Finite Group Databasebeta

← #27320644

Subgroups — #27320644

Order 13568.

2038 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133216C211yesmaximal, characteristic, complemented
2C2×#16810535C212yesmaximal
3C2×#16810481C212yesmaximal
4C2×C22⋊C4×Dic53C211yesmaximal, characteristic
5#20272985C212yesmaximal, complemented
6#20272974C211yesmaximal, characteristic, complemented
7#20272972C211yesmaximal, characteristic, complemented
8#20272960C211yesmaximal, characteristic
9#20272943C212yesmaximal, complemented
10#20272943C211yesmaximal, characteristic, complemented
11C53×#1439C211yesmaximal, characteristic
12C23×Dic106C2212yes—
13C24×Dic53C2211yescharacteristic
14C2×#13133216—28no—
15C2×#13133216—24no—
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C2212yes—
18C22×#11719919C2212yes—
19C22×C22⋊Dic53C2212yes—
20C22×C22⋊Dic53C2211yescharacteristic
21C22×C4×Dic53C2211yescharacteristic
22C22×C4×Dic53C2211yescharacteristic
23#16810535—28no—
24#16810481—28no—
25C22⋊C4×Dic53—24no—
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2212yes—
37C22×C22⋊C4×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C22×Dic106—24no—
41C22×Dic106—24no—
42C22×Dic106—24no—
43C22×Dic106—24no—
44C22×Dic106—24no—
45C22×Dic106—42no—
46C22×Dic106—42no—
47C23×Dic53—22no—
48C23×Dic53—24no—
49C23×Dic53—24no—
50C23×Dic53—22no—
51C23×Dic53Q814yescomplemented
52C23×Dic53C2312yes—
53C23×Dic53C2311yescharacteristic
54C23×Dic53C2312yes—
55C23×Dic53C2311yescharacteristic
56C23×Dic53C2311yescharacteristic
57C23×Dic53C2311yescharacteristic
58#13133216—232no—
59C2×C4⋊Dic53—22no—
60C2×C4⋊Dic53—24no—
61C2×#11719919—24no—
62C2×#11719919—24no—
63C2×#11719919—24no—
64C2×#11719919—24no—
65C2×#11719919—24no—
66C2×#11719919D418yes—
67C2×C22⋊Dic53—24no—
68C2×C22⋊Dic53—22no—
69C2×C22⋊Dic53—24no—
70C2×C22⋊Dic53—22no—
71C2×C22⋊Dic53—42no—
72C2×C22⋊Dic53D414yes—
73C2×C4×Dic53—24no—
74C2×C4×Dic53—22no—
75C2×C4×Dic53—24no—
76C2×C4×Dic53D414yes—
77#13133124—22no—
78#13133124—24no—
79#13133124—22no—
80#13133124—24no—
81#13133124—24no—
82#13133124—24no—
83#13133124—22no—
84#13133124—24no—
85#13133124—42no—
86C25×C53C2311yescharacteristic
87C23×C4×C53C2312yes—
88C23×C4×C53C2312yes—
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2311yescharacteristic
91C2×C4⋊C4×C53—24no—
92C2×C4⋊C4×C53—22no—
93C2×C22⋊C4×C53—24no—
94C2×C22⋊C4×C53—22no—
95C2×C22⋊C4×C53—24no—
96C2×C22⋊C4×C53—22no—
97C2×C22⋊C4×C53—42no—
98C2×C22⋊C4×C53Q814yescomplemented
99#95×C53—22no—
100#95×C53—42no—
101C24×C53—21no—
102C24×C53—22no—
103C24×C53—21no—
104C24×C53—21no—
105C24×C53—21no—
106C24×C53—22no—
107C24×C53—22no—
108C24×C53—22no—
109C24×C53C2×D412yes—
110C24×C53C2×Q812yes—
111C24×C53C2×Q812yes—
112C24×C53C2411yescharacteristic
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
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×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53C2×D414yes—
133C22×C4×C53C2×D412yes—
134C22×C4×C53C2×Q812yes—
135C22×C4×C53C2×Q812yes—
136C22×C4×C53#4112yes—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—22no—
141C22×Dic53—21no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—28no—
154C22×Dic53—216no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—44no—
163C22×Dic53—44no—
164C22×Dic53—48no—
165C22×Dic53C2×D412yes—
166C22×Dic53C2×D414yes—
167C22×Dic53C2×D412yes—
168C22×Dic53C2×D412yes—
169C22×Dic53C2×D412yes—
170C22×Dic53C2×D414yes—
171C22×Dic53C2×Q812yes—
172C22×Dic53C2×Q812yes—
173C22×Dic53#4112yes—
174C22×Dic53#4114yes—
175C22×Dic53#4112yes—
176C2×Dic106—216no—
177C2×Dic106—48no—
178C2×Dic106—48no—
179C2×Dic106—48no—
180C2×Dic106—48no—
181C2×Dic106—48no—
182C2×Dic106—48no—
183C4⋊C4×C53—28no—
184C22⋊C4×C53—28no—
185C22⋊C4×C53—48no—
186C22⋊Dic53—28no—
187C22⋊Dic53—48no—
188C4⋊Dic53—28no—
189#11719919—216no—
190#11719919—48no—
191C4×Dic53—28no—
192C4×Dic53—44no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—24no—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53—24no—
201C23×C53—24no—
202C23×C53—24no—
203C23×C53—24no—
204C23×C53—24no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—42no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—42no—
211C23×C53—42no—
212C23×C53—42no—
213C23×C53—42no—
214C23×C53—42no—
215C23×C53C4⋊Q814yescomplemented
216C23×C53C22×D411yescharacteristic
217C23×C53C22×D412yes—
218C23×C53C22×D411yescharacteristic
219C23×C53C22×Q811yescharacteristic
220C23×C53C22×Q811yescharacteristic
221C23×C53C2×#4111yescharacteristic
222C23×C53C2×#4111yescharacteristic
223C23×C53C2×#4112yes—
224C23×C53C2×#4111yescharacteristic
225C23×C53H(5,𝔽2)12yes—
226C23×C532−1+412yes—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—22no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—42no—
237C2×C4×C53—41no—
238C2×C4×C53—42no—
239C2×C4×C53—41no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
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×C53C22⋊Q814yes—
251C2×Dic53—22no—
252C2×Dic53—22no—
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—41no—
263C2×Dic53—42no—
264C2×Dic53—41no—
265C2×Dic53—42no—
266C2×Dic53—41no—
267C2×Dic53—42no—
268C2×Dic53—41no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—416no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53#12014yes—
281C2×Dic53#12114yes—
282C2×Dic53C22⋊Q814yes—
283C2×Dic53C22⋊Q818yes—
284C2×Dic53#12414yes—
285C2×Dic53#12418yes—
286Dic106—432no—
287#15125—531nomaximal, Sylow
288C22×C53—24no—
289C22×C53—28no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—44no—
296C22×C53—44no—
297C22×C53—44no—
298C22×C53—42no—
299C22×C53C2×#12011yescharacteristic
300C22×C53C2×#12111yescharacteristic
301C22×C53C2×C22⋊Q811yescharacteristic
302C22×C53C2×C22⋊Q812yes—
303C22×C53C2×C22⋊Q811yescharacteristic
304C22×C53C2×#12411yescharacteristic
305C22×C53C2×#12412yes—
306C22×C53C2×C4⋊Q811yescharacteristic
307C22×C53#53211yescharacteristic
308C22×C53#53311yescharacteristic
309C22×C53#53311yescharacteristic
310C22×C53#53511yescharacteristic
311C22×C53#53611yescharacteristic
312C22×C53#54212yes—
313C22×C53#54412yes—
314C22×C53D4212yes—
315C22×C53#54612yes—
316C22×C53#54712yes—
317C22×C53#54812yes—
318C22×C53D4×Q812yes—
319C22×C53#55412yes—
320C22×C53#55512yes—
321C22×C53#55612yes—
322C4×C53—24no—
323C4×C53—42no—
324C4×C53—44no—
325C4×C53—82no—
326Dic53—24no—
327Dic53—42no—
328Dic53—44no—
329Dic53—42no—
330Dic53—82no—
331Dic53#39318yescomplemented
332C22×C4⋊Q8—531no—
333C2×#393—532no—
334C2×#380—532no—
335C2×C4×C22⋊C4—531no—
336#1446—532no—
337#1442—531no—
338#1442—531no—
339#1439—531no—
340#1439D5311yescharacteristic
341C4⋊#95—532no—
342C4⋊#95—531no—
343C2×C53—44no—
344C2×C53C2×#39311yescharacteristic
345C2×C53#240811yescharacteristic
346C2×C53#240911yescharacteristic
347C2×C53#243912yes—
348C2×C53#244012yes—
349C2×C53#249112yes—
350C2×C53#249212yes—
351C2×C53#254711yescharacteristic
352C2×C53#256311yescharacteristic
353C2×C53#262811yescharacteristic
354C2×C53#285711yescharacteristic
355C23×Q8—532no—
356C24×C4—531no—
357C2×C4⋊Q8—1064no—
358C2×C4⋊Q8—1068no—
359C22×C4⋊C4—532no—
360C22×C4⋊C4—532no—
361C22×C4⋊C4—531no—
362C22×C4⋊C4C2×D5311yescharacteristic
363C22×C22⋊C4—531no—
364C22×C22⋊C4—532no—
365C22×C22⋊C4C2×D5312yes—
366C22×C22⋊C4C2×D5311yescharacteristic
367C22×C42—531no—
368C22×C42—531no—
369#393—1068no—
370#380—1068no—
371C4×C22⋊C4—1064no—
372C2×#95—532no—
373C2×#95—532no—
374C2×#95—532no—
375C2×#95—531no—
376C2×#95—531no—
377C2×#95—532no—
378C2×#95—532no—
379C2×#95—531no—
380C2×#95—532no—
381C2×#95C2×D5311yescharacteristic
382C2×#95C2×D5312yes—
383C53#1512511yescharacteristic, complemented, Sylow
384C25C22×D5311yescharacteristic
385C22×Q8—1064no—
386C22×Q8—1064no—
387C22×Q8—1064no—
388C22×Q8—1064no—
389C22×Q8—1064no—
390C22×Q8—2122no—
391C22×Q8—2122no—
392C23×C4—531no—
393C23×C4—531no—
394C23×C4—534no—
395C23×C4—532no—
396C23×C4—531no—
397C23×C4—531no—
398C23×C4—532no—
399C23×C4—1062no—
400C23×C4—1064no—
401C23×C4—1064no—
402C23×C4—1062no—
403C23×C4C22×D5312yes—
404C23×C4C22×D5311yescharacteristic
405C23×C4C22×D5312yes—
406C23×C4C22×D5311yescharacteristic
407C4⋊Q8—10632no—
408C2×C4⋊C4—24no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—538no—
411C2×C4⋊C4—1064no—
412C2×C4⋊C4—1064no—
413C2×C4⋊C4—1064no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1064no—
416C2×C4⋊C4—1064no—
417C2×C4⋊C4—1064no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—24no—
421C2×C22⋊C4—24no—
422C2×C22⋊C4—42no—
423C2×C22⋊C4—534no—
424C2×C22⋊C4—1062no—
425C2×C22⋊C4—1062no—
426C2×C22⋊C4—1064no—
427C2×C22⋊C4—1064no—
428C2×C22⋊C4—2122no—
429C2×C22⋊C4Dic10614yescomplemented
430C2×C42—534no—
431C2×C42—1064no—
432C2×C42—1062no—
433C2×C42—1064no—
434#95—22no—
435#95—42no—
436#95—1062no—
437#95—1064no—
438#95—1064no—
439#95—1064no—
440#95—1064no—
441#95—1062no—
442#95—1064no—
443#95—1062no—
444#95—2122no—
445C24—22no—
446C24—21no—
447C24—21no—
448C24—22no—
449C24—21no—
450C24—21no—
451C24—22no—
452C24—22no—
453C24C2×Dic10612yes—
454C24D4×D5312yes—
455C24Q8×D5312yes—
456C24C23×D5311yescharacteristic
457C2×Q8—10616no—
458C2×Q8—2128no—
459C2×Q8—2128no—
460C2×Q8—2128no—
461C2×Q8—2128no—
462C2×Q8—2128no—
463C2×Q8—2128no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
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—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—532no—
484C22×C4—534no—
485C22×C4—532no—
486C22×C4—534no—
487C22×C4—532no—
488C22×C4—532no—
489C22×C4—532no—
490C22×C4—532no—
491C22×C4—532no—
492C22×C4—534no—
493C22×C4—532no—
494C22×C4—1068no—
495C22×C4—10616no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1061no—
506C22×C4—1062no—
507C22×C4—1061no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4—1062no—
515C22×C4—1062no—
516C22×C4—1062no—
517C22×C4—1062no—
518C22×C4—1062no—
519C22×C4—2124no—
520C22×C4—2124no—
521C22×C4—2128no—
522C22×C4C2×Dic10612yes—
523C22×C4C2×Dic10612yes—
524C22×C4D4×D5314yes—
525C22×C4D4×D5312yes—
526C22×C4#1171994712yes—
527C4⋊C4—28no—
528C4⋊C4—10616no—
529C4⋊C4—1068no—
530C4⋊C4—2128no—
531C22⋊C4—28no—
532C22⋊C4—48no—
533C22⋊C4—1068no—
534C22⋊C4—2128no—
535C42—1068no—
536C42—2124no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—22no—
541C23—24no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—41no—
553C23—42no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23#1313321614yescomplemented
560C23C2×D4×D5311yescharacteristic
561C23C2×D4×D5312yes—
562C23C2×D4×D5311yescharacteristic
563C23C2×Q8×D5311yescharacteristic
564C23C22×Dic10611yescharacteristic
565C23#41×D5312yes—
566C23#41×D5311yescharacteristic
567C23#41×D5311yescharacteristic
568C23C2×#1171994711yescharacteristic
569C23#1313326212yes—
570C23#1313326412yes—
571Q8—21232no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—534no—
596C2×C4—538no—
597C2×C4—534no—
598C2×C4—534no—
599C2×C4—538no—
600C2×C4—534no—
601C2×C4—1062no—
602C2×C4—1062no—
603C2×C4—1064no—
604C2×C4—1064no—
605C2×C4—1064no—
606C2×C4—1064no—
607C2×C4—1062no—
608C2×C4—1064no—
609C2×C4—1062no—
610C2×C4—1064no—
611C2×C4—1064no—
612C2×C4—21216no—
613C2×C4—2122no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4—2122no—
621C2×C4—2121no—
622C2×C4—2122no—
623C2×C4—2121no—
624C2×C4—2122no—
625C2×C4—2121no—
626C2×C4—2122no—
627C2×C4—2121no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4#1313319414yes—
631C22—28no—
632C22—24no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—44no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22#120×D5311yescharacteristic
643C22#121×D5311yescharacteristic
644C22C22⋊Q8×D5312yes—
645C22C22⋊Q8×D5311yescharacteristic
646C22C2×#1313319411yescharacteristic
647C22#124×D5312yes—
648C22#124×D5311yescharacteristic
649C22C2×#1313321611yescharacteristic
650C22#1681116811yescharacteristic
651C22#1681117511yescharacteristic
652C22#1681117611yescharacteristic
653C22#1681118611yescharacteristic
654C22#1681119511yescharacteristic
655C22#1681124112yes—
656C22#1681125212yes—
657C22#1681125512yes—
658C22#1681126212yes—
659C22#1681127112yes—
660C22#1681127912yes—
661C22D4×Dic10612yes—
662C22#1681133212yes—
663C22#1681133712yes—
664C22#1681134012yes—
665C4—24no—
666C4—44no—
667C4—42no—
668C4—82no—
669C4—538no—
670C4—1064no—
671C4—4242no—
672C4—2124no—
673C4—2122no—
674C4—2122no—
675C2—44no—
676C2#393×D5311yescharacteristic
677C2#2027806411yescharacteristic
678C2#2027807711yescharacteristic
679C2#2027840112yes—
680C2#2027841612yes—
681C2#2027907112yes—
682C2#2027908612yes—
683C2#2027970611yescharacteristic
684C2#2027988611yescharacteristic
685C2#2028054111yescharacteristic
686C2#2028306511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
FittingC53×#1439C2
prime radicalC25×C53C23
hypercenter#1439D53
hypocenterC53#15125
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C22×#13133216C2semidirect
5. #20272985C2semidirect
6. #20272974C2semidirect
7. #20272972C2semidirect
9. #20272943C2semidirect
10. #20272943C2semidirect
51. C23×Dic53Q8semidirect
98. C2×C22⋊C4×C53Q8semidirect
215. C23×C53C4⋊Q8semidirect
331. Dic53#393semidirect
383. C53#15125semidirect
429. C2×C22⋊C4Dic106semidirect
559. C23#13133216semidirect

Maximal subgroups

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

subgroupindex
C22×#131332162
C2×#168105352
C2×#168104812
C2×C22⋊C4×Dic532
#202729852
#202729742
#202729722
#202729602
#202729432
C53×#14392
#1512553