Finite Group Databasebeta

← #20273884

Subgroups — #20273884

Order 6784.

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

normal subgroups only

This group has 194 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#16811347C211yesmaximal, characteristic
2#16810915C211yesmaximal, characteristic, complemented
3#16810910C211yesmaximal, characteristic
4#16810369C211yesmaximal, characteristic, complemented
5#16810355C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic
7C53×#487C211yesmaximal, characteristic, complemented
8#13133212—22no—
9#13133206—21no—
10#13133197—21no—
11Q8×Dic53—21no—
12C4×Dic106C2211yescharacteristic
13C4×D212C2211yescharacteristic
14#13133175—21no—
15#13133150—21no—
16#13133148—21no—
17C4⋊#88322C2211yescharacteristic
18C4⋊#88322C2211yescharacteristic
19#13133140—21no—
20#13133137—21no—
21C4⋊Q8×C53C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23#103×C53—22no—
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26C2×Q8×C53—21no—
27C2×D212D411yescharacteristic
28C2×C4×D53—21no—
29C2×Dic106D411yescharacteristic
30C2×C8×C53—21no—
31C4⋊C4×C53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53D412yes—
35C42×C53C2311yescharacteristic
36#11719921—22no—
37#11719921—22no—
38#11719921—21no—
39C4⋊Dic53—22no—
40C4⋊Dic53D411yescharacteristic
41C4⋊Dic53D411yescharacteristic
42#11719919—21no—
43C4×Dic53—22no—
44C4×Dic53—21no—
45C2×#88322—21no—
46C2×#88322—21no—
47C22×D53—21no—
48Q8×C53—42no—
49Q8×C53—41no—
50Q8×C53—41no—
51C2×C4×C53—22no—
52C2×C4×C53—41no—
53C2×C4×C53C2×D411yescharacteristic
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×D411yescharacteristic
56C2×Dic53—21no—
57C2×Dic53—21no—
58C2×Dic53—21no—
59D212—21no—
60D212—21no—
61C4×D53—41no—
62C4×D53—41no—
63Dic106—21no—
64Dic106—21no—
65C8×C53—41no—
66#88322—41no—
67#88322—41no—
68C22×C53#12011yescharacteristic
69C2×D53—41no—
70C2×D53—41no—
71C2×D53—41no—
72C4×C53—21no—
73C4×C53—42no—
74C4×C53—81no—
75C4×C53#13511yescharacteristic
76C4×C53#13511yescharacteristic
77C4×C53#13711yescharacteristic
78C4×C53#13711yescharacteristic
79Dic53—41no—
80Dic53—41no—
81Dic53—41no—
82#1638—531nomaximal, Sylow
83C2×C53#45211yescharacteristic
84C2×C53#45211yescharacteristic
85C2×C53#45411yescharacteristic
86D53—81no—
87#556—531no—
88#487D5311yescharacteristic, complemented
89#477—531no—
90#476—531no—
91#332—531no—
92Q8⋊C8—531no—
93D4⋊C8—531no—
94C53#163811yescharacteristic, complemented, Sylow
95C4⋊Q8C2×D5311yescharacteristic
96C4⋊Q8C2×D5311yescharacteristic
97#126—1062no—
98#124—1061no—
99C22⋊Q8—1061no—
100C4×Q8—531no—
101C4×Q8—1061no—
102C4×D4—531no—
103#117—1061no—
104#107—1061no—
105#106—1061no—
106C4⋊C8—531no—
107C4⋊C8—531no—
108#103—22no—
109#103—1061no—
110#102—1061no—
111C4×C8C2×D5311yescharacteristic
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—531no—
115C2×D4—531no—
116C22×C4—1061no—
117C2×C8—21no—
118C2×C8—1061no—
119C2×C8—1061no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—531no—
124C4⋊C4—531no—
125C4⋊C4—1062no—
126C4⋊C4—1061no—
127C4⋊C4C22⋊D5312yes—
128C22⋊C4—1062no—
129C22⋊C4—1062no—
130C22⋊C4—1061no—
131C42—1062no—
132C42—1061no—
133C42C22×D5311yescharacteristic
134C23—1061no—
135Q8—42no—
136Q8—41no—
137Q8—41no—
138Q8—1061no—
139Q8—1061no—
140D4—1061no—
141D4—1061no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—1061no—
145C2×C4—1061no—
146C2×C4—1061no—
147C2×C4—2121no—
148C2×C4—2121no—
149C2×C4D4×D5311yescharacteristic
150C2×C4D4×D5311yescharacteristic
151C2×C4C2×C22⋊D5311yescharacteristic
152C8—41no—
153C8—2121no—
154C8—2121no—
155C22—2121no—
156C22—2121no—
157C22—2121no—
158C22#1313318511yescharacteristic
159C4—21no—
160C4—42no—
161C4—81no—
162C4—2121no—
163C4—2121no—
164C4—2121no—
165C4#1313323811yescharacteristic
166C4#1313324011yescharacteristic
167C4#1313324611yescharacteristic
168C4#1313324911yescharacteristic
169C2—4241no—
170C2#1681078211yescharacteristic
171C2#1681078411yescharacteristic
172C2#1681079511yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#487C2
socleC22×C53#120
hypercenter#487D53
hypocenterC53#1638
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #16810915C2semidirect
4. #16810369C2semidirect
5. #16810355C2semidirect
7. C53×#487C2semidirect
88. #487D53semidirect
94. C53#1638semidirect