Finite Group Databasebeta

← #20285578

Subgroups — #20285578

Order 6784.

442 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811190C211yesmaximal, characteristic, complemented
2#16811189C211yesmaximal, characteristic, complemented
3#16810948C211yesmaximal, characteristic, complemented
4#16810946C211yesmaximal, characteristic, complemented
5#16810945C211yesmaximal, characteristic, complemented
6#16810944C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810873C211yesmaximal, characteristic, complemented
9#16810858C211yesmaximal, characteristic, complemented
10#16810856C211yesmaximal, characteristic, complemented
11#16810757C211yesmaximal, characteristic, complemented
12#16810755C211yesmaximal, characteristic, complemented
13#16810755C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#447C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic, complemented
18C2×#11719922C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719912C2211yescharacteristic, complemented
21C2×C8×D53C2211yescharacteristic, complemented
22#13133213—21no—
23#13133213—21no—
24#13133204—21no—
25#13133203—21no—
26#13133192C2211yescharacteristic, complemented
27#13133191C2211yescharacteristic
28#13133188C2211yescharacteristic
29#13133188C2211yescharacteristic
30#13133188C2211yescharacteristic, complemented
31#13133188C2211yescharacteristic, complemented
32#13133188C2211yescharacteristic, complemented
33#13133187—21no—
34#13133186—21no—
35#13133186—21no—
36#13133185—21no—
37#13133176C2211yescharacteristic, complemented
38#13133175C2211yescharacteristic, complemented
39#13133151C2211yescharacteristic, complemented
40#13133150C2211yescharacteristic, complemented
41#13133149C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic, complemented
46#13133138C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic, complemented
48#13133138C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52C22×D4×C53C2211yescharacteristic, complemented
53C2×D8×C53C2211yescharacteristic
54C2×D8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic, complemented
59C24×C53—21no—
60C23×D53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—21no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C2×D4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C2×C22⋊D53D411yescharacteristic
77C2×C22⋊D53D411yescharacteristic
78D4×D53—41no—
79D4×D53—41no—
80#11719945—41no—
81#11719945—41no—
82C2×D212—21no—
83C2×D212D411yescharacteristic
84C2×D212C2311yescharacteristic
85C2×C4×D53C2311yescharacteristic
86C2×C4×D53C2311yescharacteristic
87C2×Dic106D411yescharacteristic
88D8×C53—41no—
89D8×C53—41no—
90C2×C8×C53C2311yescharacteristic
91C2×C8×C53C2311yescharacteristic
92C4⋊C4×C53C2311yescharacteristic
93C22⋊C4×C53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97C22⋊Dic53—21no—
98C22⋊Dic53—21no—
99C22⋊Dic53—21no—
100#11719922—41no—
101#11719922—41no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105C4⋊Dic53C2311yescharacteristic
106C4⋊Dic53C2311yescharacteristic
107C4⋊Dic53C2311yescharacteristic
108C4×Dic53—21no—
109C4×Dic53—21no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112#11719912—41no—
113C8×D53—41no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—42no—
118C23×C53—81no—
119C23×C53—81no—
120C23×C53C2×D411yescharacteristic
121C22×D53—21no—
122C22×D53—42no—
123C22×D53—81no—
124C22×D53—81no—
125C22×D53C2×D411yescharacteristic
126C22×D53C2×D411yescharacteristic
127D4×C53—21no—
128D4×C53—21no—
129D4×C53—21no—
130D4×C53—21no—
131D4×C53—21no—
132D4×C53—21no—
133D4×C53—42no—
134D4×C53—81no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2411yescharacteristic
140C22⋊D53—41no—
141C22⋊D53—41no—
142C22⋊D53—42no—
143C22⋊D53—81no—
144C22⋊D53—81no—
145C22⋊D53—81no—
146C22⋊D53—81no—
147C22⋊D53—81no—
148C22⋊D53—81no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53C2×D411yescharacteristic
153C2×Dic53C2×D411yescharacteristic
154D212—21no—
155D212—21no—
156D212—41no—
157D212—81no—
158C4×D53—21no—
159C4×D53—21no—
160C4×D53—21no—
161C4×D53—21no—
162Dic106—41no—
163C8×C53—21no—
164C8×C53—21no—
165#88322—21no—
166#88322—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—84no—
180C22×C53C22×D411yescharacteristic
181C2×D53—21no—
182C2×D53—21no—
183C2×D53—21no—
184C2×D53—21no—
185C2×D53—21no—
186C2×D53—21no—
187C2×D53—41no—
188C2×D53—41no—
189C2×D53—41no—
190C2×D53—84no—
191C4×C53—21no—
192C4×C53—41no—
193C4×C53H(5,𝔽2)11yescharacteristic
194C4×C53H(5,𝔽2)11yescharacteristic
195Dic53—21no—
196Dic53—21no—
197Dic53—41no—
198Dic53—41no—
199Dic53—41no—
200#3199—531nomaximal, Sylow
201C2×C53—41no—
202C2×C53—81no—
203C2×C53—81no—
204C2×C53—81no—
205C2×C53#53411yescharacteristic
206C2×C53#57611yescharacteristic
207C2×C53#57611yescharacteristic
208D53—41no—
209D53—41no—
210D53—81no—
211#535—531no—
212#535—531no—
213#482—531no—
214#482—531no—
215#482—531no—
216#482—531no—
217#469—531no—
218#469—531no—
219#466—531no—
220#466—531no—
221#447—531no—
222#447—531no—
223#447—531no—
224#447D5311yescharacteristic, complemented
225#408—531no—
226C53#319911yescharacteristic, complemented, Sylow
227C2×#41—531no—
228C22×D4—531no—
229C22×D4C2×D5311yescharacteristic, complemented
230C2×D8—531no—
231C2×D8—531no—
232C2×D8C2×D5311yescharacteristic
233C2×D8C2×D5311yescharacteristic
234C2×M16—531no—
235C22×C8—531no—
236#127—1061no—
237#127—1061no—
238#124—1061no—
239#124—1061no—
240#121—531no—
241#121—531no—
242#121—531no—
243#121—531no—
244#121—531no—
245#121—531no—
246#121—531no—
247#121C2×D5311yescharacteristic
248#120—1061no—
249#120—1061no—
250#120—1061no—
251#120—1061no—
252#117—531no—
253#117—531no—
254#107—531no—
255#107—531no—
256#106—531no—
257#106—531no—
258#102—531no—
259#102—531no—
260#102—531no—
261#102—531no—
262#102—531no—
263#102—531no—
264#102C2×D5311yescharacteristic, complemented
265#102C2×D5311yescharacteristic, complemented
266C22⋊C8—531no—
267C22⋊C8—531no—
268C22⋊C8—531no—
269C22⋊C8C2×D5311yescharacteristic, complemented
270C24—21no—
271C24—1061no—
272#41—2121no—
273#41—2121no—
274C2×Q8—531no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—531no—
280C2×D4—531no—
281C2×D4—531no—
282C2×D4—531no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1061no—
289C2×D4—1061no—
290C2×D4—1061no—
291C2×D4—2121no—
292C2×D4—2121no—
293C2×D4C22×D5311yescharacteristic
294C2×D4C22×D5311yescharacteristic
295C2×D4C22×D5311yescharacteristic
296C22×C4—531no—
297C22×C4—531no—
298C22×C4C22×D5311yescharacteristic
299D8—41no—
300D8—41no—
301D8—2121no—
302D8—2121no—
303M16—2121no—
304C2×C8—531no—
305C2×C8—531no—
306C2×C8—2121no—
307C2×C8C22×D5311yescharacteristic
308C2×C8C22×D5311yescharacteristic
309C4⋊C4—531no—
310C4⋊C4—531no—
311C4⋊C4—531no—
312C4⋊C4C22×D5311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C22⋊C4—1061no—
322C22⋊C4—1061no—
323C42—1061no—
324C42—1061no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—42no—
329C23—81no—
330C23—81no—
331C23—531no—
332C23—531no—
333C23—1061no—
334C23—4241no—
335C23—4241no—
336C23—2122no—
337C23D4×D5311yescharacteristic
338Q8—2121no—
339D4—21no—
340D4—21no—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—42no—
346D4—81no—
347D4—1061no—
348D4—1061no—
349D4—4241no—
350D4—4241no—
351D4—4241no—
352D4—4241no—
353D4—4241no—
354D4—4241no—
355D4—4241no—
356D4—2121no—
357D4—2121no—
358D4—2121no—
359D4—2122no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—531no—
364C2×C4—531no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—1061no—
372C2×C4D4×D5311yescharacteristic
373C2×C4C23×D5311yescharacteristic
374C8—21no—
375C8—21no—
376C8—1061no—
377C8—1061no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—84no—
391C22—1061no—
392C22—1061no—
393C22—1061no—
394C22—1061no—
395C22—1061no—
396C22—1061no—
397C22—4244no—
398C22—2121no—
399C22—2121no—
400C22—2121no—
401C22C2×D4×D5311yescharacteristic
402C4—21no—
403C4—41no—
404C4—1061no—
405C4—1061no—
406C4—2121no—
407C4—2121no—
408C4—2121no—
409C4#1313326211yescharacteristic
410C4#1313326211yescharacteristic
411C2—41no—
412C2—81no—
413C2—81no—
414C2—81no—
415C2—4241no—
416C2—2121no—
417C2—2121no—
418C2#1681118311yescharacteristic
419C2#1681142411yescharacteristic
420C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#447C2
socleC22×C53C22×D4
hypercenter#447D53
hypocenterC53#3199
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #16811190C2semidirect
2. #16811189C2semidirect
3. #16810948C2semidirect
4. #16810946C2semidirect
5. #16810945C2semidirect
6. #16810944C2semidirect
7. #16810875C2semidirect
8. #16810873C2semidirect
9. #16810858C2semidirect
10. #16810856C2semidirect
11. #16810757C2semidirect
12. #16810755C2semidirect
13. #16810755C2semidirect
14. #16810588C2semidirect
15. C53×#447C2semidirect
17. C2×D4×D53C22semidirect
20. C2×#11719912C22semidirect
21. C2×C8×D53C22semidirect
26. #13133192C22semidirect
30. #13133188C22semidirect
31. #13133188C22semidirect
32. #13133188C22semidirect
37. #13133176C22semidirect
38. #13133175C22semidirect
39. #13133151C22semidirect
40. #13133150C22semidirect
41. #13133149C22semidirect
42. #13133148C22semidirect
43. #13133139C22semidirect
44. #13133139C22semidirect
45. #13133138C22semidirect
46. #13133138C22semidirect
47. #13133138C22semidirect
48. #13133138C22semidirect
49. C22⋊#88322C22semidirect
50. #13133129C22semidirect
51. #13133129C22semidirect
52. C22×D4×C53C22semidirect
56. #102×C53C22semidirect
57. #102×C53C22semidirect
58. C22⋊C8×C53C22semidirect
224. #447D53semidirect
226. C53#3199semidirect
229. C22×D4C2×D53semidirect
264. #102C2×D53semidirect
265. #102C2×D53semidirect
269. C22⋊C8C2×D53semidirect