Finite Group Databasebeta

← #13086384

Subgroups — #13086384

Order 1440.

168 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 24 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
1#10624285C211yesmaximal, characteristic
2C4×#4380C211yesmaximal, characteristic, complemented
3#10624114C211yesmaximal, characteristic, complemented
4C2×#4380—21no—
5C32×C5×D4C411yescharacteristic, complemented
6#67184—21no—
7C2×C3⋊Dic15—21no—
8C4×C3⋊D15C2211yescharacteristic
9#67180C411yescharacteristic, complemented
10#67127—21no—
11#63783—51nomaximal, Hall
12#6819—62no—
13C22×C32×C5—21no—
14C2×C3⋊D15D411yescharacteristic
15#4380—22no—
16C32×C4×C5C2×C411yescharacteristic
17C3⋊Dic15—21no—
18C3⋊Dic15D411yescharacteristic
19#4008—91nomaximal, Hall
20#3830—51no—
21C4×#157—51no—
22#3788—51no—
23C3×C5×D4—22no—
24#1220—64no—
25C2×Dic15—64no—
26C4×D15—62no—
27Dic30—62no—
28C2×C32×C5—41no—
29C2×C32×C5C22⋊C411yescharacteristic
30C3⋊D15—21no—
31#711—91no—
32C4×AGL(1,𝔽5)—91no—
33#700—91no—
34C2×#157—101no—
35C32×D4AGL(1,𝔽5)11yescharacteristic, complemented
36#638—101no—
37C2×C3⋊Dic3—101no—
38C4×C3⋊D3—51no—
39#634—51no—
40#622—101no—
41C22×C3×C5—24no—
42C2×D15—62no—
43C3×C4×C5—22no—
44Dic15—62no—
45Dic15—64no—
46#237—302no—
47C32×C5#10411yescharacteristic, complemented, Hall
48C2×AGL(1,𝔽5)—181no—
49C5×D4#15711yescharacteristic, complemented
50C22⋊D5—181no—
51C2×Dic5—181no—
52C4×D5—91no—
53Dic10—91no—
54#170—181no—
55C22×C32—21no—
56C2×C3⋊D3—51no—
57#157—102no—
58C32×C4C2×AGL(1,𝔽5)11yescharacteristic
59C3⋊Dic3—51no—
60C3⋊Dic3—101no—
61#104—451noSylow
62C2×C3×C5—22no—
63C2×C3×C5—44no—
64D15—122no—
65C3×D4—22no—
66#67—304no—
67C2×Dic3—304no—
68C4×D3—302no—
69Dic6—302no—
70C22×C5—21no—
71C2×D5—91no—
72AGL(1,𝔽5)—182no—
73C4×C5C2×#15711yescharacteristic
74Dic5—91no—
75Dic5—181no—
76C2×C32—41no—
77C2×C32C22⋊AGL(1,𝔽5)11yescharacteristic
78C3⋊D3—101no—
79#41—451no—
80M16—451no—
81C42—451no—
82C3×C5—22no—
83C22×C3—24no—
84C2×D3—302no—
85C3×C4—22no—
86Dic3—302no—
87Dic3—304no—
88C2×C5—41no—
89C2×C5C22⋊#15711yescharacteristic
90D5—181no—
91C32#400811yescharacteristic, complemented, Sylow
92Q8—451no—
93D4—901no—
94D4#438011yescharacteristic, complemented
95C2×C4—451no—
96C2×C4—901no—
97C2×C4—901no—
98C8—901no—
99C2×C3—22no—
100C2×C3—44no—
101D3—602no—
102C5#6378311yescharacteristic, complemented, Sylow
103C22—21no—
104C22—451no—
105C4—451no—
106C4—902no—
107C4—901no—
108C4C2×#438011yescharacteristic
109C3—22no—
110C2—41no—
111C2—901no—
112C2C22⋊#438011yescharacteristic
subgroupG/H
centerC2C22⋊#4380
commutatorC32×C4×C5C2×C4
FrattiniC4C2×#4380
FittingC32×C5×D4C4
socleC2×C32×C5C22⋊C4
prime radical#10624285C2
cosocleC4×C3⋊D15C22
hypercenterD4#4380
hypocenterC32×C5#104
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. C4×#4380C2semidirect
3. #10624114C2semidirect
5. C32×C5×D4C4semidirect
9. #67180C4semidirect
35. C32×D4AGL(1,𝔽5)semidirect
47. C32×C5#104semidirect
49. C5×D4#157semidirect
91. C32#4008semidirect
94. D4#4380semidirect
102. C5#63783semidirect

Maximal subgroups

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

subgroupindex
#106242852
C4×#43802
#106241142
#637835
#40089