Finite Group Databasebeta

← #27538925

Subgroups — #27538925

Order 13568.

580 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#467×D53C211yesmaximal, characteristic
2#427×D53C211yesmaximal, characteristic
3#20276924C211yesmaximal, characteristic, complemented
4#20276923C211yesmaximal, characteristic, complemented
5#20276920C211yesmaximal, characteristic, complemented
6#20276919C211yesmaximal, characteristic, complemented
7#20276916C211yesmaximal, characteristic, complemented
8#20276914C211yesmaximal, characteristic, complemented
9#20276911C211yesmaximal, characteristic, complemented
10#20276502C211yesmaximal, characteristic, complemented
11#20276500C211yesmaximal, characteristic, complemented
12#20276433C211yesmaximal, characteristic, complemented
13#20276433C211yesmaximal, characteristic, complemented
14#20276324C211yesmaximal, characteristic
15C53×#2237C211yesmaximal, characteristic, complemented
16C2×Q16×D53C2211yescharacteristic
17C22×C8×D53C2211yescharacteristic
18C22⋊Q8×D53—21no—
19#117×D53—21no—
20C2×#13133156C2211yescharacteristic
21C2×#13133156C2211yescharacteristic
22#16810865C2211yescharacteristic
23#16810864C2211yescharacteristic
24#16810863—21no—
25#16810862—21no—
26#16810861C2211yescharacteristic
27#16810661C2211yescharacteristic
28#16810660C2211yescharacteristic
29#16810660C2211yescharacteristic
30#16810659—21no—
31#16810658—21no—
32#107×D53C2211yescharacteristic
33#107×D53C2211yescharacteristic
34#107×D53C2211yescharacteristic
35#106×D53—21no—
36#103×D53—21no—
37#16810451C2211yescharacteristic, complemented
38#16810450C2211yescharacteristic, complemented
39#16810450C2211yescharacteristic, complemented
40#16810449C2211yescharacteristic, complemented
41#16810448C2211yescharacteristic, complemented
42#16810448C2211yescharacteristic, complemented
43#16810428C2211yescharacteristic, complemented
44#16810428C2211yescharacteristic, complemented
45#16810427C2211yescharacteristic, complemented
46#16810427C2211yescharacteristic, complemented
47#16810426C2211yescharacteristic, complemented
48#16810426C2211yescharacteristic, complemented
49C22⋊#11719908C2211yescharacteristic
50#16810402C2211yescharacteristic
51#16810402C2211yescharacteristic
52C53×#467C2211yescharacteristic
53C53×#427C2211yescharacteristic
54C53×#367C2211yescharacteristic, complemented
55C53×#366C2211yescharacteristic, complemented
56C53×#358C2211yescharacteristic, complemented
57C53×#358C2211yescharacteristic, complemented
58C53×C22⋊C16C2211yescharacteristic
59C2×Q8×D53—21no—
60C22×C4×D53D411yescharacteristic
61C2×Dic212C2311yescharacteristic
62C2×#11719925—21no—
63Q16×D53—22no—
64Q16×D53—22no—
65C22×#88322D411yescharacteristic
66C2×C8×D53—22no—
67C2×C8×D53—22no—
68C2×C8×D53D411yescharacteristic
69C2×C8×D53D411yescharacteristic
70C2×C8×D53C2311yescharacteristic
71C2×C8×D53C2311yescharacteristic
72#13133197—21no—
73#13133196—21no—
74#13133195—41no—
75#13133194—41no—
76#13133193—21no—
77#13133176—21no—
78#13133175—21no—
79#13133175—21no—
80#13133174—41no—
81#13133173—41no—
82C4⋊C4×D53—21no—
83C4⋊C4×D53—21no—
84C4⋊C4×D53—21no—
85C4⋊C4×D53—41no—
86C22⋊C4×D53—41no—
87C22⋊C4×D53—41no—
88C42×D53—41no—
89#13133156—22no—
90#13133156—22no—
91C2×#11719908C2311yescharacteristic
92C2×#11719908C2311yescharacteristic
93#13133151C2311yescharacteristic
94#13133151C2311yescharacteristic
95#13133151C2311yescharacteristic
96#13133150—21no—
97#13133150—21no—
98#13133150—21no—
99#13133149—21no—
100#13133148—21no—
101#13133148—21no—
102#13133142—21no—
103#13133141—21no—
104#13133140—21no—
105C2×Q16×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22⋊Q8×C53—21no—
108#117×C53—21no—
109C2×C16×C53C2311yescharacteristic
110C2×C16×C53C2311yescharacteristic
111#107×C53C2311yescharacteristic
112#107×C53C2311yescharacteristic
113#107×C53C2311yescharacteristic
114#106×C53—21no—
115#103×C53—21no—
116C23×D53D811yescharacteristic
117C2×Q8×C53—21no—
118C22×C4×C53C2×D411yescharacteristic
119C22×Dic53D811yescharacteristic
120Q8×D53—42no—
121Q8×D53—42no—
122C2×C4×D53—22no—
123C2×C4×D53—22no—
124C2×C4×D53—41no—
125C2×C4×D53—41no—
126C2×C4×D53—41no—
127C2×C4×D53—41no—
128C2×C4×D53D811yescharacteristic
129C2×C4×D53D811yescharacteristic
130C2×C4×D53C2×D411yescharacteristic
131C2×C4×D53C2×D411yescharacteristic
132C2×Dic106—21no—
133C2×Dic106—41no—
134Q16×C53—21no—
135Q16×C53—21no—
136C2×C8×C53—21no—
137C2×C8×C53—21no—
138C2×C8×C53C2×D411yescharacteristic
139C2×C8×C53C2411yescharacteristic
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—41no—
144C22⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C42×C53—41no—
147C22⋊Dic53—41no—
148C22⋊Dic53—41no—
149#11719925—42no—
150#11719921—41no—
151#11719921—41no—
152#11719921—41no—
153#11719921—41no—
154C4⋊Dic53—21no—
155C4⋊Dic53—21no—
156C4⋊Dic53—21no—
157C4⋊Dic53—41no—
158#11719919—41no—
159#11719919—41no—
160#11719919—41no—
161#11719919—41no—
162#11719919—41no—
163C4×Dic53—41no—
164C4×Dic53—41no—
165C4×Dic53—41no—
166C2×#88322—22no—
167C2×#88322C2×D411yescharacteristic
168C2×#88322C2×D411yescharacteristic
169Dic212—21no—
170Dic212—21no—
171C8×D53—22no—
172C8×D53—21no—
173C8×D53—21no—
174C8×D53—22no—
175C8×D53#4112yes—
176C8×D53#4112yes—
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×D53C2×D811yescharacteristic
186C22×D53C2×D811yescharacteristic
187Q8×C53—41no—
188Q8×C53—41no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53C2×D811yescharacteristic
196C2×C4×C53C22×D411yescharacteristic
197C2×Dic53—22no—
198C2×Dic53—41no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201C2×Dic53—41no—
202C2×Dic53C2×D811yescharacteristic
203C2×Dic53C2×D811yescharacteristic
204C4×D53—22no—
205C4×D53—21no—
206C4×D53—21no—
207C4×D53—22no—
208C4×D53—82no—
209C4×D53—82no—
210C4×D53—82no—
211C4×D53—82no—
212C4×D53#12312yes—
213C4×D53#13712yes—
214Dic106—41no—
215Dic106—41no—
216Dic106—82no—
217C8×C53—21no—
218C8×C53C2×#4111yescharacteristic
219C8×C53C2×#4111yescharacteristic
220#88322—21no—
221#88322#12312yes—
222#33649—531nomaximal, Sylow
223C22×C53—21no—
224C22×C53—21no—
225C22×C53—21no—
226C22×C53C22×D811yescharacteristic
227C2×D53—21no—
228C2×D53—21no—
229C2×D53—21no—
230C2×D53—21no—
231C2×D53—44no—
232C2×D53#48012yes—
233C2×D53#51012yes—
234C4×C53—21no—
235C4×C53—81no—
236C4×C53—81no—
237C4×C53—81no—
238C4×C53—81no—
239C4×C53C2×#12311yescharacteristic
240C4×C53C2×#13711yescharacteristic
241Dic53—21no—
242Dic53—81no—
243Dic53—81no—
244Dic53—81no—
245Dic53—81no—
246Dic53#48012yes—
247C2×#467—531no—
248C2×#427—531no—
249#2237—531no—
250#2237—531no—
251#2237—531no—
252#2237D5311yescharacteristic, complemented
253#2236—531no—
254#2236—531no—
255#2235—531no—
256#2235—531no—
257#2159—531no—
258#2159—531no—
259#2141—531no—
260#2141—531no—
261#2112—531no—
262C2×C53—41no—
263C2×C53C2×#48011yescharacteristic
264C2×C53C2×#51011yescharacteristic
265C2×C53#341411yescharacteristic
266D53—22no—
267D53—22no—
268C22×Q16—531no—
269C23×C8—531no—
270C2×C22⋊Q8—1061no—
271C2×#117—1061no—
272C2×M32—531no—
273C2×M32—531no—
274#467—531no—
275#467—531no—
276#467—531no—
277#467—1061no—
278#467—1061no—
279#467C2×D5311yescharacteristic
280#427—531no—
281#427—531no—
282#427—531no—
283#427—1061no—
284#427—1061no—
285#427C2×D5311yescharacteristic
286C2×#107—531no—
287C2×#107—531no—
288C2×#107—531no—
289C2×#106—1061no—
290C2×#103—1061no—
291#367—531no—
292#367—531no—
293#367—531no—
294#367C2×D5311yescharacteristic, complemented
295#366—531no—
296#366—531no—
297#366—531no—
298#366C2×D5311yescharacteristic, complemented
299#358—531no—
300#358—531no—
301#358—531no—
302#358—531no—
303#358—531no—
304#358—531no—
305#358C2×D5311yescharacteristic, complemented
306#358C2×D5311yescharacteristic, complemented
307C22⋊C16—531no—
308C22⋊C16—531no—
309C22⋊C16—531no—
310C22⋊C16C2×D5311yescharacteristic
311C53#3364911yescharacteristic, complemented, Sylow
312C22×Q8—1061no—
313C23×C4—531no—
314C2×Q16—531no—
315C2×Q16—1062no—
316C2×Q16—1062no—
317C2×Q16—1061no—
318C2×Q16C22×D5311yescharacteristic
319C22×C8—531no—
320C22×C8—531no—
321C22×C8—531no—
322C22×C8—531no—
323C22×C8—531no—
324C22×C8—1062no—
325C22×C8—1062no—
326C22×C8C22×D5311yescharacteristic
327C22⋊Q8—21no—
328C22⋊Q8—1061no—
329C22⋊Q8—1061no—
330C22⋊Q8—1061no—
331C22⋊Q8—2121no—
332C22⋊Q8—2121no—
333#117—21no—
334#117—1061no—
335#117—1061no—
336#117—1061no—
337#117—2121no—
338#117—2121no—
339C2×C4⋊C4—1061no—
340C2×C4⋊C4—1061no—
341C2×C4⋊C4—1061no—
342C2×C4⋊C4—2121no—
343C2×C22⋊C4—2121no—
344C2×C22⋊C4—2121no—
345C2×C42—2121no—
346M32—1062no—
347M32—1062no—
348C2×C16—531no—
349C2×C16—531no—
350C2×C16C22×D5311yescharacteristic
351C2×C16C22×D5311yescharacteristic
352#107—531no—
353#107—531no—
354#107—531no—
355#107—1061no—
356#107—1061no—
357#107—1061no—
358#107C22×D5311yescharacteristic
359#107C22×D5311yescharacteristic
360#107C22×D5311yescharacteristic
361#106—21no—
362#106—1061no—
363#106—1061no—
364#106—1061no—
365#103—21no—
366#103—1061no—
367#103—1061no—
368#103—1061no—
369C24—531no—
370C2×Q8—21no—
371C2×Q8—1061no—
372C2×Q8—2121no—
373C2×Q8—2122no—
374C2×Q8—2122no—
375C22×C4—531no—
376C22×C4—531no—
377C22×C4—531no—
378C22×C4—531no—
379C22×C4—531no—
380C22×C4—1062no—
381C22×C4—1062no—
382C22×C4—2121no—
383C22×C4—2121no—
384C22×C4—2121no—
385C22×C4—2121no—
386C22×C4D4×D5311yescharacteristic
387Q16—21no—
388Q16—21no—
389Q16—1061no—
390Q16—1061no—
391Q16—2122no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—531no—
395C2×C8—532no—
396C2×C8—532no—
397C2×C8—531no—
398C2×C8—1062no—
399C2×C8—1062no—
400C2×C8—1062no—
401C2×C8—1061no—
402C2×C8—1061no—
403C2×C8D4×D5311yescharacteristic
404C2×C8C23×D5311yescharacteristic
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—41no—
409C4⋊C4—1061no—
410C4⋊C4—1061no—
411C4⋊C4—1061no—
412C4⋊C4—2121no—
413C4⋊C4—2121no—
414C4⋊C4—2121no—
415C4⋊C4—2121no—
416C4⋊C4—2121no—
417C4⋊C4—2121no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—2121no—
421C22⋊C4—2121no—
422C22⋊C4—2121no—
423C22⋊C4—2121no—
424C22⋊C4—2121no—
425C22⋊C4—2121no—
426C42—41no—
427C42—2121no—
428C42—2121no—
429C42—2121no—
430C16—21no—
431C16—21no—
432C16—1061no—
433C16—1061no—
434C23—531no—
435C23—531no—
436C23—1062no—
437C23—1062no—
438C23—2121no—
439C23D8×D5311yescharacteristic
440Q8—41no—
441Q8—41no—
442Q8—4242no—
443Q8—2121no—
444Q8—2121no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—531no—
452C2×C4—532no—
453C2×C4—532no—
454C2×C4—531no—
455C2×C4—1062no—
456C2×C4—1062no—
457C2×C4—1062no—
458C2×C4—1061no—
459C2×C4—1061no—
460C2×C4—4242no—
461C2×C4—4242no—
462C2×C4—4242no—
463C2×C4—4242no—
464C2×C4—2121no—
465C2×C4—2121no—
466C2×C4—2121no—
467C2×C4—2121no—
468C2×C4D8×D5311yescharacteristic
469C2×C4C2×D4×D5311yescharacteristic
470C8—21no—
471C8—532no—
472C8—1061no—
473C8#41×D5311yescharacteristic
474C8#41×D5311yescharacteristic
475C22—21no—
476C22—21no—
477C22—21no—
478C22—532no—
479C22—532no—
480C22—1061no—
481C22—1061no—
482C22—1061no—
483C22—1061no—
484C22—2124no—
485C22C2×D8×D5311yescharacteristic
486C4—21no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—532no—
492C4—1061no—
493C4—4241no—
494C4—4241no—
495C4—4241no—
496C4—4241no—
497C4#123×D5311yescharacteristic
498C4#137×D5311yescharacteristic
499C2—41no—
500C2—1062no—
501C2—1062no—
502C2#480×D5311yescharacteristic
503C2#510×D5311yescharacteristic
504C2#2028784811yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2237C2
socleC22×C53C22×D8
prime radicalC23×D53D8
hypercenter#2237D53
hypocenterC53#33649
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2237

Complements

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

NH ≅ G/Ntype
3. #20276924C2semidirect
4. #20276923C2semidirect
5. #20276920C2semidirect
6. #20276919C2semidirect
7. #20276916C2semidirect
8. #20276914C2semidirect
9. #20276911C2semidirect
10. #20276502C2semidirect
11. #20276500C2semidirect
12. #20276433C2semidirect
13. #20276433C2semidirect
15. C53×#2237C2semidirect
37. #16810451C22semidirect
38. #16810450C22semidirect
39. #16810450C22semidirect
40. #16810449C22semidirect
41. #16810448C22semidirect
42. #16810448C22semidirect
43. #16810428C22semidirect
44. #16810428C22semidirect
45. #16810427C22semidirect
46. #16810427C22semidirect
47. #16810426C22semidirect
48. #16810426C22semidirect
54. C53×#367C22semidirect
55. C53×#366C22semidirect
56. C53×#358C22semidirect
57. C53×#358C22semidirect
252. #2237D53semidirect
294. #367C2×D53semidirect
298. #366C2×D53semidirect
305. #358C2×D53semidirect
306. #358C2×D53semidirect
311. C53#33649semidirect

Maximal subgroups

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

subgroupindex
#467×D532
#427×D532
#202769242
#202769232
#202769202
#202769192
#202769162
#202769142
#202769112
#202765022
#202765002
#202764332
#202763242
C53×#22372
#3364953