Finite Group Databasebeta

← C3⋊D3×C4⋊AGL(1,𝔽5) #13088674

Subgroups — C3⋊D3×C4⋊AGL(1,𝔽5) #13088674

Order 1440.

552 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 116 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×C3⋊D3×AGL(1,𝔽5)C212yesmaximal
2C4×D5×C3⋊D3C211yesmaximal, characteristic
3C4⋊(C3⋊ΓL(1,𝔽16))C211yesmaximal, characteristic, complemented
4C32×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
5#10623965C212yesmaximal, complemented
6D3×C4⋊AGL(1,𝔽5)—34nomaximal
7C2×D5×C3⋊D3C2211yescharacteristic
8C2×C3⋊ΓL(1,𝔽16)C2212yes—
9C2×C32×AGL(1,𝔽5)C2212yes—
10C3⋊D3×AGL(1,𝔽5)—24no—
11C4×C3⋊D15C411yescharacteristic, complemented
12C4×C5×C3⋊D3C411yescharacteristic, complemented
13C32×C4×D5C2211yescharacteristic
14#67137C411yescharacteristic, complemented
15C3⋊D3×Dic5C411yescharacteristic, complemented
16D5×C3⋊Dic3C2211yescharacteristic
17C4⋊C4×C3⋊D3—51nomaximal, Hall
18C2×D3×AGL(1,𝔽5)—38no—
19C4×D3×D5—34no—
20C4⋊ΓL(1,𝔽16)—34no—
21C3×C4⋊AGL(1,𝔽5)D314yescomplemented
22Dic3⋊AGL(1,𝔽5)—38no—
23C2×C3⋊D15C2×C411yescharacteristic
24C2×C5×C3⋊D3C2×C411yescharacteristic
25C2×C32×D5C2311yescharacteristic
26D5×C3⋊D3D412yes—
27D5×C3⋊D3Q812yes—
28C3⋊ΓL(1,𝔽16)—22no—
29C32×AGL(1,𝔽5)—22no—
30C32×C4×C5C2×C411yescharacteristic, complemented
31C3⋊Dic15C2×C411yescharacteristic, complemented
32C5×C3⋊Dic3C2×C411yescharacteristic, complemented
33C32×Dic5C2×C411yescharacteristic, complemented
34C2×C4⋊AGL(1,𝔽5)—91noHall
35C2×C4×C3⋊D3—51no—
36C2×C4×C3⋊D3—52no—
37C32×C4⋊C4—51no—
38C4⋊(C3⋊Dic3)—51no—
39#3750—52no—
40C2×D3×D5—34no—
41C2×ΓL(1,𝔽16)—38no—
42C2×C3×AGL(1,𝔽5)C2×D318yes—
43D3×AGL(1,𝔽5)—616no—
44C4×D15—34no—
45C4×C5×D3—34no—
46C3×C4×D5C2×D314yes—
47#1200—34no—
48D3×Dic5—34no—
49D5×Dic3—34no—
50D3×C4⋊C4—154no—
51C2×C32×C5C22×C411yescharacteristic
52C3⋊D15C4⋊C412yescomplemented
53C5×C3⋊D3C4⋊C412yescomplemented
54C32×D5C2×D411yescharacteristic
55C32×D5C2×Q811yescharacteristic
56C22×AGL(1,𝔽5)—92no—
57C2×C4×D5—91no—
58C4⋊AGL(1,𝔽5)—92no—
59C4⋊AGL(1,𝔽5)—91no—
60C4⋊AGL(1,𝔽5)C3⋊D311yescharacteristic, direct product factor
61C22×C3⋊D3—51no—
62C2×C32×C4—51no—
63C2×C32×C4—52no—
64C2×C3⋊Dic3—51no—
65C2×C3⋊Dic3—52no—
66C4×C3⋊D3—51no—
67C4×C3⋊D3—51no—
68C4×C3⋊D3—51no—
69C4×C3⋊D3—104no—
70C4×C3⋊D3AGL(1,𝔽5)11yescharacteristic, complemented
71C2×D15—34no—
72C2×C5×D3—34no—
73C2×C3×D5C22×D314yes—
74D3×D5—38no—
75D3×D5—38no—
76ΓL(1,𝔽16)—68no—
77C3×AGL(1,𝔽5)—28no—
78C3×C4×C5C4×D314yescomplemented
79Dic15—34no—
80C3×Dic5C4×D314yescomplemented
81C5×Dic3—34no—
82C2×C4×D3—154no—
83C2×C4×D3—158no—
84C3×C4⋊C4—54no—
85C4⋊Dic3—154no—
86Dic3⋊C4—158no—
87C32×C5C2×C4⋊C411yescharacteristic, complemented, Hall
88C22×D5—91no—
89C2×AGL(1,𝔽5)—92no—
90C2×AGL(1,𝔽5)—184no—
91C2×AGL(1,𝔽5)C2×C3⋊D312yes—
92C2×C4×C5—91no—
93C2×Dic5—91no—
94C4×D5—91no—
95C4×D5—91no—
96C4×D5—91no—
97C4×D5C2×C3⋊D311yescharacteristic
98C22×C32—51no—
99C2×C3⋊D3—51no—
100C2×C3⋊D3—52no—
101C2×C3⋊D3—52no—
102C2×C3⋊D3C2×AGL(1,𝔽5)11yescharacteristic
103C32×C4—51no—
104C32×C4—102no—
105C32×C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
106C3⋊Dic3—51no—
107C3⋊Dic3—102no—
108C3⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic, complemented
109C2×C4⋊C4—451noSylow
110C2×C3×C5C2×C4×D314yes—
111D15—38no—
112C3×D5D3×D414yes—
113C3×D5D3×Q814yes—
114C5×D3—38no—
115C22×D3—154no—
116C2×C3×C4—54no—
117C2×C3×C4—58no—
118C2×Dic3—154no—
119C2×Dic3—158no—
120C4×D3—34no—
121C4×D3—154no—
122C4×D3—154no—
123C4×D3—154no—
124C4×D3—3016no—
125C22×C5—91no—
126C2×D5—91no—
127C2×D5—92no—
128C2×D5—92no—
129C2×D5C22×C3⋊D311yescharacteristic
130AGL(1,𝔽5)—22no—
131AGL(1,𝔽5)—182no—
132C4×C5—91no—
133C4×C5C4×C3⋊D311yescharacteristic, complemented
134Dic5—91no—
135Dic5C4×C3⋊D311yescharacteristic, complemented
136C2×C32—51no—
137C2×C32—51no—
138C2×C32C22×AGL(1,𝔽5)11yescharacteristic
139C3⋊D3—52no—
140C3⋊D3C4⋊AGL(1,𝔽5)12yesdirect product factor
141C22×C4—451no—
142C22×C4—452no—
143C4⋊C4—51no—
144C4⋊C4—452no—
145C4⋊C4—451no—
146C3×C5D3×C4⋊C414yescomplemented
147C22×C3—54no—
148C2×D3—34no—
149C2×D3—158no—
150C2×D3—158no—
151C2×D3—154no—
152C3×C4—54no—
153C3×C4—108no—
154C3×C4D3×AGL(1,𝔽5)14yescomplemented
155Dic3—34no—
156Dic3—154no—
157Dic3—308no—
158C2×C5—92no—
159C2×C5C2×C4×C3⋊D311yescharacteristic
160D5—92no—
161D5D4×C3⋊D311yescharacteristic
162D5Q8×C3⋊D311yescharacteristic
163C32C2×C4⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
164C23—451no—
165C2×C4—51no—
166C2×C4—52no—
167C2×C4—91no—
168C2×C4—451no—
169C2×C4—451no—
170C2×C4—451no—
171C2×C4—451no—
172C2×C4—452no—
173C2×C4—904no—
174C2×C3—54no—
175C2×C3—54no—
176C2×C3C2×D3×AGL(1,𝔽5)14yes—
177D3—38no—
178D3—158no—
179C5C4⋊C4×C3⋊D311yescharacteristic, complemented, Sylow
180C22—51no—
181C22—91no—
182C22—452no—
183C22—452no—
184C22—451no—
185C4—51no—
186C4—91no—
187C4—102no—
188C4—451no—
189C4—902no—
190C4C3⋊D3×AGL(1,𝔽5)11yescharacteristic, complemented
191C3D3×C4⋊AGL(1,𝔽5)14yescomplemented
192C2—51no—
193C2—51no—
194C2—92no—
195C2—452no—
196C2C2×C3⋊D3×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊D3×AGL(1,𝔽5)
commutator, socleC2×C32×C5C22×C4
FittingC32×C4×C5C2×C4
prime radicalC2×D5×C3⋊D3C22
cosocleC2×C32×D5C23
hypercenterC4C3⋊D3×AGL(1,𝔽5)
hypocenterC32×C5C2×C4⋊C4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
3. C4⋊(C3⋊ΓL(1,𝔽16))C2semidirect
4. C32×C4⋊AGL(1,𝔽5)C2semidirect
5. #10623965C2semidirect
11. C4×C3⋊D15C4semidirect
12. C4×C5×C3⋊D3C4semidirect
14. #67137C4semidirect
15. C3⋊D3×Dic5C4semidirect
21. C3×C4⋊AGL(1,𝔽5)D3semidirect
30. C32×C4×C5C2×C4semidirect
31. C3⋊Dic15C2×C4semidirect
32. C5×C3⋊Dic3C2×C4semidirect
33. C32×Dic5C2×C4semidirect
52. C3⋊D15C4⋊C4semidirect
53. C5×C3⋊D3C4⋊C4semidirect
60. C4⋊AGL(1,𝔽5)C3⋊D3direct (N × H)
70. C4×C3⋊D3AGL(1,𝔽5)semidirect
78. C3×C4×C5C4×D3semidirect
80. C3×Dic5C4×D3semidirect
87. C32×C5C2×C4⋊C4semidirect
105. C32×C4C2×AGL(1,𝔽5)semidirect
108. C3⋊Dic3C2×AGL(1,𝔽5)semidirect
133. C4×C5C4×C3⋊D3semidirect
135. Dic5C4×C3⋊D3semidirect
140. C3⋊D3C4⋊AGL(1,𝔽5)direct (N × H)
146. C3×C5D3×C4⋊C4semidirect
154. C3×C4D3×AGL(1,𝔽5)semidirect
163. C32C2×C4⋊AGL(1,𝔽5)semidirect
179. C5C4⋊C4×C3⋊D3semidirect
190. C4C3⋊D3×AGL(1,𝔽5)semidirect
191. C3D3×C4⋊AGL(1,𝔽5)semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×AGL(1,𝔽5)2
C4×D5×C3⋊D32
C4⋊(C3⋊ΓL(1,𝔽16))2
C32×C4⋊AGL(1,𝔽5)2
#106239652
D3×C4⋊AGL(1,𝔽5)3
C4⋊C4×C3⋊D35