Finite Group Databasebeta

← D53⋊#1503 #28314993

Subgroups — D53⋊#1503 #28314993

Order 13568.

1194 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D53⋊#431C211yesmaximal, characteristic
2C2×C4×C4⋊#6157C211yesmaximal, characteristic
3#20290099C212yesmaximal, complemented
4#20290098C212yesmaximal, complemented
5#20290097C214yesmaximal, complemented
6D53⋊#328C214yesmaximal
7#533×D53C211yesmaximal, characteristic
8C22×C4⋊#6157—21no—
9C22×C4×#6157—21no—
10#16811642C2212yes—
11#16811641C2212yes—
12#16811640—21no—
13#16811639—21no—
14D53⋊(C4⋊C8)—21no—
15D53⋊(C22⋊C8)C2212yes—
16C4×D53⋊C8—21no—
17C4×C4⋊#6157—21no—
18C4×C4⋊#6157—21no—
19C4×C4⋊#6157C2212yes—
20C4×C4⋊#6157C2212yes—
21C2×(C2×#6157)⋊C4C2212yes—
22#16811477C2218yes—
23#16811476C2218yes—
24#16811180C411yescharacteristic, complemented
25#16811179—21no—
26#16811178C2211yescharacteristic
27#16811178C2211yescharacteristic
28#16811177—21no—
29#16811177C2212yes—
30#16811176—21no—
31#16811175—21no—
32#16811175C412yescomplemented
33C4⋊Q8×D53C411yescharacteristic, complemented
34C4⋊Q8×D53C411yescharacteristic, complemented
35#125×D53—21no—
36C22⋊Q8×D53—21no—
37C22⋊Q8×D53C2212yes—
38C4×Q8×D53—21no—
39C4×Q8×D53C412yescomplemented
40#117×D53—21no—
41C2×C42×D53C2211yescharacteristic
42C53×#533C411yescharacteristic, complemented
43C23×#6157—22no—
44C2×D53⋊C8—22no—
45C2×C4⋊#6157—28no—
46C2×C4⋊#6157—22no—
47C2×C4⋊#6157—22no—
48C2×C4×#6157—22no—
49C2×C4×#6157—22no—
50C2×C4×#6157—44no—
51#13133284—22no—
52C4⋊#88324—22no—
53#13133274C2314yes—
54C22⋊#88324—22no—
55C4×#88324—22no—
56C4×#88324—22no—
57(C2×#6157)⋊C4—22no—
58(C2×#6157)⋊C4C2314yes—
59C2×Q8×D53—21no—
60C2×Q8×D53C2×C412yes—
61C22×C4×D53D411yescharacteristic
62C22×C4×D53D411yescharacteristic
63C22×C4×D53C2311yescharacteristic
64#13133217C2×C411yescharacteristic, complemented
65#13133217C2×C411yescharacteristic, complemented
66#13133216—21no—
67#13133216—21no—
68#13133216—21no—
69#13133216C2×C412yescomplemented
70#13133215—21no—
71#13133215C2×C412yescomplemented
72#13133209—21no—
73#13133208—21no—
74#13133208—21no—
75#13133207—22no—
76#13133207—22no—
77#13133197—21no—
78#13133197C2312yes—
79#13133196—21no—
80#13133196C2×C412yes—
81#13133195—22no—
82#13133195—22no—
83#13133194—22no—
84#13133194—22no—
85#13133193—21no—
86#13133193C2312yes—
87Q8×Dic53—21no—
88Q8×Dic53C2×C412yescomplemented
89C4×Dic106—21no—
90C4×Dic106—22no—
91C4×Dic106—22no—
92C4×Dic106C2×C412yescomplemented
93#13133182—22no—
94#13133182—22no—
95#13133182—21no—
96#13133182C2×C412yescomplemented
97#13133176—21no—
98#13133175—21no—
99#13133175—21no—
100#13133174—22no—
101#13133173—22no—
102C4⋊C4×D53—21no—
103C4⋊C4×D53—21no—
104C4⋊C4×D53—21no—
105C4⋊C4×D53—22no—
106C4⋊C4×D53—22no—
107C4⋊C4×D53C2×C412yes—
108C22⋊C4×D53—22no—
109C2×C4×Dic53C2×C411yescharacteristic
110C2×C4×Dic53D411yescharacteristic
111C2×C4×Dic53D411yescharacteristic
112C42×D53—22no—
113C42×D53C2×C411yescharacteristic
114C42×D53C2×C411yescharacteristic
115C42×D53D411yescharacteristic
116C42×D53D411yescharacteristic
117C42×D53D411yescharacteristic
118C42×D53D411yescharacteristic
119C42×D53C2311yescharacteristic
120C42×D53C2311yescharacteristic
121C4⋊Q8×C53C2×C411yescharacteristic, complemented
122C4⋊Q8×C53C2×C411yescharacteristic, complemented
123#125×C53—21no—
124C22⋊Q8×C53—21no—
125C22⋊Q8×C53C2×C412yes—
126C4×Q8×C53—21no—
127C4×Q8×C53C2×C412yescomplemented
128#117×C53—21no—
129C2×C42×C53C2×C411yescharacteristic
130C23×D53C2×D411yescharacteristic
131C22×#6157—28no—
132C22×#6157—24no—
133C22×#6157—44no—
134C22×#6157—44no—
135C2×Q8×C53—21no—
136C2×Q8×C53C22×C412yes—
137C22×C4×C53C22⋊C411yescharacteristic
138C22×C4×C53C22⋊C411yescharacteristic
139C22×C4×C53C22×C411yescharacteristic
140C22×Dic53C22⋊C411yescharacteristic
141C22×Dic53C22⋊C411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143Q8×D53—44no—
144Q8×D53—44no—
145C2×C4×D53—24no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—42no—
149C2×C4×D53C22⋊C411yescharacteristic
150C2×C4×D53C22⋊C411yescharacteristic
151C2×C4×D53C22⋊C411yescharacteristic
152C2×C4×D53C22⋊C411yescharacteristic
153C2×C4×D53SD1614yes—
154C2×C4×D53Q1614yes—
155C2×C4×D53C22×C411yescharacteristic
156C2×C4×D53C2×D411yescharacteristic
157C2×C4×D53C2×D411yescharacteristic
158C2×C4×D53C2×D411yescharacteristic
159C2×C4×D53C2×D411yescharacteristic
160C2×C4×D53C2×D411yescharacteristic
161C2×C4×D53C2×D411yescharacteristic
162C2×C4×D53C2411yescharacteristic
163C2×Dic106—21no—
164C2×Dic106—22no—
165C2×Dic106—22no—
166C2×Dic106C22×C412yes—
167C2×#88324—24no—
168C4⋊#6157—28no—
169C4⋊#6157—28no—
170C4×#6157—44no—
171C4×#6157—44no—
172D53⋊C8—44no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53C22×C412yes—
179C22⋊C4×C53—22no—
180C42×C53—22no—
181C42×C53C22⋊C411yescharacteristic
182C42×C53C22⋊C411yescharacteristic
183C42×C53C22×C411yescharacteristic
184C42×C53C22×C411yescharacteristic
185C22⋊Dic53—22no—
186#11719921—22no—
187#11719921—22no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—22no—
192C4⋊Dic53—22no—
193C4⋊Dic53C22×C412yes—
194#11719919—22no—
195#11719919—22no—
196#11719919—22no—
197#11719919—22no—
198#11719919—22no—
199#11719919—22no—
200#11719919—22no—
201#11719919—22no—
202C4×Dic53—22no—
203C4×Dic53—22no—
204C4×Dic53—22no—
205C4×Dic53C22⋊C411yescharacteristic
206C4×Dic53C22⋊C411yescharacteristic
207C4×Dic53C22⋊C411yescharacteristic
208C4×Dic53C22⋊C411yescharacteristic
209C4×Dic53C22⋊C411yescharacteristic
210C4×Dic53C22⋊C411yescharacteristic
211C4×Dic53C22×C411yescharacteristic
212C4×Dic53C22×C411yescharacteristic
213C4×Dic53C2×D411yescharacteristic
214C4×Dic53C2×D411yescharacteristic
215C4×Dic53C2×D411yescharacteristic
216C4×Dic53C2×D411yescharacteristic
217C23×C53C2×C22⋊C411yescharacteristic
218C22×D53—22no—
219C22×D53—22no—
220C22×D53C2×C22⋊C411yescharacteristic
221C22×D53C2×SD1612yes—
222C22×D53C2×Q1612yes—
223C22×D53C22×D411yescharacteristic
224C2×#6157—28no—
225C2×#6157—44no—
226C2×#6157—416no—
227C2×#6157—44no—
228Q8×C53—42no—
229Q8×C53—42no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—41no—
234C2×C4×C53#10314yes—
235C2×C4×C53C2×C22⋊C411yescharacteristic
236C2×C4×C53C2×C22⋊C411yescharacteristic
237C2×C4×C53C2×C22⋊C411yescharacteristic
238C2×C4×C53C2×C22⋊C411yescharacteristic
239C2×C4×C53C2×C22⋊C411yescharacteristic
240C2×C4×C53C23×C411yescharacteristic
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—41no—
245C2×Dic53#10314yes—
246C2×Dic53C2×C22⋊C411yescharacteristic
247C2×Dic53C2×C22⋊C411yescharacteristic
248C2×Dic53C2×C22⋊C411yescharacteristic
249C2×Dic53C2×C22⋊C411yescharacteristic
250C2×Dic53C2×C22⋊C411yescharacteristic
251C2×Dic53C22×D411yescharacteristic
252C4×D53—22no—
253C4×D53—22no—
254C4×D53—22no—
255C4×D53—22no—
256C4×D53—22no—
257C4×D53—41no—
258C4×D53—41no—
259C4×D53—44no—
260C4×D53—44no—
261C4×D53#10314yes—
262C4×D53#10314yes—
263C4×D53#10414yes—
264C4×D53C2×SD1612yes—
265C4×D53C2×SD1612yes—
266C4×D53C2×Q1612yes—
267C4×D53C2×Q1612yes—
268Dic106—44no—
269Dic106—42no—
270Dic106—44no—
271Dic106—42no—
272#88324—44no—
273C2×#1503—531nomaximal, Sylow
274C22×C53—21no—
275C22×C53—21no—
276C22×C53C2×#10312yes—
277C22×C53C22×C22⋊C411yescharacteristic
278C2×D53—21no—
279C2×D53—21no—
280C2×D53—24no—
281C2×D53—24no—
282C2×D53#41012yes—
283C2×D53C2×#10312yes—
284C2×D53C2×#10412yes—
285C2×D53C22×SD1611yescharacteristic
286C2×D53C22×Q1611yescharacteristic
287#6157—48no—
288C4×C53—21no—
289C4×C53—21no—
290C4×C53—21no—
291C4×C53—42no—
292C4×C53—42no—
293C4×C53C2×#10312yes—
294C4×C53C2×#10312yes—
295C4×C53C2×#10412yes—
296Dic53—21no—
297Dic53—21no—
298Dic53—21no—
299Dic53—42no—
300Dic53—42no—
301Dic53C2×#10312yes—
302Dic53C2×#10312yes—
303Dic53C2×#10412yes—
304C2×#533—531no—
305C2×#431—531no—
306C2×C4×C4⋊C4—531no—
307#1503—534no—
308#1503—532no—
309#1503—532no—
310C2×#328—534no—
311C2×C53—22no—
312C2×C53C2×#41011yescharacteristic
313C2×C53C22×#10311yescharacteristic
314C2×C53C22×#10411yescharacteristic
315D53—21no—
316D53—21no—
317D53#150314yescomplemented
318#533—531no—
319#533—531no—
320#533—532no—
321#533—532no—
322#533—531no—
323#533—1061no—
324#533—1061no—
325#533—1061no—
326#533—1061no—
327#533#615711yescharacteristic, complemented
328C2×C4⋊Q8—531no—
329C2×C4⋊Q8—531no—
330C2×#125—1061no—
331C2×C22⋊Q8—532no—
332C2×C22⋊Q8—1061no—
333C2×C4×Q8—532no—
334C2×C4×Q8—1061no—
335C2×#117—1061no—
336C22×C4⋊C4—1061no—
337C22×C42—531no—
338C22×C42—1061no—
339#431—532no—
340#431—532no—
341#431—1061no—
342#431—1061no—
343C2×C4⋊C8—1061no—
344C2×C22⋊C8—532no—
345C2×C4×C8—1061no—
346C4×C4⋊C4—532no—
347C4×C4⋊C4—532no—
348C4×C4⋊C4—1061no—
349C4×C4⋊C4—1061no—
350C2×#95—532no—
351#328—538no—
352#328—538no—
353C53C2×#150311yescharacteristic, complemented, Sylow
354C22×Q8—532no—
355C22×Q8—1061no—
356C23×C4—531no—
357C23×C4—531no—
358C23×C4—531no—
359C23×C4—1062no—
360C22×C8—1062no—
361C4⋊Q8—532no—
362C4⋊Q8—532no—
363C4⋊Q8—531no—
364C4⋊Q8—531no—
365C4⋊Q8—1061no—
366C4⋊Q8—1061no—
367C4⋊Q8—1061no—
368C4⋊Q8—1061no—
369C4⋊Q8C2×#615711yescharacteristic, complemented
370C4⋊Q8C2×#615711yescharacteristic, complemented
371#125—21no—
372#125—1062no—
373#125—1062no—
374#125—1061no—
375#125—1061no—
376#125—1061no—
377C22⋊Q8—21no—
378C22⋊Q8—532no—
379C22⋊Q8—532no—
380C22⋊Q8—532no—
381C22⋊Q8—1062no—
382C22⋊Q8—1062no—
383C22⋊Q8—1062no—
384C22⋊Q8—1062no—
385C22⋊Q8—1061no—
386C22⋊Q8—1061no—
387C22⋊Q8—1061no—
388C22⋊Q8C2×#615712yes—
389C4×Q8—21no—
390C4×Q8—532no—
391C4×Q8—532no—
392C4×Q8—532no—
393C4×Q8—1062no—
394C4×Q8—1062no—
395C4×Q8—1062no—
396C4×Q8—1061no—
397C4×Q8—1062no—
398C4×Q8—1061no—
399C4×Q8—1061no—
400C4×Q8C2×#615712yescomplemented
401#117—21no—
402#117—1062no—
403#117—1062no—
404#117—1061no—
405#117—1061no—
406#117—1061no—
407C2×C4⋊C4—532no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1062no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1068no—
416C2×C22⋊C4—1062no—
417C2×C42—531no—
418C2×C42—531no—
419C2×C42—531no—
420C2×C42—531no—
421C2×C42—531no—
422C2×C42—531no—
423C2×C42—531no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—1062no—
429C2×C42—1062no—
430C2×C42—1062no—
431C2×C42—2124no—
432C2×C42C2×#615711yescharacteristic
433C4⋊C8—1062no—
434C4⋊C8—1062no—
435C22⋊C8—534no—
436C22⋊C8—1062no—
437C4×C8—1062no—
438C4×C8—1062no—
439#95—534no—
440#95—1062no—
441C24—531no—
442C2×Q8—21no—
443C2×Q8—532no—
444C2×Q8—1062no—
445C2×Q8—1062no—
446C2×Q8—1061no—
447C2×Q8—2124no—
448C2×Q8—2124no—
449C2×Q8C22×#615712yes—
450C22×C4—531no—
451C22×C4—534no—
452C22×C4—534no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—1064no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1064no—
471C22×C4—1068no—
472C22×C4—2122no—
473C22×C4—2124no—
474C22×C4—2124no—
475C22×C4C22⋊#615711yescharacteristic
476C22×C4C22⋊#615711yescharacteristic
477C22×C4C22×#615711yescharacteristic
478C2×C8—1064no—
479C2×C8—2124no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—532no—
486C4⋊C4—1062no—
487C4⋊C4—1062no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4—1062no—
491C4⋊C4—1062no—
492C4⋊C4—1062no—
493C4⋊C4—1062no—
494C4⋊C4—1068no—
495C4⋊C4—1068no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1062no—
499C4⋊C4—1062no—
500C4⋊C4—1061no—
501C4⋊C4C22×#615712yes—
502C22⋊C4—22no—
503C22⋊C4—1062no—
504C22⋊C4—1062no—
505C22⋊C4—1062no—
506C42—22no—
507C42—531no—
508C42—531no—
509C42—531no—
510C42—531no—
511C42—531no—
512C42—531no—
513C42—531no—
514C42—531no—
515C42—531no—
516C42—531no—
517C42—531no—
518C42—531no—
519C42—1062no—
520C42—1062no—
521C42—1062no—
522C42—2124no—
523C42—2124no—
524C42C22⋊#615711yescharacteristic
525C42C22⋊#615711yescharacteristic
526C42C22×#615711yescharacteristic
527C42C22×#615711yescharacteristic
528C23—531no—
529C23—532no—
530C23—532no—
531C23—531no—
532C23—1062no—
533C23—1062no—
534C23C2×C22⋊#615711yescharacteristic
535Q8—42no—
536Q8—42no—
537Q8—2122no—
538Q8—2124no—
539Q8—2124no—
540Q8—2122no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—534no—
546C2×C4—531no—
547C2×C4—531no—
548C2×C4—531no—
549C2×C4—534no—
550C2×C4—534no—
551C2×C4—532no—
552C2×C4—532no—
553C2×C4—532no—
554C2×C4—532no—
555C2×C4—534no—
556C2×C4—531no—
557C2×C4—531no—
558C2×C4—531no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1068no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2124no—
571C2×C4—2124no—
572C2×C4—2121no—
573C2×C4—2124no—
574C2×C4—21216no—
575C2×C4—2124no—
576C2×C4#1313328014yes—
577C2×C4C2×C22⋊#615711yescharacteristic
578C2×C4C2×C22⋊#615711yescharacteristic
579C2×C4C2×C22⋊#615711yescharacteristic
580C2×C4C2×C22⋊#615711yescharacteristic
581C2×C4C2×C22⋊#615711yescharacteristic
582C2×C4C23×#615711yescharacteristic
583C8—2124no—
584C22—21no—
585C22—21no—
586C22—532no—
587C22—532no—
588C22—532no—
589C22—531no—
590C22—531no—
591C22—1064no—
592C22—1064no—
593C22—1061no—
594C22—1061no—
595C22C2×#1313328012yes—
596C22C22×C22⋊#615711yescharacteristic
597C4—21no—
598C4—21no—
599C4—21no—
600C4—42no—
601C4—42no—
602C4—532no—
603C4—532no—
604C4—532no—
605C4—1061no—
606C4—1061no—
607C4—1061no—
608C4—2128no—
609C4—2122no—
610C4—2122no—
611C4C2×#1313328012yes—
612C4C2×#1313328012yes—
613C4D53⋊#10412yes—
614C2—22no—
615C2—534no—
616C2—1061no—
617C2—1061no—
618C2D53⋊#41011yescharacteristic
619C2C22×#1313328011yescharacteristic
620C2C2×D53⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#533C4
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC2×C4×D53C24
hypercenter#533#6157
hypocenterC53C2×#1503
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #20290099C2semidirect
4. #20290098C2semidirect
5. #20290097C2semidirect
24. #16811180C4semidirect
32. #16811175C4semidirect
33. C4⋊Q8×D53C4semidirect
34. C4⋊Q8×D53C4semidirect
39. C4×Q8×D53C4semidirect
42. C53×#533C4semidirect
64. #13133217C2×C4semidirect
65. #13133217C2×C4semidirect
69. #13133216C2×C4semidirect
71. #13133215C2×C4semidirect
88. Q8×Dic53C2×C4semidirect
92. C4×Dic106C2×C4semidirect
96. #13133182C2×C4semidirect
121. C4⋊Q8×C53C2×C4semidirect
122. C4⋊Q8×C53C2×C4semidirect
127. C4×Q8×C53C2×C4semidirect
317. D53#1503semidirect
327. #533#6157semidirect
353. C53C2×#1503semidirect
369. C4⋊Q8C2×#6157semidirect
370. C4⋊Q8C2×#6157semidirect
400. C4×Q8C2×#6157semidirect

Maximal subgroups

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

subgroupindex
D53⋊#4312
C2×C4×C4⋊#61572
#202900992
#202900982
#202900972
D53⋊#3282
#533×D532
C2×#150353