Finite Group Databasebeta

← #28302222

Subgroups — #28302222

Order 13568.

810 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 132 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#20290606C212yesmaximal
2C2×#16811468C214yesmaximal
3C22×#13133208C211yesmaximal, characteristic
4C2×C22⋊#88324C2214yes—
5#16811489—24no—
6#16811468—28no—
7C2×#13133208—28no—
8C2×#13133208—24no—
9C22×C4⋊Dic53C411yescharacteristic
10C22×C4⋊Dic53C2212yes—
11C22×#11719919C412yes—
12C22×C4×Dic53C2211yescharacteristic
13C22×C4⋊C4×C53C411yescharacteristic
14C22×#88324—24no—
15C22⋊#88324—28no—
16C23×Dic53C2×C411yescharacteristic
17C23×Dic53C2×C412yes—
18C23×Dic53C2311yescharacteristic
19#13133208—232no—
20C2×C4⋊Dic53—22no—
21C2×C4⋊Dic53—24no—
22C2×C4⋊Dic53—24no—
23C2×C4⋊Dic53—24no—
24C2×C4⋊Dic53D418yes—
25C2×#11719919—24no—
26C2×#11719919—24no—
27C2×#11719919—24no—
28C2×C4×Dic53—24no—
29C2×C4×Dic53D414yes—
30C23×C4×C53C2×C412yes—
31C23×C4×C53C2×C411yescharacteristic
32C2×C4⋊C4×C53—24no—
33C2×C4⋊C4×C53—22no—
34C24×C53C22×C411yescharacteristic
35C22×C4×C53—22no—
36C22×C4×C53—24no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—28no—
40C22×C4×C53C22⋊C414yes—
41C22×C4×C53C22⋊C412yes—
42C22×Dic53—22no—
43C22×Dic53—22no—
44C22×Dic53—22no—
45C22×Dic53—24no—
46C22×Dic53—42no—
47C22×Dic53—44no—
48C22×Dic53C22⋊C412yes—
49C22×Dic53C22⋊C414yes—
50C22×Dic53C2×D412yes—
51C22×Dic53C2×D414yes—
52C2×#88324—48no—
53C2×#88324—44no—
54C4⋊C4×C53—28no—
55C4⋊Dic53—28no—
56C4⋊Dic53—216no—
57#11719919—216no—
58C4×Dic53—28no—
59C23×C53—22no—
60C23×C53—22no—
61C23×C53C2×C22⋊C412yes—
62C23×C53C2×C22⋊C411yescharacteristic
63C23×C53#13114yes—
64C2×C4×C53—28no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—24no—
68C2×C4×C53—28no—
69C2×C4×C53—44no—
70C2×C4×C53—44no—
71C2×C4×C53#10114yes—
72C2×Dic53—24no—
73C2×Dic53—22no—
74C2×Dic53—24no—
75C2×Dic53—24no—
76C2×Dic53—44no—
77C2×Dic53—44no—
78C2×Dic53—44no—
79C2×Dic53—44no—
80C2×Dic53—44no—
81C2×Dic53#12014yes—
82#88324—84no—
83#8306—531nomaximal, Sylow
84C22×C53—28no—
85C22×C53—22no—
86C22×C53—22no—
87C22×C53#37911yescharacteristic
88C22×C53#40814yes—
89C22×C53#40812yes—
90C22×C53#41012yes—
91C22×C53C2×#10112yes—
92C4×C53—24no—
93C4×C53—44no—
94Dic53—24no—
95Dic53—42no—
96Dic53—44no—
97C22×#125—531no—
98#1817—532no—
99C2×#324—534no—
100C2×C53—24no—
101C2×C53#147014yes—
102C2×C53#178011yescharacteristic
103C2×C53#178211yescharacteristic
104C2×C53#178411yescharacteristic
105C2×#125—1064no—
106C2×#125—1068no—
107C22×C4⋊C4—532no—
108C22×C4⋊C4—532no—
109C22×C4⋊C4—531no—
110C22×C4⋊C4#615711yescharacteristic
111C22×C42—531no—
112C2×C22⋊C8—534no—
113#336—1064no—
114#324—1068no—
115C53#830611yescharacteristic, complemented, Sylow
116C23×C4—531no—
117C23×C4—531no—
118C23×C4—532no—
119C23×C4C2×#615712yes—
120C23×C4C2×#615711yescharacteristic
121C22×C8—1064no—
122#125—10632no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—538no—
126C2×C4⋊C4—1064no—
127C2×C4⋊C4—1064no—
128C2×C4⋊C4—1064no—
129C2×C4⋊C4—1064no—
130C2×C4⋊C4—1064no—
131C2×C4⋊C4—1064no—
132C2×C4⋊C4—1062no—
133C2×C42—534no—
134C2×C42—1064no—
135C22⋊C8—1068no—
136C24C22×#615711yescharacteristic
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—28no—
142C22×C4—532no—
143C22×C4—534no—
144C22×C4—532no—
145C22×C4—534no—
146C22×C4—1062no—
147C22×C4—1062no—
148C22×C4—1062no—
149C22×C4—1064no—
150C22×C4—2122no—
151C22×C4—2124no—
152C22×C4C22⋊#615714yes—
153C22×C4C22⋊#615712yes—
154C2×C8—2124no—
155C2×C8—2128no—
156C4⋊C4—28no—
157C4⋊C4—10616no—
158C4⋊C4—10616no—
159C4⋊C4—1068no—
160C42—1068no—
161C23—22no—
162C23—22no—
163C23C2×C22⋊#615712yes—
164C23C2×C22⋊#615711yescharacteristic
165C23#1313330414yes—
166C2×C4—28no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—28no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—534no—
174C2×C4—1064no—
175C2×C4—1062no—
176C2×C4—1064no—
177C2×C4—1064no—
178C2×C4—2124no—
179C2×C4—2124no—
180C2×C4—2124no—
181C2×C4—2124no—
182C2×C4—2124no—
183C2×C4#1313327814yes—
184C8—4244no—
185C22—28no—
186C22—22no—
187C22—22no—
188C22#1681156411yescharacteristic
189C22#1681159614yes—
190C22#1681159712yes—
191C22#1681160412yes—
192C22C2×#1313327812yes—
193C4—24no—
194C4—44no—
195C4—1064no—
196C4—2122no—
197C4—2124no—
198C2—24no—
199C2#2028998814yes—
200C2#2029053911yescharacteristic
201C2#2029054611yescharacteristic
202C2#2029054911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#8306
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
115. C53#8306semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202906062
C2×#168114682
C22×#131332082
#830653