Finite Group Databasebeta

← C3×C5×C31×Dic15 #42124661

Subgroups — C3×C5×C31×Dic15 #42124661

Order 27900.

160 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 72 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×C32×C52×C31C211yesmaximal, characteristic
2C5×C31×Dic15C311yesmaximal, characteristic, direct product factor
3C3×C5×Dic5×C31—31nomaximal
4C32×C52×C31C411yescharacteristic, complemented, Hall
5C3×C31×Dic15C511yesmaximal, characteristic, direct product factor
6C3×C5×Dic3×C31—51nomaximal
7C2×C3×C52×C31—21no—
8C2×C3×C52×C31D311yescharacteristic
9C2×C3×C52×C31C2×C311yescharacteristic
10C5×Dic5×C31—31noHall
11C2×C32×C5×C31—22no—
12C2×C32×C5×C31D511yescharacteristic
13C2×C32×C5×C31C2×C511yescharacteristic
14C3×C52×C31—21no—
15C3×C52×C31Dic311yescharacteristic, complemented
16C3×C52×C31C3×C411yescharacteristic, complemented
17C3×C4×C5×C31—151no—
18C31×Dic15C3×C511yescharacteristic, direct product factor
19C5×Dic3×C31—51no—
20C3×Dic5×C31—31no—
21C2×C52×C31C3×D311yescharacteristic
22C32×C5×C31—22no—
23C32×C5×C31Dic511yescharacteristic, complemented
24C32×C5×C31C4×C511yescharacteristic, complemented
25C3×Dic3×C31—51noHall
26C2×C3×C5×C31—24no—
27C2×C3×C5×C31—21no—
28C2×C3×C5×C31—21no—
29C2×C3×C5×C31—22no—
30C2×C3×C5×C31—22no—
31C2×C3×C5×C31C5×D311yescharacteristic
32C2×C3×C5×C31C3×D511yescharacteristic
33C2×C3×C5×C31D1511yescharacteristic
34C2×C3×C5×C31C2×C3×C511yescharacteristic
35C3×C5×Dic15C3111yesmaximal, characteristic, direct product factor, Hall
36C52×C31C3×Dic311yescharacteristic, complemented, Hall
37C4×C5×C31—151no—
38Dic5×C31—31no—
39C2×C32×C31C5×D511yescharacteristic
40C3×C5×C31—24no—
41C3×C5×C31—21no—
42C3×C5×C31—21no—
43C3×C5×C31—22no—
44C3×C5×C31—22no—
45C3×C5×C31C5×Dic311yescharacteristic, complemented
46C3×C5×C31C3×Dic511yescharacteristic, complemented
47C3×C5×C31Dic1511yescharacteristic, direct product factor
48C3×C5×C31C3×C4×C511yescharacteristic, complemented
49C2×C32×C52C2×C3111yescharacteristic
50C3×C4×C31—151no—
51Dic3×C31—51no—
52C2×C5×C31—22no—
53C2×C5×C31C3×C5×D311yescharacteristic
54C2×C5×C31C3×D1511yescharacteristic
55C5×Dic15C3×C3111yescharacteristic, direct product factor
56C3×C5×Dic5—31no—
57C32×C31C5×Dic511yescharacteristic, complemented, Hall
58C32×C52C4×C3111yescharacteristic, complemented, Hall
59C2×C3×C31—21no—
60C2×C3×C31C3×C5×D511yescharacteristic
61C2×C3×C31C5×D1511yescharacteristic
62C3×Dic15C5×C3111yescharacteristic, direct product factor
63C3×C5×Dic3—51no—
64C5×C31—22no—
65C5×C31C3×C5×Dic311yescharacteristic, complemented
66C5×C31C3×Dic1511yescharacteristic, direct product factor
67C2×C3×C52—21no—
68C2×C3×C52D3×C3111yescharacteristic
69C2×C3×C52C2×C3×C3111yescharacteristic
70C4×C31—151noHall
71C5×Dic5—31noHall
72C3×C31—21no—
73C3×C31C3×C5×Dic511yescharacteristic, complemented
74C3×C31C5×Dic1511yescharacteristic, direct product factor
75C2×C32×C5—22no—
76C2×C32×C5D5×C3111yescharacteristic
77C2×C32×C5C2×C5×C3111yescharacteristic
78C3×C52—21no—
79C3×C52Dic3×C3111yescharacteristic, complemented
80C3×C52C3×C4×C3111yescharacteristic, complemented
81C2×C31C3×C5×D1511yescharacteristic
82C3×C4×C5—151no—
83Dic15C3×C5×C3111yescharacteristic, direct product factor
84C3×Dic5—31no—
85C5×Dic3—51no—
86C2×C52C3×D3×C3111yescharacteristic
87C32×C5—22no—
88C32×C5Dic5×C3111yescharacteristic, complemented
89C32×C5C4×C5×C3111yescharacteristic, complemented
90C3×Dic3—51noHall
91C31C3×C5×Dic1511yescharacteristic, direct product factor, Sylow
92C2×C3×C5—24no—
93C2×C3×C5—21no—
94C2×C3×C5—21no—
95C2×C3×C5—22no—
96C2×C3×C5—22no—
97C2×C3×C5C3×D5×C3111yescharacteristic
98C2×C3×C5C5×D3×C3111yescharacteristic
99C2×C3×C5D15×C3111yescharacteristic
100C2×C3×C5C2×C3×C5×C3111yescharacteristic
101C52C3×Dic3×C3111yescharacteristic, complemented, Sylow
102C4×C5—151no—
103Dic5—31no—
104C2×C32C5×D5×C3111yescharacteristic
105C3×C5—24no—
106C3×C5—21no—
107C3×C5—21no—
108C3×C5—22no—
109C3×C5—22no—
110C3×C5C3×Dic5×C3111yescharacteristic, complemented
111C3×C5C5×Dic3×C3111yescharacteristic, complemented
112C3×C5C31×Dic1511yescharacteristic, direct product factor
113C3×C5C3×C4×C5×C3111yescharacteristic, complemented
114C3×C4—151no—
115Dic3—51no—
116C2×C5—22no—
117C2×C5C3×C5×D3×C3111yescharacteristic
118C2×C5C3×D15×C3111yescharacteristic
119C32C5×Dic5×C3111yescharacteristic, complemented, Sylow
120C2×C3—21no—
121C2×C3C3×C5×D5×C3111yescharacteristic
122C2×C3C5×D15×C3111yescharacteristic
123C5—22no—
124C5C3×C5×Dic3×C3111yescharacteristic, complemented
125C5C3×C31×Dic1511yescharacteristic, direct product factor
126C4—151noSylow
127C3—21no—
128C3C3×C5×Dic5×C3111yescharacteristic, complemented
129C3C5×C31×Dic1511yescharacteristic, direct product factor
130C2C3×C5×D15×C3111yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C5×C31D15
commutator, hypocenterC3×C5C3×C4×C5×C31
FrattiniC2C3×C5×D15×C31
Fitting, socle, prime radicalC2×C32×C52×C31C2
cosocleC2×C3×C5C2×C3×C5×C31
LCSG ▷ C3×C5
UCSC1 ◁ C2×C3×C5×C31

Complements

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

NH ≅ G/Ntype
2. C5×C31×Dic15C3direct (N × H)
4. C32×C52×C31C4semidirect
5. C3×C31×Dic15C5direct (N × H)
15. C3×C52×C31Dic3semidirect
16. C3×C52×C31C3×C4semidirect
18. C31×Dic15C3×C5direct (N × H)
23. C32×C5×C31Dic5semidirect
24. C32×C5×C31C4×C5semidirect
35. C3×C5×Dic15C31direct (N × H)
36. C52×C31C3×Dic3semidirect
45. C3×C5×C31C5×Dic3semidirect
46. C3×C5×C31C3×Dic5semidirect
47. C3×C5×C31Dic15direct (N × H)
48. C3×C5×C31C3×C4×C5semidirect
55. C5×Dic15C3×C31direct (N × H)
57. C32×C31C5×Dic5semidirect
58. C32×C52C4×C31semidirect
62. C3×Dic15C5×C31direct (N × H)
65. C5×C31C3×C5×Dic3semidirect
66. C5×C31C3×Dic15direct (N × H)
73. C3×C31C3×C5×Dic5semidirect
74. C3×C31C5×Dic15direct (N × H)
79. C3×C52Dic3×C31semidirect
80. C3×C52C3×C4×C31semidirect
83. Dic15C3×C5×C31direct (N × H)
88. C32×C5Dic5×C31semidirect
89. C32×C5C4×C5×C31semidirect
91. C31C3×C5×Dic15direct (N × H)
101. C52C3×Dic3×C31semidirect
110. C3×C5C3×Dic5×C31semidirect
111. C3×C5C5×Dic3×C31semidirect
112. C3×C5C31×Dic15direct (N × H)
113. C3×C5C3×C4×C5×C31semidirect
119. C32C5×Dic5×C31semidirect
124. C5C3×C5×Dic3×C31semidirect
125. C5C3×C31×Dic15direct (N × H)
128. C3C3×C5×Dic5×C31semidirect
129. C3C5×C31×Dic15direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C52×C312
C5×C31×Dic153
C3×C5×Dic5×C313
C3×C31×Dic155
C3×C5×Dic3×C315
C3×C5×Dic1531