Finite Group Databasebeta

← #20290180

Subgroups — #20290180

Order 6784.

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

normal subgroups only

This group has 216 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#16811645C211yesmaximal, characteristic, complemented
2C4×#11719936C211yesmaximal, characteristic, complemented
3#16811480C212yesmaximal, complemented
4#16811478C212yesmaximal, complemented
5#16811155C211yesmaximal, characteristic
6C2×#11719936—21no—
7C4⋊#88324—21no—
8#13133274—21no—
9#13133272C2212yes—
10#13133272C2212yes—
11C4×#88324—21no—
12#13133210—21no—
13#13133208C2211yescharacteristic
14#13133204C2211yescharacteristic
15#13133199—21no—
16#13133197C411yescharacteristic
17#13133188C411yescharacteristic
18C4×Dic106C411yescharacteristic
19C4×C22⋊D53—21no—
20#13133176—21no—
21C42×D53C2211yescharacteristic
22C4×D4×C53C411yescharacteristic
23C2×D4×C53C2×C411yescharacteristic
24C22×C4×C53—21no—
25C2×C22⋊D53—21no—
26C2×C4×D53C2×C411yescharacteristic
27C2×C4×D53D411yescharacteristic
28C2×C4×D53D411yescharacteristic
29C2×Dic106C2×C411yescharacteristic
30C2×#88324—22no—
31C2×#88324—22no—
32#11719936—24no—
33C4⋊C4×C53C2×C411yescharacteristic
34C22⋊C4×C53—21no—
35C42×C53C2×C411yescharacteristic
36C22⋊Dic53—21no—
37C22⋊Dic53—21no—
38C22⋊Dic53—21no—
39#11719921—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53C2×C411yescharacteristic
43#11719919—21no—
44#11719919—21no—
45C4×Dic53—21no—
46C4×Dic53D411yescharacteristic
47C4×Dic53D411yescharacteristic
48C4×Dic53C2311yescharacteristic
49C23×C53—21no—
50C22×D53C22⋊C411yescharacteristic
51D4×C53—41no—
52C2×C4×C53—21no—
53C2×C4×C53—42no—
54C2×C4×C53C22⋊C411yescharacteristic
55C2×C4×C53C22⋊C411yescharacteristic
56C2×C4×C53C22×C411yescharacteristic
57C22⋊D53—42no—
58C2×Dic53—21no—
59C2×Dic53—21no—
60C2×Dic53C22⋊C411yescharacteristic
61C2×Dic53C2×D411yescharacteristic
62C2×Dic53C2×D411yescharacteristic
63C4×D53—22no—
64C4×D53—41no—
65C4×D53—41no—
66Dic106—41no—
67#88324—24no—
68#88324—42no—
69C22×C53—41no—
70C22×C53—41no—
71C22×C53—41no—
72C22×C53C2×C22⋊C411yescharacteristic
73C2×D53—21no—
74C2×D53—21no—
75C2×D53—21no—
76C4×C53—21no—
77C4×C53—21no—
78C4×C53—41no—
79C4×C53#10412yes—
80Dic53—21no—
81Dic53—21no—
82Dic53—21no—
83Dic53—41no—
84Dic53—41no—
85#1529—531nomaximal, Sylow
86C2×C53—81no—
87C2×C53#41311yescharacteristic
88C2×C53C2×#10411yescharacteristic
89C2×C53#42111yescharacteristic
90D53—41no—
91#529—531no—
92#432—531no—
93C4×M16—531no—
94#330—532no—
95#329—532no—
96C53#152911yescharacteristic, complemented, Sylow
97C2×M16—1061no—
98#126—1061no—
99#125—531no—
100#124—531no—
101#123—1061no—
102C22⋊Q8—531no—
103#121—531no—
104C4×Q8—531no—
105C4×D4—1061no—
106C4×D4#615711yescharacteristic
107#117—1061no—
108C2×C42—531no—
109C4⋊C8—1061no—
110C22⋊C8—1061no—
111#97—532no—
112#97—532no—
113C4×C8—1061no—
114C2×Q8—531no—
115C2×D4—1061no—
116C2×D4C2×#615711yescharacteristic
117C22×C4—21no—
118C22×C4—531no—
119C22×C4—531no—
120C22×C4—531no—
121M16—1064no—
122C2×C8—1062no—
123C2×C8—1062no—
124C4⋊C4—531no—
125C4⋊C4—1061no—
126C4⋊C4—1061no—
127C4⋊C4—1061no—
128C4⋊C4—1061no—
129C4⋊C4C2×#615711yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—1061no—
132C22⋊C4—1061no—
133C22⋊C4—1061no—
134C22⋊C4—1061no—
135C42—531no—
136C42—531no—
137C42—531no—
138C42—1061no—
139C42C2×#615711yescharacteristic
140C23—21no—
141C23—531no—
142Q8—2121no—
143D4—41no—
144D4—2122no—
145C2×C4—21no—
146C2×C4—42no—
147C2×C4—531no—
148C2×C4—531no—
149C2×C4—531no—
150C2×C4—1062no—
151C2×C4—1061no—
152C2×C4—1061no—
153C2×C4—2121no—
154C2×C4—2121no—
155C2×C4C22⋊#615711yescharacteristic
156C2×C4C22⋊#615711yescharacteristic
157C2×C4C22×#615711yescharacteristic
158C8—1064no—
159C8—2122no—
160C22—41no—
161C22—41no—
162C22—41no—
163C22—1061no—
164C22—1061no—
165C22—1061no—
166C22C2×C22⋊#615711yescharacteristic
167C4—21no—
168C4—21no—
169C4—41no—
170C4—1061no—
171C4—1061no—
172C4—1061no—
173C4—2121no—
174C4—2121no—
175C4#1313328112yes—
176C2—81no—
177C2—2121no—
178C2#1681160911yescharacteristic
179C2C2×#1313328111yescharacteristic
180C2#1681162011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4×D4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4×D4#6157
hypocenterC53#1529
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #16811645C2semidirect
2. C4×#11719936C2semidirect
3. #16811480C2semidirect
4. #16811478C2semidirect
96. C53#1529semidirect