Finite Group Databasebeta

← #27351303

Subgroups — #27351303

Order 13568.

1102 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288764C211yesmaximal, characteristic
2#20277076C211yesmaximal, characteristic, complemented
3#20273241C211yesmaximal, characteristic, complemented
4#20273236C212yesmaximal, complemented
5#20273228C211yesmaximal, characteristic, complemented
6#20273220C212yesmaximal, complemented
7#20273215C211yesmaximal, characteristic, complemented
8#20273207C212yesmaximal, complemented
9#20273202C211yesmaximal, characteristic, complemented
10C2×#16810361C212yesmaximal, complemented
11C53×#2974C211yesmaximal, characteristic, complemented
12#16811343C411yescharacteristic, complemented
13#16811328—21no—
14#16811313C412yescomplemented
15#16811299—22no—
16D4×Dic106C411yescharacteristic, complemented
17#16811282C412yescomplemented
18#16811278C411yescharacteristic, complemented
19#16811277C411yescharacteristic, complemented
20#16811267—21no—
21#16811238—21no—
22#16811215—22no—
23#16811180C2211yescharacteristic
24#16811157C2211yescharacteristic
25#16811134C2212yes—
26C2×#13133196C2212yes—
27#16811102—21no—
28C2×D4×Dic53—21no—
29C4⋊C4×Dic53—21no—
30C4×C4⋊Dic53C2211yescharacteristic
31C22⋊C4×Dic53—22no—
32C4×C22⋊Dic53C2212yes—
33#16810469C2211yescharacteristic
34C2×#13133124C2212yes—
35#16810361C2211yescharacteristic
36#16810361C2212yes—
37#16810361C2212yes—
38#16810361C2212yes—
39#16810361C2211yescharacteristic
40#16810361C2212yes—
41#16810361C2212yes—
42#16810358C2211yescharacteristic
43#16810358C2211yescharacteristic
44#16810358C2212yes—
45C2×C4×D4×C53C2211yescharacteristic
46C53×#436—21no—
47C53×#435—22no—
48C8×D4×C53—21no—
49C53×#432C2211yescharacteristic
50C53×#431C2211yescharacteristic
51C53×#407C2212yes—
52C2×C22⋊C8×C53C2212yes—
53C2×#11719945C2×C412yescomplemented
54C2×#11719947—21no—
55C22×Dic106C2×C412yescomplemented
56C23×Dic53—22no—
57#13133217C2×C411yescharacteristic, complemented
58#13133217C2×C411yescharacteristic, complemented
59#13133216—21no—
60#13133211—21no—
61#13133210—22no—
62#13133210—22no—
63#13133209—21no—
64#13133209C2×C411yescharacteristic, complemented
65#13133207—22no—
66#13133205C2×C411yescharacteristic, complemented
67#13133204—21no—
68#13133203—22no—
69#13133201—22no—
70#13133200—21no—
71#13133200—22no—
72#13133199—22no—
73#13133198—22no—
74#13133197—21no—
75#13133196—21no—
76#13133196—22no—
77#13133196—22no—
78#13133196C2×C412yescomplemented
79#13133196C2×C412yescomplemented
80#13133196C2×C412yescomplemented
81#13133196C2311yescharacteristic
82#13133196C2312yes—
83#13133195—22no—
84#13133194—22no—
85#13133191C2×C412yescomplemented
86#13133191C2311yescharacteristic
87#13133189—22no—
88#13133188—21no—
89#13133186—22no—
90C4×Dic106—21no—
91C4×Dic106C2×C412yescomplemented
92C4×Dic106C2×C411yescharacteristic, complemented
93#13133182—21no—
94C4×D212C2×C411yescharacteristic, complemented
95D4×Dic53—28no—
96D4×Dic53—21no—
97C4×C22⋊D53—21no—
98C4×C22⋊D53—22no—
99C4×C22⋊D53—22no—
100#13133177—22no—
101#13133176—21no—
102#13133176C2×C412yescomplemented
103#13133174—21no—
104#13133173—22no—
105C2×C4⋊Dic53—21no—
106C2×C4⋊Dic53C2×C412yescomplemented
107C2×#11719919—22no—
108C4⋊C4×D53—21no—
109C2×C22⋊Dic53—22no—
110C2×C22⋊Dic53—22no—
111C2×C4×Dic53—21no—
112C2×C4×Dic53—22no—
113C2×C4×Dic53—21no—
114#13133124—22no—
115#13133124—21no—
116#13133124C2311yescharacteristic
117#13133124C2312yes—
118#13133124C2311yescharacteristic
119C22×D4×C53D411yescharacteristic
120C23×C4×C53C2312yes—
121C2×M16×C53—22no—
122C22×C8×C53—22no—
123C4×D4×C53C2×C411yescharacteristic
124C4×D4×C53C2×C411yescharacteristic
125C4×D4×C53C2×C412yes—
126C4×D4×C53D411yescharacteristic
127C4×D4×C53D412yes—
128C4×D4×C53D411yescharacteristic
129C2×C4⋊C4×C53D411yescharacteristic
130C2×C22⋊C4×C53D412yes—
131C2×C42×C53C2311yescharacteristic
132C4⋊C8×C53—21no—
133C4⋊C8×C53—21no—
134C22⋊C8×C53—22no—
135C22⋊C8×C53—22no—
136C22⋊C8×C53C2311yescharacteristic
137C22⋊C8×C53C2312yes—
138C22⋊C8×C53C2311yescharacteristic
139#97×C53—21no—
140C4×C8×C53—21no—
141C24×C53C2×D412yes—
142C2×D4×C53—24no—
143C2×D4×C53C22⋊C411yescharacteristic
144C2×D4×C53C22⋊C412yes—
145C2×D4×C53C22⋊C411yescharacteristic
146C22×C4×C53—82no—
147C22×C4×C53C22×C412yes—
148C22×C4×C53C22×C412yes—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D412yes—
152C22×C4×C53C2×D412yes—
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2411yescharacteristic
155C2×C22⋊D53—22no—
156C2×C22⋊D53—21no—
157C2×C22⋊D53—22no—
158C22×Dic53—22no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—22no—
162C22×Dic53—28no—
163C22×Dic53—22no—
164C22×Dic53—82no—
165C22×Dic53—82no—
166#11719947—44no—
167#11719945—82no—
168C2×D212C22×C411yescharacteristic
169C2×C4×D53—21no—
170C2×C4×D53—22no—
171C2×Dic106—21no—
172C2×Dic106—22no—
173C2×Dic106—82no—
174C2×Dic106C22×C411yescharacteristic
175C2×Dic106C22×C412yes—
176M16×C53—82no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53—22no—
180C2×C8×C53—82no—
181C4⋊C4×C53C22⋊C411yescharacteristic
182C4⋊C4×C53C22⋊C412yes—
183C4⋊C4×C53C22⋊C411yescharacteristic
184C22⋊C4×C53C22⋊C412yes—
185C22⋊C4×C53C22⋊C412yes—
186C22⋊C4×C53C22⋊C412yes—
187C22⋊C4×C53C22⋊C412yes—
188C42×C53C22×C411yescharacteristic
189C42×C53C22×C411yescharacteristic
190C42×C53C2×D411yescharacteristic
191C42×C53C2×D411yescharacteristic
192C22⋊Dic53—28no—
193C22⋊Dic53—22no—
194C22⋊Dic53—22no—
195C22⋊Dic53—22no—
196C22⋊Dic53—22no—
197C22⋊Dic53—21no—
198C22⋊Dic53—22no—
199#11719921—21no—
200#11719921—21no—
201#11719921—22no—
202#11719921—21no—
203#11719921—21no—
204C4⋊Dic53—24no—
205C4⋊Dic53—21no—
206C4⋊Dic53—22no—
207C4⋊Dic53—21no—
208C4⋊Dic53—21no—
209C4⋊Dic53C22×C411yescharacteristic
210C4⋊Dic53C22×C412yes—
211C4⋊Dic53C22×C411yescharacteristic
212#11719919—22no—
213#11719919—22no—
214#11719919—21no—
215#11719919—22no—
216#11719919—21no—
217#11719919—22no—
218#11719919—21no—
219#11719919—21no—
220#11719919—22no—
221C4×Dic53—24no—
222C4×Dic53—22no—
223C4×Dic53—22no—
224C4×Dic53—21no—
225C4×Dic53—81no—
226C4×Dic53—82no—
227C23×C53—22no—
228C23×C53—22no—
229C23×C53—24no—
230C23×C53—42no—
231C23×C53C2×C22⋊C412yes—
232C23×C53C2×C22⋊C412yes—
233C23×C53C22×D411yescharacteristic
234C22×D53—21no—
235D4×C53—24no—
236D4×C53—22no—
237D4×C53#10414yes—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—22no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—41no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53C2×C22⋊C412yes—
248C2×C4×C53C2×C22⋊C412yes—
249C2×C4×C53C2×C22⋊C411yescharacteristic
250C2×C4×C53C2×C22⋊C411yescharacteristic
251C2×C4×C53C2×C22⋊C411yescharacteristic
252C2×C4×C53C2×C22⋊C411yescharacteristic
253C2×C4×C53C23×C411yescharacteristic
254C2×C4×C53C22×D411yescharacteristic
255C22⋊D53—41no—
256C22⋊D53—41no—
257C22⋊D53—44no—
258C22⋊D53—82no—
259C2×Dic53—22no—
260C2×Dic53—24no—
261C2×Dic53—22no—
262C2×Dic53—21no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—41no—
268C2×Dic53—42no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—42no—
272C2×Dic53—41no—
273C2×Dic53—44no—
274C2×Dic53—82no—
275C2×Dic53—88no—
276D212—41no—
277C4×D53—42no—
278C4×D53—82no—
279Dic106—42no—
280Dic106—41no—
281Dic106—42no—
282Dic106—82no—
283C8×C53—41no—
284C8×C53—42no—
285C8×C53—41no—
286#17548—531nomaximal, Sylow
287C22×C53—21no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—21no—
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—44no—
295C22×C53—44no—
296C22×C53C2×#10414yes—
297C22×C53C22×C22⋊C411yescharacteristic
298C22×C53#57711yescharacteristic
299C22×C53#57811yescharacteristic
300C2×D53—41no—
301C2×D53—41no—
302C2×D53—41no—
303C4×C53—21no—
304C4×C53—21no—
305C4×C53—22no—
306C4×C53—22no—
307C4×C53—21no—
308C4×C53—21no—
309C4×C53—21no—
310C4×C53C2×#10412yes—
311Dic53—42no—
312Dic53—44no—
313Dic53—42no—
314Dic53—41no—
315Dic53—42no—
316Dic53—41no—
317Dic53—42no—
318Dic53—41no—
319#3520—531no—
320#2974D5311yescharacteristic, complemented
321#2275—531no—
322#1509—531no—
323#1508—532no—
324#1506—531no—
325#1504—532no—
326#1503—531no—
327#1501—532no—
328#1500—531no—
329C2×#328—532no—
330C2×C53—21no—
331C2×C53—21no—
332C2×C53—24no—
333C2×C53—42no—
334C2×C53#288311yescharacteristic
335C2×C53#289811yescharacteristic
336C2×C53C22×#10411yescharacteristic
337D53—81no—
338#556—531no—
339#554—1061no—
340#552—532no—
341#551—1062no—
342D4×Q8—531no—
343#548—532no—
344#547—531no—
345#547—531no—
346#546—1061no—
347#542—1061no—
348#539—1062no—
349#533—531no—
350#529—531no—
351#525—532no—
352C2×C22⋊Q8—532no—
353#520—1061no—
354C2×C4×D4—1061no—
355C2×C4×D4C2×D5311yescharacteristic
356#436—21no—
357#435—22no—
358C8×D4—21no—
359#432C2×D5311yescharacteristic
360#431C2×D5311yescharacteristic
361#407C2×D5312yes—
362C2×C22⋊C8C2×D5312yes—
363C4×C4⋊C4—531no—
364C4×C4⋊C4—1061no—
365C4×C22⋊C4—532no—
366C4×C22⋊C4—1062no—
367#376—531no—
368C2×#95—532no—
369#328—531no—
370#328—532no—
371#328—532no—
372#328—532no—
373#328—531no—
374#328—532no—
375#328—532no—
376#327—531no—
377#327—531no—
378#327—532no—
379C53#1754811yescharacteristic, complemented, Sylow
380C2×#41—532no—
381C2×#41—1061no—
382C22×Q8—532no—
383C22×D4D21211yescharacteristic
384C23×C4—1062no—
385C23×C4C22×D5312yes—
386C2×M16—22no—
387C22×C8—22no—
388C4⋊Q8—531no—
389C4⋊Q8—531no—
390C4⋊Q8—1061no—
391#126—1062no—
392#126—1062no—
393#126—1061no—
394#125—531no—
395#125—1062no—
396#125—1061no—
397#124—531no—
398#124—1061no—
399#124—1062no—
400#123—1062no—
401#123—1061no—
402#123—1062no—
403#123—1062no—
404#123—1062no—
405C22⋊Q8—532no—
406C22⋊Q8—532no—
407C22⋊Q8—532no—
408C22⋊Q8—531no—
409C22⋊Q8—532no—
410C22⋊Q8—1062no—
411C22⋊Q8—1061no—
412C22⋊Q8—1062no—
413C22⋊Q8—1062no—
414C22⋊Q8—1061no—
415C22⋊Q8—1062no—
416#121—532no—
417#121—531no—
418#121—1062no—
419#121—1061no—
420#120—1062no—
421C4×Q8—531no—
422C4×Q8—532no—
423C4×Q8—1061no—
424C4×Q8—1061no—
425C4×D4—531no—
426C4×D4—1068no—
427C4×D4—1061no—
428C4×D4—1062no—
429C4×D4—1061no—
430C4×D4—1062no—
431C4×D4—1062no—
432C4×D4C4×D5311yescharacteristic
433C4×D4C4×D5311yescharacteristic
434C4×D4C4×D5312yes—
435C4×D4C22⋊D5311yescharacteristic
436C4×D4C22⋊D5312yes—
437C4×D4C22⋊D5311yescharacteristic
438#117—532no—
439#117—1061no—
440#117—1062no—
441#117—1061no—
442C2×C4⋊C4—532no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1062no—
445C2×C4⋊C4—1061no—
446C2×C4⋊C4D21211yescharacteristic
447C2×C22⋊C4—1062no—
448C2×C22⋊C4—1062no—
449C2×C22⋊C4D21212yes—
450C2×C42—1061no—
451C2×C42—1062no—
452C2×C42—1061no—
453C2×C42C22×D5311yescharacteristic
454C4⋊C8—21no—
455C4⋊C8—21no—
456C22⋊C8—22no—
457C22⋊C8—22no—
458C22⋊C8C22×D5311yescharacteristic
459C22⋊C8C22×D5312yes—
460C22⋊C8C22×D5311yescharacteristic
461#97—21no—
462C4×C8—21no—
463#95—531no—
464#95—532no—
465#95—531no—
466#95—1062no—
467#95—1061no—
468C24C2×D21212yes—
469#41—4242no—
470#41—2124no—
471C2×Q8—531no—
472C2×Q8—532no—
473C2×Q8—1062no—
474C2×Q8—1061no—
475C2×Q8—4242no—
476C2×D4—24no—
477C2×D4—531no—
478C2×D4—1062no—
479C2×D4—1062no—
480C2×D4—1061no—
481C2×D4#1171992112yes—
482C2×D4#1171992111yescharacteristic
483C2×D4#1171992111yescharacteristic
484C22×C4—82no—
485C22×C4—1068no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1061no—
493C22×C4—4242no—
494C22×C4—4242no—
495C22×C4C2×C4×D5312yes—
496C22×C4C2×C4×D5312yes—
497C22×C4C2×D21211yescharacteristic
498C22×C4C2×D21211yescharacteristic
499C22×C4C2×D21212yes—
500C22×C4C2×C22⋊D5312yes—
501C22×C4C2×C22⋊D5312yes—
502C22×C4C23×D5311yescharacteristic
503M16—82no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—22no—
507C2×C8—82no—
508C4⋊C4—531no—
509C4⋊C4—532no—
510C4⋊C4—531no—
511C4⋊C4—1064no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—1061no—
515C4⋊C4—1062no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1062no—
520C4⋊C4—1061no—
521C4⋊C4—1062no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1062no—
525C4⋊C4#1171992111yescharacteristic
526C4⋊C4#1171992112yes—
527C4⋊C4#1171992111yescharacteristic
528C22⋊C4—1068no—
529C22⋊C4—1062no—
530C22⋊C4—1062no—
531C22⋊C4—1062no—
532C22⋊C4—1062no—
533C22⋊C4—1061no—
534C22⋊C4—1062no—
535C22⋊C4—1061no—
536C22⋊C4—1061no—
537C22⋊C4—1062no—
538C22⋊C4—1061no—
539C22⋊C4—1061no—
540C22⋊C4#1171992112yes—
541C22⋊C4#1171992112yes—
542C22⋊C4#1171992112yes—
543C22⋊C4#1171992112yes—
544C42—1064no—
545C42—1062no—
546C42—1062no—
547C42—1061no—
548C42—4241no—
549C42—4242no—
550C42C2×C4×D5311yescharacteristic
551C42C2×C4×D5311yescharacteristic
552C42C2×C22⋊D5311yescharacteristic
553C42C2×C22⋊D5311yescharacteristic
554C23—24no—
555C23—22no—
556C23—22no—
557C23—42no—
558C23—1061no—
559C23C2×#1171992112yes—
560C23C2×#1171992112yes—
561C23C22×D21211yescharacteristic
562Q8—4242no—
563Q8—2121no—
564Q8—2122no—
565Q8—2122no—
566D4—24no—
567D4—22no—
568D4—4242no—
569D4—2121no—
570D4—2124no—
571D4—2121no—
572D4—2121no—
573D4#1313314314yes—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—1062no—
584C2×C4—1062no—
585C2×C4—1064no—
586C2×C4—1061no—
587C2×C4—1062no—
588C2×C4—1061no—
589C2×C4—1062no—
590C2×C4—1061no—
591C2×C4—4242no—
592C2×C4—4242no—
593C2×C4—4248no—
594C2×C4—2121no—
595C2×C4—2122no—
596C2×C4—2121no—
597C2×C4—2124no—
598C2×C4—2122no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2122no—
602C2×C4C2×#1171992111yescharacteristic
603C2×C4C2×#1171992112yes—
604C2×C4C2×#1171992111yescharacteristic
605C2×C4C2×#1171992111yescharacteristic
606C2×C4C2×#1171992112yes—
607C2×C4C2×#1171992111yescharacteristic
608C2×C4C22×C4×D5311yescharacteristic
609C2×C4C22×C22⋊D5311yescharacteristic
610C8—41no—
611C8—42no—
612C8—41no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—21no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—44no—
621C22—44no—
622C22—2121no—
623C22—2121no—
624C22—2121no—
625C22C2×#1313314314yes—
626C22C22×#1171992111yescharacteristic
627C22#1681143511yescharacteristic
628C22#1681144011yescharacteristic
629C4—21no—
630C4—21no—
631C4—22no—
632C4—22no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—2122no—
637C4—2124no—
638C4—2122no—
639C4—2122no—
640C4—2121no—
641C4—2121no—
642C4—2121no—
643C4—2122no—
644C4C2×#1313314312yes—
645C2—21no—
646C2—21no—
647C2—24no—
648C2—42no—
649C2—4241no—
650C2#2028317511yescharacteristic
651C2#2028324911yescharacteristic
652C2C22×#1313314311yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2974C2
socleC22×C53C22×C22⋊C4
prime radical#20288764C2
cosocleC22×C4×C53C24
hypercenter#2974D53
hypocenterC53#17548
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2974

Complements

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

NH ≅ G/Ntype
2. #20277076C2semidirect
3. #20273241C2semidirect
4. #20273236C2semidirect
5. #20273228C2semidirect
6. #20273220C2semidirect
7. #20273215C2semidirect
8. #20273207C2semidirect
9. #20273202C2semidirect
10. C2×#16810361C2semidirect
11. C53×#2974C2semidirect
12. #16811343C4semidirect
14. #16811313C4semidirect
16. D4×Dic106C4semidirect
17. #16811282C4semidirect
18. #16811278C4semidirect
19. #16811277C4semidirect
53. C2×#11719945C2×C4semidirect
55. C22×Dic106C2×C4semidirect
57. #13133217C2×C4semidirect
58. #13133217C2×C4semidirect
64. #13133209C2×C4semidirect
66. #13133205C2×C4semidirect
78. #13133196C2×C4semidirect
79. #13133196C2×C4semidirect
80. #13133196C2×C4semidirect
85. #13133191C2×C4semidirect
91. C4×Dic106C2×C4semidirect
92. C4×Dic106C2×C4semidirect
94. C4×D212C2×C4semidirect
102. #13133176C2×C4semidirect
106. C2×C4⋊Dic53C2×C4semidirect
320. #2974D53semidirect
379. C53#17548semidirect

Maximal subgroups

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

subgroupindex
#202887642
#202770762
#202732412
#202732362
#202732282
#202732202
#202732152
#202732072
#202732022
C2×#168103612
C53×#29742
#1754853