Finite Group Databasebeta

← #20273902

Subgroups — #20273902

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#16811359C211yesmaximal, characteristic
2#16810969C211yesmaximal, characteristic
3#16810909C211yesmaximal, characteristic, complemented
4#16810834C211yesmaximal, characteristic, complemented
5#16810353C211yesmaximal, characteristic, complemented
6#16810353C211yesmaximal, characteristic, complemented
7C53×#331C211yesmaximal, characteristic
8C2×Dic212—21no—
9#13133217C2211yescharacteristic
10#13133204—21no—
11#13133204—21no—
12#13133198—22no—
13C4×Dic106C2211yescharacteristic
14C4×Dic106C2211yescharacteristic
15D4×Dic53—21no—
16D4×Dic53—21no—
17#13133149—21no—
18#13133142—21no—
19#13133142—21no—
20#13133138—22no—
21C4×#88322C2211yescharacteristic
22#127×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4⋊C8×C53C2211yescharacteristic
25C2×D4×C53—21no—
26C2×D4×C53—21no—
27C2×D4×C53D412yes—
28C22×Dic53—22no—
29C2×Dic106—21no—
30C2×Dic106D411yescharacteristic
31C2×Dic106D411yescharacteristic
32C2×C8×C53—21no—
33C2×C8×C53—21no—
34C42×C53C2311yescharacteristic
35C22⋊Dic53—22no—
36C22⋊Dic53—22no—
37C22⋊Dic53—22no—
38C4⋊Dic53—21no—
39C4⋊Dic53—21no—
40C4⋊Dic53D411yescharacteristic
41C4⋊Dic53D411yescharacteristic
42#11719919—21no—
43#11719919—21no—
44C4×Dic53—21no—
45C4×Dic53—21no—
46C2×#88322—21no—
47Dic212—42no—
48C23×C53—22no—
49D4×C53—42no—
50D4×C53—41no—
51D4×C53—41no—
52D4×C53—41no—
53D4×C53—41no—
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×Dic53—21no—
58C2×Dic53—21no—
59C2×Dic53—21no—
60C2×Dic53—21no—
61C2×Dic53—41no—
62C2×Dic53—82no—
63Dic106—21no—
64Dic106—21no—
65Dic106—21no—
66Dic106—21no—
67Dic106—41no—
68Dic106—41no—
69C8×C53—41no—
70C8×C53—41no—
71#88322—41no—
72C22×C53—42no—
73C22×C53—42no—
74C22×C53—42no—
75C22×C53#12011yescharacteristic
76C4×C53—21no—
77C4×C53#13511yescharacteristic
78C4×C53#13511yescharacteristic
79C4×C53AGL(1,ℤ/8)11yescharacteristic
80C4×C53#13711yescharacteristic
81Dic53—41no—
82Dic53—41no—
83Dic53—41no—
84Dic53—41no—
85Dic53—81no—
86#1640—531nomaximal, Sylow
87C2×C53—82no—
88C2×C53#44911yescharacteristic
89C2×C53#45211yescharacteristic
90C2×C53#45511yescharacteristic
91#559—531no—
92#486—531no—
93#475—531no—
94#462—531no—
95#331D5311yescharacteristic
96Q8⋊C8—531no—
97Q8⋊C8—531no—
98C53#164011yescharacteristic, complemented, Sylow
99C2×Q16—1061no—
100C4⋊Q8—531no—
101#127C2×D5311yescharacteristic
102#124—1061no—
103#124—1061no—
104#123—1062no—
105C4×Q8—531no—
106C4×Q8—531no—
107C4×D4—1061no—
108C4×D4—1061no—
109#106—1061no—
110C4⋊C8C2×D5311yescharacteristic
111C4⋊C8C2×D5311yescharacteristic
112#103—1061no—
113#103—1061no—
114#102—1062no—
115C4×C8—531no—
116C2×Q8—531no—
117C2×Q8—531no—
118C2×Q8—1061no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4D21212yes—
122C22×C4—1062no—
123Q16—2122no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—1061no—
127C4⋊C4—531no—
128C4⋊C4—531no—
129C4⋊C4—1061no—
130C4⋊C4—1061no—
131C4⋊C4—1061no—
132C4⋊C4—1061no—
133C22⋊C4—1062no—
134C22⋊C4—1062no—
135C22⋊C4—1062no—
136C42—1061no—
137C42—1061no—
138C42C22×D5311yescharacteristic
139C23—22no—
140Q8—1061no—
141Q8—1061no—
142Q8—1061no—
143Q8—1061no—
144Q8—2121no—
145Q8—2121no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—1061no—
152C2×C4—1061no—
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—42no—
164C22—42no—
165C22—42no—
166C22#1313318711yescharacteristic
167C4—21no—
168C4—4241no—
169C4—2121no—
170C4—2121no—
171C4—2121no—
172C4—2121no—
173C4#1313323611yescharacteristic
174C4#1313323811yescharacteristic
175C4#1313324111yescharacteristic
176C4#1313324611yescharacteristic
177C2—82no—
178C2#1681076911yescharacteristic
179C2#1681078611yescharacteristic
180C2#1681080311yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#331C2
socleC22×C53#120
hypercenter#331D53
hypocenterC53#1640
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
3. #16810909C2semidirect
4. #16810834C2semidirect
5. #16810353C2semidirect
6. #16810353C2semidirect
98. C53#1640semidirect