Finite Group Databasebeta

← #27358204

Subgroups — #27358204

Order 13568.

1402 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1402 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C2×#16811185C211yesmaximal, characteristic
2#20285517C212yesmaximal, complemented
3#20273700C214yesmaximal, complemented
4#20273666C214yesmaximal, complemented
5#20273230C212yesmaximal, complemented
6#20273042C211yesmaximal, characteristic, complemented
7C53×#2561C211yesmaximal, characteristic, complemented
8C23×C22⋊D53—21no—
9#16811185—24no—
10#16811185—22no—
11#16811185C2214yes—
12C2×#13133201—22no—
13C2×#13133191—21no—
14C2×#13133191C2212yes—
15C2×#13133187—22no—
16#16810956—22no—
17#16810936C2214yescomplemented
18#16810871—24no—
19#16810773—22no—
20#16810762C2214yescomplemented
21#16810587C2212yescomplemented
22#16810356C2218yescomplemented
23#16810347C2214yescomplemented
24C2×C22⋊Q8×C53C2211yescharacteristic, complemented
25C22×C22⋊C4×C53C2211yescharacteristic
26C53×#397C2212yescomplemented
27C53×#393C2212yescomplemented
28C53×#388—22no—
29C53×#381—21no—
30C53×#379—21no—
31C4×C22⋊C4×C53C2211yescharacteristic, complemented
32C24×D53—21no—
33C22×D212—21no—
34C22×D212D412yes—
35C22×C22⋊D53—22no—
36C22×C22⋊D53—24no—
37C22×C22⋊D53—22no—
38C22×C22⋊D53—24no—
39C22×C22⋊D53—48no—
40C23×Dic53—21no—
41C2×#11719924—24no—
42C2×#11719917—22no—
43#13133201—28no—
44#13133201—28no—
45#13133191—24no—
46#13133191—24no—
47#13133191—22no—
48#13133191—22no—
49#13133191D414yes—
50#13133191C2314yes—
51#13133187—28no—
52#13133187—28no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53D412yes—
55C2×#11719921—22no—
56C2×#11719921—22no—
57#13133150—24no—
58#13133141—24no—
59#13133137—24no—
60C22⋊#88322—22no—
61C22⋊#88322C2314yescomplemented
62C25×C53D411yescharacteristic
63C22×Q8×C53C2311yescharacteristic, complemented
64C23×C4×C53D411yescharacteristic
65C23×C4×C53C2311yescharacteristic
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53—22no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53C2311yescharacteristic, complemented
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—21no—
72C2×C22⋊C4×C53—22no—
73C2×C22⋊C4×C53—22no—
74C2×C22⋊C4×C53—22no—
75C2×C22⋊C4×C53—22no—
76C2×C22⋊C4×C53D412yes—
77C2×C22⋊C4×C53C2312yes—
78C2×C42×C53—21no—
79#95×C53—22no—
80#95×C53—22no—
81#95×C53C2312yescomplemented
82C24×C53—21no—
83C24×C53—21no—
84C24×C53—24no—
85C24×C53—42no—
86C24×C53—81no—
87C24×C53C2×D411yescharacteristic
88C24×C53C2×D412yes—
89C23×D53—22no—
90C23×D53—24no—
91C23×D53—44no—
92C23×D53—82no—
93C23×D53—82no—
94C2×Q8×C53—22no—
95C2×Q8×C53—81no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—42no—
102C22×C4×C53—81no—
103C22×C4×C53C2×D412yes—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D412yes—
106C22×C4×C53C2×D412yes—
107C22×C4×C53C2411yescharacteristic
108C2×C22⋊D53—28no—
109C2×C22⋊D53—24no—
110C2×C22⋊D53—24no—
111C2×C22⋊D53—48no—
112C2×C22⋊D53—416no—
113C2×C22⋊D53—48no—
114C2×C22⋊D53—48no—
115C2×C22⋊D53—48no—
116C22×Dic53—24no—
117C22×Dic53—22no—
118C22×Dic53—44no—
119C2×D212—24no—
120C2×D212—22no—
121C2×D212—82no—
122C2×D212—82no—
123C2×D212C2×D414yes—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—42no—
126C22⋊C4×C53—24no—
127C22⋊C4×C53—24no—
128C22⋊C4×C53—42no—
129C22⋊C4×C53—41no—
130C22⋊C4×C53—41no—
131C22⋊C4×C53—44no—
132C22⋊C4×C53—82no—
133C42×C53—81no—
134#11719924—84no—
135#11719921—28no—
136#11719921—24no—
137#11719921—24no—
138C4⋊Dic53—24no—
139C4⋊Dic53—22no—
140C4⋊Dic53C2×D414yes—
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—41no—
150C23×C53—42no—
151C23×C53—42no—
152C23×C53—42no—
153C23×C53—42no—
154C23×C53—81no—
155C23×C53—81no—
156C23×C53—88no—
157C23×C53#12011yescharacteristic
158C23×C53#12011yescharacteristic
159C23×C53AGL(1,ℤ/8)14yes—
160C23×C53C22×D411yescharacteristic
161C22×D53—24no—
162C22×D53—44no—
163C22×D53—44no—
164C22×D53—48no—
165C22×D53—44no—
166C22×D53—84no—
167C22×D53—84no—
168C22×D53—816no—
169Q8×C53—44no—
170C2×C4×C53—22no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—42no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53—81no—
185C2×C4×C53#12012yes—
186C2×C4×C53C22×D412yes—
187C22⋊D53—416no—
188C22⋊D53—48no—
189C22⋊D53—48no—
190C2×Dic53—24no—
191C2×Dic53—44no—
192C2×Dic53—48no—
193D212—44no—
194D212—82no—
195D212—84no—
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—82no—
212C22×C53—84no—
213C22×C53#45512yes—
214C22×C53C2×#12011yescharacteristic
215C22×C53C2×AGL(1,ℤ/8)12yes—
216C2×D53—44no—
217C2×D53—44no—
218C2×D53—44no—
219C2×D53—88no—
220C2×D53—816no—
221C4×C53—22no—
222C4×C53—22no—
223C4×C53—42no—
224C4×C53—82no—
225C4×C53—81no—
226C4×C53—81no—
227Dic53—44no—
228C2×#534—531no—
229#3193—532no—
230#2561D5311yescharacteristic, complemented
231#1607—534no—
232#1601—534no—
233#1507—532no—
234#1462—531no—
235C2×C53—22no—
236C2×C53—22no—
237C2×C53—44no—
238C2×C53—41no—
239C2×C53#300311yescharacteristic
240C2×C53C2×#45511yescharacteristic
241C2×C53#302611yescharacteristic
242D53—84no—
243C23×D4—1061no—
244#534—534no—
245#534—1064no—
246#534—1062no—
247C2×#123—1062no—
248C2×C22⋊Q8C2×D5311yescharacteristic, complemented
249C2×#121—532no—
250C2×#121—1061no—
251C2×#120—1062no—
252C22×C22⋊C4C2×D5311yescharacteristic
253#484—1062no—
254#480—534no—
255#468—1064no—
256#450—1062no—
257#448—534no—
258#407—532no—
259#397C2×D5312yescomplemented
260#393C2×D5312yescomplemented
261#388—22no—
262#381—21no—
263#379—21no—
264C4×C22⋊C4C2×D5311yescharacteristic, complemented
265#327—538no—
266#323—534no—
267C53#1831711yescharacteristic, complemented, Sylow
268C25—1061no—
269C25C22⋊D5311yescharacteristic
270C22×Q8C22×D5311yescharacteristic, complemented
271C22×D4—532no—
272C22×D4—1061no—
273C22×D4—1062no—
274C22×D4—1064no—
275C22×D4—1062no—
276C22×D4—1064no—
277C22×D4—2128no—
278C23×C4—1061no—
279C23×C4C22⋊D5311yescharacteristic
280C23×C4C22×D5311yescharacteristic
281C2×SD16—1064no—
282C2×M16—1062no—
283#123—1068no—
284#123—1068no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287#121—534no—
288#121—534no—
289#121—1064no—
290#121—1062no—
291#121—1062no—
292#121—1064no—
293#120—1068no—
294#120—1068no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—532no—
297C2×C4⋊C4—1061no—
298C2×C4⋊C4C22×D5311yescharacteristic, complemented
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—1062no—
306C2×C22⋊C4—1062no—
307C2×C22⋊C4C22⋊D5312yes—
308C2×C22⋊C4C22×D5312yes—
309C2×C42—21no—
310#107—1064no—
311#103—1064no—
312#102—1064no—
313C22⋊C8—534no—
314C22⋊C8—1062no—
315#95—22no—
316#95—22no—
317#95C22×D5312yescomplemented
318C24—21no—
319C24—21no—
320C24—24no—
321C24—42no—
322C24—81no—
323C24—1064no—
324C24—1062no—
325C24—4242no—
326C24—4242no—
327C24—2124no—
328C24C2×C22⋊D5311yescharacteristic
329C24C2×C22⋊D5312yes—
330C2×Q8—22no—
331C2×Q8—81no—
332C2×D4—534no—
333C2×D4—1064no—
334C2×D4—1068no—
335C2×D4—1064no—
336C2×D4—1064no—
337C2×D4—1062no—
338C2×D4—4242no—
339C2×D4—4242no—
340C2×D4—2128no—
341C2×D4—21216no—
342C2×D4—2128no—
343C2×D4—2128no—
344C2×D4—2128no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—42no—
351C22×C4—81no—
352C22×C4—1064no—
353C22×C4—1062no—
354C22×C4—2124no—
355C22×C4D4×D5312yes—
356C22×C4D4×D5312yes—
357C22×C4C2×C22⋊D5312yes—
358C22×C4C2×C22⋊D5311yescharacteristic
359C22×C4C23×D5311yescharacteristic
360SD16—4244no—
361M16—4242no—
362C2×C8—1064no—
363C4⋊C4—22no—
364C4⋊C4—42no—
365C4⋊C4—534no—
366C4⋊C4—1064no—
367C4⋊C4—1062no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—42no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—44no—
374C22⋊C4—82no—
375C22⋊C4—1068no—
376C22⋊C4—1064no—
377C22⋊C4—1064no—
378C42—81no—
379C23—24no—
380C23—24no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—81no—
391C23—81no—
392C23—88no—
393C23—1064no—
394C23—42416no—
395C23—4244no—
396C23—4244no—
397C23—2124no—
398C23—2124no—
399C23—2128no—
400C23—2124no—
401C23#1313318511yescharacteristic
402C23#1313318511yescharacteristic
403C23#1313324214yes—
404C23C22×C22⋊D5311yescharacteristic
405Q8—44no—
406D4—4242no—
407D4—4244no—
408D4—2124no—
409D4—21216no—
410D4—2128no—
411D4—2128no—
412C2×C4—22no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—81no—
427C2×C4—1064no—
428C2×C4—2124no—
429C2×C4—2128no—
430C2×C4#1313318512yes—
431C2×C4C2×D4×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—82no—
447C22—84no—
448C22—4248no—
449C22—42416no—
450C22—2124no—
451C22—2124no—
452C22—2124no—
453C22#1681080012yes—
454C22C2×#1313318511yescharacteristic
455C22C2×#1313324212yes—
456C4—22no—
457C4—22no—
458C4—42no—
459C4—82no—
460C4—81no—
461C4—81no—
462C4—2124no—
463C2—22no—
464C2—22no—
465C2—41no—
466C2—44no—
467C2—4244no—
468C2#2028394111yescharacteristic
469C2C2×#1681080011yescharacteristic
470C2#2028411111yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2561C2
socleC22×C53C2×#120
prime radicalC2×#16811185C2
hypercenter#2561D53
hypocenterC53#18317
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
2. #20285517C2semidirect
3. #20273700C2semidirect
4. #20273666C2semidirect
5. #20273230C2semidirect
6. #20273042C2semidirect
7. C53×#2561C2semidirect
17. #16810936C22semidirect
20. #16810762C22semidirect
21. #16810587C22semidirect
22. #16810356C22semidirect
23. #16810347C22semidirect
24. C2×C22⋊Q8×C53C22semidirect
26. C53×#397C22semidirect
27. C53×#393C22semidirect
31. C4×C22⋊C4×C53C22semidirect
61. C22⋊#88322C23semidirect
63. C22×Q8×C53C23semidirect
69. C2×C4⋊C4×C53C23semidirect
81. #95×C53C23semidirect
230. #2561D53semidirect
248. C2×C22⋊Q8C2×D53semidirect
259. #397C2×D53semidirect
260. #393C2×D53semidirect
264. C4×C22⋊C4C2×D53semidirect
267. C53#18317semidirect
270. C22×Q8C22×D53semidirect
298. C2×C4⋊C4C22×D53semidirect
317. #95C22×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111852
#202855172
#202737002
#202736662
#202732302
#202730422
C53×#25612
#1831753