Finite Group Databasebeta

← #20273890

Subgroups — #20273890

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#16811346C211yesmaximal, characteristic
2#16810974C211yesmaximal, characteristic, complemented
3#16810918C211yesmaximal, characteristic, complemented
4#16810914C211yesmaximal, characteristic
5#16810370C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic
7C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8#13133217C2211yescharacteristic
9#13133217C2211yescharacteristic
10#13133212—22no—
11#13133205—21no—
12#13133193—21no—
13C4×Dic106—21no—
14C4×D212C2211yescharacteristic
15#13133175—21no—
16#13133150—21no—
17#13133149—21no—
18C4⋊#88322C2211yescharacteristic
19#13133142—22no—
20#13133141—21no—
21#13133139—21no—
22C4×Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53D411yescharacteristic
26C2×D212D411yescharacteristic
27C2×C4×D53—21no—
28C2×Dic106—21no—
29C2×Dic106—21no—
30C2×C8×C53—21no—
31C2×C8×C53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53D411yescharacteristic
34C42×C53—21no—
35C42×C53C2311yescharacteristic
36#11719921—22no—
37#11719921—22no—
38#11719921—21no—
39C4⋊Dic53—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53D412yes—
43C4⋊Dic53D411yescharacteristic
44#11719919—22no—
45C4×Dic53—22no—
46C2×#88322—21no—
47C22×D53—21no—
48Q8×C53—21no—
49Q8×C53—21no—
50C2×C4×C53—21no—
51C2×C4×C53—21no—
52C2×C4×C53C2×D411yescharacteristic
53C2×C4×C53C2×D411yescharacteristic
54C2×C4×C53C2×D411yescharacteristic
55C2×Dic53—22no—
56C2×Dic53—21no—
57C2×Dic53—41no—
58D212—21no—
59D212—21no—
60C4×D53—41no—
61C4×D53—41no—
62Dic106—41no—
63Dic106—41no—
64Dic106—42no—
65C8×C53—41no—
66C8×C53—41no—
67#88322—41no—
68C22×C53#12011yescharacteristic
69C2×D53—41no—
70C2×D53—41no—
71C2×D53—41no—
72C4×C53—21no—
73C4×C53—41no—
74C4×C53—41no—
75C4×C53#13511yescharacteristic
76C4×C53#13511yescharacteristic
77C4×C53#13711yescharacteristic
78C4×C53#13711yescharacteristic
79Dic53—42no—
80Dic53—41no—
81Dic53—81no—
82#1638—531nomaximal, Sylow
83C2×C53#45211yescharacteristic
84C2×C53#45211yescharacteristic
85C2×C53#45411yescharacteristic
86D53—81no—
87#556—531no—
88#487—531no—
89#477—531no—
90#476—531no—
91#332—531no—
92Q8⋊C8D5311yescharacteristic, complemented
93D4⋊C8—531no—
94C53#163811yescharacteristic, complemented, Sylow
95C4⋊Q8—531no—
96C4⋊Q8—531no—
97#126—1062no—
98#124—1061no—
99C22⋊Q8—1061no—
100C4×Q8—1061no—
101C4×Q8C2×D5311yescharacteristic
102C4×D4—531no—
103#117—1061no—
104#107—1061no—
105#106—1061no—
106C4⋊C8—531no—
107C4⋊C8C2×D5311yescharacteristic
108#103—1062no—
109#103—1061no—
110#102—1061no—
111C4×C8C2×D5311yescharacteristic
112C2×Q8—1061no—
113C2×Q8—1061no—
114C2×Q8D21211yescharacteristic
115C2×D4—531no—
116C22×C4—1061no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—1061no—
120C4⋊C4—21no—
121C4⋊C4—532no—
122C4⋊C4—531no—
123C4⋊C4—1062no—
124C4⋊C4—1061no—
125C4⋊C4—1061no—
126C4⋊C4—1061no—
127C4⋊C4D21211yescharacteristic
128C22⋊C4—1062no—
129C22⋊C4—1062no—
130C22⋊C4—1061no—
131C42—21no—
132C42—1062no—
133C42C22×D5311yescharacteristic
134C23—1061no—
135Q8—21no—
136Q8—21no—
137Q8—2122no—
138Q8—2121no—
139Q8—2121no—
140D4—1061no—
141D4—1061no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—1061no—
145C2×C4—1062no—
146C2×C4—2121no—
147C2×C4—2121no—
148C2×C4—2121no—
149C2×C4C2×D21211yescharacteristic
150C2×C4D4×D5311yescharacteristic
151C2×C4D4×D5311yescharacteristic
152C8—41no—
153C8—41no—
154C8—2121no—
155C22—2121no—
156C22—2121no—
157C22—2121no—
158C22#1313318711yescharacteristic
159C4—21no—
160C4—41no—
161C4—41no—
162C4—4241no—
163C4—2121no—
164C4—2122no—
165C4#1313323811yescharacteristic
166C4#1313323911yescharacteristic
167C4#1313324911yescharacteristic
168C4#1313325011yescharacteristic
169C2—4241no—
170C2#1681078711yescharacteristic
171C2#1681078811yescharacteristic
172C2#1681079711yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×Q8⋊C8C2
socleC22×C53#120
hypercenterQ8⋊C8D53
hypocenterC53#1638
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
2. #16810974C2semidirect
3. #16810918C2semidirect
5. #16810370C2semidirect
7. C53×Q8⋊C8C2semidirect
92. Q8⋊C8D53semidirect
94. C53#1638semidirect