Finite Group Databasebeta

← #20290246

Subgroups — #20290246

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#16811646C211yesmaximal, characteristic, complemented
2#16811628C211yesmaximal, characteristic, complemented
3#16811482C212yesmaximal, complemented
4#16811481C212yesmaximal, complemented
5#16811170C211yesmaximal, characteristic
6C2×#11719936—21no—
7#13133284—21no—
8#13133284C2212yes—
9#13133284C2212yes—
10#13133274—21no—
11#13133272—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—
38#11719936—42no—
39C4⋊C4×C53C2×C411yescharacteristic
40C22⋊C4×C53—21no—
41C42×C53C2×C411yescharacteristic
42C22⋊Dic53—21no—
43#11719921—21no—
44#11719921—21no—
45C4⋊Dic53C2×C411yescharacteristic
46#11719919—21no—
47C4×Dic53D411yescharacteristic
48C4×Dic53D411yescharacteristic
49C4×Dic53C2311yescharacteristic
50C23×C53—21no—
51C22×D53—21no—
52C22×D53C22⋊C411yescharacteristic
53D4×C53—41no—
54C2×C4×C53—21no—
55C2×C4×C53—42no—
56C2×C4×C53C22⋊C411yescharacteristic
57C2×C4×C53C22⋊C411yescharacteristic
58C2×C4×C53C22×C411yescharacteristic
59C22⋊D53—42no—
60C22⋊D53—42no—
61C22⋊D53—41no—
62C22⋊D53—41no—
63C2×Dic53—21no—
64C2×Dic53C22⋊C411yescharacteristic
65C2×Dic53C2×D411yescharacteristic
66C2×Dic53C2×D411yescharacteristic
67D212—41no—
68D212—42no—
69C4×D53—21no—
70C4×D53—22no—
71C4×D53—21no—
72C4×D53—41no—
73C4×D53—42no—
74Dic106—42no—
75#88324—42no—
76#88324—42no—
77C22×C53—41no—
78C22×C53—41no—
79C22×C53—41no—
80C22×C53C2×C22⋊C411yescharacteristic
81C2×D53—21no—
82C2×D53—21no—
83C2×D53—21no—
84C2×D53—41no—
85C2×D53—41no—
86C2×D53—41no—
87C4×C53—21no—
88C4×C53—21no—
89C4×C53—21no—
90C4×C53—41no—
91Dic53—21no—
92Dic53—41no—
93Dic53AGL(1,ℤ/8)11yescharacteristic
94Dic53AGL(1,ℤ/8)12yes—
95Dic53#13711yescharacteristic
96#1557—531nomaximal, Sylow
97C2×C53—81no—
98C2×C53#41311yescharacteristic
99C2×C53#41711yescharacteristic
100C2×C53#41811yescharacteristic
101D53—41no—
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—531no—
116#121—1061no—
117C4×D4—531no—
118C4×D4—1061no—
119C4×D4#615711yescharacteristic
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—531no—
129C2×D4—1061no—
130C2×D4—1061no—
131C2×D4—1061no—
132C2×D4—1061no—
133C2×D4C2×#615711yescharacteristic
134C22×C4—21no—
135C22×C4—531no—
136C22×C4—531no—
137C22×C4—531no—
138C22×C4—1061no—
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—
152C42C2×#615711yescharacteristic
153C23—21no—
154C23—531no—
155C23—1061no—
156Q8—2122no—
157D4—41no—
158D4—2121no—
159D4—2122no—
160D4—2121no—
161D4—2121no—
162D4—2122no—
163D4—2122no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—531no—
167C2×C4—531no—
168C2×C4—531no—
169C2×C4—1061no—
170C2×C4—1061no—
171C2×C4—1062no—
172C2×C4—1061no—
173C2×C4—2121no—
174C2×C4—2122no—
175C2×C4C22⋊#615711yescharacteristic
176C2×C4C22⋊#615711yescharacteristic
177C2×C4C22×#615711yescharacteristic
178C8—2122no—
179C8—2122no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—1061no—
184C22—1061no—
185C22—1061no—
186C22—2121no—
187C22—2121no—
188C22—2121no—
189C22C2×C22⋊#615711yescharacteristic
190C4—21no—
191C4—21no—
192C4—21no—
193C4—41no—
194C4—531no—
195C4—531no—
196C4—532no—
197C4—1061no—
198C4—2121no—
199C2—81no—
200C2—4241no—
201C2—2121no—
202C2#1681160911yescharacteristic
203C2#1681161411yescharacteristic
204C2#1681161511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4×D4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4×D4#6157
hypocenterC53#1557
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. #16811628C2semidirect
3. #16811482C2semidirect
4. #16811481C2semidirect
108. C53#1557semidirect