Finite Group Databasebeta

← #20285924

Subgroups — #20285924

Order 6784.

322 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#16811179C211yesmaximal, characteristic
2#16811178C211yesmaximal, characteristic
3#16811033C211yesmaximal, characteristic, complemented
4#16811032C211yesmaximal, characteristic, complemented
5#16810991C211yesmaximal, characteristic, complemented
6#16810963C212yesmaximal
7#16810962C212yesmaximal
8#16810916C212yesmaximal, complemented
9#16810906C212yesmaximal, complemented
10#16810678C211yesmaximal, characteristic
11C53×#491C211yesmaximal, characteristic, complemented
12#13133217C2211yescharacteristic
13#13133216C2211yescharacteristic
14#13133216C2211yescharacteristic
15#13133209C2211yescharacteristic
16#13133208—21no—
17#13133197—21no—
18#13133195—21no—
19#13133195—21no—
20#13133193C2212yes—
21Q8×Dic53C2212yes—
22#13133182—21no—
23#13133182—21no—
24#13133182—21no—
25#13133175—21no—
26#13133175C2211yescharacteristic
27#13133175C2211yescharacteristic
28C42×D53C2211yescharacteristic
29#13133151C2212yes—
30#13133150C2212yes—
31#13133149C2212yes—
32#13133148C2212yes—
33#13133146C2212yes—
34#13133141C2212yes—
35#13133141C2212yes—
36#13133129C2212yes—
37#13133128C2211yescharacteristic
38C4⋊Q8×C53C2211yescharacteristic
39#125×C53C2211yescharacteristic
40#103×C53C2212yes—
41#103×C53C2212yes—
42#97×C53C2211yescharacteristic
43C2×Q8×C53C2312yes—
44C2×C4×D53D411yescharacteristic
45C2×C4×D53D411yescharacteristic
46C2×C4×D53C2311yescharacteristic
47C2×Dic106—21no—
48C2×Dic106—21no—
49C2×Dic106—21no—
50C2×C8×C53C2312yes—
51C4⋊C4×C53—21no—
52C4⋊C4×C53—21no—
53C4⋊C4×C53—21no—
54C4⋊C4×C53—21no—
55C4⋊C4×C53C2311yescharacteristic
56C4⋊C4×C53C2311yescharacteristic
57C42×C53C2311yescharacteristic
58#11719921—22no—
59#11719921—21no—
60#11719921—21no—
61C4⋊Dic53—21no—
62C4⋊Dic53—22no—
63C4⋊Dic53C2311yescharacteristic
64C4⋊Dic53C2311yescharacteristic
65C4⋊Dic53C2312yes—
66#11719919—21no—
67#11719919—21no—
68#11719919—22no—
69#11719919—21no—
70#11719919—21no—
71C4×Dic53—22no—
72C4×Dic53—21no—
73C4×Dic53—21no—
74C4×Dic53D411yescharacteristic
75C4×Dic53D411yescharacteristic
76C4×Dic53C2311yescharacteristic
77C2×#88322C2312yes—
78C22×D53C2×D411yescharacteristic
79Q8×C53—24no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2×D411yescharacteristic
85C2×C4×C53C2411yescharacteristic
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—22no—
89C2×Dic53C2×D411yescharacteristic
90C2×Dic53C2×D411yescharacteristic
91C2×Dic53C2×D411yescharacteristic
92C4×D53—22no—
93C4×D53—22no—
94C4×D53—41no—
95Dic106—42no—
96Dic106—42no—
97Dic106—42no—
98C8×C53—22no—
99#88322—22no—
100C22×C53C22×D411yescharacteristic
101C2×D53—21no—
102C2×D53—21no—
103C2×D53—21no—
104C4×C53—21no—
105C4×C53—21no—
106C4×C53—42no—
107C4×C53—41no—
108C4×C53—41no—
109C4×C532−1+412yes—
110Dic53—21no—
111Dic53—21no—
112Dic53—41no—
113Dic53—41no—
114Dic53—42no—
115Dic53#13712yes—
116#3237—531nomaximal, Sylow
117C2×C53#53611yescharacteristic
118C2×C53C2×#13711yescharacteristic
119C2×C53#57511yescharacteristic
120D53—41no—
121#533—531no—
122#533—531no—
123#501—531no—
124#501—531no—
125#491—531no—
126#491D5311yescharacteristic, complemented
127#485—532no—
128#485—532no—
129#477—532no—
130#475—532no—
131#432—531no—
132C53#323711yescharacteristic, complemented, Sylow
133C4⋊Q8—531no—
134C4⋊Q8—531no—
135C4⋊Q8—531no—
136C4⋊Q8C2×D5311yescharacteristic
137#125—531no—
138#125—1061no—
139#125C2×D5311yescharacteristic
140C22⋊Q8—532no—
141C22⋊Q8—1061no—
142C22⋊Q8—1061no—
143C22⋊Q8—1061no—
144C4×Q8—532no—
145C4×Q8—1061no—
146C4×Q8—1061no—
147C4×Q8—1061no—
148#117—531no—
149#117—531no—
150#117—1061no—
151C2×C42—531no—
152#107—532no—
153#107—532no—
154#106—532no—
155#106—532no—
156C4⋊C8—532no—
157#103—532no—
158#103—532no—
159#103C2×D5312yes—
160#103C2×D5312yes—
161C22⋊C8—532no—
162#97—531no—
163#97C2×D5311yescharacteristic
164C2×Q8—1061no—
165C2×Q8—1061no—
166C2×Q8—1061no—
167C2×Q8C22×D5312yes—
168C22×C4—531no—
169C22×C4—531no—
170C22×C4—531no—
171C2×C8—532no—
172C2×C8C22×D5312yes—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—531no—
178C4⋊C4—531no—
179C4⋊C4—532no—
180C4⋊C4—1062no—
181C4⋊C4—1062no—
182C4⋊C4—1061no—
183C4⋊C4—1061no—
184C4⋊C4—1061no—
185C4⋊C4—1061no—
186C4⋊C4—1061no—
187C4⋊C4C22×D5311yescharacteristic
188C4⋊C4C22×D5311yescharacteristic
189C22⋊C4—1061no—
190C22⋊C4—1061no—
191C22⋊C4—1062no—
192C42—531no—
193C42—531no—
194C42—531no—
195C42—1061no—
196C42—1061no—
197C42—1062no—
198C42C22×D5311yescharacteristic
199C23—531no—
200Q8—24no—
201Q8—2122no—
202Q8—2122no—
203Q8—2122no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—531no—
208C2×C4—531no—
209C2×C4—531no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—1062no—
215C2×C4—2121no—
216C2×C4D4×D5311yescharacteristic
217C2×C4D4×D5311yescharacteristic
218C2×C4C23×D5311yescharacteristic
219C8—22no—
220C8—1062no—
221C22—1061no—
222C22—1061no—
223C22—1061no—
224C22C2×D4×D5311yescharacteristic
225C4—21no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—42no—
230C4—532no—
231C4—1061no—
232C4—1061no—
233C4—2121no—
234C4—2122no—
235C4—2121no—
236C4#1313326412yes—
237C2—2121no—
238C2#1681119311yescharacteristic
239C2#137×D5311yescharacteristic
240C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#491C2
socleC22×C53C22×D4
hypercenter#491D53
hypocenterC53#3237
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
3. #16811033C2semidirect
4. #16811032C2semidirect
5. #16810991C2semidirect
8. #16810916C2semidirect
9. #16810906C2semidirect
11. C53×#491C2semidirect
126. #491D53semidirect
132. C53#3237semidirect