Finite Group Databasebeta

← #20283962

Subgroups — #20283962

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×#11719947C211yesmaximal, characteristic, complemented
2C2×#13133241C212yesmaximal, complemented
3#121×D53C211yesmaximal, characteristic, complemented
4#16810769C212yesmaximal, complemented
5#16810765C212yesmaximal, complemented
6#16810764C211yesmaximal, characteristic, complemented
7#16810761C211yesmaximal, characteristic, complemented
8#16810757C211yesmaximal, characteristic, complemented
9#16810621C212yesmaximal, complemented
10#16810585C211yesmaximal, characteristic, complemented
11C53×#447C211yesmaximal, characteristic, complemented
12C2×#11719947—28no—
13C2×#11719947—21no—
14C2×#11719947—21no—
15C2×#11719947C2212yes—
16C2×#11719947C2212yescomplemented
17C22×Dic106C2211yescharacteristic, complemented
18C22×C22⋊D53—21no—
19C2×D4×D53—21no—
20C2×D4×D53C2211yescharacteristic, complemented
21C23×Dic53—21no—
22C22×C4×D53C2211yescharacteristic
23#13133241—28no—
24C2×#11719913C2212yescomplemented
25C2×#11719923C2212yescomplemented
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#13133142C2212yescomplemented
36#13133140C2212yescomplemented
37#13133138C2212yescomplemented
38C22⋊#88322C2211yescharacteristic, complemented
39#13133129C2212yescomplemented
40C22×D4×C53C2211yescharacteristic, complemented
41C2×D8×C53C2212yescomplemented
42#121×C53C2211yescharacteristic
43#102×C53C2212yescomplemented
44C22⋊C8×C53C2211yescharacteristic, complemented
45C24×C53—21no—
46C23×D53—21no—
47C23×D53D411yescharacteristic
48C2×D4×C53—24no—
49C2×D4×C53—21no—
50C2×D4×C53—21no—
51C2×D4×C53C2311yescharacteristic
52C2×D4×C53C2312yes—
53C22×C4×C53C2311yescharacteristic
54C2×C22⋊D53—28no—
55C2×C22⋊D53—22no—
56C2×C22⋊D53—22no—
57C2×C22⋊D53—21no—
58C2×C22⋊D53—21no—
59C2×C22⋊D53—21no—
60C22×Dic53—28no—
61C22×Dic53—21no—
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53D411yescharacteristic
65#11719947—28no—
66#11719947—28no—
67#11719947—24no—
68#11719947—24no—
69#11719947—24no—
70#11719947D418yescomplemented
71D4×D53—24no—
72D4×D53—44no—
73C2×D212—21no—
74C2×D212C2311yescharacteristic
75C2×C4×D53—24no—
76C2×C4×D53—21no—
77C2×C4×D53D411yescharacteristic
78C2×C4×D53D411yescharacteristic
79C2×C4×D53C2311yescharacteristic
80C2×C4×D53C2311yescharacteristic
81C2×Dic106—21no—
82C2×Dic106—24no—
83C2×Dic106C2312yes—
84D8×C53—24no—
85C2×C8×C53C2312yes—
86C4⋊C4×C53C2311yescharacteristic
87C22⋊C4×C53—21no—
88C22⋊Dic53—21no—
89#11719923—24no—
90#11719922—24no—
91#11719921—21no—
92#11719921—21no—
93C4⋊Dic53C2311yescharacteristic
94#11719919—21no—
95C2×#88322C2312yes—
96#11719913—24no—
97#11719912—24no—
98C23×C53—24no—
99C23×C53—22no—
100C23×C53—21no—
101C23×C53—42no—
102C23×C53—42no—
103C23×C53C2×D411yescharacteristic
104C22×D53—22no—
105C22×D53—22no—
106C22×D53—21no—
107C22×D53—41no—
108C22×D53—42no—
109C22×D53—42no—
110C22×D53—42no—
111C22×D53C2×D411yescharacteristic
112C22×D53C2×D411yescharacteristic
113D4×C53—24no—
114D4×C53—22no—
115D4×C53—22no—
116D4×C53—42no—
117D4×C53C2×D414yescomplemented
118C2×C4×C53—22no—
119C2×C4×C53—21no—
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2411yescharacteristic
122C22⋊D53—28no—
123C22⋊D53—28no—
124C22⋊D53—42no—
125C22⋊D53—42no—
126C22⋊D53—42no—
127C2×Dic53—24no—
128C2×Dic53—28no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—28no—
132C2×Dic53—21no—
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135D212—22no—
136D212—42no—
137C4×D53—22no—
138C4×D53—22no—
139C4×D53—22no—
140C4×D53—42no—
141C4×D53C2×D414yes—
142Dic106—24no—
143Dic106—22no—
144Dic106C2×D414yescomplemented
145C8×C53—22no—
146#88322—22no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—24no—
150C22×C53—48no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53AGL(1,ℤ/8)12yescomplemented
157C22×C53C22×D411yescharacteristic
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—21no—
163C2×D53—44no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—84no—
168C2×D53#12012yes—
169C4×C53—21no—
170C4×C53—41no—
171C4×C53C22×D412yes—
172Dic53—24no—
173Dic53—21no—
174Dic53—41no—
175Dic53#12012yes—
176#3006—531nomaximal, Sylow
177C2×C53—22no—
178C2×C53—44no—
179C2×C53—81no—
180C2×C53C2×#12011yescharacteristic
181C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
182C2×C53#57511yescharacteristic
183D53—22no—
184D53—41no—
185D53—81no—
186C22×#41—531no—
187C2×AGL(1,ℤ/8)—532no—
188C2×#121—531no—
189#449—532no—
190#449—532no—
191#448—531no—
192#448—531no—
193#447—531no—
194#447D5311yescharacteristic, complemented
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×Q8—531no—
204C22×D4—531no—
205C22×D4—1061no—
206C22×D4—1061no—
207C22×D4C2×D5311yescharacteristic, complemented
208C23×C4—531no—
209C23×C4—1061no—
210AGL(1,ℤ/8)—1068no—
211C2×SD16—532no—
212C2×SD16—532no—
213C2×D8—532no—
214C2×D8C2×D5312yescomplemented
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#103—532no—
226#102—532no—
227#102C2×D5312yescomplemented
228C22⋊C8—531no—
229C22⋊C8—532no—
230C22⋊C8C2×D5311yescharacteristic, complemented
231C24—21no—
232C24—531no—
233C24—1061no—
234#41—538no—
235#41—1064no—
236#41—1064no—
237#41—1068no—
238#41—1068no—
239#41—1064no—
240C2×Q8—532no—
241C2×Q8—1064no—
242C2×Q8—1061no—
243C2×D4—24no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—531no—
247C2×D4—1061no—
248C2×D4—1064no—
249C2×D4—1061no—
250C2×D4—1062no—
251C2×D4—1062no—
252C2×D4—1068no—
253C2×D4—1061no—
254C2×D4—1061no—
255C2×D4—2124no—
256C2×D4C22×D5311yescharacteristic
257C2×D4C22×D5312yes—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—1064no—
264C22×C4—1061no—
265C22×C4—1062no—
266C22×C4—1062no—
267C22×C4—1068no—
268C22×C4—1061no—
269C22×C4C22×D5311yescharacteristic
270SD16—1064no—
271SD16—1064no—
272D8—24no—
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—24no—
285C23—22no—
286C23—21no—
287C23—42no—
288C23—42no—
289C23—531no—
290C23—531no—
291C23—1062no—
292C23—1062no—
293C23—1061no—
294C23—2121no—
295C23—2122no—
296C23—2122no—
297C23—2122no—
298C23D4×D5311yescharacteristic
299Q8—534no—
300Q8—1064no—
301Q8—1062no—
302D4—24no—
303D4—22no—
304D4—22no—
305D4—42no—
306D4—1068no—
307D4—1068no—
308D4—1062no—
309D4—2122no—
310D4—2122no—
311D4—2122no—
312D4—2122no—
313D4D4×D5314yescomplemented
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—531no—
317C2×C4—534no—
318C2×C4—531no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—1068no—
323C2×C4—1064no—
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—24no—
336C22—48no—
337C22—42no—
338C22—42no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—532no—
343C22—1061no—
344C22—1061no—
345C22—1061no—
346C22—1061no—
347C22—1061no—
348C22—4244no—
349C22—2124no—
350C22—2121no—
351C22—2121no—
352C22—2121no—
353C22#1313324112yescomplemented
354C22C2×D4×D5311yescharacteristic
355C4—21no—
356C4—41no—
357C4—532no—
358C4—1061no—
359C4—1064no—
360C4—2121no—
361C4C2×D4×D5312yes—
362C2—22no—
363C2—44no—
364C2—81no—
365C2—1062no—
366C2—4241no—
367C2—2121no—
368C2#120×D5311yescharacteristic
369C2C2×#1313324111yescharacteristic
370C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#447C2
socleC22×C53C22×D4
hypercenter#447D53
hypocenterC53#3006
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. C22×#11719947C2semidirect
2. C2×#13133241C2semidirect
3. #121×D53C2semidirect
4. #16810769C2semidirect
5. #16810765C2semidirect
6. #16810764C2semidirect
7. #16810761C2semidirect
8. #16810757C2semidirect
9. #16810621C2semidirect
10. #16810585C2semidirect
11. C53×#447C2semidirect
16. C2×#11719947C22semidirect
17. C22×Dic106C22semidirect
20. C2×D4×D53C22semidirect
24. C2×#11719913C22semidirect
25. C2×#11719923C22semidirect
26. C2×#11719922C22semidirect
27. C2×#11719912C22semidirect
28. #13133192C22semidirect
32. #13133188C22semidirect
33. C4⋊C4×D53C22semidirect
35. #13133142C22semidirect
36. #13133140C22semidirect
37. #13133138C22semidirect
38. C22⋊#88322C22semidirect
39. #13133129C22semidirect
40. C22×D4×C53C22semidirect
41. C2×D8×C53C22semidirect
43. #102×C53C22semidirect
44. C22⋊C8×C53C22semidirect
70. #11719947D4semidirect
117. D4×C53C2×D4semidirect
144. Dic106C2×D4semidirect
156. C22×C53AGL(1,ℤ/8)semidirect
194. #447D53semidirect
197. C53#3006semidirect
207. C22×D4C2×D53semidirect
214. C2×D8C2×D53semidirect
227. #102C2×D53semidirect
230. C22⋊C8C2×D53semidirect
313. D4D4×D53semidirect
353. C22#13133241semidirect