Finite Group Databasebeta

← #20273720

Subgroups — #20273720

Order 6784.

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

normal subgroups only

This group has 254 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#16811204C211yesmaximal, characteristic
2#16810937C212yesmaximal, complemented
3#16810553C211yesmaximal, characteristic, complemented
4#16810358C212yesmaximal, complemented
5C53×#324C211yesmaximal, characteristic, complemented
6C2×#11719945—21no—
7#13133197—22no—
8#13133192—22no—
9#13133191—21no—
10#13133191C2212yes—
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719919—21no—
13#13133151—22no—
14#13133139—22no—
15#13133124—21no—
16#13133124C2211yescharacteristic
17C2×C4⋊C4×C53C2211yescharacteristic
18C22⋊C8×C53C2212yes—
19C22×C4×C53D411yescharacteristic
20C22×C4×C53D411yescharacteristic
21C22×C4×C53C2311yescharacteristic
22C2×C22⋊D53—22no—
23C22×Dic53—21no—
24C22×Dic53—21no—
25#11719945—42no—
26#11719945—42no—
27C2×D212—21no—
28C2×D212—21no—
29C2×D212D412yes—
30C2×C4×D53—22no—
31C2×Dic106—21no—
32C2×C8×C53—22no—
33C4⋊C4×C53—22no—
34#11719921—22no—
35#11719921—22no—
36C4⋊Dic53—21no—
37C4⋊Dic53—22no—
38C4⋊Dic53D412yes—
39#11719919—42no—
40C23×C53C2×D411yescharacteristic
41C22×D53—22no—
42C2×C4×C53—21no—
43C2×C4×C53—21no—
44C2×C4×C53—21no—
45C2×C4×C53—41no—
46C2×C4×C53—41no—
47C2×C4×C53D812yes—
48C2×C4×C53C2×D412yes—
49C22⋊D53—42no—
50C22⋊D53—42no—
51C2×Dic53—22no—
52C2×Dic53—22no—
53C2×Dic53—41no—
54C2×Dic53—41no—
55C2×Dic53—41no—
56C2×Dic53—41no—
57C2×Dic53—41no—
58D212—42no—
59D212—42no—
60D212—81no—
61C4×D53—44no—
62Dic106—42no—
63C8×C53—42no—
64C22×C53—21no—
65C22×C53—21no—
66C22×C53#12011yescharacteristic
67C22×C53C2×D811yescharacteristic
68C22×C53AGL(1,ℤ/8)11yescharacteristic
69C2×D53—42no—
70C2×D53—42no—
71C2×D53—42no—
72C4×C53—22no—
73C4×C53—22no—
74C4×C53—41no—
75Dic53—42no—
76Dic53—42no—
77Dic53—81no—
78#1610—531nomaximal, Sylow
79C2×C53—21no—
80C2×C53—21no—
81C2×C53#44711yescharacteristic
82C2×C53#45611yescharacteristic
83C2×C53#45811yescharacteristic
84D53—82no—
85#537—531no—
86#480—532no—
87#398—531no—
88#327—532no—
89#324D5311yescharacteristic, complemented
90C53#161011yescharacteristic, complemented, Sylow
91C2×#41—1061no—
92C22⋊Q8—1062no—
93#121—532no—
94#121—1062no—
95#121—1061no—
96C2×C4⋊C4—531no—
97C2×C4⋊C4—1061no—
98C2×C4⋊C4C2×D5311yescharacteristic
99#107—1062no—
100#102—1062no—
101C22⋊C8C2×D5312yes—
102#95—531no—
103#95—1061no—
104#41—2122no—
105#41—2122no—
106C2×Q8—1061no—
107C2×D4—532no—
108C2×D4—1061no—
109C2×D4—1061no—
110C2×D4—1062no—
111C22×C4—1062no—
112C22×C4—1061no—
113C22×C4—1061no—
114C22×C4D21211yescharacteristic
115C22×C4D21211yescharacteristic
116C22×C4C22×D5311yescharacteristic
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—532no—
120C4⋊C4—1061no—
121C4⋊C4—1062no—
122C4⋊C4—2122no—
123C22⋊C4—1062no—
124C22⋊C4—1062no—
125C23—1062no—
126C23C2×D21211yescharacteristic
127Q8—2122no—
128D4—4241no—
129D4—2122no—
130D4—2122no—
131D4—2122no—
132D4—2122no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—1062no—
139C2×C4—1062no—
140C2×C4—2124no—
141C2×C4—2121no—
142C2×C4—2121no—
143C2×C4—2121no—
144C2×C4—2121no—
145C2×C4—2121no—
146C2×C4D42412yes—
147C2×C4D4×D5312yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—2122no—
152C22—2122no—
153C22—2122no—
154C22#1313318711yescharacteristic
155C22C2×D42411yescharacteristic
156C22#1313324511yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—4241no—
161C4—2122no—
162C4—2122no—
163C2—21no—
164C2—21no—
165C2—4242no—
166C2#1681075911yescharacteristic
167C2#1681080811yescharacteristic
168C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#324C2
socleC22×C53#120
hypercenter#324D53
hypocenterC53#1610
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #16810937C2semidirect
3. #16810553C2semidirect
4. #16810358C2semidirect
5. C53×#324C2semidirect
89. #324D53semidirect
90. C53#1610semidirect