Finite Group Databasebeta

← #27277622

Subgroups — #27277622

Order 13568.

398 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 91 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#20277318C211yesmaximal, characteristic
2#20274795C211yesmaximal, characteristic
3#20272624C214yesmaximal
4C53×#1821C211yesmaximal, characteristic
5#16810587C2212yes—
6#16810509—22no—
7#16810509—22no—
8#16810486C2212yes—
9#16810483C412yes—
10C4×C22⋊Dic53C412yes—
11#16810392—22no—
12C22×C4⋊C4×C53C2211yescharacteristic
13C53×#407C2212yes—
14C53×#343—22no—
15C2×#11719917—22no—
16C2×C4⋊Dic53—22no—
17C2×C22⋊Dic53C2×C414yes—
18C2×C4×Dic53—22no—
19C22⋊#88322—24no—
20#13133124—22no—
21#13133124—22no—
22#13133124—22no—
23#13133124—22no—
24C23×C4×C53C2×C411yescharacteristic
25C23×C4×C53C2×C411yescharacteristic
26C23×C4×C53C2311yescharacteristic
27C2×M16×C53—22no—
28C2×C4⋊C4×C53—24no—
29C2×C4⋊C4×C53D412yes—
30C2×C4⋊C4×C53D412yes—
31C22⋊C8×C53—24no—
32C24×C53C22×C411yescharacteristic
33C22×C4×C53—22no—
34C22×C4×C53—22no—
35C22×C4×C53—42no—
36C22×C4×C53—81no—
37C22×C4×C53—81no—
38C22×C4×C53C22⋊C412yes—
39C22×C4×C53C22⋊C412yes—
40C22×C4×C53C2×D411yescharacteristic
41C22×C4×C53C2×D411yescharacteristic
42C22×C4×C53#4114yes—
43C22×Dic53—24no—
44M16×C53—44no—
45C2×C8×C53—42no—
46C4⋊C4×C53—24no—
47C4⋊C4×C53—24no—
48C22⋊Dic53—44no—
49C4⋊Dic53—82no—
50C4×Dic53—82no—
51#11719917—44no—
52C2×#88322—42no—
53C23×C53—22no—
54C23×C53—41no—
55C23×C53—41no—
56C23×C53C2×C22⋊C411yescharacteristic
57C23×C53#11712yes—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—44no—
63C2×C4×C53—42no—
64C2×C4×C53—42no—
65C2×C4×C53—82no—
66C2×C4×C53—82no—
67C2×C4×C53#12312yes—
68C2×C4×C53#12312yes—
69C2×Dic53—44no—
70C2×Dic53—44no—
71C2×Dic53—44no—
72C8×C53—82no—
73#88322—82no—
74#8755—531nomaximal, Sylow
75C22×C53—21no—
76C22×C53—21no—
77C22×C53—22no—
78C22×C53—22no—
79C22×C53—22no—
80C22×C53—44no—
81C22×C53#38111yescharacteristic
82C22×C53#41012yes—
83C4×C53—22no—
84C4×C53—22no—
85C4×C53—42no—
86C4×C53—42no—
87Dic53—84no—
88#2313—531no—
89#1821—531no—
90#1821D5311yescharacteristic
91#1306—534no—
92C2×C53—22no—
93C2×C53—42no—
94C2×C53#182011yescharacteristic
95C2×C53#182911yescharacteristic
96C2×C53#183011yescharacteristic
97C22×C4⋊C4C2×D5311yescharacteristic
98#407—532no—
99#407C2×D5312yes—
100#387—1062no—
101#387—1062no—
102#381—532no—
103#380—532no—
104C4×C22⋊C4—532no—
105#343—22no—
106#343—1062no—
107C53#875511yescharacteristic, complemented, Sylow
108C23×C4C4×D5311yescharacteristic
109C23×C4C4×D5311yescharacteristic
110C23×C4C22×D5311yescharacteristic
111C2×M16—22no—
112C2×M16—1062no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—1062no—
115C2×C4⋊C4D21212yes—
116C2×C4⋊C4C22⋊D5312yes—
117C2×C22⋊C4—534no—
118C2×C42—1062no—
119C22⋊C8—24no—
120C22⋊C8—1064no—
121#95—1062no—
122#95—1062no—
123#95—1062no—
124#95—1062no—
125C24C2×C4×D5311yescharacteristic
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4—81no—
130C22×C4—81no—
131C22×C4—1064no—
132C22×C4#1171992112yes—
133C22×C4#1171992112yes—
134C22×C4C2×D21211yescharacteristic
135C22×C4#1171994714yes—
136C22×C4C2×C22⋊D5311yescharacteristic
137M16—44no—
138M16—2124no—
139C2×C8—42no—
140C2×C8—2122no—
141C4⋊C4—24no—
142C4⋊C4—24no—
143C4⋊C4—4242no—
144C22⋊C4—2124no—
145C42—4242no—
146C23—22no—
147C23—41no—
148C23—41no—
149C23C2×#1171992111yescharacteristic
150C23#1313317312yes—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—44no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4—2124no—
161C2×C4—2124no—
162C2×C4—2124no—
163C2×C4#1313319812yes—
164C2×C4#1313320112yes—
165C8—82no—
166C8—4242no—
167C22—21no—
168C22—21no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—44no—
173C22#1681048411yescharacteristic
174C22#1681059612yes—
175C4—22no—
176C4—22no—
177C4—42no—
178C4—42no—
179C4—4244no—
180C2—22no—
181C2—42no—
182C2#2027479011yescharacteristic
183C2#2027482411yescharacteristic
184C2#2027482711yescharacteristic
subgroupG/H
centerC22#16810484
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1821C2
socleC22×C53#381
cosocleC23×C4×C53C23
hypercenter#1821D53
hypocenterC53#8755
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
107. C53#8755semidirect

Maximal subgroups

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

subgroupindex
#202773182
#202747952
#202726242
C53×#18212
#875553