Finite Group Databasebeta

← #27539215

Subgroups — #27539215

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285365C211yesmaximal, characteristic, complemented
2#20276900C211yesmaximal, characteristic, complemented
3#20276895C211yesmaximal, characteristic, complemented
4#20276879C211yesmaximal, characteristic, complemented
5#20276872C211yesmaximal, characteristic, complemented
6#20276755C211yesmaximal, characteristic, complemented
7#20276750C211yesmaximal, characteristic, complemented
8#20276638C211yesmaximal, characteristic, complemented
9#20276638C211yesmaximal, characteristic, complemented
10#20276589C211yesmaximal, characteristic, complemented
11#20276577C211yesmaximal, characteristic, complemented
12C4×#13133160C211yesmaximal, characteristic, complemented
13C4×#13133158C211yesmaximal, characteristic, complemented
14#20276522C211yesmaximal, characteristic, complemented
15C53×#2951C211yesmaximal, characteristic, complemented
16#16811228—21no—
17C2×#13133160C2211yescharacteristic
18C2×#13133160C2211yescharacteristic
19C2×#13133158C2211yescharacteristic
20C2×#13133158C2211yescharacteristic
21#16811025C2211yescharacteristic
22#16811000C2211yescharacteristic
23#16810988—21no—
24#16810876—21no—
25#16810859C2211yescharacteristic
26#16810859C2211yescharacteristic
27#16810736—21no—
28C4×D424C2211yescharacteristic
29C4×D424C2211yescharacteristic
30#16810450C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810448C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35C4⋊#11719908C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810424C2211yescharacteristic
39#16810424C2211yescharacteristic
40#16810423C2211yescharacteristic
41#16810423C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44C22⋊#11719908C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46#16810399C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C2×#135×C53C2211yescharacteristic
49C53×#520—21no—
50C53×#440—21no—
51C53×#440—21no—
52C4×Q16×C53C2211yescharacteristic
53C4×Q16×C53C2211yescharacteristic
54C4×D8×C53C2211yescharacteristic
55C4×D8×C53C2211yescharacteristic
56C2×#107×C53C2211yescharacteristic
57C53×#417—21no—
58C53×#405C2211yescharacteristic
59C2×D424—21no—
60C2×D424C2311yescharacteristic
61C2×D424C2311yescharacteristic
62#13133214—21no—
63#13133209—21no—
64#13133202—41no—
65#13133202—41no—
66#13133191—21no—
67#13133191—21no—
68#13133190—41no—
69#13133190—41no—
70C4×D212—21no—
71C4×D212—21no—
72#13133160—22no—
73#13133160—22no—
74#13133158—22no—
75#13133158—22no—
76C2×#11719908C2311yescharacteristic
77C2×#11719908C2311yescharacteristic
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80#13133151C2311yescharacteristic
81#13133151C2311yescharacteristic
82#13133149—21no—
83#13133139—21no—
84#13133139—21no—
85#13133139—21no—
86#13133139—21no—
87C2×#41×C53—21no—
88#135×C53—22no—
89#135×C53—22no—
90C2×Q16×C53C2311yescharacteristic
91C2×SD16×C53—21no—
92C2×D8×C53C2311yescharacteristic
93C2×M16×C53D411yescharacteristic
94C22×C8×C53C2311yescharacteristic
95C4×Q8×C53—21no—
96C4×Q8×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—41no—
100C4×D4×C53—41no—
101#117×C53—41no—
102#117×C53D411yescharacteristic
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—41no—
105#107×C53—21no—
106#107×C53C2311yescharacteristic
107#107×C53C2311yescharacteristic
108#103×C53—21no—
109#103×C53—21no—
110#102×C53—21no—
111#102×C53—21no—
112#97×C53D411yescharacteristic
113#97×C53D411yescharacteristic
114C4×C8×C53C2311yescharacteristic
115C4×C8×C53C2311yescharacteristic
116#41×C53—24no—
117#41×C53—42no—
118C2×Q8×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—41no—
121C22×C4×C53—41no—
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53C2×D411yescharacteristic
126C2×C22⋊D53—41no—
127C2×C22⋊D53—41no—
128C2×D212—21no—
129C2×D212—21no—
130C2×D212—41no—
131C2×D212—41no—
132C2×C4×D53—41no—
133C2×C4×D53—41no—
134Q16×C53—21no—
135Q16×C53#4112yes—
136SD16×C53—24no—
137D8×C53—21no—
138D8×C53#4112yes—
139M16×C53—41no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C2×C8×C53C2×D411yescharacteristic
143C2×C8×C53C2×D411yescharacteristic
144C2×C8×C53C2×D411yescharacteristic
145C2×C8×C53C2411yescharacteristic
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—41no—
149C4⋊C4×C53—41no—
150C4⋊C4×C53—41no—
151C4⋊C4×C53D811yescharacteristic
152C4⋊C4×C53D811yescharacteristic
153C22⋊C4×C53—41no—
154C22⋊C4×C53—41no—
155C22⋊C4×C53D811yescharacteristic
156C22⋊C4×C53D811yescharacteristic
157C42×C53—41no—
158C42×C53—41no—
159C42×C53C2×D411yescharacteristic
160C42×C53C2×D411yescharacteristic
161#11719921—41no—
162#11719921—41no—
163#11719921—41no—
164#11719921—41no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167#11719919—41no—
168#11719919—41no—
169D424—21no—
170D424—21no—
171D424—21no—
172D424—21no—
173D424—42no—
174#11719908—21no—
175#11719908—21no—
176#11719908—21no—
177#11719908—21no—
178C23×C53—41no—
179C23×C53C2×D811yescharacteristic
180C22×D53—41no—
181C22×D53—41no—
182Q8×C53—22no—
183Q8×C53—41no—
184D4×C53—22no—
185D4×C53—44no—
186D4×C53—41no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—44no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—161no—
194C2×C4×C53—161no—
195C2×C4×C53—161no—
196C2×C4×C53C2×D811yescharacteristic
197C2×C4×C53C2×D811yescharacteristic
198C2×C4×C53C2×D811yescharacteristic
199C2×C4×C53C2×D811yescharacteristic
200C2×C4×C53C2×D811yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C22⋊D53—161no—
203C22⋊D53—161no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206D212—41no—
207D212—41no—
208D212—41no—
209D212—41no—
210D212—161no—
211D212—161no—
212C4×D53—161no—
213C4×D53—161no—
214C8×C53—21no—
215C8×C53—21no—
216C8×C53—21no—
217C8×C53C2×#4111yescharacteristic
218C8×C53C2×#4111yescharacteristic
219#33679—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—42no—
224C22×C53—81no—
225C22×C53—81no—
226C22×C53C22×D811yescharacteristic
227C2×D53—81no—
228C2×D53—81no—
229C2×D53—81no—
230C2×D53—81no—
231C2×D53—81no—
232C2×D53—81no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—42no—
239C4×C53—81no—
240C4×C53—81no—
241C4×C53#52511yescharacteristic
242C4×C53#57511yescharacteristic
243Dic53—81no—
244Dic53—81no—
245#3176—531no—
246#2951D5311yescharacteristic, complemented
247#2233—531no—
248#2232—531no—
249#2229—531no—
250#2228—531no—
251#2207—531no—
252#2206—531no—
253#2186—531no—
254#2186—531no—
255#2177—531no—
256#2175—531no—
257C4×SD32—531no—
258C4×D16—531no—
259#2164—531no—
260C2×C53—41no—
261C2×C53—82no—
262C2×C53#310011yescharacteristic
263C2×C53#341511yescharacteristic
264C2×C53#341611yescharacteristic
265D53—161no—
266D53—161no—
267C2×#135C2×D5311yescharacteristic
268#540—1061no—
269#520—21no—
270C2×SD32—531no—
271C2×SD32—531no—
272C2×D16—531no—
273C2×D16—531no—
274#499—531no—
275#493—531no—
276#490—1061no—
277#469—1061no—
278#466—531no—
279#466—531no—
280#442—1061no—
281#440—21no—
282#440—21no—
283C4×Q16C2×D5311yescharacteristic
284C4×Q16C2×D5311yescharacteristic
285C4×D8—531no—
286C4×D8—531no—
287C4×D8C2×D5311yescharacteristic
288C4×D8C2×D5311yescharacteristic
289C2×#107C2×D5311yescharacteristic
290#417—21no—
291#405C2×D5311yescharacteristic
292#367—531no—
293#367—531no—
294#366—531no—
295#366—531no—
296C4⋊C16—531no—
297C4⋊C16—531no—
298#358—531no—
299#358—531no—
300#357—531no—
301#357—531no—
302#357—531no—
303#357—531no—
304#357—531no—
305#357—531no—
306C22⋊C16—531no—
307C22⋊C16—531no—
308#346—531no—
309C4×C16—531no—
310C53#3367911yescharacteristic, complemented, Sylow
311C2×#41—21no—
312#135—22no—
313#135—22no—
314C2×Q16C22×D5311yescharacteristic
315C2×SD16—21no—
316C2×D8—531no—
317C2×D8—531no—
318C2×D8—1061no—
319C2×D8C22×D5311yescharacteristic
320C2×M16C22⋊D5311yescharacteristic
321C22×C8C22×D5311yescharacteristic
322#127—1061no—
323#125—1061no—
324#123—2121no—
325#123—2121no—
326#121—1061no—
327#121—1061no—
328#121—2121no—
329#121—2121no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—41no—
335C4×D4—41no—
336C4×D4—1061no—
337C4×D4—1061no—
338#117—41no—
339#117C22⋊D5311yescharacteristic
340C2×C4⋊C4—21no—
341C2×C4⋊C4—41no—
342SD32—1062no—
343SD32—1062no—
344D16—1062no—
345D16—1062no—
346C2×C16—531no—
347C2×C16—531no—
348C2×C16—531no—
349C2×C16—531no—
350#107—21no—
351#107—531no—
352#107—531no—
353#107C22×D5311yescharacteristic
354#107C22×D5311yescharacteristic
355#106—1061no—
356#103—21no—
357#103—21no—
358#102—21no—
359#102—21no—
360#102—1061no—
361#102—1061no—
362#102—1061no—
363#102—1061no—
364#97C22⋊D5311yescharacteristic
365#97C22⋊D5311yescharacteristic
366C4×C8C22×D5311yescharacteristic
367C4×C8C22×D5311yescharacteristic
368#41—24no—
369#41—42no—
370C2×Q8—21no—
371C2×D4—21no—
372C2×D4—41no—
373C2×D4—1061no—
374C2×D4—1061no—
375C2×D4—2121no—
376C2×D4—2121no—
377C2×D4—2121no—
378C2×D4—2121no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—2121no—
384C22×C4—2121no—
385C22×C4C2×C22⋊D5311yescharacteristic
386Q16—21no—
387Q16#1171994512yes—
388SD16—24no—
389D8—21no—
390D8—1061no—
391D8—1061no—
392D8—1061no—
393D8—1061no—
394D8—2122no—
395D8#1171994512yes—
396M16—41no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8C2×C22⋊D5311yescharacteristic
400C2×C8C2×C22⋊D5311yescharacteristic
401C2×C8C2×C22⋊D5311yescharacteristic
402C2×C8C23×D5311yescharacteristic
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—1061no—
409C4⋊C4—1061no—
410C4⋊C4—2121no—
411C4⋊C4—2121no—
412C4⋊C4#1171992211yescharacteristic
413C4⋊C4#1171992211yescharacteristic
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—2121no—
417C22⋊C4—2121no—
418C22⋊C4—2121no—
419C22⋊C4—2121no—
420C22⋊C4#1171992211yescharacteristic
421C22⋊C4#1171992211yescharacteristic
422C42—41no—
423C42—41no—
424C42C2×C22⋊D5311yescharacteristic
425C42C2×C22⋊D5311yescharacteristic
426C16—1061no—
427C16—1061no—
428C16—1061no—
429C16—1061no—
430C23—41no—
431C23—2121no—
432C23—2121no—
433C23C2×#1171992211yescharacteristic
434Q8—22no—
435Q8—41no—
436D4—22no—
437D4—44no—
438D4—41no—
439D4—8481no—
440D4—8481no—
441D4—8481no—
442D4—8481no—
443D4—2121no—
444D4—2121no—
445D4—2121no—
446D4—2121no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—44no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—161no—
454C2×C4—161no—
455C2×C4—161no—
456C2×C4—8481no—
457C2×C4—8481no—
458C2×C4—2121no—
459C2×C4—2121no—
460C2×C4C2×#1171992211yescharacteristic
461C2×C4C2×#1171992211yescharacteristic
462C2×C4C2×#1171992211yescharacteristic
463C2×C4C2×#1171992211yescharacteristic
464C2×C4C2×#1171992211yescharacteristic
465C2×C4C22×C22⋊D5311yescharacteristic
466C8—21no—
467C8—21no—
468C8—21no—
469C8C2×#1171994511yescharacteristic
470C8C2×#1171994511yescharacteristic
471C22—21no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—81no—
476C22—81no—
477C22—4241no—
478C22—4241no—
479C22—4241no—
480C22—4241no—
481C22—4241no—
482C22—4241no—
483C22C22×#1171992211yescharacteristic
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—42no—
490C4—81no—
491C4—81no—
492C4—4241no—
493C4—4241no—
494C4#1681113411yescharacteristic
495C4#1681142111yescharacteristic
496C2—41no—
497C2—82no—
498C2—8481no—
499C2—8481no—
500C2#2028471511yescharacteristic
501C2#2028785411yescharacteristic
502C2#2028785911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2951C2
socleC22×C53C22×D8
hypercenter#2951D53
hypocenterC53#33679
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. #20285365C2semidirect
2. #20276900C2semidirect
3. #20276895C2semidirect
4. #20276879C2semidirect
5. #20276872C2semidirect
6. #20276755C2semidirect
7. #20276750C2semidirect
8. #20276638C2semidirect
9. #20276638C2semidirect
10. #20276589C2semidirect
11. #20276577C2semidirect
12. C4×#13133160C2semidirect
13. C4×#13133158C2semidirect
14. #20276522C2semidirect
15. C53×#2951C2semidirect
246. #2951D53semidirect
310. C53#33679semidirect

Maximal subgroups

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

subgroupindex
#202853652
#202769002
#202768952
#202768792
#202768722
#202767552
#202767502
#202766382
#202765892
#202765772
C4×#131331602
C4×#131331582
#202765222
C53×#29512
#3367953