Finite Group Databasebeta

← #28255656

Subgroups — #28255656

Order 13568.

2098 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811244C211yesmaximal, characteristic, complemented
2#20288048C211yesmaximal, characteristic, complemented
3#20285826C218yesmaximal, complemented
4#20285811C218yesmaximal, complemented
5#20285181C212yesmaximal, complemented
6#20285109C212yesmaximal, complemented
7#20284189C212yesmaximal, complemented
8#20284181C212yesmaximal, complemented
9#20283977C212yesmaximal, complemented
10#20283969C212yesmaximal, complemented
11C53×#2921C211yesmaximal, characteristic, complemented
12C22×#11719945C2212yes—
13C2×#13133250C2214yescomplemented
14C2×#13133245C2214yescomplemented
15#16811244—24no—
16#16811244C2218yescomplemented
17#16811204—21no—
18#16811204—21no—
19#16811199C2214yescomplemented
20#16811192C2214yescomplemented
21#16811185—21no—
22#16811185—21no—
23#16811173C2212yes—
24#16811173C2212yes—
25C2×#13133217C2211yescharacteristic, complemented
26C2×#13133212—21no—
27C2×#13133205C2211yescharacteristic, complemented
28C2×#13133203—21no—
29#16811135—24no—
30C2×#13133199—21no—
31C2×#13133196C2211yescharacteristic
32C2×#13133195—21no—
33C2×#13133191C2211yescharacteristic
34C2×C4×Dic106C2211yescharacteristic, complemented
35C2×C4×D212C2211yescharacteristic, complemented
36#16811017C2214yescomplemented
37#16811012C2214yescomplemented
38#16811007C2218yescomplemented
39#16810983C2218yescomplemented
40#16810974C2214yescomplemented
41#16810970C2214yescomplemented
42#16810846C2218yescomplemented
43#16810839C2218yescomplemented
44#16810834C2214yescomplemented
45#16810829C2214yescomplemented
46#16810824C2214yescomplemented
47#16810819C2214yescomplemented
48#16810787C2218yescomplemented
49#16810781C2214yescomplemented
50#16810776C2214yescomplemented
51#16810770C2218yes—
52#16810764C2214yescomplemented
53#16810759C2214yescomplemented
54#16810625C2214yescomplemented
55#16810625C2214yescomplemented
56C2×#117×C53C2211yescharacteristic
57C53×#433C2218yescomplemented
58C53×#423C2212yescomplemented
59C53×#407C2212yes—
60C4×M16×C53C2212yescomplemented
61C2×#11719945—22no—
62C2×#11719945—22no—
63C2×#11719945—48no—
64C2×#11719945C2314yes—
65C2×#11719945C2314yes—
66C22×Dic106—21no—
67C22×Dic106C2311yescharacteristic
68C22×Dic106C2312yes—
69C22×D212C2311yescharacteristic
70C22×D212C2312yes—
71C22×C22⋊D53—22no—
72C22×C22⋊D53—21no—
73C22×C4×D53—22no—
74C22×C4×D53—21no—
75#13133250—48no—
76#13133245—48no—
77C2×Dic212C2318yes—
78C2×#11719913C2318yes—
79C2×#11719913C2318yes—
80C2×D424C2318yes—
81#13133217—22no—
82#13133217C2314yes—
83#13133217C2312yes—
84#13133217C2312yes—
85#13133214C2312yes—
86#13133214C2312yes—
87#13133212—28no—
88#13133205—22no—
89#13133205C2314yes—
90#13133205C2314yes—
91#13133205C2314yes—
92#13133203—28no—
93#13133201—24no—
94#13133201—24no—
95#13133199—28no—
96#13133197—24no—
97#13133197—24no—
98#13133196—22no—
99#13133196C2314yes—
100#13133195—28no—
101#13133192—24no—
102#13133192—24no—
103#13133191—22no—
104#13133191—22no—
105#13133191—22no—
106#13133191—22no—
107#13133191—22no—
108#13133191C2314yes—
109#13133187—24no—
110#13133187—24no—
111C4×Dic106—22no—
112C4×Dic106C2314yes—
113C4×D212—22no—
114C4×D212—22no—
115C4×D212—22no—
116C4×D212—22no—
117C4×D212—22no—
118C4×D212C2314yes—
119C2×C4⋊Dic53—21no—
120C2×C4⋊Dic53C2311yescharacteristic
121C2×C4⋊Dic53C2311yescharacteristic
122C2×#11719919—21no—
123C2×#11719919—21no—
124C2×C22⋊Dic53—21no—
125C2×#11719921—21no—
126C2×#11719921—21no—
127C2×#11719921—21no—
128C2×#11719921—21no—
129C2×C4×Dic53—21no—
130#13133142C2318yes—
131#13133142C2318yes—
132#13133139C2318yes—
133#13133139C2318yes—
134C23×C4×C53C2311yescharacteristic
135C2×M16×C53C2314yes—
136#117×C53D414yes—
137#117×C53C2314yes—
138C2×C4⋊C4×C53D411yescharacteristic
139C2×C4⋊C4×C53D411yescharacteristic
140C2×C22⋊C4×C53D411yescharacteristic
141C2×C22⋊C4×C53D411yescharacteristic
142C2×C42×C53C2311yescharacteristic
143C2×C42×C53C2311yescharacteristic
144C4⋊C8×C53C2314yes—
145C4⋊C8×C53C2314yes—
146C22⋊C8×C53C2314yes—
147C22⋊C8×C53C2314yes—
148#97×C53C2314yes—
149C4×C8×C53C2314yes—
150C24×C53C2×D411yescharacteristic
151C23×D53—21no—
152C23×D53—22no—
153C22×C4×C53—24no—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D412yes—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2412yes—
161C22×C4×C53C2411yescharacteristic
162C2×C22⋊D53—24no—
163C2×C22⋊D53—24no—
164C2×C22⋊D53—24no—
165C2×C22⋊D53—48no—
166C2×C22⋊D53—82no—
167C22×Dic53—21no—
168C22×Dic53—21no—
169C22×Dic53—21no—
170C22×Dic53—22no—
171#11719945—28no—
172#11719945—28no—
173#11719945—44no—
174#11719945—44no—
175#11719945—48no—
176#11719945—48no—
177C2×D212—24no—
178C2×D212—21no—
179C2×D212—22no—
180C2×D212—22no—
181C2×D212—22no—
182C2×D212—22no—
183C2×D212—22no—
184C2×D212—44no—
185C2×D212C2414yes—
186C2×D212C2412yes—
187C2×C4×D53—24no—
188C2×C4×D53—24no—
189C2×C4×D53—24no—
190C2×C4×D53—48no—
191C2×C4×D53—82no—
192C2×Dic106—24no—
193C2×Dic106—21no—
194C2×Dic106—22no—
195C2×Dic106—24no—
196C2×Dic106—44no—
197C2×Dic106—82no—
198C2×Dic106C2414yes—
199C2×Dic106C2412yes—
200M16×C53—28no—
201C2×C8×C53C2418yes—
202C4⋊C4×C53C2×D414yes—
203C4⋊C4×C53C2×D414yes—
204C22⋊C4×C53C2×D414yes—
205C22⋊C4×C53C2×D414yes—
206C42×C53C2×D412yes—
207C42×C53C2×D412yes—
208C42×C53C2412yes—
209C42×C53C2412yes—
210C22⋊Dic53—24no—
211#11719921—24no—
212#11719921—24no—
213#11719921—24no—
214#11719921—24no—
215#11719921—24no—
216#11719921—24no—
217#11719921—24no—
218#11719921—24no—
219C4⋊Dic53—24no—
220C4⋊Dic53—21no—
221C4⋊Dic53—21no—
222C4⋊Dic53—22no—
223C4⋊Dic53—22no—
224C4⋊Dic53—22no—
225C4⋊Dic53—22no—
226C4⋊Dic53C2412yes—
227C4⋊Dic53C2412yes—
228#11719919—24no—
229#11719919—24no—
230C4×Dic53—24no—
231Dic212—48no—
232D424—48no—
233#11719913—48no—
234#11719913—48no—
235C23×C53—22no—
236C23×C53—41no—
237C23×C53—41no—
238C23×C53C22×D412yes—
239C23×C53C22×D411yescharacteristic
240C22×D53—24no—
241C22×D53—22no—
242C22×D53—44no—
243C22×D53—42no—
244C22×D53—42no—
245C22×D53—42no—
246C22×D53—82no—
247C22×D53—82no—
248C2×C4×C53—22no—
249C2×C4×C53—24no—
250C2×C4×C53—24no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53C22×D412yes—
256C2×C4×C53C22×D412yes—
257C2×C4×C53C22×D412yes—
258C2×C4×C53C22×D412yes—
259C2×C4×C53C22×D412yes—
260C2×C4×C53C22×D411yescharacteristic
261C2×C4×C53H(5,𝔽2)14yes—
262C2×C4×C53C2511yescharacteristic
263C22⋊D53—48no—
264C22⋊D53—48no—
265C22⋊D53—84no—
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—24no—
269C2×Dic53—22no—
270C2×Dic53—44no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274D212—24no—
275D212—24no—
276D212—24no—
277D212—44no—
278D212—42no—
279D212—82no—
280D212—82no—
281D212—82no—
282D212—82no—
283C4×D53—48no—
284C4×D53—48no—
285C4×D53—84no—
286Dic106—24no—
287Dic106—24no—
288Dic106—24no—
289Dic106—44no—
290Dic106—42no—
291Dic106—84no—
292C8×C53—28no—
293#60810—531nomaximal, Sylow
294C22×C53—21no—
295C22×C53—21no—
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—22no—
299C22×C53—82no—
300C22×C53#53512yes—
301C22×C53C23×D411yescharacteristic
302C2×D53—42no—
303C2×D53—44no—
304C2×D53—44no—
305C2×D53—42no—
306C2×D53—44no—
307C2×D53—42no—
308C2×D53—84no—
309C2×D53—88no—
310C4×C53—22no—
311C4×C53—22no—
312C4×C53—22no—
313C4×C53—22no—
314C4×C53—22no—
315C4×C53#53512yes—
316C4×C53C2×H(5,𝔽2)12yes—
317Dic53—42no—
318Dic53—42no—
319Dic53—44no—
320Dic53—42no—
321C2×#542—531no—
322#3449—531no—
323#3225—538no—
324#3224—538no—
325#3153—532no—
326#3141—532no—
327#3037—532no—
328#3036—532no—
329#3007—532no—
330#3006—532no—
331#2921D5311yescharacteristic, complemented
332C2×C53—22no—
333C2×C53—42no—
334C2×C53C2×#53511yescharacteristic
335C2×C53#358511yescharacteristic
336C2×C53#358611yescharacteristic
337D53—82no—
338D53—84no—
339C22×#41—532no—
340C2×#137—534no—
341C2×AGL(1,ℤ/8)—534no—
342#542—538no—
343#542—1064no—
344#537—1061no—
345#537—1061no—
346#536—534no—
347#535—534no—
348#534—1061no—
349#534—1061no—
350#532—532no—
351#532—532no—
352C2×C4⋊Q8—531no—
353C2×#126—1061no—
354C2×#124—531no—
355C2×#124—1061no—
356#525—1064no—
357C2×#123—1061no—
358C2×C22⋊Q8—531no—
359C2×C22⋊Q8—1061no—
360C2×#121—531no—
361C2×C4×Q8—531no—
362C2×C4×D4—531no—
363C2×#117C2×D5311yescharacteristic
364#497—534no—
365#496—534no—
366#495—538no—
367#489—538no—
368#487—534no—
369#486—534no—
370#464—538no—
371#463—538no—
372#462—534no—
373#461—534no—
374#460—534no—
375#459—534no—
376#452—538no—
377#451—534no—
378#450—534no—
379#449—538no—
380#448—534no—
381#447—534no—
382#433C2×D5318yescomplemented
383#423C2×D5312yescomplemented
384#417—534no—
385#417—534no—
386#407C2×D5312yes—
387C4×M16C2×D5312yescomplemented
388C53#6081011yescharacteristic, complemented, Sylow
389C2×#41—534no—
390C2×#41—534no—
391C2×#41—1062no—
392C2×#41—1062no—
393C2×#41—2128no—
394C22×Q8—531no—
395C22×Q8—532no—
396C22×Q8—1061no—
397C22×D4—531no—
398C22×D4—532no—
399C22×D4—1061no—
400C22×D4—1062no—
401C23×C4—1061no—
402C23×C4—1062no—
403C23×C4C22×D5311yescharacteristic
404#137—2128no—
405AGL(1,ℤ/8)—2128no—
406C2×Q16—538no—
407C2×SD16—538no—
408C2×SD16—538no—
409C2×D8—538no—
410C2×M16C22×D5314yes—
411C4⋊Q8—534no—
412C4⋊Q8—532no—
413C4⋊Q8—532no—
414C4⋊Q8—1062no—
415#127—532no—
416#127—532no—
417#126—1068no—
418#124—534no—
419#124—534no—
420#124—534no—
421#124—1068no—
422#124—1062no—
423#123—1068no—
424#123—1064no—
425#123—1064no—
426C22⋊Q8—534no—
427C22⋊Q8—1062no—
428C22⋊Q8—1068no—
429C22⋊Q8—1064no—
430C22⋊Q8—1064no—
431#121—534no—
432#121—1062no—
433#121—1062no—
434#121—1062no—
435#121—1062no—
436#121—1062no—
437#121—1064no—
438#121—1064no—
439#120—1064no—
440#120—1064no—
441C4×Q8—534no—
442C4×Q8—1062no—
443C4×D4—534no—
444C4×D4—1062no—
445C4×D4—1062no—
446C4×D4—1062no—
447C4×D4—1062no—
448C4×D4—1062no—
449#117D21214yes—
450#117C22×D5314yes—
451C2×C4⋊C4—531no—
452C2×C4⋊C4—531no—
453C2×C4⋊C4—1061no—
454C2×C4⋊C4—1061no—
455C2×C4⋊C4—1061no—
456C2×C4⋊C4D21211yescharacteristic
457C2×C4⋊C4D21211yescharacteristic
458C2×C22⋊C4—1061no—
459C2×C22⋊C4—1061no—
460C2×C22⋊C4—1061no—
461C2×C22⋊C4—1061no—
462C2×C22⋊C4—1061no—
463C2×C22⋊C4D21211yescharacteristic
464C2×C22⋊C4D21211yescharacteristic
465C2×C42—1061no—
466C2×C42C22×D5311yescharacteristic
467C2×C42C22×D5311yescharacteristic
468C4⋊C8C22×D5314yes—
469C4⋊C8C22×D5314yes—
470#103—538no—
471#103—538no—
472#102—538no—
473#102—538no—
474C22⋊C8C22×D5314yes—
475C22⋊C8C22×D5314yes—
476#97C22×D5314yes—
477C4×C8C22×D5314yes—
478C24—1062no—
479C24—1061no—
480C24C2×D21211yescharacteristic
481#41—1068no—
482#41—1068no—
483#41—2124no—
484#41—2124no—
485#41—2128no—
486#41—2128no—
487C2×Q8—534no—
488C2×Q8—532no—
489C2×Q8—1064no—
490C2×Q8—1061no—
491C2×Q8—1062no—
492C2×Q8—1064no—
493C2×Q8—4242no—
494C2×Q8—2124no—
495C2×D4—534no—
496C2×D4—532no—
497C2×D4—1064no—
498C2×D4—1064no—
499C2×D4—1064no—
500C2×D4—1062no—
501C2×D4—1062no—
502C2×D4—1062no—
503C2×D4—1062no—
504C2×D4—1061no—
505C2×D4—1062no—
506C2×D4—1064no—
507C2×D4—4242no—
508C2×D4—2124no—
509C2×D4—2128no—
510C22×C4—24no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—1061no—
514C22×C4—1062no—
515C22×C4—1064no—
516C22×C4—1064no—
517C22×C4—1064no—
518C22×C4—4242no—
519C22×C4—2128no—
520C22×C4C2×D21211yescharacteristic
521C22×C4C2×D21212yes—
522C22×C4C2×D21211yescharacteristic
523C22×C4C2×D21211yescharacteristic
524C22×C4C2×D21211yescharacteristic
525C22×C4C2×D21211yescharacteristic
526C22×C4C23×D5312yes—
527C22×C4C23×D5311yescharacteristic
528Q16—2128no—
529SD16—2128no—
530SD16—2128no—
531D8—2128no—
532M16—28no—
533C2×C8C23×D5318yes—
534C4⋊C4—532no—
535C4⋊C4—532no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—1064no—
539C4⋊C4—1064no—
540C4⋊C4—1064no—
541C4⋊C4—1062no—
542C4⋊C4—1062no—
543C4⋊C4—1062no—
544C4⋊C4—1062no—
545C4⋊C4C2×D21214yes—
546C4⋊C4C2×D21214yes—
547C22⋊C4—1064no—
548C22⋊C4—1064no—
549C22⋊C4—1064no—
550C22⋊C4—1064no—
551C22⋊C4—1064no—
552C22⋊C4—1064no—
553C22⋊C4—1064no—
554C22⋊C4—1064no—
555C22⋊C4—1064no—
556C22⋊C4C2×D21214yes—
557C22⋊C4C2×D21214yes—
558C42—1064no—
559C42C2×D21212yes—
560C42C2×D21212yes—
561C42C23×D5312yes—
562C42C23×D5312yes—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—1062no—
567C23—1064no—
568C23—4242no—
569C23—4242no—
570C23—2122no—
571C23—2124no—
572C23—2122no—
573C23—2122no—
574C23C22×D21212yes—
575C23C22×D21211yescharacteristic
576Q8—1064no—
577Q8—1064no—
578Q8—1064no—
579Q8—4244no—
580Q8—2124no—
581Q8—2122no—
582D4—1064no—
583D4—1064no—
584D4—1064no—
585D4—4242no—
586D4—4242no—
587D4—4242no—
588D4—4242no—
589D4—4244no—
590D4—2124no—
591D4—2128no—
592D4—2128no—
593D4—2122no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—1062no—
602C2×C4—1062no—
603C2×C4—1062no—
604C2×C4—1064no—
605C2×C4—4244no—
606C2×C4—2128no—
607C2×C4—2122no—
608C2×C4—2124no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2128no—
612C2×C4C22×D21212yes—
613C2×C4C22×D21212yes—
614C2×C4C22×D21212yes—
615C2×C4C22×D21212yes—
616C2×C4C22×D21212yes—
617C2×C4C22×D21211yescharacteristic
618C2×C4#1313326314yes—
619C2×C4C24×D5311yescharacteristic
620C8—28no—
621C22—21no—
622C22—21no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—82no—
627C22—4244no—
628C22—4248no—
629C22—2124no—
630C22—2122no—
631C22—2122no—
632C22—2124no—
633C22—2124no—
634C22—2122no—
635C22#1681119212yes—
636C22C23×D21211yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—2124no—
643C4—2122no—
644C4—2122no—
645C4—2122no—
646C4#1681119212yes—
647C4C2×#1313326312yes—
648C2—22no—
649C2—42no—
650C2—4242no—
651C2—4244no—
652C2C2×#1681119211yescharacteristic
653C2#2028949611yescharacteristic
654C2#2028950311yescharacteristic
subgroupG/H
centerC22C23×D212
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC53×#2921C2
socleC22×C53C23×D4
hypercenter#2921D53
hypocenterC53#60810
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#117 ◁ #2921

Complements

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

NH ≅ G/Ntype
1. C2×#16811244C2semidirect
2. #20288048C2semidirect
3. #20285826C2semidirect
4. #20285811C2semidirect
5. #20285181C2semidirect
6. #20285109C2semidirect
7. #20284189C2semidirect
8. #20284181C2semidirect
9. #20283977C2semidirect
10. #20283969C2semidirect
11. C53×#2921C2semidirect
13. C2×#13133250C22semidirect
14. C2×#13133245C22semidirect
16. #16811244C22semidirect
19. #16811199C22semidirect
20. #16811192C22semidirect
25. C2×#13133217C22semidirect
27. C2×#13133205C22semidirect
34. C2×C4×Dic106C22semidirect
35. C2×C4×D212C22semidirect
36. #16811017C22semidirect
37. #16811012C22semidirect
38. #16811007C22semidirect
39. #16810983C22semidirect
40. #16810974C22semidirect
41. #16810970C22semidirect
42. #16810846C22semidirect
43. #16810839C22semidirect
44. #16810834C22semidirect
45. #16810829C22semidirect
46. #16810824C22semidirect
47. #16810819C22semidirect
48. #16810787C22semidirect
49. #16810781C22semidirect
50. #16810776C22semidirect
52. #16810764C22semidirect
53. #16810759C22semidirect
54. #16810625C22semidirect
55. #16810625C22semidirect
57. C53×#433C22semidirect
58. C53×#423C22semidirect
60. C4×M16×C53C22semidirect
331. #2921D53semidirect
382. #433C2×D53semidirect
383. #423C2×D53semidirect
387. C4×M16C2×D53semidirect
388. C53#60810semidirect

Maximal subgroups

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

subgroupindex
C2×#168112442
#202880482
#202858262
#202858112
#202851812
#202851092
#202841892
#202841812
#202839772
#202839692
C53×#29212
#6081053