Finite Group Databasebeta

← #20273903

Subgroups — #20273903

Order 6784.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#16811358C211yesmaximal, characteristic
2#16810970C211yesmaximal, characteristic
3#16810909C211yesmaximal, characteristic, complemented
4#16810833C211yesmaximal, characteristic, complemented
5#16810366C211yesmaximal, characteristic
6#16810355C211yesmaximal, characteristic, complemented
7C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#11719925—21no—
9#13133217C2211yescharacteristic
10#13133214C2211yescharacteristic
11#13133206—21no—
12#13133205—21no—
13#13133202—22no—
14C4×Dic106C2211yescharacteristic
15C4×D212—21no—
16#13133179—21no—
17#13133149—21no—
18C4⋊#88322C2211yescharacteristic
19#13133142—21no—
20#13133140—21no—
21#13133139—22no—
22C4×Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53D411yescharacteristic
26C2×D212—21no—
27C2×D212—21no—
28C2×D212D412yes—
29C2×C4×D53—22no—
30C2×Dic106—21no—
31C2×Dic106D411yescharacteristic
32C2×C8×C53—21no—
33C2×C8×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53D411yescharacteristic
36C42×C53—21no—
37C42×C53C2311yescharacteristic
38#11719925—42no—
39#11719921—22no—
40#11719921—22no—
41#11719921—22no—
42C4⋊Dic53—21no—
43C4⋊Dic53—21no—
44C4⋊Dic53D411yescharacteristic
45#11719919—21no—
46C4×Dic53—21no—
47C2×#88322—21no—
48C22×D53—22no—
49Q8×C53—21no—
50Q8×C53—21no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53C2×D411yescharacteristic
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×D411yescharacteristic
56C2×Dic53—21no—
57C2×Dic53—21no—
58C2×Dic53—41no—
59D212—42no—
60D212—41no—
61D212—41no—
62D212—41no—
63D212—41no—
64C4×D53—82no—
65Dic106—21no—
66Dic106—21no—
67Dic106—41no—
68Dic106—41no—
69C8×C53—41no—
70C8×C53—41no—
71#88322—41no—
72C22×C53#12011yescharacteristic
73C2×D53—42no—
74C2×D53—42no—
75C2×D53—42no—
76C4×C53—21no—
77C4×C53—41no—
78C4×C53—41no—
79C4×C53#13511yescharacteristic
80C4×C53#13511yescharacteristic
81C4×C53AGL(1,ℤ/8)11yescharacteristic
82C4×C53#13711yescharacteristic
83Dic53—41no—
84Dic53—41no—
85Dic53—81no—
86#1640—531nomaximal, Sylow
87C2×C53#44911yescharacteristic
88C2×C53#45211yescharacteristic
89C2×C53#45511yescharacteristic
90D53—82no—
91#559—531no—
92#486—531no—
93#475—531no—
94#462—531no—
95#331—531no—
96Q8⋊C8—531no—
97Q8⋊C8D5311yescharacteristic, complemented
98C53#164011yescharacteristic, complemented, Sylow
99C2×Q16—1061no—
100C4⋊Q8—531no—
101#127—531no—
102#124—1061no—
103#124—1061no—
104#123—1062no—
105C4×Q8—531no—
106C4×Q8C2×D5311yescharacteristic
107C4×D4—1061no—
108C4×D4—1061no—
109#106—1061no—
110C4⋊C8—531no—
111C4⋊C8C2×D5311yescharacteristic
112#103—1061no—
113#103—1061no—
114#102—1062no—
115C4×C8C2×D5311yescharacteristic
116C2×Q8—531no—
117C2×Q8—1061no—
118C2×Q8D21211yescharacteristic
119C2×D4—532no—
120C2×D4—1061no—
121C2×D4—1061no—
122C22×C4—1062no—
123Q16—2122no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—1061no—
127C4⋊C4—21no—
128C4⋊C4—531no—
129C4⋊C4—1061no—
130C4⋊C4—1061no—
131C4⋊C4—1061no—
132C4⋊C4D21211yescharacteristic
133C22⋊C4—1062no—
134C22⋊C4—1062no—
135C22⋊C4—1062no—
136C42—21no—
137C42—1061no—
138C42C22×D5311yescharacteristic
139C23—1062no—
140Q8—21no—
141Q8—21no—
142Q8—1061no—
143Q8—1061no—
144Q8—2121no—
145Q8—2121no—
146D4—2122no—
147D4—2121no—
148D4—2121no—
149D4—2121no—
150D4—2121no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—1061no—
154C2×C4—1061no—
155C2×C4—4242no—
156C2×C4—2121no—
157C2×C4C2×D21211yescharacteristic
158C2×C4D4×D5311yescharacteristic
159C2×C4D4×D5311yescharacteristic
160C8—41no—
161C8—41no—
162C8—2121no—
163C22—2122no—
164C22—2122no—
165C22—2122no—
166C22#1313318711yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—4241no—
171C4—2121no—
172C4—2121no—
173C4#1313323811yescharacteristic
174C4#1313323911yescharacteristic
175C4#1313324511yescharacteristic
176C4#1313324911yescharacteristic
177C2—4242no—
178C2#1681077011yescharacteristic
179C2#1681078811yescharacteristic
180C2#1681080211yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×Q8⋊C8C2
socleC22×C53#120
hypercenterQ8⋊C8D53
hypocenterC53#1640
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #16810909C2semidirect
4. #16810833C2semidirect
6. #16810355C2semidirect
7. C53×Q8⋊C8C2semidirect
97. Q8⋊C8D53semidirect
98. C53#1640semidirect