Finite Group Databasebeta

← D53×#2233 #27538874

Subgroups — D53×#2233 #27538874

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#466×D53C211yesmaximal, characteristic
2C2×#107×D53C211yesmaximal, characteristic, complemented
3#20276902C212yesmaximal, complemented
4#20276901C211yesmaximal, characteristic, complemented
5#20276900C211yesmaximal, characteristic, complemented
6#20276899C212yesmaximal, complemented
7#20276898C211yesmaximal, characteristic, complemented
8#367×D53C212yesmaximal, complemented
9#357×D53C212yesmaximal, complemented
10C22⋊C16×D53C211yesmaximal, characteristic, complemented
11C53×#2233C211yesmaximal, characteristic, complemented
12C2×D8×D53C2211yescharacteristic
13C22×C8×D53C2211yescharacteristic
14#121×D53—21no—
15C2×C4⋊C4×D53—21no—
16C2×C16×D53C2212yes—
17#16810860C2211yescharacteristic
18#16810859C2211yescharacteristic
19#16810858—21no—
20#16810857—21no—
21#16810856C2211yescharacteristic
22C2×#13133151C2211yescharacteristic, complemented
23C2×#13133150—21no—
24#107×D53—21no—
25#107×D53—21no—
26#107×D53C2211yescharacteristic
27#107×D53C2212yes—
28#107×D53C2212yescomplemented
29#102×D53—21no—
30#16810451C2212yescomplemented
31#16810450C2212yescomplemented
32#16810450C2212yescomplemented
33#16810425C2212yescomplemented
34#16810424C2212yescomplemented
35#16810423C2212yescomplemented
36C22⋊#11719908C2211yescharacteristic, complemented
37#16810402C2212yescomplemented
38C53×#466C2211yescharacteristic
39C2×#107×C53C2211yescharacteristic, complemented
40C53×#367C2212yescomplemented
41C53×#357C2212yescomplemented
42C53×C22⋊C16C2211yescharacteristic, complemented
43C2×D4×D53—21no—
44C2×D4×D53—41no—
45C22×C4×D53—41no—
46C22×C4×D53D411yescharacteristic
47C2×D424C2311yescharacteristic
48C2×#11719922—21no—
49D8×D53—24no—
50C22×#88322D411yescharacteristic
51C2×C8×D53—24no—
52C2×C8×D53D411yescharacteristic
53C2×C8×D53D411yescharacteristic
54C2×C8×D53C2311yescharacteristic
55C2×C8×D53C2311yescharacteristic
56#13133192—21no—
57#13133191—21no—
58#13133190—41no—
59#13133189—41no—
60#13133188—21no—
61C2×C4⋊Dic53—21no—
62C2×#11719919—41no—
63C4⋊C4×D53—22no—
64C4⋊C4×D53—22no—
65C4⋊C4×D53—21no—
66C4⋊C4×D53—41no—
67C4⋊C4×D53—41no—
68C22⋊C4×D53—41no—
69C2×#11719908C2312yes—
70C16×D53—24no—
71#13133151—21no—
72#13133151C2311yescharacteristic
73#13133151C2312yes—
74#13133150—21no—
75#13133150—22no—
76#13133150—22no—
77#13133139—21no—
78#13133138—21no—
79#13133137—21no—
80C2×D8×C53C2311yescharacteristic
81C22×C8×C53C2311yescharacteristic
82#121×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C16×C53C2312yes—
85#107×C53—21no—
86#107×C53C2311yescharacteristic
87#107×C53C2312yes—
88#102×C53—21no—
89C23×D53—41no—
90C23×D53D811yescharacteristic
91C2×D4×C53—21no—
92C2×D4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53C2×D411yescharacteristic
95C2×C22⋊D53—41no—
96C2×C22⋊D53—41no—
97C2×C22⋊D53—41no—
98C22×Dic53—41no—
99C22×Dic53D811yescharacteristic
100D4×D53—42no—
101D4×D53—42no—
102D4×D53—84no—
103C2×D212—21no—
104C2×D212—41no—
105C2×C4×D53—22no—
106C2×C4×D53—22no—
107C2×C4×D53—41no—
108C2×C4×D53—42no—
109C2×C4×D53—42no—
110C2×C4×D53—84no—
111C2×C4×D53D811yescharacteristic
112C2×C4×D53D811yescharacteristic
113C2×C4×D53C2×D411yescharacteristic
114C2×C4×D53C2×D411yescharacteristic
115D8×C53—22no—
116C2×C8×C53—22no—
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53C2411yescharacteristic
119C4⋊C4×C53—21no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C22⋊Dic53—41no—
124#11719922—42no—
125#11719921—41no—
126#11719921—41no—
127C4⋊Dic53—21no—
128C4⋊Dic53—22no—
129C4⋊Dic53—41no—
130#11719919—42no—
131#11719919—42no—
132#11719919—41no—
133C2×#88322—22no—
134C2×#88322C2×D411yescharacteristic
135C2×#88322C2×D411yescharacteristic
136D424—22no—
137C8×D53—22no—
138C8×D53—22no—
139C8×D53—22no—
140C8×D53#4114yes—
141C16×C53—22no—
142#11719908—22no—
143C23×C53—41no—
144C23×C53C2×D811yescharacteristic
145C22×D53—22no—
146C22×D53—22no—
147C22×D53—41no—
148C22×D53—82no—
149C22×D53—82no—
150C22×D53—82no—
151C22×D53SD3214yes—
152C22×D53C2×D811yescharacteristic
153C22×D53C2×D811yescharacteristic
154D4×C53—41no—
155D4×C53—41no—
156D4×C53—82no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—42no—
160C2×C4×C53—41no—
161C2×C4×C53—82no—
162C2×C4×C53C2×D811yescharacteristic
163C2×C4×C53C22×D411yescharacteristic
164C22⋊D53—82no—
165C22⋊D53—82no—
166C22⋊D53—82no—
167C2×Dic53—22no—
168C2×Dic53—42no—
169C2×Dic53—41no—
170C2×Dic53—82no—
171C2×Dic53C2×D811yescharacteristic
172C2×Dic53C2×D811yescharacteristic
173D212—41no—
174D212—41no—
175D212—82no—
176C4×D53—22no—
177C4×D53—21no—
178C4×D53—21no—
179C4×D53—22no—
180C4×D53—84no—
181C4×D53—82no—
182C4×D53—84no—
183C4×D53#12312yes—
184C4×D53#13712yes—
185C8×C53—21no—
186C8×C53C2×#4112yes—
187#88322—21no—
188#88322#12312yes—
189C2×#2233—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53C2×SD3212yes—
196C22×C53C22×D811yescharacteristic
197C2×D53—21no—
198C2×D53—21no—
199C2×D53—28no—
200C2×D53—81no—
201C2×D53—81no—
202C2×D53—81no—
203C2×D53—164no—
204C2×D53#48012yes—
205C2×D53C2×SD3212yes—
206C2×D53C2×SD3212yes—
207C2×D53#50912yes—
208C4×C53—21no—
209C4×C53—81no—
210C4×C53—82no—
211C4×C53C2×#12311yescharacteristic
212C4×C53C2×#13711yescharacteristic
213Dic53—21no—
214Dic53—81no—
215Dic53—82no—
216Dic53#48012yes—
217C2×#466—531no—
218C22×#107—531no—
219#2233—531no—
220#2233—531no—
221#2233—532no—
222#2233—532no—
223#2233—531no—
224#2233D5311yescharacteristic, direct product factor
225C2×#367—532no—
226C2×#357—532no—
227C2×C22⋊C16—531no—
228C2×C53—22no—
229C2×C53—161no—
230C2×C53C2×#48011yescharacteristic
231C2×C53C22×SD3211yescharacteristic
232C2×C53C2×#50911yescharacteristic
233D53—22no—
234D53—161no—
235D53#223314yesdirect product factor
236C22×D8—531no—
237C23×C8—531no—
238C2×#121—1061no—
239C22×C4⋊C4—1061no—
240C22×C16—532no—
241#466—531no—
242#466—531no—
243#466—531no—
244#466—1061no—
245#466—1061no—
246#466C2×D5311yescharacteristic
247C2×#107—531no—
248C2×#107—532no—
249C2×#107—532no—
250C2×#107—531no—
251C2×#107—1061no—
252C2×#107—1061no—
253C2×#107—1061no—
254C2×#107C2×D5311yescharacteristic, complemented
255C2×#102—1061no—
256#367—532no—
257#367—532no—
258#367—532no—
259#367C2×D5312yescomplemented
260#357—532no—
261#357—532no—
262#357—532no—
263#357C2×D5312yescomplemented
264C22⋊C16—531no—
265C22⋊C16—532no—
266C22⋊C16C2×D5311yescharacteristic, complemented
267C53C2×#223311yescharacteristic, complemented, Sylow
268C22×D4—1061no—
269C22×D4—2121no—
270C23×C4—531no—
271C23×C4—2121no—
272C2×D8—531no—
273C2×D8—1064no—
274C2×D8—1061no—
275C2×D8C22×D5311yescharacteristic
276C22×C8—531no—
277C22×C8—531no—
278C22×C8—531no—
279C22×C8—531no—
280C22×C8—531no—
281C22×C8—1064no—
282C22×C8C22×D5311yescharacteristic
283#121—21no—
284#121—1061no—
285#121—1061no—
286#121—1061no—
287#121—2121no—
288#121—2121no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—1062no—
291C2×C4⋊C4—1062no—
292C2×C4⋊C4—1061no—
293C2×C4⋊C4—1061no—
294C2×C4⋊C4—2121no—
295C2×C4⋊C4—2121no—
296C2×C4⋊C4—2121no—
297C2×C22⋊C4—2121no—
298C2×C16—532no—
299C2×C16—1064no—
300C2×C16C22×D5312yes—
301#107—21no—
302#107—532no—
303#107—531no—
304#107—1062no—
305#107—1062no—
306#107—1061no—
307#107—1061no—
308#107C22×D5312yes—
309#107C22×D5311yescharacteristic
310#102—21no—
311#102—1061no—
312#102—1061no—
313#102—1061no—
314C24—531no—
315C24—2121no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—1061no—
319C2×D4—4244no—
320C2×D4—2121no—
321C2×D4—2122no—
322C2×D4—2122no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C22×C4—41no—
327C22×C4—531no—
328C22×C4—531no—
329C22×C4—531no—
330C22×C4—531no—
331C22×C4—531no—
332C22×C4—1062no—
333C22×C4—1062no—
334C22×C4—4244no—
335C22×C4—2121no—
336C22×C4—2122no—
337C22×C4—2122no—
338C22×C4—2121no—
339C22×C4D4×D5311yescharacteristic
340D8—22no—
341D8—1062no—
342D8—2122no—
343C2×C8—22no—
344C2×C8—531no—
345C2×C8—534no—
346C2×C8—531no—
347C2×C8—1062no—
348C2×C8—1062no—
349C2×C8—1062no—
350C2×C8—1062no—
351C2×C8D4×D5311yescharacteristic
352C2×C8C23×D5311yescharacteristic
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—41no—
356C4⋊C4—1061no—
357C4⋊C4—1062no—
358C4⋊C4—2122no—
359C4⋊C4—2122no—
360C4⋊C4—2121no—
361C4⋊C4—2121no—
362C22⋊C4—41no—
363C22⋊C4—2121no—
364C22⋊C4—2121no—
365C22⋊C4—2121no—
366C16—22no—
367C16—1062no—
368C23—41no—
369C23—531no—
370C23—534no—
371C23—531no—
372C23—1062no—
373C23—1062no—
374C23—4242no—
375C23—4242no—
376C23—4242no—
377C23—2121no—
378C23D8×D5311yescharacteristic
379D4—41no—
380D4—41no—
381D4—82no—
382D4—4242no—
383D4—4242no—
384D4—4242no—
385D4—4242no—
386D4—2121no—
387D4—2121no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—82no—
393C2×C4—531no—
394C2×C4—532no—
395C2×C4—532no—
396C2×C4—531no—
397C2×C4—1062no—
398C2×C4—1062no—
399C2×C4—1062no—
400C2×C4—1061no—
401C2×C4—1061no—
402C2×C4—4244no—
403C2×C4—4242no—
404C2×C4—4244no—
405C2×C4—4242no—
406C2×C4—2121no—
407C2×C4—2122no—
408C2×C4D8×D5311yescharacteristic
409C2×C4C2×D4×D5311yescharacteristic
410C8—21no—
411C8—532no—
412C8—1061no—
413C8#41×D5312yes—
414C22—21no—
415C22—21no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—532no—
420C22—532no—
421C22—532no—
422C22—532no—
423C22—8484no—
424C22—1068no—
425C22—1061no—
426C22—1061no—
427C22—4241no—
428C22—4241no—
429C22—4241no—
430C22SD32×D5312yes—
431C22C2×D8×D5311yescharacteristic
432C4—21no—
433C4—81no—
434C4—82no—
435C4—532no—
436C4—1061no—
437C4—4241no—
438C4—4242no—
439C4#123×D5311yescharacteristic
440C4#137×D5311yescharacteristic
441C2—22no—
442C2—161no—
443C2—534no—
444C2—8481no—
445C2—1062no—
446C2#480×D5311yescharacteristic
447C2C2×SD32×D5311yescharacteristic
448C2#509×D5311yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2233C2
socleC22×C53C22×D8
prime radical#466×D53C2
hypercenter#2233D53
hypocenterC53C2×#2233
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2233

Complements

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

NH ≅ G/Ntype
2. C2×#107×D53C2semidirect
3. #20276902C2semidirect
4. #20276901C2semidirect
5. #20276900C2semidirect
6. #20276899C2semidirect
7. #20276898C2semidirect
8. #367×D53C2semidirect
9. #357×D53C2semidirect
10. C22⋊C16×D53C2semidirect
11. C53×#2233C2semidirect
22. C2×#13133151C22semidirect
28. #107×D53C22semidirect
30. #16810451C22semidirect
31. #16810450C22semidirect
32. #16810450C22semidirect
33. #16810425C22semidirect
34. #16810424C22semidirect
35. #16810423C22semidirect
36. C22⋊#11719908C22semidirect
37. #16810402C22semidirect
39. C2×#107×C53C22semidirect
40. C53×#367C22semidirect
41. C53×#357C22semidirect
42. C53×C22⋊C16C22semidirect
224. #2233D53direct (N × H)
235. D53#2233direct (N × H)
254. C2×#107C2×D53semidirect
259. #367C2×D53semidirect
263. #357C2×D53semidirect
266. C22⋊C16C2×D53semidirect
267. C53C2×#2233semidirect

Maximal subgroups

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

subgroupindex
#466×D532
C2×#107×D532
#202769022
#202769012
#202769002
#202768992
#202768982
#367×D532
#357×D532
C22⋊C16×D532
C53×#22332
C2×#223353