Finite Group Databasebeta

← C3×AGL(1,𝔽5)×#157 #72216970

Subgroups — C3×AGL(1,𝔽5)×#157 #72216970

Order 2160.

208 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 44 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×D5×#157C211yesmaximal, characteristic
2C3×D5⋊#157C211yesmaximal, characteristic
3C3×C3⋊D3×AGL(1,𝔽5)C211yesmaximal, characteristic
4AGL(1,𝔽5)×#157C311yesmaximal, characteristic, direct product factor
5C3×D5×C3⋊D3C2211yescharacteristic
6C3×#4379C411yescharacteristic, complemented
7C3×#4380C412yescomplemented
8C3×C5×#157C411yescharacteristic, complemented
9C3×C3⋊ΓL(1,𝔽16)C411yescharacteristic, complemented
10C33×AGL(1,𝔽5)C411yescharacteristic, complemented
11C3×C4×#157—51nomaximal, Hall
12D5⋊#157C2×C311yescharacteristic
13D5×#157C2×C311yescharacteristic
14C3⋊D3×AGL(1,𝔽5)C2×C311yescharacteristic
15C3×D3×AGL(1,𝔽5)—62no—
16C3×C3⋊D15C2×C411yescharacteristic
17C3×C5×C3⋊D3C2×C411yescharacteristic
18C33×D5C2×C411yescharacteristic
19C3×C4×AGL(1,𝔽5)—91nomaximal
20C2×C3×#157—51no—
21C2×C3×#157—51no—
22C3×C4×C3⋊D3—51no—
23D5×C3⋊D3C22×C311yescharacteristic
24C3×D3×D5—62no—
25#4380C3×C412yescomplemented
26#4379C3×C411yescharacteristic, complemented
27C5×#157C3×C411yescharacteristic, complemented
28C3⋊ΓL(1,𝔽16)C3×C411yescharacteristic, complemented
29C3×ΓL(1,𝔽16)—62no—
30C32×AGL(1,𝔽5)—22no—
31C32×AGL(1,𝔽5)—42no—
32C32×AGL(1,𝔽5)C3×C411yescharacteristic, complemented
33C4×#157—51no—
34C33×C5C4211yescharacteristic, complemented, Hall
35C2×C3×AGL(1,𝔽5)—91no—
36C2×C3×AGL(1,𝔽5)—91no—
37D3×AGL(1,𝔽5)—62no—
38C3×C4×D5—91no—
39C2×C3×C3⋊D3—51no—
40C3×#157—52no—
41C3×#157—51no—
42C3×#157AGL(1,𝔽5)11yescharacteristic, direct product factor
43C33×C4—51no—
44C3×C3⋊Dic3—51no—
45C3⋊D15C2×C3×C411yescharacteristic
46C5×C3⋊D3C2×C3×C411yescharacteristic
47C3×D15—62no—
48C3×C5×D3—62no—
49C32×D5—22no—
50C32×D5—42no—
51C32×D5C2×C3×C411yescharacteristic
52C4×AGL(1,𝔽5)—91noHall
53C2×#157—51no—
54C2×#157—51no—
55C4×C3⋊D3—51no—
56C3×C4×D3—302no—
57C2×C3×D5—91no—
58D3×D5—62no—
59ΓL(1,𝔽16)—62no—
60C3×AGL(1,𝔽5)—22no—
61C3×AGL(1,𝔽5)—42no—
62C3×AGL(1,𝔽5)—92no—
63C3×AGL(1,𝔽5)—91no—
64C3×AGL(1,𝔽5)#15711yescharacteristic, direct product factor
65C3×C4×C5—91no—
66C3×Dic5—91no—
67C2×C33—51no—
68C3×C3⋊D3—51no—
69C3×C3⋊D3C2×AGL(1,𝔽5)11yescharacteristic
70C3×C42—451no—
71C32×C5—22no—
72C32×C5—42no—
73C32×C5C3×C4211yescharacteristic, complemented
74C2×AGL(1,𝔽5)—91no—
75C2×AGL(1,𝔽5)—91no—
76C4×D5—91no—
77C2×C3⋊D3—51no—
78C2×C3×D3—302no—
79#157—52no—
80#157—51no—
81#157C3×AGL(1,𝔽5)11yescharacteristic, direct product factor
82C32×C4—51no—
83C32×C4—102no—
84C32×C4—202no—
85C3⋊Dic3—51no—
86C3×Dic3—302no—
87C2×C3×C5—91no—
88D15—62no—
89C3×D5—22no—
90C3×D5—42no—
91C3×D5—91no—
92C3×D5C2×#15711yescharacteristic
93C5×D3—62no—
94C33C4×AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
95C2×C3×C4—451no—
96C2×C3×C4—451no—
97C2×C3×C4—451no—
98C4×D3—302no—
99C2×D5—91no—
100AGL(1,𝔽5)—92no—
101AGL(1,𝔽5)—91no—
102AGL(1,𝔽5)C3×#15711yescharacteristic, direct product factor
103C4×C5—91no—
104Dic5—91no—
105C2×C32—51no—
106C2×C32—102no—
107C2×C32—202no—
108C3⋊D3—51no—
109C3⋊D3C2×C3×AGL(1,𝔽5)11yescharacteristic
110C3×D3—62no—
111C3×D3—302no—
112C42—451noSylow
113C3×C5—22no—
114C3×C5—42no—
115C3×C5C4×#15711yescharacteristic, complemented
116C22×C3—451no—
117C2×D3—302no—
118C3×C4—51no—
119C3×C4—91no—
120C3×C4—102no—
121C3×C4—202no—
122C3×C4—451no—
123C3×C4—451no—
124C3×C4—452no—
125Dic3—302no—
126C2×C5—91no—
127D5—91no—
128D5C2×C3×#15711yescharacteristic
129C32—22no—
130C32—42no—
131C32C3×C4×AGL(1,𝔽5)11yescharacteristic, complemented
132C2×C4—451no—
133C2×C4—451no—
134C2×C4—451no—
135C2×C3—51no—
136C2×C3—91no—
137C2×C3—102no—
138C2×C3—202no—
139C2×C3—451no—
140D3—62no—
141D3—302no—
142C5C3×C4×#15711yescharacteristic, complemented, Sylow
143C22—451no—
144C4—51no—
145C4—91no—
146C4—451no—
147C4—451no—
148C4—452no—
149C3—22no—
150C3—42no—
151C3AGL(1,𝔽5)×#15711yescharacteristic, direct product factor
152C2—51no—
153C2—91no—
154C2—451no—
subgroupG/H
center, hypercenterC3AGL(1,𝔽5)×#157
commutator, hypocenterC32×C5C3×C42
FrattiniC1G
Fitting, socleC33×C5C42
prime radicalC3×D5×C3⋊D3C22
cosocleD5×C3⋊D3C22×C3
LCSG ▷ C32×C5
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
4. AGL(1,𝔽5)×#157C3direct (N × H)
6. C3×#4379C4semidirect
7. C3×#4380C4semidirect
8. C3×C5×#157C4semidirect
9. C3×C3⋊ΓL(1,𝔽16)C4semidirect
10. C33×AGL(1,𝔽5)C4semidirect
25. #4380C3×C4semidirect
26. #4379C3×C4semidirect
27. C5×#157C3×C4semidirect
28. C3⋊ΓL(1,𝔽16)C3×C4semidirect
32. C32×AGL(1,𝔽5)C3×C4semidirect
34. C33×C5C42semidirect
42. C3×#157AGL(1,𝔽5)direct (N × H)
64. C3×AGL(1,𝔽5)#157direct (N × H)
73. C32×C5C3×C42semidirect
81. #157C3×AGL(1,𝔽5)direct (N × H)
94. C33C4×AGL(1,𝔽5)semidirect
102. AGL(1,𝔽5)C3×#157direct (N × H)
115. C3×C5C4×#157semidirect
131. C32C3×C4×AGL(1,𝔽5)semidirect
142. C5C3×C4×#157semidirect
151. C3AGL(1,𝔽5)×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D5×#1572
C3×D5⋊#1572
C3×C3⋊D3×AGL(1,𝔽5)2
AGL(1,𝔽5)×#1573
C3×C4×#1575
C3×C4×AGL(1,𝔽5)9