Finite Group Databasebeta

← #20283988

Subgroups — #20283988

Order 6784.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#13133262C211yesmaximal, characteristic
2C2×#13133248C211yesmaximal, characteristic, complemented
3#16811197C211yesmaximal, characteristic, complemented
4#16810782C214yesmaximal, complemented
5#16810772C214yesmaximal, complemented
6#16810628C211yesmaximal, characteristic, complemented
7#16810620C211yesmaximal, characteristic, complemented
8#16810589C211yesmaximal, characteristic, complemented
9C22×SD16×C53C211yesmaximal, characteristic, complemented
10#13133262—24no—
11#13133262—24no—
12C2×#11719945C2211yescharacteristic
13C2×#11719945C2211yescharacteristic
14C2×#11719947—21no—
15C2×#11719947C2212yes—
16C22×C22⋊D53—22no—
17C2×D4×D53—21no—
18C2×D4×D53C2212yes—
19#13133248—24no—
20C2×#11719925C2212yescomplemented
21C2×#11719924C2212yescomplemented
22C2×#11719917C2211yescharacteristic, complemented
23#13133215—21no—
24#13133206—21no—
25#13133200—22no—
26#13133193C2214yescomplemented
27#13133176C2211yescharacteristic, complemented
28#13133142C2212yescomplemented
29#13133139C2212yescomplemented
30#13133138C2214yescomplemented
31#13133129C2214yescomplemented
32C22×Q8×C53C2211yescharacteristic, complemented
33C22×D4×C53C2211yescharacteristic
34C2×SD16×C53—24no—
35C2×SD16×C53C2214yescomplemented
36C22×C8×C53C2211yescharacteristic, complemented
37C24×C53—21no—
38C23×D53—22no—
39C2×Q8×C53—21no—
40C2×Q8×C53—22no—
41C2×Q8×C53—22no—
42C2×Q8×C53C2312yes—
43C2×D4×C53—21no—
44C2×D4×C53—22no—
45C2×D4×C53D414yes—
46C2×D4×C53C2312yes—
47C22×C4×C53—21no—
48C22×C4×C53C2311yescharacteristic
49C2×C22⋊D53—216no—
50C2×C22⋊D53—21no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53C2×C22⋊D53D412yes—
54C22×Dic53—22no—
55#11719947—28no—
56#11719947—24no—
57#11719947—24no—
58D4×D53—28no—
59D4×D53—24no—
60D4×D53—24no—
61#11719945—22no—
62#11719945—22no—
63#11719945—22no—
64#11719945D414yes—
65C2×D212D411yescharacteristic
66C2×D212C2311yescharacteristic
67C2×C4×D53—21no—
68C2×C4×D53C2312yes—
69C2×Dic106D411yescharacteristic
70C2×Dic106C2311yescharacteristic
71SD16×C53—28no—
72C2×C8×C53—22no—
73C2×C8×C53C2312yes—
74C22⋊Dic53—21no—
75#11719925—24no—
76#11719924—24no—
77#11719921—24no—
78C4⋊Dic53C2312yes—
79#11719919—24no—
80C4×Dic53—21no—
81#11719917—22no—
82C2×#88322C2312yes—
83C23×C53—22no—
84C23×C53—24no—
85C23×C53—82no—
86C23×C53C2×D411yescharacteristic
87C22×D53—21no—
88C22×D53—28no—
89C22×D53—84no—
90C22×D53C2×D412yes—
91Q8×C53—22no—
92Q8×C53—22no—
93Q8×C53—42no—
94D4×C53—24no—
95D4×C53—22no—
96D4×C53C2×D414yes—
97C2×C4×C53—22no—
98C2×C4×C53—42no—
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53C2×D412yes—
101C2×C4×C53C2×D412yes—
102C2×C4×C53C2411yescharacteristic
103C22⋊D53—28no—
104C22⋊D53—28no—
105C22⋊D53—24no—
106C22⋊D53—24no—
107C2×Dic53—28no—
108C2×Dic53—21no—
109C2×Dic53—22no—
110C2×Dic53C2×D412yes—
111D212—22no—
112D212—21no—
113D212C2×D412yes—
114C4×D53—24no—
115C4×D53—22no—
116C4×D53—22no—
117Dic106—22no—
118Dic106—21no—
119Dic106C2×D412yes—
120C8×C53—22no—
121#88322—22no—
122C22×C53—22no—
123C22×C53—24no—
124C22×C53—44no—
125C22×C53—84no—
126C22×C53#12012yes—
127C22×C53C22×D411yescharacteristic
128C2×D53—22no—
129C2×D53—22no—
130C2×D53—24no—
131C2×D53—42no—
132C2×D53—88no—
133C4×C53—42no—
134C4×C53#12012yes—
135C4×C53C22×D411yescharacteristic
136C4×C53C22×D411yescharacteristic
137Dic53—22no—
138Dic53—22no—
139Dic53—42no—
140#3009—531nomaximal, Sylow
141C2×C53—22no—
142C2×C53—44no—
143C2×C53C2×#12011yescharacteristic
144C2×C53#57712yes—
145D53—42no—
146D53—42no—
147C2×H(5,𝔽2)—531no—
148C2×#137—531no—
149C22×SD16D5311yescharacteristic, complemented
150#536—531no—
151#452—534no—
152#450—534no—
153#418—531no—
154#416—531no—
155#408—531no—
156C53#300911yescharacteristic, complemented, Sylow
157H(5,𝔽2)—1064no—
158H(5,𝔽2)—1064no—
159C2×#41—531no—
160C2×#41—532no—
161C2×#41—531no—
162C2×#41—1061no—
163C22×Q8C2×D5311yescharacteristic, complemented
164C22×D4—532no—
165C22×D4—1061no—
166C22×D4—1062no—
167C22×D4C2×D5311yescharacteristic
168#137—1064no—
169C2×Q16—532no—
170C2×SD16—24no—
171C2×SD16—532no—
172C2×SD16C2×D5314yescomplemented
173C2×M16—531no—
174C22×C8C2×D5311yescharacteristic, complemented
175C4⋊Q8—1061no—
176#124—1061no—
177#123—1062no—
178C22⋊Q8—534no—
179#117—531no—
180#103—532no—
181#102—534no—
182#102—532no—
183C22⋊C8—534no—
184C24—21no—
185C24—1062no—
186#41—534no—
187#41—1064no—
188#41—1068no—
189#41—1064no—
190#41—1062no—
191#41—1062no—
192#41—1062no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—531no—
197C2×Q8—531no—
198C2×Q8C22×D5312yes—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—532no—
202C2×D4—531no—
203C2×D4—531no—
204C2×D4—1062no—
205C2×D4—1064no—
206C2×D4—1068no—
207C2×D4—1062no—
208C2×D4—10616no—
209C2×D4—1061no—
210C2×D4—1064no—
211C2×D4C22⋊D5314yes—
212C2×D4C22×D5312yes—
213C22×C4—21no—
214C22×C4—532no—
215C22×C4—1062no—
216C22×C4—1061no—
217C22×C4C22×D5311yescharacteristic
218Q16—1064no—
219SD16—28no—
220SD16—1064no—
221M16—1062no—
222C2×C8—22no—
223C2×C8—532no—
224C2×C8C22×D5312yes—
225C4⋊C4—532no—
226C4⋊C4—1064no—
227C22⋊C4—1064no—
228C22⋊C4—1061no—
229C42—1061no—
230C23—22no—
231C23—24no—
232C23—82no—
233C23—532no—
234C23—1068no—
235C23—1061no—
236C23—4244no—
237C23D4×D5311yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—42no—
241Q8—532no—
242Q8—1062no—
243Q8—1061no—
244D4—24no—
245D4—22no—
246D4—532no—
247D4—1068no—
248D4—1068no—
249D4—1064no—
250D4—1064no—
251D4—1062no—
252D4—1061no—
253D4C2×C22⋊D5314yes—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—532no—
257C2×C4—1068no—
258C2×C4—1061no—
259C2×C4—1064no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4D4×D5311yescharacteristic
264C2×C4D4×D5312yes—
265C2×C4C2×C22⋊D5312yes—
266C2×C4C23×D5311yescharacteristic
267C8—22no—
268C8—1062no—
269C22—22no—
270C22—24no—
271C22—44no—
272C22—84no—
273C22—1062no—
274C22—1062no—
275C22—1064no—
276C22—4248no—
277C22—2122no—
278C22#1313318512yes—
279C22C2×D4×D5311yescharacteristic
280C4—42no—
281C4—1062no—
282C4—1062no—
283C4—2122no—
284C4#1313318512yes—
285C4C2×D4×D5311yescharacteristic
286C4C22×C22⋊D5311yescharacteristic
287C2—22no—
288C2—44no—
289C2—2122no—
290C2—2122no—
291C2C2×#1313318511yescharacteristic
292C2#1681142912yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC22×SD16×C53C2
socleC22×C53C22×D4
hypercenterC22×SD16D53
hypocenterC53#3009
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#13133248C2semidirect
3. #16811197C2semidirect
4. #16810782C2semidirect
5. #16810772C2semidirect
6. #16810628C2semidirect
7. #16810620C2semidirect
8. #16810589C2semidirect
9. C22×SD16×C53C2semidirect
20. C2×#11719925C22semidirect
21. C2×#11719924C22semidirect
22. C2×#11719917C22semidirect
26. #13133193C22semidirect
27. #13133176C22semidirect
28. #13133142C22semidirect
29. #13133139C22semidirect
30. #13133138C22semidirect
31. #13133129C22semidirect
32. C22×Q8×C53C22semidirect
35. C2×SD16×C53C22semidirect
36. C22×C8×C53C22semidirect
149. C22×SD16D53semidirect
156. C53#3009semidirect
163. C22×Q8C2×D53semidirect
172. C2×SD16C2×D53semidirect
174. C22×C8C2×D53semidirect