Finite Group Databasebeta

← #20290241

Subgroups — #20290241

Order 6784.

198 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 58 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#16811643C211yesmaximal, characteristic
2#16811640C211yesmaximal, characteristic
3#16811483C212yesmaximal, complemented
4#16811482C212yesmaximal, complemented
5#16811175C211yesmaximal, characteristic
6#13133284C2212yes—
7#13133284C2212yes—
8C22⋊#88324—21no—
9C22⋊#88324—21no—
10#13133272—21no—
11C4×#88324—21no—
12#13133216C2211yescharacteristic
13#13133215C2211yescharacteristic
14#13133207—21no—
15#13133196C411yescharacteristic
16#13133194—21no—
17Q8×Dic53C411yescharacteristic
18#13133182—21no—
19#13133182C411yescharacteristic
20#13133173—21no—
21C2×C4×Dic53C2211yescharacteristic
22C22⋊Q8×C53C411yescharacteristic
23C2×Q8×C53C2×C411yescharacteristic
24C22×C4×C53C2×C411yescharacteristic
25C22×Dic53D411yescharacteristic
26C22×Dic53D411yescharacteristic
27C2×Dic106—21no—
28C2×Dic106C2×C411yescharacteristic
29C2×#88324—22no—
30C2×#88324—22no—
31C4⋊C4×C53—21no—
32C4⋊C4×C53C2×C411yescharacteristic
33C22⋊C4×C53—21no—
34C22⋊Dic53—21no—
35C4⋊Dic53—21no—
36C4⋊Dic53C2×C411yescharacteristic
37#11719919—21no—
38#11719919—21no—
39#11719919—21no—
40#11719919—21no—
41#11719919—21no—
42C4×Dic53—21no—
43C4×Dic53—21no—
44C4×Dic53C2×C411yescharacteristic
45C4×Dic53D411yescharacteristic
46C4×Dic53D411yescharacteristic
47C4×Dic53C2311yescharacteristic
48C23×C53C22⋊C411yescharacteristic
49Q8×C53—41no—
50C2×C4×C53—21no—
51C2×C4×C53—21no—
52C2×C4×C53—41no—
53C2×C4×C53C22⋊C411yescharacteristic
54C2×C4×C53C22×C411yescharacteristic
55C2×Dic53—22no—
56C2×Dic53—22no—
57C2×Dic53—21no—
58C2×Dic53—21no—
59C2×Dic53C22⋊C411yescharacteristic
60C2×Dic53C22⋊C411yescharacteristic
61C2×Dic53C2×D411yescharacteristic
62C2×Dic53C2×D411yescharacteristic
63Dic106—42no—
64Dic106—41no—
65#88324—42no—
66#88324—42no—
67C22×C53—21no—
68C22×C53—21no—
69C22×C53—21no—
70C22×C53C2×C22⋊C411yescharacteristic
71C4×C53—21no—
72C4×C53—21no—
73C4×C53—41no—
74C4×C53—41no—
75Dic53—21no—
76Dic53—21no—
77Dic53—41no—
78Dic53—41no—
79Dic53#13512yes—
80Dic53#13712yes—
81#1554—531nomaximal, Sylow
82C2×C53—41no—
83C2×C53#41311yescharacteristic
84C2×C53#41611yescharacteristic
85C2×C53#41911yescharacteristic
86#533—531no—
87#432—531no—
88#431—531no—
89#333—532no—
90#332—532no—
91C53#155411yescharacteristic, complemented, Sylow
92C4⋊Q8—531no—
93C4⋊Q8—531no—
94#125—1061no—
95C22⋊Q8—531no—
96C22⋊Q8—1061no—
97C22⋊Q8#615711yescharacteristic
98C4×Q8—531no—
99C4×Q8—531no—
100C4×Q8—1061no—
101#117—1061no—
102C2×C42—531no—
103C4⋊C8—532no—
104C4⋊C8—532no—
105C22⋊C8—1061no—
106C22⋊C8—1061no—
107#97—1061no—
108C4×C8—1061no—
109C2×Q8—531no—
110C2×Q8—1061no—
111C2×Q8C2×#615711yescharacteristic
112C22×C4—531no—
113C22×C4—531no—
114C22×C4C2×#615711yescharacteristic
115C2×C8—1062no—
116C2×C8—1062no—
117C4⋊C4—21no—
118C4⋊C4—531no—
119C4⋊C4—1061no—
120C4⋊C4—1061no—
121C4⋊C4—1061no—
122C4⋊C4—1061no—
123C4⋊C4—1061no—
124C4⋊C4—1061no—
125C4⋊C4C2×#615711yescharacteristic
126C22⋊C4—21no—
127C22⋊C4—1061no—
128C42—531no—
129C42—531no—
130C42—531no—
131C42—531no—
132C42—1061no—
133C42—1061no—
134C23C22⋊#615711yescharacteristic
135Q8—41no—
136Q8—2121no—
137Q8—2122no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—531no—
142C2×C4—531no—
143C2×C4—531no—
144C2×C4—531no—
145C2×C4—1062no—
146C2×C4—1062no—
147C2×C4—1061no—
148C2×C4—1061no—
149C2×C4C22⋊#615711yescharacteristic
150C2×C4C22×#615711yescharacteristic
151C8—2122no—
152C8—2122no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22C2×C22⋊#615711yescharacteristic
157C4—21no—
158C4—21no—
159C4—41no—
160C4—41no—
161C4—532no—
162C4—532no—
163C4—1061no—
164C4—1061no—
165C4—2121no—
166C4—2121no—
167C2—41no—
168C2#1681161011yescharacteristic
169C2#1681161311yescharacteristic
170C2#1681161611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22⋊Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22⋊Q8#6157
hypocenterC53#1554
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #16811483C2semidirect
4. #16811482C2semidirect
91. C53#1554semidirect