Finite Group Databasebeta

← #20273773

Subgroups — #20273773

Order 6784.

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

normal subgroups only

This group has 240 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#16811358C211yesmaximal, characteristic, complemented
2#16810999C211yesmaximal, characteristic
3#16810820C211yesmaximal, characteristic, complemented
4#16810366C211yesmaximal, characteristic, complemented
5Q8⋊#88322C211yesmaximal, characteristic, complemented
6#16810353C211yesmaximal, characteristic, complemented
7C53×#460C211yesmaximal, characteristic, complemented
8C2×#11719923—21no—
9C2×#11719922—22no—
10#13133214C2211yescharacteristic
11#13133206—21no—
12#13133205—21no—
13#13133202—22no—
14C4×Dic106C2211yescharacteristic
15C4×D212—21no—
16#13133179—21no—
17C4⋊#88322C2211yescharacteristic
18#13133138—21no—
19C4×#88322C2211yescharacteristic
20C2×SD16×C53—21no—
21#127×C53C2211yescharacteristic
22C4×Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24#102×C53—21no—
25C2×Q8×C53D411yescharacteristic
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29C2×D212—21no—
30C2×D212—21no—
31C2×D212D412yes—
32C2×C4×D53—22no—
33C2×Dic106D411yescharacteristic
34SD16×C53—42no—
35C2×C8×C53—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53D411yescharacteristic
38C42×C53—21no—
39C42×C53C2311yescharacteristic
40#11719923—42no—
41#11719922—44no—
42#11719921—22no—
43#11719921—22no—
44#11719921—22no—
45C4⋊Dic53D411yescharacteristic
46#11719919—21no—
47C4×Dic53—21no—
48C2×#88322—21no—
49C2×#88322—21no—
50C23×C53—41no—
51C22×D53—22no—
52Q8×C53—21no—
53Q8×C53—21no—
54D4×C53—42no—
55D4×C53—41no—
56D4×C53—41no—
57D4×C53—41no—
58D4×C53—41no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53C2×D411yescharacteristic
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53C2×D411yescharacteristic
64C2×Dic53—21no—
65C2×Dic53—21no—
66D212—42no—
67D212—41no—
68D212—41no—
69D212—41no—
70D212—41no—
71C4×D53—82no—
72Dic106—21no—
73Dic106—21no—
74C8×C53—41no—
75#88322—22no—
76#88322—41no—
77C22×C53—81no—
78C22×C53—81no—
79C22×C53—81no—
80C22×C53#12011yescharacteristic
81C2×D53—42no—
82C2×D53—42no—
83C2×D53—42no—
84C4×C53—21no—
85C4×C53—41no—
86C4×C53—41no—
87C4×C53#13511yescharacteristic
88C4×C53#13511yescharacteristic
89C4×C53AGL(1,ℤ/8)11yescharacteristic
90C4×C53AGL(1,ℤ/8)11yescharacteristic
91Dic53—41no—
92Dic53—41no—
93#1620—531nomaximal, Sylow
94C2×C53—161no—
95C2×C53#44911yescharacteristic
96C2×C53#44911yescharacteristic
97C2×C53#45311yescharacteristic
98D53—82no—
99#559—531no—
100#493—531no—
101#460—531no—
102#460D5311yescharacteristic, complemented
103#331—531no—
104Q8⋊C8—531no—
105Q8⋊C8—531no—
106C53#162011yescharacteristic, complemented, Sylow
107C2×SD16—21no—
108C2×SD16—1061no—
109C2×D8—1062no—
110#127—531no—
111#127C2×D5311yescharacteristic
112#124—1061no—
113#124—1061no—
114#123—1062no—
115C4×Q8—531no—
116C4×Q8C2×D5311yescharacteristic
117C4×D4—1061no—
118C4×D4—1061no—
119C4⋊C8—531no—
120C4⋊C8C2×D5311yescharacteristic
121#102—21no—
122#102—1061no—
123C4×C8—531no—
124C2×Q8—531no—
125C2×Q8C22⋊D5311yescharacteristic
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—532no—
130C2×D4—1061no—
131C2×D4—1061no—
132C22×C4—1062no—
133SD16—42no—
134SD16—2122no—
135D8—2124no—
136C2×C8—21no—
137C2×C8—1061no—
138C2×C8—1061no—
139C4⋊C4—21no—
140C4⋊C4—531no—
141C4⋊C4—1061no—
142C4⋊C4C22⋊D5311yescharacteristic
143C22⋊C4—1062no—
144C22⋊C4—1062no—
145C22⋊C4—1062no—
146C42—21no—
147C42—1061no—
148C42C22×D5311yescharacteristic
149C23—41no—
150C23—1062no—
151Q8—21no—
152Q8—21no—
153Q8—1061no—
154Q8—1061no—
155D4—42no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—2122no—
161D4—2121no—
162D4—2121no—
163D4—2121no—
164D4—2121no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—1061no—
168C2×C4—1061no—
169C2×C4—4242no—
170C2×C4D4×D5311yescharacteristic
171C2×C4D4×D5311yescharacteristic
172C2×C4C2×C22⋊D5311yescharacteristic
173C8—41no—
174C8—1062no—
175C8—2121no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—2122no—
180C22—2122no—
181C22—2122no—
182C22#1313318511yescharacteristic
183C4—21no—
184C4—41no—
185C4—41no—
186C4—2121no—
187C4—2121no—
188C4#1313323711yescharacteristic
189C4#1313323811yescharacteristic
190C4#1313324311yescharacteristic
191C4#1313324411yescharacteristic
192C2—161no—
193C2—4242no—
194C2#1681076611yescharacteristic
195C2#1681076711yescharacteristic
196C2#1681078911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#460C2
socleC22×C53#120
hypercenter#460D53
hypocenterC53#1620
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #16811358C2semidirect
3. #16810820C2semidirect
4. #16810366C2semidirect
5. Q8⋊#88322C2semidirect
6. #16810353C2semidirect
7. C53×#460C2semidirect
102. #460D53semidirect
106. C53#1620semidirect