Finite Group Databasebeta

← C5×D5×D12 #11925577

Subgroups — C5×D5×D12 #11925577

Order 1200.

248 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 72 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×C5×D3×D5C212yesmaximal, complemented
2C5×D60C211yesmaximal, characteristic, complemented
3C52×D12C211yesmaximal, characteristic, complemented
4C3×C4×C5×D5C211yesmaximal, characteristic, complemented
5C5×#1203C212yesmaximal, complemented
6C5×D4×D5—31nomaximal, Hall
7C2×C5×D15C2212yescomplemented
8C2×C52×D3C2212yescomplemented
9C2×C3×C5×D5C2211yescharacteristic
10C5×D3×D5—24no—
11C3×C4×C52C2211yescharacteristic, complemented
12C3×C5×Dic5C2211yescharacteristic, complemented
13C2×C5×D12—51nomaximal
14D5×D12C511yesmaximal, characteristic, direct product factor
15C22×C5×D5—32no—
16C52×D4—31no—
17C5×C22⋊D5—32no—
18C5×D20—31no—
19C4×C5×D5D311yescharacteristic, complemented
20C2×C3×C52C2311yescharacteristic
21C5×D15—22no—
22C52×D3—22no—
23C3×C5×D5D412yescomplemented
24C22×C5×D3—52no—
25C2×D3×D5C2×C512yescomplemented
26C2×C3×C4×C5—51no—
27D60C2×C511yescharacteristic, complemented
28C5×D12—22no—
29C5×D12—52no—
30C5×D12—51no—
31C5×D12D511yescharacteristic, direct product factor
32C5×D12C2×C511yescharacteristic, complemented
33C3×C4×D5C2×C511yescharacteristic, complemented
34#1203C2×C512yescomplemented
35C22×C52—32no—
36C2×C5×D5—32no—
37C2×C5×D5—64no—
38C2×C5×D5C2×D311yescharacteristic
39C4×C52C2×D311yescharacteristic, complemented
40C5×Dic5C2×D311yescharacteristic, complemented
41C2×C5×D4—151no—
42D4×D5—31no—
43C3×C52C2×D411yescharacteristic, complemented, Hall
44C22×C3×C5—51no—
45C2×D15C22×C512yescomplemented
46C2×C5×D3—24no—
47C2×C5×D3—52no—
48C2×C5×D3—104no—
49C2×C5×D3C2×D512yescomplemented
50C2×C5×D3C22×C512yescomplemented
51C2×C3×D5C22×C511yescharacteristic
52D3×D5—24no—
53C3×C4×C5—22no—
54C3×C4×C5—51no—
55C3×C4×C5C2×D511yescharacteristic, complemented
56C3×C4×C5C22×C511yescharacteristic, complemented
57C3×Dic5C22×C511yescharacteristic, complemented
58C2×C52—62no—
59C2×C52C22×D311yescharacteristic
60C5×D5—62no—
61C5×D5D1212yesdirect product factor
62C2×D12—51noHall
63C23×C5—152no—
64C22×D5—32no—
65C5×D4—31no—
66C5×D4—31no—
67C5×D4—62no—
68C5×D4—152no—
69C5×D4—151no—
70C2×C4×C5—51no—
71C22⋊D5—32no—
72D20—31no—
73C4×D5C5×D311yescharacteristic, complemented
74C2×C3×C5—22no—
75C2×C3×C5—52no—
76C2×C3×C5C22×D511yescharacteristic
77C2×C3×C5C23×C511yescharacteristic
78D15—22no—
79C3×D5C5×D412yescomplemented
80C5×D3—22no—
81C5×D3—22no—
82C5×D3—44no—
83C5×D3—102no—
84C52C2×D1211yescharacteristic, complemented, Sylow
85C22×D3—52no—
86C2×C3×C4—51no—
87D12—52no—
88D12—51no—
89D12C5×D511yescharacteristic, direct product factor
90C22×C5—32no—
91C22×C5—32no—
92C22×C5—51no—
93C22×C5—64no—
94C22×C5—152no—
95C22×C5—304no—
96C2×D5—32no—
97C2×D5—64no—
98C2×D5C2×C5×D311yescharacteristic
99C4×C5—22no—
100C4×C5—51no—
101C4×C5D3×D511yescharacteristic, complemented
102C4×C5C2×C5×D311yescharacteristic, complemented
103Dic5C2×C5×D311yescharacteristic, complemented
104C2×D4—151noSylow
105C3×C5—22no—
106C3×C5D4×D511yescharacteristic, complemented
107C3×C5C2×C5×D411yescharacteristic, complemented
108C22×C3—51no—
109C2×D3—52no—
110C2×D3—104no—
111C2×D3C2×C5×D512yescomplemented
112C3×C4—51no—
113C3×C4C2×C5×D511yescharacteristic, complemented
114C2×C5—22no—
115C2×C5—52no—
116C2×C5—62no—
117C2×C5—62no—
118C2×C5—124no—
119C2×C5—302no—
120C2×C5C2×D3×D511yescharacteristic
121C2×C5C22×C5×D311yescharacteristic
122D5—62no—
123D5C5×D1212yesdirect product factor
124C23—152no—
125D4—31no—
126D4—152no—
127D4—151no—
128C2×C4—51no—
129C2×C3—52no—
130C2×C3C22×C5×D511yescharacteristic
131D3—22no—
132D3—102no—
133C5—22no—
134C5D5×D1211yescharacteristic, direct product factor
135C5C2×C5×D1211yescharacteristic, complemented
136C22—32no—
137C22—51no—
138C22—152no—
139C22—304no—
140C4—51no—
141C4C5×D3×D511yescharacteristic, complemented
142C3C5×D4×D511yescharacteristic, complemented, Sylow
143C2—52no—
144C2—62no—
145C2—302no—
146C2C2×C5×D3×D511yescharacteristic
subgroupG/H
centerC2×C5C2×D3×D5
commutator, cosocleC2×C3×C5C23×C5
FrattiniC2C2×C5×D3×D5
FittingC3×C4×C52C22
socleC2×C3×C52C23
hypercenterC4×C5D3×D5
hypocenterC3×C5C2×C5×D4
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C5 ◁ C4×C5

Complements

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

NH ≅ G/Ntype
1. C2×C5×D3×D5C2semidirect
2. C5×D60C2semidirect
3. C52×D12C2semidirect
4. C3×C4×C5×D5C2semidirect
5. C5×#1203C2semidirect
7. C2×C5×D15C22semidirect
8. C2×C52×D3C22semidirect
11. C3×C4×C52C22semidirect
12. C3×C5×Dic5C22semidirect
14. D5×D12C5direct (N × H)
19. C4×C5×D5D3semidirect
23. C3×C5×D5D4semidirect
25. C2×D3×D5C2×C5semidirect
27. D60C2×C5semidirect
31. C5×D12D5direct (N × H)
32. C5×D12C2×C5semidirect
33. C3×C4×D5C2×C5semidirect
34. #1203C2×C5semidirect
39. C4×C52C2×D3semidirect
40. C5×Dic5C2×D3semidirect
43. C3×C52C2×D4semidirect
45. C2×D15C22×C5semidirect
49. C2×C5×D3C2×D5semidirect
50. C2×C5×D3C22×C5semidirect
55. C3×C4×C5C2×D5semidirect
56. C3×C4×C5C22×C5semidirect
57. C3×Dic5C22×C5semidirect
61. C5×D5D12direct (N × H)
73. C4×D5C5×D3semidirect
79. C3×D5C5×D4semidirect
84. C52C2×D12semidirect
89. D12C5×D5direct (N × H)
101. C4×C5D3×D5semidirect
102. C4×C5C2×C5×D3semidirect
103. Dic5C2×C5×D3semidirect
106. C3×C5D4×D5semidirect
107. C3×C5C2×C5×D4semidirect
111. C2×D3C2×C5×D5semidirect
113. C3×C4C2×C5×D5semidirect
123. D5C5×D12direct (N × H)
134. C5D5×D12direct (N × H)
135. C5C2×C5×D12semidirect
141. C4C5×D3×D5semidirect
142. C3C5×D4×D5semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D3×D52
C5×D602
C52×D122
C3×C4×C5×D52
C5×#12032
C5×D4×D53
C2×C5×D125
D5×D125