Finite Group Databasebeta

← #20275156

Subgroups — #20275156

Order 6784.

442 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 97 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#16811134C211yesmaximal, characteristic
2C22×C4×Dic53C211yesmaximal, characteristic, complemented
3C2×#13133143C212yesmaximal, complemented
4#16810384C212yesmaximal, complemented
5C53×#407C211yesmaximal, characteristic, complemented
6C2×#11719945C2211yescharacteristic
7C23×Dic53—21no—
8#13133200—21no—
9#13133196C412yescomplemented
10#13133191C412yescomplemented
11#13133186—21no—
12C4×C22⋊D53—22no—
13#13133176C2211yescharacteristic
14C2×C4×Dic53—24no—
15C2×C4×Dic53—21no—
16C2×C4×Dic53C2212yes—
17#13133143—24no—
18C23×C4×C53C2211yescharacteristic
19C2×M16×C53C2212yes—
20C22⋊C8×C53—22no—
21C24×C53D411yescharacteristic
22C22×C4×C53—22no—
23C22×C4×C53—22no—
24C22×C4×C53C2×C412yes—
25C22×C4×C53D411yescharacteristic
26C22×C4×C53D412yes—
27C22×C4×C53C2311yescharacteristic
28C2×C22⋊D53—22no—
29C2×C22⋊D53—21no—
30C22×Dic53—24no—
31C22×Dic53—28no—
32C22×Dic53—22no—
33#11719945—22no—
34#11719945—22no—
35C2×D212C2×C411yescharacteristic, complemented
36C2×C4×D53—21no—
37C2×Dic106C2×C411yescharacteristic, complemented
38M16×C53—22no—
39M16×C53—22no—
40C2×C8×C53—22no—
41C22⋊Dic53—22no—
42C22⋊Dic53—21no—
43#11719921—22no—
44C4⋊Dic53C2×C412yescomplemented
45#11719919—22no—
46C4×Dic53—24no—
47C4×Dic53—22no—
48C4×Dic53—21no—
49C4×Dic53D414yescomplemented
50C23×C53—24no—
51C23×C53—21no—
52C23×C53—21no—
53C23×C53C22⋊C412yes—
54C23×C53C2×D411yescharacteristic
55C22×D53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—24no—
59C2×C4×C53—24no—
60C2×C4×C53C22⋊C412yes—
61C2×C4×C53C22×C411yescharacteristic
62C2×C4×C53C2×D412yes—
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53#4112yes—
65C22⋊D53—44no—
66C22⋊D53—41no—
67C22⋊D53—41no—
68C2×Dic53—216no—
69C2×Dic53—28no—
70C2×Dic53—24no—
71C2×Dic53—21no—
72C2×Dic53—22no—
73D212—22no—
74C4×D53—42no—
75Dic106—22no—
76C8×C53—42no—
77C22×C53—24no—
78C22×C53—24no—
79C22×C53—22no—
80C22×C53—22no—
81C22×C53—22no—
82C22×C53#10414yescomplemented
83C22×C53C2×C22⋊C411yescharacteristic
84C22×C53#12011yescharacteristic
85C22×C53#12311yescharacteristic
86C2×D53—41no—
87C2×D53—42no—
88C4×C53—22no—
89C4×C53C4×D412yes—
90C4×C53#12112yes—
91Dic53—28no—
92Dic53—41no—
93Dic53—42no—
94#1896—531nomaximal, Sylow
95C2×C53—21no—
96C2×C53—24no—
97C2×C53—21no—
98C2×C53#38611yescharacteristic
99C2×C53C2×#10412yes—
100D53—81no—
101#525—531no—
102C22×C42—531no—
103C2×#104—532no—
104#407D5311yescharacteristic, complemented
105#339—532no—
106C53#189611yescharacteristic, complemented, Sylow
107C2×#41—531no—
108C23×C4—1061no—
109C23×C4C2×D5311yescharacteristic
110C2×M16C2×D5312yes—
111#123—1061no—
112C22⋊Q8—532no—
113#121—532no—
114#120—1061no—
115C4×D4—1062no—
116#117—531no—
117C2×C42—532no—
118C2×C42—1064no—
119C2×C42—1061no—
120#104—1064no—
121C22⋊C8—22no—
122C24D21211yescharacteristic
123#41—1062no—
124#41—1062no—
125C2×Q8—531no—
126C2×D4—531no—
127C2×D4—1061no—
128C2×D4—1062no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—1064no—
132C22×C4—1068no—
133C22×C4—1062no—
134C22×C4—1061no—
135C22×C4C4×D5312yes—
136C22×C4D21211yescharacteristic
137C22×C4C22⋊D5312yes—
138C22×C4C22×D5311yescharacteristic
139M16—22no—
140M16—22no—
141C2×C8—22no—
142C4⋊C4—532no—
143C4⋊C4—1062no—
144C22⋊C4—1061no—
145C22⋊C4—1062no—
146C22⋊C4—1062no—
147C42—534no—
148C42—1064no—
149C42—1062no—
150C42—1061no—
151C23—24no—
152C23—21no—
153C23—21no—
154C23—1061no—
155C23#1171992112yes—
156C23C2×D21211yescharacteristic
157Q8—1062no—
158D4—1062no—
159D4—2124no—
160D4—2121no—
161D4—2121no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—10616no—
167C2×C4—1068no—
168C2×C4—1064no—
169C2×C4—1062no—
170C2×C4—1061no—
171C2×C4—2122no—
172C2×C4#1171992112yes—
173C2×C4C2×C4×D5311yescharacteristic
174C2×C4D4×D5312yes—
175C2×C4#1171994712yes—
176C2×C4C2×C22⋊D5311yescharacteristic
177C8—42no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—2121no—
184C22—2122no—
185C22#1313314314yescomplemented
186C22C2×#1171992111yescharacteristic
187C22#1313318711yescharacteristic
188C22#1313320111yescharacteristic
189C4—22no—
190C4—1068no—
191C4—2121no—
192C4—2122no—
193C4#1313317712yes—
194C4#1313318912yes—
195C2—21no—
196C2—24no—
197C2—21no—
198C2—4241no—
199C2#1681050011yescharacteristic
200C2C2×#1313314312yes—
subgroupG/H
centerC2×C4C2×C22⋊D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#407C2
socleC22×C53C2×C22⋊C4
hypercenter#407D53
hypocenterC53#1896
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic53C2semidirect
3. C2×#13133143C2semidirect
4. #16810384C2semidirect
5. C53×#407C2semidirect
9. #13133196C4semidirect
10. #13133191C4semidirect
35. C2×D212C2×C4semidirect
37. C2×Dic106C2×C4semidirect
44. C4⋊Dic53C2×C4semidirect
49. C4×Dic53D4semidirect
82. C22×C53#104semidirect
104. #407D53semidirect
106. C53#1896semidirect
185. C22#13133143semidirect