Finite Group Databasebeta

← #27539202

Subgroups — #27539202

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285354C211yesmaximal, characteristic, complemented
2#20283568C211yesmaximal, characteristic, complemented
3#20276924C211yesmaximal, characteristic, complemented
4#20276907C211yesmaximal, characteristic, complemented
5#20276863C211yesmaximal, characteristic, complemented
6#20276853C211yesmaximal, characteristic, complemented
7#20276771C211yesmaximal, characteristic, complemented
8#20276771C211yesmaximal, characteristic, complemented
9#20276646C211yesmaximal, characteristic, complemented
10#20276629C211yesmaximal, characteristic
11#20276592C211yesmaximal, characteristic, complemented
12#20276581C211yesmaximal, characteristic, complemented
13C4×#13133162C211yesmaximal, characteristic, complemented
14C4×D848C211yesmaximal, characteristic, complemented
15C53×#2164C211yesmaximal, characteristic, complemented
16C22×D424C2211yescharacteristic
17#16811244—21no—
18#16811097—21no—
19C2×#13133162C2211yescharacteristic
20C2×#13133162C2211yescharacteristic
21C2×D848C2211yescharacteristic
22C2×D848C2211yescharacteristic
23#16811028C2211yescharacteristic, complemented
24#16811007C2211yescharacteristic, complemented
25#16810983—21no—
26#16810869—21no—
27#16810864C2211yescharacteristic
28#16810859C2211yescharacteristic
29#16810736—21no—
30#16810736—21no—
31#16810725—21no—
32C4×Dic212C2211yescharacteristic, complemented
33C4×D424C2211yescharacteristic
34C4×D424C2211yescharacteristic
35C4×D424C2211yescharacteristic
36C4×D424C2211yescharacteristic
37C4×D424C2211yescharacteristic, complemented
38#16810661C2211yescharacteristic
39#16810629—21no—
40#16810451C2211yescharacteristic, complemented
41#16810451C2211yescharacteristic, complemented
42#16810449C2211yescharacteristic, complemented
43#16810449C2211yescharacteristic, complemented
44#16810428C2211yescharacteristic, complemented
45#16810428C2211yescharacteristic, complemented
46#16810425C2211yescharacteristic
47#16810425C2211yescharacteristic
48#16810425C2211yescharacteristic
49#16810425C2211yescharacteristic
50#16810425C2211yescharacteristic, complemented
51#16810425C2211yescharacteristic, complemented
52C53×#405C2211yescharacteristic
53C53×C4⋊C16C2211yescharacteristic, complemented
54C53×C4⋊C16C2211yescharacteristic, complemented
55C53×C22⋊C16C2211yescharacteristic
56C53×C22⋊C16C2211yescharacteristic
57C53×#346C2211yescharacteristic, complemented
58C4×C16×C53C2211yescharacteristic, complemented
59C22×D212—21no—
60C2×Dic212C2311yescharacteristic
61C2×#11719913—21no—
62C2×D424—21no—
63C2×D424—22no—
64C2×D424—22no—
65C2×D424C2311yescharacteristic
66C2×D424C2311yescharacteristic
67C2×D424C2311yescharacteristic
68#13133217—21no—
69#13133212—41no—
70#13133205—21no—
71#13133203—41no—
72#13133199—41no—
73#13133196—21no—
74#13133195—41no—
75#13133191—21no—
76C4×Dic106—21no—
77C4×D212—21no—
78C4×D212—21no—
79C4×D212—21no—
80C4×D212—21no—
81C4×D212—21no—
82#13133179—41no—
83#13133179—41no—
84#13133176—21no—
85C22⋊C4×D53—41no—
86C22⋊C4×D53—41no—
87#13133162—22no—
88#13133162—22no—
89D848—22no—
90D848—22no—
91#13133151—21no—
92#13133151C2311yescharacteristic
93#13133151C2311yescharacteristic
94#13133151C2311yescharacteristic
95#13133151C2311yescharacteristic
96#13133149—21no—
97#13133142—21no—
98#13133142—21no—
99#13133139—21no—
100#13133139—21no—
101#13133139—21no—
102#13133139—21no—
103#13133139—21no—
104#13133139—21no—
105C2×M16×C53D411yescharacteristic
106C22×C8×C53C2311yescharacteristic
107#117×C53D411yescharacteristic
108C2×C16×C53C2311yescharacteristic
109C2×C16×C53C2311yescharacteristic
110C2×C16×C53C2311yescharacteristic
111C2×C16×C53C2311yescharacteristic
112#97×C53D411yescharacteristic
113#97×C53D411yescharacteristic
114C4×C8×C53C2311yescharacteristic
115C4×C8×C53C2311yescharacteristic
116C23×D53—41no—
117C22×C4×C53C2×D411yescharacteristic
118C2×C22⋊D53—41no—
119C2×D212—24no—
120C2×D212—21no—
121C2×D212—21no—
122C2×D212—21no—
123C2×D212—42no—
124C2×D212—41no—
125C2×C4×D53—41no—
126C2×C4×D53—41no—
127C2×C4×D53—41no—
128C2×C4×D53—41no—
129C2×C4×D53—41no—
130C2×Dic106—21no—
131C2×Dic106—41no—
132M16×C53—41no—
133C2×C8×C53—21no—
134C2×C8×C53—21no—
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53C2×D411yescharacteristic
137C2×C8×C53C2×D411yescharacteristic
138C2×C8×C53C2411yescharacteristic
139C4⋊C4×C53D811yescharacteristic
140C4⋊C4×C53D811yescharacteristic
141C22⋊C4×C53D811yescharacteristic
142C22⋊C4×C53D811yescharacteristic
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C22⋊Dic53—41no—
146C22⋊Dic53—41no—
147#11719921—41no—
148#11719921—41no—
149#11719921—41no—
150#11719921—41no—
151#11719921—41no—
152#11719921—41no—
153#11719921—41no—
154#11719921—41no—
155C4⋊Dic53—21no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—41no—
160#11719919—41no—
161#11719919—41no—
162C4×Dic53—41no—
163C4×Dic53—41no—
164Dic212—21no—
165Dic212—21no—
166D424—24no—
167D424—21no—
168D424—21no—
169D424—21no—
170D424—21no—
171D424#4112yes—
172D424#4112yes—
173#11719913—42no—
174C16×C53—21no—
175C16×C53—21no—
176C16×C53—21no—
177C16×C53—21no—
178C23×C53C2×D811yescharacteristic
179C22×D53—41no—
180C22×D53—41no—
181C22×D53—44no—
182C22×D53—41no—
183C22×D53—161no—
184C22×D53—161no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53C2×D811yescharacteristic
188C2×C4×C53C2×D811yescharacteristic
189C2×C4×C53C2×D811yescharacteristic
190C2×C4×C53C2×D811yescharacteristic
191C2×C4×C53C2×D811yescharacteristic
192C2×C4×C53C22×D411yescharacteristic
193C22⋊D53—161no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—41no—
197C2×Dic53—41no—
198C2×Dic53—41no—
199D212—22no—
200D212—22no—
201D212—44no—
202D212—41no—
203D212—41no—
204D212—41no—
205D212—41no—
206C4×D53—161no—
207C4×D53—161no—
208C4×D53—161no—
209C4×D53—161no—
210C4×D53—161no—
211Dic106—41no—
212Dic106—41no—
213Dic106—161no—
214C8×C53—21no—
215C8×C53—21no—
216C8×C53—21no—
217C8×C53C2×#4111yescharacteristic
218C8×C53C2×#4111yescharacteristic
219#33678—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—21no—
223C22×C53C22×D811yescharacteristic
224C2×D53—42no—
225C2×D53—42no—
226C2×D53—81no—
227C2×D53—81no—
228C2×D53—81no—
229C2×D53—81no—
230C2×D53—81no—
231C2×D53—81no—
232C2×D53—81no—
233C2×D53—164no—
234C4×C53—21no—
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—21no—
239C4×C53#52511yescharacteristic
240C4×C53#57511yescharacteristic
241Dic53—81no—
242Dic53—81no—
243Dic53—81no—
244Dic53—81no—
245Dic53—81no—
246#3175—531no—
247#2948—531no—
248#2237—531no—
249#2234—531no—
250#2226—531no—
251#2224—531no—
252#2210—531no—
253#2210—531no—
254#2187—531no—
255#2184—531no—
256#2177—531no—
257#2175—531no—
258C4×SD32—531no—
259C4×D16—531no—
260#2164D5311yescharacteristic, complemented
261C2×C53—41no—
262C2×C53#310011yescharacteristic
263C2×C53#341511yescharacteristic
264C2×C53#341611yescharacteristic
265D53—82no—
266D53—82no—
267D53—161no—
268C22×D8—531no—
269#542—1061no—
270#518—1061no—
271C2×SD32—531no—
272C2×SD32—531no—
273C2×D16—531no—
274C2×D16—531no—
275#500—531no—
276#495—531no—
277#489—1061no—
278#468—1061no—
279#467—531no—
280#466—531no—
281#442—1061no—
282#442—1061no—
283#440—1061no—
284C4×Q16—531no—
285C4×D8—531no—
286C4×D8—531no—
287C4×D8—531no—
288C4×D8—531no—
289C4×D8—531no—
290#427—531no—
291#418—1061no—
292#405C2×D5311yescharacteristic
293#367—531no—
294#367—531no—
295#366—531no—
296#366—531no—
297C4⋊C16C2×D5311yescharacteristic, complemented
298C4⋊C16C2×D5311yescharacteristic, complemented
299#358—531no—
300#358—531no—
301#357—531no—
302#357—531no—
303#357—531no—
304#357—531no—
305#357—531no—
306#357—531no—
307C22⋊C16C2×D5311yescharacteristic
308C22⋊C16C2×D5311yescharacteristic
309#346C2×D5311yescharacteristic, complemented
310C4×C16C2×D5311yescharacteristic, complemented
311C53#3367811yescharacteristic, complemented, Sylow
312C22×D4—1061no—
313C2×Q16—531no—
314C2×SD16—1061no—
315C2×D8—531no—
316C2×D8—531no—
317C2×D8—531no—
318C2×D8—1062no—
319C2×D8—1062no—
320C2×D8—1061no—
321C2×M16D21211yescharacteristic
322C22×C8C22×D5311yescharacteristic
323C4⋊Q8—1061no—
324#126—2121no—
325#124—1061no—
326#124—2121no—
327#123—2121no—
328C22⋊Q8—1061no—
329C22⋊Q8—2121no—
330#121—1061no—
331C4×Q8—1061no—
332C4×D4—1061no—
333C4×D4—1061no—
334C4×D4—1061no—
335C4×D4—1061no—
336C4×D4—1061no—
337C4×D4—2121no—
338C4×D4—2121no—
339#117—1061no—
340#117D21211yescharacteristic
341C2×C22⋊C4—2121no—
342C2×C22⋊C4—2121no—
343SD32—1062no—
344SD32—1062no—
345D16—1062no—
346D16—1062no—
347C2×C16C22×D5311yescharacteristic
348C2×C16C22×D5311yescharacteristic
349C2×C16C22×D5311yescharacteristic
350C2×C16C22×D5311yescharacteristic
351#107—531no—
352#107—531no—
353#107—531no—
354#107—531no—
355#107—1061no—
356#106—1061no—
357#103—1061no—
358#103—1061no—
359#102—1061no—
360#102—1061no—
361#102—1061no—
362#102—1061no—
363#102—1061no—
364#102—1061no—
365#97D21211yescharacteristic
366#97D21211yescharacteristic
367C4×C8C22×D5311yescharacteristic
368C4×C8C22×D5311yescharacteristic
369C24—2121no—
370C2×Q8—1061no—
371C2×Q8—2121no—
372C2×D4—1064no—
373C2×D4—1061no—
374C2×D4—1061no—
375C2×D4—1061no—
376C2×D4—2122no—
377C2×D4—2121no—
378C2×D4—2121no—
379C22×C4—2121no—
380C22×C4—2121no—
381C22×C4—2121no—
382C22×C4—2121no—
383C22×C4—2121no—
384C22×C4C2×D21211yescharacteristic
385Q16—1061no—
386Q16—1061no—
387SD16—2122no—
388D8—532no—
389D8—532no—
390D8—1064no—
391D8—1061no—
392D8—1061no—
393D8—1061no—
394D8—1061no—
395M16—41no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8C2×D21211yescharacteristic
399C2×C8C2×D21211yescharacteristic
400C2×C8C2×D21211yescharacteristic
401C2×C8C23×D5311yescharacteristic
402C4⋊C4—1061no—
403C4⋊C4—1061no—
404C4⋊C4—1061no—
405C4⋊C4—1061no—
406C4⋊C4—2121no—
407C4⋊C4—2121no—
408C4⋊C4—2121no—
409C4⋊C4D42411yescharacteristic
410C4⋊C4D42411yescharacteristic
411C22⋊C4—2121no—
412C22⋊C4—2121no—
413C22⋊C4—2121no—
414C22⋊C4—2121no—
415C22⋊C4—2121no—
416C22⋊C4—2121no—
417C22⋊C4—2121no—
418C22⋊C4—2121no—
419C22⋊C4—2121no—
420C22⋊C4—2121no—
421C22⋊C4D42411yescharacteristic
422C22⋊C4D42411yescharacteristic
423C42—2121no—
424C42—2121no—
425C42C2×D21211yescharacteristic
426C42C2×D21211yescharacteristic
427C16—21no—
428C16—21no—
429C16—21no—
430C16—21no—
431C23—8481no—
432C23—8481no—
433C23—2124no—
434C23—2121no—
435C23—2121no—
436C23—2121no—
437C23C2×D42411yescharacteristic
438Q8—8481no—
439Q8—2121no—
440Q8—2121no—
441D4—8481no—
442D4—1062no—
443D4—1062no—
444D4—2124no—
445D4—2121no—
446D4—2121no—
447D4—2121no—
448D4—2121no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—8481no—
452C2×C4—8481no—
453C2×C4—8481no—
454C2×C4—8481no—
455C2×C4—8481no—
456C2×C4—2121no—
457C2×C4—2121no—
458C2×C4—2121no—
459C2×C4—2121no—
460C2×C4—2121no—
461C2×C4C2×D42411yescharacteristic
462C2×C4C2×D42411yescharacteristic
463C2×C4C2×D42411yescharacteristic
464C2×C4C2×D42411yescharacteristic
465C2×C4C2×D42411yescharacteristic
466C2×C4C22×D21211yescharacteristic
467C8—21no—
468C8—21no—
469C8—21no—
470C8#41×D5311yescharacteristic
471C8#41×D5311yescharacteristic
472C22—21no—
473C22—21no—
474C22—21no—
475C22—8484no—
476C22—4241no—
477C22—4241no—
478C22—4241no—
479C22—4241no—
480C22—4241no—
481C22—4241no—
482C22—4241no—
483C22—2122no—
484C22—2122no—
485C22C22×D42411yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—4241no—
492C4—4241no—
493C4—4241no—
494C4—4241no—
495C4—4241no—
496C4#1681113511yescharacteristic
497C4#1681142311yescharacteristic
498C2—41no—
499C2—8481no—
500C2—4242no—
501C2—4242no—
502C2#2028471611yescharacteristic
503C2#2028785611yescharacteristic
504C2#2028786311yescharacteristic
subgroupG/H
centerC22C22×D424
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2164C2
socleC22×C53C22×D8
prime radical#20276629C2
hypercenter#2164D53
hypocenterC53#33678
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2164

Complements

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

NH ≅ G/Ntype
1. #20285354C2semidirect
2. #20283568C2semidirect
3. #20276924C2semidirect
4. #20276907C2semidirect
5. #20276863C2semidirect
6. #20276853C2semidirect
7. #20276771C2semidirect
8. #20276771C2semidirect
9. #20276646C2semidirect
11. #20276592C2semidirect
12. #20276581C2semidirect
13. C4×#13133162C2semidirect
14. C4×D848C2semidirect
15. C53×#2164C2semidirect
23. #16811028C22semidirect
24. #16811007C22semidirect
32. C4×Dic212C22semidirect
37. C4×D424C22semidirect
40. #16810451C22semidirect
41. #16810451C22semidirect
42. #16810449C22semidirect
43. #16810449C22semidirect
44. #16810428C22semidirect
45. #16810428C22semidirect
50. #16810425C22semidirect
51. #16810425C22semidirect
53. C53×C4⋊C16C22semidirect
54. C53×C4⋊C16C22semidirect
57. C53×#346C22semidirect
58. C4×C16×C53C22semidirect
260. #2164D53semidirect
297. C4⋊C16C2×D53semidirect
298. C4⋊C16C2×D53semidirect
309. #346C2×D53semidirect
310. C4×C16C2×D53semidirect
311. C53#33678semidirect

Maximal subgroups

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

subgroupindex
#202853542
#202835682
#202769242
#202769072
#202768632
#202768532
#202767712
#202766462
#202766292
#202765922
#202765812
C4×#131331622
C4×D8482
C53×#21642
#3367853