Finite Group Databasebeta

← #20273944

Subgroups — #20273944

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#16811339C211yesmaximal, characteristic
2#16810973C211yesmaximal, characteristic, complemented
3#16810904C211yesmaximal, characteristic, complemented
4#16810829C211yesmaximal, characteristic, complemented
5#16810350C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×#333C211yesmaximal, characteristic, complemented
8C2×#11719913—21no—
9#13133217C2211yescharacteristic
10#13133206—22no—
11#13133192—21no—
12#13133192—21no—
13C4×D212C2211yescharacteristic
14C4×D212C2211yescharacteristic
15#13133175—21no—
16#13133175—21no—
17#13133151—21no—
18#13133142—21no—
19#13133141—22no—
20#13133139—21no—
21C4×#88322C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4⋊C8×C53C2211yescharacteristic
25C2×Q8×C53D412yes—
26C2×D212—22no—
27C2×D212D411yescharacteristic
28C2×D212D411yescharacteristic
29C2×C4×D53—21no—
30C2×C4×D53—21no—
31C2×Dic106—21no—
32C2×C8×C53—21no—
33C2×C8×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53—21no—
36C42×C53C2311yescharacteristic
37#11719921—22no—
38#11719921—22no—
39#11719921—21no—
40#11719921—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53—21no—
43C4⋊Dic53D411yescharacteristic
44C4⋊Dic53D411yescharacteristic
45C4×Dic53—22no—
46C2×#88322—21no—
47#11719913—42no—
48C22×D53—21no—
49C22×D53—21no—
50Q8×C53—42no—
51C2×C4×C53—22no—
52C2×C4×C53C2×D411yescharacteristic
53C2×C4×C53C2×D411yescharacteristic
54C2×C4×C53C2×D411yescharacteristic
55C2×Dic53—21no—
56C2×Dic53—21no—
57C2×Dic53—41no—
58D212—21no—
59D212—21no—
60D212—21no—
61D212—21no—
62D212—82no—
63C4×D53—41no—
64C4×D53—41no—
65C4×D53—41no—
66C4×D53—41no—
67Dic106—41no—
68Dic106—41no—
69C8×C53—41no—
70C8×C53—41no—
71#88322—41no—
72C22×C53#12011yescharacteristic
73C2×D53—41no—
74C2×D53—41no—
75C2×D53—41no—
76C2×D53—41no—
77C2×D53—41no—
78C2×D53—41no—
79C4×C53—21no—
80C4×C53—42no—
81C4×C53#13511yescharacteristic
82C4×C53#13511yescharacteristic
83C4×C53AGL(1,ℤ/8)11yescharacteristic
84C4×C53#13711yescharacteristic
85Dic53—41no—
86Dic53—41no—
87Dic53—81no—
88#1646—531nomaximal, Sylow
89C2×C53#44911yescharacteristic
90C2×C53#45211yescharacteristic
91C2×C53#45511yescharacteristic
92D53—81no—
93D53—81no—
94#555—531no—
95#487—531no—
96#474—531no—
97#461—531no—
98#333D5311yescharacteristic, complemented
99D4⋊C8—531no—
100D4⋊C8—531no—
101C53#164611yescharacteristic, complemented, Sylow
102C2×SD16—1061no—
103C4⋊Q8—531no—
104C4⋊Q8C2×D5311yescharacteristic
105#124—1062no—
106#121—1061no—
107#121—1061no—
108C4×D4—531no—
109C4×D4—531no—
110#117—1061no—
111#117—1061no—
112#107—1061no—
113C4⋊C8C2×D5311yescharacteristic
114C4⋊C8C2×D5311yescharacteristic
115#103—1061no—
116#103—1062no—
117#102—1061no—
118C4×C8—531no—
119C2×Q8—1061no—
120C2×Q8D21212yes—
121C2×D4—531no—
122C2×D4—531no—
123C2×D4—1062no—
124C22×C4—1061no—
125C22×C4—1061no—
126SD16—2122no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—1061no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—531no—
133C4⋊C4—531no—
134C4⋊C4—1061no—
135C4⋊C4—1061no—
136C22⋊C4—1062no—
137C22⋊C4—1062no—
138C22⋊C4—1061no—
139C22⋊C4—1061no—
140C42—1062no—
141C42C22×D5311yescharacteristic
142C23—1061no—
143C23—1061no—
144Q8—42no—
145Q8—2121no—
146Q8—2121no—
147D4—1061no—
148D4—1061no—
149D4—1061no—
150D4—1061no—
151D4—4242no—
152C2×C4—22no—
153C2×C4—1061no—
154C2×C4—1061no—
155C2×C4—2121no—
156C2×C4—2121no—
157C2×C4—2121no—
158C2×C4—2121no—
159C2×C4—2121no—
160C2×C4C2×D21211yescharacteristic
161C2×C4D4×D5311yescharacteristic
162C2×C4D4×D5311yescharacteristic
163C8—41no—
164C8—41no—
165C8—2121no—
166C22—2121no—
167C22—2121no—
168C22—2121no—
169C22—2121no—
170C22—2121no—
171C22—2121no—
172C22#1313318711yescharacteristic
173C4—21no—
174C4—42no—
175C4—4241no—
176C4—2121no—
177C4—2121no—
178C4#1313323811yescharacteristic
179C4#1313324011yescharacteristic
180C4#1313324411yescharacteristic
181C4#1313324911yescharacteristic
182C2—4241no—
183C2—4241no—
184C2#1681077111yescharacteristic
185C2#1681078811yescharacteristic
186C2#1681080311yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#333C2
socleC22×C53#120
hypercenter#333D53
hypocenterC53#1646
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #16810973C2semidirect
3. #16810904C2semidirect
4. #16810829C2semidirect
5. #16810350C2semidirect
6. #16810350C2semidirect
7. C53×#333C2semidirect
98. #333D53semidirect
101. C53#1646semidirect