Finite Group Databasebeta

← #27406342

Subgroups — #27406342

Order 13568.

1030 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810826C211yesmaximal, characteristic
2C2×#16810714C211yesmaximal, characteristic
3#20277769C211yesmaximal, characteristic, complemented
4D4×#11719919C211yesmaximal, characteristic
5#20275515C211yesmaximal, characteristic, complemented
6#20275475C211yesmaximal, characteristic, complemented
7#20275391C211yesmaximal, characteristic, complemented
8#20275388C211yesmaximal, characteristic, complemented
9#20275051C211yesmaximal, characteristic
10#20275049C211yesmaximal, characteristic
11#20274983C211yesmaximal, characteristic, complemented
12#20274980C211yesmaximal, characteristic, complemented
13C8×#11719919C211yesmaximal, characteristic, complemented
14#102×Dic53C211yesmaximal, characteristic
15C2×C4×SD16×C53C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×#13133215C2211yescharacteristic
18C2×#13133182C2211yescharacteristic
19C2×D4×Dic53C2211yescharacteristic
20C2×D4×Dic53C2211yescharacteristic
21C22×#11719919—21no—
22#16810826C418yescomplemented
23#16810714—24no—
24C2×#13133148C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33C2×C8×Dic53C2211yescharacteristic
34#16810530C2211yescharacteristic
35#16810529C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810519C2211yescharacteristic
38#16810518—21no—
39#16810507—21no—
40#16810504—21no—
41#16810496—21no—
42#16810495—21no—
43#16810492C2211yescharacteristic
44#16810481—21no—
45C4⋊C4×Dic53C2211yescharacteristic
46C4×#11719919—21no—
47C4×#11719919C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810378C2211yescharacteristic
53#16810377C2211yescharacteristic
54C22×SD16×C53C2211yescharacteristic
55C2×C4×Q8×C53C2211yescharacteristic
56C2×C4×D4×C53C2211yescharacteristic
57C4×SD16×C53—24no—
58C2×#106×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#102×C53C2211yescharacteristic
61C2×C4×C8×C53C2211yescharacteristic
62C22×Dic106C2311yescharacteristic
63C23×Dic53—21no—
64C23×Dic53—21no—
65C2×#11719923—22no—
66C2×#11719923—22no—
67C2×#11719923C2×C414yes—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133215—22no—
71#13133215C2×C414yescomplemented
72#13133182—22no—
73#13133182—41no—
74D4×Dic53—22no—
75D4×Dic53—22no—
76D4×Dic53—41no—
77D4×Dic53C2×C414yes—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—44no—
89C2×#11719919D411yescharacteristic
90C2×#11719919D411yescharacteristic
91C2×C22⋊Dic53—21no—
92C2×C22⋊Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53C2311yescharacteristic
96C2×C4×Dic53C2311yescharacteristic
97#13133148—22no—
98#13133146C2×C414yescomplemented
99#13133142—22no—
100#13133142C2×C414yescomplemented
101#13133141—41no—
102#13133141C2×C414yescomplemented
103#13133140—41no—
104#13133138—22no—
105C8×Dic53—22no—
106C8×Dic53—22no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124D411yescharacteristic
115#13133124D411yescharacteristic
116C22×Q8×C53C2311yescharacteristic
117C22×D4×C53C2311yescharacteristic
118C23×C4×C53—21no—
119C2×SD16×C53—22no—
120C2×SD16×C53—22no—
121C2×SD16×C53C2×C414yes—
122C22×C8×C53C2311yescharacteristic
123C22×C8×C53C2311yescharacteristic
124C4×Q8×C53—22no—
125C4×Q8×C53—41no—
126C4×D4×C53—22no—
127C4×D4×C53—41no—
128C2×C4⋊C4×C53—21no—
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C22⋊C4×C53—21no—
132C2×C42×C53—21no—
133C2×C42×C53C2311yescharacteristic
134#106×C53—22no—
135#103×C53—22no—
136#102×C53D414yes—
137C4×C8×C53—22no—
138C24×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53C22×C412yes—
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—21no—
148C2×D4×C53C22×C412yes—
149C2×D4×C53C2×D412yes—
150C2×D4×C53#4112yes—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—42no—
155C22×C4×C53—42no—
156C22×C4×C53—81no—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2411yescharacteristic
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—24no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—42no—
166C22×Dic53—42no—
167C22×Dic53—42no—
168C22×Dic53—42no—
169C22×Dic53—81no—
170C22×Dic53C2×D411yescharacteristic
171C22×Dic53C2×D411yescharacteristic
172C22×Dic53C2×D411yescharacteristic
173C22×Dic53C2×D411yescharacteristic
174C2×Dic106—21no—
175C2×Dic106—21no—
176C2×Dic106—21no—
177C2×Dic106—21no—
178C2×Dic106—21no—
179C2×Dic106C22×C412yes—
180SD16×C53—28no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C2×C8×C53C22×C412yes—
186C2×C8×C53C2×D412yes—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53—42no—
191C4⋊C4×C53C2×D412yes—
192C22⋊C4×C53—42no—
193C42×C53—21no—
194C42×C53—21no—
195C42×C53—42no—
196C22⋊Dic53—24no—
197C22⋊Dic53—42no—
198#11719923—28no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53C22×C412yes—
205#11719919—24no—
206#11719919—24no—
207#11719919—22no—
208#11719919—42no—
209#11719919—42no—
210#11719919—42no—
211#11719919—42no—
212#11719919—81no—
213#11719919SD1614yescomplemented
214C4×Dic53—21no—
215C4×Dic53—21no—
216C4×Dic53—42no—
217C4×Dic53—42no—
218C4×Dic53C22×C412yes—
219C4×Dic53#4112yes—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322C22×C412yes—
226C23×C53—22no—
227C23×C53—22no—
228C23×C53—22no—
229C23×C53—41no—
230C23×C53—41no—
231C23×C53—41no—
232C23×C53—41no—
233C23×C53C22×D411yescharacteristic
234Q8×C53—22no—
235Q8×C53—22no—
236Q8×C53—41no—
237Q8×C53—41no—
238D4×C53—22no—
239D4×C53—22no—
240D4×C53—22no—
241D4×C53C4×D414yes—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—84no—
257C2×C4×C53C2×SD1612yes—
258C2×C4×C53C23×C411yescharacteristic
259C2×C4×C53C22×D411yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53H(5,𝔽2)11yescharacteristic
263C2×C4×C532−1+411yescharacteristic
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—22no—
269C2×Dic53—22no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—48no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—84no—
285C2×Dic53C4×D412yes—
286C2×Dic53C4×D412yes—
287C2×Dic53C2×SD1612yes—
288C2×Dic53C2×SD1612yes—
289C2×Dic53#13512yes—
290Dic106—22no—
291Dic106—22no—
292Dic106—41no—
293Dic106—41no—
294C8×C53—22no—
295C8×C53—22no—
296#88322—22no—
297#88322—41no—
298#22996—531nomaximal, Sylow
299C22×C53—24no—
300C22×C53—42no—
301C22×C53—42no—
302C22×C53—42no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53—42no—
306C22×C53C2×C4×D411yescharacteristic
307C22×C53D4211yescharacteristic
308C22×C53#54811yescharacteristic
309C22×C53C22×SD1611yescharacteristic
310C22×C53C2×#13511yescharacteristic
311C22×C53#57711yescharacteristic
312C22×C53#57811yescharacteristic
313C4×C53—22no—
314C4×C53—42no—
315C4×C53—41no—
316C4×C53—42no—
317C4×C53—81no—
318C4×C53C2×C4×D411yescharacteristic
319C4×C53#52011yescharacteristic
320C4×C53#54711yescharacteristic
321C4×C53#55311yescharacteristic
322Dic53—22no—
323Dic53—22no—
324Dic53—42no—
325Dic53—42no—
326Dic53—41no—
327Dic53—81no—
328Dic53—81no—
329Dic53C4×SD1614yescomplemented
330C2×#461—531no—
331C2×C4×SD16—531no—
332C2×C4×SD16D5311yescharacteristic
333#2365—531no—
334D4×C4⋊C4—531no—
335#1956—531no—
336#1948—531no—
337#1933—531no—
338#1933—531no—
339#1877—531no—
340#1877—531no—
341#1865—531no—
342#1865—531no—
343C8×C4⋊C4—531no—
344C4×#102—531no—
345C2×C53—44no—
346C2×C53#236111yescharacteristic
347C2×C53C2×C4×SD1611yescharacteristic
348C2×C53#294711yescharacteristic
349C2×C53D4×SD1611yescharacteristic
350C2×C53#328311yescharacteristic
351C2×C53#335911yescharacteristic
352C2×C53#336011yescharacteristic
353C22×SD16—531no—
354C22×SD16C2×D5311yescharacteristic
355C2×C4⋊Q8—531no—
356C2×C4×Q8—531no—
357C2×C4×Q8C2×D5311yescharacteristic
358C2×C4×D4—531no—
359C2×C4×D4—531no—
360C2×C4×D4C2×D5311yescharacteristic
361C22×C4⋊C4—1061no—
362#461—538no—
363C4×SD16—24no—
364C4×SD16—1064no—
365C2×#106—531no—
366C2×#106C2×D5311yescharacteristic
367C2×C4⋊C8—531no—
368C2×#103—531no—
369C2×#103—531no—
370C2×#103—531no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#103C2×D5311yescharacteristic
374C2×#102—531no—
375C2×#102C2×D5311yescharacteristic
376C2×C4×C8—531no—
377C2×C4×C8—531no—
378C2×C4×C8C2×D5311yescharacteristic
379#391—531no—
380#391—531no—
381#389—531no—
382#389—531no—
383#389—1061no—
384#387—1061no—
385#387—1061no—
386#385—1061no—
387#385—1061no—
388#384—531no—
389#380—1061no—
390C4×C4⋊C4—531no—
391C4×C4⋊C4—531no—
392C4×C4⋊C4—1061no—
393#340—531no—
394#340—531no—
395#340—531no—
396#340—531no—
397#336—531no—
398#336—531no—
399C53#2299611yescharacteristic, complemented, Sylow
400C22×Q8—531no—
401C22×Q8C22×D5311yescharacteristic
402C22×D4C22×D5311yescharacteristic
403C23×C4—21no—
404C23×C4—1061no—
405C23×C4—1061no—
406C2×SD16—22no—
407C2×SD16—22no—
408C2×SD16—534no—
409C2×SD16—1062no—
410C2×SD16—1062no—
411C2×SD16C4×D5314yes—
412C22×C8—531no—
413C22×C8—531no—
414C22×C8C22×D5311yescharacteristic
415C22×C8C22×D5311yescharacteristic
416C4⋊Q8—534no—
417C4⋊Q8—1062no—
418C4×Q8—22no—
419C4×Q8—41no—
420C4×Q8—1062no—
421C4×Q8—2121no—
422C4×D4—22no—
423C4×D4—41no—
424C4×D4—534no—
425C4×D4—1062no—
426C4×D4—1062no—
427C4×D4—2121no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—531no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—2124no—
442C2×C4⋊C4C22×D5311yescharacteristic
443C2×C4⋊C4C22×D5311yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—1061no—
446C2×C22⋊C4—1061no—
447C2×C42—21no—
448C2×C42—531no—
449C2×C42—531no—
450C2×C42—1061no—
451C2×C42—1061no—
452C2×C42C22×D5311yescharacteristic
453#106—22no—
454#106—1062no—
455C4⋊C8—534no—
456#103—22no—
457#103—534no—
458#103—534no—
459#103—1062no—
460#103—2121no—
461#103—2121no—
462#102—1062no—
463#102C22⋊D5314yes—
464C4×C8—22no—
465C4×C8—1062no—
466C4×C8—1062no—
467#95—531no—
468#95—531no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473#95—1061no—
474#95—1061no—
475#95—1061no—
476C24—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—532no—
483C2×Q8—1061no—
484C2×Q8—1061no—
485C2×Q8—1061no—
486C2×Q8—1061no—
487C2×Q8—1061no—
488C2×Q8C2×C4×D5312yes—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4C2×C4×D5312yes—
493C2×D4#1171994512yes—
494C2×D4C2×C22⋊D5312yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—81no—
501C22×C4—531no—
502C22×C4—531no—
503C22×C4—531no—
504C22×C4—531no—
505C22×C4—1064no—
506C22×C4—1061no—
507C22×C4—1061no—
508C22×C4—1061no—
509C22×C4—1061no—
510C22×C4—4241no—
511C22×C4—2122no—
512C22×C4—2122no—
513C22×C4—2122no—
514C22×C4—2122no—
515C22×C4D4×D5311yescharacteristic
516C22×C4D4×D5311yescharacteristic
517C22×C4C23×D5311yescharacteristic
518SD16—28no—
519SD16—1068no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—532no—
525C2×C8—1061no—
526C2×C8—1061no—
527C2×C8—1061no—
528C2×C8—1061no—
529C2×C8—1061no—
530C2×C8C2×C4×D5312yes—
531C2×C8C2×C22⋊D5312yes—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—42no—
536C4⋊C4—534no—
537C4⋊C4—532no—
538C4⋊C4—1062no—
539C4⋊C4—1064no—
540C4⋊C4—1064no—
541C4⋊C4—1061no—
542C4⋊C4—1061no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4—4241no—
547C4⋊C4—2122no—
548C4⋊C4—2122no—
549C4⋊C4—2122no—
550C4⋊C4—2122no—
551C4⋊C4C2×C22⋊D5312yes—
552C22⋊C4—42no—
553C22⋊C4—1064no—
554C22⋊C4—2122no—
555C42—21no—
556C42—21no—
557C42—42no—
558C42—532no—
559C42—532no—
560C42—1061no—
561C42—1061no—
562C42—2122no—
563C42—2122no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23C2×D4×D5311yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—41no—
575Q8—41no—
576Q8—1062no—
577Q8—1062no—
578Q8—2121no—
579Q8—2121no—
580D4—22no—
581D4—22no—
582D4—22no—
583D4C4×C22⋊D5314yes—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—532no—
600C2×C4—532no—
601C2×C4—532no—
602C2×C4—532no—
603C2×C4—532no—
604C2×C4—1061no—
605C2×C4—1061no—
606C2×C4—1061no—
607C2×C4—1061no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—1061no—
611C2×C4—1061no—
612C2×C4—1061no—
613C2×C4—4244no—
614C2×C4—2128no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4—2121no—
624C2×C4—2121no—
625C2×C4SD16×D5312yes—
626C2×C4C22×C4×D5311yescharacteristic
627C2×C4C22×C22⋊D5311yescharacteristic
628C2×C4#41×D5311yescharacteristic
629C2×C4C2×#1171994511yescharacteristic
630C2×C4#1313326211yescharacteristic
631C2×C4#1313326411yescharacteristic
632C8—22no—
633C8—22no—
634C8—1062no—
635C8—2121no—
636C22—24no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22C4×D4×D5311yescharacteristic
644C22D4×C22⋊D5311yescharacteristic
645C22#1681128211yescharacteristic
646C22C2×SD16×D5311yescharacteristic
647C22#135×D5311yescharacteristic
648C22#1681142911yescharacteristic
649C22#1681143711yescharacteristic
650C4—22no—
651C4—41no—
652C4—42no—
653C4—42no—
654C4—81no—
655C4—534no—
656C4—1062no—
657C4—1062no—
658C4—4241no—
659C4—4241no—
660C4—2122no—
661C4—2121no—
662C4—2122no—
663C4C2×C4×C22⋊D5311yescharacteristic
664C4#1681110211yescharacteristic
665C4#1681127211yescharacteristic
666C4#1681132211yescharacteristic
667C2—44no—
668C2#2027773211yescharacteristic
669C2C4×SD16×D5311yescharacteristic
670C2#2028355111yescharacteristic
671C2SD16×C22⋊D5311yescharacteristic
672C2#2028643111yescharacteristic
673C2#2028735511yescharacteristic
674C2#2028736511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C4×SD16×C53C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenterC2×C4×SD16D53
hypocenterC53#22996
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
3. #20277769C2semidirect
5. #20275515C2semidirect
6. #20275475C2semidirect
7. #20275391C2semidirect
8. #20275388C2semidirect
11. #20274983C2semidirect
12. #20274980C2semidirect
13. C8×#11719919C2semidirect
22. #16810826C4semidirect
71. #13133215C2×C4semidirect
98. #13133146C2×C4semidirect
100. #13133142C2×C4semidirect
102. #13133141C2×C4semidirect
213. #11719919SD16semidirect
329. Dic53C4×SD16semidirect
399. C53#22996semidirect

Maximal subgroups

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

subgroupindex
C2×#168108262
C2×#168107142
#202777692
D4×#117199192
#202755152
#202754752
#202753912
#202753882
#202750512
#202750492
#202749832
#202749802
C8×#117199192
#102×Dic532
C2×C4×SD16×C532
#2299653