Finite Group Databasebeta

← D53×#1824 #27383875

Subgroups — D53×#1824 #27383875

Order 13568.

1878 conjugacy classes of subgroups.

The subgroup lattice has 498 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#20274806C211yesmaximal, characteristic, complemented
5#20274805C211yesmaximal, characteristic, complemented
6#20274804C214yesmaximal, complemented
7#20274803C211yesmaximal, characteristic, complemented
8#340×D53C214yesmaximal, complemented
9C53×#1824C211yesmaximal, 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—48no—
47C22×C4×D53C2×C414yes—
48C22×C4×D53D411yescharacteristic
49C22×C4×D53D411yescharacteristic
50C22×C4×D53D414yes—
51C22×C4×D53C2311yescharacteristic
52C22×C4×D53C2311yescharacteristic
53C22×#88322C2312yes—
54C2×C8×D53—24no—
55C2×C8×D53—28no—
56C2×C4⋊Dic53—22no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53C2×C414yescomplemented
59C2×C4⋊Dic53C2312yes—
60C2×C4⋊Dic53C2312yes—
61C2×#11719919—28no—
62C2×#11719919—22no—
63C2×#11719919—22no—
64C2×#11719919—22no—
65C4⋊C4×D53—28no—
66C4⋊C4×D53—28no—
67C4⋊C4×D53—24no—
68C4⋊C4×D53—24no—
69C4⋊C4×D53—28no—
70C4⋊C4×D53C2×C418yes—
71C4⋊C4×D53C2×C418yescomplemented
72C2×C22⋊Dic53—21no—
73C2×#11719921—22no—
74C22⋊C4×D53—44no—
75C22⋊#88322—22no—
76#13133129—24no—
77#13133124—21no—
78#13133124—22no—
79C23×C4×C53—21no—
80C23×C4×C53C2311yescharacteristic
81C22×C8×C53C2312yes—
82C2×C4⋊C4×C53—22no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53C2×C414yescomplemented
85C2×C4⋊C4×C53C2312yes—
86C2×C4⋊C4×C53C2312yes—
87C2×C22⋊C4×C53—21no—
88C22⋊C8×C53—22no—
89#95×C53—21no—
90C24×C53C2×D411yescharacteristic
91C23×D53—24no—
92C23×D53—24no—
93C23×D53C22⋊C414yes—
94C23×D53SD1614yes—
95C23×D53Q1614yes—
96C23×D53C2×D411yescharacteristic
97C23×D53C2×D411yescharacteristic
98C22×C4×C53—24no—
99C22×C4×C53—24no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—44no—
103C22×C4×C53C22×C412yes—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D412yes—
106C22×C4×C53C2411yescharacteristic
107C22×Dic53—24no—
108C22×Dic53—24no—
109C22×Dic53—22no—
110C22×Dic53—22no—
111C22×Dic53—44no—
112C22×Dic53C22⋊C414yes—
113C22×Dic53C2×D411yescharacteristic
114C22×Dic53C2×D411yescharacteristic
115C2×C4×D53—24no—
116C2×C4×D53—24no—
117C2×C4×D53—24no—
118C2×C4×D53—216no—
119C2×C4×D53—216no—
120C2×C4×D53—28no—
121C2×C4×D53—28no—
122C2×C4×D53—48no—
123C2×C4×D53—432no—
124C2×C4×D53—48no—
125C2×C4×D53—44no—
126C2×C4×D53—44no—
127C2×C4×D53—44no—
128C2×C4×D53C22⋊C414yes—
129C2×C4×D53C22⋊C414yes—
130C2×C4×D53C22×C412yes—
131C2×C4×D53C22×C412yes—
132C2×C4×D53C2×D412yes—
133C2×C4×D53C2×D412yes—
134C2×C4×D53#4118yes—
135C2×C8×C53—22no—
136C2×C8×C53—24no—
137C4⋊C4×C53—24no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—24no—
140C4⋊C4×C53C22×C414yes—
141C22⋊C4×C53—42no—
142C22⋊Dic53—42no—
143#11719921—44no—
144C4⋊Dic53—24no—
145C4⋊Dic53—22no—
146C4⋊Dic53—24no—
147C4⋊Dic53C22×C414yes—
148#11719919—28no—
149#11719919—28no—
150#11719919—44no—
151C2×#88322—22no—
152C2×#88322—24no—
153C8×D53—48no—
154C23×C53—22no—
155C23×C53—22no—
156C23×C53C2×C22⋊C412yes—
157C23×C53C2×SD1612yes—
158C23×C53C2×Q1612yes—
159C23×C53C22×D411yescharacteristic
160C22×D53—22no—
161C22×D53—22no—
162C22×D53—28no—
163C22×D53—28no—
164C22×D53—216no—
165C22×D53—216no—
166C22×D53#103116yes—
167C22×D53C2×C22⋊C412yes—
168C22×D53C2×C22⋊C412yes—
169C22×D53#12314yes—
170C22×D53C2×SD1612yes—
171C22×D53C2×SD1612yes—
172C22×D53C2×Q1612yes—
173C22×D53C2×Q1612yes—
174C22×D53AGL(1,ℤ/8)14yes—
175C2×C4×C53—22no—
176C2×C4×C53—24no—
177C2×C4×C53—24no—
178C2×C4×C53—24no—
179C2×C4×C53—48no—
180C2×C4×C53—44no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53C2×C22⋊C412yes—
185C2×C4×C53C23×C411yescharacteristic
186C2×C4×C53C22×D411yescharacteristic
187C2×C4×C53C2×#4114yes—
188C2×Dic53—22no—
189C2×Dic53—28no—
190C2×Dic53—24no—
191C2×Dic53—48no—
192C2×Dic53—44no—
193C2×Dic53—42no—
194C2×Dic53—42no—
195C2×Dic53—42no—
196C2×Dic53C2×C22⋊C412yes—
197C2×Dic53C2×C22⋊C412yes—
198C2×Dic53#12314yes—
199C4×D53—28no—
200C4×D53—24no—
201C4×D53—24no—
202C4×D53—28no—
203C4×D53—416no—
204C4×D53—416no—
205C4×D53—88no—
206C4×D53#11718yes—
207C4×D53#12318yes—
208C8×C53—42no—
209#88322—42no—
210C2×#1824—531nomaximal, Sylow
211C22×C53—24no—
212C22×C53—24no—
213C22×C53—22no—
214C22×C53—22no—
215C22×C53C2×#10314yes—
216C22×C53C22×C22⋊C411yescharacteristic
217C22×C53C2×#12312yes—
218C22×C53C22×SD1611yescharacteristic
219C22×C53C22×Q1611yescharacteristic
220C22×C53C2×AGL(1,ℤ/8)12yes—
221C2×D53—24no—
222C2×D53—24no—
223C2×D53—22no—
224C2×D53—22no—
225C2×D53—232no—
226C2×D53#38114yes—
227C2×D53C2×#10314yes—
228C2×D53C2×#10314yes—
229C2×D53#41814yes—
230C2×D53#48118yes—
231C2×D53#48418yes—
232C4×C53—22no—
233C4×C53—44no—
234C4×C53—82no—
235C4×C53C2×#11712yes—
236C4×C53C2×#12312yes—
237Dic53—22no—
238Dic53—44no—
239Dic53—82no—
240Dic53#38114yes—
241C23×C4⋊C4—531no—
242C22×C22⋊C8—531no—
243C2×#380—531no—
244#1824—531no—
245#1824—531no—
246#1824—534no—
247#1824—531no—
248#1824D5311yescharacteristic, direct product factor
249C2×#340—534no—
250C2×C53—24no—
251C2×C53C2×#38111yescharacteristic
252C2×C53C22×#10311yescharacteristic
253C2×C53C2×#41811yescharacteristic
254C2×C53C2×#48112yes—
255C2×C53C2×#48412yes—
256D53—24no—
257D53#182418yesdirect product factor
258C24×C4—531no—
259C24×C4—1061no—
260C23×C8—532no—
261C22×C4⋊C4—531no—
262C22×C4⋊C4—538no—
263C22×C4⋊C4—532no—
264C22×C4⋊C4—532no—
265C22×C4⋊C4—532no—
266C22×C4⋊C4—1061no—
267C22×C4⋊C4—1061no—
268C22×C4⋊C4—1064no—
269C22×C4⋊C4—1061no—
270C22×C4⋊C4C2×D5311yescharacteristic
271C22×C22⋊C4—1061no—
272C2×C22⋊C8—531no—
273C2×C22⋊C8—532no—
274C2×C22⋊C8—1064no—
275C2×C22⋊C8C2×D5311yescharacteristic
276#380—531no—
277#380—532no—
278#380—1061no—
279#380—1061no—
280#380C2×D5311yescharacteristic
281C2×#95—1061no—
282#340—534no—
283#340—534no—
284#340—534no—
285#340C2×D5314yescomplemented
286C53C2×#182411yescharacteristic, complemented, Sylow
287C25—531no—
288C23×C4—21no—
289C23×C4—531no—
290C23×C4—531no—
291C23×C4—534no—
292C23×C4—531no—
293C23×C4—531no—
294C23×C4—534no—
295C23×C4—531no—
296C23×C4—1068no—
297C23×C4—1068no—
298C23×C4—1062no—
299C23×C4—1062no—
300C23×C4—1062no—
301C23×C4—1061no—
302C23×C4—2128no—
303C23×C4C22×D5311yescharacteristic
304C22×C8—532no—
305C22×C8—1064no—
306C22×C8—1068no—
307C22×C8C22×D5312yes—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—534no—
311C2×C4⋊C4—532no—
312C2×C4⋊C4—538no—
313C2×C4⋊C4—538no—
314C2×C4⋊C4—532no—
315C2×C4⋊C4—1068no—
316C2×C4⋊C4—1068no—
317C2×C4⋊C4—1064no—
318C2×C4⋊C4—1064no—
319C2×C4⋊C4—1068no—
320C2×C4⋊C4—1068no—
321C2×C4⋊C4—1062no—
322C2×C4⋊C4—1062no—
323C2×C4⋊C4—1061no—
324C2×C4⋊C4—1062no—
325C2×C4⋊C4—1062no—
326C2×C4⋊C4C4×D5314yescomplemented
327C2×C4⋊C4C22×D5312yes—
328C2×C4⋊C4C22×D5312yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—1062no—
331C2×C22⋊C4—1061no—
332C2×C22⋊C4—2124no—
333C22⋊C8—22no—
334C22⋊C8—1064no—
335C22⋊C8—1062no—
336#95—21no—
337#95—1062no—
338#95—1061no—
339C24—531no—
340C24—534no—
341C24—534no—
342C24—534no—
343C24—531no—
344C24—1064no—
345C24—1064no—
346C24D4×D5311yescharacteristic
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—44no—
352C22×C4—531no—
353C22×C4—532no—
354C22×C4—532no—
355C22×C4—538no—
356C22×C4—534no—
357C22×C4—531no—
358C22×C4—534no—
359C22×C4—534no—
360C22×C4—532no—
361C22×C4—532no—
362C22×C4—1064no—
363C22×C4—10616no—
364C22×C4—10616no—
365C22×C4—1064no—
366C22×C4—1068no—
367C22×C4—1068no—
368C22×C4—1064no—
369C22×C4—1064no—
370C22×C4—1064no—
371C22×C4—1062no—
372C22×C4—1062no—
373C22×C4—2124no—
374C22×C4—2124no—
375C22×C4—2124no—
376C22×C4—2128no—
377C22×C4—21232no—
378C22×C4—2128no—
379C22×C4—2124no—
380C22×C4C2×C4×D5312yes—
381C22×C4D4×D5311yescharacteristic
382C22×C4D4×D5312yes—
383C22×C4C23×D5311yescharacteristic
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—1062no—
387C2×C8—1064no—
388C2×C8—2128no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—534no—
393C4⋊C4—1068no—
394C4⋊C4—1068no—
395C4⋊C4—1064no—
396C4⋊C4—1062no—
397C4⋊C4—1064no—
398C4⋊C4—2124no—
399C4⋊C4C2×C4×D5314yes—
400C22⋊C4—42no—
401C22⋊C4—2124no—
402C22⋊C4—2122no—
403C23—22no—
404C23—22no—
405C23—532no—
406C23—534no—
407C23—532no—
408C23—532no—
409C23—534no—
410C23—532no—
411C23—5316no—
412C23—532no—
413C23—532no—
414C23—1068no—
415C23—1068no—
416C23—10616no—
417C23—10616no—
418C23—1062no—
419C23—1062no—
420C23C22⋊C4×D5312yes—
421C23SD16×D5312yes—
422C23Q16×D5312yes—
423C23C2×D4×D5311yescharacteristic
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—48no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—532no—
434C2×C4—534no—
435C2×C4—538no—
436C2×C4—538no—
437C2×C4—532no—
438C2×C4—1068no—
439C2×C4—1068no—
440C2×C4—1068no—
441C2×C4—1062no—
442C2×C4—1064no—
443C2×C4—1064no—
444C2×C4—1064no—
445C2×C4—4248no—
446C2×C4—21216no—
447C2×C4—21216no—
448C2×C4—2122no—
449C2×C4—2122no—
450C2×C4—2122no—
451C2×C4—2128no—
452C2×C4—2124no—
453C2×C4C22⋊C4×D5312yes—
454C2×C4C22×C4×D5311yescharacteristic
455C2×C4C2×D4×D5311yescharacteristic
456C2×C4#41×D5314yes—
457C8—42no—
458C8—2122no—
459C22—22no—
460C22—22no—
461C22—24no—
462C22—24no—
463C22—534no—
464C22—534no—
465C22—534no—
466C22—538no—
467C22—538no—
468C22—534no—
469C22—10632no—
470C22—1062no—
471C22—1062no—
472C22—1064no—
473C22—1064no—
474C22#103×D5314yes—
475C22C2×C22⋊C4×D5311yescharacteristic
476C22#123×D5312yes—
477C22C2×SD16×D5311yescharacteristic
478C22C2×Q16×D5311yescharacteristic
479C22AGL(1,ℤ/8)×D5312yes—
480C4—22no—
481C4—44no—
482C4—82no—
483C4—534no—
484C4—1062no—
485C4—4242no—
486C4—2124no—
487C4#117×D5312yes—
488C4#123×D5312yes—
489C2—24no—
490C2—538no—
491C2—1064no—
492C2#381×D5311yescharacteristic
493C2C2×#103×D5311yescharacteristic
494C2#418×D5311yescharacteristic
495C2#481×D5312yes—
496C2#484×D5312yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1824C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenter#1824D53
hypocenterC53C2×#1824
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
4. #20274806C2semidirect
5. #20274805C2semidirect
6. #20274804C2semidirect
7. #20274803C2semidirect
8. #340×D53C2semidirect
9. C53×#1824C2semidirect
18. C2×C4⋊C4×D53C4semidirect
31. #16810386C22semidirect
32. #16810385C22semidirect
33. #16810385C22semidirect
37. C53×#340C22semidirect
58. C2×C4⋊Dic53C2×C4semidirect
71. C4⋊C4×D53C2×C4semidirect
84. C2×C4⋊C4×C53C2×C4semidirect
248. #1824D53direct (N × H)
257. D53#1824direct (N × H)
285. #340C2×D53semidirect
286. C53C2×#1824semidirect
326. 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
#202748062
#202748052
#202748042
#202748032
#340×D532
C53×#18242
C2×#182453