Finite Group Databasebeta

← #27394658

Subgroups — #27394658

Order 13568.

902 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810679C211yesmaximal, characteristic
2#384×D53C211yesmaximal, characteristic
3#20277111C211yesmaximal, characteristic
4#20275479C211yesmaximal, characteristic, complemented
5#20275476C211yesmaximal, characteristic, complemented
6#20275473C211yesmaximal, characteristic, complemented
7#20274909C212yesmaximal
8#20274901C212yesmaximal
9#20274882C212yesmaximal, complemented
10#20274847C212yesmaximal, complemented
11C53×#1947C211yesmaximal, characteristic, complemented
12C2×C4⋊C4×D53—21no—
13C2×C4⋊C4×D53—21no—
14C2×C4⋊C4×D53C2212yes—
15C2×C42×D53C2211yescharacteristic
16#16810679C418yescomplemented
17C2×#13133150C2212yes—
18C2×#13133149C2212yes—
19C2×#13133148C2212yes—
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133129C2212yes—
22C2×#13133127C2212yes—
23#16810493C2211yescharacteristic
24#16810492—21no—
25#16810492—21no—
26#16810492C2212yes—
27#16810490—21no—
28#16810488C2211yescharacteristic
29#16810488C2211yescharacteristic
30#16810485—21no—
31#16810480C2211yescharacteristic
32#16810480C2211yescharacteristic
33C4×C4⋊Dic53C2211yescharacteristic
34C4×#11719921—21no—
35#16810469—21no—
36#16810386C2212yes—
37#16810385C2212yes—
38#16810385C2212yes—
39C2×#107×C53C2212yes—
40C2×C4⋊C8×C53C2211yescharacteristic
41C53×#384C2211yescharacteristic
42C4×C4⋊C4×C53C2211yescharacteristic
43C53×#340C2212yes—
44C22×C4×D53—22no—
45C22×C4×D53D411yescharacteristic
46C22×C4×D53D411yescharacteristic
47C22×C4×D53C2311yescharacteristic
48C22×#88322C2312yes—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53C2312yes—
52C2×C4⋊Dic53C2311yescharacteristic
53C2×C4⋊Dic53C2311yescharacteristic
54C2×#11719919—22no—
55C2×#11719919—22no—
56C2×#11719919—22no—
57C4⋊C4×D53—42no—
58C4⋊C4×D53—42no—
59C4⋊C4×D53—44no—
60C4⋊C4×D53—44no—
61C2×#11719921—21no—
62C2×#11719921—21no—
63C2×C4×Dic53—21no—
64C2×C4×Dic53D411yescharacteristic
65C2×C4×Dic53D411yescharacteristic
66C2×C4×Dic53C2311yescharacteristic
67C42×D53C2×C414yes—
68C42×D53Q814yes—
69#13133150—42no—
70#13133149—42no—
71#13133148—42no—
72C4⋊#88322C2×C414yescomplemented
73#13133129C2×C418yes—
74#13133127C2×C418yescomplemented
75#13133124—21no—
76#13133124—21no—
77#13133124—21no—
78#13133124—21no—
79#13133124—21no—
80#13133124—21no—
81#13133124—21no—
82C22×C8×C53C2312yes—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53C2312yes—
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C4⋊C4×C53C2311yescharacteristic
88C2×C42×C53—21no—
89C2×C42×C53C2311yescharacteristic
90#107×C53—42no—
91C4⋊C8×C53C2×C414yescomplemented
92#95×C53—21no—
93C23×D53C2×D411yescharacteristic
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—22no—
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2411yescharacteristic
100C22×Dic53—21no—
101C22×Dic53—21no—
102C22×Dic53—22no—
103C22×Dic53C2×D411yescharacteristic
104C22×Dic53C2×D411yescharacteristic
105C22×Dic53C2×D411yescharacteristic
106C2×C4×D53—22no—
107C2×C4×D53—22no—
108C2×C4×D53—22no—
109C2×C4×D53—22no—
110C2×C4×D53—21no—
111C2×C4×D53—21no—
112C2×C4×D53—21no—
113C2×C4×D53—21no—
114C2×C4×D53—44no—
115C2×C4×D53—44no—
116C2×C4×D53—44no—
117C2×C4×D53C4⋊C414yes—
118C2×C4×D53C4⋊C414yes—
119C2×C4×D53C22×C412yes—
120C2×C4×D53C2×Q812yes—
121C2×C8×C53—22no—
122C2×C8×C53—22no—
123C2×C8×C53C22×C414yes—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—42no—
131C4⋊C4×C53—42no—
132C42×C53—42no—
133C42×C53C22×C412yes—
134C42×C53C2×Q812yes—
135#11719921—42no—
136#11719921—42no—
137C4⋊Dic53—21no—
138C4⋊Dic53—21no—
139C4⋊Dic53—22no—
140C4⋊Dic53—22no—
141C4⋊Dic53—21no—
142C4⋊Dic53—21no—
143C4⋊Dic53—42no—
144C4⋊Dic53—42no—
145#11719919—44no—
146#11719919—44no—
147#11719919—44no—
148C4×Dic53—42no—
149C4×Dic53C4⋊C414yes—
150C4×Dic53C4⋊C414yes—
151C4×Dic53C22×C412yes—
152C4×Dic53C2×Q812yes—
153C2×#88322—22no—
154C2×#88322—22no—
155C2×#88322C22×C414yes—
156C23×C53C22×D411yescharacteristic
157C22×D53—21no—
158C22×D53—21no—
159C22×D53—21no—
160C22×D53—21no—
161C22×D53C2×C4⋊C412yes—
162C22×D53AGL(1,ℤ/8)12yes—
163C22×D53#13712yes—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—42no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—42no—
170C2×C4×C53—42no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53C2×C4⋊C412yes—
176C2×C4×C53C2×C4⋊C412yes—
177C2×C4×C53AGL(1,ℤ/8)12yes—
178C2×C4×C53#13712yes—
179C2×C4×C53C23×C411yescharacteristic
180C2×C4×C53C22×Q811yescharacteristic
181C2×C4×C53H(5,𝔽2)11yescharacteristic
182C2×C4×C53H(5,𝔽2)11yescharacteristic
183C2×C4×C532−1+411yescharacteristic
184C2×C4×C532−1+411yescharacteristic
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—21no—
190C2×Dic53—21no—
191C2×Dic53—42no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—42no—
195C2×Dic53—42no—
196C2×Dic53—41no—
197C2×Dic53—41no—
198C2×Dic53—41no—
199C2×Dic53—41no—
200C2×Dic53C2×C4⋊C412yes—
201C2×Dic53C2×C4⋊C412yes—
202C2×Dic53C2×C4⋊C412yes—
203C4×D53—28no—
204C4×D53—22no—
205C4×D53—22no—
206C4×D53—22no—
207C4×D53—22no—
208C4×D53—28no—
209C4×D53—88no—
210C8×C53—24no—
211#88322—24no—
212#21987—531nomaximal, Sylow
213C22×C53C22×C4⋊C411yescharacteristic
214C22×C53#53511yescharacteristic
215C22×C53#53611yescharacteristic
216C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
217C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
218C22×C53C2×#13711yescharacteristic
219C22×C53C2×#13711yescharacteristic
220C2×D53—22no—
221C2×D53—22no—
222C2×D53—22no—
223C2×D53—22no—
224C2×D53—22no—
225C2×D53—22no—
226C2×D53#42814yes—
227C4×C53—22no—
228C4×C53—82no—
229C4×C53—81no—
230C4×C53—81no—
231C4×C53#42814yes—
232C4×C53#52011yescharacteristic
233C4×C53#52011yescharacteristic
234C4×C53#54411yescharacteristic
235C4×C53#54411yescharacteristic
236Dic53—22no—
237Dic53—22no—
238Dic53—22no—
239Dic53—82no—
240Dic53—81no—
241Dic53—81no—
242C2×#432—531no—
243C2×#384—531no—
244#2283—531no—
245#1948—531no—
246#1948—531no—
247#1947—531no—
248#1947D5311yescharacteristic, complemented
249#1853—532no—
250#1852—532no—
251#1847—532no—
252#1836—532no—
253C2×C53#231711yescharacteristic
254C2×C53C2×#42811yescharacteristic
255C2×C53C2×#42811yescharacteristic
256C2×C53#321711yescharacteristic
257C2×C53#321811yescharacteristic
258C2×C53#323811yescharacteristic
259C2×C53#323911yescharacteristic
260D53—24no—
261C22×C4⋊C4—532no—
262C22×C4⋊C4—1061no—
263C22×C4⋊C4—1061no—
264C22×C42—531no—
265#432—538no—
266C2×#107—532no—
267C2×#107C2×D5312yes—
268C2×#106—532no—
269C2×#106—532no—
270C2×C4⋊C8—531no—
271C2×C4⋊C8C2×D5311yescharacteristic
272C2×C22⋊C8—532no—
273C2×#97—532no—
274#384—532no—
275#384—531no—
276#384—1061no—
277#384—1061no—
278#384C2×D5311yescharacteristic
279#383—1061no—
280#382—531no—
281#382—531no—
282#381—1061no—
283#380—531no—
284#380—531no—
285C4×C4⋊C4—531no—
286C4×C4⋊C4C2×D5311yescharacteristic
287C4×C22⋊C4—1061no—
288#376—1061no—
289#340—532no—
290#340—532no—
291#340—532no—
292#340C2×D5312yes—
293C53#2198711yescharacteristic, complemented, Sylow
294C23×C4—531no—
295C23×C4—531no—
296C23×C4—531no—
297C23×C4—1062no—
298C22×C8—532no—
299C22×C8C22×D5312yes—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—532no—
303C2×C4⋊C4—531no—
304C2×C4⋊C4—531no—
305C2×C4⋊C4—1062no—
306C2×C4⋊C4—1062no—
307C2×C4⋊C4—1062no—
308C2×C4⋊C4—1061no—
309C2×C4⋊C4—1061no—
310C2×C4⋊C4—2122no—
311C2×C4⋊C4—2122no—
312C2×C4⋊C4—2124no—
313C2×C4⋊C4—2124no—
314C2×C4⋊C4C22×D5312yes—
315C2×C4⋊C4C22×D5311yescharacteristic
316C2×C4⋊C4C22×D5311yescharacteristic
317C2×C22⋊C4—1061no—
318C2×C22⋊C4—1061no—
319C2×C42—21no—
320C2×C42—531no—
321C2×C42—531no—
322C2×C42—531no—
323C2×C42—534no—
324C2×C42—534no—
325C2×C42—1061no—
326C2×C42C22×D5311yescharacteristic
327#107—42no—
328#107—2122no—
329#106—2122no—
330#106—2122no—
331C4⋊C8—534no—
332C4⋊C8C4×D5314yescomplemented
333C22⋊C8—538no—
334#97—538no—
335#95—21no—
336#95—1061no—
337#95—1061no—
338#95—1061no—
339#95—1061no—
340#95—1061no—
341#95—1061no—
342#95—1061no—
343C24—531no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—531no—
348C22×C4—531no—
349C22×C4—531no—
350C22×C4—534no—
351C22×C4—534no—
352C22×C4—532no—
353C22×C4—532no—
354C22×C4—1062no—
355C22×C4—1062no—
356C22×C4—1062no—
357C22×C4—1062no—
358C22×C4—1061no—
359C22×C4—1061no—
360C22×C4—1061no—
361C22×C4—1061no—
362C22×C4—1061no—
363C22×C4—1061no—
364C22×C4—1062no—
365C22×C4—2124no—
366C22×C4—2124no—
367C22×C4—2124no—
368C22×C4D4×D5311yescharacteristic
369C22×C4D4×D5311yescharacteristic
370C22×C4C23×D5311yescharacteristic
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—534no—
374C2×C8—1062no—
375C2×C8—1062no—
376C2×C8C2×C4×D5314yes—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—1061no—
386C4⋊C4—1062no—
387C4⋊C4—1061no—
388C4⋊C4—1062no—
389C4⋊C4—1061no—
390C4⋊C4—1061no—
391C4⋊C4—2122no—
392C4⋊C4—2124no—
393C4⋊C4—2124no—
394C4⋊C4—2124no—
395C4⋊C4—2122no—
396C22⋊C4—2122no—
397C22⋊C4—2122no—
398C42—42no—
399C42—534no—
400C42—534no—
401C42—532no—
402C42—532no—
403C42—2122no—
404C42C2×C4×D5312yes—
405C42Q8×D5312yes—
406C23—532no—
407C23—532no—
408C23—532no—
409C23—1061no—
410C23—1061no—
411C23—1061no—
412C23—1061no—
413C23C2×D4×D5311yescharacteristic
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—532no—
426C2×C4—532no—
427C2×C4—532no—
428C2×C4—1068no—
429C2×C4—1061no—
430C2×C4—1061no—
431C2×C4—1061no—
432C2×C4—1061no—
433C2×C4—1061no—
434C2×C4—1061no—
435C2×C4—1062no—
436C2×C4—1062no—
437C2×C4—1068no—
438C2×C4—1062no—
439C2×C4—1062no—
440C2×C4—4248no—
441C2×C4—2121no—
442C2×C4—2122no—
443C2×C4—2122no—
444C2×C4—2121no—
445C2×C4—2121no—
446C2×C4—2121no—
447C2×C4—2122no—
448C2×C4—2121no—
449C2×C4—2121no—
450C2×C4C4⋊C4×D5312yes—
451C2×C4C4⋊C4×D5312yes—
452C2×C4#1313324112yes—
453C2×C4#1313324912yes—
454C2×C4C22×C4×D5311yescharacteristic
455C2×C4C2×Q8×D5311yescharacteristic
456C2×C4#1313326211yescharacteristic
457C2×C4#1313326311yescharacteristic
458C2×C4#1313326411yescharacteristic
459C2×C4#1313326511yescharacteristic
460C8—24no—
461C8—1064no—
462C22—534no—
463C22—1062no—
464C22—1062no—
465C22—1062no—
466C22—1062no—
467C22—1062no—
468C22—1062no—
469C22C2×C4⋊C4×D5311yescharacteristic
470C22#1681118911yescharacteristic
471C22#1681119611yescharacteristic
472C22AGL(1,ℤ/8)×D5311yescharacteristic
473C22C2×#1313324111yescharacteristic
474C22#137×D5311yescharacteristic
475C22C2×#1313324911yescharacteristic
476C4—22no—
477C4—82no—
478C4—81no—
479C4—81no—
480C4—1062no—
481C4—1062no—
482C4—1062no—
483C4—4242no—
484C4—4241no—
485C4—4241no—
486C4#1681066414yes—
487C4#1681110211yescharacteristic
488C4#1681110511yescharacteristic
489C4#1681125011yescharacteristic
490C4#1681125311yescharacteristic
491C2—1064no—
492C2#2027733711yescharacteristic
493C2#428×D5311yescharacteristic
494C2C2×#1681066411yescharacteristic
495C2#2028573211yescharacteristic
496C2#2028574611yescharacteristic
497C2#2028594011yescharacteristic
498C2#2028594411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1947C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1947D53
hypocenterC53#21987
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
4. #20275479C2semidirect
5. #20275476C2semidirect
6. #20275473C2semidirect
9. #20274882C2semidirect
10. #20274847C2semidirect
11. C53×#1947C2semidirect
16. #16810679C4semidirect
72. C4⋊#88322C2×C4semidirect
74. #13133127C2×C4semidirect
91. C4⋊C8×C53C2×C4semidirect
248. #1947D53semidirect
293. C53#21987semidirect
332. C4⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106792
#384×D532
#202771112
#202754792
#202754762
#202754732
#202749092
#202749012
#202748822
#202748472
C53×#19472
#2198753