Finite Group Databasebeta

← #20286482

Subgroups — #20286482

Order 6784.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811435C211yesmaximal, characteristic, complemented
2#16811430C211yesmaximal, characteristic, complemented
3#16811428C211yesmaximal, characteristic, complemented
4#16811425C211yesmaximal, characteristic, complemented
5#16811012C211yesmaximal, characteristic, complemented
6#16810894C211yesmaximal, characteristic, complemented
7#16810893C211yesmaximal, characteristic, complemented
8#16810887C211yesmaximal, characteristic, complemented
9#16810886C211yesmaximal, characteristic, complemented
10#16810802C211yesmaximal, characteristic, complemented
11#16810797C211yesmaximal, characteristic, complemented
12#16810792C211yesmaximal, characteristic, complemented
13#16810792C211yesmaximal, characteristic, complemented
14#16810746C211yesmaximal, characteristic, complemented
15C53×#444C211yesmaximal, characteristic, complemented
16#13133264C2211yescharacteristic, complemented
17#13133263C2211yescharacteristic
18#13133263C2211yescharacteristic, complemented
19#13133263C2211yescharacteristic, complemented
20#13133250—21no—
21#13133250C2211yescharacteristic, complemented
22#13133249—21no—
23#13133247C2211yescharacteristic, complemented
24#13133245—21no—
25#13133245—21no—
26#13133245—21no—
27#13133245C2211yescharacteristic
28#13133245C2211yescharacteristic, complemented
29#13133245C2211yescharacteristic, complemented
30#13133244—21no—
31#13133244—21no—
32#13133242C2211yescharacteristic
33#13133242C2211yescharacteristic, complemented
34#13133242C2211yescharacteristic, complemented
35#13133241—21no—
36#13133240—21no—
37#13133239—21no—
38#13133239—21no—
39#13133239C2211yescharacteristic
40#13133238—21no—
41C2×#11719913—21no—
42C2×#11719913C2211yescharacteristic, complemented
43C2×#11719913C2211yescharacteristic, complemented
44SD16×D53—21no—
45C2×D424—21no—
46C2×D424C2211yescharacteristic, complemented
47C2×D424C2211yescharacteristic, complemented
48D8×D53—21no—
49#13133223C2211yescharacteristic, complemented
50#13133223C2211yescharacteristic, complemented
51#13133214C2211yescharacteristic, complemented
52#13133205C2211yescharacteristic, complemented
53#13133153C2211yescharacteristic, complemented
54#13133145C2211yescharacteristic, complemented
55#13133145C2211yescharacteristic, complemented
56#13133135C2211yescharacteristic, complemented
57#13133133C2211yescharacteristic, complemented
58#13133133C2211yescharacteristic, complemented
59#13133133C2211yescharacteristic, complemented
60#135×C53C2211yescharacteristic
61#131×C53C2211yescharacteristic, complemented
62#131×C53C2211yescharacteristic, complemented
63#108×C53C2211yescharacteristic, complemented
64#104×C53C2211yescharacteristic, complemented
65#104×C53C2211yescharacteristic, complemented
66#97×C53C2211yescharacteristic, complemented
67#41×C53C2311yescharacteristic
68#41×C53C2311yescharacteristic
69#11719949—21no—
70#11719949—21no—
71#11719949—21no—
72Q8×D53—21no—
73#11719947—21no—
74#11719947—41no—
75D4×D53—21no—
76D4×D53—21no—
77D4×D53—21no—
78D4×D53—41no—
79D4×D53—41no—
80D4×D53—41no—
81#11719945—21no—
82#11719945—21no—
83#11719945—21no—
84#11719945—21no—
85#11719945C2311yescharacteristic
86#11719945C2311yescharacteristic
87C2×D212—21no—
88C2×D212—21no—
89C2×D212—21no—
90C2×D212—41no—
91C2×D212C2311yescharacteristic
92C2×D212C2311yescharacteristic
93C2×D212C2311yescharacteristic
94C2×Dic106—21no—
95C2×Dic106C2311yescharacteristic
96Q16×C53D411yescharacteristic, complemented
97SD16×C53D411yescharacteristic, complemented
98SD16×C53D411yescharacteristic, complemented
99D8×C53D411yescharacteristic, complemented
100M16×C53—21no—
101M16×C53—21no—
102M16×C53C2311yescharacteristic
103M16×C53C2311yescharacteristic
104C2×C8×C53—21no—
105C2×C8×C53—21no—
106C2×C8×C53C2311yescharacteristic
107C2×C8×C53C2311yescharacteristic
108C42×C53C2311yescharacteristic
109#11719925—21no—
110#11719924—21no—
111#11719924—21no—
112#11719924—21no—
113#11719923—21no—
114#11719922—21no—
115#11719922—21no—
116#11719922—21no—
117#11719921—41no—
118#11719917C2311yescharacteristic
119#11719917C2311yescharacteristic
120Dic212—21no—
121Dic212D411yescharacteristic, complemented
122D424—21no—
123D424—21no—
124D424—21no—
125D424—21no—
126D424—21no—
127D424—21no—
128D424—21no—
129D424D411yescharacteristic, complemented
130#11719913—21no—
131#11719913—21no—
132#11719913—21no—
133#11719913—21no—
134#11719913—21no—
135#11719913—21no—
136#11719913—21no—
137#11719913—21no—
138#11719913D411yescharacteristic, complemented
139#11719913D411yescharacteristic, complemented
140#11719912—21no—
141#11719912—21no—
142C8×D53—21no—
143C8×D53—21no—
144C22×D53—21no—
145C22×D53—21no—
146C22×D53—21no—
147C22×D53—41no—
148C22×D53—41no—
149C22×D53—41no—
150Q8×C53C2×D411yescharacteristic
151Q8×C53C2×D411yescharacteristic
152D4×C53—21no—
153D4×C53—21no—
154D4×C53C2×D411yescharacteristic
155D4×C53C2×D411yescharacteristic
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53C2411yescharacteristic
160C22⋊D53—21no—
161C22⋊D53—21no—
162C22⋊D53—41no—
163C22⋊D53—41no—
164C22⋊D53—41no—
165C22⋊D53—41no—
166C2×Dic53—21no—
167C2×Dic53—41no—
168D212—21no—
169D212—21no—
170D212—21no—
171D212—21no—
172D212—21no—
173D212—21no—
174D212—41no—
175D212—41no—
176D212—41no—
177D212—82no—
178D212C2×D411yescharacteristic
179D212C2×D411yescharacteristic
180C4×D53—21no—
181C4×D53—21no—
182C4×D53—41no—
183C4×D53—41no—
184C4×D53—41no—
185C4×D53—41no—
186Dic106—21no—
187Dic106—21no—
188Dic106—41no—
189Dic106C2×D411yescharacteristic
190Dic106C2×D411yescharacteristic
191C8×C53—21no—
192C8×C53—21no—
193C8×C53C2×D411yescharacteristic
194C8×C53C2×D411yescharacteristic
195C8×C53C2×D411yescharacteristic
196C8×C53C2×D411yescharacteristic
197#88322—21no—
198#88322—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53H(5,𝔽2)11yescharacteristic
202C2×D53—21no—
203C2×D53—21no—
204C2×D53—41no—
205C2×D53—41no—
206C2×D53—41no—
207C2×D53—81no—
208C2×D53—81no—
209C2×D53—81no—
210C2×D53—161no—
211C2×D53—161no—
212C2×D53—161no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—41no—
216C4×C53C22×D411yescharacteristic
217C4×C53C22×D411yescharacteristic
218Dic53—21no—
219Dic53—21no—
220Dic53—41no—
221#3288—531nomaximal, Sylow
222C2×C53—21no—
223C2×C53—41no—
224C2×C53—41no—
225C2×C53D4211yescharacteristic
226D53—41no—
227D53—41no—
228D53—81no—
229D53—81no—
230D53—81no—
231#577—531no—
232#577—531no—
233#576—531no—
234#576—531no—
235#496—531no—
236#472—531no—
237#472—531no—
238#471—531no—
239#471—531no—
240#455—531no—
241#454—531no—
242#453—531no—
243#453—531no—
244#444—531no—
245#444D5311yescharacteristic, complemented
246C53#328811yescharacteristic, complemented, Sylow
2472−1+4—531no—
248H(5,𝔽2)—531no—
249H(5,𝔽2)—531no—
250H(5,𝔽2)—531no—
251#137—531no—
252#137—531no—
253#137—1061no—
254#137—1061no—
255AGL(1,ℤ/8)—531no—
256AGL(1,ℤ/8)—531no—
257AGL(1,ℤ/8)—531no—
258AGL(1,ℤ/8)—531no—
259AGL(1,ℤ/8)—531no—
260AGL(1,ℤ/8)—531no—
261AGL(1,ℤ/8)—1061no—
262AGL(1,ℤ/8)—1061no—
263AGL(1,ℤ/8)—1061no—
264AGL(1,ℤ/8)—1061no—
265AGL(1,ℤ/8)—1061no—
266AGL(1,ℤ/8)—1061no—
267#135—531no—
268#135—1061no—
269#135—1061no—
270#135—1061no—
271#135—1061no—
272#135C2×D5311yescharacteristic
273C2×SD16—531no—
274C2×SD16—531no—
275C2×SD16—1061no—
276C2×SD16—1061no—
277C2×D8—531no—
278C2×D8—531no—
279C2×D8—1061no—
280C2×D8—1061no—
281#131—531no—
282#131—531no—
283#131C2×D5311yescharacteristic, complemented
284#131C2×D5311yescharacteristic, complemented
285#127—531no—
286#124—531no—
287#108—531no—
288#108C2×D5311yescharacteristic, complemented
289#104—531no—
290#104—531no—
291#104C2×D5311yescharacteristic, complemented
292#104C2×D5311yescharacteristic, complemented
293#101—531no—
294#100—531no—
295#100—531no—
296#100—531no—
297#97C2×D5311yescharacteristic, complemented
298#41—531no—
299#41—531no—
300#41—1061no—
301#41—1061no—
302#41—1061no—
303#41—1061no—
304#41—1061no—
305#41—1061no—
306#41—1061no—
307#41—1061no—
308#41—2121no—
309#41C22×D5311yescharacteristic
310#41C22×D5311yescharacteristic
311C2×Q8—531no—
312C2×Q8—1061no—
313C2×Q8—1061no—
314C2×D4—531no—
315C2×D4—531no—
316C2×D4—531no—
317C2×D4—1061no—
318C2×D4—1061no—
319C2×D4—1061no—
320C2×D4—1061no—
321C2×D4—1061no—
322C2×D4—1061no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C2×D4—2121no—
327Q16—531no—
328Q16—1061no—
329Q16—1061no—
330Q16D21211yescharacteristic, complemented
331SD16—531no—
332SD16—531no—
333SD16—1061no—
334SD16—1061no—
335SD16—1061no—
336SD16—1061no—
337SD16—1061no—
338SD16—1061no—
339SD16—1061no—
340SD16—1061no—
341SD16—1061no—
342SD16—1061no—
343SD16—1061no—
344SD16—1061no—
345SD16D21211yescharacteristic, complemented
346SD16D21211yescharacteristic, complemented
347D8—531no—
348D8—1061no—
349D8—1061no—
350D8—1061no—
351D8—1061no—
352D8—1061no—
353D8—1061no—
354D8—1061no—
355D8—1061no—
356D8—1061no—
357D8—1061no—
358D8D21211yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—531no—
362M16—531no—
363M16—1061no—
364M16—1061no—
365M16C22×D5311yescharacteristic
366M16C22×D5311yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—1061no—
370C2×C8—1061no—
371C2×C8C22×D5311yescharacteristic
372C2×C8C22×D5311yescharacteristic
373C22⋊C4—2121no—
374C42C22×D5311yescharacteristic
375C23—1061no—
376C23—1061no—
377C23—1061no—
378C23—2121no—
379C23—2121no—
380C23—2121no—
381Q8—531no—
382Q8—531no—
383Q8—1061no—
384Q8—1061no—
385Q8—2121no—
386Q8C2×D21211yescharacteristic
387Q8C2×D21211yescharacteristic
388D4—21no—
389D4—21no—
390D4—531no—
391D4—531no—
392D4—1061no—
393D4—1061no—
394D4—1061no—
395D4—1061no—
396D4—1061no—
397D4—1061no—
398D4—1061no—
399D4—1061no—
400D4—4242no—
401D4—2121no—
402D4—2121no—
403D4—2121no—
404D4—2121no—
405D4—2121no—
406D4—2121no—
407D4—2121no—
408D4C2×D21211yescharacteristic
409D4C2×D21211yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—1061no—
414C2×C4—1061no—
415C2×C4—1061no—
416C2×C4—2121no—
417C2×C4—2121no—
418C2×C4—2121no—
419C2×C4—2121no—
420C2×C4—2121no—
421C2×C4C23×D5311yescharacteristic
422C8—21no—
423C8—21no—
424C8—1061no—
425C8—1061no—
426C8C2×D21211yescharacteristic
427C8C2×D21211yescharacteristic
428C8D4×D5311yescharacteristic
429C8D4×D5311yescharacteristic
430C22—21no—
431C22—21no—
432C22—8481no—
433C22—8481no—
434C22—8481no—
435C22—1061no—
436C22—1061no—
437C22—4241no—
438C22—4241no—
439C22—4241no—
440C22—2121no—
441C22—2121no—
442C22—2121no—
443C22#1313326311yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—1061no—
448C4—1061no—
449C4—2121no—
450C4C2×D4×D5311yescharacteristic
451C4C22×D21211yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—4241no—
456C2—4241no—
457C2—4241no—
458C2—2121no—
459C2—2121no—
460C2D4×D21211yescharacteristic
subgroupG/H
centerC2D4×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#444C2
socleC2×C53D42
hypercenter#444D53
hypocenterC53#3288
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #16811435C2semidirect
2. #16811430C2semidirect
3. #16811428C2semidirect
4. #16811425C2semidirect
5. #16811012C2semidirect
6. #16810894C2semidirect
7. #16810893C2semidirect
8. #16810887C2semidirect
9. #16810886C2semidirect
10. #16810802C2semidirect
11. #16810797C2semidirect
12. #16810792C2semidirect
13. #16810792C2semidirect
14. #16810746C2semidirect
15. C53×#444C2semidirect
16. #13133264C22semidirect
18. #13133263C22semidirect
19. #13133263C22semidirect
21. #13133250C22semidirect
23. #13133247C22semidirect
28. #13133245C22semidirect
29. #13133245C22semidirect
33. #13133242C22semidirect
34. #13133242C22semidirect
42. C2×#11719913C22semidirect
43. C2×#11719913C22semidirect
46. C2×D424C22semidirect
47. C2×D424C22semidirect
49. #13133223C22semidirect
50. #13133223C22semidirect
51. #13133214C22semidirect
52. #13133205C22semidirect
53. #13133153C22semidirect
54. #13133145C22semidirect
55. #13133145C22semidirect
56. #13133135C22semidirect
57. #13133133C22semidirect
58. #13133133C22semidirect
59. #13133133C22semidirect
61. #131×C53C22semidirect
62. #131×C53C22semidirect
63. #108×C53C22semidirect
64. #104×C53C22semidirect
65. #104×C53C22semidirect
66. #97×C53C22semidirect
96. Q16×C53D4semidirect
97. SD16×C53D4semidirect
98. SD16×C53D4semidirect
99. D8×C53D4semidirect
121. Dic212D4semidirect
129. D424D4semidirect
138. #11719913D4semidirect
139. #11719913D4semidirect
245. #444D53semidirect
246. C53#3288semidirect
283. #131C2×D53semidirect
284. #131C2×D53semidirect
288. #108C2×D53semidirect
291. #104C2×D53semidirect
292. #104C2×D53semidirect
297. #97C2×D53semidirect
330. Q16D212semidirect
345. SD16D212semidirect
346. SD16D212semidirect
358. D8D212semidirect