Finite Group Databasebeta

← C5×D3×ΓL(1,𝔽16) #14268431

Subgroups — C5×D3×ΓL(1,𝔽16) #14268431

Order 1800.

149 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 149 conjugacy classes of subgroups — too many to draw. Only its 48 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
1C3×C5×D3×D5C211yesmaximal, characteristic
2C5×C3⋊ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
3C3×C5×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
4C2×C5×ΓL(1,𝔽16)—31nomaximal
5C5×D3×AGL(1,𝔽5)—31nomaximal
6C3×C5×D15C411yescharacteristic, complemented
7C3×C52×D3C411yescharacteristic, complemented
8C32×C5×D5C2211yescharacteristic
9D3×ΓL(1,𝔽16)C511yesmaximal, characteristic, direct product factor
10C5×D3×Dic3—51nomaximal
11C2×C3×C5×D5—31no—
12C5×D3×D5D311yescharacteristic
13C5×ΓL(1,𝔽16)—31no—
14C5×ΓL(1,𝔽16)—31no—
15C5×ΓL(1,𝔽16)—61no—
16C5×ΓL(1,𝔽16)D311yescharacteristic, direct product factor
17C3×C5×AGL(1,𝔽5)—31no—
18C32×C52C2×C411yescharacteristic, complemented, Hall
19C2×C5×AGL(1,𝔽5)—91noHall
20C2×C3×C5×D3—51no—
21C3×D3×D5C2×C511yescharacteristic
22C3⋊ΓL(1,𝔽16)C2×C511yescharacteristic, complemented
23C3×ΓL(1,𝔽16)C2×C511yescharacteristic, complemented
24C5×C3⋊Dic3—51no—
25C3×C5×Dic3—51no—
26C2×C3×C52—31no—
27C5×D15Dic311yescharacteristic, complemented
28C52×D3Dic311yescharacteristic, complemented
29C3×C5×D5—21no—
30C3×C5×D5—31no—
31C3×C5×D5C2×D311yescharacteristic
32C3×C5×D5C2×D311yescharacteristic
33C2×ΓL(1,𝔽16)—31no—
34D3×AGL(1,𝔽5)—31no—
35C2×C5×Dic3—151no—
36C4×C5×D3—151no—
37C2×C5×D5—31no—
38C5×AGL(1,𝔽5)—31no—
39C5×AGL(1,𝔽5)—91no—
40C2×C32×C5—51no—
41C3×D15C4×C511yescharacteristic, complemented
42C3×C5×D3—41no—
43C3×C5×D3—51no—
44C3×C5×D3C4×C511yescharacteristic, complemented
45C3×C5×D3AGL(1,𝔽5)11yescharacteristic, complemented
46C32×D5C22×C511yescharacteristic
47C3×C52—21no—
48C3×C52C4×D311yescharacteristic, complemented
49C3×C52C2×Dic311yescharacteristic, complemented
50D3×Dic3—51noHall
51C22×C3×C5—151no—
52C2×C5×D3—51no—
53C2×C3×D5—31no—
54D3×D5C5×D311yescharacteristic
55ΓL(1,𝔽16)—31no—
56ΓL(1,𝔽16)—31no—
57ΓL(1,𝔽16)—61no—
58ΓL(1,𝔽16)C5×D311yescharacteristic, direct product factor
59C3×AGL(1,𝔽5)—31no—
60C3×C4×C5—151no—
61C5×Dic3—51no—
62C5×Dic3—151no—
63C5×Dic3—151no—
64C5×Dic3—301no—
65C2×C52—31no—
66C5×D5—31no—
67C5×D5D3211yescharacteristic
68C32×C5—41no—
69C32×C5C2×C4×C511yescharacteristic, complemented
70C32×C5C2×AGL(1,𝔽5)11yescharacteristic, complemented
71C2×AGL(1,𝔽5)—91no—
72C2×C4×C5—451no—
73C2×C3×D3—51no—
74C3⋊Dic3—51no—
75C3×Dic3—51no—
76C2×C3×C5—31no—
77C2×C3×C5—31no—
78C2×C3×C5—51no—
79C2×C3×C5—51no—
80C2×C3×C5—101no—
81C2×C3×C5—121no—
82C2×C3×C5—151no—
83D15C5×Dic311yescharacteristic, complemented
84C3×D5—21no—
85C3×D5—31no—
86C3×D5C2×C5×D311yescharacteristic
87C3×D5C2×C5×D311yescharacteristic
88C5×D3—41no—
89C5×D3—51no—
90C5×D3C5×Dic311yescharacteristic, complemented
91C5×D3ΓL(1,𝔽16)11yescharacteristic, direct product factor
92C52D3×Dic311yescharacteristic, complemented, Sylow
93C2×Dic3—151no—
94C4×D3—151no—
95C22×C5—151no—
96C2×D5—31no—
97AGL(1,𝔽5)—31no—
98AGL(1,𝔽5)—91no—
99C4×C5—151no—
100C4×C5—451no—
101C2×C32—51no—
102C3×D3—51no—
103C3×D3C5×AGL(1,𝔽5)11yescharacteristic, complemented
104C3×C5—21no—
105C3×C5—21no—
106C3×C5—41no—
107C3×C5—41no—
108C3×C5—81no—
109C3×C5C4×C5×D311yescharacteristic, complemented
110C3×C5C2×C5×Dic311yescharacteristic, complemented
111C3×C5D3×AGL(1,𝔽5)11yescharacteristic, complemented
112C3×C5C2×ΓL(1,𝔽16)11yescharacteristic, complemented
113C22×C3—151no—
114C2×D3—51no—
115C3×C4—151no—
116Dic3—51no—
117Dic3—151no—
118Dic3—151no—
119Dic3—301no—
120C2×C5—31no—
121C2×C5—31no—
122C2×C5—51no—
123C2×C5—121no—
124C2×C5—151no—
125D5—31no—
126D5C5×D3211yescharacteristic
127C32C2×C5×AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
128C2×C4—451noSylow
129C2×C3—31no—
130C2×C3—51no—
131C2×C3—51no—
132C2×C3—101no—
133C2×C3—151no—
134D3—51no—
135D3C5×ΓL(1,𝔽16)11yescharacteristic, direct product factor
136C5—41no—
137C5C5×D3×Dic311yescharacteristic, complemented
138C5D3×ΓL(1,𝔽16)11yescharacteristic, direct product factor
139C22—151no—
140C4—151no—
141C4—451no—
142C3—21no—
143C3C5×D3×AGL(1,𝔽5)11yescharacteristic, complemented
144C3C2×C5×ΓL(1,𝔽16)11yescharacteristic, complemented
145C2—31no—
146C2—51no—
147C2—151no—
subgroupG/H
center, hypercenterC5D3×ΓL(1,𝔽16)
commutator, hypocenterC32×C5C2×C4×C5
FrattiniC1G
Fitting, socleC32×C52C2×C4
prime radicalC3×C5×D3×D5C2
cosocleC32×D5C22×C5
LCSG ▷ C32×C5
UCSC1 ◁ C5

Complements

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

NH ≅ G/Ntype
2. C5×C3⋊ΓL(1,𝔽16)C2semidirect
3. C3×C5×ΓL(1,𝔽16)C2semidirect
6. C3×C5×D15C4semidirect
7. C3×C52×D3C4semidirect
9. D3×ΓL(1,𝔽16)C5direct (N × H)
16. C5×ΓL(1,𝔽16)D3direct (N × H)
18. C32×C52C2×C4semidirect
22. C3⋊ΓL(1,𝔽16)C2×C5semidirect
23. C3×ΓL(1,𝔽16)C2×C5semidirect
27. C5×D15Dic3semidirect
28. C52×D3Dic3semidirect
41. C3×D15C4×C5semidirect
44. C3×C5×D3C4×C5semidirect
45. C3×C5×D3AGL(1,𝔽5)semidirect
48. C3×C52C4×D3semidirect
49. C3×C52C2×Dic3semidirect
58. ΓL(1,𝔽16)C5×D3direct (N × H)
69. C32×C5C2×C4×C5semidirect
70. C32×C5C2×AGL(1,𝔽5)semidirect
83. D15C5×Dic3semidirect
90. C5×D3C5×Dic3semidirect
91. C5×D3ΓL(1,𝔽16)direct (N × H)
92. C52D3×Dic3semidirect
103. C3×D3C5×AGL(1,𝔽5)semidirect
109. C3×C5C4×C5×D3semidirect
110. C3×C5C2×C5×Dic3semidirect
111. C3×C5D3×AGL(1,𝔽5)semidirect
112. C3×C5C2×ΓL(1,𝔽16)semidirect
127. C32C2×C5×AGL(1,𝔽5)semidirect
135. D3C5×ΓL(1,𝔽16)direct (N × H)
137. C5C5×D3×Dic3semidirect
138. C5D3×ΓL(1,𝔽16)direct (N × H)
143. C3C5×D3×AGL(1,𝔽5)semidirect
144. C3C2×C5×ΓL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
C3×C5×D3×D52
C5×C3⋊ΓL(1,𝔽16)2
C3×C5×ΓL(1,𝔽16)2
C2×C5×ΓL(1,𝔽16)3
C5×D3×AGL(1,𝔽5)3
D3×ΓL(1,𝔽16)5
C5×D3×Dic35