Finite Group Databasebeta

← #20286498

Subgroups — #20286498

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#16811435C211yesmaximal, characteristic, complemented
3#16811429C211yesmaximal, characteristic, complemented
4#16811427C211yesmaximal, characteristic, complemented
5#16811004C211yesmaximal, characteristic, complemented
6#16810897C211yesmaximal, characteristic, complemented
7#16810892C211yesmaximal, characteristic, complemented
8#16810887C211yesmaximal, characteristic, complemented
9#16810803C211yesmaximal, characteristic, complemented
10#16810799C211yesmaximal, characteristic, complemented
11#16810798C211yesmaximal, characteristic, complemented
12#16810794C211yesmaximal, characteristic, complemented
13#16810739C211yesmaximal, characteristic, complemented
14#16810738C211yesmaximal, characteristic, complemented
15C53×#471C211yesmaximal, characteristic, complemented
16#13133265C2211yescharacteristic, complemented
17#13133264C2211yescharacteristic, complemented
18#13133263C2211yescharacteristic
19#13133262C2211yescharacteristic
20#13133250—21no—
21#13133250C2211yescharacteristic
22#13133250C2211yescharacteristic, complemented
23#13133249—21no—
24#13133248C2211yescharacteristic
25#13133246—21no—
26#13133246—21no—
27#13133245—21no—
28#13133245C2211yescharacteristic
29#13133245C2211yescharacteristic, complemented
30#13133244—21no—
31#13133244—21no—
32#13133243C2211yescharacteristic, complemented
33#13133241—21no—
34#13133239—21no—
35#13133239C2211yescharacteristic
36#13133238—21no—
37#13133238—21no—
38#13133238—21no—
39#13133237C2211yescharacteristic
40C2×#11719925C2211yescharacteristic, complemented
41Q16×D53—21no—
42C2×#11719913—21no—
43C2×#11719913C2211yescharacteristic, complemented
44SD16×D53—21no—
45C2×#11719922C2211yescharacteristic
46D8×D53—21no—
47#13133225C2211yescharacteristic
48#13133225C2211yescharacteristic, complemented
49#13133223C2211yescharacteristic, complemented
50#13133206C2211yescharacteristic, complemented
51#13133204C2211yescharacteristic, complemented
52#13133152C2211yescharacteristic, complemented
53#13133144C2211yescharacteristic, complemented
54#13133143C2211yescharacteristic
55#13133143C2211yescharacteristic, complemented
56#13133143C2211yescharacteristic, complemented
57#13133135C2211yescharacteristic, complemented
58#13133133C2211yescharacteristic, complemented
59C8×Dic53C2211yescharacteristic, 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
67#41×C53C2311yescharacteristic
68C2×Q8×C53C2311yescharacteristic
69C2×D4×C53C2311yescharacteristic
70C2×C22⋊D53—41no—
71#11719949—21no—
72#11719949—21no—
73#11719949—21no—
74Q8×D53—21no—
75Q8×D53—21no—
76Q8×D53—21no—
77#11719947—21no—
78#11719947—21no—
79#11719947—21no—
80#11719947—41no—
81D4×D53—21no—
82D4×D53—21no—
83D4×D53—21no—
84D4×D53—41no—
85#11719945—21no—
86#11719945—21no—
87#11719945—41no—
88#11719945C2311yescharacteristic
89#11719945C2311yescharacteristic
90#11719945C2311yescharacteristic
91C2×D212—21no—
92C2×D212C2311yescharacteristic
93C2×Dic106—21no—
94C2×Dic106C2311yescharacteristic
95Q16×C53—21no—
96SD16×C53—21no—
97SD16×C53—21no—
98SD16×C53—21no—
99SD16×C53—21no—
100D8×C53—21no—
101M16×C53—21no—
102M16×C53C2311yescharacteristic
103M16×C53C2311yescharacteristic
104M16×C53C2311yescharacteristic
105C2×C8×C53—21no—
106C2×C8×C53C2311yescharacteristic
107C22⋊Dic53—41no—
108#11719925—21no—
109#11719925—21no—
110#11719925—21no—
111#11719925D411yescharacteristic
112#11719924—21no—
113#11719924D411yescharacteristic
114#11719923—21no—
115#11719923D411yescharacteristic
116#11719922—21no—
117#11719922—21no—
118#11719922—21no—
119#11719922D411yescharacteristic
120#11719921—41no—
121C4×Dic53C2311yescharacteristic
122#11719917C2311yescharacteristic
123C2×#88322C2311yescharacteristic
124Dic212—21no—
125Dic212—21no—
126Dic212D411yescharacteristic
127D424—21no—
128D424—21no—
129D424D411yescharacteristic
130#11719913—21no—
131#11719913—21no—
132#11719913—21no—
133#11719913—21no—
134#11719913—21no—
135#11719913—21no—
136#11719913D411yescharacteristic
137#11719913D411yescharacteristic
138#11719912—21no—
139#11719912—21no—
140#11719912—21no—
141C8×D53—21no—
142C8×D53—21no—
143C8×D53—21no—
144C23×C53—21no—
145C22×D53—21no—
146C22×D53—41no—
147C22×D53—41no—
148Q8×C53—21no—
149Q8×C53—21no—
150Q8×C53C2×D411yescharacteristic
151D4×C53—21no—
152D4×C53—21no—
153D4×C53—21no—
154D4×C53C2×D411yescharacteristic
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53C2411yescharacteristic
158C22⋊D53—21no—
159C22⋊D53—21no—
160C22⋊D53—21no—
161C22⋊D53—41no—
162C22⋊D53—41no—
163C22⋊D53—41no—
164C22⋊D53—41no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169D212—21no—
170D212—21no—
171D212—41no—
172D212—41no—
173D212C2×D411yescharacteristic
174D212C2×D411yescharacteristic
175D212C2×D411yescharacteristic
176C4×D53—21no—
177C4×D53—21no—
178C4×D53—21no—
179C4×D53—41no—
180C4×D53—41no—
181C4×D53—41no—
182C4×D53—41no—
183Dic106—21no—
184Dic106—21no—
185Dic106—41no—
186Dic106—41no—
187Dic106C2×D411yescharacteristic
188Dic106C2×D411yescharacteristic
189Dic106C2×D411yescharacteristic
190C8×C53—21no—
191C8×C53—21no—
192C8×C53—21no—
193C8×C53C2×D411yescharacteristic
194C8×C53C2×D411yescharacteristic
195#88322—21no—
196#88322C2×D411yescharacteristic
197#88322C2×D411yescharacteristic
198C22×C53—21no—
199C22×C53—41no—
200C22×C53—81no—
201C22×C53H(5,𝔽2)11yescharacteristic
202C2×D53—21no—
203C2×D53—21no—
204C2×D53—21no—
205C2×D53—41no—
206C2×D53—81no—
207C2×D53—161no—
208C2×D53—161no—
209C4×C53—21no—
210C4×C53—41no—
211C4×C53C22×D411yescharacteristic
212C4×C53C22×D411yescharacteristic
213Dic53—21no—
214Dic53—21no—
215Dic53—21no—
216Dic53—41no—
217Dic53—41no—
218#3290—531nomaximal, Sylow
219C2×C53—21no—
220C2×C53—41no—
221C2×C53—81no—
222C2×C53D4211yescharacteristic
223D53—41no—
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#471D5311yescharacteristic, complemented
236#455—531no—
237#455—531no—
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)—531no—
247#137—531no—
248#137—531no—
249#137—531no—
250#137—1061no—
251#137—1061no—
252#137—1061no—
253#137—1061no—
254#137C2×D5311yescharacteristic, complemented
255AGL(1,ℤ/8)—531no—
256AGL(1,ℤ/8)—531no—
257AGL(1,ℤ/8)—531no—
258AGL(1,ℤ/8)—1061no—
259AGL(1,ℤ/8)—1061no—
260AGL(1,ℤ/8)—1061no—
261AGL(1,ℤ/8)—1061no—
262AGL(1,ℤ/8)C2×D5311yescharacteristic
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—1061no—
273C2×SD16—1061no—
274C2×SD16C2×D5311yescharacteristic, complemented
275C2×D8—531no—
276C2×D8—1061no—
277#131—531no—
278#131—531no—
279#131—531no—
280#131C2×D5311yescharacteristic, complemented
281#124—531no—
282#124—531no—
283#108—531no—
284#108C2×D5311yescharacteristic, complemented
285#104—531no—
286#104—531no—
287#104—531no—
288#104—531no—
289#101—531no—
290#101C2×D5311yescharacteristic, complemented
291#100—531no—
292#100C2×D5311yescharacteristic, complemented
293C4×C8—531no—
294#41—531no—
295#41—531no—
296#41—531no—
297#41—1061no—
298#41—1061no—
299#41—1061no—
300#41—1061no—
301#41—1061no—
302#41—1061no—
303#41—1061no—
304#41—1061no—
305#41—2121no—
306#41—2121no—
307#41C22×D5311yescharacteristic
308C2×Q8—531no—
309C2×Q8—1061no—
310C2×Q8—1061no—
311C2×Q8—1061no—
312C2×Q8—1061no—
313C2×Q8C22×D5311yescharacteristic
314C2×D4—531no—
315C2×D4—1061no—
316C2×D4—1061no—
317C2×D4—1061no—
318C2×D4—1061no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4C22×D5311yescharacteristic
322Q16—21no—
323Q16—531no—
324Q16—531no—
325Q16—1061no—
326Q16—1061no—
327Q16—1061no—
328Q16—1061no—
329Q16—1061no—
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—531no—
335SD16—531no—
336SD16—531no—
337SD16—531no—
338SD16—1061no—
339SD16—1061no—
340SD16—1061no—
341SD16—1061no—
342SD16—1061no—
343SD16—1061no—
344SD16—1061no—
345SD16—1061no—
346D8—21no—
347D8—531no—
348D8—531no—
349D8—1061no—
350D8—1061no—
351D8—1061no—
352D8—1061no—
353D8—1061no—
354M16—21no—
355M16—531no—
356M16—1061no—
357M16—1061no—
358M16—1061no—
359M16C22×D5311yescharacteristic
360M16C22×D5311yescharacteristic
361M16C22×D5311yescharacteristic
362C2×C8—21no—
363C2×C8—531no—
364C2×C8—1061no—
365C2×C8—1061no—
366C2×C8—1061no—
367C2×C8C22×D5311yescharacteristic
368C22⋊C4—2121no—
369C22⋊C4—2121no—
370C42—531no—
371C23—21no—
372C23—1061no—
373C23—2121no—
374C23—2121no—
375Q8—21no—
376Q8—21no—
377Q8—531no—
378Q8—531no—
379Q8—531no—
380Q8—1061no—
381Q8—1061no—
382Q8—2121no—
383Q8—2121no—
384Q8D4×D5311yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—531no—
389D4—531no—
390D4—531no—
391D4—1061no—
392D4—1061no—
393D4—1061no—
394D4—1061no—
395D4—1061no—
396D4—2121no—
397D4—2121no—
398D4—2121no—
399D4—2121no—
400D4—2121no—
401D4—2121no—
402D4D4×D5311yescharacteristic
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—1061no—
406C2×C4—1061no—
407C2×C4—1061no—
408C2×C4—1061no—
409C2×C4—1061no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4—2121no—
414C2×C4—2121no—
415C2×C4—2121no—
416C2×C4C23×D5311yescharacteristic
417C8—21no—
418C8—21no—
419C8—21no—
420C8—531no—
421C8—531no—
422C8—1061no—
423C8D4×D5311yescharacteristic
424C8D4×D5311yescharacteristic
425C22—21no—
426C22—41no—
427C22—81no—
428C22—8481no—
429C22—8481no—
430C22—1061no—
431C22—1061no—
432C22—1061no—
433C22—4241no—
434C22—2121no—
435C22#1313326311yescharacteristic
436C4—21no—
437C4—41no—
438C4—1061no—
439C4—1061no—
440C4—1061no—
441C4—2121no—
442C4—2121no—
443C4C2×D4×D5311yescharacteristic
444C4C2×D4×D5311yescharacteristic
445C2—21no—
446C2—41no—
447C2—81no—
448C2—4241no—
449C2—2121no—
450C2—2121no—
451C2—2121no—
452C2#1681125711yescharacteristic
subgroupG/H
centerC2#16811257
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#471C2
socleC2×C53D42
hypercenter#471D53
hypocenterC53#3290
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. #16811439C2semidirect
2. #16811435C2semidirect
3. #16811429C2semidirect
4. #16811427C2semidirect
5. #16811004C2semidirect
6. #16810897C2semidirect
7. #16810892C2semidirect
8. #16810887C2semidirect
9. #16810803C2semidirect
10. #16810799C2semidirect
11. #16810798C2semidirect
12. #16810794C2semidirect
13. #16810739C2semidirect
14. #16810738C2semidirect
15. C53×#471C2semidirect
16. #13133265C22semidirect
17. #13133264C22semidirect
22. #13133250C22semidirect
29. #13133245C22semidirect
32. #13133243C22semidirect
40. C2×#11719925C22semidirect
43. C2×#11719913C22semidirect
48. #13133225C22semidirect
49. #13133223C22semidirect
50. #13133206C22semidirect
51. #13133204C22semidirect
52. #13133152C22semidirect
53. #13133144C22semidirect
55. #13133143C22semidirect
56. #13133143C22semidirect
57. #13133135C22semidirect
58. #13133133C22semidirect
59. C8×Dic53C22semidirect
60. #137×C53C22semidirect
62. C2×SD16×C53C22semidirect
63. #131×C53C22semidirect
64. #108×C53C22semidirect
65. #101×C53C22semidirect
66. #100×C53C22semidirect
235. #471D53semidirect
242. C53#3290semidirect
254. #137C2×D53semidirect
274. C2×SD16C2×D53semidirect
280. #131C2×D53semidirect
284. #108C2×D53semidirect
290. #101C2×D53semidirect
292. #100C2×D53semidirect