Finite Group Databasebeta

← C3×C5×#711 #11925680

Subgroups — C3×C5×#711 #11925680

Order 1200.

192 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C3×C52×D4C211yesmaximal, characteristic, complemented
2C3×C5×C22⋊D5C212yesmaximal, complemented
3C2×C3×C5×Dic5C212yesmaximal, complemented
4C3×C4×C5×D5C211yesmaximal, characteristic, complemented
5C3×C5×Dic10C211yesmaximal, characteristic, complemented
6C5×#711C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C52C2212yes—
8C2×C3×C5×D5C2211yescharacteristic
9C3×C4×C52C2211yescharacteristic
10C3×C5×Dic5C2212yes—
11C3×C5×Dic5C2211yescharacteristic
12C3×C5×#41—51nomaximal
13C3×#711C511yesmaximal, characteristic, direct product factor
14C52×D4C2×C311yescharacteristic, complemented
15C5×C22⋊D5C2×C312yescomplemented
16C2×C5×Dic5C2×C312yescomplemented
17C4×C5×D5C2×C311yescharacteristic, complemented
18C5×Dic10C2×C311yescharacteristic, complemented
19C2×C3×C52—22no—
20C2×C3×C52C2311yescharacteristic
21C3×C5×D5—21no—
22C3×C5×Q8—51no—
23C3×C5×D4—22no—
24C3×C5×D4—52no—
25C3×C5×D4D511yescharacteristic, complemented
26C3×C5×D4C2×C511yescharacteristic, complemented
27C2×C3×C4×C5—52no—
28C2×C3×C4×C5—51no—
29C3×C22⋊D5C2×C512yescomplemented
30C2×C3×Dic5C2×C512yescomplemented
31C3×C4×D5C2×C511yescharacteristic, complemented
32C3×Dic10C2×C511yescharacteristic, complemented
33C22×C52C22×C312yes—
34C2×C5×D5C22×C311yescharacteristic
35C4×C52C22×C311yescharacteristic
36C5×Dic5C22×C312yes—
37C5×Dic5C22×C311yescharacteristic
38C5×#41—51no—
39#711C3×C511yescharacteristic, direct product factor
40C3×C52#4111yescharacteristic, complemented, Hall
41C22×C3×C5—24no—
42C22×C3×C5—51no—
43C22×C3×C5C2×D512yes—
44C22×C3×C5C22×C512yes—
45C2×C3×D5C22×C511yescharacteristic
46C3×C4×C5—22no—
47C3×C4×C5—51no—
48C3×C4×C5—52no—
49C3×C4×C5C2×D511yescharacteristic
50C3×C4×C5C22×C511yescharacteristic
51C3×Dic5C22×C512yes—
52C3×Dic5C22×C511yescharacteristic
53C2×C52—22no—
54C2×C52C23×C311yescharacteristic
55C5×D5—21no—
56C3×#41—51noHall
57C5×Q8—51no—
58C5×D4—22no—
59C5×D4—52no—
60C5×D4C3×D511yescharacteristic, complemented
61C5×D4C2×C3×C511yescharacteristic, complemented
62C2×C4×C5—52no—
63C2×C4×C5—51no—
64C22⋊D5C2×C3×C512yescomplemented
65C2×Dic5C2×C3×C512yescomplemented
66C4×D5C2×C3×C511yescharacteristic, complemented
67Dic10C2×C3×C511yescharacteristic, complemented
68C2×C3×C5—22no—
69C2×C3×C5—22no—
70C2×C3×C5—22no—
71C2×C3×C5—44no—
72C2×C3×C5—101no—
73C2×C3×C5C22×D511yescharacteristic
74C2×C3×C5C23×C511yescharacteristic
75C3×D5—21no—
76C52C3×#4111yescharacteristic, complemented, Sylow
77C3×Q8—51no—
78C3×D4—52no—
79C3×D4C5×D511yescharacteristic, complemented
80C2×C3×C4—52no—
81C2×C3×C4—51no—
82C22×C5—24no—
83C22×C5—51no—
84C22×C5C2×C3×D512yes—
85C22×C5C22×C3×C512yes—
86C2×D5C22×C3×C511yescharacteristic
87C4×C5—22no—
88C4×C5—51no—
89C4×C5—52no—
90C4×C5C2×C3×D511yescharacteristic
91C4×C5C22×C3×C511yescharacteristic
92Dic5C22×C3×C512yes—
93Dic5C22×C3×C511yescharacteristic
94#41—51noSylow
95C3×C5—22no—
96C3×C5#71111yescharacteristic, direct product factor
97C3×C5C5×#4111yescharacteristic, complemented
98C22×C3—51no—
99C22×C3C2×C5×D512yes—
100C3×C4—51no—
101C3×C4—52no—
102C3×C4C2×C5×D511yescharacteristic
103C2×C5—22no—
104C2×C5—22no—
105C2×C5—22no—
106C2×C5—44no—
107C2×C5—101no—
108C2×C5C22×C3×D511yescharacteristic
109C2×C5C23×C3×C511yescharacteristic
110D5—21no—
111Q8—51no—
112D4—52no—
113D4C3×C5×D511yescharacteristic, complemented
114C2×C4—52no—
115C2×C4—51no—
116C2×C3—22no—
117C2×C3—101no—
118C2×C3C22×C5×D511yescharacteristic
119C5—22no—
120C5C3×#71111yescharacteristic, direct product factor
121C5C3×C5×#4111yescharacteristic, complemented
122C22—51no—
123C22C2×C3×C5×D512yes—
124C4—51no—
125C4—52no—
126C4C2×C3×C5×D511yescharacteristic
127C3C5×#71111yescharacteristic, direct product factor, Sylow
128C2—22no—
129C2—101no—
130C2C22×C3×C5×D511yescharacteristic
subgroupG/H
centerC2×C3×C5C22×D5
commutator, cosocleC2×C5C23×C3×C5
FrattiniC2C22×C3×C5×D5
FittingC3×C52×D4C2
socleC2×C3×C52C23
hypercenterC3×C5×D4D5
hypocenterC5C3×C5×#41
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3×C5 ◁ C3×C5×D4

Complements

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

NH ≅ G/Ntype
1. C3×C52×D4C2semidirect
2. C3×C5×C22⋊D5C2semidirect
3. C2×C3×C5×Dic5C2semidirect
4. C3×C4×C5×D5C2semidirect
5. C3×C5×Dic10C2semidirect
6. C5×#711C3direct (N × H)
13. C3×#711C5direct (N × H)
14. C52×D4C2×C3semidirect
15. C5×C22⋊D5C2×C3semidirect
16. C2×C5×Dic5C2×C3semidirect
17. C4×C5×D5C2×C3semidirect
18. C5×Dic10C2×C3semidirect
25. C3×C5×D4D5semidirect
26. C3×C5×D4C2×C5semidirect
29. C3×C22⋊D5C2×C5semidirect
30. C2×C3×Dic5C2×C5semidirect
31. C3×C4×D5C2×C5semidirect
32. C3×Dic10C2×C5semidirect
39. #711C3×C5direct (N × H)
40. C3×C52#41semidirect
60. C5×D4C3×D5semidirect
61. C5×D4C2×C3×C5semidirect
64. C22⋊D5C2×C3×C5semidirect
65. C2×Dic5C2×C3×C5semidirect
66. C4×D5C2×C3×C5semidirect
67. Dic10C2×C3×C5semidirect
76. C52C3×#41semidirect
79. C3×D4C5×D5semidirect
96. C3×C5#711direct (N × H)
97. C3×C5C5×#41semidirect
113. D4C3×C5×D5semidirect
120. C5C3×#711direct (N × H)
121. C5C3×C5×#41semidirect
127. C3C5×#711direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C52×D42
C3×C5×C22⋊D52
C2×C3×C5×Dic52
C3×C4×C5×D52
C3×C5×Dic102
C5×#7113
C3×C5×#415
C3×#7115