Finite Group Databasebeta

← #27465747

Subgroups — #27465747

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810952C212yesmaximal
2#20277253C211yesmaximal, characteristic
3#384×D53C211yesmaximal, characteristic
4#20276193C211yesmaximal, characteristic, complemented
5#20276192C211yesmaximal, characteristic, complemented
6#20276123C211yesmaximal, characteristic, complemented
7#20275477C212yesmaximal, complemented
8#20274800C212yesmaximal
9#20274718C212yesmaximal, complemented
10#20274694C211yesmaximal, characteristic
11C53×#2072C211yesmaximal, characteristic, complemented
12C2×#13133193C2211yescharacteristic
13C2×Q8×Dic53C2211yescharacteristic
14C2×C4⋊C4×D53—21no—
15C2×C4⋊C4×D53—21no—
16C2×C4⋊C4×D53C2212yes—
17C2×C42×D53C2211yescharacteristic
18#16810952—28no—
19C2×#13133149C2212yes—
20C2×#13133148C2212yes—
21C2×#13133146C2212yes—
22C2×#13133141C2212yes—
23C2×#13133129C2212yes—
24#16810521C2211yescharacteristic
25#16810510—21no—
26#16810497—21no—
27#16810493C2211yescharacteristic
28#16810492—21no—
29#16810492—21no—
30#16810492C2212yes—
31#16810480C2211yescharacteristic
32C4⋊C4×Dic53C2211yescharacteristic
33#16810474—21no—
34#16810470—21no—
35#16810386C2212yes—
36#16810385C2212yes—
37#16810385C2212yes—
38#16810378C2211yescharacteristic
39C2×#103×C53C2212yes—
40C53×#391C2211yescharacteristic
41C53×#384C2211yescharacteristic
42C53×#340C2212yes—
43C53×#336C2211yescharacteristic
44C22×C4×D53—22no—
45C22×C4×D53D411yescharacteristic
46C22×C4×D53D411yescharacteristic
47C22×C4×D53C2311yescharacteristic
48C22×#88322C2312yes—
49#13133193—22no—
50#13133193D414yes—
51Q8×Dic53—24no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53C2311yescharacteristic
56C2×C4⋊Dic53C2312yes—
57C2×C4⋊Dic53C2311yescharacteristic
58C2×#11719919—21no—
59C2×#11719919—21no—
60C2×#11719919—22no—
61C2×#11719919—22no—
62C2×#11719919—22no—
63C4⋊C4×D53—24no—
64C4⋊C4×D53—24no—
65C4⋊C4×D53—44no—
66C4⋊C4×D53—44no—
67C2×#11719921—21no—
68C2×#11719921—21no—
69C2×C4×Dic53—21no—
70C2×C4×Dic53—21no—
71C2×C4×Dic53D411yescharacteristic
72C2×C4×Dic53D411yescharacteristic
73C2×C4×Dic53C2311yescharacteristic
74C42×D53—24no—
75#13133149—24no—
76#13133148—24no—
77#13133146—24no—
78#13133141—24no—
79#13133129—24no—
80#13133124—21no—
81#13133124—21no—
82#13133124—21no—
83C22×Q8×C53C2311yescharacteristic
84C22×C8×C53C2312yes—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53C2312yes—
88C2×C4⋊C4×C53C2311yescharacteristic
89C2×C42×C53C2311yescharacteristic
90#103×C53—24no—
91#95×C53—21no—
92#95×C53—21no—
93C23×D53C2×D411yescharacteristic
94C2×Q8×C53—21no—
95C2×Q8×C53—22no—
96C2×Q8×C53—22no—
97C2×Q8×C53C2×D412yes—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—22no—
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2411yescharacteristic
104C22×Dic53—21no—
105C22×Dic53—21no—
106C22×Dic53—22no—
107C22×Dic53C2×D411yescharacteristic
108C22×Dic53C2×D411yescharacteristic
109C22×Dic53C2×D411yescharacteristic
110C2×C4×D53—22no—
111C2×C4×D53—22no—
112C2×C4×D53—22no—
113C2×C4×D53—22no—
114C2×C4×D53—22no—
115C2×C4×D53—22no—
116C2×C4×D53—22no—
117C2×C4×D53—44no—
118C2×C4×D53—44no—
119C2×C4×D53—44no—
120C2×C4×D53SD1614yes—
121C2×C4×D53C2×D412yes—
122C2×C4×D53#4112yes—
123C2×C8×C53—22no—
124C2×C8×C53—22no—
125C2×C8×C53—22no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—42no—
130C4⋊C4×C53—42no—
131C42×C53—22no—
132#11719921—24no—
133#11719921—42no—
134C4⋊Dic53—21no—
135C4⋊Dic53—22no—
136C4⋊Dic53—22no—
137C4⋊Dic53—22no—
138C4⋊Dic53—42no—
139C4⋊Dic53—42no—
140C4⋊Dic53—42no—
141C4⋊Dic53C2×D412yes—
142#11719919—24no—
143#11719919—44no—
144#11719919—42no—
145#11719919—44no—
146#11719919—44no—
147C4×Dic53—22no—
148C4×Dic53—22no—
149C4×Dic53—42no—
150C4×Dic53—42no—
151C4×Dic53#4114yes—
152C2×#88322—22no—
153C2×#88322—22no—
154C2×#88322—22no—
155C23×C53C22×D411yescharacteristic
156C22×D53—21no—
157C22×D53—21no—
158C22×D53—21no—
159C22×D53—21no—
160C22×D53#12112yes—
161C22×D53C2×SD1612yes—
162C22×D53AGL(1,ℤ/8)12yes—
163Q8×C53—24no—
164Q8×C53—42no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—22no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—42no—
173C2×C4×C53—41no—
174C2×C4×C53—42no—
175C2×C4×C53—42no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53C2×SD1612yes—
179C2×C4×C53#13712yes—
180C2×C4×C53C22×D411yescharacteristic
181C2×C4×C53C2×#4112yes—
182C2×C4×C53C2×#4111yescharacteristic
183C2×C4×C532−1+412yes—
184C2×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53—22no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—42no—
192C2×Dic53—41no—
193C2×Dic53—42no—
194C2×Dic53—42no—
195C2×Dic53—41no—
196C2×Dic53—41no—
197C2×Dic53#12112yes—
198C2×Dic53#12312yes—
199C2×Dic53#12312yes—
200C2×Dic53C2×SD1612yes—
201C2×Dic53#13712yes—
202C4×D53—28no—
203C4×D53—24no—
204C4×D53—24no—
205C4×D53—28no—
206C4×D53—88no—
207C8×C53—42no—
208#88322—42no—
209#27647—531nomaximal, Sylow
210C22×C53C2×#12111yescharacteristic
211C22×C53C2×#12311yescharacteristic
212C22×C53#53611yescharacteristic
213C22×C53C22×SD1611yescharacteristic
214C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
215C22×C53C2×#13711yescharacteristic
216C22×C53C2×#13711yescharacteristic
217C2×D53—22no—
218C2×D53—22no—
219C2×D53—22no—
220C2×D53—22no—
221C2×D53—22no—
222C2×D53—22no—
223C2×D53#46014yes—
224C4×C53—22no—
225C4×C53—22no—
226C4×C53—42no—
227C4×C53—81no—
228C4×C53—82no—
229C4×C53#54712yes—
230C4×C53#55212yes—
231Dic53—22no—
232Dic53—22no—
233Dic53—42no—
234Dic53—81no—
235Dic53—82no—
236Dic53#48314yescomplemented
237C2×#483—532no—
238C4×C22⋊Q8—531no—
239C2×#384—531no—
240#2084—531no—
241#2084—531no—
242#2072—531no—
243#2072D5311yescharacteristic, complemented
244#1948—532no—
245#1823—532no—
246#1803—532no—
247#1797—531no—
248C2×C53#248611yescharacteristic
249C2×C53C2×#46011yescharacteristic
250C2×C53#303911yescharacteristic
251C2×C53C2×#48311yescharacteristic
252C2×C53#309211yescharacteristic
253C2×C53#322911yescharacteristic
254C2×C53#323811yescharacteristic
255D53—24no—
256C2×C22⋊Q8—531no—
257C2×C4×Q8—531no—
258C22×C4⋊C4—532no—
259C22×C4⋊C4—1061no—
260C22×C4⋊C4—1061no—
261C22×C42—531no—
262#483—1068no—
263C2×#106—532no—
264C2×#106—532no—
265C2×C4⋊C8—532no—
266C2×#103—532no—
267C2×#103C2×D5312yes—
268C2×C22⋊C8—532no—
269#391C2×D5311yescharacteristic
270#389—531no—
271#387—1061no—
272#385—1061no—
273#384—532no—
274#384—531no—
275#384—1061no—
276#384—1061no—
277#384C2×D5311yescharacteristic
278#380—531no—
279C4×C4⋊C4—531no—
280C4×C4⋊C4—1061no—
281C4×C22⋊C4—1061no—
282#340—532no—
283#340—532no—
284#340—532no—
285#340C2×D5312yes—
286#336—531no—
287#336C2×D5311yescharacteristic
288C53#2764711yescharacteristic, complemented, Sylow
289C22×Q8C22×D5311yescharacteristic
290C23×C4—531no—
291C23×C4—531no—
292C23×C4—531no—
293C23×C4—1062no—
294C22×C8—532no—
295C22×C8C22×D5312yes—
296C22⋊Q8—534no—
297C22⋊Q8—1062no—
298C4×Q8—1064no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—531no—
302C2×C4⋊C4—532no—
303C2×C4⋊C4—531no—
304C2×C4⋊C4—1064no—
305C2×C4⋊C4—1064no—
306C2×C4⋊C4—1061no—
307C2×C4⋊C4—1062no—
308C2×C4⋊C4—1062no—
309C2×C4⋊C4—1062no—
310C2×C4⋊C4—1061no—
311C2×C4⋊C4—1061no—
312C2×C4⋊C4—1061no—
313C2×C4⋊C4—1061no—
314C2×C4⋊C4—2124no—
315C2×C4⋊C4—2124no—
316C2×C4⋊C4C22×D5312yes—
317C2×C4⋊C4C22×D5311yescharacteristic
318C2×C22⋊C4—1061no—
319C2×C22⋊C4—1061no—
320C2×C42—531no—
321C2×C42—531no—
322C2×C42—531no—
323C2×C42—1064no—
324C2×C42—1061no—
325C2×C42—1061no—
326C2×C42C22×D5311yescharacteristic
327#106—1064no—
328#106—1064no—
329C4⋊C8—1064no—
330#103—24no—
331#103—1064no—
332C22⋊C8—1064no—
333#95—21no—
334#95—21no—
335#95—1061no—
336#95—1061no—
337#95—1061no—
338C24—531no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8D4×D5312yes—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—531no—
347C22×C4—531no—
348C22×C4—531no—
349C22×C4—534no—
350C22×C4—532no—
351C22×C4—532no—
352C22×C4—1062no—
353C22×C4—1062no—
354C22×C4—1062no—
355C22×C4—1062no—
356C22×C4—1062no—
357C22×C4—1062no—
358C22×C4—1062no—
359C22×C4—1061no—
360C22×C4—1061no—
361C22×C4—1062no—
362C22×C4—2124no—
363C22×C4—2124no—
364C22×C4—2124no—
365C22×C4D4×D5311yescharacteristic
366C22×C4D4×D5311yescharacteristic
367C22×C4C23×D5311yescharacteristic
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—1062no—
372C2×C8—1062no—
373C2×C8—1062no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—532no—
380C4⋊C4—1064no—
381C4⋊C4—1061no—
382C4⋊C4—1062no—
383C4⋊C4—1062no—
384C4⋊C4—1062no—
385C4⋊C4—2122no—
386C4⋊C4—2124no—
387C4⋊C4—2124no—
388C4⋊C4—2122no—
389C4⋊C4—2124no—
390C4⋊C4—2122no—
391C4⋊C4—2122no—
392C22⋊C4—1064no—
393C22⋊C4—2122no—
394C42—22no—
395C42—534no—
396C42—1062no—
397C42—1062no—
398C42—2122no—
399C42—2122no—
400C23—532no—
401C23—532no—
402C23—532no—
403C23—1061no—
404C23—1061no—
405C23—1061no—
406C23—1061no—
407C23C2×D4×D5311yescharacteristic
408Q8—24no—
409Q8—42no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—532no—
424C2×C4—532no—
425C2×C4—532no—
426C2×C4—532no—
427C2×C4—532no—
428C2×C4—1068no—
429C2×C4—1068no—
430C2×C4—1061no—
431C2×C4—1061no—
432C2×C4—1061no—
433C2×C4—1061no—
434C2×C4—1064no—
435C2×C4—1064no—
436C2×C4—1062no—
437C2×C4—4248no—
438C2×C4—2122no—
439C2×C4—2121no—
440C2×C4—2122no—
441C2×C4—2121no—
442C2×C4—2122no—
443C2×C4—2121no—
444C2×C4—2121no—
445C2×C4—2121no—
446C2×C4SD16×D5312yes—
447C2×C4#1313324912yes—
448C2×C4C2×D4×D5311yescharacteristic
449C2×C4#41×D5312yes—
450C2×C4#41×D5311yescharacteristic
451C2×C4#1313326412yes—
452C8—42no—
453C8—2122no—
454C22—534no—
455C22—1062no—
456C22—1062no—
457C22—1062no—
458C22—1062no—
459C22—1062no—
460C22—1062no—
461C22#121×D5311yescharacteristic
462C22#123×D5311yescharacteristic
463C22#1681119311yescharacteristic
464C22C2×SD16×D5311yescharacteristic
465C22AGL(1,ℤ/8)×D5311yescharacteristic
466C22#137×D5311yescharacteristic
467C22C2×#1313324911yescharacteristic
468C4—22no—
469C4—22no—
470C4—42no—
471C4—81no—
472C4—82no—
473C4—534no—
474C4—1062no—
475C4—1062no—
476C4—4241no—
477C4—4242no—
478C4—2122no—
479C4#1681127112yes—
480C4#1681130912yes—
481C2—1064no—
482C2#2027901611yescharacteristic
483C2#460×D5311yescharacteristic
484C2#2028419911yescharacteristic
485C2#483×D5311yescharacteristic
486C2#2028464211yescharacteristic
487C2#2028585511yescharacteristic
488C2#2028593211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2072C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2072D53
hypocenterC53#27647
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2072

Complements

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

NH ≅ G/Ntype
4. #20276193C2semidirect
5. #20276192C2semidirect
6. #20276123C2semidirect
7. #20275477C2semidirect
9. #20274718C2semidirect
11. C53×#2072C2semidirect
236. Dic53#483semidirect
243. #2072D53semidirect
288. C53#27647semidirect

Maximal subgroups

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

subgroupindex
C2×#168109522
#202772532
#384×D532
#202761932
#202761922
#202761232
#202754772
#202748002
#202747182
#202746942
C53×#20722
#2764753