Finite Group Databasebeta

← C23×#8545 #19156171

Subgroups — C23×#8545 #19156171

Order 5888.

910 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C22×C23×#126C211yesmaximal, characteristic
2C23×#1819C216yesmaximal
3C2×C23×#126—212no—
4C22×C4⋊C4×C23C413yescomplemented
5C22×C22⋊C4×C23C2213yes—
6C22×C42×C23C411yescharacteristic, complemented
7C23×#379C2214yes—
8C23×#342—212no—
9C25×C23C2311yescharacteristic
10C23×C4×C23C2×C413yes—
11C23×C4×C23C2×C413yes—
12C23×#126—232no—
13C2×C4⋊C4×C23—26no—
14C2×C4⋊C4×C23—26no—
15C2×C4⋊C4×C23—26no—
16C2×C22⋊C4×C23—26no—
17C2×C22⋊C4×C23—26no—
18C2×C22⋊C4×C23—212no—
19C2×C22⋊C4×C23D4112yes—
20C2×C42×C23—26no—
21C24×C23—46no—
22C24×C23C22×C411yescharacteristic
23C24×C23C2×D416yes—
24C22×C4×C23—23no—
25C22×C4×C23—23no—
26C22×C4×C23—23no—
27C22×C4×C23—23no—
28C22×C4×C23—46no—
29C22×C4×C23—46no—
30C22×C4×C23—44no—
31C22×C4×C23C22⋊C416yes—
32C22×C4×C23C22⋊C416yes—
33C4⋊C4×C23—224no—
34C22⋊C4×C23—224no—
35C22⋊C4×C23—424no—
36C42×C23—28no—
37#8545C2311yesmaximal, characteristic, direct product factor, Sylow
38C23×C23—26no—
39C23×C23—212no—
40C23×C23—46no—
41C23×C23—46no—
42C23×C23—88no—
43C23×C23C2×C22⋊C413yes—
44C23×C23#12014yes—
45C2×C4×C23—26no—
46C2×C4×C23—26no—
47C2×C4×C23—46no—
48C2×C4×C23—46no—
49C2×C4×C23—46no—
50C2×C4×C23—46no—
51C2×C4×C23—46no—
52C2×C4×C23—46no—
53C2×C4×C23—812no—
54C22×#126C2×C2311yescharacteristic
55#1819C2×C2316yes—
56C22×C23—28no—
57C22×C23—23no—
58C22×C23—23no—
59C22×C23—212no—
60C22×C23—812no—
61C22×C23#37911yescharacteristic
62C22×C23#41016yes—
63C4×C23—46no—
64C4×C23—46no—
65C4×C23—164no—
66C2×#126—212no—
67C22×C4⋊C4C4×C2313yescomplemented
68C22×C22⋊C4C22×C2313yes—
69C22×C42C4×C2311yescharacteristic, complemented
70#379C22×C2314yes—
71#342—212no—
72C2×C23—26no—
73C2×C23—44no—
74C2×C23#178213yes—
75C25C23×C2311yescharacteristic
76C23×C4C2×C4×C2313yes—
77C23×C4C2×C4×C2313yes—
78#126—232no—
79C2×C4⋊C4—26no—
80C2×C4⋊C4—26no—
81C2×C4⋊C4—26no—
82C2×C22⋊C4—26no—
83C2×C22⋊C4—26no—
84C2×C22⋊C4—212no—
85C2×C22⋊C4D4×C23112yes—
86C2×C42—26no—
87C23#854511yescharacteristic, direct product factor, Sylow
88C24—46no—
89C24C22×C4×C2311yescharacteristic
90C24C2×D4×C2316yes—
91C22×C4—23no—
92C22×C4—23no—
93C22×C4—23no—
94C22×C4—23no—
95C22×C4—46no—
96C22×C4—46no—
97C22×C4—44no—
98C22×C4C22⋊C4×C2316yes—
99C22×C4C22⋊C4×C2316yes—
100C4⋊C4—224no—
101C22⋊C4—224no—
102C22⋊C4—424no—
103C42—28no—
104C23—26no—
105C23—212no—
106C23—46no—
107C23—46no—
108C23—88no—
109C23C2×C22⋊C4×C2313yes—
110C23C23×#12014yes—
111C2×C4—26no—
112C2×C4—26no—
113C2×C4—46no—
114C2×C4—46no—
115C2×C4—46no—
116C2×C4—46no—
117C2×C4—46no—
118C2×C4—46no—
119C2×C4—812no—
120C22—28no—
121C22—23no—
122C22—23no—
123C22—212no—
124C22—812no—
125C22C23×#37911yescharacteristic
126C22C23×#41016yes—
127C4—46no—
128C4—46no—
129C4—164no—
130C2—26no—
131C2—44no—
132C2C23×#178213yes—
subgroupG/H
center, socleC22×C23#379
commutatorC24C22×C4×C23
Frattini, cosocleC25C23×C23
prime radicalC25×C23C23
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C25×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C22×C4⋊C4×C23C4semidirect
6. C22×C42×C23C4semidirect
37. #8545C23direct (N × H)
67. C22×C4⋊C4C4×C23semidirect
69. C22×C42C4×C23semidirect
87. C23#8545direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C23×#1262
C23×#18192
#854523