Finite Group Databasebeta

← D53⋊#1491 #28314514

Subgroups — D53⋊#1491 #28314514

Order 13568.

1302 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
1D53⋊#432C211yesmaximal, characteristic
2C4×D53⋊M16C211yesmaximal, characteristic
3#20290059C212yesmaximal, complemented
4#20290058C214yesmaximal, complemented
5#20290057C212yesmaximal, complemented
6D53⋊(Q8⋊C8)C214yesmaximal, complemented
7C2×C4×Q8×D53C211yesmaximal, characteristic
8C2×D53⋊M16—21no—
9#16811646—21no—
10#16811645C2212yes—
11#16811644C2212yes—
12#16811643—21no—
13D53⋊(C4⋊C8)C2212yes—
14D53⋊(C22⋊C8)—21no—
15C4×#11719936—21no—
16C4×#11719936C2212yes—
17C4×#11719940—21no—
18C4×#11719940C2212yes—
19#16811582—21no—
20#16811582—21no—
21C4×D53⋊C8C2212yes—
22#16811473C2218yescomplemented
23Q8⋊#88324C2218yescomplemented
24C22×Q8×D53C411yescharacteristic
25C2×Q8×Dic53C2211yescharacteristic
26C2×C4×Dic106—21no—
27C2×C4×Dic106C411yescharacteristic
28C2×#13133182—21no—
29C2×#13133182C2211yescharacteristic
30C4×Q8×D53—21no—
31C4×Q8×D53—21no—
32C4×Q8×D53—21no—
33C4×Q8×D53—21no—
34C4×Q8×D53C412yes—
35C4×Q8×D53C412yes—
36C4×Q8×D53C2212yes—
37C4×Q8×D53C2212yes—
38C2×C4⋊C4×D53—21no—
39C2×C4⋊C4×D53C411yescharacteristic
40C2×C42×D53—21no—
41C2×C42×D53C2211yescharacteristic
42C2×C4×Q8×C53C411yescharacteristic
43C2×#11719936—22no—
44D53⋊M16—28no—
45C2×#11719940—22no—
46C2×D53⋊C8—22no—
47C2×D53⋊C8—22no—
48#13133284—22no—
49C4⋊#88324C2314yes—
50#13133274—22no—
51C22⋊#88324—22no—
52#13133272—22no—
53#13133272—22no—
54#13133272—22no—
55#13133272—22no—
56C4×#88324—22no—
57C4×#88324C2314yes—
58C22×Dic106—21no—
59C22×Dic106C2×C411yescharacteristic
60C2×Q8×D53—28no—
61C2×Q8×D53—21no—
62C2×Q8×D53—21no—
63C2×Q8×D53C2×C412yes—
64C2×Q8×D53C2×C412yes—
65C22×C4×D53—21no—
66C22×C4×D53—21no—
67C22×C4×D53C2×C411yescharacteristic
68C22×C4×D53D411yescharacteristic
69C22×C4×D53D411yescharacteristic
70Q8×Dic53—21no—
71Q8×Dic53—21no—
72Q8×Dic53C2×C412yes—
73Q8×Dic53C2312yes—
74C4×Dic106—21no—
75C4×Dic106—21no—
76C4×Dic106—22no—
77C4×Dic106—22no—
78C4×Dic106—22no—
79C4×Dic106—22no—
80C4×Dic106C2×C412yes—
81C4×Dic106C2×C412yes—
82#13133182—22no—
83#13133182—22no—
84#13133182—21no—
85#13133182—21no—
86#13133182—22no—
87#13133182—22no—
88#13133182C2×C412yes—
89#13133182C2312yes—
90C2×C4⋊Dic53—21no—
91C2×C4⋊Dic53C2×C411yescharacteristic
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C4⋊C4×D53—22no—
96C4⋊C4×D53—22no—
97C4⋊C4×D53—21no—
98C4⋊C4×D53—21no—
99C4⋊C4×D53—22no—
100C4⋊C4×D53—22no—
101C4⋊C4×D53C2×C412yes—
102C4⋊C4×D53C2×C412yes—
103C2×C4×Dic53—21no—
104C2×C4×Dic53—21no—
105C2×C4×Dic53—21no—
106C2×C4×Dic53D411yescharacteristic
107C2×C4×Dic53D411yescharacteristic
108C2×C4×Dic53C2311yescharacteristic
109C42×D53—22no—
110C42×D53—22no—
111C42×D53—22no—
112C42×D53—22no—
113C42×D53C2×C411yescharacteristic
114C42×D53C2×C411yescharacteristic
115C42×D53D411yescharacteristic
116C42×D53D411yescharacteristic
117C42×D53D411yescharacteristic
118C42×D53D411yescharacteristic
119C42×D53C2311yescharacteristic
120C42×D53C2311yescharacteristic
121C22×Q8×C53C2×C411yescharacteristic
122C4×Q8×C53—21no—
123C4×Q8×C53—21no—
124C4×Q8×C53C2×C412yes—
125C4×Q8×C53C2×C412yes—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53C2×C411yescharacteristic
128C2×C42×C53—21no—
129C2×C42×C53C2×C411yescharacteristic
130C23×D53C22⋊C411yescharacteristic
131C2×Q8×C53—24no—
132C2×Q8×C53—21no—
133C2×Q8×C53C22×C412yes—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C22⋊C411yescharacteristic
137C22×C4×C53C22⋊C411yescharacteristic
138C22×C4×C53C22×C411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53C22⋊C411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144Q8×D53—28no—
145Q8×D53—28no—
146Q8×D53—24no—
147Q8×D53—24no—
148Q8×D53—24no—
149Q8×D53M1618yescomplemented
150C2×C4×D53—28no—
151C2×C4×D53—24no—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—24no—
155C2×C4×D53—24no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—44no—
159C2×C4×D53C22⋊C411yescharacteristic
160C2×C4×D53C22⋊C411yescharacteristic
161C2×C4×D53C22⋊C411yescharacteristic
162C2×C4×D53C22⋊C411yescharacteristic
163C2×C4×D53C22⋊C411yescharacteristic
164C2×C4×D53C22⋊C411yescharacteristic
165C2×C4×D53C22×C411yescharacteristic
166C2×C4×D53C22×C411yescharacteristic
167C2×C4×D53C2×D411yescharacteristic
168C2×C4×D53C2×D411yescharacteristic
169C2×C4×D53C2×D411yescharacteristic
170C2×C4×D53C2×D411yescharacteristic
171C2×Dic106—28no—
172C2×Dic106—21no—
173C2×Dic106—22no—
174C2×Dic106—22no—
175C2×Dic106—24no—
176C2×Dic106C22×C412yes—
177#11719940—28no—
178C2×#88324—24no—
179C2×#88324—24no—
180#11719936—28no—
181D53⋊C8—28no—
182D53⋊C8—44no—
183C4⋊C4×C53—22no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53C22×C412yes—
187C42×C53—22no—
188C42×C53—22no—
189C42×C53C22⋊C411yescharacteristic
190C42×C53C22⋊C411yescharacteristic
191C42×C53C22×C411yescharacteristic
192C42×C53C22×C411yescharacteristic
193C4⋊Dic53—21no—
194C4⋊Dic53—22no—
195C4⋊Dic53—22no—
196C4⋊Dic53C22×C412yes—
197#11719919—22no—
198#11719919—22no—
199#11719919—22no—
200#11719919—22no—
201#11719919—22no—
202#11719919—22no—
203C4×Dic53—22no—
204C4×Dic53—22no—
205C4×Dic53—22no—
206C4×Dic53—22no—
207C4×Dic53—22no—
208C4×Dic53—22no—
209C4×Dic53C22⋊C411yescharacteristic
210C4×Dic53C22⋊C411yescharacteristic
211C4×Dic53C22⋊C411yescharacteristic
212C4×Dic53C22⋊C411yescharacteristic
213C4×Dic53C22×C411yescharacteristic
214C4×Dic53C2×D411yescharacteristic
215C4×Dic53C2×D411yescharacteristic
216C4×Dic53C2×D411yescharacteristic
217C4×Dic53C2×D411yescharacteristic
218C4×Dic53C2×D411yescharacteristic
219C4×Dic53C2×D411yescharacteristic
220C4×Dic53C2411yescharacteristic
221C23×C53C2×C22⋊C411yescharacteristic
222C22×D53—22no—
223C22×D53—22no—
224C22×D53#10414yes—
225C22×D53C2×C22⋊C411yescharacteristic
226C22×D53C2×C22⋊C411yescharacteristic
227Q8×C53—24no—
228Q8×C53—22no—
229Q8×C53C2×M1614yescomplemented
230C2×C4×C53—24no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—42no—
237C2×C4×C53C2×C22⋊C411yescharacteristic
238C2×C4×C53C2×C22⋊C411yescharacteristic
239C2×C4×C53C2×C22⋊C411yescharacteristic
240C2×C4×C53C2×C22⋊C411yescharacteristic
241C2×C4×C53C2×C22⋊C411yescharacteristic
242C2×C4×C53C23×C411yescharacteristic
243C2×Dic53—24no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—42no—
250C2×Dic53C2×C22⋊C411yescharacteristic
251C2×Dic53C2×C22⋊C411yescharacteristic
252C2×Dic53C2×C22⋊C411yescharacteristic
253C2×Dic53C2×C22⋊C411yescharacteristic
254C2×Dic53C22×D411yescharacteristic
255C2×Dic53C22×D411yescharacteristic
256C4×D53—28no—
257C4×D53—28no—
258C4×D53—22no—
259C4×D53—22no—
260C4×D53—22no—
261C4×D53—22no—
262C4×D53—22no—
263C4×D53—22no—
264C4×D53—22no—
265C4×D53—22no—
266C4×D53—22no—
267C4×D53—22no—
268C4×D53—44no—
269C4×D53—44no—
270C4×D53C2×M1614yes—
271C4×D53#13714yes—
272Dic106—28no—
273Dic106—24no—
274Dic106—22no—
275Dic106—28no—
276Dic106C2×M1614yescomplemented
277#88324—28no—
278#88324—44no—
279C2×#1491—531nomaximal, Sylow
280C22×C53—21no—
281C22×C53—21no—
282C22×C53C2×#10412yes—
283C22×C53C22×C22⋊C411yescharacteristic
284C2×D53—21no—
285C2×D53—21no—
286C2×D53—28no—
287C2×D53#40712yes—
288C2×D53#41912yes—
289C2×D53C2×#10412yes—
290C2×D53C2×#10412yes—
291C4×C53—24no—
292C4×C53—21no—
293C4×C53—21no—
294C4×C53—21no—
295C4×C53—21no—
296C4×C53—42no—
297C4×C53#41912yes—
298C4×C53C22×M1612yes—
299Dic53—21no—
300Dic53—24no—
301Dic53—21no—
302Dic53—21no—
303Dic53—21no—
304Dic53—42no—
305Dic53#40712yes—
306Dic53C2×#13712yes—
307C22×C4×Q8—531no—
308C2×#432—531no—
309C2×C4×M16—531no—
310#1491—532no—
311#1491—534no—
312#1491—532no—
313C2×Q8⋊C8—534no—
314C2×C53—22no—
315C2×C53C2×#40711yescharacteristic
316C2×C53C2×#41911yescharacteristic
317C2×C53C22×#10411yescharacteristic
318D53—22no—
319D53#149114yescomplemented
320C23×Q8—531no—
321C22×M16—1061no—
322C2×C4×Q8—531no—
323C2×C4×Q8—532no—
324C2×C4×Q8—532no—
325C2×C4×Q8—532no—
326C2×C4×Q8—532no—
327C2×C4×Q8—531no—
328C2×C4×Q8—531no—
329C2×C4×Q8—1061no—
330C2×C4×Q8—1061no—
331C2×C4×Q8—1061no—
332C2×C4×Q8—1061no—
333C2×C4×Q8—1061no—
334C2×C4×Q8—1061no—
335C2×C4×Q8#615711yescharacteristic
336C22×C4⋊C4—531no—
337C22×C4⋊C4—1061no—
338C22×C42—531no—
339C22×C42—1061no—
340#432—532no—
341#432—532no—
342#432—1061no—
343#432—1061no—
344C2×C4⋊C8—532no—
345C2×C22⋊C8—1061no—
346C4×M16—532no—
347C4×M16—532no—
348C4×M16—1061no—
349C4×M16—1061no—
350C2×#97—1061no—
351C2×#97—1061no—
352C2×C4×C8—532no—
353Q8⋊C8—538no—
354Q8⋊C8—538no—
355C53C2×#149111yescharacteristic, complemented, Sylow
356C22×Q8—532no—
357C22×Q8—532no—
358C22×Q8—531no—
359C22×Q8—1061no—
360C22×Q8—1061no—
361C22×Q8—1061no—
362C22×Q8—1068no—
363C22×Q8C2×#615711yescharacteristic
364C23×C4—531no—
365C23×C4—531no—
366C23×C4—531no—
367C23×C4—1061no—
368C23×C4—1061no—
369C2×M16—1062no—
370C2×M16—1062no—
371C2×M16—1068no—
372C22×C8—1062no—
373C22×C8—1062no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8—532no—
377C4×Q8—532no—
378C4×Q8—532no—
379C4×Q8—532no—
380C4×Q8—532no—
381C4×Q8—532no—
382C4×Q8—1062no—
383C4×Q8—1062no—
384C4×Q8—1062no—
385C4×Q8—1062no—
386C4×Q8—1062no—
387C4×Q8—1061no—
388C4×Q8—1061no—
389C4×Q8—1062no—
390C4×Q8—1062no—
391C4×Q8—1062no—
392C4×Q8—1061no—
393C4×Q8—1061no—
394C4×Q8—1061no—
395C4×Q8—1061no—
396C4×Q8C2×#615712yes—
397C4×Q8C2×#615712yes—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—532no—
401C2×C4⋊C4—532no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1062no—
405C2×C4⋊C4—1062no—
406C2×C4⋊C4—1062no—
407C2×C4⋊C4—1062no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4C2×#615711yescharacteristic
413C2×C42—21no—
414C2×C42—531no—
415C2×C42—531no—
416C2×C42—531no—
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—1062no—
426C2×C42—1062no—
427C2×C42—1062no—
428C2×C42—1062no—
429C2×C42—1061no—
430C2×C42—1061no—
431C2×C42—1061no—
432C2×C42C2×#615711yescharacteristic
433C4⋊C8—534no—
434C4⋊C8—1062no—
435C22⋊C8—1062no—
436C22⋊C8—1062no—
437#97—1062no—
438#97—1062no—
439#97—1062no—
440#97—1062no—
441C4×C8—534no—
442C4×C8—1062no—
443C24—531no—
444C2×Q8—24no—
445C2×Q8—21no—
446C2×Q8—538no—
447C2×Q8—532no—
448C2×Q8—1062no—
449C2×Q8—1062no—
450C2×Q8—1064no—
451C2×Q8—1068no—
452C2×Q8—1064no—
453C2×Q8—1064no—
454C2×Q8—1068no—
455C2×Q8—1068no—
456C2×Q8—1064no—
457C2×Q8—1061no—
458C2×Q8C22×#615712yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—531no—
472C22×C4—531no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—1064no—
477C22×C4—1064no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1064no—
481C22×C4—1068no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—2124no—
487C22×C4C22⋊#615711yescharacteristic
488C22×C4C22⋊#615711yescharacteristic
489C22×C4C22×#615711yescharacteristic
490M16—1068no—
491M16—1068no—
492C2×C8—1064no—
493C2×C8—1068no—
494C2×C8—1064no—
495C2×C8—2124no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—532no—
500C4⋊C4—1062no—
501C4⋊C4—1062no—
502C4⋊C4—1062no—
503C4⋊C4—1062no—
504C4⋊C4—1062no—
505C4⋊C4—1062no—
506C4⋊C4—1061no—
507C4⋊C4—1062no—
508C4⋊C4—1062no—
509C4⋊C4C22×#615712yes—
510C42—22no—
511C42—22no—
512C42—531no—
513C42—531no—
514C42—531no—
515C42—531no—
516C42—531no—
517C42—531no—
518C42—531no—
519C42—531no—
520C42—531no—
521C42—531no—
522C42—531no—
523C42—531no—
524C42—1062no—
525C42—1062no—
526C42—1062no—
527C42—1062no—
528C42—1062no—
529C42—1062no—
530C42C22⋊#615711yescharacteristic
531C42C22⋊#615711yescharacteristic
532C42C22×#615711yescharacteristic
533C42C22×#615711yescharacteristic
534C23—531no—
535C23—534no—
536C23—531no—
537C23—1062no—
538C23—1062no—
539C23C2×C22⋊#615711yescharacteristic
540Q8—24no—
541Q8—22no—
542Q8—534no—
543Q8—1068no—
544Q8—1068no—
545Q8—1064no—
546Q8—1062no—
547Q8D53⋊M1614yescomplemented
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—42no—
555C2×C4—531no—
556C2×C4—531no—
557C2×C4—534no—
558C2×C4—534no—
559C2×C4—531no—
560C2×C4—531no—
561C2×C4—531no—
562C2×C4—531no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1068no—
572C2×C4—1068no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1064no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—2124no—
582C2×C4—2124no—
583C2×C4—2122no—
584C2×C4C2×C22⋊#615711yescharacteristic
585C2×C4C2×C22⋊#615711yescharacteristic
586C2×C4C2×C22⋊#615711yescharacteristic
587C2×C4C2×C22⋊#615711yescharacteristic
588C2×C4C2×C22⋊#615711yescharacteristic
589C2×C4C23×#615711yescharacteristic
590C8—1068no—
591C8—2124no—
592C22—21no—
593C22—21no—
594C22—532no—
595C22—532no—
596C22—532no—
597C22—532no—
598C22—1068no—
599C22—1061no—
600C22—1061no—
601C22D53⋊#10412yes—
602C22C22×C22⋊#615711yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—24no—
607C4—21no—
608C4—42no—
609C4—532no—
610C4—532no—
611C4—1061no—
612C4—1061no—
613C4—1061no—
614C4—1061no—
615C4—1064no—
616C4—2122no—
617C4#1681161612yes—
618C4C2×D53⋊M1612yes—
619C2—22no—
620C2—534no—
621C2—1062no—
622C2D53⋊#40711yescharacteristic
623C2C2×#1681161611yescharacteristic
624C2C2×D53⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC2×C4×Q8×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC4×Dic53C24
hypercenterC2×C4×Q8#6157
hypocenterC53C2×#1491
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #20290059C2semidirect
4. #20290058C2semidirect
5. #20290057C2semidirect
6. D53⋊(Q8⋊C8)C2semidirect
22. #16811473C22semidirect
23. Q8⋊#88324C22semidirect
149. Q8×D53M16semidirect
229. Q8×C53C2×M16semidirect
276. Dic106C2×M16semidirect
319. D53#1491semidirect
355. C53C2×#1491semidirect
547. Q8D53⋊M16semidirect

Maximal subgroups

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

subgroupindex
D53⋊#4322
C4×D53⋊M162
#202900592
#202900582
#202900572
D53⋊(Q8⋊C8)2
C2×C4×Q8×D532
C2×#149153