Finite Group Databasebeta

← #27469599

Subgroups — #27469599

Order 13568.

842 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283024C211yesmaximal, characteristic
2#384×D53C211yesmaximal, characteristic
3#20276193C211yesmaximal, characteristic, complemented
4#20276192C211yesmaximal, characteristic, complemented
5#20276186C211yesmaximal, characteristic, complemented
6#20275682C212yesmaximal, complemented
7#20274909C212yesmaximal
8#20274901C212yesmaximal
9#20274882C212yesmaximal, complemented
10#20274476C211yesmaximal, characteristic
11C53×#2083C211yesmaximal, characteristic, complemented
12C2×#13133208C2211yescharacteristic
13C2×C4⋊C4×D53—21no—
14C2×C4⋊C4×D53—21no—
15C2×C4⋊C4×D53C2212yes—
16C2×C42×D53C2211yescharacteristic
17C2×#13133150C2212yes—
18C2×#13133149C2212yes—
19C2×#13133148C2212yes—
20C2×#13133129C2212yes—
21C2×#13133127C2212yes—
22#16810521C2211yescharacteristic
23#16810509—21no—
24#16810505—21no—
25#16810497—21no—
26#16810494—21no—
27#16810493C2211yescharacteristic
28#16810492—21no—
29#16810492—21no—
30#16810492C2212yes—
31#16810488C2211yescharacteristic
32#16810480C2211yescharacteristic
33#16810480C2211yescharacteristic
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810378C2211yescharacteristic
38C2×#107×C53C2212yes—
39C53×#389C2211yescharacteristic
40C53×#384C2211yescharacteristic
41C53×#340C2212yes—
42C53×#336C2211yescharacteristic
43C22×C4×D53—22no—
44C22×C4×D53D411yescharacteristic
45C22×C4×D53D411yescharacteristic
46C22×C4×D53C2311yescharacteristic
47C22×#88322C2312yes—
48#13133208—24no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53C2311yescharacteristic
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2312yes—
55C2×#11719919—21no—
56C2×#11719919—21no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C4⋊C4×D53—42no—
61C4⋊C4×D53—42no—
62C4⋊C4×D53—44no—
63C4⋊C4×D53—44no—
64C2×#11719921—21no—
65C2×#11719921—21no—
66C2×C4×Dic53D411yescharacteristic
67C2×C4×Dic53D411yescharacteristic
68C2×C4×Dic53C2311yescharacteristic
69C42×D53—24no—
70#13133150—42no—
71#13133149—42no—
72#13133148—42no—
73#13133129Q818yes—
74#13133127—24no—
75#13133124—21no—
76#13133124—21no—
77#13133124—21no—
78#13133124—21no—
79#13133124—21no—
80C22×C8×C53C2312yes—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53C2312yes—
85C2×C4⋊C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C42×C53C2311yescharacteristic
88#107×C53—42no—
89#95×C53—21no—
90C23×D53C2×D411yescharacteristic
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2411yescharacteristic
97C22×Dic53—21no—
98C22×Dic53—21no—
99C22×Dic53—22no—
100C22×Dic53C2×D411yescharacteristic
101C22×Dic53C2×D411yescharacteristic
102C22×Dic53C2×D411yescharacteristic
103C2×C4×D53—22no—
104C2×C4×D53—22no—
105C2×C4×D53—22no—
106C2×C4×D53—22no—
107C2×C4×D53—22no—
108C2×C4×D53—21no—
109C2×C4×D53—21no—
110C2×C4×D53—22no—
111C2×C4×D53—22no—
112C2×C4×D53—44no—
113C2×C4×D53—44no—
114C2×C4×D53—44no—
115C2×C4×D53C2×Q814yes—
116C2×C8×C53—22no—
117C2×C8×C53—22no—
118C2×C8×C53C2×Q814yes—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—42no—
125C4⋊C4×C53—42no—
126C4⋊C4×C53—42no—
127C42×C53—22no—
128#11719921—42no—
129#11719921—42no—
130C4⋊Dic53—22no—
131C4⋊Dic53—21no—
132C4⋊Dic53—21no—
133C4⋊Dic53—22no—
134C4⋊Dic53—22no—
135C4⋊Dic53—42no—
136C4⋊Dic53—42no—
137C4⋊Dic53—42no—
138#11719919—44no—
139#11719919—44no—
140#11719919—44no—
141#11719919—42no—
142#11719919—42no—
143C4×Dic53—22no—
144C4×Dic53—22no—
145C4×Dic53#4114yes—
146C2×#88322—22no—
147C2×#88322—22no—
148C2×#88322C2×Q814yes—
149C23×C53C22×D411yescharacteristic
150C22×D53—21no—
151C22×D53—21no—
152C22×D53—21no—
153C22×D53—21no—
154C22×D53C4⋊Q812yes—
155C22×D53AGL(1,ℤ/8)12yes—
156C22×D53#13712yes—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—41no—
162C2×C4×C53—42no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53AGL(1,ℤ/8)12yes—
171C2×C4×C53#13712yes—
172C2×C4×C53C22×Q812yes—
173C2×C4×C53C2×#4112yes—
174C2×C4×C53H(5,𝔽2)11yescharacteristic
175C2×C4×C53H(5,𝔽2)11yescharacteristic
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53—41no—
183C2×Dic53—42no—
184C2×Dic53—41no—
185C2×Dic53—42no—
186C2×Dic53—42no—
187C2×Dic53—41no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53#12312yes—
192C2×Dic53#12312yes—
193C2×Dic53C4⋊Q812yes—
194C4×D53—28no—
195C4×D53—28no—
196C4×D53—24no—
197C4×D53—24no—
198C4×D53—88no—
199C8×C53—24no—
200#88322—24no—
201#27953—531nomaximal, Sylow
202C22×C53C2×#12311yescharacteristic
203C22×C53C2×C4⋊Q811yescharacteristic
204C22×C53#53511yescharacteristic
205C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
206C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
207C22×C53C2×#13711yescharacteristic
208C22×C53C2×#13711yescharacteristic
209C2×D53—22no—
210C2×D53—22no—
211C2×D53—22no—
212C2×D53—22no—
213C2×D53—22no—
214C2×D53—22no—
215C2×D53#50114yes—
216C4×C53—22no—
217C4×C53—22no—
218C4×C53—81no—
219C4×C53—82no—
220C4×C53—81no—
221C4×C53#55412yes—
222C4×C53#55412yes—
223Dic53—22no—
224Dic53—22no—
225Dic53—22no—
226Dic53—81no—
227Dic53—82no—
228Dic53—81no—
229#2851—531no—
230C2×#384—531no—
231#2084—531no—
232#2084—531no—
233#2083—531no—
234#2083D5311yescharacteristic, complemented
235#1992—532no—
236#1853—532no—
237#1852—532no—
238#1847—532no—
239#1749—531no—
240C2×C53#250211yescharacteristic
241C2×C53#309211yescharacteristic
242C2×C53#309211yescharacteristic
243C2×C53C2×#50111yescharacteristic
244C2×C53#316511yescharacteristic
245C2×C53#321711yescharacteristic
246C2×C53#321811yescharacteristic
247D53—24no—
248C2×#125—531no—
249C22×C4⋊C4—532no—
250C22×C4⋊C4—1061no—
251C22×C4⋊C4—1061no—
252C22×C42—531no—
253C2×#107—532no—
254C2×#107C2×D5312yes—
255C2×#106—532no—
256C2×#106—532no—
257C2×C22⋊C8—532no—
258C2×#97—532no—
259#389—531no—
260#389C2×D5311yescharacteristic
261#387—1061no—
262#387—1061no—
263#385—1061no—
264#385—1061no—
265#384—532no—
266#384—531no—
267#384—1061no—
268#384—1061no—
269#384C2×D5311yescharacteristic
270#382—531no—
271#380—531no—
272#380—531no—
273#340—532no—
274#340—532no—
275#340—532no—
276#340C2×D5312yes—
277#336—531no—
278#336C2×D5311yescharacteristic
279C53#2795311yescharacteristic, complemented, Sylow
280C23×C4—531no—
281C23×C4—531no—
282C23×C4—531no—
283C23×C4—1062no—
284C22×C8—532no—
285C22×C8C22×D5312yes—
286#125—1064no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—532no—
291C2×C4⋊C4—531no—
292C2×C4⋊C4—531no—
293C2×C4⋊C4—1062no—
294C2×C4⋊C4—1062no—
295C2×C4⋊C4—1062no—
296C2×C4⋊C4—1061no—
297C2×C4⋊C4—1061no—
298C2×C4⋊C4—1061no—
299C2×C4⋊C4—1061no—
300C2×C4⋊C4—1061no—
301C2×C4⋊C4—2122no—
302C2×C4⋊C4—2122no—
303C2×C4⋊C4—2124no—
304C2×C4⋊C4—2124no—
305C2×C4⋊C4C22×D5312yes—
306C2×C4⋊C4C22×D5311yescharacteristic
307C2×C4⋊C4C22×D5311yescharacteristic
308C2×C22⋊C4—1061no—
309C2×C22⋊C4—1061no—
310C2×C42—531no—
311C2×C42—531no—
312C2×C42—531no—
313C2×C42—1064no—
314C2×C42C22×D5311yescharacteristic
315#107—42no—
316#107—2122no—
317#106—2122no—
318#106—2122no—
319C22⋊C8—538no—
320#97—1064no—
321#95—21no—
322#95—1061no—
323#95—1061no—
324#95—1061no—
325#95—1061no—
326#95—1061no—
327C24—531no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—531no—
332C22×C4—531no—
333C22×C4—531no—
334C22×C4—534no—
335C22×C4—1062no—
336C22×C4—1062no—
337C22×C4—1062no—
338C22×C4—1062no—
339C22×C4—1062no—
340C22×C4—1062no—
341C22×C4—1062no—
342C22×C4—1061no—
343C22×C4—1061no—
344C22×C4—1061no—
345C22×C4—1061no—
346C22×C4—1062no—
347C22×C4—2124no—
348C22×C4—2124no—
349C22×C4—2124no—
350C22×C4D4×D5311yescharacteristic
351C22×C4D4×D5311yescharacteristic
352C22×C4C23×D5311yescharacteristic
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—534no—
356C2×C8—1062no—
357C2×C8—1062no—
358C2×C8Q8×D5314yes—
359C4⋊C4—22no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—1062no—
368C4⋊C4—1061no—
369C4⋊C4—1062no—
370C4⋊C4—1061no—
371C4⋊C4—1062no—
372C4⋊C4—2122no—
373C4⋊C4—2122no—
374C4⋊C4—2122no—
375C4⋊C4—2124no—
376C4⋊C4—2124no—
377C4⋊C4—2124no—
378C4⋊C4—2122no—
379C4⋊C4—2122no—
380C22⋊C4—2122no—
381C22⋊C4—2122no—
382C42—22no—
383C42—534no—
384C42—1062no—
385C42—1062no—
386C23—532no—
387C23—532no—
388C23—532no—
389C23—1061no—
390C23—1061no—
391C23—1061no—
392C23—1061no—
393C23C2×D4×D5311yescharacteristic
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—532no—
408C2×C4—532no—
409C2×C4—532no—
410C2×C4—1068no—
411C2×C4—1068no—
412C2×C4—1061no—
413C2×C4—1061no—
414C2×C4—1061no—
415C2×C4—1061no—
416C2×C4—1061no—
417C2×C4—1061no—
418C2×C4—1064no—
419C2×C4—1064no—
420C2×C4—4248no—
421C2×C4—2121no—
422C2×C4—2121no—
423C2×C4—2122no—
424C2×C4—2122no—
425C2×C4—2121no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4—2122no—
429C2×C4—2121no—
430C2×C4#1313324112yes—
431C2×C4#1313324912yes—
432C2×C4C2×Q8×D5312yes—
433C2×C4#41×D5312yes—
434C2×C4#1313326211yescharacteristic
435C2×C4#1313326311yescharacteristic
436C8—24no—
437C8—1064no—
438C22—534no—
439C22—1062no—
440C22—1062no—
441C22—1062no—
442C22—1062no—
443C22—1062no—
444C22—1062no—
445C22#123×D5311yescharacteristic
446C22C4⋊Q8×D5311yescharacteristic
447C22#1681118911yescharacteristic
448C22AGL(1,ℤ/8)×D5311yescharacteristic
449C22C2×#1313324111yescharacteristic
450C22#137×D5311yescharacteristic
451C22C2×#1313324911yescharacteristic
452C4—22no—
453C4—22no—
454C4—82no—
455C4—81no—
456C4—81no—
457C4—1062no—
458C4—1062no—
459C4—1062no—
460C4—4242no—
461C4—4241no—
462C4—4241no—
463C4#1681132712yes—
464C4#1681132912yes—
465C2—1064no—
466C2#2027918811yescharacteristic
467C2#2028463411yescharacteristic
468C2#2028464211yescharacteristic
469C2#501×D5311yescharacteristic
470C2#2028525811yescharacteristic
471C2#2028573211yescharacteristic
472C2#2028574611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2083C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2083D53
hypocenterC53#27953
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2083

Complements

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

NH ≅ G/Ntype
3. #20276193C2semidirect
4. #20276192C2semidirect
5. #20276186C2semidirect
6. #20275682C2semidirect
9. #20274882C2semidirect
11. C53×#2083C2semidirect
234. #2083D53semidirect
279. C53#27953semidirect

Maximal subgroups

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

subgroupindex
#202830242
#384×D532
#202761932
#202761922
#202761862
#202756822
#202749092
#202749012
#202748822
#202744762
C53×#20832
#2795353