Finite Group Databasebeta

← #20283966

Subgroups — #20283966

Order 6784.

806 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#11719949C211yesmaximal, characteristic, complemented
2C2×#13133244C212yesmaximal, complemented
3#121×D53C211yesmaximal, characteristic, complemented
4#16810771C212yesmaximal, complemented
5#16810767C212yesmaximal, complemented
6#16810762C211yesmaximal, characteristic, complemented
7#16810759C211yesmaximal, characteristic, complemented
8#16810756C211yesmaximal, characteristic, complemented
9#16810623C212yesmaximal, complemented
10#16810585C211yesmaximal, characteristic, complemented
11C53×#448C211yesmaximal, characteristic, complemented
12C2×#11719949—28no—
13C2×#11719949—21no—
14C2×#11719949—21no—
15C2×#11719949C2212yes—
16C2×#11719949C2212yescomplemented
17C22×D212—21no—
18C22×D212C2211yescharacteristic, complemented
19C2×D4×D53—21no—
20C2×D4×D53C2211yescharacteristic, complemented
21C22×C4×D53—21no—
22C22×C4×D53C2211yescharacteristic
23#13133244—28no—
24C2×#11719924C2212yescomplemented
25C2×D424C2212yescomplemented
26C2×#11719922C2212yescomplemented
27C2×#11719912C2212yescomplemented
28#13133192C2211yescharacteristic, complemented
29#13133191C2211yescharacteristic
30#13133190—21no—
31#13133189—21no—
32#13133188C2211yescharacteristic, complemented
33C4⋊C4×D53C2211yescharacteristic, complemented
34C22⋊C4×D53—21no—
35#13133141C2212yescomplemented
36#13133139C2212yescomplemented
37#13133137C2212yescomplemented
38C22⋊#88322C2211yescharacteristic, complemented
39#13133129C2212yescomplemented
40C22×Q8×C53C2211yescharacteristic, complemented
41C2×SD16×C53C2212yescomplemented
42#121×C53C2211yescharacteristic
43#103×C53C2212yescomplemented
44C22⋊C8×C53C2211yescharacteristic, complemented
45C23×D53—21no—
46C23×D53—21no—
47C23×D53D411yescharacteristic
48C2×Q8×C53—24no—
49C2×Q8×C53—21no—
50C2×Q8×C53C2312yes—
51C2×D4×C53—21no—
52C2×D4×C53C2311yescharacteristic
53C22×C4×C53—21no—
54C22×C4×C53C2311yescharacteristic
55C2×C22⋊D53—21no—
56C2×C22⋊D53—21no—
57C2×C22⋊D53—21no—
58C22×Dic53D411yescharacteristic
59#11719949—28no—
60#11719949—28no—
61#11719949—24no—
62#11719949—24no—
63#11719949—24no—
64#11719949D418yescomplemented
65D4×D53—24no—
66D4×D53—44no—
67C2×D212—28no—
68C2×D212—22no—
69C2×D212—22no—
70C2×D212—21no—
71C2×D212—24no—
72C2×D212—21no—
73C2×D212C2311yescharacteristic
74C2×D212C2312yes—
75C2×C4×D53—28no—
76C2×C4×D53—22no—
77C2×C4×D53—22no—
78C2×C4×D53—24no—
79C2×C4×D53—21no—
80C2×C4×D53D411yescharacteristic
81C2×C4×D53D411yescharacteristic
82C2×C4×D53C2311yescharacteristic
83C2×C4×D53C2311yescharacteristic
84SD16×C53—24no—
85C2×C8×C53C2312yes—
86C4⋊C4×C53C2311yescharacteristic
87C22⋊C4×C53—21no—
88C22⋊Dic53—21no—
89#11719924—24no—
90#11719922—24no—
91#11719921—21no—
92#11719921—21no—
93C4⋊Dic53C2311yescharacteristic
94#11719919—21no—
95C2×#88322C2312yes—
96D424—24no—
97#11719912—24no—
98C23×C53—21no—
99C23×C53C2×D411yescharacteristic
100C22×D53—24no—
101C22×D53—22no—
102C22×D53—22no—
103C22×D53—22no—
104C22×D53—21no—
105C22×D53—41no—
106C22×D53—42no—
107C22×D53—42no—
108C22×D53—42no—
109C22×D53—42no—
110C22×D53—42no—
111C22×D53C2×D411yescharacteristic
112C22×D53C2×D411yescharacteristic
113Q8×C53—24no—
114Q8×C53—22no—
115Q8×C53C2×D414yescomplemented
116D4×C53—22no—
117D4×C53—42no—
118C2×C4×C53—24no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—21no—
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2411yescharacteristic
124C22⋊D53—42no—
125C22⋊D53—42no—
126C22⋊D53—42no—
127C2×Dic53—22no—
128C2×Dic53—21no—
129C2×Dic53C2×D411yescharacteristic
130C2×Dic53C2×D411yescharacteristic
131D212—28no—
132D212—24no—
133D212—28no—
134D212—22no—
135D212—22no—
136D212—42no—
137D212C2×D414yescomplemented
138C4×D53—28no—
139C4×D53—22no—
140C4×D53—22no—
141C4×D53—28no—
142C4×D53—22no—
143C4×D53—42no—
144C4×D53C2×D414yes—
145C8×C53—22no—
146#88322—22no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53AGL(1,ℤ/8)12yescomplemented
153C22×C53C22×D411yescharacteristic
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—24no—
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—44no—
161C2×D53—48no—
162C2×D53—42no—
163C2×D53—42no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—84no—
168C2×D53#12012yes—
169C4×C53—24no—
170C4×C53—21no—
171C4×C53—41no—
172C4×C53C22×D412yes—
173Dic53—21no—
174Dic53—41no—
175Dic53#12012yes—
176#3006—531nomaximal, Sylow
177C2×C53—22no—
178C2×C53—81no—
179C2×C53C2×#12011yescharacteristic
180C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C53#57511yescharacteristic
182D53—22no—
183D53—41no—
184D53—44no—
185D53—81no—
186C22×#41—531no—
187C2×AGL(1,ℤ/8)—532no—
188C2×#121—531no—
189#449—532no—
190#449—532no—
191#448—531no—
192#448D5311yescharacteristic, complemented
193#447—531no—
194#447—531no—
195#417—532no—
196#407—531no—
197C53#300611yescharacteristic, complemented, Sylow
198C2×#41—532no—
199C2×#41—532no—
200C2×#41—1061no—
201C2×#41—1061no—
202C2×#41—1068no—
203C22×Q8C2×D5311yescharacteristic, complemented
204C22×D4—531no—
205C22×D4—531no—
206C22×D4—1061no—
207C22×D4—1061no—
208C23×C4—531no—
209C23×C4—1061no—
210AGL(1,ℤ/8)—1068no—
211C2×SD16—532no—
212C2×SD16C2×D5312yescomplemented
213C2×D8—532no—
214C2×D8—532no—
215C2×M16—532no—
216#121—531no—
217#121—531no—
218#121—531no—
219#121—1061no—
220#121—1061no—
221#121C2×D5311yescharacteristic
222C2×C4⋊C4—531no—
223C2×C22⋊C4—1061no—
224#103—532no—
225#103C2×D5312yescomplemented
226#102—532no—
227#102—532no—
228C22⋊C8—531no—
229C22⋊C8—532no—
230C22⋊C8C2×D5311yescharacteristic, complemented
231C24—531no—
232C24—1061no—
233C24—1061no—
234#41—538no—
235#41—1064no—
236#41—1064no—
237#41—1068no—
238#41—1068no—
239#41—1064no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8C22×D5312yes—
243C2×D4—21no—
244C2×D4—532no—
245C2×D4—531no—
246C2×D4—1064no—
247C2×D4—1062no—
248C2×D4—1062no—
249C2×D4—1068no—
250C2×D4—1061no—
251C2×D4—1064no—
252C2×D4—1061no—
253C2×D4—1061no—
254C2×D4—1061no—
255C2×D4—1061no—
256C2×D4—2124no—
257C2×D4C22×D5311yescharacteristic
258C22×C4—21no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—531no—
264C22×C4—1064no—
265C22×C4—1062no—
266C22×C4—1062no—
267C22×C4—1068no—
268C22×C4—1061no—
269C22×C4C22×D5311yescharacteristic
270SD16—24no—
271SD16—1064no—
272D8—1064no—
273D8—1064no—
274M16—1064no—
275C2×C8—532no—
276C2×C8C22×D5312yes—
277C4⋊C4—531no—
278C4⋊C4—1061no—
279C4⋊C4C22×D5311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C23—21no—
285C23—531no—
286C23—531no—
287C23—1062no—
288C23—1064no—
289C23—1062no—
290C23—1062no—
291C23—1061no—
292C23—2121no—
293C23—2122no—
294C23—2122no—
295C23—2122no—
296C23—2122no—
297C23—2122no—
298C23D4×D5311yescharacteristic
299Q8—24no—
300Q8—22no—
301Q8D4×D5314yescomplemented
302D4—22no—
303D4—42no—
304D4—534no—
305D4—1068no—
306D4—1068no—
307D4—1064no—
308D4—1062no—
309D4—1062no—
310D4—2122no—
311D4—2122no—
312D4—2122no—
313D4—2122no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—531no—
319C2×C4—534no—
320C2×C4—531no—
321C2×C4—1062no—
322C2×C4—1068no—
323C2×C4—1062no—
324C2×C4—1068no—
325C2×C4—1062no—
326C2×C4—1062no—
327C2×C4—1061no—
328C2×C4—2122no—
329C2×C4D4×D5311yescharacteristic
330C2×C4C23×D5311yescharacteristic
331C8—22no—
332C8—1062no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—532no—
339C22—1061no—
340C22—1061no—
341C22—1064no—
342C22—1061no—
343C22—1061no—
344C22—1061no—
345C22—4244no—
346C22—2124no—
347C22—2128no—
348C22—2121no—
349C22—2121no—
350C22—2121no—
351C22—2122no—
352C22—2122no—
353C22#1313324412yescomplemented
354C22C2×D4×D5311yescharacteristic
355C4—24no—
356C4—21no—
357C4—41no—
358C4—532no—
359C4—1061no—
360C4—2121no—
361C4C2×D4×D5312yes—
362C2—22no—
363C2—81no—
364C2—1062no—
365C2—4241no—
366C2—2121no—
367C2—2124no—
368C2#120×D5311yescharacteristic
369C2C2×#1313324411yescharacteristic
370C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#448C2
socleC22×C53C22×D4
hypercenter#448D53
hypocenterC53#3006
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#11719949C2semidirect
2. C2×#13133244C2semidirect
3. #121×D53C2semidirect
4. #16810771C2semidirect
5. #16810767C2semidirect
6. #16810762C2semidirect
7. #16810759C2semidirect
8. #16810756C2semidirect
9. #16810623C2semidirect
10. #16810585C2semidirect
11. C53×#448C2semidirect
16. C2×#11719949C22semidirect
18. C22×D212C22semidirect
20. C2×D4×D53C22semidirect
24. C2×#11719924C22semidirect
25. C2×D424C22semidirect
26. C2×#11719922C22semidirect
27. C2×#11719912C22semidirect
28. #13133192C22semidirect
32. #13133188C22semidirect
33. C4⋊C4×D53C22semidirect
35. #13133141C22semidirect
36. #13133139C22semidirect
37. #13133137C22semidirect
38. C22⋊#88322C22semidirect
39. #13133129C22semidirect
40. C22×Q8×C53C22semidirect
41. C2×SD16×C53C22semidirect
43. #103×C53C22semidirect
44. C22⋊C8×C53C22semidirect
64. #11719949D4semidirect
115. Q8×C53C2×D4semidirect
137. D212C2×D4semidirect
152. C22×C53AGL(1,ℤ/8)semidirect
192. #448D53semidirect
197. C53#3006semidirect
203. C22×Q8C2×D53semidirect
212. C2×SD16C2×D53semidirect
225. #103C2×D53semidirect
230. C22⋊C8C2×D53semidirect
301. Q8D4×D53semidirect
353. C22#13133244semidirect