Finite Group Databasebeta

← #20275159

Subgroups — #20275159

Order 6784.

442 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×C4×Dic53C211yesmaximal, characteristic, complemented
2C2×#13133144C212yesmaximal, complemented
3#16810587C211yesmaximal, characteristic, complemented
4#16810382C212yesmaximal, complemented
5C53×#525C211yesmaximal, characteristic
6C23×Dic53—21no—
7C2×#11719917C2212yes—
8C2×C4×Dic53—24no—
9C2×C4×Dic53—21no—
10C2×C4×Dic53C2212yes—
11#13133144—24no—
12C22⋊#88322—22no—
13C2×#41×C53C2211yescharacteristic
14C23×C4×C53C2211yescharacteristic
15#123×C53—21no—
16C22⋊Q8×C53C412yescomplemented
17#121×C53C412yescomplemented
18#120×C53—21no—
19C4×D4×C53—22no—
20#117×C53C2211yescharacteristic
21C24×C53D411yescharacteristic
22#41×C53—22no—
23#41×C53—22no—
24C2×Q8×C53C2×C411yescharacteristic, complemented
25C2×D4×C53—22no—
26C2×D4×C53—21no—
27C2×D4×C53C2×C411yescharacteristic, complemented
28C22×C4×C53—22no—
29C22×C4×C53—22no—
30C22×C4×C53—21no—
31C22×C4×C53C2×C412yes—
32C22×C4×C53D411yescharacteristic
33C22×C4×C53D412yes—
34C22×C4×C53C2311yescharacteristic
35C22×Dic53—24no—
36C22×Dic53—28no—
37C22×Dic53—22no—
38C4⋊C4×C53—22no—
39C4⋊C4×C53C2×C412yescomplemented
40C22⋊C4×C53—22no—
41C22⋊C4×C53—21no—
42C22⋊C4×C53—22no—
43C42×C53—21no—
44C4×Dic53—24no—
45C4×Dic53—22no—
46C4×Dic53D414yescomplemented
47#11719917—22no—
48#11719917—22no—
49C2×#88322—22no—
50C23×C53—24no—
51C23×C53—21no—
52C23×C53—21no—
53C23×C53—21no—
54C23×C53C22⋊C412yes—
55C23×C53C2×D411yescharacteristic
56Q8×C53—22no—
57D4×C53—22no—
58D4×C53—44no—
59D4×C53—41no—
60D4×C53—41no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—24no—
64C2×C4×C53—24no—
65C2×C4×C53—21no—
66C2×C4×C53—22no—
67C2×C4×C53—42no—
68C2×C4×C53C22⋊C412yes—
69C2×C4×C53C22×C411yescharacteristic
70C2×C4×C53C2×D412yes—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53#4112yes—
73C2×Dic53—216no—
74C2×Dic53—28no—
75C2×Dic53—24no—
76#88322—42no—
77C22×C53—24no—
78C22×C53—24no—
79C22×C53—22no—
80C22×C53—22no—
81C22×C53—22no—
82C22×C53—41no—
83C22×C53—42no—
84C22×C53#10414yescomplemented
85C22×C53C2×C22⋊C411yescharacteristic
86C22×C53#12011yescharacteristic
87C22×C53#12311yescharacteristic
88C4×C53—22no—
89C4×C53—41no—
90C4×C53—42no—
91C4×C53C4×D412yes—
92C4×C53#12112yes—
93Dic53—28no—
94#1896—531nomaximal, Sylow
95C2×C53—21no—
96C2×C53—24no—
97C2×C53—21no—
98C2×C53—81no—
99C2×C53#38611yescharacteristic
100C2×C53C2×#10412yes—
101#525D5311yescharacteristic
102C22×C42—531no—
103C2×#104—532no—
104#407—531no—
105#339—532no—
106C53#189611yescharacteristic, complemented, Sylow
107C2×#41C2×D5311yescharacteristic
108C23×C4—1061no—
109C23×C4C2×D5311yescharacteristic
110C2×M16—532no—
111#123—21no—
112C22⋊Q8Dic5312yescomplemented
113#121Dic5312yescomplemented
114#120—21no—
115C4×D4—22no—
116#117C2×D5311yescharacteristic
117C2×C42—532no—
118C2×C42—1064no—
119C2×C42—1061no—
120#104—1064no—
121C22⋊C8—1062no—
122C24C22⋊D5311yescharacteristic
123#41—22no—
124#41—22no—
125C2×Q8C2×Dic5311yescharacteristic, complemented
126C2×D4—21no—
127C2×D4—22no—
128C2×D4C2×Dic5311yescharacteristic, complemented
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—1064no—
133C22×C4—1068no—
134C22×C4—1062no—
135C22×C4C2×Dic5312yes—
136C22×C4C22⋊D5312yes—
137C22×C4C22⋊D5311yescharacteristic
138C22×C4C22×D5311yescharacteristic
139M16—1062no—
140M16—1062no—
141C2×C8—1062no—
142C4⋊C4—22no—
143C4⋊C4C2×Dic5312yescomplemented
144C22⋊C4—21no—
145C22⋊C4—22no—
146C22⋊C4—22no—
147C42—21no—
148C42—534no—
149C42—1064no—
150C42—1062no—
151C23—24no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23C22⋊Dic5312yes—
156C23C2×C22⋊D5311yescharacteristic
157Q8—22no—
158D4—22no—
159D4—44no—
160D4—41no—
161D4—41no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—10616no—
170C2×C4—1068no—
171C2×C4—1064no—
172C2×C4C22⋊Dic5312yes—
173C2×C4D4×D5312yes—
174C2×C4#1171994712yes—
175C2×C4C22×Dic5311yescharacteristic
176C2×C4C2×C22⋊D5311yescharacteristic
177C8—2122no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—41no—
184C22—42no—
185C22#1313314414yescomplemented
186C22C2×C22⋊Dic5311yescharacteristic
187C22#1313318511yescharacteristic
188C22#1313319811yescharacteristic
189C4—22no—
190C4—41no—
191C4—42no—
192C4—1068no—
193C4D4×Dic5312yes—
194C4#1313318912yes—
195C2—21no—
196C2—24no—
197C2—21no—
198C2—81no—
199C2#1681050311yescharacteristic
200C2C2×#1313314412yes—
subgroupG/H
centerC2×C4C2×C22⋊D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#525C2
socleC22×C53C2×C22⋊C4
hypercenter#525D53
hypocenterC53#1896
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic53C2semidirect
2. C2×#13133144C2semidirect
3. #16810587C2semidirect
4. #16810382C2semidirect
16. C22⋊Q8×C53C4semidirect
17. #121×C53C4semidirect
24. C2×Q8×C53C2×C4semidirect
27. C2×D4×C53C2×C4semidirect
39. C4⋊C4×C53C2×C4semidirect
46. C4×Dic53D4semidirect
84. C22×C53#104semidirect
106. C53#1896semidirect
112. C22⋊Q8Dic53semidirect
113. #121Dic53semidirect
125. C2×Q8C2×Dic53semidirect
128. C2×D4C2×Dic53semidirect
143. C4⋊C4C2×Dic53semidirect
185. C22#13133144semidirect