Finite Group Databasebeta

← #27484095

Subgroups — #27484095

Order 13568.

828 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283068C211yesmaximal, characteristic
2#20277253C211yesmaximal, characteristic
3#20276272C211yesmaximal, characteristic, complemented
4#20276049C211yesmaximal, characteristic, complemented
5#20275856C211yesmaximal, characteristic, complemented
6#20275295C212yesmaximal, complemented
7#20275135C212yesmaximal
8#20274985C212yesmaximal, complemented
9#20274737C212yesmaximal
10#336×D53C211yesmaximal, characteristic
11C53×#2096C211yesmaximal, characteristic, complemented
12C22×C8×D53C2212yes—
13C2×#13133217C2211yescharacteristic
14C2×#13133197—21no—
15C2×#13133193C2211yescharacteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×C42×D53C2211yescharacteristic
18C2×#13133149C2212yes—
19C2×#13133141C2212yes—
20#16810528—21no—
21#16810522—21no—
22#16810522C2212yes—
23#16810521C2211yescharacteristic
24#16810510—21no—
25#16810497—21no—
26#16810480—21no—
27#16810480C2211yescharacteristic
28#16810480C2212yes—
29C4⋊C4×Dic53C2211yescharacteristic
30#16810474—21no—
31#16810470—21no—
32#16810386C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810385C2212yes—
36#16810378C2211yescharacteristic
37#16810377C2212yes—
38C2×#106×C53C2212yes—
39C2×#103×C53C2212yes—
40C53×#391C2211yescharacteristic
41C53×#384C2211yescharacteristic
42C53×#336C2211yescharacteristic
43C22×Dic106—21no—
44C22×C4×D53D411yescharacteristic
45C22×C4×D53D411yescharacteristic
46C22×C4×D53C2311yescharacteristic
47C22×#88322C2312yes—
48C2×C8×D53—24no—
49C2×C8×D53—24no—
50C2×C8×D53—42no—
51#13133217—42no—
52#13133197—44no—
53#13133193—22no—
54#13133193Q814yescomplemented
55Q8×Dic53—42no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53C2311yescharacteristic
60C2×C4⋊Dic53C2312yes—
61C2×C4⋊Dic53C2311yescharacteristic
62C2×#11719919—21no—
63C2×#11719919—21no—
64C2×#11719921—21no—
65C2×#11719921—21no—
66C2×#11719921—22no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53—21no—
69C2×C4×Dic53D411yescharacteristic
70C2×C4×Dic53D411yescharacteristic
71C2×C4×Dic53C2311yescharacteristic
72C42×D53—42no—
73#13133149—42no—
74#13133141—42no—
75#13133124—21no—
76#13133124—21no—
77#13133124—21no—
78#13133124—22no—
79#13133124—22no—
80C22×Q8×C53C2311yescharacteristic
81C22×C8×C53C2312yes—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53C2312yes—
85C2×C4⋊C4×C53C2311yescharacteristic
86C2×C42×C53C2311yescharacteristic
87#106×C53—24no—
88#103×C53—24no—
89#95×C53—21no—
90#95×C53—21no—
91C23×D53C2×D411yescharacteristic
92C2×Q8×C53—21no—
93C2×Q8×C53—22no—
94C2×Q8×C53—22no—
95C2×Q8×C53C2×Q812yes—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2411yescharacteristic
102C22×Dic53—21no—
103C22×Dic53—21no—
104C22×Dic53—22no—
105C22×Dic53C2×D411yescharacteristic
106C22×Dic53C2×D411yescharacteristic
107C22×Dic53C2×D411yescharacteristic
108C2×C4×D53—22no—
109C2×C4×D53—22no—
110C2×C4×D53—22no—
111C2×C4×D53—22no—
112C2×C4×D53—41no—
113C2×C4×D53—41no—
114C2×C4×D53—41no—
115C2×C4×D53SD1614yes—
116C2×C4×D53C2×Q812yes—
117C2×C4×D53#4112yes—
118C2×Dic106—42no—
119C2×Dic106—42no—
120C2×Dic106—42no—
121C2×C8×C53—22no—
122C2×C8×C53—22no—
123C2×C8×C53#4114yes—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—42no—
128C4⋊C4×C53—42no—
129C42×C53—22no—
130#11719921—24no—
131#11719921—44no—
132#11719921—42no—
133C4⋊Dic53—21no—
134C4⋊Dic53—22no—
135C4⋊Dic53—22no—
136C4⋊Dic53—22no—
137C4⋊Dic53—42no—
138C4⋊Dic53—42no—
139C4⋊Dic53—44no—
140C4⋊Dic53C2×Q812yes—
141#11719919—24no—
142#11719919—42no—
143C4×Dic53—22no—
144C4×Dic53—41no—
145C4×Dic53—41no—
146C4×Dic53—42no—
147C4×Dic53—42no—
148C2×#88322—22no—
149C2×#88322—22no—
150C2×#88322—22no—
151C8×D53—48no—
152C23×C53C22×D411yescharacteristic
153C22×D53—21no—
154C22×D53—21no—
155C22×D53—21no—
156C22×D53—21no—
157C22×D53C22⋊Q812yes—
158C22×D53C2×SD1612yes—
159C22×D53#13712yes—
160Q8×C53—24no—
161Q8×C53—42no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—22no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—42no—
170C2×C4×C53—41no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53C2×SD1612yes—
176C2×C4×C53#13512yes—
177C2×C4×C53C22×Q811yescharacteristic
178C2×C4×C53C2×#4111yescharacteristic
179C2×C4×C53C2×#4112yes—
180C2×C4×C53H(5,𝔽2)12yes—
181C2×Dic53—21no—
182C2×Dic53—21no—
183C2×Dic53—21no—
184C2×Dic53—21no—
185C2×Dic53—22no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—42no—
192C2×Dic53—41no—
193C2×Dic53—42no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—42no—
197C2×Dic53C22⋊Q812yes—
198C2×Dic53C2×SD1612yes—
199C4×D53—24no—
200C4×D53—24no—
201C4×D53—44no—
202C4×D53—44no—
203Dic106—84no—
204C8×C53—24no—
205#88322—42no—
206#29159—531nomaximal, Sylow
207C22×C53C2×C22⋊Q811yescharacteristic
208C22×C53#53211yescharacteristic
209C22×C53#53411yescharacteristic
210C22×C53C22×SD1611yescharacteristic
211C22×C53C2×#13511yescharacteristic
212C22×C53C2×#13711yescharacteristic
213C22×C53#57511yescharacteristic
214C2×D53—22no—
215C2×D53—22no—
216C2×D53—22no—
217C2×D53—22no—
218C2×D53—22no—
219C2×D53—22no—
220C2×D53#47514yes—
221C4×C53—22no—
222C4×C53—22no—
223C4×C53—42no—
224C4×C53—82no—
225C4×C53—81no—
226C4×C53#55112yes—
227C4×C53#55412yes—
228Dic53—22no—
229Dic53—22no—
230Dic53—42no—
231Dic53—41no—
232Dic53—82no—
233Dic53—81no—
234#2857—531no—
235C4×C22⋊Q8—531no—
236#2099—531no—
237#2096D5311yescharacteristic, complemented
238#2058—531no—
239#2024—531no—
240#1919—532no—
241#1892—532no—
242#1865—532no—
243#1807—532no—
244C2×#336—531no—
245C2×C53#256411yescharacteristic
246C2×C53C2×#47511yescharacteristic
247C2×C53#307611yescharacteristic
248C2×C53#316711yescharacteristic
249C2×C53#318111yescharacteristic
250C2×C53#318711yescharacteristic
251C2×C53#319011yescharacteristic
252D53—24no—
253C23×C8—532no—
254C2×C4⋊Q8—531no—
255C2×C22⋊Q8—531no—
256C2×C22⋊Q8—1061no—
257C2×C4×Q8—531no—
258C22×C42—531no—
259C2×#106—532no—
260C2×#106C2×D5312yes—
261C2×#103—532no—
262C2×#103C2×D5312yes—
263#391—1061no—
264#391C2×D5311yescharacteristic
265#389—532no—
266#389—531no—
267#389—1061no—
268#387—1061no—
269#385—1061no—
270#384C2×D5311yescharacteristic
271#380—531no—
272#380—532no—
273#380—1061no—
274C4×C4⋊C4—531no—
275C4×C4⋊C4—1061no—
276C4×C22⋊C4—1061no—
277#340—532no—
278#340—532no—
279#340—532no—
280#340—532no—
281#336—532no—
282#336—531no—
283#336C2×D5311yescharacteristic
284C53#2915911yescharacteristic, complemented, Sylow
285C22×Q8—1061no—
286C22×Q8C22×D5311yescharacteristic
287C23×C4—531no—
288C23×C4—531no—
289C23×C4—531no—
290C22×C8—532no—
291C22×C8—1064no—
292C22×C8—1064no—
293C22×C8—2122no—
294C22×C8C22×D5312yes—
295C4⋊Q8—2122no—
296C22⋊Q8—534no—
297C22⋊Q8—1062no—
298C22⋊Q8—2124no—
299C4×Q8—2122no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—531no—
303C2×C4⋊C4—531no—
304C2×C4⋊C4—532no—
305C2×C4⋊C4—1061no—
306C2×C4⋊C4—1061no—
307C2×C4⋊C4—1062no—
308C2×C4⋊C4—1061no—
309C2×C4⋊C4—1061no—
310C2×C4⋊C4C22×D5311yescharacteristic
311C2×C4⋊C4C22×D5312yes—
312C2×C22⋊C4—1061no—
313C2×C22⋊C4—1062no—
314C2×C22⋊C4—1061no—
315C2×C42—531no—
316C2×C42—531no—
317C2×C42—531no—
318C2×C42—1061no—
319C2×C42—1061no—
320C2×C42—2122no—
321C2×C42C22×D5311yescharacteristic
322#106—24no—
323#106—2122no—
324#103—24no—
325#103—2122no—
326#95—21no—
327#95—21no—
328#95—1061no—
329#95—1061no—
330#95—1062no—
331#95—1062no—
332#95—1061no—
333C24—531no—
334C2×Q8—21no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—2122no—
338C2×Q8—2122no—
339C2×Q8—2122no—
340C2×Q8Q8×D5312yes—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—531no—
345C22×C4—531no—
346C22×C4—532no—
347C22×C4—532no—
348C22×C4—534no—
349C22×C4—531no—
350C22×C4—1062no—
351C22×C4—1062no—
352C22×C4—1062no—
353C22×C4—1061no—
354C22×C4—1062no—
355C22×C4—1061no—
356C22×C4—1062no—
357C22×C4—2121no—
358C22×C4—2121no—
359C22×C4—2121no—
360C22×C4D4×D5311yescharacteristic
361C22×C4D4×D5311yescharacteristic
362C22×C4C23×D5311yescharacteristic
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—1062no—
366C2×C8—1062no—
367C2×C8—1062no—
368C2×C8—2128no—
369C2×C8#1171994714yes—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—532no—
376C4⋊C4—1064no—
377C4⋊C4—1061no—
378C4⋊C4—1062no—
379C4⋊C4—1062no—
380C4⋊C4—1062no—
381C4⋊C4—2122no—
382C4⋊C4—2124no—
383C4⋊C4—2122no—
384C4⋊C4—2122no—
385C22⋊C4—1064no—
386C22⋊C4—2124no—
387C22⋊C4—2122no—
388C42—22no—
389C42—1062no—
390C42—2121no—
391C42—2121no—
392C42—2122no—
393C42—2122no—
394C23—532no—
395C23—532no—
396C23—532no—
397C23—1061no—
398C23—1061no—
399C23—1061no—
400C23—1061no—
401C23C2×D4×D5311yescharacteristic
402Q8—24no—
403Q8—42no—
404Q8—4244no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—532no—
419C2×C4—532no—
420C2×C4—1061no—
421C2×C4—1061no—
422C2×C4—1061no—
423C2×C4—1061no—
424C2×C4—1064no—
425C2×C4—1064no—
426C2×C4—1062no—
427C2×C4—1061no—
428C2×C4—1061no—
429C2×C4—1061no—
430C2×C4—2124no—
431C2×C4—2124no—
432C2×C4—2122no—
433C2×C4—2121no—
434C2×C4—2121no—
435C2×C4—2122no—
436C2×C4—2121no—
437C2×C4—2122no—
438C2×C4—2121no—
439C2×C4—2121no—
440C2×C4SD16×D5312yes—
441C2×C4#1313323812yes—
442C2×C4C2×Q8×D5311yescharacteristic
443C2×C4#41×D5311yescharacteristic
444C2×C4C2×#1171994712yes—
445C2×C4#1313326312yes—
446C8—24no—
447C8—2122no—
448C22—534no—
449C22—1062no—
450C22—1062no—
451C22—1062no—
452C22—1062no—
453C22—1062no—
454C22—1062no—
455C22C22⋊Q8×D5311yescharacteristic
456C22#1681117111yescharacteristic
457C22#1681118411yescharacteristic
458C22C2×SD16×D5311yescharacteristic
459C22C2×#1313323811yescharacteristic
460C22#137×D5311yescharacteristic
461C22#1681142211yescharacteristic
462C4—22no—
463C4—22no—
464C4—42no—
465C4—82no—
466C4—81no—
467C4—1062no—
468C4—1062no—
469C4—4242no—
470C4—4241no—
471C4—2121no—
472C4—2122no—
473C4#1681129612yes—
474C4#1681133112yes—
475C2—1064no—
476C2#2027989011yescharacteristic
477C2#475×D5311yescharacteristic
478C2#2028449811yescharacteristic
479C2#2028527211yescharacteristic
480C2#2028542111yescharacteristic
481C2#2028547111yescharacteristic
482C2#2028550111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2096C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2096D53
hypocenterC53#29159
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2096

Complements

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

NH ≅ G/Ntype
3. #20276272C2semidirect
4. #20276049C2semidirect
5. #20275856C2semidirect
6. #20275295C2semidirect
8. #20274985C2semidirect
11. C53×#2096C2semidirect
54. #13133193Q8semidirect
237. #2096D53semidirect
284. C53#29159semidirect

Maximal subgroups

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

subgroupindex
#202830682
#202772532
#202762722
#202760492
#202758562
#202752952
#202751352
#202749852
#202747372
#336×D532
C53×#20962
#2915953