Finite Group Databasebeta

← C2×C3×C5×D255 #29506079

Subgroups — C2×C3×C5×D255 #29506079

Order 15300.

224 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 76 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×C5×D255C212yesmaximal, direct product factor
2C2×C32×C52×C17C211yesmaximal, characteristic, complemented
3C2×C5×D255C311yesmaximal, characteristic, direct product factor
4C2×C3×C5×D85—31nomaximal
5C32×C52×C17C2211yescharacteristic, complemented, Hall
6C2×C3×C5×D51—51nomaximal
7C2×C3×D255C511yesmaximal, characteristic, direct product factor
8C5×D255C2×C312yesdirect product factor
9C3×C5×D85—32no—
10C2×C3×C52×C17—21no—
11C2×C3×C52×C17D311yescharacteristic, complemented
12C2×C3×C52×C17C2×C311yescharacteristic, complemented
13C2×C5×D85—31noHall
14C2×C32×C5×C17—22no—
15C2×C32×C5×C17D511yescharacteristic, complemented
16C2×C32×C5×C17C2×C511yescharacteristic, complemented
17C3×D255C2×C512yesdirect product factor
18C3×C5×D51—52no—
19C3×C52×C17—21no—
20C3×C52×C17C2×D311yescharacteristic, complemented
21C3×C52×C17C22×C311yescharacteristic, complemented
22C2×D255C3×C511yescharacteristic, direct product factor
23C2×C5×D51—51no—
24C2×C3×D85—31no—
25C2×C3×C5×D17—151no—
26C2×C3×C5×D15—171nomaximal, Hall
27C2×C52×C17C3×D311yescharacteristic, complemented
28C5×D85—32no—
29C32×C5×C17—22no—
30C32×C5×C17C2×D511yescharacteristic, complemented
31C32×C5×C17C22×C511yescharacteristic, complemented
32C2×C3×D51—51noHall
33C2×C3×C5×C17—24no—
34C2×C3×C5×C17—21no—
35C2×C3×C5×C17—21no—
36C2×C3×C5×C17—22no—
37C2×C3×C5×C17—22no—
38C2×C3×C5×C17C5×D311yescharacteristic, complemented
39C2×C3×C5×C17C3×D511yescharacteristic, complemented
40C2×C3×C5×C17D1511yescharacteristic, complemented
41C2×C3×C5×C17C2×C3×C511yescharacteristic, complemented
42D255C2×C3×C512yesdirect product factor
43C5×D51—52no—
44C3×D85—32no—
45C3×C5×D17—152no—
46C2×C32×C52D1711yescharacteristic, complemented
47C3×C5×D15—172no—
48C52×C17C2×C3×D311yescharacteristic, complemented, Hall
49C2×D85—31no—
50C2×C5×D17—151no—
51C2×C32×C17C5×D511yescharacteristic, complemented
52C3×D51—52no—
53C2×C5×D15—171no—
54C2×C3×C5×D5—511no—
55C3×C5×C17—24no—
56C3×C5×C17—21no—
57C3×C5×C17—21no—
58C3×C5×C17—22no—
59C3×C5×C17—22no—
60C3×C5×C17C2×C3×D511yescharacteristic, complemented
61C3×C5×C17C2×C5×D311yescharacteristic, complemented
62C3×C5×C17C2×D1511yescharacteristic, complemented
63C3×C5×C17C22×C3×C511yescharacteristic, complemented
64C32×C52C2×D1711yescharacteristic, complemented, Hall
65C2×D51—51no—
66C2×C3×D17—151no—
67C2×C3×D15—171no—
68C2×C3×C5×D3—851no—
69C2×C5×C17—22no—
70C2×C5×C17C3×C5×D311yescharacteristic, complemented
71C2×C5×C17C3×D1511yescharacteristic, complemented
72D85—32no—
73C5×D17—152no—
74C32×C17C2×C5×D511yescharacteristic, complemented, Hall
75C2×C3×C52—21no—
76C2×C3×C52C3×D1711yescharacteristic, complemented
77C2×C3×C52D5111yescharacteristic, complemented
78C5×D15—172no—
79C3×C5×D5—512no—
80C2×C3×C17—21no—
81C2×C3×C17C3×C5×D511yescharacteristic, complemented
82C2×C3×C17C5×D1511yescharacteristic, complemented
83D51—52no—
84C3×D17—152no—
85C2×C5×D5—511noHall
86C2×C32×C5—22no—
87C2×C32×C5C5×D1711yescharacteristic, complemented
88C2×C32×C5D8511yescharacteristic, complemented
89C3×D15—172no—
90C3×C5×D3—852no—
91C5×C17—22no—
92C5×C17C2×C3×C5×D311yescharacteristic, complemented
93C5×C17C2×C3×D1511yescharacteristic, complemented
94C3×C52—21no—
95C3×C52C2×C3×D1711yescharacteristic, complemented
96C3×C52C2×D5111yescharacteristic, complemented
97C2×D17—151noHall
98C22×C3×C5—2551no—
99C2×D15—171no—
100C2×C5×D3—851no—
101C2×C3×D5—511no—
102C3×C17—21no—
103C3×C17C2×C3×C5×D511yescharacteristic, complemented
104C3×C17C2×C5×D1511yescharacteristic, complemented
105C2×C52C3×D5111yescharacteristic, complemented
106C5×D5—512no—
107C32×C5—22no—
108C32×C5C2×C5×D1711yescharacteristic, complemented
109C32×C5C2×D8511yescharacteristic, complemented
110C2×C3×D3—851noHall
111C2×C17C3×C5×D1511yescharacteristic, complemented
112D17—152no—
113C2×C3×C5—24no—
114C2×C3×C5—21no—
115C2×C3×C5—21no—
116C2×C3×C5—22no—
117C2×C3×C5—22no—
118C2×C3×C5—2552no—
119C2×C3×C5C3×C5×D1711yescharacteristic, complemented
120C2×C3×C5C3×D8511yescharacteristic, complemented
121C2×C3×C5C5×D5111yescharacteristic, complemented
122C2×C3×C5D25511yescharacteristic, direct product factor
123D15—172no—
124C3×D5—512no—
125C5×D3—852no—
126C52C2×C3×D5111yescharacteristic, complemented, Sylow
127C22×C5—2551no—
128C2×D5—511no—
129C2×C32C5×D8511yescharacteristic, complemented
130C3×D3—852no—
131C17C2×C3×C5×D1511yescharacteristic, complemented, Sylow
132C3×C5—24no—
133C3×C5—21no—
134C3×C5—21no—
135C3×C5—22no—
136C3×C5—22no—
137C3×C5C2×C3×C5×D1711yescharacteristic, complemented
138C3×C5C2×C3×D8511yescharacteristic, complemented
139C3×C5C2×C5×D5111yescharacteristic, complemented
140C3×C5C2×D25511yescharacteristic, direct product factor
141C22×C3—2551no—
142C2×D3—851no—
143C2×C5—22no—
144C2×C5—2552no—
145C2×C5C3×C5×D5111yescharacteristic, complemented
146C2×C5C3×D25511yescharacteristic, direct product factor
147D5—512no—
148C32C2×C5×D8511yescharacteristic, complemented, Sylow
149C2×C3—21no—
150C2×C3—2552no—
151C2×C3C3×C5×D8511yescharacteristic, complemented
152C2×C3C5×D25511yescharacteristic, direct product factor
153D3—852no—
154C5—22no—
155C5C2×C3×D25511yescharacteristic, direct product factor
156C5C2×C3×C5×D5111yescharacteristic, complemented
157C22—2551noSylow
158C3—21no—
159C3C2×C3×C5×D8511yescharacteristic, complemented
160C3C2×C5×D25511yescharacteristic, direct product factor
161C2—2552no—
162C2C3×C5×D25511yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5D255
commutator, cosocle, hypocenterC3×C5×C17C22×C3×C5
FrattiniC1G
Fitting, socleC2×C32×C52×C17C2
LCSG ▷ C3×C5×C17
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
1. C3×C5×D255C2direct (N × H)
2. C2×C32×C52×C17C2semidirect
3. C2×C5×D255C3direct (N × H)
5. C32×C52×C17C22semidirect
7. C2×C3×D255C5direct (N × H)
8. C5×D255C2×C3direct (N × H)
11. C2×C3×C52×C17D3semidirect
12. C2×C3×C52×C17C2×C3semidirect
15. C2×C32×C5×C17D5semidirect
16. C2×C32×C5×C17C2×C5semidirect
17. C3×D255C2×C5direct (N × H)
20. C3×C52×C17C2×D3semidirect
21. C3×C52×C17C22×C3semidirect
22. C2×D255C3×C5direct (N × H)
27. C2×C52×C17C3×D3semidirect
30. C32×C5×C17C2×D5semidirect
31. C32×C5×C17C22×C5semidirect
38. C2×C3×C5×C17C5×D3semidirect
39. C2×C3×C5×C17C3×D5semidirect
40. C2×C3×C5×C17D15semidirect
41. C2×C3×C5×C17C2×C3×C5semidirect
42. D255C2×C3×C5direct (N × H)
46. C2×C32×C52D17semidirect
48. C52×C17C2×C3×D3semidirect
51. C2×C32×C17C5×D5semidirect
60. C3×C5×C17C2×C3×D5semidirect
61. C3×C5×C17C2×C5×D3semidirect
62. C3×C5×C17C2×D15semidirect
63. C3×C5×C17C22×C3×C5semidirect
64. C32×C52C2×D17semidirect
70. C2×C5×C17C3×C5×D3semidirect
71. C2×C5×C17C3×D15semidirect
74. C32×C17C2×C5×D5semidirect
76. C2×C3×C52C3×D17semidirect
77. C2×C3×C52D51semidirect
81. C2×C3×C17C3×C5×D5semidirect
82. C2×C3×C17C5×D15semidirect
87. C2×C32×C5C5×D17semidirect
88. C2×C32×C5D85semidirect
92. C5×C17C2×C3×C5×D3semidirect
93. C5×C17C2×C3×D15semidirect
95. C3×C52C2×C3×D17semidirect
96. C3×C52C2×D51semidirect
103. C3×C17C2×C3×C5×D5semidirect
104. C3×C17C2×C5×D15semidirect
105. C2×C52C3×D51semidirect
108. C32×C5C2×C5×D17semidirect
109. C32×C5C2×D85semidirect
111. C2×C17C3×C5×D15semidirect
119. C2×C3×C5C3×C5×D17semidirect
120. C2×C3×C5C3×D85semidirect
121. C2×C3×C5C5×D51semidirect
122. C2×C3×C5D255direct (N × H)
126. C52C2×C3×D51semidirect
129. C2×C32C5×D85semidirect
131. C17C2×C3×C5×D15semidirect
137. C3×C5C2×C3×C5×D17semidirect
138. C3×C5C2×C3×D85semidirect
139. C3×C5C2×C5×D51semidirect
140. C3×C5C2×D255direct (N × H)
145. C2×C5C3×C5×D51semidirect
146. C2×C5C3×D255direct (N × H)
148. C32C2×C5×D85semidirect
151. C2×C3C3×C5×D85semidirect
152. C2×C3C5×D255direct (N × H)
155. C5C2×C3×D255direct (N × H)
156. C5C2×C3×C5×D51semidirect
159. C3C2×C3×C5×D85semidirect
160. C3C2×C5×D255direct (N × H)
162. C2C3×C5×D255direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×D2552
C2×C32×C52×C172
C2×C5×D2553
C2×C3×C5×D853
C2×C3×C5×D515
C2×C3×D2555
C2×C3×C5×D1517