Finite Group Databasebeta

← #20273812

Subgroups — #20273812

Order 6784.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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#16811348C211yesmaximal, characteristic, complemented
2#16810971C211yesmaximal, characteristic
3#16810824C211yesmaximal, characteristic
4#16810824C211yesmaximal, characteristic
5#16810353C211yesmaximal, characteristic, complemented
6#16810353C211yesmaximal, characteristic, complemented
7C53×#332C211yesmaximal, characteristic, complemented
8C2×#11719913—21no—
9C2×#11719913—21no—
10#13133215—21no—
11#13133214C2211yescharacteristic
12#13133208—21no—
13#13133207—22no—
14Q8×Dic53—21no—
15C4×Dic106C2211yescharacteristic
16C4×Dic106C2211yescharacteristic
17#13133182—21no—
18#13133139—21no—
19#13133139—21no—
20#13133137—22no—
21C4×#88322C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4⋊C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26C2×D212—21no—
27C2×D212—21no—
28C2×D212—21no—
29C2×Dic106D411yescharacteristic
30C2×Dic106D411yescharacteristic
31C2×C8×C53—21no—
32C2×C8×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53D412yes—
35C42×C53C2311yescharacteristic
36C4⋊Dic53—22no—
37C4⋊Dic53D411yescharacteristic
38C4⋊Dic53D411yescharacteristic
39#11719919—22no—
40#11719919—22no—
41#11719919—21no—
42#11719919—21no—
43C4×Dic53—22no—
44C4×Dic53—21no—
45C4×Dic53—21no—
46C2×#88322—21no—
47#11719913—42no—
48#11719913—42no—
49C22×D53—41no—
50Q8×C53—41no—
51Q8×C53—41no—
52C2×C4×C53—22no—
53C2×C4×C53C2×D411yescharacteristic
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×D411yescharacteristic
56C2×Dic53—21no—
57C2×Dic53—21no—
58C2×Dic53—21no—
59C2×Dic53—21no—
60D212—42no—
61D212—41no—
62D212—41no—
63D212—41no—
64D212—41no—
65Dic106—21no—
66Dic106—21no—
67Dic106—21no—
68Dic106SD1612yescomplemented
69C8×C53—41no—
70C8×C53—41no—
71#88322—41no—
72C22×C53#12011yescharacteristic
73C2×D53—81no—
74C2×D53—81no—
75C2×D53—81no—
76C4×C53—21no—
77C4×C53—42no—
78C4×C53C2×SD1611yescharacteristic
79C4×C53#13511yescharacteristic
80C4×C53AGL(1,ℤ/8)11yescharacteristic
81C4×C53#13711yescharacteristic
82Dic53—22no—
83Dic53—41no—
84Dic53—41no—
85Dic53—41no—
86#1626—531nomaximal, Sylow
87C2×C53#44811yescharacteristic
88C2×C53#44911yescharacteristic
89C2×C53#45411yescharacteristic
90D53—161no—
91#557—531no—
92#486—531no—
93#460—531no—
94#460—531no—
95#332D5311yescharacteristic, complemented
96Q8⋊C8—531no—
97Q8⋊C8—531no—
98C53#162611yescharacteristic, complemented, Sylow
99C2×SD16—1061no—
100C2×SD16—1061no—
101C4⋊Q8—1061no—
102C4⋊Q8C2×D5311yescharacteristic
103#127—531no—
104#125—1061no—
105#125—1062no—
106C4×Q8—531no—
107C4×Q8—531no—
108C4×Q8—1061no—
109C4×Q8—1061no—
110C4⋊C8C2×D5311yescharacteristic
111C4⋊C8C2×D5311yescharacteristic
112#102—1061no—
113#102—1061no—
114#102—1062no—
115C4×C8—531no—
116C2×Q8—21no—
117C2×Q8—531no—
118C2×Q8—531no—
119C2×D4—1061no—
120C2×D4—1061no—
121C2×D4—1061no—
122SD16—2122no—
123SD16—2122no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—1061no—
127C4⋊C4—21no—
128C4⋊C4—531no—
129C4⋊C4—531no—
130C4⋊C4—1062no—
131C4⋊C4—1062no—
132C4⋊C4—1062no—
133C4⋊C4—1061no—
134C4⋊C4—1061no—
135C4⋊C4D21212yes—
136C42—1062no—
137C42—1061no—
138C42—1061no—
139C42C22×D5311yescharacteristic
140C23—2121no—
141Q8—41no—
142Q8—41no—
143Q8—532no—
144Q8—1061no—
145Q8—1061no—
146Q8—1061no—
147D4—2122no—
148D4—2121no—
149D4—2121no—
150D4—2121no—
151D4—2121no—
152C2×C4—22no—
153C2×C4—1061no—
154C2×C4—1061no—
155C2×C4—1061no—
156C2×C4—1061no—
157C2×C4C2×D21211yescharacteristic
158C2×C4D4×D5311yescharacteristic
159C2×C4D4×D5311yescharacteristic
160C8—41no—
161C8—41no—
162C8—2121no—
163C22—4241no—
164C22—4241no—
165C22—4241no—
166C22#1313318711yescharacteristic
167C4—21no—
168C4—42no—
169C4—1062no—
170C4—2121no—
171C4—2121no—
172C4—2121no—
173C4SD16×D5311yescharacteristic
174C4#1313323811yescharacteristic
175C4#1313324111yescharacteristic
176C4#1313324911yescharacteristic
177C2—8481no—
178C2#1681076311yescharacteristic
179C2#1681076911yescharacteristic
180C2#1681079811yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#332C2
socleC22×C53#120
hypercenter#332D53
hypocenterC53#1626
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #16811348C2semidirect
5. #16810353C2semidirect
6. #16810353C2semidirect
7. C53×#332C2semidirect
68. Dic106SD16semidirect
95. #332D53semidirect
98. C53#1626semidirect