Finite Group Databasebeta

← D53×#1823 #27383864

Subgroups — D53×#1823 #27383864

Order 13568.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D53C211yesmaximal, characteristic
2C2×C22⋊C8×D53C211yesmaximal, characteristic
3#380×D53C211yesmaximal, characteristic
4#20274802C211yesmaximal, characteristic, complemented
5#20274801C211yesmaximal, characteristic, complemented
6#20274800C214yesmaximal, complemented
7#20274799C211yesmaximal, characteristic, complemented
8#340×D53C214yesmaximal, complemented
9C53×#1823C211yesmaximal, characteristic, complemented
10C23×C4×D53—21no—
11C23×C4×D53C2211yescharacteristic
12C22×C8×D53C2212yes—
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919—21no—
15C2×C4⋊C4×D53—21no—
16C2×C4⋊C4×D53—21no—
17C2×C4⋊C4×D53—24no—
18C2×C4⋊C4×D53C418yescomplemented
19C2×C4⋊C4×D53C2212yes—
20C2×C4⋊C4×D53C2212yes—
21C2×C4⋊C4×D53C2212yes—
22C2×C22⋊C4×D53—21no—
23C2×C22⋊#88322C2211yescharacteristic
24C2×#13133129C2212yes—
25C22⋊C8×D53—24no—
26#16810483C2211yescharacteristic
27#16810482—21no—
28#16810481—21no—
29#16810480C2212yes—
30#95×D53—21no—
31#16810386C2214yescomplemented
32#16810385C2214yescomplemented
33#16810385C2214yescomplemented
34C22×C4⋊C4×C53C2211yescharacteristic
35C2×C22⋊C8×C53C2211yescharacteristic
36C53×#380C2211yescharacteristic
37C53×#340C2214yescomplemented
38C24×D53D411yescharacteristic
39C23×Dic53—21no—
40C23×Dic53D411yescharacteristic
41C22×C4×D53—28no—
42C22×C4×D53—28no—
43C22×C4×D53—22no—
44C22×C4×D53—22no—
45C22×C4×D53—22no—
46C22×C4×D53—44no—
47C22×C4×D53—44no—
48C22×C4×D53C2×C414yes—
49C22×C4×D53D411yescharacteristic
50C22×C4×D53D411yescharacteristic
51C22×C4×D53D414yes—
52C22×C4×D53C2311yescharacteristic
53C22×C4×D53C2311yescharacteristic
54C22×#88322C2312yes—
55C2×C8×D53—24no—
56C2×C8×D53—28no—
57C2×C4⋊Dic53—22no—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53C2×C414yescomplemented
60C2×C4⋊Dic53C2312yes—
61C2×C4⋊Dic53C2312yes—
62C2×#11719919—28no—
63C2×#11719919—22no—
64C2×#11719919—22no—
65C2×#11719919—22no—
66C4⋊C4×D53—28no—
67C4⋊C4×D53—28no—
68C4⋊C4×D53—24no—
69C4⋊C4×D53—24no—
70C4⋊C4×D53—28no—
71C4⋊C4×D53C2×C418yes—
72C4⋊C4×D53C2×C418yescomplemented
73C2×C22⋊Dic53—21no—
74C2×#11719921—22no—
75C22⋊C4×D53—44no—
76C22⋊#88322—22no—
77#13133129—24no—
78#13133124—21no—
79#13133124—22no—
80C23×C4×C53—21no—
81C23×C4×C53C2311yescharacteristic
82C22×C8×C53C2312yes—
83C2×C4⋊C4×C53—22no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53C2×C414yescomplemented
86C2×C4⋊C4×C53C2312yes—
87C2×C4⋊C4×C53C2312yes—
88C2×C22⋊C4×C53—21no—
89C22⋊C8×C53—22no—
90#95×C53—21no—
91C24×C53C2×D411yescharacteristic
92C23×D53—24no—
93C23×D53—24no—
94C23×D53C22⋊C414yes—
95C23×D53D814yes—
96C23×D53SD1614yes—
97C23×D53C2×D411yescharacteristic
98C23×D53C2×D411yescharacteristic
99C22×C4×C53—24no—
100C22×C4×C53—24no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—42no—
104C22×C4×C53—42no—
105C22×C4×C53C22×C412yes—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D412yes—
108C22×C4×C53C2411yescharacteristic
109C22×Dic53—24no—
110C22×Dic53—24no—
111C22×Dic53—22no—
112C22×Dic53—22no—
113C22×Dic53—42no—
114C22×Dic53—42no—
115C22×Dic53C22⋊C414yes—
116C22×Dic53C2×D411yescharacteristic
117C22×Dic53C2×D411yescharacteristic
118C2×C4×D53—24no—
119C2×C4×D53—24no—
120C2×C4×D53—24no—
121C2×C4×D53—216no—
122C2×C4×D53—216no—
123C2×C4×D53—28no—
124C2×C4×D53—28no—
125C2×C4×D53—432no—
126C2×C4×D53—44no—
127C2×C4×D53—44no—
128C2×C4×D53—44no—
129C2×C4×D53—44no—
130C2×C4×D53—44no—
131C2×C4×D53—44no—
132C2×C4×D53—44no—
133C2×C4×D53C22⋊C414yes—
134C2×C4×D53C22⋊C414yes—
135C2×C4×D53C22×C412yes—
136C2×C4×D53C22×C412yes—
137C2×C4×D53C2×D412yes—
138C2×C4×D53C2×D412yes—
139C2×C4×D53#4118yes—
140C2×C8×C53—22no—
141C2×C8×C53—24no—
142C4⋊C4×C53—24no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—24no—
145C4⋊C4×C53C22×C414yes—
146C22⋊C4×C53—42no—
147C22⋊Dic53—42no—
148#11719921—44no—
149C4⋊Dic53—24no—
150C4⋊Dic53—22no—
151C4⋊Dic53—24no—
152C4⋊Dic53C22×C414yes—
153#11719919—28no—
154#11719919—28no—
155#11719919—44no—
156C2×#88322—22no—
157C2×#88322—24no—
158C8×D53—48no—
159C23×C53—22no—
160C23×C53—22no—
161C23×C53C2×C22⋊C412yes—
162C23×C53C2×D812yes—
163C23×C53C2×SD1612yes—
164C23×C53C22×D411yescharacteristic
165C22×D53—22no—
166C22×D53—22no—
167C22×D53—28no—
168C22×D53—28no—
169C22×D53—216no—
170C22×D53—216no—
171C22×D53#102116yes—
172C22×D53C2×C22⋊C412yes—
173C22×D53C2×C22⋊C412yes—
174C22×D53#12314yes—
175C22×D53C2×D812yes—
176C22×D53C2×D812yes—
177C22×D53C2×SD1612yes—
178C22×D53C2×SD1612yes—
179C22×D53#13714yes—
180C2×C4×C53—22no—
181C2×C4×C53—24no—
182C2×C4×C53—24no—
183C2×C4×C53—24no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—48no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53C2×C22⋊C412yes—
191C2×C4×C53C23×C411yescharacteristic
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C2×#4114yes—
194C2×Dic53—22no—
195C2×Dic53—28no—
196C2×Dic53—24no—
197C2×Dic53—42no—
198C2×Dic53—42no—
199C2×Dic53—48no—
200C2×Dic53—42no—
201C2×Dic53—42no—
202C2×Dic53—42no—
203C2×Dic53C2×C22⋊C412yes—
204C2×Dic53C2×C22⋊C412yes—
205C2×Dic53#12314yes—
206C4×D53—28no—
207C4×D53—24no—
208C4×D53—24no—
209C4×D53—28no—
210C4×D53—416no—
211C4×D53—416no—
212C4×D53—88no—
213C4×D53#11718yes—
214C4×D53#12318yes—
215C8×C53—42no—
216#88322—42no—
217C2×#1823—531nomaximal, Sylow
218C22×C53—24no—
219C22×C53—24no—
220C22×C53—22no—
221C22×C53—22no—
222C22×C53C2×#10214yes—
223C22×C53C22×C22⋊C411yescharacteristic
224C22×C53C2×#12312yes—
225C22×C53C22×D811yescharacteristic
226C22×C53C22×SD1611yescharacteristic
227C22×C53C2×#13712yes—
228C2×D53—24no—
229C2×D53—24no—
230C2×D53—22no—
231C2×D53—22no—
232C2×D53—232no—
233C2×D53#38114yes—
234C2×D53C2×#10214yes—
235C2×D53C2×#10214yes—
236C2×D53#41914yes—
237C2×D53#48018yes—
238C2×D53#48318yes—
239C4×C53—22no—
240C4×C53—44no—
241C4×C53—82no—
242C4×C53C2×#11712yes—
243C4×C53C2×#12312yes—
244Dic53—22no—
245Dic53—44no—
246Dic53—82no—
247Dic53#38114yes—
248C23×C4⋊C4—531no—
249C22×C22⋊C8—531no—
250C2×#380—531no—
251#1823—531no—
252#1823—531no—
253#1823—534no—
254#1823—531no—
255#1823D5311yescharacteristic, direct product factor
256C2×#340—534no—
257C2×C53—24no—
258C2×C53C2×#38111yescharacteristic
259C2×C53C22×#10211yescharacteristic
260C2×C53C2×#41911yescharacteristic
261C2×C53C2×#48012yes—
262C2×C53C2×#48312yes—
263D53—24no—
264D53#182318yesdirect product factor
265C24×C4—531no—
266C24×C4—1061no—
267C23×C8—532no—
268C22×C4⋊C4—531no—
269C22×C4⋊C4—538no—
270C22×C4⋊C4—532no—
271C22×C4⋊C4—532no—
272C22×C4⋊C4—532no—
273C22×C4⋊C4—1061no—
274C22×C4⋊C4—1061no—
275C22×C4⋊C4—1064no—
276C22×C4⋊C4—1061no—
277C22×C4⋊C4C2×D5311yescharacteristic
278C22×C22⋊C4—1061no—
279C2×C22⋊C8—531no—
280C2×C22⋊C8—532no—
281C2×C22⋊C8—1064no—
282C2×C22⋊C8C2×D5311yescharacteristic
283#380—531no—
284#380—532no—
285#380—1061no—
286#380—1061no—
287#380C2×D5311yescharacteristic
288C2×#95—1061no—
289#340—534no—
290#340—534no—
291#340—534no—
292#340C2×D5314yescomplemented
293C53C2×#182311yescharacteristic, complemented, Sylow
294C25—531no—
295C23×C4—21no—
296C23×C4—531no—
297C23×C4—531no—
298C23×C4—534no—
299C23×C4—531no—
300C23×C4—531no—
301C23×C4—534no—
302C23×C4—531no—
303C23×C4—1068no—
304C23×C4—1068no—
305C23×C4—1062no—
306C23×C4—1062no—
307C23×C4—1062no—
308C23×C4—1061no—
309C23×C4—2124no—
310C23×C4—2124no—
311C23×C4C22×D5311yescharacteristic
312C22×C8—532no—
313C22×C8—1064no—
314C22×C8—1068no—
315C22×C8C22×D5312yes—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—534no—
319C2×C4⋊C4—532no—
320C2×C4⋊C4—538no—
321C2×C4⋊C4—538no—
322C2×C4⋊C4—532no—
323C2×C4⋊C4—1068no—
324C2×C4⋊C4—1068no—
325C2×C4⋊C4—1064no—
326C2×C4⋊C4—1064no—
327C2×C4⋊C4—1068no—
328C2×C4⋊C4—1068no—
329C2×C4⋊C4—1062no—
330C2×C4⋊C4—1062no—
331C2×C4⋊C4—1061no—
332C2×C4⋊C4—1062no—
333C2×C4⋊C4—1062no—
334C2×C4⋊C4C4×D5314yescomplemented
335C2×C4⋊C4C22×D5312yes—
336C2×C4⋊C4C22×D5312yes—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—1062no—
339C2×C22⋊C4—1061no—
340C2×C22⋊C4—2124no—
341C22⋊C8—22no—
342C22⋊C8—1064no—
343C22⋊C8—1062no—
344#95—21no—
345#95—1062no—
346#95—1061no—
347C24—531no—
348C24—534no—
349C24—534no—
350C24—534no—
351C24—531no—
352C24—1064no—
353C24—1064no—
354C24D4×D5311yescharacteristic
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—531no—
362C22×C4—532no—
363C22×C4—532no—
364C22×C4—538no—
365C22×C4—534no—
366C22×C4—531no—
367C22×C4—534no—
368C22×C4—534no—
369C22×C4—532no—
370C22×C4—532no—
371C22×C4—1064no—
372C22×C4—10616no—
373C22×C4—10616no—
374C22×C4—1064no—
375C22×C4—1068no—
376C22×C4—1068no—
377C22×C4—1064no—
378C22×C4—1064no—
379C22×C4—1064no—
380C22×C4—1062no—
381C22×C4—1062no—
382C22×C4—2124no—
383C22×C4—2124no—
384C22×C4—2124no—
385C22×C4—21232no—
386C22×C4—2124no—
387C22×C4—2124no—
388C22×C4—2124no—
389C22×C4—2124no—
390C22×C4—2122no—
391C22×C4—2122no—
392C22×C4C2×C4×D5312yes—
393C22×C4D4×D5311yescharacteristic
394C22×C4D4×D5312yes—
395C22×C4C23×D5311yescharacteristic
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—1062no—
399C2×C8—1064no—
400C2×C8—2128no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—534no—
405C4⋊C4—1068no—
406C4⋊C4—1068no—
407C4⋊C4—1064no—
408C4⋊C4—1062no—
409C4⋊C4—1064no—
410C4⋊C4—2124no—
411C4⋊C4C2×C4×D5314yes—
412C22⋊C4—42no—
413C22⋊C4—2124no—
414C22⋊C4—2122no—
415C23—22no—
416C23—22no—
417C23—532no—
418C23—534no—
419C23—532no—
420C23—532no—
421C23—534no—
422C23—532no—
423C23—5316no—
424C23—532no—
425C23—532no—
426C23—1068no—
427C23—1068no—
428C23—10616no—
429C23—10616no—
430C23—1062no—
431C23—1062no—
432C23C22⋊C4×D5312yes—
433C23D8×D5312yes—
434C23SD16×D5312yes—
435C23C2×D4×D5311yescharacteristic
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—532no—
447C2×C4—534no—
448C2×C4—538no—
449C2×C4—538no—
450C2×C4—532no—
451C2×C4—1068no—
452C2×C4—1068no—
453C2×C4—1068no—
454C2×C4—1062no—
455C2×C4—1064no—
456C2×C4—1064no—
457C2×C4—1064no—
458C2×C4—4248no—
459C2×C4—21216no—
460C2×C4—21216no—
461C2×C4—2122no—
462C2×C4—2122no—
463C2×C4—2122no—
464C2×C4—2128no—
465C2×C4—2122no—
466C2×C4—2122no—
467C2×C4C22⋊C4×D5312yes—
468C2×C4C22×C4×D5311yescharacteristic
469C2×C4C2×D4×D5311yescharacteristic
470C2×C4#41×D5314yes—
471C8—42no—
472C8—2122no—
473C22—22no—
474C22—22no—
475C22—24no—
476C22—24no—
477C22—534no—
478C22—534no—
479C22—534no—
480C22—538no—
481C22—538no—
482C22—534no—
483C22—10632no—
484C22—1062no—
485C22—1062no—
486C22—1064no—
487C22—1064no—
488C22#102×D5314yes—
489C22C2×C22⋊C4×D5311yescharacteristic
490C22#123×D5312yes—
491C22C2×D8×D5311yescharacteristic
492C22C2×SD16×D5311yescharacteristic
493C22#137×D5312yes—
494C4—22no—
495C4—44no—
496C4—82no—
497C4—534no—
498C4—1062no—
499C4—4242no—
500C4—2124no—
501C4#117×D5312yes—
502C4#123×D5312yes—
503C2—24no—
504C2—538no—
505C2—1064no—
506C2#381×D5311yescharacteristic
507C2C2×#102×D5311yescharacteristic
508C2#419×D5311yescharacteristic
509C2#480×D5312yes—
510C2#483×D5312yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1823C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenter#1823D53
hypocenterC53C2×#1823
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. #20274802C2semidirect
5. #20274801C2semidirect
6. #20274800C2semidirect
7. #20274799C2semidirect
8. #340×D53C2semidirect
9. C53×#1823C2semidirect
18. C2×C4⋊C4×D53C4semidirect
31. #16810386C22semidirect
32. #16810385C22semidirect
33. #16810385C22semidirect
37. C53×#340C22semidirect
59. C2×C4⋊Dic53C2×C4semidirect
72. C4⋊C4×D53C2×C4semidirect
85. C2×C4⋊C4×C53C2×C4semidirect
255. #1823D53direct (N × H)
264. D53#1823direct (N × H)
292. #340C2×D53semidirect
293. C53C2×#1823semidirect
334. C2×C4⋊C4C4×D53semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D532
C2×C22⋊C8×D532
#380×D532
#202748022
#202748012
#202748002
#202747992
#340×D532
C53×#18232
C2×#182353