Finite Group Databasebeta

← ΓL(1,𝔽32)×(C3⋊D3)⋊D5 #42124724

Subgroups — ΓL(1,𝔽32)×(C3⋊D3)⋊D5 #42124724

Order 27900.

224 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C5×C3⋊D3×ΓL(1,𝔽32)C211yesmaximal, characteristic, complemented
2C3×D15×ΓL(1,𝔽32)C212yesmaximal, complemented
3D3×D5×ΓL(1,𝔽32)—32nomaximal
4C32×C5×ΓL(1,𝔽32)C2211yescharacteristic, complemented, Hall
5C31×(C3⋊D3)⋊D5C511yesmaximal, characteristic, complemented
6D32×ΓL(1,𝔽32)—51nomaximal
7D15×ΓL(1,𝔽32)D312yescomplemented
8C5×D3×ΓL(1,𝔽32)—32no—
9C5×D3×ΓL(1,𝔽32)—61no—
10C3×D5×ΓL(1,𝔽32)—32no—
11C2×D5×ΓL(1,𝔽32)—91noHall
12C5×C3⋊D3×C31C2×C511yescharacteristic, complemented
13C3×D15×C31C2×C512yescomplemented
14C3⋊D3×ΓL(1,𝔽32)—22no—
15C3⋊D3×ΓL(1,𝔽32)D511yescharacteristic, complemented
16C3×D3×ΓL(1,𝔽32)—52no—
17C3×C5×ΓL(1,𝔽32)—21no—
18C3×C5×ΓL(1,𝔽32)C2×D312yescomplemented
19D3×D5×C31—32no—
20C2×D3×ΓL(1,𝔽32)—152no—
21C2×C5×ΓL(1,𝔽32)—91no—
22D5×ΓL(1,𝔽32)—32no—
23C32×C5×C31C22×C511yescharacteristic, complemented
24C32×ΓL(1,𝔽32)—22no—
25C32×ΓL(1,𝔽32)C2×D511yescharacteristic, complemented
26D32×C31—51noHall
27D15×C31C5×D312yescomplemented
28C5×D3×C31—32no—
29C5×D3×C31—61no—
30C3×D5×C31—32no—
31C2×C3×ΓL(1,𝔽32)—152no—
32D3×ΓL(1,𝔽32)—32no—
33D3×ΓL(1,𝔽32)—52no—
34D3×ΓL(1,𝔽32)—64no—
35D3×ΓL(1,𝔽32)—61no—
36D3×ΓL(1,𝔽32)—64no—
37C5×(C3⋊D3)⋊D5—311nomaximal, Hall
38C5×ΓL(1,𝔽32)D3211yescharacteristic, complemented, Hall
39C2×D5×C31—91no—
40C22×ΓL(1,𝔽32)—451no—
41C3⋊D3×C31C5×D511yescharacteristic, complemented
42C3×D3×C31—52no—
43C3×C5×C31—21no—
44C3×C5×C31C2×C5×D312yescomplemented
45C3×ΓL(1,𝔽32)—24no—
46C3×ΓL(1,𝔽32)—21no—
47C3×ΓL(1,𝔽32)—24no—
48C3×ΓL(1,𝔽32)D3×D512yescomplemented
49C52×C3⋊D3—311no—
50C3×C5×D15—312no—
51C2×D3×C31—152no—
52C2×C5×C31—91no—
53D5×C31—32no—
54C2×ΓL(1,𝔽32)—91no—
55C2×ΓL(1,𝔽32)—152no—
56C2×ΓL(1,𝔽32)—182no—
57C5×D3×D5—932no—
58C32×C31C2×C5×D511yescharacteristic, complemented, Hall
59C32×C52—311noHall
60C2×C3×C31—152no—
61D3×C31—32no—
62D3×C31—52no—
63D3×C31—61no—
64(C3⋊D3)⋊D5ΓL(1,𝔽32)11yescharacteristic, direct product factor
65C5×D32—1551no—
66C5×C31C5×D3211yescharacteristic, complemented
67ΓL(1,𝔽32)—22no—
68ΓL(1,𝔽32)(C3⋊D3)⋊D511yescharacteristic, direct product factor
69C5×D15—312no—
70C52×D3—932no—
71C52×D3—1861no—
72C3×C5×D5—932no—
73C22×C31—451noHall
74C2×C5×D5—2791noHall
75C3×C31—21no—
76C3×C31C5×D3×D512yescomplemented
77C5×C3⋊D3—311no—
78C5×C3⋊D3—622no—
79C5×C3⋊D3C2×ΓL(1,𝔽32)11yescharacteristic, complemented
80C3×D15C2×ΓL(1,𝔽32)12yescomplemented
81C3×C5×D3—1552no—
82C3×C52—312no—
83C3×C52—621no—
84C2×C31—91no—
85C2×C31—152no—
86C2×C5×D3—4652no—
87D3×D5—32no—
88C2×C52—2791no—
89C5×D5—932no—
90C32×C5—311no—
91C32×C5—622no—
92C32×C5C22×ΓL(1,𝔽32)11yescharacteristic, complemented
93D32—51noHall
94C31C5×(C3⋊D3)⋊D511yescharacteristic, complemented, Sylow
95C2×C3×C5—4652no—
96D15D3×ΓL(1,𝔽32)12yescomplemented
97C3×D5—32no—
98C5×D3—32no—
99C5×D3—61no—
100C5×D3—932no—
101C5×D3—1552no—
102C5×D3—1864no—
103C5×D3—1861no—
104C5×D3—1864no—
105C52—311noSylow
106C22×C5—13951no—
107C2×D5—91no—
108C3⋊D3D5×ΓL(1,𝔽32)11yescharacteristic, complemented
109C3×D3—52no—
110C3×C5—21no—
111C3×C5—312no—
112C3×C5—621no—
113C3×C5—624no—
114C3×C5—624no—
115C3×C5C2×D3×ΓL(1,𝔽32)12yescomplemented
116C2×D3—152no—
117C2×C5—91no—
118C2×C5—2791no—
119C2×C5—5582no—
120C2×C5—4652no—
121D5—32no—
122C32C2×D5×ΓL(1,𝔽32)11yescharacteristic, complemented, Sylow
123C2×C3—152no—
124D3—32no—
125D3—52no—
126D3—61no—
127C5—311no—
128C5—622no—
129C5D32×ΓL(1,𝔽32)11yescharacteristic, complemented
130C22—451noSylow
131C3—21no—
132C3D3×D5×ΓL(1,𝔽32)12yescomplemented
133C2—91no—
134C2—152no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C5×C31C22×C5
LCSG ▷ C32×C5×C31
UCSC1

Complements

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

NH ≅ G/Ntype
1. C5×C3⋊D3×ΓL(1,𝔽32)C2semidirect
2. C3×D15×ΓL(1,𝔽32)C2semidirect
4. C32×C5×ΓL(1,𝔽32)C22semidirect
5. C31×(C3⋊D3)⋊D5C5semidirect
7. D15×ΓL(1,𝔽32)D3semidirect
12. C5×C3⋊D3×C31C2×C5semidirect
13. C3×D15×C31C2×C5semidirect
15. C3⋊D3×ΓL(1,𝔽32)D5semidirect
18. C3×C5×ΓL(1,𝔽32)C2×D3semidirect
23. C32×C5×C31C22×C5semidirect
25. C32×ΓL(1,𝔽32)C2×D5semidirect
27. D15×C31C5×D3semidirect
38. C5×ΓL(1,𝔽32)D32semidirect
41. C3⋊D3×C31C5×D5semidirect
44. C3×C5×C31C2×C5×D3semidirect
48. C3×ΓL(1,𝔽32)D3×D5semidirect
58. C32×C31C2×C5×D5semidirect
64. (C3⋊D3)⋊D5ΓL(1,𝔽32)direct (N × H)
66. C5×C31C5×D32semidirect
68. ΓL(1,𝔽32)(C3⋊D3)⋊D5direct (N × H)
76. C3×C31C5×D3×D5semidirect
79. C5×C3⋊D3C2×ΓL(1,𝔽32)semidirect
80. C3×D15C2×ΓL(1,𝔽32)semidirect
92. C32×C5C22×ΓL(1,𝔽32)semidirect
94. C31C5×(C3⋊D3)⋊D5semidirect
96. D15D3×ΓL(1,𝔽32)semidirect
108. C3⋊D3D5×ΓL(1,𝔽32)semidirect
115. C3×C5C2×D3×ΓL(1,𝔽32)semidirect
122. C32C2×D5×ΓL(1,𝔽32)semidirect
129. C5D32×ΓL(1,𝔽32)semidirect
132. C3D3×D5×ΓL(1,𝔽32)semidirect

Maximal subgroups

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

subgroupindex
C5×C3⋊D3×ΓL(1,𝔽32)2
C3×D15×ΓL(1,𝔽32)2
D3×D5×ΓL(1,𝔽32)3
C31×(C3⋊D3)⋊D55
D32×ΓL(1,𝔽32)5
C5×(C3⋊D3)⋊D531