Finite Group Databasebeta

← #20284438

Subgroups — #20284438

Order 6784.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811422C211yesmaximal, characteristic, complemented
2C2×#13133244C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D53C211yesmaximal, characteristic, complemented
4#16811388C211yesmaximal, characteristic, complemented
5#16810894C211yesmaximal, characteristic, complemented
6#16810893C211yesmaximal, characteristic, complemented
7#16810892C211yesmaximal, characteristic, complemented
8#16810891C211yesmaximal, characteristic, complemented
9#16810889C211yesmaximal, characteristic, complemented
10#16810888C211yesmaximal, characteristic, complemented
11#16810883C211yesmaximal, characteristic, complemented
12#16810670C211yesmaximal, characteristic, complemented
13#16810606C211yesmaximal, characteristic, complemented
14#100×D53C211yesmaximal, characteristic, complemented
15C53×#471C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17#41×D53C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19#13133249—21no—
20#13133248C2211yescharacteristic, complemented
21#13133245C2211yescharacteristic
22#13133245C2211yescharacteristic, complemented
23#13133244—21no—
24#13133244—21no—
25#13133244—21no—
26#13133244—21no—
27#13133244—21no—
28#13133244—21no—
29#13133243C2211yescharacteristic
30#13133242C2211yescharacteristic
31#13133242C2211yescharacteristic, complemented
32#13133241—21no—
33#13133240—21no—
34#13133238—21no—
35C2×#11719924C2211yescharacteristic, complemented
36SD16×D53—21no—
37C2×D424C2211yescharacteristic, complemented
38C2×#11719922C2211yescharacteristic, complemented
39D8×D53—21no—
40#13133225—21no—
41#13133225C2211yescharacteristic, complemented
42#13133224C2211yescharacteristic, complemented
43#13133223—21no—
44#13133223C2211yescharacteristic, complemented
45C2×#11719912—21no—
46C2×#11719912C2211yescharacteristic
47M16×D53—21no—
48M16×D53C2211yescharacteristic
49M16×D53C2211yescharacteristic
50M16×D53C2211yescharacteristic
51#13133153C2211yescharacteristic, complemented
52#13133152C2211yescharacteristic, complemented
53#13133152C2211yescharacteristic, complemented
54#13133136C2211yescharacteristic, complemented
55#13133134C2211yescharacteristic, complemented
56#13133133C2211yescharacteristic, complemented
57#13133133C2211yescharacteristic, complemented
58#13133133C2211yescharacteristic, complemented
59#13133133C2211yescharacteristic, complemented
60#137×C53C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C53C2211yescharacteristic
62C2×SD16×C53C2211yescharacteristic, complemented
63#131×C53C2211yescharacteristic, complemented
64#108×C53C2211yescharacteristic, complemented
65#101×C53C2211yescharacteristic, complemented
66#100×C53C2211yescharacteristic, complemented
67C23×D53—21no—
68#41×C53C2311yescharacteristic
69C2×Q8×C53C2311yescharacteristic
70C2×D4×C53C2311yescharacteristic
71C2×C22⋊D53—21no—
72#11719949—21no—
73#11719949—21no—
74#11719949—21no—
75#11719949—21no—
76#11719949D411yescharacteristic
77Q8×D53D411yescharacteristic
78#11719947—21no—
79#11719947D411yescharacteristic
80D4×D53—21no—
81D4×D53—21no—
82D4×D53—21no—
83D4×D53—21no—
84D4×D53—21no—
85D4×D53D411yescharacteristic
86#11719945—21no—
87#11719945C2311yescharacteristic
88C2×D212—21no—
89C2×D212C2311yescharacteristic
90C2×D212C2311yescharacteristic
91C2×C4×D53—21no—
92C2×C4×D53—21no—
93C2×C4×D53C2311yescharacteristic
94Q16×C53—21no—
95SD16×C53—21no—
96SD16×C53—21no—
97SD16×C53—21no—
98SD16×C53—21no—
99D8×C53—21no—
100M16×C53—21no—
101M16×C53C2311yescharacteristic
102M16×C53C2311yescharacteristic
103M16×C53C2311yescharacteristic
104C2×C8×C53—21no—
105C2×C8×C53C2311yescharacteristic
106#11719925—21no—
107#11719924—21no—
108#11719924—21no—
109#11719924—21no—
110#11719924—21no—
111#11719924—21no—
112#11719923—21no—
113#11719922—21no—
114#11719922—21no—
115#11719922—21no—
116#11719922—21no—
117#11719922—21no—
118#11719917—21no—
119#11719917C2311yescharacteristic
120#11719917C2311yescharacteristic
121#11719917C2311yescharacteristic
122C2×#88322—21no—
123C2×#88322C2311yescharacteristic
124D424—21no—
125D424—21no—
126D424—21no—
127D424—21no—
128#11719913—21no—
129#11719913—21no—
130#11719912—21no—
131#11719912—21no—
132#11719912—21no—
133#11719912—21no—
134#11719912—21no—
135#11719912D411yescharacteristic
136#11719912D411yescharacteristic
137#11719912D411yescharacteristic
138#11719912D411yescharacteristic
139C8×D53—21no—
140C8×D53—21no—
141C8×D53—21no—
142C8×D53—21no—
143C8×D53—21no—
144C23×C53—21no—
145C22×D53—21no—
146C22×D53—21no—
147C22×D53—21no—
148C22×D53—41no—
149C22×D53—41no—
150C22×D53—82no—
151C22×D53#4111yescharacteristic
152Q8×C53—21no—
153Q8×C53—21no—
154Q8×C53C2×D411yescharacteristic
155D4×C53—21no—
156D4×C53—21no—
157D4×C53—21no—
158D4×C53C2×D411yescharacteristic
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53C2411yescharacteristic
162C22⋊D53—21no—
163C22⋊D53—21no—
164C22⋊D53—21no—
165C22⋊D53—41no—
166C22⋊D53—41no—
167C2×Dic53—21no—
168C2×Dic53#4111yescharacteristic
169D212—21no—
170D212—21no—
171D212—21no—
172D212—21no—
173D212—21no—
174D212—41no—
175D212—41no—
176D212C2×D411yescharacteristic
177C4×D53—21no—
178C4×D53—21no—
179C4×D53—21no—
180C4×D53—41no—
181C4×D53—41no—
182C4×D53C2×D411yescharacteristic
183C4×D53C2×D411yescharacteristic
184C4×D53C2×D411yescharacteristic
185C4×D53C2×D411yescharacteristic
186Dic106—21no—
187Dic106C2×D411yescharacteristic
188C8×C53—21no—
189C8×C53—21no—
190C8×C53—21no—
191C8×C53C2×D411yescharacteristic
192C8×C53C2×D411yescharacteristic
193#88322—21no—
194#88322—21no—
195#88322—21no—
196#88322C2×D411yescharacteristic
197#88322C2×D411yescharacteristic
198C22×C53—21no—
199C22×C53—41no—
200C22×C53—81no—
201C22×C53C2×#4111yescharacteristic
202C2×D53—21no—
203C2×D53—41no—
204C2×D53—41no—
205C2×D53—41no—
206C2×D53—81no—
207C2×D53—82no—
208C2×D53—82no—
209C2×D53—81no—
210C2×D53—81no—
211C2×D53#12111yescharacteristic
212C2×D53#12111yescharacteristic
213C4×C53—21no—
214C4×C53—41no—
215C4×C53C22×D411yescharacteristic
216C4×C53C22×D411yescharacteristic
217Dic53—21no—
218Dic53#12111yescharacteristic
219Dic53#12111yescharacteristic
220#3068—531nomaximal, Sylow
221C2×C53—21no—
222C2×C53—41no—
223C2×C53—81no—
224C2×C53C2×#12111yescharacteristic
225D53—21no—
226D53—21no—
227D53—41no—
228D53—81no—
229D53—81no—
230#575—531no—
231C2×AGL(1,ℤ/8)—531no—
232C2×AGL(1,ℤ/8)—531no—
233#568—531no—
234#472—531no—
235#472—531no—
236#472—531no—
237#472—531no—
238#471—531no—
239#471—531no—
240#471—531no—
241#471D5311yescharacteristic, complemented
242#430—531no—
243#413—531no—
244C2×#100—531no—
245C53#306811yescharacteristic, complemented, Sylow
246C2×#41—531no—
247C2×#41—531no—
248C22×D4—531no—
249#137—531no—
250#137—1061no—
251#137C2×D5311yescharacteristic, complemented
252AGL(1,ℤ/8)—531no—
253AGL(1,ℤ/8)—531no—
254AGL(1,ℤ/8)—531no—
255AGL(1,ℤ/8)—531no—
256AGL(1,ℤ/8)—531no—
257AGL(1,ℤ/8)—1061no—
258AGL(1,ℤ/8)—1061no—
259AGL(1,ℤ/8)—1061no—
260AGL(1,ℤ/8)—1061no—
261AGL(1,ℤ/8)—1061no—
262AGL(1,ℤ/8)—1061no—
263AGL(1,ℤ/8)—1061no—
264AGL(1,ℤ/8)C2×D5311yescharacteristic
265#135—1061no—
266#135—1061no—
267C2×SD16—531no—
268C2×SD16—1061no—
269C2×SD16C2×D5311yescharacteristic, complemented
270C2×D8—531no—
271C2×D8—531no—
272C2×D8—1061no—
273#131—531no—
274#131—531no—
275#131—531no—
276#131—1061no—
277#131—1061no—
278#131C2×D5311yescharacteristic, complemented
279C2×M16—531no—
280C2×M16—531no—
281C2×M16—531no—
282C2×M16—531no—
283C2×M16—1061no—
284C2×M16—1061no—
285#108—531no—
286#108—531no—
287#108—531no—
288#108C2×D5311yescharacteristic, complemented
289#101—531no—
290#101C2×D5311yescharacteristic, complemented
291#100—531no—
292#100—531no—
293#100—531no—
294#100—531no—
295#100—531no—
296#100C2×D5311yescharacteristic, complemented
297C24—1061no—
298#41—531no—
299#41—531no—
300#41—531no—
301#41—1061no—
302#41—1061no—
303#41—1061no—
304#41—1061no—
305#41—1061no—
306#41—1061no—
307#41C22×D5311yescharacteristic
308C2×Q8—531no—
309C2×Q8C22×D5311yescharacteristic
310C2×D4—531no—
311C2×D4—531no—
312C2×D4—531no—
313C2×D4—1061no—
314C2×D4—1061no—
315C2×D4—1061no—
316C2×D4—1061no—
317C2×D4—1061no—
318C2×D4—1061no—
319C2×D4—1061no—
320C2×D4C22×D5311yescharacteristic
321C22×C4—531no—
322C22×C4—1061no—
323C22×C4—1061no—
324Q16—21no—
325Q16—1061no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—21no—
330SD16—1061no—
331SD16—1061no—
332SD16—1061no—
333SD16—1061no—
334SD16—1061no—
335SD16—1061no—
336SD16—1061no—
337SD16—1061no—
338D8—21no—
339D8—1061no—
340D8—1061no—
341D8—1061no—
342D8—1061no—
343D8—1061no—
344D8—1061no—
345D8—1061no—
346D8—1061no—
347D8—1061no—
348M16—21no—
349M16—531no—
350M16—531no—
351M16—531no—
352M16—531no—
353M16—531no—
354M16—531no—
355M16—531no—
356M16—1061no—
357M16—1061no—
358M16—1061no—
359M16—1061no—
360M16—1061no—
361M16—1061no—
362M16C22×D5311yescharacteristic
363M16C22×D5311yescharacteristic
364M16C22×D5311yescharacteristic
365C2×C8—21no—
366C2×C8—531no—
367C2×C8—1061no—
368C2×C8—1061no—
369C2×C8—1061no—
370C2×C8—1061no—
371C2×C8—1061no—
372C2×C8—1061no—
373C2×C8C22×D5311yescharacteristic
374C23—21no—
375C23—531no—
376C23—1061no—
377C23—1061no—
378C23—1061no—
379C23—4242no—
380C23—2121no—
381C23—2121no—
382Q8—21no—
383Q8—21no—
384Q8—531no—
385Q8—1061no—
386Q8D4×D5311yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—531no—
391D4—1061no—
392D4—1061no—
393D4—1061no—
394D4—1061no—
395D4—1061no—
396D4—1061no—
397D4—1061no—
398D4—1061no—
399D4—2121no—
400D4—2121no—
401D4—2121no—
402D4—2121no—
403D4D4×D5311yescharacteristic
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—531no—
407C2×C4—531no—
408C2×C4—531no—
409C2×C4—531no—
410C2×C4—531no—
411C2×C4—1061no—
412C2×C4—1061no—
413C2×C4—1061no—
414C2×C4—1061no—
415C2×C4—2121no—
416C2×C4—2121no—
417C2×C4C23×D5311yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—531no—
422C8—531no—
423C8—1061no—
424C8—1061no—
425C8—1061no—
426C8D4×D5311yescharacteristic
427C8D4×D5311yescharacteristic
428C22—21no—
429C22—41no—
430C22—81no—
431C22—531no—
432C22—531no—
433C22—1061no—
434C22—4241no—
435C22—4241no—
436C22—4241no—
437C22—4242no—
438C22—4242no—
439C22—2121no—
440C22—2121no—
441C22—2121no—
442C22#41×D5311yescharacteristic
443C4—21no—
444C4—41no—
445C4—531no—
446C4—531no—
447C4—1061no—
448C4C2×D4×D5311yescharacteristic
449C4C2×D4×D5311yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—1061no—
454C2—1061no—
455C2—4241no—
456C2—4241no—
457C2—2121no—
458C2#121×D5311yescharacteristic
subgroupG/H
centerC2#121×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#471C2
socleC2×C53C2×#121
hypercenter#471D53
hypocenterC53#3068
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #16811422C2semidirect
2. C2×#13133244C2semidirect
3. AGL(1,ℤ/8)×D53C2semidirect
4. #16811388C2semidirect
5. #16810894C2semidirect
6. #16810893C2semidirect
7. #16810892C2semidirect
8. #16810891C2semidirect
9. #16810889C2semidirect
10. #16810888C2semidirect
11. #16810883C2semidirect
12. #16810670C2semidirect
13. #16810606C2semidirect
14. #100×D53C2semidirect
15. C53×#471C2semidirect
20. #13133248C22semidirect
22. #13133245C22semidirect
31. #13133242C22semidirect
35. C2×#11719924C22semidirect
37. C2×D424C22semidirect
38. C2×#11719922C22semidirect
41. #13133225C22semidirect
42. #13133224C22semidirect
44. #13133223C22semidirect
51. #13133153C22semidirect
52. #13133152C22semidirect
53. #13133152C22semidirect
54. #13133136C22semidirect
55. #13133134C22semidirect
56. #13133133C22semidirect
57. #13133133C22semidirect
58. #13133133C22semidirect
59. #13133133C22semidirect
60. #137×C53C22semidirect
62. C2×SD16×C53C22semidirect
63. #131×C53C22semidirect
64. #108×C53C22semidirect
65. #101×C53C22semidirect
66. #100×C53C22semidirect
241. #471D53semidirect
245. C53#3068semidirect
251. #137C2×D53semidirect
269. C2×SD16C2×D53semidirect
278. #131C2×D53semidirect
288. #108C2×D53semidirect
290. #101C2×D53semidirect
296. #100C2×D53semidirect