Finite Group Databasebeta

← #20286499

Subgroups — #20286499

Order 6784.

454 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811439C211yesmaximal, characteristic, complemented
2#16811434C211yesmaximal, characteristic, complemented
3#16811432C211yesmaximal, characteristic, complemented
4#16811426C211yesmaximal, characteristic, complemented
5#16811005C211yesmaximal, characteristic, complemented
6#16810896C211yesmaximal, characteristic, complemented
7#16810892C211yesmaximal, characteristic, complemented
8#16810888C211yesmaximal, characteristic, complemented
9#16810885C211yesmaximal, characteristic, complemented
10#16810802C211yesmaximal, characteristic, complemented
11#16810797C211yesmaximal, characteristic, complemented
12#16810795C211yesmaximal, characteristic, complemented
13#16810737C211yesmaximal, characteristic, complemented
14#16810737C211yesmaximal, characteristic, complemented
15C53×#455C211yesmaximal, characteristic, complemented
16#13133265C2211yescharacteristic, complemented
17#13133264C2211yescharacteristic, complemented
18#13133263C2211yescharacteristic
19#13133250C2211yescharacteristic, complemented
20#13133249—21no—
21#13133249—21no—
22#13133247—21no—
23#13133247C2211yescharacteristic
24#13133246—21no—
25#13133245C2211yescharacteristic, complemented
26#13133244—21no—
27#13133243—21no—
28#13133243—21no—
29#13133243C2211yescharacteristic
30#13133243C2211yescharacteristic, complemented
31#13133242—21no—
32#13133242C2211yescharacteristic
33#13133240—21no—
34#13133238—21no—
35#13133237—21no—
36#13133237—21no—
37#13133237C2211yescharacteristic
38#13133237C2211yescharacteristic
39C2×#11719925C2211yescharacteristic, complemented
40Q16×D53—21no—
41C2×#11719923—21no—
42C2×#11719923C2211yescharacteristic, complemented
43C2×#11719924C2211yescharacteristic, complemented
44SD16×D53—21no—
45C2×#11719922—21no—
46C2×#11719922C2211yescharacteristic
47#13133225C2211yescharacteristic, complemented
48#13133225C2211yescharacteristic, complemented
49#13133224C2211yescharacteristic
50#13133224C2211yescharacteristic, complemented
51#13133205C2211yescharacteristic, complemented
52#13133152C2211yescharacteristic, complemented
53#13133152C2211yescharacteristic, complemented
54#13133145C2211yescharacteristic, complemented
55#13133145C2211yescharacteristic, complemented
56#13133136C2211yescharacteristic, complemented
57#13133135C2211yescharacteristic, complemented
58#13133133C2211yescharacteristic, complemented
59C4×#88322C2211yescharacteristic, complemented
60H(5,𝔽2)×C53C2211yescharacteristic
61#137×C53C2211yescharacteristic
62#137×C53C2211yescharacteristic, complemented
63#124×C53C2211yescharacteristic, complemented
64#104×C53C2211yescharacteristic
65#104×C53C2211yescharacteristic, complemented
66#100×C53C2211yescharacteristic, complemented
67#41×C53—21no—
68#41×C53—21no—
69#41×C53C2311yescharacteristic
70#41×C53C2311yescharacteristic
71C2×Q8×C53C2311yescharacteristic
72C2×D4×C53—21no—
73C2×D4×C53—21no—
74C2×D4×C53—41no—
75C2×D4×C53C2311yescharacteristic
76#11719949—21no—
77#11719949—21no—
78#11719949—21no—
79Q8×D53—21no—
80Q8×D53—21no—
81Q8×D53—21no—
82#11719947—21no—
83#11719947—41no—
84D4×D53—21no—
85D4×D53—41no—
86#11719945—21no—
87#11719945—21no—
88#11719945—41no—
89#11719945C2311yescharacteristic
90#11719945C2311yescharacteristic
91C2×D212—21no—
92C2×D212C2311yescharacteristic
93C2×Dic106—21no—
94C2×Dic106C2311yescharacteristic
95Q16×C53—21no—
96Q16×C53—21no—
97SD16×C53—21no—
98SD16×C53—21no—
99M16×C53C2311yescharacteristic
100M16×C53C2311yescharacteristic
101C22⋊C4×C53—41no—
102C42×C53C2311yescharacteristic
103#11719925—21no—
104#11719925—21no—
105#11719925—21no—
106#11719925D411yescharacteristic
107#11719925D411yescharacteristic
108#11719924—21no—
109#11719924—21no—
110#11719924—21no—
111#11719924D411yescharacteristic
112#11719924D411yescharacteristic
113#11719923—21no—
114#11719923—21no—
115#11719923—21no—
116#11719923—21no—
117#11719923—21no—
118#11719923D411yescharacteristic
119#11719923D411yescharacteristic
120#11719922—21no—
121#11719922—21no—
122#11719922—21no—
123#11719922—21no—
124#11719922—21no—
125#11719922D411yescharacteristic
126#11719922D411yescharacteristic
127#11719921—41no—
128#11719917—21no—
129#11719917—21no—
130#11719917C2311yescharacteristic
131#11719917C2311yescharacteristic
132C2×#88322—21no—
133C2×#88322—21no—
134C2×#88322C2311yescharacteristic
135C2×#88322C2311yescharacteristic
136Dic212—21no—
137D424—21no—
138#11719913—21no—
139#11719913—21no—
140#11719912—21no—
141#11719912—21no—
142C8×D53—21no—
143C8×D53—21no—
144C23×C53—21no—
145C23×C53—41no—
146C22×D53—21no—
147C22×D53—41no—
148Q8×C53—21no—
149Q8×C53—21no—
150Q8×C53C2×D411yescharacteristic
151Q8×C53C2×D411yescharacteristic
152D4×C53—21no—
153D4×C53—21no—
154D4×C53—21no—
155D4×C53—21no—
156D4×C53—41no—
157D4×C53—41no—
158D4×C53C2×D411yescharacteristic
159D4×C53C2×D411yescharacteristic
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—41no—
165C2×C4×C53C2411yescharacteristic
166C22⋊D53—21no—
167C22⋊D53—21no—
168C22⋊D53—41no—
169C22⋊D53—41no—
170C2×Dic53—21no—
171C2×Dic53—41no—
172D212—21no—
173D212—21no—
174D212—41no—
175D212—41no—
176D212C2×D411yescharacteristic
177D212C2×D411yescharacteristic
178C4×D53—21no—
179C4×D53—21no—
180C4×D53—41no—
181C4×D53—41no—
182C4×D53—41no—
183C4×D53—41no—
184Dic106—21no—
185Dic106—21no—
186Dic106—41no—
187Dic106—41no—
188Dic106C2×D411yescharacteristic
189Dic106C2×D411yescharacteristic
190C8×C53—21no—
191C8×C53—21no—
192#88322—21no—
193#88322—21no—
194#88322C2×D411yescharacteristic
195#88322C2×D411yescharacteristic
196#88322C2×D411yescharacteristic
197#88322C2×D411yescharacteristic
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—41no—
201C22×C53—81no—
202C22×C53—161no—
203C22×C53H(5,𝔽2)11yescharacteristic
204C2×D53—21no—
205C2×D53—21no—
206C2×D53—41no—
207C2×D53—81no—
208C2×D53—161no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—41no—
212C4×C53—41no—
213C4×C53C22×D411yescharacteristic
214C4×C53C22×D411yescharacteristic
215Dic53—21no—
216Dic53—21no—
217Dic53—41no—
218#3290—531nomaximal, Sylow
219C2×C53—21no—
220C2×C53—41no—
221C2×C53—41no—
222C2×C53—81no—
223C2×C53D4211yescharacteristic
224D53—41no—
225D53—41no—
226D53—81no—
227#578—531no—
228#577—531no—
229#577—531no—
230#576—531no—
231#495—531no—
232#473—531no—
233#472—531no—
234#471—531no—
235#471—531no—
236#455—531no—
237#455D5311yescharacteristic, complemented
238#454—531no—
239#454—531no—
240#443—531no—
241#443—531no—
242C53#329011yescharacteristic, complemented, Sylow
2432−1+4—531no—
2442−1+4—531no—
245H(5,𝔽2)—531no—
246H(5,𝔽2)C2×D5311yescharacteristic
247#137—531no—
248#137—531no—
249#137—1061no—
250#137—1061no—
251#137—1061no—
252#137—1061no—
253#137C2×D5311yescharacteristic
254#137C2×D5311yescharacteristic, complemented
255AGL(1,ℤ/8)—531no—
256AGL(1,ℤ/8)—531no—
257AGL(1,ℤ/8)—531no—
258AGL(1,ℤ/8)—531no—
259AGL(1,ℤ/8)—1061no—
260AGL(1,ℤ/8)—1061no—
261AGL(1,ℤ/8)—1061no—
262AGL(1,ℤ/8)—1061no—
263#135—531no—
264#135—531no—
265#135—1061no—
266#135—1061no—
267#135—1061no—
268#135—1061no—
269C2×Q16—531no—
270C2×Q16—1061no—
271C2×SD16—531no—
272C2×SD16—531no—
273C2×SD16—1061no—
274C2×SD16—1061no—
275C2×D8—531no—
276C2×D8—1061no—
277#131—531no—
278#131—531no—
279#131—531no—
280#131—531no—
281#124—531no—
282#124C2×D5311yescharacteristic, complemented
283#108—531no—
284#108—531no—
285#104—531no—
286#104—531no—
287#104C2×D5311yescharacteristic
288#104C2×D5311yescharacteristic, complemented
289#101—531no—
290#101—531no—
291#100—531no—
292#100C2×D5311yescharacteristic, complemented
293C4×C8—531no—
294#41—21no—
295#41—21no—
296#41—531no—
297#41—531no—
298#41—1061no—
299#41—1061no—
300#41—1061no—
301#41—1061no—
302#41—1061no—
303#41—1061no—
304#41—2121no—
305#41—2121no—
306#41C22×D5311yescharacteristic
307#41C22×D5311yescharacteristic
308C2×Q8—531no—
309C2×Q8—1061no—
310C2×Q8—1061no—
311C2×Q8—1061no—
312C2×Q8—1061no—
313C2×Q8C22×D5311yescharacteristic
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—531no—
318C2×D4—1061no—
319C2×D4—1061no—
320C2×D4—2121no—
321C2×D4C22×D5311yescharacteristic
322Q16—21no—
323Q16—21no—
324Q16—531no—
325Q16—531no—
326Q16—1061no—
327Q16—1061no—
328Q16—1061no—
329Q16—1061no—
330SD16—21no—
331SD16—21no—
332SD16—531no—
333SD16—531no—
334SD16—531no—
335SD16—531no—
336SD16—1061no—
337SD16—1061no—
338SD16—1061no—
339SD16—1061no—
340SD16—1061no—
341SD16—1061no—
342SD16—1061no—
343SD16—1061no—
344SD16—1061no—
345SD16—1061no—
346D8—531no—
347D8—531no—
348D8—1061no—
349D8—1061no—
350D8—1061no—
351D8—1061no—
352D8—1061no—
353D8—1061no—
354M16—531no—
355M16—531no—
356M16—1061no—
357M16—1061no—
358M16—1061no—
359M16—1061no—
360M16C22×D5311yescharacteristic
361M16C22×D5311yescharacteristic
362C2×C8—531no—
363C2×C8—531no—
364C2×C8—1061no—
365C2×C8—1061no—
366C2×C8—1061no—
367C2×C8—1061no—
368C22⋊C4—41no—
369C22⋊C4—2121no—
370C42C22×D5311yescharacteristic
371C23—21no—
372C23—41no—
373C23—1061no—
374C23—2121no—
375Q8—21no—
376Q8—21no—
377Q8—531no—
378Q8—531no—
379Q8—1061no—
380Q8—1061no—
381Q8—2121no—
382Q8—2121no—
383Q8D4×D5311yescharacteristic
384Q8D4×D5311yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—21no—
389D4—41no—
390D4—41no—
391D4—531no—
392D4—531no—
393D4—1061no—
394D4—1061no—
395D4—1061no—
396D4—1061no—
397D4—2121no—
398D4—2121no—
399D4—2121no—
400D4—2121no—
401D4D4×D5311yescharacteristic
402D4D4×D5311yescharacteristic
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—1061no—
409C2×C4—1061no—
410C2×C4—1061no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4—2121no—
414C2×C4—2121no—
415C2×C4—2121no—
416C2×C4C23×D5311yescharacteristic
417C8—21no—
418C8—21no—
419C8—531no—
420C8—531no—
421C8—531no—
422C8—531no—
423C8—1061no—
424C8—1061no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—81no—
429C22—161no—
430C22—8481no—
431C22—1061no—
432C22—1061no—
433C22—4241no—
434C22—2121no—
435C22#1313326211yescharacteristic
436C4—21no—
437C4—21no—
438C4—41no—
439C4—41no—
440C4—1061no—
441C4—1061no—
442C4—2121no—
443C4C2×D4×D5311yescharacteristic
444C4C2×D4×D5311yescharacteristic
445C2—21no—
446C2—41no—
447C2—41no—
448C2—81no—
449C2—4241no—
450C2—2121no—
451C2—2121no—
452C2#1681125511yescharacteristic
subgroupG/H
centerC2#16811255
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#455C2
socleC2×C53D42
hypercenter#455D53
hypocenterC53#3290
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #16811439C2semidirect
2. #16811434C2semidirect
3. #16811432C2semidirect
4. #16811426C2semidirect
5. #16811005C2semidirect
6. #16810896C2semidirect
7. #16810892C2semidirect
8. #16810888C2semidirect
9. #16810885C2semidirect
10. #16810802C2semidirect
11. #16810797C2semidirect
12. #16810795C2semidirect
13. #16810737C2semidirect
14. #16810737C2semidirect
15. C53×#455C2semidirect
16. #13133265C22semidirect
17. #13133264C22semidirect
19. #13133250C22semidirect
25. #13133245C22semidirect
30. #13133243C22semidirect
39. C2×#11719925C22semidirect
42. C2×#11719923C22semidirect
43. C2×#11719924C22semidirect
47. #13133225C22semidirect
48. #13133225C22semidirect
50. #13133224C22semidirect
51. #13133205C22semidirect
52. #13133152C22semidirect
53. #13133152C22semidirect
54. #13133145C22semidirect
55. #13133145C22semidirect
56. #13133136C22semidirect
57. #13133135C22semidirect
58. #13133133C22semidirect
59. C4×#88322C22semidirect
62. #137×C53C22semidirect
63. #124×C53C22semidirect
65. #104×C53C22semidirect
66. #100×C53C22semidirect
237. #455D53semidirect
242. C53#3290semidirect
254. #137C2×D53semidirect
282. #124C2×D53semidirect
288. #104C2×D53semidirect
292. #100C2×D53semidirect