Finite Group Databasebeta

← #27277573

Subgroups — #27277573

Order 13568.

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

normal subgroups only

This group has 418 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#20277213C211yesmaximal, characteristic
2#20274798C211yesmaximal, characteristic
3#20272567C214yesmaximal
4C53×#1820C211yesmaximal, characteristic
5#16810587C2212yes—
6#16810516—22no—
7#16810509—22no—
8C4×C4⋊Dic53C412yescomplemented
9C4×C22⋊Dic53C2212yes—
10#16810469C411yescharacteristic, complemented
11C42×Dic53C411yescharacteristic, complemented
12#16810392—22no—
13C2×#117×C53C2211yescharacteristic
14C53×#381C2212yes—
15C53×#342—22no—
16C2×#11719917—22no—
17C2×C4⋊Dic53—22no—
18C2×C22⋊Dic53—22no—
19C2×C4×Dic53—22no—
20C2×C4×Dic53C2×C412yes—
21C2×C4×Dic53C2×C412yes—
22C22⋊#88322—24no—
23#13133124—22no—
24#13133124—22no—
25#13133124—22no—
26C23×C4×C53C2311yescharacteristic
27#117×C53—24no—
28C2×C4⋊C4×C53D412yes—
29C2×C22⋊C4×C53—22no—
30C2×C22⋊C4×C53D412yes—
31C2×C42×C53C2×C411yescharacteristic
32C2×C42×C53C2×C411yescharacteristic
33#95×C53—24no—
34C24×C53C2×D411yescharacteristic
35C22×C4×C53—42no—
36C22×C4×C53—42no—
37C22×C4×C53C22⋊C412yes—
38C22×C4×C53C22⋊C412yes—
39C22×C4×C53C22×C411yescharacteristic
40C22×C4×C53C2×D411yescharacteristic
41C22×C4×C53#4114yes—
42C22×Dic53—22no—
43C22×Dic53—22no—
44C4⋊C4×C53—24no—
45C22⋊C4×C53—24no—
46C22⋊C4×C53—44no—
47C42×C53—22no—
48C42×C53—22no—
49C22⋊Dic53—44no—
50C4⋊Dic53—44no—
51C4×Dic53—44no—
52C4×Dic53—42no—
53C4×Dic53—42no—
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—22no—
63C2×C4×C53—22no—
64C2×C4×C53—44no—
65C2×C4×C53—42no—
66C2×C4×C53—42no—
67C2×C4×C53—84no—
68C2×C4×C53—82no—
69C2×C4×C53#11712yes—
70C2×C4×C53#12312yes—
71C2×Dic53—24no—
72C2×Dic53—42no—
73C2×Dic53—42no—
74C2×Dic53—42no—
75C2×Dic53—42no—
76C2×Dic53—42no—
77#88322—82no—
78#8741—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—42no—
88C4×C53—42no—
89C4×C53—162no—
90Dic53—44no—
91Dic53—82no—
92#2298—531no—
93#1822—531no—
94#1820D5311yescharacteristic
95#1284—534no—
96C2×C53—22no—
97C2×C53—42no—
98C2×C53#182011yescharacteristic
99C2×C53#182712yes—
100C2×#117C2×D5311yescharacteristic
101#407—532no—
102#388—1062no—
103#387—1062no—
104#381C2×D5312yes—
105C4×C4⋊C4—532no—
106C4×C22⋊C4—532no—
107#376—531no—
108C43—531no—
109#343—1062no—
110#342—22no—
111C53#874111yescharacteristic, complemented, Sylow
112C23×C4C22×D5311yescharacteristic
113C2×M16—1062no—
114#117—24no—
115C2×C4⋊C4—1062no—
116C2×C4⋊C4D21212yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—1062no—
119C2×C22⋊C4C22⋊D5312yes—
120C2×C42—532no—
121C2×C42—532no—
122C2×C42—1062no—
123C2×C42C4×D5311yescharacteristic
124C2×C42C4×D5311yescharacteristic
125C22⋊C8—1064no—
126#95—24no—
127#95—1062no—
128#95—1062no—
129#95—1062no—
130C24C2×C22⋊D5311yescharacteristic
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—1062no—
134C22×C4—1062no—
135C22×C4#1171992112yes—
136C22×C4#1171992112yes—
137C22×C4C2×C4×D5311yescharacteristic
138C22×C4C2×D21211yescharacteristic
139C22×C4#1171994714yes—
140M16—2124no—
141C2×C8—2122no—
142C4⋊C4—24no—
143C4⋊C4—2124no—
144C22⋊C4—24no—
145C22⋊C4—44no—
146C22⋊C4—2124no—
147C42—22no—
148C42—22no—
149C42—2122no—
150C42—2122no—
151C42—2124no—
152C23—22no—
153C23—42no—
154C23C2×#1171992111yescharacteristic
155C23#1313319812yes—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—44no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—82no—
164C2×C4—84no—
165C2×C4—1064no—
166C2×C4—2122no—
167C2×C4—2122no—
168C2×C4—2122no—
169C2×C4—2122no—
170C2×C4—2122no—
171C2×C4#1313317312yes—
172C2×C4#1313320112yes—
173C8—4242no—
174C22—22no—
175C22—22no—
176C22—24no—
177C22—82no—
178C22#1681048411yescharacteristic
179C22#1681059612yes—
180C4—22no—
181C4—22no—
182C4—42no—
183C4—42no—
184C4—162no—
185C4—4242no—
186C4—2124no—
187C2—22no—
188C2—42no—
189C2#2027479011yescharacteristic
190C2#2027481612yes—
subgroupG/H
centerC22#16810484
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1820C2
socleC22×C53#381
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1820D53
hypocenterC53#8741
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
8. C4×C4⋊Dic53C4semidirect
10. #16810469C4semidirect
11. C42×Dic53C4semidirect
111. C53#8741semidirect

Maximal subgroups

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

subgroupindex
#202772132
#202747982
#202725672
C53×#18202
#874153