Finite Group Databasebeta

← C2×C5×C5⋊(#4294⋊C2) #30685560

Subgroups — C2×C5×C5⋊(#4294⋊C2) #30685560

Order 17100.

144 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 58 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×C52×#4294C211yesmaximal, characteristic, complemented
2C5×C5⋊(#4294⋊C2)C211yesmaximal, characteristic, direct product factor
3C5×C5⋊(#4294⋊C2)C211yesmaximal, characteristic, direct product factor
4C2×C3×C5×D95C311yesmaximal, characteristic
5C52×#4294C2211yescharacteristic, complemented, Hall
6C2×C5⋊(#4294⋊C2)C511yesmaximal, characteristic, direct product factor
7C2×C5×#4294⋊C2—51nomaximal
8C3×C5×D95C2×C311yescharacteristic
9C3×C5×D95C2×C311yescharacteristic
10C2×C3×C52×C19C2×C311yescharacteristic
11C2×C5×D95C911yescharacteristic, complemented, Hall
12C2×C5×#4294—22no—
13C2×C5×#4294D511yescharacteristic, complemented
14C2×C5×#4294C2×C511yescharacteristic, complemented
15C5⋊(#4294⋊C2)C2×C511yescharacteristic, direct product factor
16C5⋊(#4294⋊C2)C2×C511yescharacteristic, direct product factor
17C5×#4294⋊C2—51no—
18C5×#4294⋊C2—51no—
19C3×C52×C19C22×C311yescharacteristic
20C2×C3×D95C3×C511yescharacteristic
21C2×C3×C5×D19—51no—
22C2×C52×C19C2×C911yescharacteristic, complemented
23C5×D95C2×C911yescharacteristic, complemented
24C5×D95C2×C911yescharacteristic, complemented
25C2×C5×C9×D5—191nomaximal, Hall
26C5×#4294—22no—
27C5×#4294C2×D511yescharacteristic, complemented
28C5×#4294C22×C511yescharacteristic, complemented
29C2×#4294⋊C2—51noHall
30C2×C3×C5×C19—22no—
31C2×C3×C5×C19C3×D511yescharacteristic
32C2×C3×C5×C19C2×C3×C511yescharacteristic
33C3×D95C2×C3×C511yescharacteristic
34C3×D95C2×C3×C511yescharacteristic
35C3×C5×D19—51no—
36C3×C5×D19—51no—
37C52×C19C22×C911yescharacteristic, complemented, Hall
38C2×C52×C9—191no—
39C5×C9×D5—191no—
40C5×C9×D5—191no—
41C2×D95C5×C911yescharacteristic, complemented
42C2×C5×D19—51no—
43C2×#4294C5×D511yescharacteristic, complemented
44#4294⋊C2—51no—
45#4294⋊C2—51no—
46C2×C3×C5×D5—571no—
47C3×C5×C19—22no—
48C3×C5×C19C2×C3×D511yescharacteristic
49C3×C5×C19C22×C3×C511yescharacteristic
50C2×C3×D19—51no—
51C52×C9—191noHall
52C2×C5×C19—22no—
53C2×C5×C19C9×D511yescharacteristic, complemented
54C2×C5×C19C2×C5×C911yescharacteristic, complemented
55D95C2×C5×C911yescharacteristic, complemented
56D95C2×C5×C911yescharacteristic, complemented
57C5×D19—51no—
58C5×D19—51no—
59C22×C5×C9—951no—
60C2×C9×D5—191no—
61#4294C2×C5×D511yescharacteristic, complemented, Hall
62C2×C3×C52D19⋊C311yescharacteristic
63C3×C5×D5—571no—
64C3×C5×D5—571no—
65C2×C3×C19C3×C5×D511yescharacteristic
66C3×D19—51no—
67C3×D19—51no—
68C2×C5×D5—571noHall
69C5×C19—22no—
70C5×C19C2×C9×D511yescharacteristic, complemented
71C5×C19C22×C5×C911yescharacteristic, complemented
72C2×C5×C9—191no—
73C2×C5×C9—191no—
74C2×C5×C9—382no—
75C2×C5×C9—951no—
76C2×C5×C9—951no—
77C9×D5—191no—
78C9×D5—191no—
79C2×D19—51noHall
80C3×C52C2×D19⋊C311yescharacteristic
81C22×C3×C5—2851no—
82C2×C3×D5—571no—
83C3×C19C2×C3×C5×D511yescharacteristic
84C2×C52#4294⋊C211yescharacteristic, complemented
85C5×D5—571no—
86C5×D5—571no—
87C5×C9—191no—
88C5×C9—191no—
89C5×C9—382no—
90C2×C19C5×C9×D511yescharacteristic, complemented
91D19—51no—
92D19—51no—
93C22×C9—951noHall
94C2×C3×C5—22no—
95C2×C3×C5—2851no—
96C2×C3×C5—2851no—
97C2×C3×C5C5×D19⋊C311yescharacteristic
98C2×C3×C5C5⋊(D19⋊C3)11yescharacteristic
99C3×D5—191no—
100C3×D5—571no—
101C52C2×#4294⋊C211yescharacteristic, complemented, Sylow
102C22×C5—2851no—
103C2×D5—571no—
104C19C2×C5×C9×D511yescharacteristic, complemented, Sylow
105C2×C9—191no—
106C2×C9—951no—
107C2×C9—951no—
108C3×C5—22no—
109C3×C5C2×C5×D19⋊C311yescharacteristic
110C3×C5C2×C5⋊(D19⋊C3)11yescharacteristic
111C22×C3—2851no—
112C2×C5—22no—
113C2×C5—2851no—
114C2×C5—2851no—
115C2×C5C5×#4294⋊C211yescharacteristic, complemented
116C2×C5C5⋊(#4294⋊C2)11yescharacteristic, direct product factor
117D5—191no—
118D5—571no—
119C9—191noSylow
120C2×C3—2851no—
121C2×C3—2851no—
122C2×C3C5×C5⋊(D19⋊C3)11yescharacteristic
123C5—22no—
124C5C2×C5×#4294⋊C211yescharacteristic, complemented
125C5C2×C5⋊(#4294⋊C2)11yescharacteristic, direct product factor
126C22—2851noSylow
127C3C2×C5×C5⋊(D19⋊C3)11yescharacteristic
128C2—2851no—
129C2—2851no—
130C2C5×C5⋊(#4294⋊C2)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5C5⋊(D19⋊C3)
commutator, hypocenterC5×C19C22×C5×C9
FrattiniC3C2×C5×C5⋊(D19⋊C3)
Fitting, socleC2×C3×C52×C19C2×C3
prime radicalC2×C3×C5×D95C3
cosocleC3×C5×C19C22×C3×C5
LCSG ▷ C5×C19
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
1. C2×C52×#4294C2semidirect
2. C5×C5⋊(#4294⋊C2)C2direct (N × H)
3. C5×C5⋊(#4294⋊C2)C2direct (N × H)
5. C52×#4294C22semidirect
6. C2×C5⋊(#4294⋊C2)C5direct (N × H)
11. C2×C5×D95C9semidirect
13. C2×C5×#4294D5semidirect
14. C2×C5×#4294C2×C5semidirect
15. C5⋊(#4294⋊C2)C2×C5direct (N × H)
16. C5⋊(#4294⋊C2)C2×C5direct (N × H)
22. C2×C52×C19C2×C9semidirect
23. C5×D95C2×C9semidirect
24. C5×D95C2×C9semidirect
27. C5×#4294C2×D5semidirect
28. C5×#4294C22×C5semidirect
37. C52×C19C22×C9semidirect
41. C2×D95C5×C9semidirect
43. C2×#4294C5×D5semidirect
53. C2×C5×C19C9×D5semidirect
54. C2×C5×C19C2×C5×C9semidirect
55. D95C2×C5×C9semidirect
56. D95C2×C5×C9semidirect
61. #4294C2×C5×D5semidirect
70. C5×C19C2×C9×D5semidirect
71. C5×C19C22×C5×C9semidirect
84. C2×C52#4294⋊C2semidirect
90. C2×C19C5×C9×D5semidirect
101. C52C2×#4294⋊C2semidirect
104. C19C2×C5×C9×D5semidirect
115. C2×C5C5×#4294⋊C2semidirect
116. C2×C5C5⋊(#4294⋊C2)direct (N × H)
124. C5C2×C5×#4294⋊C2semidirect
125. C5C2×C5⋊(#4294⋊C2)direct (N × H)
130. C2C5×C5⋊(#4294⋊C2)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C52×#42942
C5×C5⋊(#4294⋊C2)2
C2×C3×C5×D953
C2×C5⋊(#4294⋊C2)5
C2×C5×#4294⋊C25
C2×C5×C9×D519