Finite Group Databasebeta

← #27394606

Subgroups — #27394606

Order 13568.

966 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
1C2×#16810675C211yesmaximal, characteristic
2#20277119C211yesmaximal, characteristic
3#384×D53C211yesmaximal, characteristic
4#20275479C211yesmaximal, characteristic, complemented
5#20275478C211yesmaximal, characteristic, complemented
6#20275476C211yesmaximal, characteristic, complemented
7#20274909C212yesmaximal
8#20274906C212yesmaximal
9#20274870C212yesmaximal, complemented
10#20274841C212yesmaximal, complemented
11C53×#1948C211yesmaximal, characteristic, complemented
12C2×#13133174—21no—
13C2×C4⋊C4×D53—21no—
14C2×C4⋊C4×D53—21no—
15C2×C4⋊C4×D53C2212yes—
16C2×C42×D53C2211yescharacteristic
17#16810675C418yescomplemented
18C2×#13133149C2212yes—
19C2×#13133148C2212yes—
20C2×#13133148C2212yes—
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133129C2212yes—
23C2×C8×Dic53C2212yes—
24#16810493C2211yescharacteristic
25#16810492—21no—
26#16810492—21no—
27#16810492C2212yes—
28#16810492C2211yescharacteristic
29#16810492C2211yescharacteristic
30#16810487—21no—
31#16810482—21no—
32#16810480C2211yescharacteristic
33#16810480C2211yescharacteristic
34C4×#11719919—21no—
35C4×C4⋊Dic53C2211yescharacteristic
36#16810386C2212yes—
37#16810385C2212yes—
38#16810385C2212yes—
39C2×#106×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—
49#13133174—44no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53C2312yes—
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2311yescharacteristic
55C2×#11719919—22no—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×#11719919—21no—
59C2×#11719919—21no—
60C2×#11719919—21no—
61C2×#11719919—21no—
62C4⋊C4×D53—42no—
63C4⋊C4×D53—42no—
64C4⋊C4×D53—44no—
65C4⋊C4×D53—44no—
66C2×#11719921—21no—
67C2×#11719921—21no—
68C2×C4×Dic53—21no—
69C2×C4×Dic53D411yescharacteristic
70C2×C4×Dic53D411yescharacteristic
71C2×C4×Dic53C2311yescharacteristic
72C42×D53C2×C414yes—
73C42×D53Q814yes—
74#13133149—42no—
75#13133148—42no—
76#13133148—42no—
77C4⋊#88322C2×C414yescomplemented
78#13133129C2×C418yes—
79C8×Dic53C2×C418yescomplemented
80#13133124—21no—
81#13133124—21no—
82#13133124—21no—
83C22×C8×C53C2312yes—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53C2312yes—
87C2×C4⋊C4×C53C2311yescharacteristic
88C2×C4⋊C4×C53C2311yescharacteristic
89C2×C42×C53—21no—
90C2×C42×C53C2311yescharacteristic
91#106×C53—42no—
92C4⋊C8×C53C2×C414yescomplemented
93#95×C53—21no—
94C23×D53C2×D411yescharacteristic
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—22no—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2411yescharacteristic
101C22×Dic53—21no—
102C22×Dic53—21no—
103C22×Dic53—22no—
104C22×Dic53C2×D411yescharacteristic
105C22×Dic53C2×D411yescharacteristic
106C22×Dic53C2×D411yescharacteristic
107C2×C4×D53—21no—
108C2×C4×D53—21no—
109C2×C4×D53—21no—
110C2×C4×D53—21no—
111C2×C4×D53—22no—
112C2×C4×D53—22no—
113C2×C4×D53—44no—
114C2×C4×D53—44no—
115C2×C4×D53—44no—
116C2×C4×D53C4⋊C414yes—
117C2×C4×D53C4⋊C414yes—
118C2×C4×D53SD1614yes—
119C2×C4×D53SD1614yes—
120C2×C4×D53C22×C412yes—
121C2×C4×D53C2×Q812yes—
122C2×C8×C53—22no—
123C2×C8×C53—22no—
124C2×C8×C53C22×C414yes—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—42no—
132C4⋊C4×C53—42no—
133C42×C53—42no—
134C42×C53C22×C412yes—
135C42×C53C2×Q812yes—
136#11719921—42no—
137#11719921—42no—
138C4⋊Dic53—21no—
139C4⋊Dic53—22no—
140C4⋊Dic53—21no—
141C4⋊Dic53—22no—
142C4⋊Dic53—21no—
143C4⋊Dic53—21no—
144C4⋊Dic53—42no—
145C4⋊Dic53—42no—
146#11719919—44no—
147#11719919—44no—
148#11719919—42no—
149#11719919—42no—
150#11719919—44no—
151#11719919—42no—
152#11719919—42no—
153C4×Dic53—42no—
154C4×Dic53C4⋊C414yes—
155C4×Dic53C4⋊C414yes—
156C4×Dic53C22×C412yes—
157C4×Dic53C2×Q812yes—
158C2×#88322—22no—
159C2×#88322—22no—
160C2×#88322C22×C414yes—
161C23×C53C22×D411yescharacteristic
162C22×D53—21no—
163C22×D53—21no—
164C22×D53—21no—
165C22×D53—21no—
166C22×D53C2×C4⋊C412yes—
167C22×D53C2×SD1612yes—
168C22×D53C2×SD1612yes—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—42no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—42no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—42no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53C2×C4⋊C412yes—
181C2×C4×C53C2×C4⋊C412yes—
182C2×C4×C53C2×SD1612yes—
183C2×C4×C53C2×SD1612yes—
184C2×C4×C53C23×C411yescharacteristic
185C2×C4×C53C22×Q811yescharacteristic
186C2×C4×C53H(5,𝔽2)11yescharacteristic
187C2×C4×C53H(5,𝔽2)11yescharacteristic
188C2×C4×C532−1+411yescharacteristic
189C2×C4×C532−1+411yescharacteristic
190C2×Dic53—21no—
191C2×Dic53—21no—
192C2×Dic53—42no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—42no—
196C2×Dic53—41no—
197C2×Dic53—41no—
198C2×Dic53—42no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201C2×Dic53C2×C4⋊C412yes—
202C2×Dic53C2×C4⋊C412yes—
203C2×Dic53C2×C4⋊C412yes—
204C2×Dic53C2×SD1612yes—
205C2×Dic53C2×SD1612yes—
206C2×Dic53AGL(1,ℤ/8)12yes—
207C2×Dic53#13712yes—
208C4×D53—22no—
209C4×D53—22no—
210C4×D53—22no—
211C4×D53—22no—
212C4×D53—28no—
213C4×D53—88no—
214C4×D53#106116yescomplemented
215C8×C53—24no—
216#88322—24no—
217#21981—531nomaximal, Sylow
218C22×C53C22×C4⋊C411yescharacteristic
219C22×C53#53511yescharacteristic
220C22×C53#53611yescharacteristic
221C22×C53C22×SD1611yescharacteristic
222C22×C53C22×SD1611yescharacteristic
223C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
224C22×C53C2×#13711yescharacteristic
225C2×D53—22no—
226C2×D53—22no—
227C2×D53—22no—
228C2×D53—22no—
229C2×D53—22no—
230C2×D53—22no—
231C2×D53C2×#10614yes—
232C4×C53—22no—
233C4×C53—81no—
234C4×C53—82no—
235C4×C53—81no—
236C4×C53C2×#10614yes—
237C4×C53#52011yescharacteristic
238C4×C53#52011yescharacteristic
239C4×C53#54411yescharacteristic
240C4×C53#54411yescharacteristic
241Dic53—22no—
242Dic53—81no—
243Dic53—82no—
244Dic53—81no—
245Dic53C2×#10614yes—
246Dic53#42814yes—
247C2×#431—531no—
248#2285—531no—
249C2×#384—531no—
250#1948—531no—
251#1948—531no—
252#1948—531no—
253#1948D5311yescharacteristic, complemented
254#1853—532no—
255#1853—532no—
256#1844—532no—
257#1834—532no—
258C2×C53#231711yescharacteristic
259C2×C53C22×#10611yescharacteristic
260C2×C53C2×#42811yescharacteristic
261C2×C53#320611yescharacteristic
262C2×C53#320811yescharacteristic
263C2×C53#322711yescharacteristic
264C2×C53#322911yescharacteristic
265D53—24no—
266C2×#117—1061no—
267C22×C4⋊C4—532no—
268C22×C4⋊C4—1061no—
269C22×C4⋊C4—1061no—
270C22×C42—531no—
271#431—538no—
272C2×#106—532no—
273C2×#106—532no—
274C2×#106—532no—
275C2×#106C2×D5312yes—
276C2×C4⋊C8—531no—
277C2×C4⋊C8C2×D5311yescharacteristic
278C2×C22⋊C8—532no—
279C2×C4×C8—532no—
280#384—531no—
281#384—532no—
282#384—531no—
283#384—531no—
284#384—1061no—
285#384—1061no—
286#384C2×D5311yescharacteristic
287#382—1061no—
288#380—531no—
289#380—531no—
290#380—1061no—
291C4×C4⋊C4—531no—
292C4×C4⋊C4—1061no—
293C4×C4⋊C4C2×D5311yescharacteristic
294#340—532no—
295#340—532no—
296#340—532no—
297#340C2×D5312yes—
298C53#2198111yescharacteristic, complemented, Sylow
299C23×C4—531no—
300C23×C4—531no—
301C23×C4—531no—
302C23×C4—1062no—
303C22×C8—532no—
304C22×C8C22×D5312yes—
305#117—2124no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—532no—
309C2×C4⋊C4—531no—
310C2×C4⋊C4—531no—
311C2×C4⋊C4—1062no—
312C2×C4⋊C4—1062no—
313C2×C4⋊C4—1062no—
314C2×C4⋊C4—1061no—
315C2×C4⋊C4—1061no—
316C2×C4⋊C4—1061no—
317C2×C4⋊C4—1061no—
318C2×C4⋊C4—1061no—
319C2×C4⋊C4—1061no—
320C2×C4⋊C4—2122no—
321C2×C4⋊C4—2122no—
322C2×C4⋊C4—2124no—
323C2×C4⋊C4—2124no—
324C2×C4⋊C4C22×D5312yes—
325C2×C4⋊C4C22×D5311yescharacteristic
326C2×C4⋊C4C22×D5311yescharacteristic
327C2×C22⋊C4—1061no—
328C2×C22⋊C4—1061no—
329C2×C42—21no—
330C2×C42—531no—
331C2×C42—531no—
332C2×C42—531no—
333C2×C42—534no—
334C2×C42—534no—
335C2×C42—1061no—
336C2×C42C22×D5311yescharacteristic
337#106—42no—
338#106—2122no—
339#106—2122no—
340#106—2122no—
341C4⋊C8—534no—
342C4⋊C8C4×D5314yescomplemented
343C22⋊C8—538no—
344C4×C8—538no—
345#95—21no—
346#95—1061no—
347#95—1061no—
348#95—1061no—
349C24—531no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—531no—
354C22×C4—531no—
355C22×C4—534no—
356C22×C4—534no—
357C22×C4—534no—
358C22×C4—531no—
359C22×C4—534no—
360C22×C4—532no—
361C22×C4—532no—
362C22×C4—1062no—
363C22×C4—1062no—
364C22×C4—1061no—
365C22×C4—1061no—
366C22×C4—1061no—
367C22×C4—1061no—
368C22×C4—1061no—
369C22×C4—1061no—
370C22×C4—1062no—
371C22×C4—2124no—
372C22×C4—2124no—
373C22×C4—2124no—
374C22×C4D4×D5311yescharacteristic
375C22×C4D4×D5311yescharacteristic
376C22×C4C23×D5311yescharacteristic
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—534no—
380C2×C8—1062no—
381C2×C8—1062no—
382C2×C8C2×C4×D5314yes—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—1061no—
392C4⋊C4—1062no—
393C4⋊C4—1061no—
394C4⋊C4—1062no—
395C4⋊C4—1061no—
396C4⋊C4—1061no—
397C4⋊C4—2122no—
398C4⋊C4—2124no—
399C4⋊C4—2124no—
400C4⋊C4—2122no—
401C4⋊C4—2122no—
402C4⋊C4—2122no—
403C4⋊C4—2122no—
404C4⋊C4—2124no—
405C4⋊C4—2122no—
406C22⋊C4—2122no—
407C22⋊C4—2122no—
408C42—42no—
409C42—534no—
410C42—534no—
411C42—532no—
412C42—532no—
413C42—2122no—
414C42C2×C4×D5312yes—
415C42Q8×D5312yes—
416C23—532no—
417C23—532no—
418C23—532no—
419C23—1061no—
420C23—1061no—
421C23—1061no—
422C23—1061no—
423C23C2×D4×D5311yescharacteristic
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—532no—
436C2×C4—532no—
437C2×C4—532no—
438C2×C4—532no—
439C2×C4—532no—
440C2×C4—532no—
441C2×C4—5316no—
442C2×C4—532no—
443C2×C4—1068no—
444C2×C4—1061no—
445C2×C4—1061no—
446C2×C4—1062no—
447C2×C4—1062no—
448C2×C4—1062no—
449C2×C4—1062no—
450C2×C4—4248no—
451C2×C4—2121no—
452C2×C4—2121no—
453C2×C4—2122no—
454C2×C4—2122no—
455C2×C4—2121no—
456C2×C4—2121no—
457C2×C4—2122no—
458C2×C4—2121no—
459C2×C4—2121no—
460C2×C4C4⋊C4×D5312yes—
461C2×C4C4⋊C4×D5312yes—
462C2×C4SD16×D5312yes—
463C2×C4SD16×D5312yes—
464C2×C4C22×C4×D5311yescharacteristic
465C2×C4C2×Q8×D5311yescharacteristic
466C2×C4#1313326211yescharacteristic
467C2×C4#1313326311yescharacteristic
468C2×C4#1313326411yescharacteristic
469C2×C4#1313326511yescharacteristic
470C8—24no—
471C8—1064no—
472C22—534no—
473C22—1062no—
474C22—1062no—
475C22—1062no—
476C22—1062no—
477C22—1062no—
478C22—1062no—
479C22C2×C4⋊C4×D5311yescharacteristic
480C22#1681118911yescharacteristic
481C22#1681119611yescharacteristic
482C22C2×SD16×D5311yescharacteristic
483C22C2×SD16×D5311yescharacteristic
484C22AGL(1,ℤ/8)×D5311yescharacteristic
485C22#137×D5311yescharacteristic
486C4—22no—
487C4—81no—
488C4—81no—
489C4—82no—
490C4—534no—
491C4—534no—
492C4—1062no—
493C4—4241no—
494C4—4241no—
495C4—4242no—
496C4#106×D5314yes—
497C4#1681110211yescharacteristic
498C4#1681110511yescharacteristic
499C4#1681125011yescharacteristic
500C4#1681125311yescharacteristic
501C2—1064no—
502C2#2027733711yescharacteristic
503C2C2×#106×D5311yescharacteristic
504C2#428×D5311yescharacteristic
505C2#2028563911yescharacteristic
506C2#2028565511yescharacteristic
507C2#2028584211yescharacteristic
508C2#2028585811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1948C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1948D53
hypocenterC53#21981
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
4. #20275479C2semidirect
5. #20275478C2semidirect
6. #20275476C2semidirect
9. #20274870C2semidirect
10. #20274841C2semidirect
11. C53×#1948C2semidirect
17. #16810675C4semidirect
77. C4⋊#88322C2×C4semidirect
79. C8×Dic53C2×C4semidirect
92. C4⋊C8×C53C2×C4semidirect
214. C4×D53#106semidirect
253. #1948D53semidirect
298. C53#21981semidirect
342. C4⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106752
#202771192
#384×D532
#202754792
#202754782
#202754762
#202749092
#202749062
#202748702
#202748412
C53×#19482
#2198153