Finite Group Databasebeta

← C5×#67152 #14268282

Subgroups — C5×#67152 #14268282

Order 1800.

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

normal subgroups only

This group has 192 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
1C2×C5×C3⋊D15C211yesmaximal, characteristic, complemented
2C2×C3×C52×D3C211yesmaximal, characteristic, complemented
3C3×C5×Dic15C211yesmaximal, characteristic, complemented
4C5×#1220—31nomaximal
5C5×#1203—31nomaximal
6C2×C32×C52C2211yescharacteristic
7C5×C3⋊D15—21no—
8C3×C52×D3—21no—
9#67152C511yesmaximal, characteristic, direct product factor
10C5×#626—51nomaximal
11C22×C3×C52—31no—
12C2×C5×D15—31no—
13C2×C5×D15—31no—
14C2×C5×D15—61no—
15C2×C52×D3D311yescharacteristic, complemented
16C5×Dic15D311yescharacteristic, complemented
17C3×C5×Dic5—31no—
18C32×C52D411yescharacteristic, complemented, Hall
19C5×C22⋊D5—91noHall
20C2×C3⋊D15C2×C511yescharacteristic, complemented
21C2×C5×C3⋊D3—51no—
22C2×C3×C5×D3—22no—
23C2×C3×C5×D3D511yescharacteristic, complemented
24C2×C3×C5×D3C2×C511yescharacteristic, complemented
25C3×Dic15C2×C511yescharacteristic, complemented
26C3×C5×Dic3—51no—
27C2×C3×C52—21no—
28C2×C3×C52—61no—
29C2×C3×C52C2×D311yescharacteristic
30C2×C3×C52C2×D311yescharacteristic
31C5×D15—62no—
32C5×D15—61no—
33C5×D15—61no—
34C52×D3—21no—
35#1220—31no—
36C5×#67—151no—
37C5×D12—151no—
38#1203—31no—
39C22×C52—31no—
40C2×C5×D5—91no—
41C5×Dic5—31no—
42C2×C32×C5—22no—
43C2×C32×C5C2×D511yescharacteristic
44C2×C32×C5C22×C511yescharacteristic
45C3⋊D15—21no—
46C5×C3⋊D3—101no—
47C3×C5×D3—24no—
48C3×C5×D3—21no—
49C3×C5×D3—21no—
50C3×C52—21no—
51C3×C52D1211yescharacteristic, complemented
52C3×C52#6711yescharacteristic, complemented
53#626—51noHall
54C22×C3×C5—31no—
55C22×C3×C5—31no—
56C22×C3×C5—62no—
57C2×D15—31no—
58C2×D15—31no—
59C2×D15—61no—
60C2×C5×D3—22no—
61C2×C5×D3—151no—
62C2×C5×D3—151no—
63C2×C5×D3—301no—
64C2×C5×D3C5×D311yescharacteristic, complemented
65C2×C5×D3D1511yescharacteristic, complemented
66C3×C4×C5—151no—
67Dic15C5×D311yescharacteristic, complemented
68C3×Dic5—31no—
69C5×Dic3—51no—
70C2×C52—61no—
71C2×C52D3211yescharacteristic
72C5×D5—181no—
73C32×C5—22no—
74C32×C5C22⋊D511yescharacteristic, complemented
75C32×C5C5×D411yescharacteristic, complemented
76C5×D4—451no—
77C22⋊D5—91no—
78C2×C3⋊D3—51no—
79C2×C3×D3C5×D511yescharacteristic, complemented
80C3×Dic3—51no—
81C2×C3×C5—21no—
82C2×C3×C5—21no—
83C2×C3×C5—22no—
84C2×C3×C5—22no—
85C2×C3×C5—42no—
86C2×C3×C5—64no—
87C2×C3×C5—61no—
88C2×C3×C5—61no—
89C2×C3×C5D3×D511yescharacteristic
90C2×C3×C5C2×C5×D311yescharacteristic
91C2×C3×C5C2×C5×D311yescharacteristic
92C2×C3×C5C2×D1511yescharacteristic
93D15—62no—
94D15—61no—
95D15—61no—
96C5×D3—24no—
97C5×D3—21no—
98C5×D3—21no—
99C5×D3—302no—
100C5×D3—301no—
101C5×D3—301no—
102C52#62611yescharacteristic, complemented, Sylow
103#67—151no—
104D12—151no—
105C22×C5—31no—
106C22×C5—31no—
107C22×C5—62no—
108C22×C5—451no—
109C2×D5—91no—
110C4×C5—151no—
111Dic5—31no—
112C2×C32C2×C5×D511yescharacteristic
113C3⋊D3—101no—
114C3×D3—21no—
115C3×C5—21no—
116C3×C5—21no—
117C3×C5—22no—
118C3×C5—22no—
119C3×C5—42no—
120C3×C5#120311yescharacteristic, complemented
121C3×C5C5×D1211yescharacteristic, complemented
122C3×C5C5×#6711yescharacteristic, complemented
123C3×C5#122011yescharacteristic, complemented
124C22×C3—31no—
125C2×D3—151no—
126C2×D3—151no—
127C2×D3—301no—
128C2×D3C5×D1511yescharacteristic, complemented
129C3×C4—151no—
130Dic3—51no—
131C2×C5—22no—
132C2×C5—64no—
133C2×C5—61no—
134C2×C5—61no—
135C2×C5—901no—
136C2×C5C5×D3211yescharacteristic
137C2×C5D3×D1511yescharacteristic
138D5—181no—
139C32C5×C22⋊D511yescharacteristic, complemented, Sylow
140D4—451noSylow
141C2×C3—21no—
142C2×C3—61no—
143C2×C3C5×D3×D511yescharacteristic
144C2×C3C2×C5×D1511yescharacteristic
145D3—21no—
146D3—302no—
147D3—301no—
148D3—301no—
149C5—22no—
150C5C5×#62611yescharacteristic, complemented
151C5#6715211yescharacteristic, direct product factor
152C22—31no—
153C22—451no—
154C4—151no—
155C3—21no—
156C3C5×#120311yescharacteristic, complemented
157C3C5×#122011yescharacteristic, complemented
158C2—61no—
159C2—901no—
160C2C5×D3×D1511yescharacteristic
subgroupG/H
center, hypercenterC2×C5D3×D15
commutator, cosocleC2×C32×C5C22×C5
FrattiniC2C5×D3×D15
Fitting, socleC2×C32×C52C22
hypocenterC32×C5C5×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2×C5

Complements

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

NH ≅ G/Ntype
1. C2×C5×C3⋊D15C2semidirect
2. C2×C3×C52×D3C2semidirect
3. C3×C5×Dic15C2semidirect
9. #67152C5direct (N × H)
15. C2×C52×D3D3semidirect
16. C5×Dic15D3semidirect
18. C32×C52D4semidirect
20. C2×C3⋊D15C2×C5semidirect
23. C2×C3×C5×D3D5semidirect
24. C2×C3×C5×D3C2×C5semidirect
25. C3×Dic15C2×C5semidirect
51. C3×C52D12semidirect
52. C3×C52#67semidirect
64. C2×C5×D3C5×D3semidirect
65. C2×C5×D3D15semidirect
67. Dic15C5×D3semidirect
74. C32×C5C22⋊D5semidirect
75. C32×C5C5×D4semidirect
79. C2×C3×D3C5×D5semidirect
102. C52#626semidirect
120. C3×C5#1203semidirect
121. C3×C5C5×D12semidirect
122. C3×C5C5×#67semidirect
123. C3×C5#1220semidirect
128. C2×D3C5×D15semidirect
139. C32C5×C22⋊D5semidirect
150. C5C5×#626semidirect
151. C5#67152direct (N × H)
156. C3C5×#1203semidirect
157. C3C5×#1220semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C3⋊D152
C2×C3×C52×D32
C3×C5×Dic152
C5×#12203
C5×#12033
#671525
C5×#6265