Finite Group Databasebeta

← D53⋊#1515 #28315117

Subgroups — D53⋊#1515 #28315117

Order 13568.

1022 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D53⋊#432C211yesmaximal, characteristic
2C2×#16811568C211yesmaximal, characteristic
3#20290139C212yesmaximal, complemented
4#20290138C212yesmaximal, complemented
5#20290137C214yesmaximal, complemented
6D53⋊#328C214yesmaximal
7#533×D53C211yesmaximal, characteristic
8#16811646C2212yes—
9#16811645C2212yes—
10#16811644—21no—
11#16811643—21no—
12D53⋊(C4⋊C8)—21no—
13D53⋊(C22⋊C8)C2212yes—
14#16811582—21no—
15#16811568—21no—
16#16811568—21no—
17#16811568C2212yes—
18#16811568C2212yes—
19C2×(C2×#6157)⋊C4—21no—
20C2×(C2×#6157)⋊C4—21no—
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—
45#13133284—22no—
46C4⋊#88324—22no—
47#13133274C2314yes—
48C22⋊#88324—22no—
49#13133272—22no—
50#13133272—22no—
51(C2×#6157)⋊C4—22no—
52(C2×#6157)⋊C4—22no—
53(C2×#6157)⋊C4—22no—
54(C2×#6157)⋊C4—22no—
55(C2×#6157)⋊C4—22no—
56(C2×#6157)⋊C4C2314yes—
57C2×Q8×D53—21no—
58C2×Q8×D53C2×C412yes—
59C22×C4×D53D411yescharacteristic
60C22×C4×D53D411yescharacteristic
61C22×C4×D53C2311yescharacteristic
62#13133217C2×C411yescharacteristic, complemented
63#13133217C2×C411yescharacteristic, complemented
64#13133216—21no—
65#13133216—21no—
66#13133216—21no—
67#13133216C2×C412yescomplemented
68#13133215—21no—
69#13133215C2×C412yescomplemented
70#13133209—21no—
71#13133208—21no—
72#13133208—21no—
73#13133207—22no—
74#13133207—22no—
75#13133197—21no—
76#13133197C2312yes—
77#13133196—21no—
78#13133196C2×C412yes—
79#13133195—22no—
80#13133195—22no—
81#13133194—22no—
82#13133194—22no—
83#13133193—21no—
84#13133193C2312yes—
85Q8×Dic53—21no—
86Q8×Dic53C2×C412yescomplemented
87C4×Dic106—21no—
88C4×Dic106—22no—
89C4×Dic106—22no—
90C4×Dic106C2×C412yescomplemented
91#13133182—22no—
92#13133182—22no—
93#13133182—21no—
94#13133182C2×C412yescomplemented
95#13133176—21no—
96#13133175—21no—
97#13133175—21no—
98#13133174—22no—
99#13133173—22no—
100C4⋊C4×D53—21no—
101C4⋊C4×D53—21no—
102C4⋊C4×D53—21no—
103C4⋊C4×D53—22no—
104C4⋊C4×D53—22no—
105C4⋊C4×D53C2×C412yes—
106C22⋊C4×D53—22no—
107C2×C4×Dic53C2×C411yescharacteristic
108C2×C4×Dic53D411yescharacteristic
109C2×C4×Dic53D411yescharacteristic
110C42×D53—22no—
111C42×D53C2×C411yescharacteristic
112C42×D53C2×C411yescharacteristic
113C42×D53D411yescharacteristic
114C42×D53D411yescharacteristic
115C42×D53D411yescharacteristic
116C42×D53D411yescharacteristic
117C42×D53C2311yescharacteristic
118C42×D53C2311yescharacteristic
119C4⋊Q8×C53C2×C411yescharacteristic, complemented
120C4⋊Q8×C53C2×C411yescharacteristic, complemented
121#125×C53—21no—
122C22⋊Q8×C53—21no—
123C22⋊Q8×C53C2×C412yes—
124C4×Q8×C53—21no—
125C4×Q8×C53C2×C412yescomplemented
126#117×C53—21no—
127C2×C42×C53C2×C411yescharacteristic
128C23×D53C2×D411yescharacteristic
129C22×#6157—24no—
130C22×#6157—44no—
131C22×#6157—44no—
132C22×#6157—44no—
133C2×Q8×C53—21no—
134C2×Q8×C53C22×C412yes—
135C22×C4×C53C22⋊C411yescharacteristic
136C22×C4×C53C22⋊C411yescharacteristic
137C22×C4×C53C22×C411yescharacteristic
138C22×Dic53C22⋊C411yescharacteristic
139C22×Dic53C22⋊C411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141Q8×D53—44no—
142Q8×D53—44no—
143C2×C4×D53—24no—
144C2×C4×D53—22no—
145C2×C4×D53—22no—
146C2×C4×D53—42no—
147C2×C4×D53—42no—
148C2×C4×D53C22⋊C411yescharacteristic
149C2×C4×D53C22⋊C411yescharacteristic
150C2×C4×D53C22⋊C411yescharacteristic
151C2×C4×D53C22⋊C411yescharacteristic
152C2×C4×D53C22×C411yescharacteristic
153C2×C4×D53C2×D411yescharacteristic
154C2×C4×D53C2×D411yescharacteristic
155C2×C4×D53C2×D411yescharacteristic
156C2×C4×D53C2×D411yescharacteristic
157C2×C4×D53C2×D411yescharacteristic
158C2×C4×D53C2×D411yescharacteristic
159C2×C4×D53C2411yescharacteristic
160C2×Dic106—21no—
161C2×Dic106—22no—
162C2×Dic106—22no—
163C2×Dic106C22×C412yes—
164C2×#88324—24no—
165D53⋊C8—44no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53C22×C412yes—
172C22⋊C4×C53—22no—
173C42×C53—22no—
174C42×C53C22⋊C411yescharacteristic
175C42×C53C22⋊C411yescharacteristic
176C42×C53C22×C411yescharacteristic
177C42×C53C22×C411yescharacteristic
178C22⋊Dic53—22no—
179#11719921—22no—
180#11719921—22no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—22no—
185C4⋊Dic53—22no—
186C4⋊Dic53C22×C412yes—
187#11719919—22no—
188#11719919—22no—
189#11719919—22no—
190#11719919—22no—
191#11719919—22no—
192#11719919—22no—
193#11719919—22no—
194#11719919—22no—
195C4×Dic53—22no—
196C4×Dic53—22no—
197C4×Dic53—22no—
198C4×Dic53C22⋊C411yescharacteristic
199C4×Dic53C22⋊C411yescharacteristic
200C4×Dic53C22⋊C411yescharacteristic
201C4×Dic53C22⋊C411yescharacteristic
202C4×Dic53C22⋊C411yescharacteristic
203C4×Dic53C22⋊C411yescharacteristic
204C4×Dic53C22×C411yescharacteristic
205C4×Dic53C22×C411yescharacteristic
206C4×Dic53C2×D411yescharacteristic
207C4×Dic53C2×D411yescharacteristic
208C4×Dic53C2×D411yescharacteristic
209C4×Dic53C2×D411yescharacteristic
210C23×C53C2×C22⋊C411yescharacteristic
211C22×D53—22no—
212C22×D53—22no—
213C22×D53C2×C22⋊C411yescharacteristic
214C22×D53#13714yes—
215C22×D53C22×D411yescharacteristic
216C2×#6157—44no—
217C2×#6157—44no—
218C2×#6157—44no—
219C2×#6157—88no—
220Q8×C53—42no—
221Q8×C53—42no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53C2×C22⋊C411yescharacteristic
228C2×C4×C53C2×C22⋊C411yescharacteristic
229C2×C4×C53C2×C22⋊C411yescharacteristic
230C2×C4×C53C2×C22⋊C411yescharacteristic
231C2×C4×C53C2×C22⋊C411yescharacteristic
232C2×C4×C53C23×C411yescharacteristic
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53C2×C22⋊C411yescharacteristic
239C2×Dic53C2×C22⋊C411yescharacteristic
240C2×Dic53C2×C22⋊C411yescharacteristic
241C2×Dic53C2×C22⋊C411yescharacteristic
242C2×Dic53C2×C22⋊C411yescharacteristic
243C2×Dic53C22×D411yescharacteristic
244C4×D53—22no—
245C4×D53—22no—
246C4×D53—22no—
247C4×D53—22no—
248C4×D53—22no—
249C4×D53—22no—
250C4×D53—22no—
251C4×D53—22no—
252C4×D53—41no—
253C4×D53—41no—
254C4×D53—41no—
255C4×D53—41no—
256C4×D53—44no—
257C4×D53—44no—
258Dic106—44no—
259Dic106—44no—
260Dic106—42no—
261Dic106—42no—
262#88324—44no—
263C2×#1515—531nomaximal, Sylow
264C22×C53—21no—
265C22×C53—21no—
266C22×C53#41912yes—
267C22×C53C22×C22⋊C411yescharacteristic
268C2×D53—21no—
269C2×D53—21no—
270C2×D53—28no—
271C2×D53#41012yes—
272C2×D53#41912yes—
273C2×D53#42112yes—
274C2×D53C2×#13712yes—
275#6157—84no—
276C4×C53—21no—
277C4×C53—21no—
278C4×C53—21no—
279C4×C53—21no—
280C4×C53—21no—
281C4×C53—21no—
282C4×C53—42no—
283C4×C53—42no—
284Dic53—21no—
285Dic53—21no—
286Dic53—21no—
287Dic53—21no—
288Dic53—21no—
289Dic53—21no—
290Dic53—42no—
291Dic53—42no—
292C2×#533—531no—
293C2×#432—531no—
294C2×#383—531no—
295#1515—534no—
296#1515—532no—
297#1515—532no—
298C2×#328—534no—
299C2×C53—22no—
300C2×C53C2×#41011yescharacteristic
301C2×C53C2×#41911yescharacteristic
302C2×C53C2×#42111yescharacteristic
303D53—22no—
304D53#151514yescomplemented
305#533—531no—
306#533—531no—
307#533—532no—
308#533—532no—
309#533—531no—
310#533—1061no—
311#533—1061no—
312#533—1061no—
313#533—1061no—
314#533#615711yescharacteristic, complemented
315C2×C4⋊Q8—531no—
316C2×C4⋊Q8—531no—
317C2×#125—1061no—
318C2×C22⋊Q8—532no—
319C2×C22⋊Q8—1061no—
320C2×C4×Q8—532no—
321C2×C4×Q8—1061no—
322C2×#117—1061no—
323C22×C42—531no—
324#432—532no—
325#432—532no—
326#432—1061no—
327#432—1061no—
328C2×C4⋊C8—1061no—
329C2×C22⋊C8—532no—
330C2×#97—1061no—
331#383—532no—
332#383—532no—
333#383—1061no—
334#383—1061no—
335C2×#95—532no—
336C2×#95—1061no—
337C2×#95—1061no—
338#328—538no—
339#328—538no—
340C53C2×#151511yescharacteristic, complemented, Sylow
341C22×Q8—532no—
342C22×Q8—1061no—
343C23×C4—531no—
344C23×C4—531no—
345C23×C4—531no—
346C23×C4—1062no—
347C22×C8—1062no—
348C4⋊Q8—532no—
349C4⋊Q8—532no—
350C4⋊Q8—531no—
351C4⋊Q8—531no—
352C4⋊Q8—1061no—
353C4⋊Q8—1061no—
354C4⋊Q8—1061no—
355C4⋊Q8—1061no—
356C4⋊Q8C2×#615711yescharacteristic, complemented
357C4⋊Q8C2×#615711yescharacteristic, complemented
358#125—21no—
359#125—1062no—
360#125—1062no—
361#125—1061no—
362#125—1061no—
363#125—1061no—
364C22⋊Q8—21no—
365C22⋊Q8—532no—
366C22⋊Q8—532no—
367C22⋊Q8—532no—
368C22⋊Q8—1062no—
369C22⋊Q8—1062no—
370C22⋊Q8—1062no—
371C22⋊Q8—1062no—
372C22⋊Q8—1061no—
373C22⋊Q8—1061no—
374C22⋊Q8—1061no—
375C22⋊Q8C2×#615712yes—
376C4×Q8—21no—
377C4×Q8—532no—
378C4×Q8—532no—
379C4×Q8—532no—
380C4×Q8—1062no—
381C4×Q8—1062no—
382C4×Q8—1062no—
383C4×Q8—1061no—
384C4×Q8—1062no—
385C4×Q8—1061no—
386C4×Q8—1061no—
387C4×Q8C2×#615712yescomplemented
388#117—21no—
389#117—1062no—
390#117—1062no—
391#117—1061no—
392#117—1061no—
393#117—1061no—
394C2×C4⋊C4—532no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1061no—
400C2×C22⋊C4—1062no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—531no—
406C2×C42—531no—
407C2×C42—531no—
408C2×C42—531no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—1062no—
413C2×C42C2×#615711yescharacteristic
414C4⋊C8—1062no—
415C4⋊C8—1062no—
416C22⋊C8—534no—
417C22⋊C8—1062no—
418#97—1062no—
419#97—1062no—
420#95—534no—
421#95—1062no—
422#95—1062no—
423#95—1062no—
424#95—1062no—
425#95—1062no—
426C24—531no—
427C2×Q8—21no—
428C2×Q8—532no—
429C2×Q8—1062no—
430C2×Q8—1062no—
431C2×Q8—1061no—
432C2×Q8—2124no—
433C2×Q8—2124no—
434C2×Q8C22×#615712yes—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—531no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—1064no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1064no—
454C22×C4—2122no—
455C22×C4—2122no—
456C22×C4—2124no—
457C22×C4—2124no—
458C22×C4—2124no—
459C22×C4C22⋊#615711yescharacteristic
460C22×C4C22⋊#615711yescharacteristic
461C22×C4C22×#615711yescharacteristic
462C2×C8—1064no—
463C2×C8—2124no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—532no—
470C4⋊C4—1062no—
471C4⋊C4—1062no—
472C4⋊C4—1062no—
473C4⋊C4—1062no—
474C4⋊C4—1062no—
475C4⋊C4—1062no—
476C4⋊C4—1062no—
477C4⋊C4—1062no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1062no—
481C4⋊C4—1062no—
482C4⋊C4—1061no—
483C4⋊C4C22×#615712yes—
484C22⋊C4—22no—
485C22⋊C4—1062no—
486C22⋊C4—1062no—
487C22⋊C4—1062no—
488C42—22no—
489C42—531no—
490C42—531no—
491C42—531no—
492C42—531no—
493C42—531no—
494C42—531no—
495C42—531no—
496C42—531no—
497C42—531no—
498C42—531no—
499C42—531no—
500C42—531no—
501C42—1062no—
502C42—1062no—
503C42—1062no—
504C42C22⋊#615711yescharacteristic
505C42C22⋊#615711yescharacteristic
506C42C22×#615711yescharacteristic
507C42C22×#615711yescharacteristic
508C23—531no—
509C23—534no—
510C23—531no—
511C23—1062no—
512C23—1062no—
513C23C2×C22⋊#615711yescharacteristic
514Q8—42no—
515Q8—42no—
516Q8—2122no—
517Q8—2122no—
518Q8—2124no—
519Q8—2124no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—531no—
526C2×C4—531no—
527C2×C4—531no—
528C2×C4—531no—
529C2×C4—531no—
530C2×C4—531no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—1062no—
536C2×C4—1062no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—1062no—
540C2×C4—1062no—
541C2×C4—1062no—
542C2×C4—4248no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2124no—
548C2×C4—2124no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2124no—
552C2×C4—2124no—
553C2×C4—2124no—
554C2×C4C2×C22⋊#615711yescharacteristic
555C2×C4C2×C22⋊#615711yescharacteristic
556C2×C4C2×C22⋊#615711yescharacteristic
557C2×C4C2×C22⋊#615711yescharacteristic
558C2×C4C2×C22⋊#615711yescharacteristic
559C2×C4C23×#615711yescharacteristic
560C8—2124no—
561C22—21no—
562C22—21no—
563C22—532no—
564C22—532no—
565C22—532no—
566C22—532no—
567C22—1068no—
568C22—1061no—
569C22—1061no—
570C22#1681161612yes—
571C22C22×C22⋊#615711yescharacteristic
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—42no—
579C4—42no—
580C4—1061no—
581C4—1061no—
582C4—1061no—
583C4—1061no—
584C4—1061no—
585C4—1061no—
586C4—4244no—
587C4—2122no—
588C4—2122no—
589C2—22no—
590C2—534no—
591C2—1062no—
592C2D53⋊#41011yescharacteristic
593C2C2×#1681161611yescharacteristic
594C2D53⋊#42111yescharacteristic
subgroupG/H
centerC22C22×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×#1515
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #20290139C2semidirect
4. #20290138C2semidirect
5. #20290137C2semidirect
24. #16811180C4semidirect
32. #16811175C4semidirect
33. C4⋊Q8×D53C4semidirect
34. C4⋊Q8×D53C4semidirect
39. C4×Q8×D53C4semidirect
42. C53×#533C4semidirect
62. #13133217C2×C4semidirect
63. #13133217C2×C4semidirect
67. #13133216C2×C4semidirect
69. #13133215C2×C4semidirect
86. Q8×Dic53C2×C4semidirect
90. C4×Dic106C2×C4semidirect
94. #13133182C2×C4semidirect
119. C4⋊Q8×C53C2×C4semidirect
120. C4⋊Q8×C53C2×C4semidirect
125. C4×Q8×C53C2×C4semidirect
304. D53#1515semidirect
314. #533#6157semidirect
340. C53C2×#1515semidirect
356. C4⋊Q8C2×#6157semidirect
357. C4⋊Q8C2×#6157semidirect
387. C4×Q8C2×#6157semidirect

Maximal subgroups

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

subgroupindex
D53⋊#4322
C2×#168115682
#202901392
#202901382
#202901372
D53⋊#3282
#533×D532
C2×#151553