Finite Group Databasebeta

← C2×C3×C5×#11925890 #49137626

Subgroups — C2×C3×C5×#11925890 #49137626

Order 36300.

168 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 36 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×#10597245C211yesmaximal, characteristic, complemented
2C3×C5×#11925890C212yesmaximal, direct product factor
3C2×C5×#11925890C311yesmaximal, characteristic, direct product factor, Hall
4C3×C5×#10597245C2211yescharacteristic, complemented, Hall
5C2×C3×C5×C11×D11—51nomaximal
6C2×C3×#11925890C511yesmaximal, characteristic, direct product factor
7C2×C5×#10597245C2×C311yescharacteristic, complemented
8C5×#11925890C2×C312yesdirect product factor
9C3×C5×C11×D11—52no—
10C2×C3×#10597245—22no—
11C2×C3×#10597245C2×C511yescharacteristic, complemented
12C3×#11925890C2×C512yesdirect product factor
13C2×C3×C5×C112D511yescharacteristic, complemented
14C5×#10597245C22×C311yescharacteristic, complemented, Hall
15C2×C5×C11×D11—51no—
16C2×#11925890C3×C511yescharacteristic, direct product factor
17C3×C5×C112C2×D511yescharacteristic, complemented
18C3×#10597245—22no—
19C3×#10597245C22×C511yescharacteristic, complemented
20C2×C3×C5×C11⋊C5—221no—
21C2×C3×C11×D11—51noHall
22C2×C5×C112C3×D511yescharacteristic, complemented
23C5×C11×D11—52no—
24C2×#10597245—22no—
25C2×#10597245C2×C3×C511yescharacteristic, complemented
26#11925890C2×C3×C512yesdirect product factor
27C3×C5×C11⋊C5—221no—
28C2×C3×C112C5×D511yescharacteristic, complemented
29C3×C11×D11—52no—
30C22×C3×C5×C11—551no—
31C2×C3×C5×D11—51no—
32C5×C112C2×C3×D511yescharacteristic, complemented
33#10597245—22no—
34#10597245C22×C3×C511yescharacteristic, complemented
35C2×C5×C11⋊C5—221no—
36C2×C11×D11—51noHall
37C3×C112C2×C5×D511yescharacteristic, complemented, Hall
38C2×C3×C5×C11—21no—
39C2×C3×C5×C11—101no—
40C2×C3×C5×C11—101no—
41C2×C3×C5×C11—552no—
42C3×C5×D11—52no—
43C2×C3×C11⋊C5—224no—
44C2×C3×C11⋊C5—221no—
45C2×C3×C5×D5—1211nomaximal, Hall
46C5×C11⋊C5—221no—
47C2×C112C3×C5×D511yescharacteristic, complemented
48C11×D11—52no—
49C22×C5×C11—551no—
50C2×C5×D11—51no—
51C3×C5×C11—21no—
52C3×C5×C11—101no—
53C3×C5×C11—101no—
54C3×C11⋊C5—224no—
55C3×C11⋊C5—221no—
56C2×C3×C52—2421no—
57C3×C5×D5—1212no—
58C22×C3×C11—551no—
59C2×C3×D11—51no—
60C112C2×C3×C5×D511yescharacteristic, complemented, Sylow
61C2×C5×C11—21no—
62C2×C5×C11—101no—
63C2×C5×C11—101no—
64C2×C5×C11—552no—
65C5×D11—52no—
66C2×C11⋊C5—224no—
67C2×C11⋊C5—221no—
68C2×C5×D5—1211noHall
69C3×C52—2421noHall
70C2×C3×C11—21no—
71C2×C3×C11—101no—
72C2×C3×C11—101no—
73C2×C3×C11—552no—
74C3×D11—52no—
75C22×C3×C5—551no—
76C2×C3×D5—1211no—
77C5×C11—21no—
78C5×C11—101no—
79C5×C11—101no—
80C11⋊C5—224no—
81C11⋊C5—221no—
82C2×C52—2421no—
83C5×D5—1212no—
84C22×C11—551no—
85C2×D11—51no—
86C3×C11—21no—
87C3×C11—101no—
88C3×C11—101no—
89C2×C3×C5—552no—
90C2×C3×C5—2421no—
91C2×C3×C5—2422no—
92C2×C3×C5#1192589011yescharacteristic, direct product factor
93C3×D5—1212no—
94C52—2421noSylow
95C2×C11—21no—
96C2×C11—101no—
97C2×C11—101no—
98C2×C11—552no—
99D11—52no—
100C22×C5—551no—
101C2×D5—1211no—
102C3×C5—2421no—
103C3×C5—2422no—
104C3×C5C2×#1192589011yescharacteristic, direct product factor
105C22×C3—551noHall
106C11—21no—
107C11—101no—
108C11—101no—
109C2×C5—552no—
110C2×C5—2421no—
111C2×C5—2422no—
112C2×C5C3×#1192589011yescharacteristic, direct product factor
113D5—1212no—
114C2×C3—552no—
115C2×C3C5×#1192589011yescharacteristic, direct product factor
116C5—2421no—
117C5—2422no—
118C5C2×C3×#1192589011yescharacteristic, direct product factor
119C22—551noSylow
120C3C2×C5×#1192589011yescharacteristic, direct product factor, Sylow
121C2—552no—
122C2C3×C5×#1192589011yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5#11925890
commutator, cosocle, hypocenter#10597245C22×C3×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C112D5
LCSG ▷ #10597245
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#10597245C2semidirect
2. C3×C5×#11925890C2direct (N × H)
3. C2×C5×#11925890C3direct (N × H)
4. C3×C5×#10597245C22semidirect
6. C2×C3×#11925890C5direct (N × H)
7. C2×C5×#10597245C2×C3semidirect
8. C5×#11925890C2×C3direct (N × H)
11. C2×C3×#10597245C2×C5semidirect
12. C3×#11925890C2×C5direct (N × H)
13. C2×C3×C5×C112D5semidirect
14. C5×#10597245C22×C3semidirect
16. C2×#11925890C3×C5direct (N × H)
17. C3×C5×C112C2×D5semidirect
19. C3×#10597245C22×C5semidirect
22. C2×C5×C112C3×D5semidirect
25. C2×#10597245C2×C3×C5semidirect
26. #11925890C2×C3×C5direct (N × H)
28. C2×C3×C112C5×D5semidirect
32. C5×C112C2×C3×D5semidirect
34. #10597245C22×C3×C5semidirect
37. C3×C112C2×C5×D5semidirect
47. C2×C112C3×C5×D5semidirect
60. C112C2×C3×C5×D5semidirect
92. C2×C3×C5#11925890direct (N × H)
104. C3×C5C2×#11925890direct (N × H)
112. C2×C5C3×#11925890direct (N × H)
115. C2×C3C5×#11925890direct (N × H)
118. C5C2×C3×#11925890direct (N × H)
120. C3C2×C5×#11925890direct (N × H)
122. C2C3×C5×#11925890direct (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×#105972452
C3×C5×#119258902
C2×C5×#119258903
C2×C3×C5×C11×D115
C2×C3×#119258905
C2×C3×C5×D5121