D4×D5314yes—
472C22×C4D4×D5312yes—
473C22×C4D4×D5312yes—
474C22×C4#1171994712yes—
475C22×C4#1171994712yes—
476C22×C4C2×C22⋊D5312yes—
477C4⋊C4—44no—
478C4⋊C4—10616no—
479C4⋊C4—1068no—
480C4⋊C4—2128no—
481C4⋊C4—2128no—
482C4⋊C4—2124no—
483C22⋊C4—416no—
484C22⋊C4—10616no—
485C22⋊C4—21216no—
486C42—44no—
487C42—5316no—
488C42—2124no—
489C23—28no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23C2×#1171991918yes—
512C23C22×C4×D5311yescharacteristic
513C23C2×D4×D5312yes—
514C23C2×D4×D5311yescharacteristic
515C23C2×D4×D5311yescharacteristic
516C23C22×C22⋊D5311yescharacteristic
517C23C22×Dic10611yescharacteristic
518C23#41×D5312yes—
519C23C2×#1171994711yescharacteristic
520C23C2×#1171994711yescharacteristic
521C23#1313326212yes—
522C23#1313326412yes—
523D4—232no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—534no—
537C2×C4—538no—
538C2×C4—538no—
539C2×C4—534no—
540C2×C4—534no—
541C2×C4—534no—
542C2×C4—1064no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1064no—
546C2×C4—1064no—
547C2×C4—1064no—
548C2×C4—1068no—
549C2×C4—1064no—
550C2×C4—1064no—
551C2×C4—21232no—
552C2×C4—2124no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2124no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4—2124no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4C22⋊C4×D5314yes—
565C2×C4C2×#1171991914yes—
566C2×C4#1313317314yes—
567C2×C4#1313318914yes—
568C2×C4#1313318914yes—
569C2×C4#1313319414yes—
570C2×C4#1313319414yes—
571C22—28no—
572C22—216no—
573C22—44no—
574C22—48no—
575C22—44no—
576C22—48no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—48no—
581C22—44no—
582C22C2×C22⋊C4×D5311yescharacteristic
583C22C22×#1171991911yescharacteristic
584C22C2×#1313317311yescharacteristic
585C22C4×D4×D5312yes—
586C22#1681110212yes—
587C22#121×D5312yes—
588C22C2×#1313318911yescharacteristic
589C22C2×#1313318911yescharacteristic
590C22C2×#1313319411yescharacteristic
591C22C2×#1313319411yescharacteristic
592C22#124×D5311yescharacteristic
593C22#127×D5311yescharacteristic
594C22#1681118611yescharacteristic
595C22#1681119311yescharacteristic
596C22#1681120212yes—
597C22D4×C22⋊D5312yes—
598C22#1681127212yes—
599C22#1681127312yes—
600C22D4×Dic10612yes—
601C22#1681129312yes—
602C22#1681133212yes—
603C22#1681133712yes—
604C22#1681134212yes—
605C4—44no—
606C4—82no—
607C4—538no—
608C4—1064no—
609C4—4242no—
610C4—2124no—
611C4—2124no—
612C4#1681048118yescomplemented
613C2—48no—
614C2C2×#1681048111yescharacteristic
615C2#390×D5311yescharacteristic
616C2#2027730511yescharacteristic
617C2D4×#1171991912yes—
618C2#2027766312yes—
619C2#2027962912yes—
620C2#2027963512yes—
621C2#2027965911yescharacteristic
622C2#2027966611yescharacteristic
623C2#2027970811yescharacteristic
624C2#2027988611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#390C2
socle, cosocleC24×C53C24
prime radicalC23×D4×C53C22
hypercenterC2×#390D53
hypocenterC53#13906
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
3. C2×#16810492C2semidirect
4. C2×#16810481C2semidirect
5. C2×C4×#11719919C2semidirect
6. #20272958C2semidirect
7. #20272945C2semidirect
8. #20272944C2semidirect
13. C2×D4×Dic53C4semidirect
58. C2×#11719919D4semidirect
112. C2×D4×C53C4⋊C4semidirect
175. C4×Dic53C22⋊C4semidirect
295. C4×C53#380semidirect
300. Dic53#390semidirect
347. C53#13906semidirect
425. C2×D4#11719919semidirect
612. C4#16810481semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic532
C2×#168105262
C2×#168104922
C2×#168104812
C2×C4×#117199192
#202729582
#202729452
#202729442
C2×C53×#3902
#1390653