Finite Group Databasebeta

← #27357940

Subgroups — #27357940

Order 13568.

1142 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285468C212yesmaximal, complemented
2#20280449C211yesmaximal, characteristic
3#20273706C214yesmaximal, complemented
4#20273638C214yesmaximal, complemented
5#20273287C212yesmaximal, complemented
6C2×#16810347C211yesmaximal, characteristic, complemented
7C2×C53×#543C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9C2×#13133186—21no—
10#16810951C2214yescomplemented
11#16810941—22no—
12#16810854—24no—
13#16810773C2214yescomplemented
14#16810762—22no—
15C2×C22⋊#88322C2212yes—
16#16810548C2212yes—
17#16810539—22no—
18#16810532C2212yes—
19#16810502—21no—
20#16810483C2211yescharacteristic
21#16810360C2218yescomplemented
22#16810347C2214yes—
23C53×#543—24no—
24C53×#543—22no—
25C53×#543C2214yes—
26C2×C22⋊Q8×C53—22no—
27C2×C22⋊Q8×C53—21no—
28C2×C22⋊Q8×C53—22no—
29C2×C22⋊Q8×C53C2212yes—
30C22×C22⋊C4×C53—21no—
31C22×C22⋊C4×C53C2211yescharacteristic
32C22×D212C2311yescharacteristic
33C22×C22⋊D53—21no—
34C22×C22⋊D53—22no—
35C2×#11719924—24no—
36C22×#88322—22no—
37#13133191—22no—
38#13133191—22no—
39#13133186—44no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53C2311yescharacteristic
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—22no—
44C2×#11719921—21no—
45C2×#11719921—22no—
46#13133148—24no—
47#13133141—24no—
48#13133137—24no—
49C22⋊#88322—22no—
50C22⋊#88322C2314yes—
51#13133124—22no—
52#13133124C2312yes—
53C25×C53D411yescharacteristic
54C22×Q8×C53—21no—
55C22×Q8×C53D412yescomplemented
56C23×C4×C53—21no—
57C23×C4×C53—21no—
58C23×C4×C53D411yescharacteristic
59C23×C4×C53C2311yescharacteristic
60C22⋊Q8×C53—28no—
61C22⋊Q8×C53—28no—
62C22⋊Q8×C53—24no—
63C22⋊Q8×C53—28no—
64C22⋊Q8×C53—28no—
65C22⋊Q8×C53—24no—
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53—22no—
68C22⋊Q8×C53D414yescomplemented
69C22⋊Q8×C53C2314yes—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—22no—
72C2×C4⋊C4×C53—22no—
73C2×C4⋊C4×C53D412yescomplemented
74C2×C22⋊C4×C53—22no—
75C2×C22⋊C4×C53—24no—
76C2×C22⋊C4×C53—22no—
77C2×C22⋊C4×C53—24no—
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—22no—
85C24×C53—81no—
86C24×C53—81no—
87C24×C53SD1614yescomplemented
88C24×C53C2×D412yes—
89C24×C53C2×D411yescharacteristic
90C23×D53—21no—
91C2×Q8×C53—24no—
92C2×Q8×C53—22no—
93C2×Q8×C53—82no—
94C2×Q8×C53—82no—
95C2×Q8×C53C2×D414yes—
96C22×C4×C53—22no—
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—24no—
100C22×C4×C53—24no—
101C22×C4×C53—21no—
102C22×C4×C53—81no—
103C22×C4×C53—81no—
104C22×C4×C53—82no—
105C22×C4×C53—82no—
106C22×C4×C53C2×D412yes—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D412yes—
109C22×C4×C53C2×D412yes—
110C22×C4×C53C2411yescharacteristic
111C2×C22⋊D53—41no—
112C2×C22⋊D53—41no—
113C2×C22⋊D53—44no—
114C2×C22⋊D53—42no—
115C2×C22⋊D53—42no—
116C2×C22⋊D53—82no—
117C2×C22⋊D53—82no—
118C22×Dic53—21no—
119C22×Dic53—22no—
120C2×D212—22no—
121C2×D212—81no—
122C4⋊C4×C53—28no—
123C4⋊C4×C53—24no—
124C4⋊C4×C53—24no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—24no—
127C4⋊C4×C53C2×D414yes—
128C22⋊C4×C53—28no—
129C22⋊C4×C53—24no—
130C22⋊C4×C53—24no—
131C22⋊C4×C53—24no—
132C22⋊C4×C53—24no—
133C22⋊Dic53—44no—
134C22⋊Dic53—42no—
135#11719924—84no—
136#11719921—42no—
137#11719921—82no—
138C4⋊Dic53—22no—
139C4⋊Dic53—81no—
140C2×#88322—24no—
141C2×#88322—82no—
142C23×C53—22no—
143C23×C53—24no—
144C23×C53—22no—
145C23×C53—24no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—42no—
149C23×C53—42no—
150C23×C53—42no—
151C23×C53—42no—
152C23×C53—81no—
153C23×C53—81no—
154C23×C53—84no—
155C23×C53—84no—
156C23×C53#12011yescharacteristic
157C23×C53#12011yescharacteristic
158C23×C53C2×SD1614yes—
159C23×C53C2×SD1612yes—
160C23×C53#13712yes—
161C23×C53C22×D411yescharacteristic
162C22×D53—42no—
163C22×D53—42no—
164C22×D53—41no—
165C22×D53—41no—
166C22×D53—41no—
167Q8×C53—44no—
168Q8×C53—82no—
169Q8×C53—84no—
170C2×C4×C53—24no—
171C2×C4×C53—24no—
172C2×C4×C53—22no—
173C2×C4×C53—41no—
174C2×C4×C53—42no—
175C2×C4×C53—42no—
176C2×C4×C53—41no—
177C2×C4×C53—42no—
178C2×C4×C53—44no—
179C2×C4×C53—44no—
180C2×C4×C53—42no—
181C2×C4×C53—81no—
182C2×C4×C53—84no—
183C2×C4×C53—84no—
184C2×C4×C53#12012yes—
185C2×C4×C53C22×D412yes—
186C22⋊D53—88no—
187C22⋊D53—82no—
188C22⋊D53—82no—
189C2×Dic53—42no—
190C2×Dic53—42no—
191C2×Dic53—41no—
192C2×Dic53—42no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195D212—44no—
196#88322—44no—
197#18286—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—24no—
205C22×C53—42no—
206C22×C53—42no—
207C22×C53—44no—
208C22×C53—48no—
209C22×C53—42no—
210C22×C53—42no—
211C22×C53—82no—
212C22×C53—84no—
213C22×C53#44812yes—
214C22×C53#44812yes—
215C22×C53C2×#12011yescharacteristic
216C22×C53C22×SD1611yescharacteristic
217C22×C53C2×#13711yescharacteristic
218C2×D53—82no—
219C2×D53—82no—
220C2×D53—81no—
221C2×D53—81no—
222C2×D53—81no—
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—42no—
226C4×C53—44no—
227C4×C53—44no—
228Dic53—82no—
229Dic53—81no—
230C2×#543D5311yescharacteristic, complemented
231#3187—532no—
232#2617—531no—
233#1608—534no—
234#1596—534no—
235#1520—532no—
236C2×#323—531no—
237C2×C53—21no—
238C2×C53—21no—
239C2×C53—22no—
240C2×C53—22no—
241C2×C53—44no—
242C2×C53C2×#44811yescharacteristic
243C2×C53#301911yescharacteristic
244C2×C53#302611yescharacteristic
245D53—161no—
246#543—24no—
247#543—22no—
248#543C2×D5314yes—
249C2×C22⋊Q8—22no—
250C2×C22⋊Q8—21no—
251C2×C22⋊Q8—22no—
252C2×C22⋊Q8C2×D5312yes—
253C2×#121—531no—
254C2×#120—1061no—
255C22×C22⋊C4—21no—
256C22×C22⋊C4C2×D5311yescharacteristic
257#483—534no—
258#481—1062no—
259#465—1064no—
260#450—534no—
261#448—1062no—
262C2×C22⋊C8—532no—
263#397—532no—
264#394—1062no—
265#392—532no—
266#386—1061no—
267#380—531no—
268#328—538no—
269#323—534no—
270C53#1828611yescharacteristic, complemented, Sylow
271C25C22⋊D5311yescharacteristic
272C22×Q8—21no—
273C22×Q8C22⋊D5312yescomplemented
274C22×D4—531no—
275C22×D4—1062no—
276C22×D4—1061no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4C22⋊D5311yescharacteristic
280C23×C4C22×D5311yescharacteristic
281C2×SD16—1064no—
282C22×C8—1062no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—24no—
286C22⋊Q8—28no—
287C22⋊Q8—24no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—28no—
291C22⋊Q8C22⋊D5314yescomplemented
292C22⋊Q8C22×D5314yes—
293#121—1062no—
294#121—1062no—
295#120—2124no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—531no—
300C2×C4⋊C4—1061no—
301C2×C4⋊C4C22⋊D5312yescomplemented
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—1062no—
309C2×C22⋊C4—1061no—
310C2×C22⋊C4—1062no—
311C2×C22⋊C4—1061no—
312C2×C22⋊C4C22⋊D5312yes—
313C2×C22⋊C4C22×D5312yes—
314#106—1064no—
315#103—1064no—
316#102—1064no—
317C22⋊C8—534no—
318C22⋊C8—1062no—
319#95—532no—
320#95—1062no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—81no—
325C24—81no—
326C24—1061no—
327C24#1171992414yescomplemented
328C24C2×C22⋊D5312yes—
329C24C2×C22⋊D5311yescharacteristic
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—82no—
333C2×Q8—82no—
334C2×Q8C2×C22⋊D5314yes—
335C2×D4—1062no—
336C2×D4—4241no—
337C2×D4—4242no—
338C2×D4—4242no—
339C2×D4—2121no—
340C2×D4—2121no—
341C2×D4—2124no—
342C2×D4—2122no—
343C2×D4—2122no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—1062no—
355C22×C4—1061no—
356C22×C4C2×C22⋊D5312yes—
357C22×C4C2×C22⋊D5311yescharacteristic
358C22×C4C2×C22⋊D5312yes—
359C22×C4C2×C22⋊D5312yes—
360C22×C4C23×D5311yescharacteristic
361SD16—4244no—
362C2×C8—1064no—
363C2×C8—4242no—
364C4⋊C4—24no—
365C4⋊C4—28no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—1062no—
370C4⋊C4—4241no—
371C4⋊C4C2×C22⋊D5314yes—
372C22⋊C4—28no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—4242no—
378C22⋊C4—2124no—
379C22⋊C4—2122no—
380C22⋊C4—2122no—
381C23—24no—
382C23—22no—
383C23—24no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—81no—
392C23—81no—
393C23—84no—
394C23—84no—
395C23—2122no—
396C23—2122no—
397C23—2121no—
398C23—2121no—
399C23—2121no—
400C23#1313318611yescharacteristic
401C23#1313318611yescharacteristic
402C23C2×#1171992414yes—
403C23C2×#1171992412yes—
404C23#1313324812yes—
405C23C22×C22⋊D5311yescharacteristic
406Q8—44no—
407Q8—82no—
408Q8—84no—
409D4—4248no—
410D4—4242no—
411D4—4242no—
412D4—2124no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—81no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—2122no—
428C2×C4—2122no—
429C2×C4—2121no—
430C2×C4—2122no—
431C2×C4—2121no—
432C2×C4—2121no—
433C2×C4#1313318612yes—
434C2×C4C22×C22⋊D5312yes—
435C8—2124no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—48no—
447C22—42no—
448C22—42no—
449C22—82no—
450C22—84no—
451C22—4242no—
452C22—4242no—
453C22—4241no—
454C22—4241no—
455C22—4241no—
456C22#1681076212yes—
457C22#1681076212yes—
458C22C2×#1313318611yescharacteristic
459C22C22×#1171992411yescharacteristic
460C22C2×#1313324811yescharacteristic
461C4—22no—
462C4—22no—
463C4—42no—
464C4—44no—
465C4—44no—
466C4—4242no—
467C4—4241no—
468C2—21no—
469C2—22no—
470C2—21no—
471C2—22no—
472C2—44no—
473C2—8481no—
474C2C2×#1681076211yescharacteristic
475C2#2028407211yescharacteristic
476C2#2028411211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#543C2
socleC22×C53C2×#120
prime radical#20280449C2
hypercenterC2×#543D53
hypocenterC53#18286
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #20285468C2semidirect
3. #20273706C2semidirect
4. #20273638C2semidirect
5. #20273287C2semidirect
6. C2×#16810347C2semidirect
7. C2×C53×#543C2semidirect
10. #16810951C22semidirect
13. #16810773C22semidirect
21. #16810360C22semidirect
55. C22×Q8×C53D4semidirect
68. C22⋊Q8×C53D4semidirect
73. C2×C4⋊C4×C53D4semidirect
87. C24×C53SD16semidirect
230. C2×#543D53semidirect
270. C53#18286semidirect
273. C22×Q8C22⋊D53semidirect
291. C22⋊Q8C22⋊D53semidirect
301. C2×C4⋊C4C22⋊D53semidirect
327. C24#11719924semidirect

Maximal subgroups

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

subgroupindex
#202854682
#202804492
#202737062
#202736382
#202732872
C2×#168103472
C2×C53×#5432
#1828653