Finite Group Databasebeta

← C3×C5×#3777 #72219422

Subgroups — C3×C5×#3777 #72219422

Order 2160.

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

normal subgroups only

This group has 132 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×#157C212yesmaximal
2C2×C3×C5×#157C211yesmaximal, characteristic
3C5×#3777C311yesmaximal, characteristic, direct product factor
4C2×C3×C5×C3⋊D3C2211yescharacteristic
5C3×C5×#157—22no—
6C3×C5×#157C412yescomplemented
7C3×#3777C511yesmaximal, characteristic, direct product factor, Hall
8C2×C5×#157C2×C312yes—
9C2×C5×#157C2×C311yescharacteristic
10C2×C33×C5Q811yescharacteristic
11C3×C5×C3⋊D3C2×C411yescharacteristic
12C3×C5×C3⋊D3D411yescharacteristic
13C3×C5×C4⋊C4—91nomaximal
14C2×C3×#157C2×C512yes—
15C2×C3×#157C2×C511yescharacteristic
16C2×C5×C3⋊D3C22×C311yescharacteristic
17C2×C3×C5×D3—121no—
18C5×#157—22no—
19C5×#157C3×C412yescomplemented
20#3777C3×C511yescharacteristic, direct product factor
21C33×C5C4⋊C411yescharacteristic, complemented, Hall
22C2×C3×C4×C5—91no—
23C2×C3×C4×C5—92no—
24C2×C3×C3⋊D3C22×C511yescharacteristic
25C3×#157—22no—
26C3×#157C4×C512yescomplemented
27C2×C32×C5—41no—
28C2×C32×C5—81no—
29C2×C32×C5C3×Q811yescharacteristic
30C5×C3⋊D3C2×C3×C411yescharacteristic
31C5×C3⋊D3C3×D411yescharacteristic
32C3×C5×D3—121no—
33C3×C5×D3—121no—
34C5×C4⋊C4—91noHall
35C2×#157C2×C3×C512yes—
36C2×#157C2×C3×C511yescharacteristic
37C22×C3×C5—91no—
38C2×C5×D3—121no—
39C3×C4×C5—92no—
40C3×C4×C5—182no—
41C2×C33C5×Q811yescharacteristic
42C3×C3⋊D3C2×C4×C511yescharacteristic
43C3×C3⋊D3C5×D411yescharacteristic
44C3×C4⋊C4—91no—
45C32×C5—41no—
46C32×C5—81no—
47C32×C5C3×C4⋊C411yescharacteristic, complemented
48C2×C4×C5—91no—
49C2×C4×C5—92no—
50C2×C3⋊D3C22×C3×C511yescharacteristic
51C2×C3×D3—121no—
52#157—22no—
53#157C3×C4×C512yescomplemented
54C2×C3×C5—41no—
55C2×C3×C5—81no—
56C2×C3×C5—91no—
57C2×C3×C5—91no—
58C2×C3×C5PSU(3,𝔽2)11yescharacteristic
59C5×D3—121no—
60C5×D3—121no—
61C33C5×C4⋊C411yescharacteristic, complemented, Sylow
62C2×C3×C4—91no—
63C2×C3×C4—92no—
64C22×C5—91no—
65C4×C5—92no—
66C4×C5—182no—
67C2×C32—41no—
68C2×C32—81no—
69C2×C32C3×C5×Q811yescharacteristic
70C3⋊D3C2×C3×C4×C511yescharacteristic
71C3⋊D3C3×C5×D411yescharacteristic
72C3×D3—121no—
73C3×D3—121no—
74C4⋊C4—91noSylow
75C3×C5—41no—
76C3×C5—81no—
77C3×C5#377711yescharacteristic, direct product factor
78C22×C3—91no—
79C2×D3—121no—
80C3×C4—92no—
81C3×C4—182no—
82C2×C5—91no—
83C2×C5—91no—
84C2×C5C3×PSU(3,𝔽2)11yescharacteristic
85C32—41no—
86C32—81no—
87C32C3×C5×C4⋊C411yescharacteristic, complemented
88C2×C4—91no—
89C2×C4—92no—
90C2×C3—41no—
91C2×C3—81no—
92C2×C3—91no—
93C2×C3—91no—
94C2×C3C5×PSU(3,𝔽2)11yescharacteristic
95D3—121no—
96D3—121no—
97C5C3×#377711yescharacteristic, direct product factor, Sylow
98C22—91no—
99C4—92no—
100C4—182no—
101C3—41no—
102C3—81no—
103C3C5×#377711yescharacteristic, direct product factor
104C2—91no—
105C2—91no—
106C2C3×C5×PSU(3,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C5PSU(3,𝔽2)
commutatorC3⋊D3C2×C3×C4×C5
FrattiniC2C3×C5×PSU(3,𝔽2)
Fitting, socleC2×C33×C5Q8
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×#3777C3direct (N × H)
6. C3×C5×#157C4semidirect
7. C3×#3777C5direct (N × H)
19. C5×#157C3×C4semidirect
20. #3777C3×C5direct (N × H)
21. C33×C5C4⋊C4semidirect
26. C3×#157C4×C5semidirect
47. C32×C5C3×C4⋊C4semidirect
53. #157C3×C4×C5semidirect
61. C33C5×C4⋊C4semidirect
77. C3×C5#3777direct (N × H)
87. C32C3×C5×C4⋊C4semidirect
97. C5C3×#3777direct (N × H)
103. C3C5×#3777direct (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
C5×#37773
C3×#37775
C3×C5×C4⋊C49