Finite Group Databasebeta

← C3×C5×#3773 #72219430

Subgroups — C3×C5×#3773 #72219430

Order 2160.

192 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 52 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×C3×C5×#157C211yesmaximal, characteristic
2C3×C5×#624C212yesmaximal
3C5×#3773C311yesmaximal, characteristic, direct product factor
4C2×C3×C5×C3⋊D3C2211yescharacteristic
5C3×C5×#157C412yescomplemented
6C32×C5×Dic3—22no—
7C3×#3773C511yesmaximal, characteristic, direct product factor, Hall
8C2×C5×#157C2×C311yescharacteristic
9C3×C4×C5×D3—62no—
10C5×#624C2×C312yes—
11C2×C33×C5D411yescharacteristic
12C3×C5×C3⋊D3C2×C411yescharacteristic
13C3×C5×C3⋊D3Q811yescharacteristic
14C3×C5×C4⋊C4—91nomaximal
15C2×C3×#157C2×C511yescharacteristic
16C3×#624C2×C512yes—
17C2×C5×C3⋊D3C22×C311yescharacteristic
18C2×C3×C5×D3—62no—
19C5×#157C3×C412yescomplemented
20C32×C4×C5—62no—
21C3×C5×Dic3—22no—
22C3×C5×Dic3—22no—
23C3×C5×Dic3—42no—
24#3773C3×C511yescharacteristic, direct product factor
25C33×C5C4⋊C411yescharacteristic, complemented, Hall
26C2×C3×C4×C5—92no—
27C2×C3×C4×C5—91no—
28C4×C5×D3—62no—
29C2×C3×C3⋊D3C22×C511yescharacteristic
30C3×#157C4×C512yescomplemented
31C32×Dic3—22no—
32C2×C32×C5—22no—
33C2×C32×C5—42no—
34C2×C32×C5C3×D411yescharacteristic
35C5×C3⋊D3C2×C3×C411yescharacteristic
36C5×C3⋊D3C3×Q811yescharacteristic
37C3×C5×D3—62no—
38C3×C5×D3—62no—
39C5×C4⋊C4—91noHall
40C2×#157C2×C3×C511yescharacteristic
41C3×C4×D3—62no—
42#624C2×C3×C512yes—
43C22×C3×C5—91no—
44C2×C5×D3—62no—
45C3×C4×C5—62no—
46C3×C4×C5—62no—
47C3×C4×C5—92no—
48C3×C4×C5—122no—
49C5×Dic3—22no—
50C2×C33C5×D411yescharacteristic
51C3×C3⋊D3C2×C4×C511yescharacteristic
52C3×C3⋊D3C5×Q811yescharacteristic
53C3×C4⋊C4—91no—
54C32×C5—22no—
55C32×C5—42no—
56C32×C5C3×C4⋊C411yescharacteristic, complemented
57C2×C4×C5—92no—
58C2×C4×C5—91no—
59C2×C3⋊D3C22×C3×C511yescharacteristic
60C2×C3×D3—62no—
61#157C3×C4×C512yescomplemented
62C32×C4—62no—
63C3×Dic3—22no—
64C3×Dic3—22no—
65C3×Dic3—42no—
66C2×C3×C5—22no—
67C2×C3×C5—42no—
68C2×C3×C5—91no—
69C2×C3×C5—91no—
70C2×C3×C5GO+(4,𝔽2)11yescharacteristic
71C5×D3—62no—
72C5×D3—62no—
73C33C5×C4⋊C411yescharacteristic, complemented, Sylow
74C2×C3×C4—92no—
75C2×C3×C4—91no—
76C4×D3—62no—
77C22×C5—91no—
78C4×C5—62no—
79C4×C5—92no—
80C2×C32—22no—
81C2×C32—42no—
82C2×C32C3×C5×D411yescharacteristic
83C3⋊D3C2×C3×C4×C511yescharacteristic
84C3⋊D3C3×C5×Q811yescharacteristic
85C3×D3—62no—
86C3×D3—62no—
87C4⋊C4—91noSylow
88C3×C5—22no—
89C3×C5—42no—
90C3×C5#377311yescharacteristic, direct product factor
91C22×C3—91no—
92C2×D3—62no—
93C3×C4—62no—
94C3×C4—62no—
95C3×C4—92no—
96C3×C4—122no—
97Dic3—22no—
98C2×C5—91no—
99C2×C5—91no—
100C2×C5C3×GO+(4,𝔽2)11yescharacteristic
101C32—22no—
102C32—42no—
103C32C3×C5×C4⋊C411yescharacteristic, complemented
104C2×C4—92no—
105C2×C4—91no—
106C2×C3—22no—
107C2×C3—42no—
108C2×C3—91no—
109C2×C3—91no—
110C2×C3C5×GO+(4,𝔽2)11yescharacteristic
111D3—62no—
112D3—62no—
113C5C3×#377311yescharacteristic, direct product factor, Sylow
114C22—91no—
115C4—62no—
116C4—92no—
117C3—22no—
118C3—42no—
119C3C5×#377311yescharacteristic, direct product factor
120C2—91no—
121C2—91no—
122C2C3×C5×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C5GO+(4,𝔽2)
commutatorC3⋊D3C2×C3×C4×C5
FrattiniC2C3×C5×GO+(4,𝔽2)
Fitting, socleC2×C33×C5D4
prime radicalC2×C3×C5×C3⋊D3C22
cosocleC2×C3⋊D3C22×C3×C5
hypocenterC32C3×C5×C4⋊C4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
3. C5×#3773C3direct (N × H)
5. C3×C5×#157C4semidirect
7. C3×#3773C5direct (N × H)
19. C5×#157C3×C4semidirect
24. #3773C3×C5direct (N × H)
25. C33×C5C4⋊C4semidirect
30. C3×#157C4×C5semidirect
56. C32×C5C3×C4⋊C4semidirect
61. #157C3×C4×C5semidirect
73. C33C5×C4⋊C4semidirect
90. C3×C5#3773direct (N × H)
103. C32C3×C5×C4⋊C4semidirect
113. C5C3×#3773direct (N × H)
119. C3C5×#3773direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1572
C3×C5×#6242
C5×#37733
C3×#37735
C3×C5×C4⋊C49