Finite Group Databasebeta

← #20290238

Subgroups — #20290238

Order 6784.

242 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 58 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#16811646C211yesmaximal, characteristic, complemented
2#16811641C211yesmaximal, characteristic, complemented
3#16811482C212yesmaximal, complemented
4#16811481C212yesmaximal, complemented
5#16811170C211yesmaximal, characteristic
6#13133284C2212yes—
7#13133284C2212yes—
8#13133274—21no—
9#13133274—21no—
10#13133272—21no—
11C4×#88324—21no—
12C2×#11719945—21no—
13#13133216C2211yescharacteristic
14#13133213C2211yescharacteristic
15#13133203—21no—
16#13133192C411yescharacteristic
17#13133190—21no—
18#13133188C411yescharacteristic
19C4×D212C411yescharacteristic
20C4×C22⋊D53—21no—
21C42×D53C2211yescharacteristic
22C4×D4×C53C411yescharacteristic
23C2×D4×C53C2×C411yescharacteristic
24C22×C4×C53—21no—
25C2×C22⋊D53—21no—
26C2×C22⋊D53—21no—
27C2×C22⋊D53—21no—
28#11719945—44no—
29C2×D212—21no—
30C2×D212C2×C411yescharacteristic
31C2×C4×D53—21no—
32C2×C4×D53C2×C411yescharacteristic
33C2×C4×D53D411yescharacteristic
34C2×C4×D53D411yescharacteristic
35C2×Dic106—21no—
36C2×#88324—22no—
37C2×#88324—22no—
38C4⋊C4×C53C2×C411yescharacteristic
39C22⋊C4×C53—21no—
40C42×C53C2×C411yescharacteristic
41C22⋊Dic53—21no—
42#11719921—21no—
43#11719921—21no—
44C4⋊Dic53C2×C411yescharacteristic
45#11719919—21no—
46C4×Dic53D411yescharacteristic
47C4×Dic53D411yescharacteristic
48C4×Dic53C2311yescharacteristic
49C23×C53—21no—
50C22×D53—21no—
51C22×D53C22⋊C411yescharacteristic
52D4×C53—41no—
53C2×C4×C53—21no—
54C2×C4×C53—42no—
55C2×C4×C53C22⋊C411yescharacteristic
56C2×C4×C53C22⋊C411yescharacteristic
57C2×C4×C53C22×C411yescharacteristic
58C22⋊D53—42no—
59C22⋊D53—42no—
60C22⋊D53—42no—
61C2×Dic53—21no—
62C2×Dic53C22⋊C411yescharacteristic
63C2×Dic53C2×D411yescharacteristic
64C2×Dic53C2×D411yescharacteristic
65D212—41no—
66D212—42no—
67C4×D53—22no—
68C4×D53—22no—
69C4×D53—41no—
70C4×D53—42no—
71Dic106—42no—
72#88324—42no—
73#88324—42no—
74C22×C53—41no—
75C22×C53—41no—
76C22×C53—41no—
77C22×C53C2×C22⋊C411yescharacteristic
78C2×D53—21no—
79C2×D53—21no—
80C2×D53—21no—
81C2×D53—41no—
82C2×D53—41no—
83C2×D53—41no—
84C4×C53—21no—
85C4×C53—21no—
86C4×C53—21no—
87C4×C53—41no—
88Dic53—21no—
89Dic53—41no—
90Dic53#13512yes—
91Dic53AGL(1,ℤ/8)12yes—
92#1553—531nomaximal, Sylow
93C2×C53—81no—
94C2×C53#41311yescharacteristic
95C2×C53#41611yescharacteristic
96C2×C53#41811yescharacteristic
97D53—41no—
98D53—81no—
99#532—531no—
100#432—531no—
101#431—531no—
102#332—532no—
103#331—532no—
104C53#155311yescharacteristic, complemented, Sylow
105C2×#41—1061no—
106C4⋊Q8—531no—
107#127—531no—
108#124—1061no—
109#121—531no—
110#121—531no—
111#121—1061no—
112C4×D4—531no—
113C4×D4—1061no—
114C4×D4#615711yescharacteristic
115C2×C42—531no—
116C4⋊C8—532no—
117C4⋊C8—532no—
118C22⋊C8—1061no—
119C22⋊C8—1061no—
120#97—1061no—
121C4×C8—1061no—
122#41—2124no—
123C2×Q8—1061no—
124C2×D4—531no—
125C2×D4—1061no—
126C2×D4—1061no—
127C2×D4—1061no—
128C2×D4—1061no—
129C2×D4C2×#615711yescharacteristic
130C22×C4—21no—
131C22×C4—531no—
132C22×C4—531no—
133C22×C4—531no—
134C22×C4—1061no—
135C2×C8—1062no—
136C2×C8—1062no—
137C4⋊C4—531no—
138C4⋊C4—1061no—
139C4⋊C4C2×#615711yescharacteristic
140C22⋊C4—21no—
141C22⋊C4—1061no—
142C22⋊C4—1061no—
143C22⋊C4—1061no—
144C42—531no—
145C42—531no—
146C42—531no—
147C42C2×#615711yescharacteristic
148C23—21no—
149C23—531no—
150C23—1061no—
151Q8—2122no—
152D4—41no—
153D4—2121no—
154D4—2122no—
155D4—2122no—
156D4—2122no—
157D4—2122no—
158C2×C4—21no—
159C2×C4—42no—
160C2×C4—531no—
161C2×C4—531no—
162C2×C4—531no—
163C2×C4—1062no—
164C2×C4—1062no—
165C2×C4—1061no—
166C2×C4—2121no—
167C2×C4—2122no—
168C2×C4C22⋊#615711yescharacteristic
169C2×C4C22⋊#615711yescharacteristic
170C2×C4C22×#615711yescharacteristic
171C8—2122no—
172C8—2122no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—1061no—
177C22—1061no—
178C22—1061no—
179C22—2121no—
180C22—2121no—
181C22—2121no—
182C22C2×C22⋊#615711yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—41no—
187C4—532no—
188C4—532no—
189C4—1061no—
190C4—2121no—
191C2—81no—
192C2—4241no—
193C2—2121no—
194C2#1681160911yescharacteristic
195C2#1681161311yescharacteristic
196C2#1681161511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4×D4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4×D4#6157
hypocenterC53#1553
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #16811646C2semidirect
2. #16811641C2semidirect
3. #16811482C2semidirect
4. #16811481C2semidirect
104. C53#1553semidirect