Finite Group Databasebeta

← #20285921

Subgroups — #20285921

Order 6784.

324 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 121 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#16811180C211yesmaximal, characteristic
2#16811033C212yesmaximal, complemented
3#16810993C212yesmaximal, complemented
4#16810958C212yesmaximal
5#16810953C212yesmaximal
6#16810927C214yesmaximal, complemented
7#16810680C211yesmaximal, characteristic
8C2×#125×C53C211yesmaximal, characteristic
9#13133217C2211yescharacteristic
10#13133217C2211yescharacteristic
11#13133209—21no—
12#13133196—21no—
13#13133196C2212yes—
14C4×Dic106—21no—
15C4×Dic106C2212yes—
16#13133176—21no—
17#13133150C2214yes—
18#13133148C2214yes—
19C4⋊#88322C2212yes—
20#13133140C2214yes—
21#13133140C2214yes—
22C22⋊#88322C2212yes—
23#13133128C2212yes—
24#125×C53—21no—
25#125×C53—21no—
26#125×C53C2212yes—
27#125×C53C2212yes—
28C2×C4⋊C4×C53—21no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53C2211yescharacteristic
31C2×C4⋊C4×C53C2211yescharacteristic
32C2×C42×C53C2211yescharacteristic
33C22×C4×C53—21no—
34C22×C4×C53—21no—
35C22×C4×C53D411yescharacteristic
36C22×C4×C53D411yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C2×Dic106—21no—
39C2×Dic106C2312yes—
40C4⋊C4×C53—22no—
41C4⋊C4×C53—22no—
42C4⋊C4×C53—22no—
43C4⋊C4×C53—22no—
44C4⋊C4×C53—21no—
45C4⋊C4×C53—21no—
46C4⋊C4×C53C2312yes—
47C4⋊C4×C53C2312yes—
48C42×C53D411yescharacteristic
49C42×C53D411yescharacteristic
50C42×C53C2311yescharacteristic
51C42×C53C2311yescharacteristic
52C22⋊Dic53—22no—
53C4⋊Dic53—21no—
54C4⋊Dic53—21no—
55C4⋊Dic53—21no—
56C4⋊Dic53C2312yes—
57#11719919—22no—
58#11719919—22no—
59C4×Dic53—22no—
60C2×#88322C2314yes—
61C23×C53C2×D411yescharacteristic
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—41no—
67C2×C4×C53—42no—
68C2×C4×C53—42no—
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2411yescharacteristic
75C2×Dic53—22no—
76C2×Dic53—22no—
77Dic106—24no—
78Dic106—42no—
79#88322—24no—
80C22×C53—21no—
81C22×C53—21no—
82C22×C53#13712yes—
83C22×C53C22×D411yescharacteristic
84C4×C53—21no—
85C4×C53—21no—
86C4×C53—21no—
87C4×C53—21no—
88C4×C53—21no—
89C4×C53—42no—
90C4×C53—42no—
91C4×C532−1+412yes—
92Dic53—42no—
93Dic53—42no—
94#3236—531nomaximal, Sylow
95C2×C53—22no—
96C2×C53#53611yescharacteristic
97C2×C53C2×#13711yescharacteristic
98C2×C53#57511yescharacteristic
99#533—531no—
100C2×#125D5311yescharacteristic
101#501—532no—
102#491—532no—
103#484—532no—
104#483—532no—
105#479—534no—
106#432—531no—
107C53#323611yescharacteristic, complemented, Sylow
108C4⋊Q8—531no—
109C4⋊Q8—531no—
110#125—21no—
111#125—21no—
112#125—1061no—
113#125C2×D5312yes—
114#125C2×D5312yes—
115C22⋊Q8—532no—
116C22⋊Q8—1061no—
117C4×Q8—532no—
118C4×Q8—1061no—
119#117—1061no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4C2×D5311yescharacteristic
123C2×C4⋊C4C2×D5311yescharacteristic
124C2×C42C2×D5311yescharacteristic
125#107—534no—
126#106—534no—
127C4⋊C8—532no—
128#103—534no—
129#103—534no—
130C22⋊C8—532no—
131#97—532no—
132C2×Q8—532no—
133C2×Q8—1061no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4C22⋊D5311yescharacteristic
137C22×C4C22⋊D5311yescharacteristic
138C22×C4C22×D5311yescharacteristic
139C2×C8—534no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—532no—
147C4⋊C4—1061no—
148C4⋊C4—1061no—
149C4⋊C4—1062no—
150C4⋊C4—1062no—
151C4⋊C4—1061no—
152C4⋊C4C22×D5312yes—
153C4⋊C4C22×D5312yes—
154C22⋊C4—1062no—
155C42—1062no—
156C42C22⋊D5311yescharacteristic
157C42C22⋊D5311yescharacteristic
158C42C22×D5311yescharacteristic
159C42C22×D5311yescharacteristic
160C23C2×C22⋊D5311yescharacteristic
161Q8—1064no—
162Q8—2122no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—1062no—
171C2×C4—1062no—
172C2×C4C2×C22⋊D5311yescharacteristic
173C2×C4C2×C22⋊D5311yescharacteristic
174C2×C4C2×C22⋊D5311yescharacteristic
175C2×C4C2×C22⋊D5311yescharacteristic
176C2×C4C2×C22⋊D5311yescharacteristic
177C2×C4C23×D5311yescharacteristic
178C8—1064no—
179C22—21no—
180C22—21no—
181C22#1313324712yes—
182C22C22×C22⋊D5311yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—21no—
187C4—21no—
188C4—42no—
189C4—42no—
190C4—2122no—
191C4—2122no—
192C4#1313326512yes—
193C2—22no—
194C2#1681119711yescharacteristic
195C2C2×#1313324711yescharacteristic
196C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#125×C53C2
socleC22×C53C22×D4
hypercenterC2×#125D53
hypocenterC53#3236
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #16811033C2semidirect
3. #16810993C2semidirect
6. #16810927C2semidirect
107. C53#3236semidirect