Finite Group Databasebeta

← #27395265

Subgroups — #27395265

Order 13568.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810773C211yesmaximal, characteristic
2C22⋊Q8×Dic53C211yesmaximal, characteristic, complemented
3C2×#16810524C211yesmaximal, characteristic
4#20275566C212yesmaximal, complemented
5#20275399C212yesmaximal, complemented
6#20275340C212yesmaximal, complemented
7#20275068C212yesmaximal, complemented
8#20274908C211yesmaximal, characteristic, complemented
9#20274647C211yesmaximal, characteristic
10C22⋊C8×Dic53C211yesmaximal, characteristic, complemented
11C53×#1872C211yesmaximal, characteristic, complemented
12C23×D212C2211yescharacteristic
13C22×#11719924C2212yes—
14C2×Q8×Dic53C2211yescharacteristic, complemented
15C22×#11719921—21no—
16C22×#11719921—21no—
17C22×C4×Dic53C2211yescharacteristic
18#16810773C418yescomplemented
19C2×#13133148C2212yescomplemented
20C2×#13133139C2212yes—
21C2×#13133139C2212yes—
22C2×#13133137C2212yes—
23C2×C22⋊#88322C2211yescharacteristic
24C2×C8×Dic53C2212yescomplemented
25#16810530C2211yescharacteristic, complemented
26#16810524—21no—
27#16810524—21no—
28#16810524C2212yes—
29#16810524C2212yes—
30#16810522C2211yescharacteristic, complemented
31#16810509—21no—
32#16810498—21no—
33#16810483C2211yescharacteristic
34C4⋊C4×Dic53—21no—
35C4⋊C4×Dic53C2211yescharacteristic, complemented
36C22⋊C4×Dic53—21no—
37#16810385C2212yescomplemented
38#16810377C2212yescomplemented
39C2×C22⋊Q8×C53C2211yescharacteristic
40C22×C4⋊C4×C53C2211yescharacteristic
41C2×#103×C53C2212yescomplemented
42C2×C22⋊C8×C53C2211yescharacteristic
43C53×#340C2212yescomplemented
44C24×D53—21no—
45C22×D212—21no—
46C22×D212—22no—
47C22×D212—42no—
48C22×D212—42no—
49C22×D212C2×C414yes—
50C22×D212C2312yes—
51C23×Dic53D411yescharacteristic
52C23×Dic53D411yescharacteristic
53C2×#11719924—42no—
54C2×#11719924—42no—
55C2×#11719924C2×C418yescomplemented
56C22×#88322C2312yes—
57Q8×Dic53—42no—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53C2311yescharacteristic
61C2×C4⋊Dic53C2311yescharacteristic
62C2×C22⋊Dic53—21no—
63C2×#11719921—22no—
64C2×#11719921—22no—
65C2×#11719921—22no—
66C2×#11719921—22no—
67C2×#11719921—44no—
68C2×#11719921—44no—
69C2×C4×Dic53—21no—
70C2×C4×Dic53—21no—
71C2×C4×Dic53—42no—
72C2×C4×Dic53D411yescharacteristic
73C2×C4×Dic53D411yescharacteristic
74C2×C4×Dic53C2311yescharacteristic
75C2×C4×Dic53C2311yescharacteristic
76#13133148—42no—
77#13133139—42no—
78#13133139—42no—
79#13133137C2×C418yescomplemented
80C22⋊#88322C2×C414yes—
81C8×Dic53—42no—
82#13133124—21no—
83#13133124—21no—
84#13133124—21no—
85#13133124—21no—
86#13133124—21no—
87C22×Q8×C53C2311yescharacteristic
88C23×C4×C53—21no—
89C23×C4×C53C2311yescharacteristic
90C22×C8×C53C2312yes—
91C22⋊Q8×C53—22no—
92C22⋊Q8×C53C2×C414yes—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—42no—
96C2×C4⋊C4×C53C2312yes—
97C2×C4⋊C4×C53C2311yescharacteristic
98C2×C22⋊C4×C53—21no—
99#103×C53—42no—
100C22⋊C8×C53—22no—
101C24×C53C2×D411yescharacteristic
102C23×D53—24no—
103C23×D53—22no—
104C23×D53—42no—
105C23×D53—42no—
106C23×D53—42no—
107C23×D53—42no—
108C23×D53—42no—
109C23×D53—42no—
110C2×Q8×C53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53—22no—
113C2×Q8×C53C22×C412yes—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53—42no—
122C22×C4×C53C22×C412yes—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53#4112yes—
125C22×C4×C53C2411yescharacteristic
126C22×Dic53—22no—
127C22×Dic53—22no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—41no—
131C22×Dic53—41no—
132C22×Dic53—41no—
133C22×Dic53SD1614yescomplemented
134C22×Dic53C2×D411yescharacteristic
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C2×D212—22no—
139C2×D212—22no—
140C2×D212—22no—
141C2×D212—22no—
142C2×D212—24no—
143C2×D212—22no—
144C2×D212—48no—
145C2×D212—42no—
146C2×D212—42no—
147C2×D212—48no—
148C2×D212C22×C414yes—
149C2×C8×C53—22no—
150C2×C8×C53—22no—
151C2×C8×C53—22no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—42no—
157C4⋊C4×C53C22×C412yes—
158C22⋊C4×C53—24no—
159C22⋊Dic53—42no—
160#11719924—48no—
161#11719921—44no—
162#11719921—44no—
163#11719921—44no—
164#11719921—44no—
165C4⋊Dic53—22no—
166C4⋊Dic53—22no—
167C4⋊Dic53—42no—
168C4⋊Dic53—42no—
169C4×Dic53—22no—
170C4×Dic53—22no—
171C4×Dic53—41no—
172C4×Dic53—41no—
173C4×Dic53—42no—
174C4×Dic53—42no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322C22×C414yes—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53C4×D412yes—
183C23×C53C2×SD1612yes—
184C23×C53#13512yes—
185C23×C53C22×D411yescharacteristic
186C22×D53—24no—
187C22×D53—48no—
188C22×D53—48no—
189C22×D53—42no—
190C22×D53—42no—
191C22×D53—42no—
192C22×D53—42no—
193C22×D53—42no—
194C22×D53—42no—
195C22×D53—48no—
196C22×D53—84no—
197C22×D53—84no—
198C22×D53—84no—
199C22×D53—84no—
200Q8×C53—24no—
201Q8×C53—42no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—84no—
216C2×C4×C53C4×D412yes—
217C2×C4×C53C23×C411yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C53H(5,𝔽2)12yes—
220C2×C4×C53H(5,𝔽2)12yes—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—44no—
231C2×Dic53—44no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53C2×SD1612yes—
239C2×Dic53C2×SD1612yes—
240C2×Dic53AGL(1,ℤ/8)12yes—
241D212—24no—
242D212—24no—
243D212—48no—
244D212—42no—
245D212—42no—
246C8×C53—42no—
247#88322—24no—
248#22043—531nomaximal, Sylow
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53C4×SD1614yes—
256C22×C53C2×C4×D411yescharacteristic
257C22×C53#53411yescharacteristic
258C22×C53#53511yescharacteristic
259C22×C53C22×SD1611yescharacteristic
260C22×C53C2×#13511yescharacteristic
261C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C53#57511yescharacteristic
263C2×D53—44no—
264C2×D53—44no—
265C2×D53—44no—
266C2×D53—816no—
267C2×D53—82no—
268C2×D53—82no—
269C2×D53—82no—
270C2×D53—82no—
271C4×C53—22no—
272C4×C53—42no—
273C4×C53—42no—
274C4×C53—82no—
275C4×C53#51812yes—
276C4×C53#53812yes—
277Dic53—22no—
278Dic53—22no—
279Dic53—22no—
280Dic53—41no—
281Dic53—81no—
282Dic53—81no—
283C2×#450—531no—
284C4×C22⋊Q8—531no—
285C2×#390—531no—
286#1968—532no—
287#1934—532no—
288#1926—532no—
289#1880—532no—
290#1872D5311yescharacteristic, complemented
291#1853—531no—
292#1786—531no—
293C4×C22⋊C8—531no—
294C2×C53—24no—
295C2×C53#232111yescharacteristic
296C2×C53C2×C4×SD1611yescharacteristic
297C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
298C2×C53#318711yescharacteristic
299C2×C53#318811yescharacteristic
300C2×C53#320811yescharacteristic
301C2×C53#321111yescharacteristic
302D53—84no—
303C23×D4—531no—
304C22×SD16—532no—
305C2×C22⋊Q8C2×D5311yescharacteristic
306C2×C4×Q8—531no—
307C22×C4⋊C4C2×D5311yescharacteristic
308C22×C22⋊C4—1061no—
309C22×C22⋊C4—1061no—
310C22×C42—531no—
311#450—538no—
312C2×#106—532no—
313C2×#103C2×D5312yescomplemented
314C2×#102—532no—
315C2×#102—532no—
316C2×#102—532no—
317C2×C22⋊C8—531no—
318C2×C22⋊C8C2×D5311yescharacteristic
319C2×C4×C8—532no—
320#391—531no—
321#390—532no—
322#390—532no—
323#390—1061no—
324#390—1061no—
325#389—531no—
326#387—1061no—
327#385—1061no—
328#380—531no—
329C4×C4⋊C4—531no—
330C4×C4⋊C4—1061no—
331C4×C22⋊C4—1061no—
332#340—532no—
333#340C2×D5312yescomplemented
334#336—532no—
335C53#2204311yescharacteristic, complemented, Sylow
336C25—1061no—
337C22×Q8C22×D5311yescharacteristic
338C22×D4—534no—
339C22×D4—532no—
340C22×D4—1061no—
341C22×D4—1062no—
342C22×D4—2122no—
343C22×D4—2122no—
344C23×C4—21no—
345C23×C4—531no—
346C23×C4—531no—
347C23×C4C22×D5311yescharacteristic
348C2×SD16—538no—
349C2×SD16—2122no—
350C2×SD16—2122no—
351C22×C8—532no—
352C22×C8C22×D5312yes—
353C22⋊Q8—22no—
354C22⋊Q8C4×D5314yes—
355C4×Q8—2122no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—42no—
359C2×C4⋊C4—531no—
360C2×C4⋊C4—531no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4C22×D5312yes—
364C2×C4⋊C4C22×D5311yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—1062no—
367C2×C22⋊C4—1062no—
368C2×C22⋊C4—1062no—
369C2×C22⋊C4—1062no—
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—2124no—
372C2×C22⋊C4—2124no—
373C2×C42—531no—
374C2×C42—531no—
375C2×C42—531no—
376C2×C42—531no—
377C2×C42—1061no—
378C2×C42—1061no—
379C2×C42—2122no—
380#106—2122no—
381#103—42no—
382#102—538no—
383#102—2122no—
384#102—2122no—
385C22⋊C8—22no—
386C22⋊C8—534no—
387C4×C8—2122no—
388#95—1061no—
389#95—1061no—
390#95—1061no—
391#95—1061no—
392#95—1061no—
393C24—1064no—
394C24—1062no—
395C24—2122no—
396C24—2122no—
397C24—2122no—
398C24—2122no—
399C24—2122no—
400C24—2122no—
401C24D4×D5311yescharacteristic
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C2×C4×D5312yes—
406C2×D4—534no—
407C2×D4—1064no—
408C2×D4—1062no—
409C2×D4—1062no—
410C2×D4—1062no—
411C2×D4—1062no—
412C2×D4—1062no—
413C2×D4—2128no—
414C2×D4—2122no—
415C2×D4—2122no—
416C2×D4—2128no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—531no—
426C22×C4—531no—
427C22×C4—531no—
428C22×C4—534no—
429C22×C4—531no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—2121no—
435C22×C4—2121no—
436C22×C4—2121no—
437C22×C4C2×C4×D5312yes—
438C22×C4D4×D5311yescharacteristic
439C22×C4#1171994712yes—
440C22×C4C23×D5311yescharacteristic
441SD16—2128no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—534no—
446C2×C8—1062no—
447C2×C8—1062no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—1062no—
454C4⋊C4—1062no—
455C4⋊C4—2122no—
456C4⋊C4—2122no—
457C4⋊C4C2×C4×D5312yes—
458C22⋊C4—24no—
459C22⋊C4—2124no—
460C22⋊C4—2124no—
461C22⋊C4—2124no—
462C22⋊C4—2124no—
463C22⋊C4—2122no—
464C42—1062no—
465C42—1062no—
466C42—2121no—
467C42—2121no—
468C42—2122no—
469C42—2122no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—1064no—
475C23—4244no—
476C23—4244no—
477C23—4244no—
478C23—4244no—
479C23—2128no—
480C23—2128no—
481C23—2122no—
482C23—2122no—
483C23—2122no—
484C23—2122no—
485C23—2122no—
486C23—2122no—
487C23—2128no—
488C23#1313317712yes—
489C23SD16×D5312yes—
490C23#1313324012yes—
491C23C2×D4×D5311yescharacteristic
492Q8—24no—
493Q8—42no—
494D4—1064no—
495D4—1064no—
496D4—2128no—
497D4—2122no—
498D4—2122no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—84no—
513C2×C4—532no—
514C2×C4—532no—
515C2×C4—532no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—2124no—
526C2×C4—2124no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4#1313317712yes—
534C2×C4C22×C4×D5311yescharacteristic
535C2×C4C2×#1171994711yescharacteristic
536C2×C4#1313326212yes—
537C2×C4#1313326312yes—
538C8—42no—
539C8—1064no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—42416no—
547C22—4242no—
548C22—4242no—
549C22—4242no—
550C22—4242no—
551C22—2124no—
552C22—2124no—
553C22—2124no—
554C22#1681070814yes—
555C22C2×#1313317711yescharacteristic
556C22#1681118411yescharacteristic
557C22#1681118811yescharacteristic
558C22C2×SD16×D5311yescharacteristic
559C22C2×#1313324011yescharacteristic
560C22AGL(1,ℤ/8)×D5311yescharacteristic
561C22#1681142211yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—82no—
566C4—1062no—
567C4—1062no—
568C4—1062no—
569C4—4241no—
570C4—4241no—
571C4—2121no—
572C4#1681109512yes—
573C4#1681121012yes—
574C2—24no—
575C2—4244no—
576C2#2027737811yescharacteristic
577C2C2×#1681070811yescharacteristic
578C2#2028352811yescharacteristic
579C2#2028547111yescharacteristic
580C2#2028548511yescharacteristic
581C2#2028565411yescharacteristic
582C2#2028568511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1872C2
socleC23×C53C22×D4
prime radicalC23×D212C22
cosocleC22×C4×C53C24
hypercenter#1872D53
hypocenterC53#22043
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C22⋊Q8×Dic53C2semidirect
4. #20275566C2semidirect
5. #20275399C2semidirect
6. #20275340C2semidirect
7. #20275068C2semidirect
8. #20274908C2semidirect
10. C22⋊C8×Dic53C2semidirect
11. C53×#1872C2semidirect
14. C2×Q8×Dic53C22semidirect
18. #16810773C4semidirect
19. C2×#13133148C22semidirect
24. C2×C8×Dic53C22semidirect
25. #16810530C22semidirect
30. #16810522C22semidirect
35. C4⋊C4×Dic53C22semidirect
37. #16810385C22semidirect
38. #16810377C22semidirect
41. C2×#103×C53C22semidirect
43. C53×#340C22semidirect
55. C2×#11719924C2×C4semidirect
79. #13133137C2×C4semidirect
133. C22×Dic53SD16semidirect
290. #1872D53semidirect
313. C2×#103C2×D53semidirect
333. #340C2×D53semidirect
335. C53#22043semidirect

Maximal subgroups

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

subgroupindex
C2×#168107732
C22⋊Q8×Dic532
C2×#168105242
#202755662
#202753992
#202753402
#202750682
#202749082
#202746472
C22⋊C8×Dic532
C53×#18722
#2204353