Finite Group Databasebeta

← #27358205

Subgroups — #27358205

Order 13568.

1402 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1402 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285521C212yesmaximal, complemented
2#20279873C211yesmaximal, characteristic, complemented
3#20273701C214yesmaximal, complemented
4#20273667C214yesmaximal, complemented
5#20273231C212yesmaximal, complemented
6#20273042C211yesmaximal, characteristic, complemented
7C2×C53×#534C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic, complemented
9#16810958—22no—
10#16810934C2214yescomplemented
11#16810867—24no—
12#16810774—22no—
13#16810761C2214yescomplemented
14#16810587C2212yescomplemented
15#16810548C2212yescomplemented
16#16810535C2212yescomplemented
17#16810517—22no—
18#16810486—21no—
19#16810479—21no—
20C4×C22⋊Dic53C2211yescharacteristic, complemented
21#16810357C2218yescomplemented
22#16810347C2214yescomplemented
23C23×D4×C53—21no—
24C53×#534—24no—
25C53×#534—22no—
26C53×#534C2214yes—
27C2×#123×C53—22no—
28C2×#121×C53—21no—
29C2×#121×C53C2212yes—
30C2×#120×C53—22no—
31C22×C22⋊C4×C53C2211yescharacteristic
32C22×Dic106C2311yescharacteristic, complemented
33C2×#11719923—24no—
34C2×#11719917—22no—
35#13133196—22no—
36#13133196—22no—
37C2×C4⋊Dic53C2311yescharacteristic, complemented
38C2×#11719919—21no—
39C2×C22⋊Dic53—22no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—22no—
43C2×C4×Dic53—21no—
44#13133150—24no—
45#13133140—24no—
46#13133138—24no—
47C22⋊#88322—22no—
48C22⋊#88322C2314yescomplemented
49#13133124—22no—
50#13133124—22no—
51#13133124C2312yescomplemented
52C25×C53—21no—
53C25×C53D411yescharacteristic
54C22×D4×C53—22no—
55C22×D4×C53—24no—
56C22×D4×C53—22no—
57C22×D4×C53—24no—
58C22×D4×C53—21no—
59C22×D4×C53—48no—
60C22×D4×C53D412yes—
61C23×C4×C53—21no—
62C23×C4×C53D411yescharacteristic
63C23×C4×C53C2311yescharacteristic
64#123×C53—28no—
65#123×C53—28no—
66#121×C53—24no—
67#121×C53—24no—
68#121×C53—22no—
69#121×C53—22no—
70#121×C53D414yes—
71#121×C53C2314yes—
72#120×C53—28no—
73#120×C53—28no—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53D412yes—
76C2×C22⋊C4×C53—22no—
77C2×C22⋊C4×C53—22no—
78C2×C22⋊C4×C53—22no—
79C2×C22⋊C4×C53—22no—
80C2×C22⋊C4×C53D412yes—
81C2×C22⋊C4×C53C2312yes—
82C24×C53—21no—
83C24×C53—21no—
84C24×C53—24no—
85C24×C53—24no—
86C24×C53—22no—
87C24×C53—44no—
88C24×C53—42no—
89C24×C53—81no—
90C24×C53—82no—
91C24×C53—82no—
92C24×C53C2×D411yescharacteristic
93C24×C53C2×D412yes—
94C2×D4×C53—28no—
95C2×D4×C53—24no—
96C2×D4×C53—24no—
97C2×D4×C53—24no—
98C2×D4×C53—22no—
99C2×D4×C53—48no—
100C2×D4×C53—416no—
101C2×D4×C53—48no—
102C2×D4×C53—48no—
103C2×D4×C53—48no—
104C2×D4×C53—82no—
105C2×D4×C53—82no—
106C2×D4×C53C2×D414yes—
107C22×C4×C53—24no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—44no—
112C22×C4×C53—42no—
113C22×C4×C53—81no—
114C22×C4×C53C2×D412yes—
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D412yes—
117C22×C4×C53C2×D412yes—
118C22×C4×C53C2411yescharacteristic
119C22×Dic53—22no—
120C22×Dic53—21no—
121C22×Dic53—21no—
122C2×Dic106—22no—
123C2×Dic106—81no—
124C4⋊C4×C53—24no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53C2×D414yes—
127C22⋊C4×C53—28no—
128C22⋊C4×C53—24no—
129C22⋊C4×C53—24no—
130C22⋊C4×C53—24no—
131C22⋊C4×C53—24no—
132C22⋊Dic53—42no—
133C22⋊Dic53—41no—
134C22⋊Dic53—41no—
135C22⋊Dic53—44no—
136C22⋊Dic53—82no—
137#11719923—84no—
138C4⋊Dic53—22no—
139#11719919—42no—
140C4×Dic53—81no—
141#11719917—82no—
142C2×#88322—24no—
143C23×C53—24no—
144C23×C53—24no—
145C23×C53—22no—
146C23×C53—22no—
147C23×C53—22no—
148C23×C53—22no—
149C23×C53—24no—
150C23×C53—41no—
151C23×C53—42no—
152C23×C53—42no—
153C23×C53—42no—
154C23×C53—44no—
155C23×C53—48no—
156C23×C53—44no—
157C23×C53—42no—
158C23×C53—44no—
159C23×C53—81no—
160C23×C53—81no—
161C23×C53—88no—
162C23×C53—84no—
163C23×C53—84no—
164C23×C53—816no—
165C23×C53#12011yescharacteristic
166C23×C53#12011yescharacteristic
167C23×C53AGL(1,ℤ/8)14yes—
168C23×C53C22×D411yescharacteristic
169D4×C53—44no—
170D4×C53—416no—
171D4×C53—48no—
172D4×C53—48no—
173D4×C53—82no—
174D4×C53—84no—
175C2×C4×C53—24no—
176C2×C4×C53—22no—
177C2×C4×C53—41no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—41no—
181C2×C4×C53—44no—
182C2×C4×C53—48no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—81no—
186C2×C4×C53#12012yes—
187C2×C4×C53C22×D412yes—
188C2×Dic53—42no—
189C2×Dic53—42no—
190C2×Dic53—42no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—42no—
194C2×Dic53—42no—
195Dic106—44no—
196#88322—44no—
197#18317—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—24no—
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—44no—
207C22×C53—44no—
208C22×C53—48no—
209C22×C53—42no—
210C22×C53—42no—
211C22×C53—44no—
212C22×C53—44no—
213C22×C53—44no—
214C22×C53—82no—
215C22×C53—88no—
216C22×C53—84no—
217C22×C53—816no—
218C22×C53#45512yes—
219C22×C53C2×#12011yescharacteristic
220C22×C53C2×AGL(1,ℤ/8)12yes—
221C4×C53—22no—
222C4×C53—22no—
223C4×C53—44no—
224C4×C53—42no—
225Dic53—82no—
226Dic53—81no—
227Dic53—81no—
228C2×#534D5311yescharacteristic
229#3193—532no—
230#2561—531no—
231#1607—534no—
232#1601—534no—
233#1507—532no—
234#1462—531no—
235C2×C53—22no—
236C2×C53—22no—
237C2×C53—41no—
238C2×C53—44no—
239C2×C53—84no—
240C2×C53#300311yescharacteristic
241C2×C53C2×#45511yescharacteristic
242C2×C53#302611yescharacteristic
243C23×D4—21no—
244#534—24no—
245#534—22no—
246#534C2×D5314yes—
247C2×#123—22no—
248C2×C22⋊Q8—531no—
249C2×#121—21no—
250C2×#121C2×D5312yes—
251C2×#120—22no—
252C22×C22⋊C4C2×D5311yescharacteristic
253#484—1062no—
254#480—534no—
255#468—1064no—
256#450—1062no—
257#448—534no—
258#407—532no—
259#397—532no—
260#393—532no—
261#388—1062no—
262#381—1061no—
263#379—1061no—
264C4×C22⋊C4—531no—
265#327—538no—
266#323—534no—
267C53#1831711yescharacteristic, complemented, Sylow
268C25—21no—
269C25C22⋊D5311yescharacteristic
270C22×Q8—531no—
271C22×D4—22no—
272C22×D4—24no—
273C22×D4—22no—
274C22×D4—24no—
275C22×D4—21no—
276C22×D4—48no—
277C22×D4C22⋊D5312yes—
278C23×C4—21no—
279C23×C4C22⋊D5311yescharacteristic
280C23×C4C22×D5311yescharacteristic
281C2×SD16—1064no—
282C2×M16—1062no—
283#123—28no—
284#123—28no—
285C22⋊Q8—1062no—
286C22⋊Q8—1062no—
287#121—24no—
288#121—24no—
289#121—22no—
290#121—22no—
291#121C22⋊D5314yes—
292#121C22×D5314yes—
293#120—28no—
294#120—28no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—531no—
297C2×C4⋊C4—1061no—
298C2×C4⋊C4C22⋊D5312yes—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—1062no—
304C2×C22⋊C4—1061no—
305C2×C22⋊C4—1061no—
306C2×C22⋊C4—1062no—
307C2×C22⋊C4C22⋊D5312yes—
308C2×C22⋊C4C22×D5312yes—
309C2×C42—1061no—
310#107—1064no—
311#103—1064no—
312#102—1064no—
313C22⋊C8—534no—
314C22⋊C8—1062no—
315#95—532no—
316#95—1062no—
317#95—1062no—
318C24—21no—
319C24—21no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—44no—
324C24—42no—
325C24—81no—
326C24—82no—
327C24—82no—
328C24C2×C22⋊D5311yescharacteristic
329C24C2×C22⋊D5312yes—
330C2×Q8—1062no—
331C2×Q8—4241no—
332C2×D4—28no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—48no—
338C2×D4—416no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4C2×C22⋊D5314yes—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—81no—
352C22×C4—1062no—
353C22×C4—1061no—
354C22×C4—1061no—
355C22×C4C2×C22⋊D5312yes—
356C22×C4C2×C22⋊D5311yescharacteristic
357C22×C4C2×C22⋊D5312yes—
358C22×C4C2×C22⋊D5312yes—
359C22×C4C23×D5311yescharacteristic
360SD16—4244no—
361M16—4242no—
362C2×C8—1064no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—1062no—
366C4⋊C4—2122no—
367C4⋊C4C2×C22⋊D5314yes—
368C22⋊C4—28no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—4242no—
374C22⋊C4—2122no—
375C22⋊C4—2121no—
376C22⋊C4—2121no—
377C22⋊C4—2124no—
378C42—4241no—
379C23—24no—
380C23—24no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—24no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—44no—
391C23—48no—
392C23—44no—
393C23—42no—
394C23—44no—
395C23—81no—
396C23—81no—
397C23—88no—
398C23—84no—
399C23—84no—
400C23—816no—
401C23#1313318611yescharacteristic
402C23#1313318611yescharacteristic
403C23#1313324314yes—
404C23C22×C22⋊D5311yescharacteristic
405Q8—2124no—
406D4—44no—
407D4—416no—
408D4—48no—
409D4—48no—
410D4—82no—
411D4—84no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—81no—
423C2×C4—2122no—
424C2×C4—2122no—
425C2×C4—2122no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4—2122no—
429C2×C4—2122no—
430C2×C4#1313318612yes—
431C2×C4C22×C22⋊D5312yes—
432C8—2124no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—24no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—44no—
442C22—44no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22—44no—
447C22—44no—
448C22—44no—
449C22—82no—
450C22—88no—
451C22—84no—
452C22—816no—
453C22#1681080112yes—
454C22C2×#1313318611yescharacteristic
455C22C2×#1313324312yes—
456C4—22no—
457C4—22no—
458C4—44no—
459C4—42no—
460C4—4242no—
461C4—4241no—
462C4—4241no—
463C2—22no—
464C2—22no—
465C2—41no—
466C2—44no—
467C2—84no—
468C2#2028394211yescharacteristic
469C2C2×#1681080111yescharacteristic
470C2#2028411211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#534C2
socleC22×C53C2×#120
hypercenterC2×#534D53
hypocenterC53#18317
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
1. #20285521C2semidirect
2. #20279873C2semidirect
3. #20273701C2semidirect
4. #20273667C2semidirect
5. #20273231C2semidirect
6. #20273042C2semidirect
8. C2×#13133196C22semidirect
10. #16810934C22semidirect
13. #16810761C22semidirect
14. #16810587C22semidirect
15. #16810548C22semidirect
16. #16810535C22semidirect
20. C4×C22⋊Dic53C22semidirect
21. #16810357C22semidirect
22. #16810347C22semidirect
32. C22×Dic106C23semidirect
37. C2×C4⋊Dic53C23semidirect
48. C22⋊#88322C23semidirect
51. #13133124C23semidirect
267. C53#18317semidirect

Maximal subgroups

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

subgroupindex
#202855212
#202798732
#202737012
#202736672
#202732312
#202730422
C2×C53×#5342
#1831753