Finite Group Databasebeta

← #20290247

Subgroups — #20290247

Order 6784.

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

normal subgroups only

This group has 246 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#16811643C211yesmaximal, characteristic, complemented
2#16811626C211yesmaximal, characteristic, complemented
3#16811482C212yesmaximal, complemented
4#16811481C212yesmaximal, complemented
5#16811168C211yesmaximal, characteristic
6C2×#11719940—21no—
7#13133284C2212yes—
8#13133284C2212yes—
9C4⋊#88324—21no—
10C22⋊#88324—21no—
11#13133272—21no—
12C2×#11719947—21no—
13#13133216C2211yescharacteristic
14#13133213C2211yescharacteristic
15#13133203—21no—
16#13133191C411yescharacteristic
17#13133189—21no—
18D4×Dic53C411yescharacteristic
19#13133179C411yescharacteristic
20#13133177—21no—
21C2×C4×Dic53C2211yescharacteristic
22#121×C53C411yescharacteristic
23C2×D4×C53—21no—
24C2×D4×C53C2×C411yescharacteristic
25C22×C4×C53C2×C411yescharacteristic
26C2×C22⋊D53—21no—
27C2×C22⋊D53—21no—
28C2×C22⋊D53—21no—
29C22×Dic53—21no—
30C22×Dic53D411yescharacteristic
31C22×Dic53D411yescharacteristic
32#11719947—44no—
33C2×D212C2×C411yescharacteristic
34C2×C4×D53—21no—
35C2×Dic106—21no—
36#11719940—42no—
37C2×#88324—22no—
38C2×#88324—22no—
39C4⋊C4×C53C2×C411yescharacteristic
40C22⋊C4×C53—21no—
41C22⋊Dic53—21no—
42#11719921—21no—
43#11719921—21no—
44C4⋊Dic53C2×C411yescharacteristic
45#11719919—21no—
46C4×Dic53C2×C411yescharacteristic
47C4×Dic53D411yescharacteristic
48C4×Dic53D411yescharacteristic
49C4×Dic53C2311yescharacteristic
50C23×C53—21no—
51C23×C53C22⋊C411yescharacteristic
52C22×D53—21no—
53D4×C53—41no—
54D4×C53—42no—
55C2×C4×C53—21no—
56C2×C4×C53—41no—
57C2×C4×C53C22⋊C411yescharacteristic
58C2×C4×C53C22×C411yescharacteristic
59C22⋊D53—42no—
60C22⋊D53—42no—
61C22⋊D53—41no—
62C22⋊D53—41no—
63C2×Dic53—21no—
64C2×Dic53—22no—
65C2×Dic53—21no—
66C2×Dic53—21no—
67C2×Dic53—42no—
68C2×Dic53C22⋊C411yescharacteristic
69C2×Dic53C22⋊C411yescharacteristic
70C2×Dic53C2×D411yescharacteristic
71C2×Dic53C2×D411yescharacteristic
72D212—41no—
73C4×D53—42no—
74Dic106—42no—
75#88324—42no—
76#88324—42no—
77C22×C53—21no—
78C22×C53—21no—
79C22×C53—21no—
80C22×C53—41no—
81C22×C53—41no—
82C22×C53—41no—
83C22×C53C2×C22⋊C411yescharacteristic
84C2×D53—41no—
85C2×D53—41no—
86C2×D53—41no—
87C4×C53—21no—
88C4×C53—21no—
89C4×C53—41no—
90Dic53—21no—
91Dic53—21no—
92Dic53—41no—
93Dic53AGL(1,ℤ/8)11yescharacteristic
94Dic53AGL(1,ℤ/8)12yes—
95Dic53#13711yescharacteristic
96#1557—531nomaximal, Sylow
97C2×C53—41no—
98C2×C53—81no—
99C2×C53#41311yescharacteristic
100C2×C53#41711yescharacteristic
101C2×C53#41811yescharacteristic
102D53—81no—
103#532—531no—
104#432—531no—
105#423—531no—
106#332—532no—
107#331—532no—
108C53#155711yescharacteristic, complemented, Sylow
109C2×#41—1061no—
110C2×M16—1061no—
111C4⋊Q8—531no—
112#127—531no—
113#124—1061no—
114#121—531no—
115#121—1061no—
116#121#615711yescharacteristic
117C4×D4—531no—
118C4×D4—531no—
119C4×D4—1061no—
120C2×C42—531no—
121C4⋊C8—532no—
122C4⋊C8—532no—
123C4⋊C8—1061no—
124C22⋊C8—1061no—
125#97—1061no—
126#41—2124no—
127C2×Q8—1061no—
128C2×D4—21no—
129C2×D4—531no—
130C2×D4—1061no—
131C2×D4—1061no—
132C2×D4—1061no—
133C2×D4C2×#615711yescharacteristic
134C22×C4—531no—
135C22×C4—531no—
136C22×C4—1061no—
137C22×C4—1061no—
138C22×C4C2×#615711yescharacteristic
139M16—2122no—
140C2×C8—1062no—
141C2×C8—1062no—
142C4⋊C4—531no—
143C4⋊C4—1061no—
144C4⋊C4C2×#615711yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—1061no—
147C22⋊C4—1061no—
148C22⋊C4—1061no—
149C42—531no—
150C42—531no—
151C42—531no—
152C42—531no—
153C23—21no—
154C23—1061no—
155C23C22⋊#615711yescharacteristic
156Q8—2122no—
157D4—41no—
158D4—42no—
159D4—2121no—
160D4—2121no—
161D4—2121no—
162D4—2122no—
163D4—2122no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—531no—
167C2×C4—531no—
168C2×C4—531no—
169C2×C4—531no—
170C2×C4—1061no—
171C2×C4—1061no—
172C2×C4—1062no—
173C2×C4—1061no—
174C2×C4—2122no—
175C2×C4—2122no—
176C2×C4C22⋊#615711yescharacteristic
177C2×C4C22×#615711yescharacteristic
178C8—2122no—
179C8—2122no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—41no—
184C22—41no—
185C22—41no—
186C22—2121no—
187C22—2121no—
188C22—2121no—
189C22C2×C22⋊#615711yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—531no—
194C4—531no—
195C4—532no—
196C4—1061no—
197C4—1061no—
198C4—2121no—
199C2—41no—
200C2—81no—
201C2—4241no—
202C2#1681161011yescharacteristic
203C2#1681161411yescharacteristic
204C2#1681161511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#121×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#121#6157
hypocenterC53#1557
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #16811643C2semidirect
2. #16811626C2semidirect
3. #16811482C2semidirect
4. #16811481C2semidirect
108. C53#1557semidirect