Finite Group Databasebeta

← #27538917

Subgroups — #27538917

Order 13568.

632 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#466×D53C211yesmaximal, characteristic
2#427×D53C211yesmaximal, characteristic, complemented
3#20276907C211yesmaximal, characteristic, complemented
4#20276906C211yesmaximal, characteristic, complemented
5#20276903C211yesmaximal, characteristic, complemented
6#20276902C211yesmaximal, characteristic, complemented
7#20276899C211yesmaximal, characteristic, complemented
8#20276897C211yesmaximal, characteristic, complemented
9#20276894C211yesmaximal, characteristic, complemented
10#20276502C211yesmaximal, characteristic, complemented
11#20276500C211yesmaximal, characteristic, complemented
12#20276429C211yesmaximal, characteristic, complemented
13#20276429C211yesmaximal, characteristic, complemented
14#20276324C211yesmaximal, characteristic, complemented
15C53×#2234C211yesmaximal, characteristic, complemented
16C2×D8×D53C2211yescharacteristic
17C22×C8×D53C2211yescharacteristic
18#121×D53—21no—
19#117×D53—21no—
20C2×#13133156C2211yescharacteristic
21C2×#13133156C2211yescharacteristic
22#16810860C2211yescharacteristic
23#16810859C2211yescharacteristic
24#16810858—21no—
25#16810857—21no—
26#16810856C2211yescharacteristic
27#16810661C2211yescharacteristic, complemented
28#16810660C2211yescharacteristic
29#16810660C2211yescharacteristic
30#16810659—21no—
31#16810658—21no—
32#107×D53C2211yescharacteristic
33#107×D53C2211yescharacteristic, complemented
34#107×D53C2211yescharacteristic, complemented
35#106×D53—21no—
36#102×D53—21no—
37#16810451C2211yescharacteristic, complemented
38#16810450C2211yescharacteristic, complemented
39#16810450C2211yescharacteristic, complemented
40#16810449C2211yescharacteristic, complemented
41#16810448C2211yescharacteristic, complemented
42#16810448C2211yescharacteristic, complemented
43#16810425C2211yescharacteristic, complemented
44#16810425C2211yescharacteristic, complemented
45#16810424C2211yescharacteristic, complemented
46#16810424C2211yescharacteristic, complemented
47#16810423C2211yescharacteristic, complemented
48#16810423C2211yescharacteristic, complemented
49C22⋊#11719908C2211yescharacteristic, complemented
50#16810402C2211yescharacteristic, complemented
51#16810402C2211yescharacteristic, complemented
52C53×#466C2211yescharacteristic
53C53×#427C2211yescharacteristic, complemented
54C53×#367C2211yescharacteristic, complemented
55C53×#366C2211yescharacteristic, complemented
56C53×#357C2211yescharacteristic, complemented
57C53×#357C2211yescharacteristic, complemented
58C53×C22⋊C16C2211yescharacteristic, complemented
59C2×D4×D53—21no—
60C2×D4×D53—41no—
61C22×C4×D53D411yescharacteristic
62C2×D424C2311yescharacteristic
63C2×#11719922—21no—
64D8×D53—22no—
65D8×D53—22no—
66C22×#88322D411yescharacteristic
67C2×C8×D53—22no—
68C2×C8×D53—22no—
69C2×C8×D53D411yescharacteristic
70C2×C8×D53D411yescharacteristic
71C2×C8×D53C2311yescharacteristic
72C2×C8×D53C2311yescharacteristic
73#13133192—21no—
74#13133191—21no—
75#13133190—41no—
76#13133189—41no—
77#13133188—21no—
78#13133176—21no—
79#13133175—21no—
80#13133175—21no—
81#13133174—41no—
82#13133173—41no—
83C4⋊C4×D53—21no—
84C4⋊C4×D53—21no—
85C4⋊C4×D53—21no—
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#13133139—21no—
103#13133138—21no—
104#13133137—21no—
105C2×D8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107#121×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#102×C53—21no—
116C23×D53—41no—
117C23×D53D811yescharacteristic
118C2×D4×C53—21no—
119C2×D4×C53—41no—
120C22×C4×C53C2×D411yescharacteristic
121C2×C22⋊D53—41no—
122C2×C22⋊D53—41no—
123C2×C22⋊D53—41no—
124C22×Dic53D811yescharacteristic
125D4×D53—42no—
126D4×D53—42no—
127D4×D53—84no—
128C2×D212—21no—
129C2×D212—41no—
130C2×C4×D53—22no—
131C2×C4×D53—22no—
132C2×C4×D53—41no—
133C2×C4×D53—41no—
134C2×C4×D53—41no—
135C2×C4×D53D811yescharacteristic
136C2×C4×D53D811yescharacteristic
137C2×C4×D53C2×D411yescharacteristic
138C2×C4×D53C2×D411yescharacteristic
139D8×C53—21no—
140D8×C53—21no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53C2×D411yescharacteristic
144C2×C8×C53C2411yescharacteristic
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C42×C53—41no—
151C22⋊Dic53—41no—
152C22⋊Dic53—41no—
153#11719922—42no—
154#11719921—41no—
155#11719921—41no—
156#11719921—41no—
157#11719921—41no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161#11719919—41no—
162#11719919—41no—
163#11719919—41no—
164C4×Dic53—41no—
165C4×Dic53—41no—
166C4×Dic53—41no—
167C2×#88322—22no—
168C2×#88322C2×D411yescharacteristic
169C2×#88322C2×D411yescharacteristic
170D424—21no—
171D424—21no—
172C8×D53—22no—
173C8×D53—21no—
174C8×D53—21no—
175C8×D53—22no—
176C8×D53#4112yes—
177C8×D53#4112yes—
178C16×C53—21no—
179C16×C53—21no—
180#11719908—21no—
181#11719908—21no—
182C23×C53—41no—
183C23×C53C2×D811yescharacteristic
184C22×D53—22no—
185C22×D53—22no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—82no—
189C22×D53—82no—
190C22×D53—82no—
191C22×D53C2×D811yescharacteristic
192C22×D53C2×D811yescharacteristic
193D4×C53—41no—
194D4×C53—41no—
195D4×C53—82no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53C2×D811yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C22⋊D53—82no—
204C22⋊D53—82no—
205C22⋊D53—82no—
206C2×Dic53—22no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53C2×D811yescharacteristic
211C2×Dic53C2×D811yescharacteristic
212D212—41no—
213D212—41no—
214D212—82no—
215C4×D53—22no—
216C4×D53—21no—
217C4×D53—21no—
218C4×D53—22no—
219C4×D53—82no—
220C4×D53—82no—
221C4×D53—82no—
222C4×D53#12312yes—
223C4×D53#13712yes—
224C8×C53—21no—
225C8×C53C2×#4111yescharacteristic
226C8×C53C2×#4111yescharacteristic
227#88322—21no—
228#88322#12312yes—
229#33648—531nomaximal, Sylow
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—81no—
234C22×C53—81no—
235C22×C53—81no—
236C22×C53C22×D811yescharacteristic
237C2×D53—21no—
238C2×D53—21no—
239C2×D53—21no—
240C2×D53—21no—
241C2×D53—44no—
242C2×D53—81no—
243C2×D53—81no—
244C2×D53—81no—
245C2×D53—164no—
246C2×D53#48012yes—
247C2×D53#50912yes—
248C4×C53—21no—
249C4×C53—81no—
250C4×C53—81no—
251C4×C53—81no—
252C4×C53C2×#12311yescharacteristic
253C4×C53C2×#13711yescharacteristic
254Dic53—21no—
255Dic53—81no—
256Dic53—81no—
257Dic53—81no—
258Dic53#48012yes—
259C2×#466—531no—
260C2×#427—531no—
261#2234—531no—
262#2234—531no—
263#2234—531no—
264#2234D5311yescharacteristic, complemented
265#2233—531no—
266#2233—531no—
267#2232—531no—
268#2232—531no—
269#2159—531no—
270#2159—531no—
271#2140—531no—
272#2140—531no—
273#2112—531no—
274C2×C53—41no—
275C2×C53—161no—
276C2×C53C2×#48011yescharacteristic
277C2×C53C2×#50911yescharacteristic
278C2×C53#341411yescharacteristic
279D53—22no—
280D53—22no—
281D53—161no—
282C22×D8—531no—
283C23×C8—531no—
284C2×#121—1061no—
285C2×#117—1061no—
286C2×M32—531no—
287C2×M32—531no—
288#466—531no—
289#466—531no—
290#466—531no—
291#466—1061no—
292#466—1061no—
293#466C2×D5311yescharacteristic
294#427—531no—
295#427—531no—
296#427—531no—
297#427—1061no—
298#427—1061no—
299#427C2×D5311yescharacteristic, complemented
300C2×#107—531no—
301C2×#107—531no—
302C2×#107—531no—
303C2×#106—1061no—
304C2×#102—1061no—
305#367—531no—
306#367—531no—
307#367—531no—
308#367C2×D5311yescharacteristic, complemented
309#366—531no—
310#366—531no—
311#366—531no—
312#366C2×D5311yescharacteristic, complemented
313#357—531no—
314#357—531no—
315#357—531no—
316#357—531no—
317#357—531no—
318#357—531no—
319#357C2×D5311yescharacteristic, complemented
320#357C2×D5311yescharacteristic, complemented
321C22⋊C16—531no—
322C22⋊C16—531no—
323C22⋊C16—531no—
324C22⋊C16C2×D5311yescharacteristic, complemented
325C53#3364811yescharacteristic, complemented, Sylow
326C22×D4—1061no—
327C22×D4—2121no—
328C23×C4—531no—
329C2×D8—531no—
330C2×D8—1062no—
331C2×D8—1062no—
332C2×D8—1061no—
333C2×D8C22×D5311yescharacteristic
334C22×C8—531no—
335C22×C8—531no—
336C22×C8—531no—
337C22×C8—531no—
338C22×C8—531no—
339C22×C8—1062no—
340C22×C8—1062no—
341C22×C8C22×D5311yescharacteristic
342#121—21no—
343#121—1061no—
344#121—1061no—
345#121—1061no—
346#121—2121no—
347#121—2121no—
348#117—21no—
349#117—1061no—
350#117—1061no—
351#117—1061no—
352#117—2121no—
353#117—2121no—
354C2×C4⋊C4—1061no—
355C2×C4⋊C4—1061no—
356C2×C4⋊C4—1061no—
357C2×C22⋊C4—2121no—
358C2×C22⋊C4—2121no—
359C2×C42—2121no—
360M32—1062no—
361M32—1062no—
362C2×C16—531no—
363C2×C16—531no—
364C2×C16C22×D5311yescharacteristic
365C2×C16C22×D5311yescharacteristic
366#107—531no—
367#107—531no—
368#107—531no—
369#107—1061no—
370#107—1061no—
371#107—1061no—
372#107C22×D5311yescharacteristic
373#107C22×D5311yescharacteristic
374#107C22×D5311yescharacteristic
375#106—21no—
376#106—1061no—
377#106—1061no—
378#106—1061no—
379#102—21no—
380#102—1061no—
381#102—1061no—
382#102—1061no—
383C24—531no—
384C24—2121no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—1061no—
388C2×D4—4244no—
389C2×D4—2121no—
390C2×D4—2122no—
391C2×D4—2122no—
392C2×D4—2121no—
393C2×D4—2121no—
394C2×D4—2121no—
395C22×C4—531no—
396C22×C4—531no—
397C22×C4—531no—
398C22×C4—531no—
399C22×C4—531no—
400C22×C4—1062no—
401C22×C4—1062no—
402C22×C4—2121no—
403C22×C4—2121no—
404C22×C4—2121no—
405C22×C4D4×D5311yescharacteristic
406D8—21no—
407D8—21no—
408D8—1061no—
409D8—1061no—
410D8—2122no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—531no—
414C2×C8—532no—
415C2×C8—532no—
416C2×C8—531no—
417C2×C8—1062no—
418C2×C8—1062no—
419C2×C8—1062no—
420C2×C8—1061no—
421C2×C8—1061no—
422C2×C8D4×D5311yescharacteristic
423C2×C8C23×D5311yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—1061no—
428C4⋊C4—1061no—
429C4⋊C4—1061no—
430C4⋊C4—2121no—
431C4⋊C4—2121no—
432C4⋊C4—2121no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—2121no—
436C22⋊C4—2121no—
437C22⋊C4—2121no—
438C22⋊C4—2121no—
439C22⋊C4—2121no—
440C22⋊C4—2121no—
441C42—41no—
442C42—2121no—
443C42—2121no—
444C42—2121no—
445C16—21no—
446C16—21no—
447C16—1061no—
448C16—1061no—
449C23—41no—
450C23—531no—
451C23—531no—
452C23—1062no—
453C23—1062no—
454C23—4242no—
455C23—4242no—
456C23—4242no—
457C23—2121no—
458C23—2121no—
459C23D8×D5311yescharacteristic
460D4—41no—
461D4—41no—
462D4—82no—
463D4—4242no—
464D4—4242no—
465D4—4242no—
466D4—4242no—
467D4—2121no—
468D4—2121no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—531no—
475C2×C4—532no—
476C2×C4—532no—
477C2×C4—531no—
478C2×C4—1062no—
479C2×C4—1062no—
480C2×C4—1062no—
481C2×C4—1061no—
482C2×C4—1061no—
483C2×C4—4242no—
484C2×C4—4242no—
485C2×C4—4242no—
486C2×C4—2121no—
487C2×C4—2121no—
488C2×C4—2121no—
489C2×C4D8×D5311yescharacteristic
490C2×C4C2×D4×D5311yescharacteristic
491C8—21no—
492C8—532no—
493C8—1061no—
494C8#41×D5311yescharacteristic
495C8#41×D5311yescharacteristic
496C22—21no—
497C22—21no—
498C22—21no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—532no—
503C22—532no—
504C22—8484no—
505C22—1061no—
506C22—1061no—
507C22—1061no—
508C22—1061no—
509C22—4241no—
510C22—4241no—
511C22—4241no—
512C22—2124no—
513C22C2×D8×D5311yescharacteristic
514C4—21no—
515C4—81no—
516C4—81no—
517C4—81no—
518C4—532no—
519C4—1061no—
520C4—4241no—
521C4—4241no—
522C4—4241no—
523C4#123×D5311yescharacteristic
524C4#137×D5311yescharacteristic
525C2—41no—
526C2—161no—
527C2—8481no—
528C2—1062no—
529C2—1062no—
530C2#480×D5311yescharacteristic
531C2#509×D5311yescharacteristic
532C2#2028784811yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2234C2
socleC22×C53C22×D8
prime radical#466×D53C2
hypercenter#2234D53
hypocenterC53#33648
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2234

Complements

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

NH ≅ G/Ntype
2. #427×D53C2semidirect
3. #20276907C2semidirect
4. #20276906C2semidirect
5. #20276903C2semidirect
6. #20276902C2semidirect
7. #20276899C2semidirect
8. #20276897C2semidirect
9. #20276894C2semidirect
10. #20276502C2semidirect
11. #20276500C2semidirect
12. #20276429C2semidirect
13. #20276429C2semidirect
14. #20276324C2semidirect
15. C53×#2234C2semidirect
27. #16810661C22semidirect
33. #107×D53C22semidirect
34. #107×D53C22semidirect
37. #16810451C22semidirect
38. #16810450C22semidirect
39. #16810450C22semidirect
40. #16810449C22semidirect
41. #16810448C22semidirect
42. #16810448C22semidirect
43. #16810425C22semidirect
44. #16810425C22semidirect
45. #16810424C22semidirect
46. #16810424C22semidirect
47. #16810423C22semidirect
48. #16810423C22semidirect
49. C22⋊#11719908C22semidirect
50. #16810402C22semidirect
51. #16810402C22semidirect
53. C53×#427C22semidirect
54. C53×#367C22semidirect
55. C53×#366C22semidirect
56. C53×#357C22semidirect
57. C53×#357C22semidirect
58. C53×C22⋊C16C22semidirect
264. #2234D53semidirect
299. #427C2×D53semidirect
308. #367C2×D53semidirect
312. #366C2×D53semidirect
319. #357C2×D53semidirect
320. #357C2×D53semidirect
324. C22⋊C16C2×D53semidirect
325. C53#33648semidirect

Maximal subgroups

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

subgroupindex
#466×D532
#427×D532
#202769072
#202769062
#202769032
#202769022
#202768992
#202768972
#202768942
#202765022
#202765002
#202764292
#202763242
C53×#22342
#3364853