Finite Group Databasebeta

← #28303439

Subgroups — #28303439

Order 13568.

566 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 76 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#20290549C212yesmaximal
2#20289618C214yesmaximal
3#20277300C211yesmaximal, characteristic
4C2×#13133278—24no—
5#16811593C2214yes—
6C23×Dic106C2211yescharacteristic
7C2×#13133173C412yes—
8#16810527—24no—
9#16810486C411yescharacteristic
10#16810484C2212yes—
11C53×#381C411yescharacteristic
12C2×#11719940—24no—
13#13133278—48no—
14C22⋊#88324—28no—
15C22×Dic106—24no—
16C22×Dic106—44no—
17C22×Dic106D414yes—
18C23×Dic53C2×C411yescharacteristic
19C23×Dic53C2311yescharacteristic
20#13133173—44no—
21C2×#11719919—22no—
22C2×C22⋊Dic53C2×C412yes—
23C2×C4×Dic53—22no—
24#13133124—22no—
25#13133124—24no—
26C23×C4×C53C2×C411yescharacteristic
27C2×C22⋊C4×C53C2×C412yes—
28#95×C53—22no—
29C24×C53C22×C411yescharacteristic
30C22×C4×C53—22no—
31C22×C4×C53—22no—
32C22×C4×C53—42no—
33C22×C4×C53C22⋊C412yes—
34C22×Dic53—22no—
35C22×Dic53—24no—
36C22×Dic53—22no—
37C22×Dic53—42no—
38C22×Dic53C22⋊C412yes—
39C22×Dic53C2×D412yes—
40C22×Dic53#4114yes—
41C2×Dic106—28no—
42C2×Dic106—28no—
43C2×Dic106—44no—
44C2×Dic106—416no—
45#11719940—48no—
46C2×#88324—44no—
47C22⋊C4×C53—42no—
48C22⋊Dic53—42no—
49#11719919—44no—
50C4×Dic53—44no—
51C23×C53—22no—
52C23×C53—42no—
53C23×C53C2×C22⋊C411yescharacteristic
54C23×C53#11712yes—
55C2×C4×C53—22no—
56C2×C4×C53—24no—
57C2×C4×C53—44no—
58C2×C4×C53—44no—
59C2×C4×C53—42no—
60C2×Dic53—28no—
61C2×Dic53—22no—
62C2×Dic53—24no—
63C2×Dic53—22no—
64C2×Dic53—44no—
65C2×Dic53—44no—
66C2×Dic53—42no—
67C2×Dic53#9914yes—
68C2×Dic53#12314yes—
69Dic106—416no—
70#88324—84no—
71#8754—531nomaximal, Sylow
72C22×C53—22no—
73C22×C53—22no—
74C22×C53—24no—
75C22×C53—44no—
76C22×C53#38111yescharacteristic
77C22×C53C2×#9912yes—
78C4×C53—42no—
79C4×C53—82no—
80Dic53—24no—
81Dic53—24no—
82Dic53—82no—
83#2310—531no—
84#1784—532no—
85#1306—534no—
86C2×C53—22no—
87C2×C53—42no—
88C2×C53#181911yescharacteristic
89C2×C53#183012yes—
90C23×Q8—531no—
91C2×#117—532no—
92C2×#101—1064no—
93#407—534no—
94#391—1064no—
95#381—531no—
96#381—532no—
97#381#615711yescharacteristic
98C53#875411yescharacteristic, complemented, Sylow
99C22×Q8—534no—
100C22×Q8—1064no—
101C22×Q8—2124no—
102C23×C4—531no—
103C23×C4—531no—
104C23×C4C2×#615711yescharacteristic
105C2×M16—1064no—
106#117—2124no—
107C2×C4⋊C4—1062no—
108C2×C22⋊C4—532no—
109C2×C22⋊C4C2×#615712yes—
110C2×C42—1062no—
111#101—2128no—
112C22⋊C8—1068no—
113#95—22no—
114#95—1064no—
115#95—1062no—
116C24C22×#615711yescharacteristic
117C2×Q8—1068no—
118C2×Q8—1068no—
119C2×Q8—2124no—
120C2×Q8—21216no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—42no—
124C22×C4—532no—
125C22×C4—532no—
126C22×C4—534no—
127C22×C4—1064no—
128C22×C4—1062no—
129C22×C4—1062no—
130C22×C4—2122no—
131C22×C4C22⋊#615712yes—
132M16—2128no—
133C2×C8—2124no—
134C4⋊C4—2124no—
135C22⋊C4—42no—
136C22⋊C4—2122no—
137C42—2124no—
138C23—22no—
139C23—42no—
140C23C2×C22⋊#615711yescharacteristic
141C23#1313329612yes—
142Q8—21216no—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—44no—
146C2×C4—44no—
147C2×C4—42no—
148C2×C4—534no—
149C2×C4—534no—
150C2×C4—1068no—
151C2×C4—1062no—
152C2×C4—1062no—
153C2×C4—1064no—
154C2×C4—2124no—
155C2×C4—2124no—
156C2×C4—2122no—
157C8—4244no—
158C22—22no—
159C22—22no—
160C22—24no—
161C22—44no—
162C22#1681156611yescharacteristic
163C22D53⋊#9912yes—
164C4—42no—
165C4—82no—
166C4—1064no—
167C4—1064no—
168C4—4242no—
169C2—22no—
170C2—42no—
171C2#2029061111yescharacteristic
172C2#2029062912yes—
subgroupG/H
centerC22#16811566
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC53×#381C4
socleC22×C53#381
cosocleC23×Dic53C23
hypercenter#381#6157
hypocenterC53#8754
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
98. C53#8754semidirect

Maximal subgroups

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

subgroupindex
#202905492
#202896182
#202773002
#875453