Finite Group Databasebeta

← #20290109

Subgroups — #20290109

Order 6784.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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
2#16811559C211yesmaximal, characteristic
3#16811477C214yesmaximal
4#16811178C211yesmaximal, characteristic
5C2×C4×#6157—21no—
6C4⋊#88324—21no—
7#13133274C2212yes—
8#13133272—21no—
9(C2×#6157)⋊C4—21no—
10(C2×#6157)⋊C4C2212yes—
11#13133217C411yescharacteristic, complemented
12#13133208—21no—
13#13133197—21no—
14#13133193C2212yes—
15Q8×Dic53C412yescomplemented
16#13133182—21no—
17#13133175—21no—
18C42×D53C2211yescharacteristic
19C4⋊Q8×C53C411yescharacteristic, complemented
20C22×#6157—22no—
21C2×Q8×C53C2×C412yes—
22C2×C4×D53D411yescharacteristic
23C2×C4×D53D411yescharacteristic
24C2×C4×D53C2311yescharacteristic
25C2×Dic106—21no—
26C2×#88324—22no—
27C4×#6157—24no—
28C4⋊C4×C53—21no—
29C4⋊C4×C53—21no—
30C42×C53C2×C411yescharacteristic
31#11719921—22no—
32C4⋊Dic53—21no—
33C4⋊Dic53—22no—
34C4⋊Dic53C2×C412yes—
35#11719919—22no—
36C4×Dic53—22no—
37C4×Dic53C2×C411yescharacteristic
38C4×Dic53D411yescharacteristic
39C4×Dic53D411yescharacteristic
40C22×D53C2×D411yescharacteristic
41C2×#6157—24no—
42C2×#6157—42no—
43C2×#6157—42no—
44Q8×C53—42no—
45C2×C4×C53—22no—
46C2×C4×C53C22⋊C411yescharacteristic
47C2×C4×C53C22⋊C411yescharacteristic
48C2×C4×C53C22×C411yescharacteristic
49C2×Dic53—22no—
50C2×Dic53C22⋊C411yescharacteristic
51C2×Dic53C22⋊C411yescharacteristic
52C2×Dic53C2×D411yescharacteristic
53C4×D53—22no—
54C4×D53—22no—
55C4×D53—41no—
56Dic106—42no—
57#88324—42no—
58C22×C53C2×C22⋊C411yescharacteristic
59C2×D53—21no—
60C2×D53—21no—
61C2×D53#13712yes—
62#6157—44no—
63C4×C53—21no—
64C4×C53—21no—
65C4×C53—42no—
66C4×C53#10412yes—
67Dic53—21no—
68Dic53—21no—
69Dic53—21no—
70Dic53—42no—
71#1506—531nomaximal, Sylow
72C2×C53#41011yescharacteristic
73C2×C53#41911yescharacteristic
74C2×C53C2×#10411yescharacteristic
75D53—22no—
76#533—531no—
77#432—531no—
78#376—531no—
79#328—534no—
80C53#150611yescharacteristic, complemented, Sylow
81C4⋊Q8—531no—
82C4⋊Q8#615711yescharacteristic, complemented
83#125—1061no—
84C22⋊Q8—532no—
85C22⋊Q8—1061no—
86C4×Q8—532no—
87C4×Q8—1061no—
88#117—1061no—
89C2×C42—531no—
90C2×C42—1061no—
91C4⋊C8—1061no—
92C22⋊C8—532no—
93#97—1061no—
94#95—532no—
95#95—1061no—
96C2×Q8—1061no—
97C2×Q8C2×#615712yes—
98C22×C4—531no—
99C22×C4—531no—
100C22×C4—531no—
101C22×C4—1062no—
102C2×C8—1062no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—532no—
106C4⋊C4—1062no—
107C4⋊C4—1062no—
108C4⋊C4—1061no—
109C22⋊C4—1062no—
110C42—531no—
111C42—531no—
112C42—531no—
113C42—1062no—
114C42—1064no—
115C42C2×#615711yescharacteristic
116C23—531no—
117Q8—42no—
118Q8—2122no—
119C2×C4—22no—
120C2×C4—531no—
121C2×C4—531no—
122C2×C4—531no—
123C2×C4—1062no—
124C2×C4—1062no—
125C2×C4—1064no—
126C2×C4—1062no—
127C2×C4—2121no—
128C2×C4—2122no—
129C2×C4—2122no—
130C2×C4C22⋊#615711yescharacteristic
131C2×C4C22⋊#615711yescharacteristic
132C2×C4C22×#615711yescharacteristic
133C8—2122no—
134C22—532no—
135C22—1061no—
136C22—1061no—
137C22C2×C22⋊#615711yescharacteristic
138C4—21no—
139C4—21no—
140C4—42no—
141C4—1061no—
142C4—1061no—
143C4—1061no—
144C4—2124no—
145C4—2122no—
146C4#1313328212yes—
147C2—1062no—
148C2#1681160311yescharacteristic
149C2#1681161611yescharacteristic
150C2C2×#1313328211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4⋊Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4⋊Q8#6157
hypocenterC53#1506
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
11. #13133217C4semidirect
15. Q8×Dic53C4semidirect
19. C4⋊Q8×C53C4semidirect
80. C53#1506semidirect
82. C4⋊Q8#6157semidirect