Finite Group Databasebeta

← #20290251

Subgroups — #20290251

Order 6784.

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

normal subgroups only

This group has 202 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#16811626C211yesmaximal, characteristic
3#16811483C212yesmaximal, complemented
4#16811482C212yesmaximal, complemented
5#16811175C211yesmaximal, characteristic
6C2×#11719940—21no—
7#13133284C2212yes—
8#13133284C2212yes—
9C4⋊#88324—21no—
10C22⋊#88324—21no—
11#13133272—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
29#11719940—42no—
30C2×#88324—22no—
31C2×#88324—22no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53C2×C411yescharacteristic
34C22⋊C4×C53—21no—
35C22⋊Dic53—21no—
36C4⋊Dic53—21no—
37C4⋊Dic53C2×C411yescharacteristic
38#11719919—21no—
39#11719919—21no—
40#11719919—21no—
41#11719919—21no—
42#11719919—21no—
43C4×Dic53—21no—
44C4×Dic53—21no—
45C4×Dic53C2×C411yescharacteristic
46C4×Dic53D411yescharacteristic
47C4×Dic53D411yescharacteristic
48C4×Dic53C2311yescharacteristic
49C23×C53C22⋊C411yescharacteristic
50Q8×C53—41no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—41no—
54C2×C4×C53C22⋊C411yescharacteristic
55C2×C4×C53C22×C411yescharacteristic
56C2×Dic53—21no—
57C2×Dic53—21no—
58C2×Dic53—22no—
59C2×Dic53—21no—
60C2×Dic53—21no—
61C2×Dic53C22⋊C411yescharacteristic
62C2×Dic53C22⋊C411yescharacteristic
63C2×Dic53C2×D411yescharacteristic
64C2×Dic53C2×D411yescharacteristic
65Dic106—41no—
66Dic106—41no—
67Dic106—41no—
68#88324—42no—
69#88324—42no—
70C22×C53—21no—
71C22×C53—21no—
72C22×C53—21no—
73C22×C53C2×C22⋊C411yescharacteristic
74C4×C53—21no—
75C4×C53—21no—
76C4×C53—41no—
77C4×C53—41no—
78Dic53—21no—
79Dic53—21no—
80Dic53—41no—
81Dic53—41no—
82Dic53AGL(1,ℤ/8)11yescharacteristic
83Dic53#13711yescharacteristic
84Dic53#13712yes—
85#1559—531nomaximal, Sylow
86C2×C53—41no—
87C2×C53#41311yescharacteristic
88C2×C53#41711yescharacteristic
89C2×C53#41911yescharacteristic
90#533—531no—
91#432—531no—
92#423—531no—
93#333—532no—
94#332—532no—
95C53#155911yescharacteristic, complemented, Sylow
96C2×M16—1061no—
97C4⋊Q8—531no—
98C4⋊Q8—531no—
99#125—1061no—
100C22⋊Q8—531no—
101C22⋊Q8—1061no—
102C22⋊Q8#615711yescharacteristic
103C4×Q8—531no—
104C4×Q8—531no—
105C4×Q8—1061no—
106#117—1061no—
107C2×C42—531no—
108C4⋊C8—532no—
109C4⋊C8—532no—
110C4⋊C8—1061no—
111C22⋊C8—1061no—
112#97—1061no—
113C2×Q8—531no—
114C2×Q8—1061no—
115C2×Q8C2×#615711yescharacteristic
116C22×C4—531no—
117C22×C4—531no—
118C22×C4C2×#615711yescharacteristic
119M16—2122no—
120C2×C8—1062no—
121C2×C8—1062no—
122C4⋊C4—21no—
123C4⋊C4—531no—
124C4⋊C4—1061no—
125C4⋊C4—1061no—
126C4⋊C4—1061no—
127C4⋊C4—1061no—
128C4⋊C4—1061no—
129C4⋊C4—1061no—
130C4⋊C4C2×#615711yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—1061no—
133C42—531no—
134C42—531no—
135C42—531no—
136C42—531no—
137C42—1061no—
138C42—1061no—
139C23C22⋊#615711yescharacteristic
140Q8—41no—
141Q8—2121no—
142Q8—2121no—
143Q8—2121no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—531no—
148C2×C4—531no—
149C2×C4—531no—
150C2×C4—531no—
151C2×C4—1061no—
152C2×C4—1061no—
153C2×C4—1062no—
154C2×C4—1061no—
155C2×C4—1061no—
156C2×C4C22⋊#615711yescharacteristic
157C2×C4C22×#615711yescharacteristic
158C8—2122no—
159C8—2122no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22C2×C22⋊#615711yescharacteristic
164C4—21no—
165C4—21no—
166C4—41no—
167C4—41no—
168C4—531no—
169C4—531no—
170C4—532no—
171C4—1061no—
172C4—1061no—
173C4—2121no—
174C4—2121no—
175C2—41no—
176C2#1681161011yescharacteristic
177C2#1681161411yescharacteristic
178C2#1681161611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22⋊Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22⋊Q8#6157
hypocenterC53#1559
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
95. C53#1559semidirect