Finite Group Databasebeta

← #28320773

Subgroups — #28320773

Order 13568.

1258 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⋊#431C211yesmaximal, characteristic
2C2×C4×C4⋊#6157C211yesmaximal, characteristic
3#20290893C212yesmaximal, complemented
4#20290881C212yesmaximal, complemented
5#20290700C214yesmaximal
6C4×#13133280C214yesmaximal, complemented
7#533×D53C211yesmaximal, characteristic
8C22×C4⋊#6157C2212yes—
9C22×C4×#6157—21no—
10#16811642—21no—
11#16811641—21no—
12#16811640—21no—
13#16811639—21no—
14D53⋊(C4⋊C8)C2212yes—
15C2×#13133280C2218yes—
16D53⋊(C22⋊C8)C2212yes—
17C4×D53⋊C8C2212yes—
18C4×C4⋊#6157—21no—
19C4×C4⋊#6157—21no—
20C4×C4⋊#6157C2212yes—
21C4×C4⋊#6157C2212yes—
22C2×(C2×#6157)⋊C4—21no—
23#16811496C2218yes—
24#16811180C411yescharacteristic, complemented
25#16811179—21no—
26#16811178C411yescharacteristic, complemented
27#16811178C411yescharacteristic, complemented
28#16811177—21no—
29#16811177C412yescomplemented
30#16811176—21no—
31#16811175—21no—
32#16811175C412yescomplemented
33C4⋊Q8×D53C2211yescharacteristic
34C4⋊Q8×D53C2211yescharacteristic
35#125×D53—21no—
36C22⋊Q8×D53—21no—
37C22⋊Q8×D53C2212yes—
38C4×Q8×D53—21no—
39C4×Q8×D53C2212yes—
40#117×D53—21no—
41C2×C42×D53C2211yescharacteristic
42C53×#533C411yescharacteristic, complemented
43C23×#6157—22no—
44C2×D53⋊C8C2314yes—
45C2×C4⋊#6157—22no—
46C2×C4⋊#6157—28no—
47C2×C4⋊#6157C2314yes—
48C2×C4×#6157—22no—
49C2×C4×#6157—22no—
50C2×C4×#6157—44no—
51#13133284—22no—
52C4⋊#88324—22no—
53#13133280—216no—
54#13133274—22no—
55C22⋊#88324—22no—
56C4×#88324—22no—
57C4×#88324—22no—
58(C2×#6157)⋊C4—22no—
59(C2×#6157)⋊C4—22no—
60C2×Q8×D53—21no—
61C2×Q8×D53C2312yes—
62C22×C4×D53D411yescharacteristic
63C22×C4×D53D411yescharacteristic
64C22×C4×D53C2311yescharacteristic
65#13133217C2×C411yescharacteristic, complemented
66#13133217C2×C411yescharacteristic, complemented
67#13133216—21no—
68#13133216—21no—
69#13133216—21no—
70#13133216C2×C412yescomplemented
71#13133215—21no—
72#13133215C2×C412yescomplemented
73#13133209—21no—
74#13133208—21no—
75#13133208—21no—
76#13133207—22no—
77#13133207—22no—
78#13133197—21no—
79#13133197C2×C412yes—
80#13133196—21no—
81#13133196C2×C412yes—
82#13133195—22no—
83#13133195—22no—
84#13133194—22no—
85#13133194—22no—
86#13133193—21no—
87#13133193C2×C412yes—
88Q8×Dic53—21no—
89Q8×Dic53C2×C412yescomplemented
90C4×Dic106—21no—
91C4×Dic106—22no—
92C4×Dic106—22no—
93C4×Dic106C2×C412yescomplemented
94#13133182—21no—
95#13133182—22no—
96#13133182—22no—
97#13133182C2×C412yescomplemented
98#13133176—21no—
99#13133175—21no—
100#13133175—21no—
101#13133174—22no—
102#13133173—22no—
103C4⋊C4×D53—21no—
104C4⋊C4×D53—21no—
105C4⋊C4×D53—21no—
106C4⋊C4×D53—22no—
107C4⋊C4×D53—22no—
108C4⋊C4×D53C2312yes—
109C22⋊C4×D53—22no—
110C2×C4×Dic53C2×C411yescharacteristic
111C2×C4×Dic53D411yescharacteristic
112C2×C4×Dic53D411yescharacteristic
113C42×D53—22no—
114C42×D53C2×C411yescharacteristic
115C42×D53C2×C411yescharacteristic
116C42×D53D411yescharacteristic
117C42×D53D411yescharacteristic
118C42×D53D411yescharacteristic
119C42×D53D411yescharacteristic
120C42×D53C2311yescharacteristic
121C42×D53C2311yescharacteristic
122C4⋊Q8×C53C2×C411yescharacteristic, complemented
123C4⋊Q8×C53C2×C411yescharacteristic, complemented
124#125×C53—21no—
125C22⋊Q8×C53—21no—
126C22⋊Q8×C53C2×C412yes—
127C4×Q8×C53—21no—
128C4×Q8×C53C2×C412yescomplemented
129#117×C53—21no—
130C2×C42×C53C2×C411yescharacteristic
131C23×D53C2×D411yescharacteristic
132C22×#6157—28no—
133C22×#6157—24no—
134C22×#6157—44no—
135C22×#6157—44no—
136C2×Q8×C53—21no—
137C2×Q8×C53C22×C412yes—
138C22×C4×C53C22⋊C411yescharacteristic
139C22×C4×C53C22⋊C411yescharacteristic
140C22×C4×C53C22×C411yescharacteristic
141C22×Dic53C22⋊C411yescharacteristic
142C22×Dic53C22⋊C411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144Q8×D53—28no—
145Q8×D53—44no—
146C2×C4×D53—24no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—42no—
150C2×C4×D53C22⋊C411yescharacteristic
151C2×C4×D53C22⋊C411yescharacteristic
152C2×C4×D53C22⋊C411yescharacteristic
153C2×C4×D53C22⋊C411yescharacteristic
154C2×C4×D53SD1614yes—
155C2×C4×D53Q1614yes—
156C2×C4×D53C22×C411yescharacteristic
157C2×C4×D53C2×D411yescharacteristic
158C2×C4×D53C2×D411yescharacteristic
159C2×C4×D53C2×D411yescharacteristic
160C2×C4×D53C2×D411yescharacteristic
161C2×C4×D53C2×D411yescharacteristic
162C2×C4×D53C2×D411yescharacteristic
163C2×C4×D53C2411yescharacteristic
164C2×Dic106—21no—
165C2×Dic106—22no—
166C2×Dic106—22no—
167C2×Dic106C22×C412yes—
168C2×#88324—24no—
169C4⋊#6157—28no—
170C4⋊#6157—24no—
171C4⋊#6157#4118yes—
172C4×#6157—44no—
173C4×#6157—44no—
174D53⋊C8—28no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—22no—
179C4⋊C4×C53—22no—
180C4⋊C4×C53C22×C412yes—
181C22⋊C4×C53—22no—
182C42×C53—22no—
183C42×C53C22⋊C411yescharacteristic
184C42×C53C22⋊C411yescharacteristic
185C42×C53C22×C411yescharacteristic
186C42×C53C22×C411yescharacteristic
187C22⋊Dic53—22no—
188#11719921—22no—
189#11719921—22no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—22no—
194C4⋊Dic53—22no—
195C4⋊Dic53C22×C412yes—
196#11719919—22no—
197#11719919—22no—
198#11719919—22no—
199#11719919—22no—
200#11719919—22no—
201#11719919—22no—
202#11719919—22no—
203#11719919—22no—
204C4×Dic53—22no—
205C4×Dic53—22no—
206C4×Dic53—22no—
207C4×Dic53C22⋊C411yescharacteristic
208C4×Dic53C22⋊C411yescharacteristic
209C4×Dic53C22⋊C411yescharacteristic
210C4×Dic53C22⋊C411yescharacteristic
211C4×Dic53C22⋊C411yescharacteristic
212C4×Dic53C22⋊C411yescharacteristic
213C4×Dic53C22×C411yescharacteristic
214C4×Dic53C22×C411yescharacteristic
215C4×Dic53C2×D411yescharacteristic
216C4×Dic53C2×D411yescharacteristic
217C4×Dic53C2×D411yescharacteristic
218C4×Dic53C2×D411yescharacteristic
219C23×C53C2×C22⋊C411yescharacteristic
220C22×D53—22no—
221C22×D53—22no—
222C22×D53C2×C22⋊C411yescharacteristic
223C22×D53C2×SD1612yes—
224C22×D53C2×Q1612yes—
225C22×D53C22×D411yescharacteristic
226C2×#6157—28no—
227C2×#6157—416no—
228C2×#6157—44no—
229C2×#6157—44no—
230Q8×C53—24no—
231Q8×C53—42no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—41no—
236C2×C4×C53#10314yescomplemented
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—22no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—41no—
247C2×Dic53#10314yescomplemented
248C2×Dic53C2×C22⋊C411yescharacteristic
249C2×Dic53C2×C22⋊C411yescharacteristic
250C2×Dic53C2×C22⋊C411yescharacteristic
251C2×Dic53C2×C22⋊C411yescharacteristic
252C2×Dic53C2×C22⋊C411yescharacteristic
253C2×Dic53C22×D411yescharacteristic
254C4×D53—22no—
255C4×D53—22no—
256C4×D53—22no—
257C4×D53—22no—
258C4×D53—22no—
259C4×D53—41no—
260C4×D53—41no—
261C4×D53—44no—
262C4×D53—44no—
263C4×D53#10314yescomplemented
264C4×D53#10314yescomplemented
265C4×D53C2×SD1612yes—
266C4×D53C2×SD1612yes—
267C4×D53C2×Q1612yes—
268C4×D53C2×Q1612yes—
269C4×D53C2×#4114yes—
270Dic106—24no—
271Dic106—44no—
272Dic106—42no—
273Dic106—44no—
274#88324—44no—
275#21385—531nomaximal, Sylow
276C22×C53—21no—
277C22×C53—21no—
278C22×C53C2×#10312yes—
279C22×C53C22×C22⋊C411yescharacteristic
280C2×D53—21no—
281C2×D53—21no—
282C2×D53—24no—
283C2×D53—24no—
284C2×D53C2×#10312yes—
285C2×D53#52512yes—
286C2×D53C22×SD1611yescharacteristic
287C2×D53C22×Q1611yescharacteristic
288C2×D53#57512yes—
289#6157—48no—
290C4×C53—21no—
291C4×C53—21no—
292C4×C53—21no—
293C4×C53—42no—
294C4×C53—42no—
295C4×C53C2×#10312yes—
296C4×C53C2×#10312yes—
297C4×C53C4×#4112yes—
298Dic53—21no—
299Dic53—21no—
300Dic53—21no—
301Dic53—42no—
302Dic53—42no—
303Dic53C2×#10312yes—
304Dic53C2×#10312yes—
305Dic53#53212yes—
306C2×#533—531no—
307C2×#431—531no—
308C2×C4×C4⋊C4—531no—
309#1987—532no—
310#1981—532no—
311#1872—534no—
312C4×#103—534no—
313C2×C53—22no—
314C2×C53#229711yescharacteristic
315C2×C53C22×#10311yescharacteristic
316C2×C53#289611yescharacteristic
317D53—22no—
318D53#310112yes—
319D53#310412yes—
320#533—531no—
321#533—531no—
322#533—532no—
323#533—532no—
324#533—531no—
325#533—1061no—
326#533—1061no—
327#533—1061no—
328#533—1061no—
329#533#615711yescharacteristic, complemented
330C2×C4⋊Q8—531no—
331C2×C4⋊Q8—531no—
332C2×#125—1061no—
333C2×C22⋊Q8—532no—
334C2×C22⋊Q8—1061no—
335C2×C4×Q8—532no—
336C2×C4×Q8—1061no—
337C2×#117—1061no—
338C22×C4⋊C4—532no—
339C22×C42—531no—
340C22×C42—1061no—
341#431—1061no—
342#431—1061no—
343#431—1061no—
344#431—1061no—
345C2×C4⋊C8—532no—
346C2×#103—538no—
347C2×C22⋊C8—532no—
348C2×C4×C8—532no—
349C4×C4⋊C4—532no—
350C4×C4⋊C4—532no—
351C4×C4⋊C4—1061no—
352C4×C4⋊C4—1061no—
353C2×#95—1061no—
354#340—538no—
355C53#2138511yescharacteristic, complemented, Sylow
356C22×Q8—532no—
357C22×Q8—1061no—
358C23×C4—531no—
359C23×C4—531no—
360C23×C4—531no—
361C23×C4—1062no—
362C22×C8—534no—
363C4⋊Q8—532no—
364C4⋊Q8—532no—
365C4⋊Q8—531no—
366C4⋊Q8—531no—
367C4⋊Q8—1061no—
368C4⋊Q8—1061no—
369C4⋊Q8—1061no—
370C4⋊Q8—1061no—
371C4⋊Q8C2×#615711yescharacteristic, complemented
372C4⋊Q8C2×#615711yescharacteristic, complemented
373#125—21no—
374#125—1062no—
375#125—1062no—
376#125—1061no—
377#125—1061no—
378#125—1061no—
379C22⋊Q8—21no—
380C22⋊Q8—532no—
381C22⋊Q8—532no—
382C22⋊Q8—532no—
383C22⋊Q8—1062no—
384C22⋊Q8—1062no—
385C22⋊Q8—1062no—
386C22⋊Q8—1062no—
387C22⋊Q8—1061no—
388C22⋊Q8—1061no—
389C22⋊Q8—1061no—
390C22⋊Q8C2×#615712yes—
391C4×Q8—21no—
392C4×Q8—532no—
393C4×Q8—532no—
394C4×Q8—532no—
395C4×Q8—1062no—
396C4×Q8—1062no—
397C4×Q8—1062no—
398C4×Q8—1061no—
399C4×Q8—1062no—
400C4×Q8—1061no—
401C4×Q8—1061no—
402C4×Q8C2×#615712yescomplemented
403#117—21no—
404#117—1062no—
405#117—1062no—
406#117—1061no—
407#117—1061no—
408#117—1061no—
409C2×C4⋊C4—534no—
410C2×C4⋊C4—532no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1068no—
417C2×C4⋊C4—1062no—
418C2×C22⋊C4—1062no—
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—531no—
429C2×C42—531no—
430C2×C42—1062no—
431C2×C42—1062no—
432C2×C42—1062no—
433C2×C42—2124no—
434C2×C42C2×#615711yescharacteristic
435C4⋊C8—1062no—
436C4⋊C8—1062no—
437#103—10616no—
438C22⋊C8—1062no—
439C22⋊C8—1062no—
440C4×C8—1062no—
441C4×C8—1062no—
442#95—1062no—
443#95—1062no—
444C24—531no—
445C2×Q8—21no—
446C2×Q8—532no—
447C2×Q8—1062no—
448C2×Q8—1062no—
449C2×Q8—1068no—
450C2×Q8—1061no—
451C2×Q8—2124no—
452C2×Q8C22×#615712yes—
453C22×C4—531no—
454C22×C4—534no—
455C22×C4—534no—
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—531no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—1064no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1064no—
474C22×C4—1068no—
475C22×C4—2122no—
476C22×C4—2124no—
477C22×C4—2124no—
478C22×C4C22⋊#615711yescharacteristic
479C22×C4C22⋊#615711yescharacteristic
480C22×C4C22×#615711yescharacteristic
481C2×C8—1064no—
482C2×C8—1068no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—538no—
489C4⋊C4—532no—
490C4⋊C4—1062no—
491C4⋊C4—1062no—
492C4⋊C4—1062no—
493C4⋊C4—1062no—
494C4⋊C4—1062no—
495C4⋊C4—1062no—
496C4⋊C4—1062no—
497C4⋊C4—1062no—
498C4⋊C4—1064no—
499C4⋊C4—1068no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1062no—
503C4⋊C4—1062no—
504C4⋊C4—1061no—
505C4⋊C4C22×#615712yes—
506C22⋊C4—22no—
507C22⋊C4—1062no—
508C22⋊C4—1062no—
509C22⋊C4—1062no—
510C42—22no—
511C42—531no—
512C42—531no—
513C42—531no—
514C42—531no—
515C42—531no—
516C42—531no—
517C42—531no—
518C42—531no—
519C42—531no—
520C42—531no—
521C42—531no—
522C42—531no—
523C42—1062no—
524C42—1062no—
525C42—1062no—
526C42—2124no—
527C42—2124no—
528C42C22⋊#615711yescharacteristic
529C42C22⋊#615711yescharacteristic
530C42C22×#615711yescharacteristic
531C42C22×#615711yescharacteristic
532C23—531no—
533C23—532no—
534C23—532no—
535C23—531no—
536C23—1062no—
537C23—1062no—
538C23C2×C22⋊#615711yescharacteristic
539Q8—24no—
540Q8—42no—
541Q8—1064no—
542Q8—2122no—
543Q8—2124no—
544Q8—2124no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—534no—
550C2×C4—531no—
551C2×C4—531no—
552C2×C4—531no—
553C2×C4—534no—
554C2×C4—534no—
555C2×C4—532no—
556C2×C4—532no—
557C2×C4—532no—
558C2×C4—532no—
559C2×C4—534no—
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—1068no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2124no—
575C2×C4—2124no—
576C2×C4—2121no—
577C2×C4—2124no—
578C2×C4—2124no—
579C2×C4—21216no—
580C2×C4#1313328014yescomplemented
581C2×C4C2×C22⋊#615711yescharacteristic
582C2×C4C2×C22⋊#615711yescharacteristic
583C2×C4C2×C22⋊#615711yescharacteristic
584C2×C4C2×C22⋊#615711yescharacteristic
585C2×C4C2×C22⋊#615711yescharacteristic
586C2×C4C23×#615711yescharacteristic
587C8—2124no—
588C22—21no—
589C22—21no—
590C22—531no—
591C22—531no—
592C22—532no—
593C22—532no—
594C22—532no—
595C22—1064no—
596C22—1064no—
597C22—1061no—
598C22—1061no—
599C22C2×#1313328012yes—
600C22C22×C22⋊#615711yescharacteristic
601C4—21no—
602C4—21no—
603C4—21no—
604C4—42no—
605C4—42no—
606C4—532no—
607C4—532no—
608C4—532no—
609C4—1061no—
610C4—1061no—
611C4—1061no—
612C4—2128no—
613C4—2122no—
614C4—2122no—
615C4C2×#1313328012yes—
616C4C2×#1313328012yes—
617C4#41×#615712yes—
618C2—22no—
619C2—532no—
620C2—532no—
621C2—1062no—
622C2#2029117611yescharacteristic
623C2C22×#1313328011yescharacteristic
624C2#2029133511yescharacteristic
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
hypocenterC53#21385
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #20290893C2semidirect
4. #20290881C2semidirect
6. C4×#13133280C2semidirect
24. #16811180C4semidirect
26. #16811178C4semidirect
27. #16811178C4semidirect
29. #16811177C4semidirect
32. #16811175C4semidirect
42. C53×#533C4semidirect
65. #13133217C2×C4semidirect
66. #13133217C2×C4semidirect
70. #13133216C2×C4semidirect
72. #13133215C2×C4semidirect
89. Q8×Dic53C2×C4semidirect
93. C4×Dic106C2×C4semidirect
97. #13133182C2×C4semidirect
122. C4⋊Q8×C53C2×C4semidirect
123. C4⋊Q8×C53C2×C4semidirect
128. C4×Q8×C53C2×C4semidirect
236. C2×C4×C53#103semidirect
247. C2×Dic53#103semidirect
263. C4×D53#103semidirect
264. C4×D53#103semidirect
329. #533#6157semidirect
355. C53#21385semidirect
371. C4⋊Q8C2×#6157semidirect
372. C4⋊Q8C2×#6157semidirect
402. C4×Q8C2×#6157semidirect
580. C2×C4#13133280semidirect

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
#202908932
#202908812
#202907002
C4×#131332802
#533×D532
#2138553