Finite Group Databasebeta

← #27275446

Subgroups — #27275446

Order 13568.

810 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 171 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#20274783C212yesmaximal
2C2×#16810349C214yesmaximal
3C22×#125×C53C211yesmaximal, characteristic
4C2×C22⋊#88322C2214yes—
5#16810378—24no—
6#16810349—28no—
7C2×#125×C53—28no—
8C2×#125×C53—24no—
9C22×C4⋊C4×C53C411yescharacteristic
10C22×C4⋊C4×C53C412yes—
11C22×C4⋊C4×C53C411yescharacteristic
12C22×C4⋊C4×C53C2212yes—
13C22×C42×C53C2211yescharacteristic
14C22×#88322—24no—
15C22⋊#88322—28no—
16C23×C4×C53C2×C411yescharacteristic
17C23×C4×C53C2×C411yescharacteristic
18C23×C4×C53C2×C412yes—
19C23×C4×C53C2×C412yes—
20C23×C4×C53C2311yescharacteristic
21#125×C53—232no—
22C2×C4⋊C4×C53—24no—
23C2×C4⋊C4×C53—24no—
24C2×C4⋊C4×C53—24no—
25C2×C4⋊C4×C53—24no—
26C2×C4⋊C4×C53—24no—
27C2×C4⋊C4×C53—22no—
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53—24no—
30C2×C4⋊C4×C53—24no—
31C2×C4⋊C4×C53D418yes—
32C2×C42×C53—24no—
33C2×C42×C53D414yes—
34C24×C53C22×C411yescharacteristic
35C22×C4×C53—22no—
36C22×C4×C53—24no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53—24no—
43C22×C4×C53—28no—
44C22×C4×C53—42no—
45C22×C4×C53—44no—
46C22×C4×C53C22⋊C412yes—
47C22×C4×C53C22⋊C414yes—
48C22×C4×C53C22⋊C412yes—
49C22×C4×C53C22⋊C414yes—
50C22×C4×C53C2×D412yes—
51C22×C4×C53C2×D414yes—
52C4⋊C4×C53—28no—
53C4⋊C4×C53—28no—
54C4⋊C4×C53—216no—
55C4⋊C4×C53—216no—
56C42×C53—28no—
57C2×#88322—48no—
58C2×#88322—44no—
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—24no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—24no—
70C2×C4×C53—24no—
71C2×C4×C53—28no—
72C2×C4×C53—24no—
73C2×C4×C53—44no—
74C2×C4×C53—44no—
75C2×C4×C53—44no—
76C2×C4×C53—44no—
77C2×C4×C53—44no—
78C2×C4×C53—44no—
79C2×C4×C53—44no—
80C2×C4×C53#10114yes—
81C2×C4×C53#12014yes—
82#88322—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—24no—
94C4×C53—42no—
95C4×C53—44no—
96C4×C53—44no—
97C22×#125D5311yescharacteristic
98#1817—532no—
99C2×#324—534no—
100C2×C53—24no—
101C2×C53#147014yes—
102C2×C53#178011yescharacteristic
103C2×C53#178211yescharacteristic
104C2×C53#178411yescharacteristic
105C2×#125—28no—
106C2×#125—24no—
107C22×C4⋊C4Dic5311yescharacteristic
108C22×C4⋊C4Dic5312yes—
109C22×C4⋊C4Dic5311yescharacteristic
110C22×C4⋊C4C2×D5312yes—
111C22×C42C2×D5311yescharacteristic
112C2×C22⋊C8—534no—
113#336—1064no—
114#324—1068no—
115C53#830611yescharacteristic, complemented, Sylow
116C23×C4C2×Dic5312yes—
117C23×C4C2×Dic5311yescharacteristic
118C23×C4C2×Dic5311yescharacteristic
119C23×C4C2×Dic5312yes—
120C23×C4C22×D5311yescharacteristic
121C22×C8—1064no—
122#125—232no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4C22⋊D5318yes—
133C2×C42—24no—
134C2×C42C22⋊D5314yes—
135C22⋊C8—1068no—
136C24C22×Dic5311yescharacteristic
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—28no—
145C22×C4—24no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4C22⋊Dic5314yes—
149C22×C4C22⋊Dic5312yes—
150C22×C4C22⋊Dic5312yes—
151C22×C4C22⋊Dic5314yes—
152C22×C4C2×C22⋊D5312yes—
153C22×C4C2×C22⋊D5314yes—
154C2×C8—2128no—
155C2×C8—2124no—
156C4⋊C4—28no—
157C4⋊C4—28no—
158C4⋊C4—216no—
159C4⋊C4—216no—
160C42—28no—
161C23—22no—
162C23—22no—
163C23C2×C22⋊Dic5312yes—
164C23C2×C22⋊Dic5311yescharacteristic
165C23#1313322414yes—
166C2×C4—28no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—28no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—44no—
176C2×C4—44no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4#1313313614yes—
183C2×C4#1313318614yes—
184C8—4244no—
185C22—28no—
186C22—22no—
187C22—22no—
188C22#1681047911yescharacteristic
189C22#1681059014yes—
190C22#1681059012yes—
191C22#1681059712yes—
192C22C2×#1313313612yes—
193C4—24no—
194C4—24no—
195C4—42no—
196C4—44no—
197C4—44no—
198C2—24no—
199C2#2027307114yes—
200C2#2027462711yescharacteristic
201C2#2027463311yescharacteristic
202C2#2027463911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC22×#125×C53C2
socleC23×C53C2×C22⋊C4
cosocleC23×C4×C53C23
hypercenterC22×#125D53
hypocenterC53#8306
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#125

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
#202747832
C2×#168103492
C22×#125×C532
#830653