Finite Group Databasebeta

← Dic3×D5⋊AGL(1,𝔽5) #72230593

Subgroups — Dic3×D5⋊AGL(1,𝔽5) #72230593

Order 2400.

492 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 112 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×D5⋊ΓL(1,𝔽16)C211yesmaximal, characteristic
2C2×C3×D5⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3D52×Dic3C211yesmaximal, characteristic
4#11925359C211yesmaximal, characteristic, complemented
5#11925354C211yesmaximal, characteristic, complemented
6Dic3×#1044C211yesmaximal, characteristic, complemented
7Dic3×#1043C211yesmaximal, characteristic, complemented
8C4×D5⋊AGL(1,𝔽5)—31nomaximal, Hall
9C2×C3×D52C2211yescharacteristic
10C2×C3⋊#1044C2211yescharacteristic
11C2×C3⋊#1043C2211yescharacteristic
12C2×C3×#1044C2211yescharacteristic
13C2×C3×#1043C2211yescharacteristic
14D5⋊ΓL(1,𝔽16)C414yescomplemented
15C3×D5⋊AGL(1,𝔽5)C414yescomplemented
16#10597126C2211yescharacteristic
17D5×Dic15C412yescomplemented
18Dic3×C5⋊D5C2211yescharacteristic
19C5×D5×Dic3C412yescomplemented
20C2×Dic3×AGL(1,𝔽5)—52nomaximal
21C2×D5⋊AGL(1,𝔽5)—31no—
22C2×D5⋊AGL(1,𝔽5)D311yescharacteristic
23C4×D52—31no—
24C4×#1044—31no—
25C4×#1043—31no—
26#87830—31no—
27#87830—31no—
28C2×C3×C5⋊D5C2311yescharacteristic
29C2×C3×C5×D5C2×C412yes—
30C3×D52C2×C412yes—
31C3×D52C2×C412yes—
32C3⋊#1044C2×C412yescomplemented
33C3⋊#1043C2×C412yescomplemented
34C3×#1044C2×C412yescomplemented
35C3×#1043C2×C412yescomplemented
36C3⋊#1039C2×C411yescharacteristic, complemented
37C5×Dic15C2×C412yescomplemented
38C52×Dic3C2×C411yescharacteristic, complemented
39C22×ΓL(1,𝔽16)—52no—
40C22×C3×AGL(1,𝔽5)—52no—
41C2×D5×Dic3—52no—
42Dic3×AGL(1,𝔽5)—52no—
43Dic3×AGL(1,𝔽5)—52no—
44Dic3×AGL(1,𝔽5)—52no—
45Dic3×AGL(1,𝔽5)—52no—
46Dic3×AGL(1,𝔽5)—102no—
47C2×D52C2×D311yescharacteristic
48C2×#1044—31no—
49C2×#1044C2×D311yescharacteristic
50C2×#1043—31no—
51C2×#1043C2×D311yescharacteristic
52D5⋊AGL(1,𝔽5)—34no—
53D5⋊AGL(1,𝔽5)Dic314yesdirect product factor
54C4×C5⋊D5—31no—
55C4×C5×D5—32no—
56#6036—31no—
57D5×Dic5—32no—
58C2×C4×AGL(1,𝔽5)—152no—
59C2×C3×C52C22×C411yescharacteristic
60C3×C5⋊D5C22×C411yescharacteristic
61C3×C5⋊D5C22×C411yescharacteristic
62C3×C5×D5C4214yescomplemented
63C22×C3×D5—52no—
64C2×ΓL(1,𝔽16)—58no—
65C2×ΓL(1,𝔽16)—52no—
66C2×ΓL(1,𝔽16)—52no—
67C2×ΓL(1,𝔽16)—102no—
68C2×C3×AGL(1,𝔽5)—58no—
69C2×C3×AGL(1,𝔽5)—52no—
70C2×C3×AGL(1,𝔽5)—52no—
71C2×C3×AGL(1,𝔽5)—102no—
72C2×Dic15—52no—
73C2×C5×Dic3—52no—
74D5×Dic3—52no—
75D5×Dic3—52no—
76D5×Dic3—52no—
77D5×Dic3—102no—
78D5×Dic3AGL(1,𝔽5)12yescomplemented
79C2×C5⋊D5C22×D311yescharacteristic
80C2×C5×D5C4×D312yes—
81D52C4×D312yes—
82D52C2×Dic312yes—
83#1044—32no—
84#1044C2×Dic312yescomplemented
85#1043—32no—
86#1043C2×Dic312yescomplemented
87C4×C52—31no—
88#1039—31no—
89C5×Dic5—32no—
90C2×C4×Dic3—251noHall
91C22×AGL(1,𝔽5)—52no—
92C22×AGL(1,𝔽5)—152no—
93C2×C4×D5—152no—
94C4×AGL(1,𝔽5)—152no—
95C4×AGL(1,𝔽5)—152no—
96C4×AGL(1,𝔽5)—152no—
97C4×AGL(1,𝔽5)—152no—
98C4×AGL(1,𝔽5)—302no—
99C3×C52C2×C4211yescharacteristic, complemented, Hall
100C22×C3×C5—52no—
101C2×C3×D5—52no—
102C2×C3×D5—54no—
103C2×C3×D5—54no—
104C2×C3×D5—102no—
105C2×C3×D5C2×AGL(1,𝔽5)12yes—
106ΓL(1,𝔽16)—54no—
107ΓL(1,𝔽16)—54no—
108ΓL(1,𝔽16)—104no—
109C3×AGL(1,𝔽5)—54no—
110C3×AGL(1,𝔽5)—54no—
111C3×AGL(1,𝔽5)—104no—
112Dic15—52no—
113Dic15—102no—
114Dic15C2×AGL(1,𝔽5)12yescomplemented
115C5×Dic3—22no—
116C5×Dic3—52no—
117C5×Dic3C2×AGL(1,𝔽5)12yescomplemented
118C2×C52C2×C4×D311yescharacteristic
119C5⋊D5C2×C4×D311yescharacteristic
120C5⋊D5C22×Dic311yescharacteristic
121C5×D5C4×Dic314yescomplemented
122C22×C3×C4—251no—
123C22×Dic3—251no—
124C22×Dic3—251no—
125C4×Dic3—251no—
126C4×Dic3—251no—
127C4×Dic3—251no—
128C4×Dic3—251no—
129C22×D5—52no—
130C2×AGL(1,𝔽5)—58no—
131C2×AGL(1,𝔽5)—52no—
132C2×AGL(1,𝔽5)—52no—
133C2×AGL(1,𝔽5)—102no—
134C2×AGL(1,𝔽5)—158no—
135C2×AGL(1,𝔽5)—152no—
136C2×AGL(1,𝔽5)—152no—
137C2×AGL(1,𝔽5)—302no—
138C2×C4×C5—152no—
139C2×Dic5—152no—
140C4×D5—32no—
141C4×D5—152no—
142C4×D5—152no—
143C4×D5—152no—
144C4×D5—302no—
145C2×C42—751noSylow
146C2×C3×C5—22no—
147C2×C3×C5—54no—
148C2×C3×C5C22×AGL(1,𝔽5)12yes—
149C3×D5—52no—
150C3×D5—52no—
151C3×D5—102no—
152C3×D5—102no—
153C3×D5C4×AGL(1,𝔽5)14yescomplemented
154C52C2×C4×Dic311yescharacteristic, complemented, Sylow
155C23×C3—251no—
156C2×C3×C4—251no—
157C2×C3×C4—251no—
158C2×C3×C4—254no—
159C2×Dic3—52no—
160C2×Dic3—251no—
161C2×Dic3—252no—
162C2×Dic3—251no—
163C2×Dic3—251no—
164C2×Dic3—254no—
165C2×Dic3—251no—
166C22×C5—52no—
167C2×D5—52no—
168C2×D5—54no—
169C2×D5—54no—
170C2×D5—102no—
171C2×D5D3×AGL(1,𝔽5)12yes—
172AGL(1,𝔽5)—54no—
173AGL(1,𝔽5)—54no—
174AGL(1,𝔽5)—104no—
175AGL(1,𝔽5)—154no—
176AGL(1,𝔽5)—154no—
177AGL(1,𝔽5)—304no—
178C4×C5—32no—
179C4×C5—62no—
180C4×C5—152no—
181Dic5—32no—
182Dic5—152no—
183Dic5—302no—
184C22×C4—251no—
185C22×C4—751no—
186C22×C4—751no—
187C42—751no—
188C42—751no—
189C42—751no—
190C42—751no—
191C3×C5—22no—
192C3×C5C2×C4×AGL(1,𝔽5)12yescomplemented
193C22×C3—52no—
194C22×C3—252no—
195C22×C3—252no—
196C22×C3—251no—
197C3×C4—252no—
198C3×C4—252no—
199Dic3—52no—
200Dic3—252no—
201Dic3—252no—
202Dic3—251no—
203Dic3D5⋊AGL(1,𝔽5)11yescharacteristic, direct product factor
204C2×C5—22no—
205C2×C5—54no—
206C2×C5C2×D3×AGL(1,𝔽5)12yes—
207D5—52no—
208D5—52no—
209D5—102no—
210D5—102no—
211D5Dic3×AGL(1,𝔽5)14yescomplemented
212C23—251no—
213C2×C4—152no—
214C2×C4—251no—
215C2×C4—251no—
216C2×C4—254no—
217C2×C4—751no—
218C2×C4—751no—
219C2×C4—752no—
220C2×C4—751no—
221C2×C4—751no—
222C2×C4—754no—
223C2×C3—54no—
224C2×C3—251no—
225C2×C3—251no—
226C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
227C5—22no—
228C5C2×Dic3×AGL(1,𝔽5)12yescomplemented
229C22—52no—
230C22—252no—
231C22—252no—
232C22—251no—
233C4—31no—
234C4—152no—
235C4—252no—
236C4—252no—
237C4—752no—
238C4—752no—
239C4—751no—
240C3C4×D5⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
241C2—54no—
242C2—251no—
243C2—251no—
244C2D3×D5⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×D5⋊AGL(1,𝔽5)
commutator, hypocenterC3×C52C2×C42
Fitting, socleC2×C3×C52C22×C4
prime radicalC2×C3×D52C22
cosocleC2×C3×C5⋊D5C23
LCSG ▷ C3×C52
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
4. #11925359C2semidirect
5. #11925354C2semidirect
6. Dic3×#1044C2semidirect
7. Dic3×#1043C2semidirect
14. D5⋊ΓL(1,𝔽16)C4semidirect
15. C3×D5⋊AGL(1,𝔽5)C4semidirect
17. D5×Dic15C4semidirect
19. C5×D5×Dic3C4semidirect
32. C3⋊#1044C2×C4semidirect
33. C3⋊#1043C2×C4semidirect
34. C3×#1044C2×C4semidirect
35. C3×#1043C2×C4semidirect
36. C3⋊#1039C2×C4semidirect
37. C5×Dic15C2×C4semidirect
38. C52×Dic3C2×C4semidirect
53. D5⋊AGL(1,𝔽5)Dic3direct (N × H)
62. C3×C5×D5C42semidirect
78. D5×Dic3AGL(1,𝔽5)semidirect
84. #1044C2×Dic3semidirect
86. #1043C2×Dic3semidirect
99. C3×C52C2×C42semidirect
114. Dic15C2×AGL(1,𝔽5)semidirect
117. C5×Dic3C2×AGL(1,𝔽5)semidirect
121. C5×D5C4×Dic3semidirect
153. C3×D5C4×AGL(1,𝔽5)semidirect
154. C52C2×C4×Dic3semidirect
192. C3×C5C2×C4×AGL(1,𝔽5)semidirect
203. Dic3D5⋊AGL(1,𝔽5)direct (N × H)
211. D5Dic3×AGL(1,𝔽5)semidirect
228. C5C2×Dic3×AGL(1,𝔽5)semidirect
240. C3C4×D5⋊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×D5⋊ΓL(1,𝔽16)2
C2×C3×D5⋊AGL(1,𝔽5)2
D52×Dic32
#119253592
#119253542
Dic3×#10442
Dic3×#10432
C4×D5⋊AGL(1,𝔽5)3
C2×Dic3×AGL(1,𝔽5)5