Finite Group Databasebeta

← #72229676

Subgroups — #72229676

Order 2400.

196 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 43 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
1C4⋊(C3⋊#1044)C211yesmaximal, characteristic
2C3×C4⋊#1043C211yesmaximal, characteristic
3#11925150C211yesmaximal, characteristic
4#11716771—31nomaximal, Hall
5C2×C3⋊#1044—21no—
6C2×C3×#1043—21no—
7C3×C4×C5⋊D5C2211yescharacteristic
8C5×C3⋊#168C412yescomplemented
9#91159—52nomaximal
10C4⋊#1044—31no—
11C4⋊#1043D311yescharacteristic
12#87767—31no—
13C2×C3×C5⋊D5D411yescharacteristic
14C3⋊#1044—41no—
15C3×#1043—41no—
16C3×C4×C52C2×C411yescharacteristic
17C3×#1039Q811yescharacteristic
18C4⋊ΓL(1,𝔽16)—52no—
19C3×C4⋊AGL(1,𝔽5)—52no—
20C3×C4⋊AGL(1,𝔽5)—102no—
21D5×C3⋊C8—52no—
22C2×#1044—61no—
23C2×#1043—21no—
24C4×C5⋊D5C2×D311yescharacteristic
25C5×#168—32no—
26#3993—152no—
27C2×C3×C52C4⋊C411yescharacteristic
28C3×C5⋊D5SD1611yescharacteristic
29C3×C5⋊D5SD1611yescharacteristic
30C2×ΓL(1,𝔽16)—102no—
31C2×C3×AGL(1,𝔽5)—102no—
32C2×C3×AGL(1,𝔽5)—104no—
33C3×C4×D5—52no—
34C3×C4×D5—102no—
35C3⋊#168AGL(1,𝔽5)12yescomplemented
36C5×C3⋊C8—52no—
37C2×C5⋊D5#6711yescharacteristic
38#1044—121no—
39#1043—41no—
40C4×C52C4×D311yescharacteristic
41#1039Dic611yescharacteristic
42#808—251noHall
43C4⋊AGL(1,𝔽5)—52no—
44C4⋊AGL(1,𝔽5)—102no—
45C4⋊AGL(1,𝔽5)—152no—
46C8×D5—152no—
47C3×C52#10611yescharacteristic, complemented, Hall
48C2×C3×D5—52no—
49C2×C3×D5—102no—
50ΓL(1,𝔽16)—202no—
51C3×AGL(1,𝔽5)—202no—
52C3×AGL(1,𝔽5)—204no—
53C3×C4×C5—22no—
54C3×C4×C5C2×AGL(1,𝔽5)12yes—
55C3×Dic5—52no—
56C3×Dic5—102no—
57C2×C52Dic3⋊C411yescharacteristic
58C5⋊D5#21411yescharacteristic
59C5⋊D5#21511yescharacteristic
60C3×C4⋊C4—251no—
61C4⋊Dic3—251no—
62C2×C3⋊C8—251no—
63C2×AGL(1,𝔽5)—102no—
64C2×AGL(1,𝔽5)—104no—
65C2×AGL(1,𝔽5)—302no—
66C4×D5—52no—
67C4×D5—102no—
68C5×C8—152no—
69#168—32no—
70#106—751noSylow
71C2×C3×C5—22no—
72C2×C3×C5C4⋊AGL(1,𝔽5)12yes—
73C3×D5—52no—
74C3×D5—52no—
75C3×D5—102no—
76C3×D5—102no—
77C52#80811yescharacteristic, complemented, Sylow
78C2×C3×C4—251no—
79C2×C3×C4—501no—
80C2×Dic3—501no—
81C3⋊C8—52no—
82C2×D5—52no—
83C2×D5—102no—
84AGL(1,𝔽5)—202no—
85AGL(1,𝔽5)—204no—
86AGL(1,𝔽5)—602no—
87C4×C5—22no—
88C4×C5D3×AGL(1,𝔽5)12yes—
89Dic5—52no—
90Dic5—102no—
91C2×C8—751no—
92C4⋊C4—251no—
93C4⋊C4—751no—
94C3×C5—22no—
95C3×C5#399312yescomplemented
96C22×C3—251no—
97C3×C4—251no—
98C3×C4—1001no—
99C3×C4D5⋊AGL(1,𝔽5)11yescharacteristic
100Dic3—1001no—
101C2×C5—22no—
102C2×C5Dic3⋊AGL(1,𝔽5)12yes—
103D5—52no—
104D5—52no—
105D5—102no—
106D5—102no—
107C2×C4—251no—
108C2×C4—501no—
109C2×C4—1501no—
110C8—152no—
111C2×C3—251no—
112C2×C3—251no—
113C2×C3#8783211yescharacteristic
114C5—22no—
115C5#9115912yescomplemented
116C22—251no—
117C4—251no—
118C4—3001no—
119C4—1001no—
120C4#1059718211yescharacteristic
121C3#1171677111yescharacteristic, complemented, Sylow
122C2—251no—
123C2—251no—
124C2#1192535611yescharacteristic
subgroupG/H
centerC2#11925356
commutator, FittingC3×C4×C52C2×C4
Frattini, hypercenterC4#10597182
socleC2×C3×C52C4⋊C4
prime radicalC2×C3×C5⋊D5D4
cosocleC3×C4×C5⋊D5C22
hypocenterC3×C52#106
LCSG ▷ C3×C4×C52 ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
8. C5×C3⋊#168C4semidirect
35. C3⋊#168AGL(1,𝔽5)semidirect
47. C3×C52#106semidirect
77. C52#808semidirect
95. C3×C5#3993semidirect
115. C5#91159semidirect
121. C3#11716771semidirect

Maximal subgroups

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

subgroupindex
C4⋊(C3⋊#1044)2
C3×C4⋊#10432
#119251502
#117167713
#911595