Finite Group Databasebeta

← D53×#3195 #28254698

Subgroups — D53×#3195 #28254698

Order 13568.

2322 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D53C212yesmaximal
2#20285539C211yesmaximal, characteristic, complemented
3#20285538C214yesmaximal, complemented
4#20285537C214yesmaximal, complemented
5#20285536C212yesmaximal, complemented
6#20285535C214yesmaximal, complemented
7#483×D53C214yesmaximal, complemented
8#470×D53C214yesmaximal, complemented
9#451×D53C214yesmaximal, complemented
10#407×D53C211yesmaximal, characteristic
11C53×#3195C211yesmaximal, characteristic, complemented
12C22×Q8×D53C2212yes—
13C23×C4×D53C2211yescharacteristic
14C2×Q16×D53C2214yescomplemented
15C2×M16×D53C2212yes—
16C2×#13133197C2212yes—
17C2×#13133196C2212yes—
18C2×#13133195—22no—
19C2×#13133194—22no—
20C2×#13133193C2212yes—
21C22⋊Q8×D53—22no—
22C22⋊Q8×D53—22no—
23C22⋊Q8×D53—22no—
24C22⋊Q8×D53—22no—
25C22⋊Q8×D53C2214yes—
26C22⋊Q8×D53C2214yes—
27C22⋊Q8×D53C2214yes—
28C22⋊Q8×D53C2214yes—
29C2×C4⋊C4×D53—22no—
30C2×C4⋊C4×D53C2212yes—
31C2×C22⋊C4×D53—22no—
32#16810955C2218yescomplemented
33#16810954C2214yescomplemented
34#16810953C2214yescomplemented
35#16810952C2218yescomplemented
36#16810951C2214yescomplemented
37#16810882C2214yescomplemented
38#16810881C2214yescomplemented
39#16810880C2218yescomplemented
40#16810879C2218yescomplemented
41#16810878C2214yescomplemented
42#16810781C2214yescomplemented
43#16810780C2218yescomplemented
44#16810779C2214yescomplemented
45#16810778C2214yescomplemented
46#16810777C2218yescomplemented
47#106×D53C2214yescomplemented
48#103×D53C2218yescomplemented
49#16810587C2211yescharacteristic
50#16810586C2212yes—
51#16810585C2214yes—
52C22⋊C8×D53C2214yes—
53C2×C22⋊Q8×C53C2212yes—
54C53×#483C2214yescomplemented
55C53×#470C2214yescomplemented
56C53×#451C2214yescomplemented
57C53×#407C2211yescharacteristic
58C24×D53D411yescharacteristic
59C22×Dic106—22no—
60C22×Dic106C2312yes—
61C2×Q8×D53—22no—
62C2×Q8×D53—22no—
63C2×Q8×D53—48no—
64C2×Q8×D53C2314yes—
65C2×Q8×D53C2314yes—
66C23×Dic53D411yescharacteristic
67C22×C4×D53—22no—
68C22×C4×D53—22no—
69C22×C4×D53—44no—
70C22×C4×D53D412yes—
71C22×C4×D53D411yescharacteristic
72C22×C4×D53D411yescharacteristic
73C22×C4×D53D412yes—
74C22×C4×D53C2312yes—
75C22×C4×D53C2311yescharacteristic
76C22×C4×D53C2312yes—
77C22×C4×D53C2311yescharacteristic
78C2×Dic212C2314yescomplemented
79C2×#11719925C2318yescomplemented
80Q16×D53—48no—
81C2×#11719917C2312yes—
82C2×#11719912C2314yes—
83M16×D53—44no—
84C2×C8×D53C2314yes—
85#13133197—22no—
86#13133197—22no—
87#13133197C2314yes—
88#13133197C2314yes—
89#13133196—22no—
90#13133196—22no—
91#13133196C2314yes—
92#13133196C2314yes—
93#13133195—24no—
94#13133195—24no—
95#13133195—24no—
96#13133195—24no—
97#13133194—24no—
98#13133194—24no—
99#13133194—24no—
100#13133194—24no—
101#13133193—22no—
102#13133193—22no—
103#13133193C2314yes—
104#13133193C2314yes—
105C2×C4⋊Dic53—22no—
106C2×C4⋊Dic53C2312yes—
107C2×#11719919—22no—
108C2×#11719919—22no—
109C2×#11719919—22no—
110C4⋊C4×D53—24no—
111C4⋊C4×D53—24no—
112C4⋊C4×D53—24no—
113C4⋊C4×D53—24no—
114C4⋊C4×D53—22no—
115C4⋊C4×D53—22no—
116C4⋊C4×D53C2314yes—
117C4⋊C4×D53C2314yes—
118C2×C22⋊Dic53—22no—
119C2×#11719921—22no—
120C2×#11719921—22no—
121C22⋊C4×D53—24no—
122C22⋊C4×D53—24no—
123C22⋊C4×D53—24no—
124C22⋊C4×D53—24no—
125#13133149C2314yescomplemented
126#13133148C2318yescomplemented
127#13133142C2318yescomplemented
128#13133141C2318yescomplemented
129#13133140C2318yescomplemented
130C22⋊#88322C2314yes—
131#13133129C2318yes—
132C22×Q8×C53C2312yes—
133C23×C4×C53C2311yescharacteristic
134C2×Q16×C53C2314yescomplemented
135C2×M16×C53C2312yes—
136C22⋊Q8×C53—22no—
137C22⋊Q8×C53—22no—
138C22⋊Q8×C53C2314yes—
139C22⋊Q8×C53C2314yes—
140C2×C4⋊C4×C53—22no—
141C2×C4⋊C4×C53C2312yes—
142C2×C22⋊C4×C53—22no—
143#106×C53C2314yescomplemented
144#103×C53C2318yescomplemented
145C22⋊C8×C53C2314yes—
146C24×C53C2×D411yescharacteristic
147C23×D53—28no—
148C23×D53—42no—
149C23×D53C2×D411yescharacteristic
150C23×D53C2×D411yescharacteristic
151C23×D53C2×D412yes—
152C23×D53C2×D412yes—
153C2×Q8×C53—22no—
154C2×Q8×C53—44no—
155C2×Q8×C53C2414yes—
156C22×C4×C53—22no—
157C22×C4×C53—22no—
158C22×C4×C53—42no—
159C22×C4×C53C2×D412yes—
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2412yes—
162C22×C4×C53C2411yescharacteristic
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—42no—
166C22×Dic53C2×D412yes—
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D412yes—
169C22×Dic53C2×D411yescharacteristic
170Q8×D53—216no—
171Q8×D53—48no—
172Q8×D53—48no—
173Q8×D53—48no—
174C2×C4×D53—24no—
175C2×C4×D53—28no—
176C2×C4×D53—24no—
177C2×C4×D53—24no—
178C2×C4×D53—24no—
179C2×C4×D53—24no—
180C2×C4×D53—44no—
181C2×C4×D53—42no—
182C2×C4×D53—42no—
183C2×C4×D53—44no—
184C2×C4×D53—44no—
185C2×C4×D53—42no—
186C2×C4×D53—48no—
187C2×C4×D53—48no—
188C2×C4×D53C2×D411yescharacteristic
189C2×C4×D53C2×D412yes—
190C2×C4×D53C2×D412yes—
191C2×C4×D53C2×D411yescharacteristic
192C2×C4×D53C2×D412yes—
193C2×C4×D53C2×D412yes—
194C2×C4×D53C2×D412yes—
195C2×C4×D53C2412yes—
196C2×C4×D53C2411yescharacteristic
197C2×C4×D53C2411yescharacteristic
198C2×Dic106—22no—
199C2×Dic106—24no—
200C2×Dic106—24no—
201C2×Dic106—44no—
202C2×Dic106—48no—
203C2×Dic106C2414yes—
204Q16×C53—44no—
205M16×C53—42no—
206C2×C8×C53C2414yes—
207C4⋊C4×C53—24no—
208C4⋊C4×C53—24no—
209C4⋊C4×C53—22no—
210C4⋊C4×C53C2414yes—
211C22⋊C4×C53—24no—
212C22⋊C4×C53—24no—
213C22⋊Dic53—24no—
214C22⋊Dic53—24no—
215#11719925—48no—
216#11719921—24no—
217#11719921—24no—
218#11719921—24no—
219#11719921—24no—
220C4⋊Dic53—22no—
221C4⋊Dic53—24no—
222C4⋊Dic53—24no—
223C4⋊Dic53C2414yes—
224#11719919—24no—
225#11719919—24no—
226#11719919—24no—
227#11719919—24no—
228#11719919—24no—
229#11719919—24no—
230#11719917—42no—
231C2×#88322C2414yes—
232Dic212—44no—
233#11719912—44no—
234C8×D53—28no—
235C23×C53—24no—
236C23×C53—41no—
237C23×C53C22×D411yescharacteristic
238C23×C53C22×D412yes—
239C22×D53—24no—
240C22×D53—24no—
241C22×D53—24no—
242C22×D53—22no—
243C22×D53—24no—
244C22×D53—24no—
245C22×D53—24no—
246C22×D53—41no—
247C22×D53—42no—
248C22×D53—416no—
249C22×D53#13718yes—
250C22×D53C22×D411yescharacteristic
251C22×D53C22×D412yes—
252C22×D53C22×D411yescharacteristic
253Q8×C53—28no—
254Q8×C53—44no—
255C2×C4×C53—24no—
256C2×C4×C53—22no—
257C2×C4×C53—24no—
258C2×C4×C53—24no—
259C2×C4×C53—41no—
260C2×C4×C53—42no—
261C2×C4×C53—44no—
262C2×C4×C53—44no—
263C2×C4×C53C22×D412yes—
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C2511yescharacteristic
266C2×Dic53—24no—
267C2×Dic53—22no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—41no—
271C2×Dic53—42no—
272C2×Dic53—44no—
273C2×Dic53—44no—
274C2×Dic53C22×D412yes—
275C2×Dic53C22×D411yescharacteristic
276C2×Dic53C22×D411yescharacteristic
277C4×D53—22no—
278C4×D53—24no—
279C4×D53—24no—
280C4×D53—22no—
281C4×D53—22no—
282C4×D53—24no—
283C4×D53—42no—
284C4×D53—42no—
285C4×D53—42no—
286C4×D53—48no—
287C4×D53—48no—
288C4×D53—48no—
289C4×D53—48no—
290C4×D53H(5,𝔽2)14yes—
291Dic106—28no—
292Dic106—48no—
293Dic106—44no—
294Dic106—48no—
295C8×C53—24no—
296#88322—24no—
297C2×#3195—531nomaximal, Sylow
298C22×C53—22no—
299C22×C53—22no—
300C22×C53—21no—
301C22×C53—22no—
302C22×C53—44no—
303C22×C53C2×#13714yes—
304C22×C53C23×D411yescharacteristic
305C2×D53—22no—
306C2×D53—22no—
307C2×D53—216no—
308C2×D53—21no—
309C2×D53—22no—
310C2×D53—44no—
311C2×D53—48no—
312C2×D53—44no—
313C2×D53#53412yes—
314C2×D53C2×#13714yes—
315C2×D53C2×#13714yes—
316C4×C53—22no—
317C4×C53—21no—
318C4×C53—44no—
319C4×C53—44no—
320C4×C53C2×H(5,𝔽2)12yes—
321Dic53—22no—
322Dic53—21no—
323Dic53—44no—
324Dic53—44no—
325Dic53#53412yes—
326C22×C22⋊Q8—532no—
327#3195—534no—
328#3195—534no—
329#3195—534no—
330#3195—531no—
331#3195—532no—
332#3195D5311yescharacteristic, direct product factor
333C2×#483—534no—
334C2×#470—534no—
335C2×#451—534no—
336C2×#407—531no—
337C2×C53—24no—
338C2×C53—41no—
339C2×C53C2×#53411yescharacteristic
340C2×C53C22×#13712yes—
341D53—24no—
342D53—41no—
343D53#319514yesdirect product factor
344C23×Q8—532no—
345C24×C4—531no—
346C22×Q16—534no—
347C22×M16—532no—
348C2×C22⋊Q8—532no—
349C2×C22⋊Q8—532no—
350C2×C22⋊Q8—534no—
351C2×C22⋊Q8—534no—
352C2×C22⋊Q8—534no—
353C2×C22⋊Q8—534no—
354C2×C22⋊Q8—532no—
355C2×C22⋊Q8—1062no—
356C2×C22⋊Q8—1062no—
357C2×C22⋊Q8—1062no—
358C2×C22⋊Q8—1062no—
359C2×C22⋊Q8—1062no—
360C2×C22⋊Q8—1062no—
361C2×C22⋊Q8C2×D5312yes—
362C22×C4⋊C4—532no—
363C22×C4⋊C4—1062no—
364C22×C22⋊C4—1062no—
365#483—534no—
366#483—534no—
367#483—538no—
368#483—538no—
369#483—534no—
370#483C2×D5314yescomplemented
371#470—534no—
372#470—534no—
373#470—538no—
374#470—534no—
375#470—538no—
376#470C2×D5314yescomplemented
377#451—534no—
378#451—538no—
379#451—534no—
380#451—538no—
381#451—534no—
382#451C2×D5314yescomplemented
383C2×#106—534no—
384C2×#103—538no—
385#407—531no—
386#407—532no—
387#407—534no—
388#407C2×D5311yescharacteristic
389C2×C22⋊C8—534no—
390C53C2×#319511yescharacteristic, complemented, Sylow
391C25—531no—
392C22×Q8—534no—
393C22×Q8—534no—
394C22×Q8—532no—
395C22×Q8—1062no—
396C22×Q8—1062no—
397C22×Q8—1062no—
398C22×Q8—2128no—
399C22×Q8C22×D5312yes—
400C23×C4—531no—
401C23×C4—532no—
402C23×C4—531no—
403C23×C4—532no—
404C23×C4—532no—
405C23×C4—531no—
406C23×C4—531no—
407C23×C4—532no—
408C23×C4—531no—
409C23×C4—1062no—
410C23×C4—1062no—
411C23×C4—2124no—
412C23×C4C22×D5311yescharacteristic
413C2×Q16—538no—
414C2×Q16—534no—
415C2×Q16—2128no—
416C2×Q16C22×D5314yescomplemented
417C2×M16—532no—
418C2×M16—534no—
419C2×M16—2124no—
420C2×M16C22×D5312yes—
421C22×C8—534no—
422C22⋊Q8—22no—
423C22⋊Q8—22no—
424C22⋊Q8—534no—
425C22⋊Q8—534no—
426C22⋊Q8—534no—
427C22⋊Q8—534no—
428C22⋊Q8—534no—
429C22⋊Q8—534no—
430C22⋊Q8—1064no—
431C22⋊Q8—1064no—
432C22⋊Q8—1064no—
433C22⋊Q8—1064no—
434C22⋊Q8—1064no—
435C22⋊Q8—1064no—
436C22⋊Q8—1064no—
437C22⋊Q8—1064no—
438C22⋊Q8—1062no—
439C22⋊Q8—1062no—
440C22⋊Q8—1062no—
441C22⋊Q8—1062no—
442C22⋊Q8—1062no—
443C22⋊Q8—1062no—
444C22⋊Q8C22×D5314yes—
445C22⋊Q8C22×D5314yes—
446C2×C4⋊C4—22no—
447C2×C4⋊C4—532no—
448C2×C4⋊C4—534no—
449C2×C4⋊C4—534no—
450C2×C4⋊C4—1062no—
451C2×C4⋊C4—1062no—
452C2×C4⋊C4—1064no—
453C2×C4⋊C4—1064no—
454C2×C4⋊C4—1064no—
455C2×C4⋊C4—1064no—
456C2×C4⋊C4—1062no—
457C2×C4⋊C4—1062no—
458C2×C4⋊C4—1062no—
459C2×C4⋊C4—1062no—
460C2×C4⋊C4C22×D5312yes—
461C2×C22⋊C4—22no—
462C2×C22⋊C4—1064no—
463C2×C22⋊C4—1064no—
464C2×C22⋊C4—1064no—
465C2×C22⋊C4—1064no—
466C2×C22⋊C4—1062no—
467C2×C22⋊C4—1062no—
468C2×C22⋊C4—1062no—
469#106—534no—
470#106—538no—
471#106C22×D5314yescomplemented
472#103—538no—
473#103—538no—
474#103—538no—
475#103C22×D5318yescomplemented
476C22⋊C8—534no—
477C22⋊C8—538no—
478C22⋊C8C22×D5314yes—
479C24—532no—
480C24—531no—
481C24—531no—
482C24—532no—
483C24—1068no—
484C24—2122no—
485C24D4×D5311yescharacteristic
486C2×Q8—22no—
487C2×Q8—44no—
488C2×Q8—534no—
489C2×Q8—1064no—
490C2×Q8—1064no—
491C2×Q8—10616no—
492C2×Q8—1062no—
493C2×Q8—2124no—
494C2×Q8—2128no—
495C2×Q8—2128no—
496C2×Q8—2128no—
497C2×Q8—2128no—
498C2×Q8C23×D5314yes—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—42no—
502C22×C4—531no—
503C22×C4—532no—
504C22×C4—531no—
505C22×C4—532no—
506C22×C4—531no—
507C22×C4—531no—
508C22×C4—532no—
509C22×C4—532no—
510C22×C4—531no—
511C22×C4—532no—
512C22×C4—531no—
513C22×C4—532no—
514C22×C4—532no—
515C22×C4—532no—
516C22×C4—1064no—
517C22×C4—1068no—
518C22×C4—1064no—
519C22×C4—1064no—
520C22×C4—1064no—
521C22×C4—1064no—
522C22×C4—1062no—
523C22×C4—1062no—
524C22×C4—2122no—
525C22×C4—2124no—
526C22×C4—2124no—
527C22×C4—2122no—
528C22×C4—2124no—
529C22×C4—2122no—
530C22×C4—2128no—
531C22×C4—2128no—
532C22×C4—2122no—
533C22×C4D4×D5312yes—
534C22×C4D4×D5311yescharacteristic
535C22×C4C23×D5312yes—
536C22×C4C23×D5311yescharacteristic
537Q16—44no—
538Q16—2124no—
539Q16—2128no—
540M16—42no—
541M16—2124no—
542M16—2122no—
543C2×C8—534no—
544C2×C8—1068no—
545C2×C8C23×D5314yes—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—22no—
549C4⋊C4—534no—
550C4⋊C4—1064no—
551C4⋊C4—1064no—
552C4⋊C4—1064no—
553C4⋊C4—1064no—
554C4⋊C4—1064no—
555C4⋊C4—1064no—
556C4⋊C4—1062no—
557C4⋊C4—1064no—
558C4⋊C4—1064no—
559C4⋊C4C23×D5314yes—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—1064no—
563C22⋊C4—1064no—
564C22⋊C4—1064no—
565C22⋊C4—1064no—
566C22⋊C4—1064no—
567C22⋊C4—1064no—
568C23—24no—
569C23—41no—
570C23—531no—
571C23—538no—
572C23—532no—
573C23—531no—
574C23—1064no—
575C23—1064no—
576C23—1062no—
577C23—1064no—
578C23—1064no—
579C23—1064no—
580C23—1064no—
581C23—2122no—
582C23—2121no—
583C23—21216no—
584C23C2×D4×D5311yescharacteristic
585C23C2×D4×D5312yes—
586Q8—28no—
587Q8—44no—
588Q8—1068no—
589Q8—2124no—
590Q8—2128no—
591Q8—2128no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—531no—
601C2×C4—534no—
602C2×C4—531no—
603C2×C4—532no—
604C2×C4—1062no—
605C2×C4—1064no—
606C2×C4—1062no—
607C2×C4—1064no—
608C2×C4—1062no—
609C2×C4—1064no—
610C2×C4—1062no—
611C2×C4—1064no—
612C2×C4—1064no—
613C2×C4—1064no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2128no—
618C2×C4—2128no—
619C2×C4—2128no—
620C2×C4—2128no—
621C2×C4—2122no—
622C2×C4—2121no—
623C2×C4—2124no—
624C2×C4—2124no—
625C2×C4C2×D4×D5311yescharacteristic
626C2×C4C2×D4×D5312yes—
627C2×C4C24×D5311yescharacteristic
628C8—24no—
629C8—1064no—
630C22—22no—
631C22—22no—
632C22—21no—
633C22—22no—
634C22—44no—
635C22—532no—
636C22—534no—
637C22—534no—
638C22—10616no—
639C22—1062no—
640C22—1062no—
641C22—1061no—
642C22—1062no—
643C22—2128no—
644C22—2124no—
645C22—2124no—
646C22#137×D5314yes—
647C22C22×D4×D5311yescharacteristic
648C4—21no—
649C4—22no—
650C4—44no—
651C4—44no—
652C4—532no—
653C4—1061no—
654C4—1062no—
655C4—2124no—
656C4—2124no—
657C4H(5,𝔽2)×D5312yes—
658C2—24no—
659C2—41no—
660C2—534no—
661C2—1064no—
662C2—2121no—
663C2#534×D5311yescharacteristic
664C2C2×#137×D5312yes—
subgroupG/H
centerC22C22×D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC53×#3195C2
socleC22×C53C23×D4
prime radicalC24×D53D4
hypercenter#3195D53
hypocenterC53C2×#3195
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
2. #20285539C2semidirect
3. #20285538C2semidirect
4. #20285537C2semidirect
5. #20285536C2semidirect
6. #20285535C2semidirect
7. #483×D53C2semidirect
8. #470×D53C2semidirect
9. #451×D53C2semidirect
11. C53×#3195C2semidirect
14. C2×Q16×D53C22semidirect
32. #16810955C22semidirect
33. #16810954C22semidirect
34. #16810953C22semidirect
35. #16810952C22semidirect
36. #16810951C22semidirect
37. #16810882C22semidirect
38. #16810881C22semidirect
39. #16810880C22semidirect
40. #16810879C22semidirect
41. #16810878C22semidirect
42. #16810781C22semidirect
43. #16810780C22semidirect
44. #16810779C22semidirect
45. #16810778C22semidirect
46. #16810777C22semidirect
47. #106×D53C22semidirect
48. #103×D53C22semidirect
54. C53×#483C22semidirect
55. C53×#470C22semidirect
56. C53×#451C22semidirect
78. C2×Dic212C23semidirect
79. C2×#11719925C23semidirect
125. #13133149C23semidirect
126. #13133148C23semidirect
127. #13133142C23semidirect
128. #13133141C23semidirect
129. #13133140C23semidirect
134. C2×Q16×C53C23semidirect
143. #106×C53C23semidirect
144. #103×C53C23semidirect
332. #3195D53direct (N × H)
343. D53#3195direct (N × H)
370. #483C2×D53semidirect
376. #470C2×D53semidirect
382. #451C2×D53semidirect
390. C53C2×#3195semidirect
416. C2×Q16C22×D53semidirect
471. #106C22×D53semidirect
475. #103C22×D53semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D532
#202855392
#202855382
#202855372
#202855362
#202855352
#483×D532
#470×D532
#451×D532
#407×D532
C53×#31952
C2×#319553