Finite Group Databasebeta

← #27465497

Subgroups — #27465497

Order 13568.

770 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133193C211yesmaximal, characteristic
2#20277120C211yesmaximal, characteristic
3#20276150C212yesmaximal
4#20275909C212yesmaximal
5#20275479C211yesmaximal, characteristic, complemented
6#20275476C211yesmaximal, characteristic, complemented
7#20275351C212yesmaximal, complemented
8#20275298C212yesmaximal, complemented
9#20274720C211yesmaximal, characteristic, complemented
10#20274692C211yesmaximal, characteristic
11C53×#1803C211yesmaximal, characteristic, complemented
12C2×#13133193C2211yescharacteristic
13C2×#13133175—21no—
14C2×C42×D53C2211yescharacteristic
15C2×#13133149C2212yes—
16C2×#13133148C2212yes—
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133141C2212yes—
19C2×#13133129C2212yes—
20#16810530C2211yescharacteristic
21#16810519C2211yescharacteristic
22#16810504—21no—
23#16810494—21no—
24#16810493C2211yescharacteristic
25#16810488—21no—
26#16810480—21no—
27#16810480C2212yes—
28#16810480C2211yescharacteristic
29C4⋊C4×Dic53—21no—
30C4⋊C4×Dic53C2212yes—
31C4×#11719919—21no—
32C4×C4⋊Dic53C2211yescharacteristic
33C4×#11719921—21no—
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C2×C4×Q8×C53C2211yescharacteristic
39C2×C4⋊C8×C53C2211yescharacteristic
40C2×#103×C53C2212yes—
41C53×#384C2211yescharacteristic
42C53×#340C2212yes—
43C22×C4×D53D411yescharacteristic
44C22×C4×D53D411yescharacteristic
45C22×C4×D53C2311yescharacteristic
46C22×#88322C2312yes—
47#13133193—22no—
48#13133193—22no—
49#13133175—44no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53C2311yescharacteristic
53C2×C4⋊Dic53C2312yes—
54C2×C4⋊Dic53C2311yescharacteristic
55C2×#11719919—21no—
56C2×#11719919—21no—
57C2×#11719921—21no—
58C2×#11719921—22no—
59C2×#11719921—21no—
60C2×C4×Dic53—21no—
61C2×C4×Dic53—22no—
62C2×C4×Dic53D411yescharacteristic
63C2×C4×Dic53D411yescharacteristic
64C2×C4×Dic53C2311yescharacteristic
65C42×D53—22no—
66C42×D53—22no—
67#13133149—42no—
68#13133148—24no—
69C4⋊#88322—22no—
70#13133141—42no—
71#13133129—42no—
72#13133124—21no—
73#13133124—21no—
74#13133124—21no—
75#13133124—21no—
76#13133124—21no—
77#13133124—22no—
78#13133124—22no—
79C22×Q8×C53C2311yescharacteristic
80C22×C8×C53C2312yes—
81C4×Q8×C53—22no—
82C4×Q8×C53—22no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C4⋊C4×C53C2312yes—
88C2×C42×C53—21no—
89C2×C42×C53C2311yescharacteristic
90C4⋊C8×C53—22no—
91#103×C53—24no—
92C23×D53C2×D411yescharacteristic
93C2×Q8×C53—21no—
94C2×Q8×C53—21no—
95C2×Q8×C53—21no—
96C2×Q8×C53—21no—
97C2×Q8×C53—21no—
98C2×Q8×C53#4112yes—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2411yescharacteristic
105C22×Dic53—21no—
106C22×Dic53—21no—
107C22×Dic53—22no—
108C22×Dic53C2×D411yescharacteristic
109C22×Dic53C2×D411yescharacteristic
110C22×Dic53C2×D411yescharacteristic
111C2×C4×D53—22no—
112C2×C4×D53—22no—
113C2×C4×D53—22no—
114C2×C4×D53—21no—
115C2×C4×D53—21no—
116C2×C4×D53—21no—
117C2×C4×D53—21no—
118C2×C4×D53—21no—
119C2×C4×D53—21no—
120C2×C4×D53—41no—
121C2×C4×D53—41no—
122C2×C4×D53SD1614yes—
123C2×C8×C53—22no—
124C2×C8×C53—22no—
125C2×C8×C53—22no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—42no—
131C4⋊C4×C53—42no—
132C4⋊C4×C53—42no—
133C42×C53—42no—
134C42×C53#4112yes—
135C42×C53#4112yes—
136#11719921—24no—
137#11719921—42no—
138#11719921—44no—
139C4⋊Dic53—21no—
140C4⋊Dic53—22no—
141C4⋊Dic53—21no—
142C4⋊Dic53—21no—
143C4⋊Dic53—22no—
144C4⋊Dic53—42no—
145C4⋊Dic53—42no—
146C4⋊Dic53#4112yes—
147#11719919—24no—
148#11719919—42no—
149C4×Dic53—22no—
150C4×Dic53—22no—
151C4×Dic53—21no—
152C4×Dic53—21no—
153C4×Dic53—42no—
154C4×Dic53—44no—
155C2×#88322—22no—
156C2×#88322—22no—
157C2×#88322—22no—
158C23×C53C22×D411yescharacteristic
159C22×D53—21no—
160C22×D53—21no—
161C22×D53—21no—
162C22×D53—21no—
163C22×D53—21no—
164C22×D53—21no—
165C22×D53C2×SD1612yes—
166Q8×C53—22no—
167Q8×C53—22no—
168Q8×C53—41no—
169Q8×C53—41no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—22no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—42no—
181C2×C4×C53#12312yes—
182C2×C4×C53#12312yes—
183C2×C4×C53C2×SD1612yes—
184C2×C4×C53#13712yes—
185C2×C4×C53C2×#4111yescharacteristic
186C2×C4×C53C2×#4111yescharacteristic
187C2×C4×C53C2×#4111yescharacteristic
188C2×C4×C53C2×#4111yescharacteristic
189C2×C4×C53H(5,𝔽2)11yescharacteristic
190C2×C4×C532−1+411yescharacteristic
191C2×Dic53—21no—
192C2×Dic53—21no—
193C2×Dic53—21no—
194C2×Dic53—21no—
195C2×Dic53—21no—
196C2×Dic53—22no—
197C2×Dic53—21no—
198C2×Dic53—21no—
199C2×Dic53—21no—
200C2×Dic53—41no—
201C2×Dic53—41no—
202C2×Dic53—42no—
203C2×Dic53—41no—
204C2×Dic53—42no—
205C2×Dic53—41no—
206C2×Dic53—42no—
207C2×Dic53—41no—
208C2×Dic53C2×SD1612yes—
209C4×D53—28no—
210C4×D53—44no—
211C4×D53—41no—
212C4×D53—41no—
213C4×D53—41no—
214C4×D53—41no—
215C8×C53—42no—
216#88322—42no—
217#27628—531nomaximal, Sylow
218C22×C53C2×#12311yescharacteristic
219C22×C53#52511yescharacteristic
220C22×C53#53711yescharacteristic
221C22×C53C22×SD1611yescharacteristic
222C22×C53C2×#13711yescharacteristic
223C22×C53#57511yescharacteristic
224C22×C53#57511yescharacteristic
225C2×D53—22no—
226C2×D53—22no—
227C2×D53—22no—
228C2×D53—41no—
229C2×D53—41no—
230C2×D53—41no—
231C2×D53—41no—
232C4×C53—22no—
233C4×C53—42no—
234C4×C53—81no—
235C4×C53—82no—
236C4×C53#48314yes—
237C4×C53#55211yescharacteristic
238C4×C53#55311yescharacteristic
239C4×C53#55511yescharacteristic
240C4×C53#55611yescharacteristic
241Dic53—22no—
242Dic53—41no—
243Dic53—42no—
244Dic53—41no—
245Dic53—82no—
246Dic53—81no—
247C4×C22⋊Q8—531no—
248#2285—531no—
249#2077—532no—
250#2035—532no—
251#1948—531no—
252#1948—531no—
253#1927—532no—
254#1920—532no—
255#1803—531no—
256#1803D5311yescharacteristic, complemented
257#1797—531no—
258C2×C53#248511yescharacteristic
259C2×C53C2×#48311yescharacteristic
260C2×C53#309311yescharacteristic
261C2×C53#310311yescharacteristic
262C2×C53#311611yescharacteristic
263C2×C53#324611yescharacteristic
264C2×C53#325211yescharacteristic
265D53—42no—
266C2×C22⋊Q8—531no—
267C2×C4×Q8C2×D5311yescharacteristic
268C2×#117—1061no—
269C22×C42—531no—
270C2×#106—532no—
271C2×#106—532no—
272C2×C4⋊C8—531no—
273C2×C4⋊C8C2×D5311yescharacteristic
274C2×#103—532no—
275C2×#103C2×D5312yes—
276C2×C22⋊C8—532no—
277#391—531no—
278#389—531no—
279#387—1061no—
280#385—1061no—
281#384—531no—
282#384C2×D5311yescharacteristic
283#382—1061no—
284#380—531no—
285#380—532no—
286#380—1061no—
287C4×C4⋊C4—532no—
288C4×C4⋊C4—531no—
289C4×C4⋊C4—1061no—
290C4×C4⋊C4—1061no—
291C4×C22⋊C4—1061no—
292#340—532no—
293#340—532no—
294#340—532no—
295#340C2×D5312yes—
296#336—532no—
297C53#2762811yescharacteristic, complemented, Sylow
298C22×Q8C22×D5311yescharacteristic
299C23×C4—531no—
300C23×C4—531no—
301C23×C4—531no—
302C22×C8—532no—
303C22×C8C22×D5312yes—
304C22⋊Q8—1062no—
305C22⋊Q8—1062no—
306C4×Q8—22no—
307C4×Q8—22no—
308#117—2124no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—531no—
313C2×C4⋊C4—532no—
314C2×C4⋊C4—531no—
315C2×C4⋊C4—1061no—
316C2×C4⋊C4—1061no—
317C2×C4⋊C4—1061no—
318C2×C4⋊C4—1061no—
319C2×C4⋊C4C22×D5311yescharacteristic
320C2×C4⋊C4C22×D5312yes—
321C2×C22⋊C4—1061no—
322C2×C22⋊C4—1062no—
323C2×C22⋊C4—1061no—
324C2×C42—21no—
325C2×C42—531no—
326C2×C42—531no—
327C2×C42—531no—
328C2×C42—1062no—
329C2×C42—1062no—
330C2×C42—1062no—
331C2×C42—1061no—
332C2×C42C22×D5311yescharacteristic
333#106—1064no—
334#106—2122no—
335C4⋊C8—22no—
336C4⋊C8—1062no—
337#103—24no—
338#103—2122no—
339C22⋊C8—2122no—
340#95—1061no—
341#95—1061no—
342#95—1062no—
343#95—1062no—
344#95—1061no—
345#95—1061no—
346#95—1061no—
347C24—531no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8#1171994712yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—531no—
358C22×C4—534no—
359C22×C4—531no—
360C22×C4—531no—
361C22×C4—1062no—
362C22×C4—1062no—
363C22×C4—1061no—
364C22×C4—1062no—
365C22×C4—1061no—
366C22×C4—1061no—
367C22×C4—1061no—
368C22×C4—1061no—
369C22×C4—1061no—
370C22×C4—1061no—
371C22×C4—1061no—
372C22×C4—1062no—
373C22×C4—2121no—
374C22×C4—2121no—
375C22×C4D4×D5311yescharacteristic
376C22×C4D4×D5311yescharacteristic
377C22×C4C23×D5311yescharacteristic
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—1062no—
382C2×C8—1062no—
383C2×C8—1062no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—532no—
392C4⋊C4—1064no—
393C4⋊C4—1061no—
394C4⋊C4—1062no—
395C4⋊C4—1061no—
396C4⋊C4—1062no—
397C4⋊C4—1061no—
398C4⋊C4—2122no—
399C4⋊C4—2122no—
400C4⋊C4—2122no—
401C22⋊C4—1064no—
402C22⋊C4—2124no—
403C22⋊C4—2122no—
404C42—42no—
405C42—1062no—
406C42—1062no—
407C42—1061no—
408C42—1061no—
409C42—2124no—
410C42—2122no—
411C42#1171994512yes—
412C42#1171994712yes—
413C23—532no—
414C23—1061no—
415C23—1061no—
416C23—1061no—
417C23—1061no—
418C23—1061no—
419C23—1061no—
420C23C2×D4×D5311yescharacteristic
421Q8—22no—
422Q8—22no—
423Q8—41no—
424Q8—41no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—532no—
437C2×C4—1061no—
438C2×C4—1061no—
439C2×C4—1068no—
440C2×C4—1061no—
441C2×C4—1061no—
442C2×C4—1062no—
443C2×C4—1061no—
444C2×C4—1061no—
445C2×C4—1061no—
446C2×C4—1061no—
447C2×C4—2124no—
448C2×C4—2122no—
449C2×C4—2121no—
450C2×C4—2121no—
451C2×C4—2122no—
452C2×C4—2122no—
453C2×C4—2121no—
454C2×C4—2121no—
455C2×C4—2121no—
456C2×C4—2121no—
457C2×C4—2121no—
458C2×C4—2121no—
459C2×C4—2121no—
460C2×C4#1313319912yes—
461C2×C4#1313319912yes—
462C2×C4SD16×D5312yes—
463C2×C4#1313324912yes—
464C2×C4#41×D5311yescharacteristic
465C2×C4C2×#1171994711yescharacteristic
466C2×C4C2×#1171994711yescharacteristic
467C2×C4C2×#1171994511yescharacteristic
468C2×C4#1313326311yescharacteristic
469C2×C4#1313326511yescharacteristic
470C8—42no—
471C8—2122no—
472C22—1062no—
473C22—1062no—
474C22—1062no—
475C22—2121no—
476C22—2121no—
477C22—2121no—
478C22—2121no—
479C22C2×#1313319911yescharacteristic
480C22#1681113311yescharacteristic
481C22#1681120311yescharacteristic
482C22C2×SD16×D5311yescharacteristic
483C22C2×#1313324911yescharacteristic
484C22#1681141911yescharacteristic
485C22#1681142211yescharacteristic
486C4—22no—
487C4—42no—
488C4—82no—
489C4—81no—
490C4—1062no—
491C4—4242no—
492C4—4241no—
493C4—2122no—
494C4—2121no—
495C4—2121no—
496C4#1681095214yes—
497C4#1681130811yescharacteristic
498C4#1681132511yescharacteristic
499C4#1681133611yescharacteristic
500C4#1681134411yescharacteristic
501C2—2122no—
502C2#2027900811yescharacteristic
503C2C2×#1681095211yescharacteristic
504C2#2028464511yescharacteristic
505C2#2028474511yescharacteristic
506C2#2028487611yescharacteristic
507C2#2028603011yescharacteristic
508C2#2028609111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1803C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#1803D53
hypocenterC53#27628
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
5. #20275479C2semidirect
6. #20275476C2semidirect
7. #20275351C2semidirect
8. #20275298C2semidirect
9. #20274720C2semidirect
11. C53×#1803C2semidirect
256. #1803D53semidirect
297. C53#27628semidirect

Maximal subgroups

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

subgroupindex
C4×#131331932
#202771202
#202761502
#202759092
#202754792
#202754762
#202753512
#202752982
#202747202
#202746922
C53×#18032
#2762853