Finite Group Databasebeta

← #27277571

Subgroups — #27277571

Order 13568.

418 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 99 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#20274798C211yesmaximal, characteristic
2#20274792C211yesmaximal, characteristic
3#20272568C214yesmaximal
4C53×#2298C211yesmaximal, characteristic
5#16810587C2212yes—
6#16810486C2212yes—
7#16810392—22no—
8#16810390—22no—
9C2×#117×C53C2211yescharacteristic
10C53×#388—22no—
11C53×#387—22no—
12C4×C4⋊C4×C53C412yescomplemented
13C4×C22⋊C4×C53C2212yes—
14C53×#376C411yescharacteristic, complemented
15C43×C53C411yescharacteristic, complemented
16C2×#11719917—22no—
17C2×C22⋊Dic53—22no—
18C22⋊#88322—24no—
19#13133124—24no—
20C23×C4×C53C2311yescharacteristic
21#117×C53—24no—
22C2×C4⋊C4×C53—22no—
23C2×C4⋊C4×C53D412yes—
24C2×C22⋊C4×C53—22no—
25C2×C22⋊C4×C53D412yes—
26C2×C42×C53—22no—
27C2×C42×C53C2×C412yes—
28C2×C42×C53C2×C412yes—
29C2×C42×C53C2×C411yescharacteristic
30C2×C42×C53C2×C411yescharacteristic
31#95×C53—22no—
32#95×C53—22no—
33#95×C53—22no—
34C24×C53C2×D411yescharacteristic
35C22×C4×C53—22no—
36C22×C4×C53—22no—
37C22×C4×C53—42no—
38C22×C4×C53C22⋊C412yes—
39C22×C4×C53C22⋊C412yes—
40C22×C4×C53C22×C411yescharacteristic
41C22×C4×C53C2×D411yescharacteristic
42C22×C4×C53#4114yes—
43C22×Dic53—42no—
44C4⋊C4×C53—24no—
45C4⋊C4×C53—44no—
46C22⋊C4×C53—24no—
47C22⋊C4×C53—44no—
48C42×C53—22no—
49C42×C53—22no—
50C42×C53—44no—
51C42×C53—42no—
52C42×C53—42no—
53C22⋊Dic53—44no—
54#11719917—44no—
55C2×#88322—42no—
56C23×C53—22no—
57C23×C53—42no—
58C23×C53C2×C22⋊C411yescharacteristic
59C23×C53#12312yes—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—24no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—44no—
66C2×C4×C53—42no—
67C2×C4×C53—42no—
68C2×C4×C53—42no—
69C2×C4×C53—42no—
70C2×C4×C53—42no—
71C2×C4×C53—42no—
72C2×C4×C53—42no—
73C2×C4×C53#11712yes—
74C2×C4×C53#12312yes—
75C2×Dic53—84no—
76C2×Dic53—82no—
77#88322—82no—
78#8740—531nomaximal, Sylow
79C22×C53—22no—
80C22×C53—22no—
81C22×C53—24no—
82C22×C53—82no—
83C22×C53#38111yescharacteristic
84C22×C53#41012yes—
85C4×C53—22no—
86C4×C53—22no—
87C4×C53—44no—
88C4×C53—42no—
89C4×C53—42no—
90C4×C53—82no—
91Dic53—162no—
92#2298D5311yescharacteristic
93#1822—531no—
94#1820—531no—
95#1284—534no—
96C2×C53—22no—
97C2×C53—42no—
98C2×C53#182011yescharacteristic
99C2×C53#182712yes—
100C2×#117C2×D5311yescharacteristic
101#407—532no—
102#388—22no—
103#387—22no—
104#381—532no—
105C4×C4⋊C4Dic5312yescomplemented
106C4×C22⋊C4C2×D5312yes—
107#376Dic5311yescharacteristic, complemented
108C43Dic5311yescharacteristic, complemented
109#343—1062no—
110#342—1062no—
111C53#874011yescharacteristic, complemented, Sylow
112C23×C4C22×D5311yescharacteristic
113C2×M16—1062no—
114#117—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4C22⋊D5312yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—1062no—
119C2×C22⋊C4C22⋊D5312yes—
120C2×C42—22no—
121C2×C42C2×Dic5311yescharacteristic
122C2×C42C2×Dic5311yescharacteristic
123C2×C42C2×Dic5312yes—
124C2×C42C2×Dic5312yes—
125C22⋊C8—1064no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—1064no—
130C24C2×C22⋊D5311yescharacteristic
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—2122no—
135C22×C4C22⋊Dic5312yes—
136C22×C4C22⋊Dic5312yes—
137C22×C4#1171994514yes—
138C22×C4C22×Dic5311yescharacteristic
139C22×C4C2×C22⋊D5311yescharacteristic
140M16—2124no—
141C2×C8—2122no—
142C4⋊C4—24no—
143C4⋊C4—44no—
144C22⋊C4—24no—
145C22⋊C4—44no—
146C22⋊C4—2124no—
147C42—22no—
148C42—22no—
149C42—42no—
150C42—42no—
151C42—44no—
152C23—22no—
153C23—42no—
154C23C2×C22⋊Dic5311yescharacteristic
155C23#1313320012yes—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—4242no—
170C2×C4—4244no—
171C2×C4#1313317612yes—
172C2×C4#1313320012yes—
173C8—4242no—
174C22—22no—
175C22—22no—
176C22—24no—
177C22—82no—
178C22#1681048611yescharacteristic
179C22#1681059712yes—
180C4—22no—
181C4—22no—
182C4—44no—
183C4—42no—
184C4—42no—
185C4—82no—
186C4—8482no—
187C2—22no—
188C2—42no—
189C2#2027479211yescharacteristic
190C2#2027481912yes—
subgroupG/H
centerC22#16810486
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2298C2
socleC22×C53#381
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2298D53
hypocenterC53#8740
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2298

Complements

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

NH ≅ G/Ntype
12. C4×C4⋊C4×C53C4semidirect
14. C53×#376C4semidirect
15. C43×C53C4semidirect
105. C4×C4⋊C4Dic53semidirect
107. #376Dic53semidirect
108. C43Dic53semidirect
111. C53#8740semidirect

Maximal subgroups

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

subgroupindex
#202747982
#202747922
#202725682
C53×#22982
#874053