Finite Group Databasebeta

← #27469676

Subgroups — #27469676

Order 13568.

866 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283065C211yesmaximal, characteristic
2C2×#16810492C211yesmaximal, characteristic
3#20276281C212yesmaximal, complemented
4#20276280C212yesmaximal, complemented
5#20275674C212yesmaximal, complemented
6#20274841C212yesmaximal, complemented
7#20274806C211yesmaximal, characteristic
8#20274805C211yesmaximal, characteristic
9#20274803C211yesmaximal, characteristic
10C22⋊C8×Dic53C211yesmaximal, characteristic
11C53×#1824C211yesmaximal, characteristic
12C2×#13133216C2211yescharacteristic
13C2×#13133194—21no—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919—21no—
16C22×#11719919—21no—
17C22×C4×Dic53C2211yescharacteristic
18C2×C22⋊#88322C2211yescharacteristic
19C2×C8×Dic53C2212yes—
20#16810527—21no—
21#16810522C2211yescharacteristic
22#16810520—21no—
23#16810519C2211yescharacteristic
24#16810492—21no—
25#16810492—21no—
26#16810492C2212yes—
27#16810492C2212yes—
28#16810492C2211yescharacteristic
29#16810483C2211yescharacteristic
30#16810481—21no—
31#16810386C2212yes—
32#16810386C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C22×C4⋊C4×C53C2211yescharacteristic
39C2×C22⋊C8×C53C2211yescharacteristic
40C53×#380C2211yescharacteristic
41C53×#340C2212yes—
42C53×#340C2212yes—
43C22×Dic106—21no—
44C23×Dic53—21no—
45C23×Dic53D411yescharacteristic
46C23×Dic53D411yescharacteristic
47C22×#88322C2312yes—
48#13133216—24no—
49#13133194—44no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2311yescharacteristic
56C2×C4⋊Dic53C2312yes—
57C2×#11719919—21no—
58C2×#11719919—21no—
59C2×#11719919—21no—
60C2×#11719919—21no—
61C2×#11719919—22no—
62C2×#11719919—22no—
63C2×#11719919—22no—
64C2×#11719919—22no—
65C2×#11719919—44no—
66C2×#11719919—44no—
67C2×C22⋊Dic53—21no—
68C2×C4×Dic53—24no—
69C2×C4×Dic53D411yescharacteristic
70C2×C4×Dic53D411yescharacteristic
71C2×C4×Dic53C2311yescharacteristic
72C2×C4×Dic53C2311yescharacteristic
73C22⋊#88322—22no—
74C8×Dic53—24no—
75#13133124—21no—
76#13133124—21no—
77#13133124—21no—
78C23×C4×C53—21no—
79C23×C4×C53C2311yescharacteristic
80C22×C8×C53C2312yes—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53—22no—
83C2×C4⋊C4×C53—22no—
84C2×C4⋊C4×C53C2311yescharacteristic
85C2×C4⋊C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53C2312yes—
87C2×C22⋊C4×C53—21no—
88C22⋊C8×C53—22no—
89#95×C53—21no—
90C24×C53C2×D411yescharacteristic
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—22no—
96C22×C4×C53—42no—
97C22×C4×C53—42no—
98C22×C4×C53—42no—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53#4112yes—
101C22×C4×C53#4112yes—
102C22×C4×C53C2411yescharacteristic
103C22×Dic53—22no—
104C22×Dic53—22no—
105C22×Dic53—22no—
106C22×Dic53—22no—
107C22×Dic53—21no—
108C22×Dic53—21no—
109C22×Dic53—22no—
110C22×Dic53—42no—
111C22×Dic53—42no—
112C22×Dic53—42no—
113C22×Dic53SD1614yes—
114C22×Dic53Q1614yes—
115C22×Dic53C2×D411yescharacteristic
116C22×Dic53C2×D411yescharacteristic
117C22×Dic53C2×D411yescharacteristic
118C22×Dic53C2×D411yescharacteristic
119C2×Dic106—42no—
120C2×Dic106—42no—
121C2×Dic106—42no—
122C2×C8×C53—22no—
123C2×C8×C53—22no—
124C2×C8×C53—22no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—42no—
130C22⋊C4×C53—42no—
131C22⋊Dic53—42no—
132C4⋊Dic53—22no—
133C4⋊Dic53—22no—
134C4⋊Dic53—22no—
135C4⋊Dic53—22no—
136C4⋊Dic53—42no—
137C4⋊Dic53—42no—
138#11719919—44no—
139#11719919—44no—
140#11719919—42no—
141#11719919—42no—
142#11719919—42no—
143#11719919—42no—
144#11719919—44no—
145#11719919—44no—
146C4×Dic53—22no—
147C4×Dic53—22no—
148C4×Dic53—22no—
149C4×Dic53#4114yes—
150C2×#88322—22no—
151C2×#88322—22no—
152C2×#88322—22no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53#12412yes—
158C23×C53C2×SD1612yes—
159C23×C53C2×Q1612yes—
160C23×C53C22×D411yescharacteristic
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—24no—
164C2×C4×C53—24no—
165C2×C4×C53—41no—
166C2×C4×C53—42no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—41no—
174C2×C4×C53—84no—
175C2×C4×C53#12412yes—
176C2×C4×C53C2×#4112yes—
177C2×C4×C53C2×#4111yescharacteristic
178C2×C4×C53C2×#4111yescharacteristic
179C2×C4×C532−1+412yes—
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53—21no—
183C2×Dic53—21no—
184C2×Dic53—28no—
185C2×Dic53—28no—
186C2×Dic53—41no—
187C2×Dic53—42no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—42no—
193C2×Dic53—42no—
194C2×Dic53—41no—
195C2×Dic53—84no—
196C2×Dic53#12312yes—
197C2×Dic53#12312yes—
198C2×Dic53C2×SD1612yes—
199C2×Dic53C2×SD1612yes—
200C2×Dic53C2×Q1612yes—
201C2×Dic53C2×Q1612yes—
202C2×Dic53AGL(1,ℤ/8)12yes—
203C2×Dic53AGL(1,ℤ/8)12yes—
204Dic106—84no—
205C8×C53—42no—
206#88322—42no—
207#27961—531nomaximal, Sylow
208C22×C53—22no—
209C22×C53—22no—
210C22×C53—22no—
211C22×C53—22no—
212C22×C53—22no—
213C22×C53—22no—
214C22×C53#48714yes—
215C22×C53C2×#12311yescharacteristic
216C22×C53C2×#12411yescharacteristic
217C22×C53#53611yescharacteristic
218C22×C53C22×SD1611yescharacteristic
219C22×C53C22×Q1611yescharacteristic
220C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
221C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
222C4×C53—22no—
223C4×C53—81no—
224C4×C53—82no—
225C4×C53—81no—
226C4×C53#55212yes—
227C4×C53#55212yes—
228Dic53—22no—
229Dic53—22no—
230Dic53—81no—
231Dic53—82no—
232Dic53—81no—
233Dic53#48114yes—
234Dic53#48414yes—
235#2857—531no—
236C2×#384—531no—
237#2100—532no—
238#2100—532no—
239#1990—532no—
240#1834—532no—
241#1824—531no—
242#1824—531no—
243#1824—531no—
244#1824D5311yescharacteristic
245C4×C22⋊C8—531no—
246C2×C53—24no—
247C2×C53#250311yescharacteristic
248C2×C53C2×#48111yescharacteristic
249C2×C53C2×#48411yescharacteristic
250C2×C53C2×#48711yescharacteristic
251C2×C53#313811yescharacteristic
252C2×C53#322711yescharacteristic
253C2×C53#323011yescharacteristic
254C2×C4⋊Q8—531no—
255C2×C22⋊Q8—1061no—
256C22×C4⋊C4—531no—
257C22×C4⋊C4—1061no—
258C22×C4⋊C4—1061no—
259C22×C4⋊C4C2×D5311yescharacteristic
260C22×C42—531no—
261C2×C22⋊C8—531no—
262C2×C22⋊C8C2×D5311yescharacteristic
263C2×C4×C8—532no—
264#391—1061no—
265#389—531no—
266#389—531no—
267#389—1061no—
268#384—531no—
269#384—532no—
270#384—532no—
271#384—1061no—
272#384—1061no—
273#380—531no—
274#380—1061no—
275#380C2×D5311yescharacteristic
276#340—532no—
277#340—532no—
278#340—532no—
279#340—532no—
280#340—532no—
281#340—532no—
282#340C2×D5312yes—
283#340C2×D5312yes—
284#336—532no—
285C53#2796111yescharacteristic, complemented, Sylow
286C22×Q8—1061no—
287C23×C4—21no—
288C23×C4—531no—
289C23×C4—531no—
290C23×C4—1061no—
291C23×C4C22×D5311yescharacteristic
292C22×C8—532no—
293C22×C8C22×D5312yes—
294C4⋊Q8—1064no—
295C22⋊Q8—2124no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—532no—
300C2×C4⋊C4—531no—
301C2×C4⋊C4—531no—
302C2×C4⋊C4—1062no—
303C2×C4⋊C4—1062no—
304C2×C4⋊C4—1062no—
305C2×C4⋊C4—1062no—
306C2×C4⋊C4—1062no—
307C2×C4⋊C4—1062no—
308C2×C4⋊C4—1061no—
309C2×C4⋊C4—1061no—
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×C4⋊C4C22×D5311yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—1061no—
321C2×C42—531no—
322C2×C42—531no—
323C2×C42—531no—
324C2×C42—531no—
325C2×C42—1064no—
326C22⋊C8—22no—
327C22⋊C8—1062no—
328C4×C8—1064no—
329#95—21no—
330#95—1061no—
331#95—1061no—
332#95—1061no—
333C24D4×D5311yescharacteristic
334C2×Q8—2122no—
335C2×Q8—2122no—
336C2×Q8—2122no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—531no—
346C22×C4—531no—
347C22×C4—534no—
348C22×C4—534no—
349C22×C4—531no—
350C22×C4—531no—
351C22×C4—1062no—
352C22×C4—1062no—
353C22×C4—1062no—
354C22×C4—1062no—
355C22×C4—1061no—
356C22×C4—1061no—
357C22×C4—1062no—
358C22×C4—2122no—
359C22×C4—2122no—
360C22×C4—2122no—
361C22×C4#1171994512yes—
362C22×C4D4×D5311yescharacteristic
363C22×C4#1171994712yes—
364C22×C4C23×D5311yescharacteristic
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—1062no—
369C2×C8—1062no—
370C2×C8—1062no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—1062no—
377C4⋊C4—1062no—
378C4⋊C4—1062no—
379C4⋊C4—1062no—
380C4⋊C4—2124no—
381C4⋊C4—2124no—
382C4⋊C4—2122no—
383C4⋊C4—2122no—
384C4⋊C4—2122no—
385C4⋊C4—2122no—
386C4⋊C4—2122no—
387C4⋊C4—2124no—
388C4⋊C4—2124no—
389C4⋊C4—2122no—
390C22⋊C4—42no—
391C22⋊C4—2122no—
392C42—534no—
393C42—1062no—
394C42—1062no—
395C42—1062no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23#1313320312yes—
401C23SD16×D5312yes—
402C23Q16×D5312yes—
403C23C2×D4×D5311yescharacteristic
404Q8—4244no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—84no—
419C2×C4—532no—
420C2×C4—532no—
421C2×C4—532no—
422C2×C4—532no—
423C2×C4—532no—
424C2×C4—532no—
425C2×C4—532no—
426C2×C4—532no—
427C2×C4—1068no—
428C2×C4—1068no—
429C2×C4—1061no—
430C2×C4—1061no—
431C2×C4—1061no—
432C2×C4—1061no—
433C2×C4—4244no—
434C2×C4—2121no—
435C2×C4—2122no—
436C2×C4—2121no—
437C2×C4—2121no—
438C2×C4—2122no—
439C2×C4—2122no—
440C2×C4—2121no—
441C2×C4—2121no—
442C2×C4—2121no—
443C2×C4#1313320312yes—
444C2×C4#41×D5312yes—
445C2×C4C2×#1171994711yescharacteristic
446C2×C4C2×#1171994511yescharacteristic
447C2×C4#1313326412yes—
448C8—42no—
449C8—2122no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22#1681097214yes—
457C22#123×D5311yescharacteristic
458C22C2×#1313320311yescharacteristic
459C22#1681119511yescharacteristic
460C22C2×SD16×D5311yescharacteristic
461C22C2×Q16×D5311yescharacteristic
462C22AGL(1,ℤ/8)×D5311yescharacteristic
463C22AGL(1,ℤ/8)×D5311yescharacteristic
464C4—22no—
465C4—81no—
466C4—82no—
467C4—81no—
468C4—534no—
469C4—534no—
470C4—1062no—
471C4—1062no—
472C4—4241no—
473C4—4242no—
474C4—4241no—
475C4#1681130612yes—
476C4#1681131012yes—
477C2—24no—
478C2#2027919811yescharacteristic
479C2#481×D5311yescharacteristic
480C2#484×D5311yescharacteristic
481C2C2×#1681097211yescharacteristic
482C2#2028508711yescharacteristic
483C2#2028584111yescharacteristic
484C2#2028586411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1824C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1824D53
hypocenterC53#27961
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
3. #20276281C2semidirect
4. #20276280C2semidirect
5. #20275674C2semidirect
6. #20274841C2semidirect
285. C53#27961semidirect

Maximal subgroups

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

subgroupindex
#202830652
C2×#168104922
#202762812
#202762802
#202756742
#202748412
#202748062
#202748052
#202748032
C22⋊C8×Dic532
C53×#18242
#2796153