Finite Group Databasebeta

← #27538895

Subgroups — #27538895

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284297C211yesmaximal, characteristic
2#427×D53C211yesmaximal, characteristic, complemented
3#20276926C211yesmaximal, characteristic, complemented
4#20276925C211yesmaximal, characteristic, complemented
5#20276919C211yesmaximal, characteristic, complemented
6#20276914C211yesmaximal, characteristic, complemented
7#20276909C211yesmaximal, characteristic, complemented
8#20276908C211yesmaximal, characteristic, complemented
9#20276902C211yesmaximal, characteristic, complemented
10#20276897C211yesmaximal, characteristic, complemented
11#20276497C211yesmaximal, characteristic, complemented
12#20276423C211yesmaximal, characteristic, complemented
13#20276420C211yesmaximal, characteristic, complemented
14C2×#16810402C211yesmaximal, characteristic, complemented
15C53×#2158C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17C22×C8×D53C2211yescharacteristic
18#16811135—21no—
19#117×D53—21no—
20C22×#11719908C2211yescharacteristic
21#16810865C2211yescharacteristic
22#16810865C2211yescharacteristic
23#16810860C2211yescharacteristic
24#16810860C2211yescharacteristic
25#16810853—21no—
26#16810853—21no—
27#16810661C2211yescharacteristic
28#16810660C2211yescharacteristic
29#16810660C2211yescharacteristic
30#16810659—21no—
31#16810658—21no—
32#107×D53C2211yescharacteristic
33#107×D53C2211yescharacteristic
34#106×D53—21no—
35#16810617—21no—
36#16810450C2211yescharacteristic, complemented
37#16810450C2211yescharacteristic, complemented
38#16810448C2211yescharacteristic, complemented
39#16810448C2211yescharacteristic, complemented
40#16810428C2211yescharacteristic, complemented
41#16810428C2211yescharacteristic, complemented
42#16810426C2211yescharacteristic, complemented
43#16810426C2211yescharacteristic, complemented
44#16810425C2211yescharacteristic, complemented
45#16810425C2211yescharacteristic, complemented
46#16810423C2211yescharacteristic, complemented
47#16810423C2211yescharacteristic, complemented
48#16810402C2211yescharacteristic
49#16810402C2211yescharacteristic
50#16810402C2211yescharacteristic
51#16810402C2211yescharacteristic
52C22×C16×C53C2211yescharacteristic
53C53×#427C2211yescharacteristic
54C53×#427C2211yescharacteristic
55C53×#367C2211yescharacteristic, complemented
56C53×#367C2211yescharacteristic, complemented
57C53×#366C2211yescharacteristic, complemented
58C53×#366C2211yescharacteristic, complemented
59C2×#11719945—21no—
60C22×C4×D53D411yescharacteristic
61#13133239—22no—
62#13133239—22no—
63C2×Dic212C2311yescharacteristic
64C2×#11719913—21no—
65C2×D424C2311yescharacteristic
66C22×#88322D411yescharacteristic
67C2×C8×D53—22no—
68C2×C8×D53—22no—
69C2×C8×D53D411yescharacteristic
70C2×C8×D53D411yescharacteristic
71C2×C8×D53C2311yescharacteristic
72C2×C8×D53C2311yescharacteristic
73#13133201—41no—
74#13133197—21no—
75#13133197—21no—
76#13133192—21no—
77#13133192—21no—
78#13133187—41no—
79C4×D212—41no—
80C4×D212—41no—
81#13133176—21no—
82#13133175—21no—
83#13133175—21no—
84#13133174—41no—
85#13133173—41no—
86C4⋊C4×D53—21no—
87C4⋊C4×D53—21no—
88C22⋊C4×D53—41no—
89C42×D53—41no—
90C2×#11719908—22no—
91C2×#11719908C2311yescharacteristic
92C2×#11719908C2311yescharacteristic
93#13133151C2311yescharacteristic
94#13133151C2311yescharacteristic
95#13133150—21no—
96#13133150—21no—
97#13133149—21no—
98#13133148—21no—
99#13133148—21no—
100#13133140—21no—
101#13133140—21no—
102#13133137—21no—
103#13133137—21no—
104C22×C8×C53C2311yescharacteristic
105#117×C53—21no—
106#117×C53—21no—
107C2×C16×C53—22no—
108C2×C16×C53C2311yescharacteristic
109C2×C16×C53C2311yescharacteristic
110#107×C53C2311yescharacteristic
111#107×C53C2311yescharacteristic
112#107×C53C2311yescharacteristic
113#107×C53C2311yescharacteristic
114#106×C53—21no—
115#106×C53—21no—
116C23×D53D811yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C2×C22⋊D53—41no—
119C22×Dic53D811yescharacteristic
120#11719945—42no—
121#11719945—42no—
122C2×D212—21no—
123C2×D212—41no—
124C2×D212—41no—
125C2×C4×D53—22no—
126C2×C4×D53—22no—
127C2×C4×D53—41no—
128C2×C4×D53—41no—
129C2×C4×D53—41no—
130C2×C4×D53D811yescharacteristic
131C2×C4×D53D811yescharacteristic
132C2×C4×D53C2×D411yescharacteristic
133C2×C4×D53C2×D411yescharacteristic
134C2×Dic106—21no—
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53#4112yes—
137C2×C8×C53#4112yes—
138C2×C8×C53C2411yescharacteristic
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C22⋊C4×C53—41no—
144C22⋊C4×C53—41no—
145C42×C53—41no—
146C42×C53—41no—
147C22⋊Dic53—41no—
148#11719921—41no—
149#11719921—41no—
150#11719921—41no—
151#11719921—41no—
152#11719921—41no—
153#11719921—41no—
154C4⋊Dic53—21no—
155C4⋊Dic53—21no—
156C4⋊Dic53—41no—
157C4⋊Dic53—41no—
158#11719919—41no—
159#11719919—41no—
160C4×Dic53—41no—
161C4×Dic53—41no—
162C4×Dic53—41no—
163C2×#88322—22no—
164C2×#88322C2×D411yescharacteristic
165C2×#88322C2×D411yescharacteristic
166Dic212—21no—
167Dic212—21no—
168D424—21no—
169D424—21no—
170#11719913—42no—
171C8×D53—22no—
172C8×D53—22no—
173C8×D53—21no—
174C8×D53—21no—
175C8×D53—21no—
176C8×D53—21no—
177C16×C53—21no—
178C16×C53—21no—
179#11719908—21no—
180#11719908—21no—
181C23×C53C2×D811yescharacteristic
182C22×D53—22no—
183C22×D53—22no—
184C22×D53—41no—
185C22×D53—41no—
186C22×D53C2×D811yescharacteristic
187C22×D53C2×D811yescharacteristic
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53#12312yes—
193C2×C4×C53C2×D811yescharacteristic
194C2×C4×C53#13712yes—
195C2×C4×C53C22×D411yescharacteristic
196C22⋊D53—82no—
197C2×Dic53—22no—
198C2×Dic53—41no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201C2×Dic53C2×D811yescharacteristic
202C2×Dic53C2×D811yescharacteristic
203D212—41no—
204D212—41no—
205D212—82no—
206D212—82no—
207C4×D53—21no—
208C4×D53—21no—
209C4×D53—21no—
210C4×D53—21no—
211C4×D53—22no—
212C4×D53—22no—
213C4×D53—82no—
214C4×D53—82no—
215C4×D53—82no—
216Dic106—41no—
217Dic106—41no—
218C8×C53#12312yes—
219C8×C53C2×#4111yescharacteristic
220C8×C53C2×#4111yescharacteristic
221#88322—21no—
222#88322—21no—
223#33645—531nomaximal, Sylow
224C22×C53—21no—
225C22×C53—21no—
226C22×C53#48012yes—
227C22×C53C22×D811yescharacteristic
228C2×D53—21no—
229C2×D53—21no—
230C2×D53—21no—
231C2×D53—21no—
232C2×D53—21no—
233C2×D53—44no—
234C2×D53—81no—
235C2×D53—81no—
236C2×D53—81no—
237C2×D53#50812yes—
238C4×C53—81no—
239C4×C53—81no—
240C4×C53—81no—
241C4×C53—81no—
242C4×C53#48012yes—
243C4×C53C2×#12311yescharacteristic
244C4×C53C2×#13711yescharacteristic
245Dic53—21no—
246Dic53—21no—
247Dic53—81no—
248Dic53—81no—
249Dic53—81no—
250#3050—531no—
251C2×#427—531no—
252#2237—531no—
253#2237—531no—
254#2236—531no—
255#2235—531no—
256#2234—531no—
257#2234—531no—
258#2233—531no—
259#2232—531no—
260#2158—531no—
261#2158D5311yescharacteristic, complemented
262#2138—531no—
263#2138—531no—
264C2×C22⋊C16—531no—
265C2×C53—22no—
266C2×C53C2×#48011yescharacteristic
267C2×C53C2×#50811yescharacteristic
268C2×C53#341411yescharacteristic
269D53—22no—
270D53—41no—
271D53—161no—
272C2×#135—531no—
273C23×C8—531no—
274#525—1061no—
275C2×#117—1061no—
276C22×C16—531no—
277C22×C16C2×D5311yescharacteristic
278#467—531no—
279#467—531no—
280#466—531no—
281#466—531no—
282#465—1061no—
283#465—1061no—
284#427—531no—
285#427—531no—
286#427—531no—
287#427—1061no—
288#427—1061no—
289#427C2×D5311yescharacteristic
290#427C2×D5311yescharacteristic
291C2×#107—531no—
292C2×#107—531no—
293C2×#106—1061no—
294#416—1061no—
295#367—531no—
296#367—531no—
297#367C2×D5311yescharacteristic, complemented
298#367C2×D5311yescharacteristic, complemented
299#366—531no—
300#366—531no—
301#366C2×D5311yescharacteristic, complemented
302#366C2×D5311yescharacteristic, complemented
303#358—531no—
304#358—531no—
305#358—531no—
306#358—531no—
307#357—531no—
308#357—531no—
309#357—531no—
310#357—531no—
311C22⋊C16—531no—
312C22⋊C16—531no—
313C22⋊C16—531no—
314C22⋊C16—531no—
315C53#3364511yescharacteristic, complemented, Sylow
316C2×#41—1061no—
317C23×C4—531no—
318#135—1062no—
319#135—1062no—
320C2×Q16—531no—
321C2×SD16—1061no—
322C2×D8—531no—
323C22×C8—531no—
324C22×C8—531no—
325C22×C8—531no—
326C22×C8—531no—
327C22×C8—531no—
328C22×C8—1062no—
329C22×C8—1062no—
330C22×C8C22×D5311yescharacteristic
331#123—2121no—
332C22⋊Q8—1061no—
333C22⋊Q8—1061no—
334#121—1061no—
335#121—1061no—
336#120—2121no—
337C4×D4—2121no—
338C4×D4—2121no—
339#117—21no—
340#117—21no—
341#117—1061no—
342#117—1061no—
343#117—1061no—
344#117—2121no—
345#117—2121no—
346C2×C4⋊C4—1061no—
347C2×C4⋊C4—1061no—
348C2×C22⋊C4—2121no—
349C2×C42—2121no—
350C2×C16—22no—
351C2×C16—531no—
352C2×C16—531no—
353C2×C16—1062no—
354C2×C16C22×D5311yescharacteristic
355C2×C16C22×D5311yescharacteristic
356#107—531no—
357#107—531no—
358#107—1061no—
359#107—1061no—
360#107C22×D5311yescharacteristic
361#107C22×D5311yescharacteristic
362#107C22×D5311yescharacteristic
363#107C22×D5311yescharacteristic
364#106—21no—
365#106—21no—
366#106—1061no—
367#106—1061no—
368#106—1061no—
369#103—1061no—
370#103—1061no—
371#102—1061no—
372#102—1061no—
373C24—531no—
374#41—2122no—
375#41—2122no—
376C2×Q8—1061no—
377C2×D4—1061no—
378C2×D4—2121no—
379C2×D4—2121no—
380C2×D4—2121no—
381C22×C4—531no—
382C22×C4—531no—
383C22×C4—531no—
384C22×C4—531no—
385C22×C4—531no—
386C22×C4—1062no—
387C22×C4—1062no—
388C22×C4—2121no—
389C22×C4—2121no—
390C22×C4—2121no—
391C22×C4D4×D5311yescharacteristic
392Q16—1061no—
393Q16—1061no—
394SD16—2122no—
395D8—1061no—
396D8—1061no—
397C2×C8—531no—
398C2×C8—531no—
399C2×C8—1062no—
400C2×C8—1062no—
401C2×C8—1062no—
402C2×C8—1061no—
403C2×C8—1061no—
404C2×C8—1061no—
405C2×C8—1061no—
406C2×C8#1171994512yes—
407C2×C8D4×D5311yescharacteristic
408C2×C8#1171994912yes—
409C2×C8C23×D5311yescharacteristic
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—1061no—
415C4⋊C4—1061no—
416C4⋊C4—2121no—
417C4⋊C4—2121no—
418C4⋊C4—2121no—
419C4⋊C4—2121no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—2121no—
423C22⋊C4—2121no—
424C22⋊C4—2121no—
425C22⋊C4—2121no—
426C22⋊C4—2121no—
427C22⋊C4—2121no—
428C22⋊C4—2121no—
429C42—41no—
430C42—41no—
431C42—2121no—
432C42—2121no—
433C42—2121no—
434C16—21no—
435C16—21no—
436C16—1061no—
437C16—1061no—
438C23—531no—
439C23—531no—
440C23—1062no—
441C23—1062no—
442C23—2121no—
443C23—2121no—
444C23D8×D5311yescharacteristic
445Q8—2121no—
446Q8—2121no—
447D4—4242no—
448D4—4242no—
449D4—4242no—
450D4—2121no—
451D4—2121no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—531no—
457C2×C4—531no—
458C2×C4—1062no—
459C2×C4—1061no—
460C2×C4—1061no—
461C2×C4—1061no—
462C2×C4—1061no—
463C2×C4—1062no—
464C2×C4—1062no—
465C2×C4—4242no—
466C2×C4—4242no—
467C2×C4—4242no—
468C2×C4—2121no—
469C2×C4—2121no—
470C2×C4—2121no—
471C2×C4#1313320212yes—
472C2×C4D8×D5311yescharacteristic
473C2×C4#1313324912yes—
474C2×C4C2×D4×D5311yescharacteristic
475C8—1061no—
476C8—1061no—
477C8#1313320212yes—
478C8C2×#1171994511yescharacteristic
479C8C2×#1171994911yescharacteristic
480C22—21no—
481C22—21no—
482C22—532no—
483C22—1061no—
484C22—1061no—
485C22—1061no—
486C22—1061no—
487C22—1061no—
488C22—4241no—
489C22—4241no—
490C22—4241no—
491C22—2124no—
492C22#1681093812yes—
493C22C2×D8×D5311yescharacteristic
494C4—81no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—1061no—
499C4—1061no—
500C4—4241no—
501C4—4241no—
502C4—4241no—
503C4#1681093812yes—
504C4C2×#1313320211yescharacteristic
505C4C2×#1313324911yescharacteristic
506C2—22no—
507C2—8481no—
508C2—1062no—
509C2—2121no—
510C2C2×#1681093811yescharacteristic
511C2#508×D5311yescharacteristic
512C2#2028784811yescharacteristic
subgroupG/H
centerC2×C4D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2158C2
socleC22×C53C22×D8
prime radical#20284297C2
hypercenter#2158D53
hypocenterC53#33645
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2158

Complements

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

NH ≅ G/Ntype
2. #427×D53C2semidirect
3. #20276926C2semidirect
4. #20276925C2semidirect
5. #20276919C2semidirect
6. #20276914C2semidirect
7. #20276909C2semidirect
8. #20276908C2semidirect
9. #20276902C2semidirect
10. #20276897C2semidirect
11. #20276497C2semidirect
12. #20276423C2semidirect
13. #20276420C2semidirect
14. C2×#16810402C2semidirect
15. C53×#2158C2semidirect
36. #16810450C22semidirect
37. #16810450C22semidirect
38. #16810448C22semidirect
39. #16810448C22semidirect
40. #16810428C22semidirect
41. #16810428C22semidirect
42. #16810426C22semidirect
43. #16810426C22semidirect
44. #16810425C22semidirect
45. #16810425C22semidirect
46. #16810423C22semidirect
47. #16810423C22semidirect
55. C53×#367C22semidirect
56. C53×#367C22semidirect
57. C53×#366C22semidirect
58. C53×#366C22semidirect
261. #2158D53semidirect
297. #367C2×D53semidirect
298. #367C2×D53semidirect
301. #366C2×D53semidirect
302. #366C2×D53semidirect
315. C53#33645semidirect

Maximal subgroups

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

subgroupindex
#202842972
#427×D532
#202769262
#202769252
#202769192
#202769142
#202769092
#202769082
#202769022
#202768972
#202764972
#202764232
#202764202
C2×#168104022
C53×#21582
#3364553