Finite Group Databasebeta

← C4×AGL(1,𝔽5)×SL(2,𝔽3) #14512528

Subgroups — C4×AGL(1,𝔽5)×SL(2,𝔽3) #14512528

Order 1920.

458 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 98 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×AGL(1,𝔽5)×SL(2,𝔽3)C212yesmaximal
2C4×D5×SL(2,𝔽3)C211yesmaximal, characteristic
3C4×Q8×AGL(1,𝔽5)C311yesmaximal, characteristic, complemented, Hall
4C2×C3×C4×AGL(1,𝔽5)—41nomaximal
5C2×D5×SL(2,𝔽3)C2211yescharacteristic
6AGL(1,𝔽5)×SL(2,𝔽3)C414yesdirect product factor
7C4×C5×SL(2,𝔽3)C411yescharacteristic, complemented
8Dic5×SL(2,𝔽3)C411yescharacteristic, complemented
9C42×SL(2,𝔽3)—51nomaximal, Hall
10C2×Q8×AGL(1,𝔽5)C2×C312yes—
11C4×Q8×D5C2×C311yescharacteristic
12C4⋊C4×AGL(1,𝔽5)—32no—
13C4×C4⋊AGL(1,𝔽5)—31no—
14C42×AGL(1,𝔽5)—31no—
15C22×C3×AGL(1,𝔽5)—42no—
16C2×C3×C4×D5—41no—
17C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
18C3×C4×AGL(1,𝔽5)—44no—
19D5×SL(2,𝔽3)C2×C412yes—
20C2×C4×SL(2,𝔽3)—52no—
21C2×C4×SL(2,𝔽3)—51no—
22C2×Q8×D5C22×C311yescharacteristic
23Q8×AGL(1,𝔽5)—34no—
24Q8×AGL(1,𝔽5)C3×C414yescomplemented
25C2×C4⋊AGL(1,𝔽5)—32no—
26C2×C4×AGL(1,𝔽5)—32no—
27C2×C4×AGL(1,𝔽5)—32no—
28C2×C4×AGL(1,𝔽5)—31no—
29C2×C4×AGL(1,𝔽5)A411yescharacteristic
30C4×C5×Q8C3×C411yescharacteristic, complemented
31Q8×Dic5C3×C411yescharacteristic, complemented
32D5×C4⋊C4—31no—
33#4033—31no—
34C42×D5—31no—
35C4×Dic10—31no—
36(C2×AGL(1,𝔽5))⋊C4—32no—
37C42×Q8—51noSylow
38C22×C3×D5—41no—
39C2×C3×AGL(1,𝔽5)—44no—
40C2×C3×AGL(1,𝔽5)—44no—
41C2×C3×AGL(1,𝔽5)—44no—
42C2×C3×C4×C5—41no—
43C2×C3×Dic5—41no—
44C3×C4×D5—42no—
45C3×C4×D5—42no—
46C5×SL(2,𝔽3)C4211yescharacteristic, complemented
47C22×SL(2,𝔽3)—51no—
48C2×C3×C42—201no—
49C4×SL(2,𝔽3)—54no—
50C4×SL(2,𝔽3)—51no—
51C4×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic, direct product factor
52C22×AGL(1,𝔽5)—32no—
53C22×AGL(1,𝔽5)C2×A412yes—
54C2×C5×Q8C2×C3×C411yescharacteristic
55Q8×D5—32no—
56Q8×D5C2×C3×C412yes—
57C2×C4×D5—31no—
58C2×C4×D5—31no—
59C2×C4×D5C2×A411yescharacteristic
60C2×Dic10—31no—
61C4⋊AGL(1,𝔽5)—34no—
62C4⋊AGL(1,𝔽5)—34no—
63C4×AGL(1,𝔽5)—34no—
64C4×AGL(1,𝔽5)—34no—
65C4×AGL(1,𝔽5)—64no—
66C4×AGL(1,𝔽5)—62no—
67C4×AGL(1,𝔽5)SL(2,𝔽3)14yesdirect product factor
68C5×C4⋊C4—31no—
69C42×C5—31no—
70C4⋊Dic5—31no—
71#683—32no—
72C4×Dic5—31no—
73C4×Dic5—31no—
74C4×Dic5—31no—
75C2×C4×Q8—52no—
76C2×C4×Q8—51no—
77C4×C4⋊C4—151no—
78C4×C4⋊C4—152no—
79C43—151no—
80C22×C3×C5—41no—
81C2×C3×D5—42no—
82C2×C3×D5—41no—
83C2×C3×D5—41no—
84C2×C3×D5—42no—
85C3×AGL(1,𝔽5)—48no—
86C3×C4×C5—42no—
87C3×Dic5—42no—
88C22×C3×C4—201no—
89C22×C3×C4—202no—
90C2×SL(2,𝔽3)—52no—
91C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
92C3×C42—204no—
93C22×D5C22×A411yescharacteristic
94C2×AGL(1,𝔽5)—34no—
95C2×AGL(1,𝔽5)—62no—
96C2×AGL(1,𝔽5)—62no—
97C2×AGL(1,𝔽5)C4×A414yes—
98C2×AGL(1,𝔽5)C2×SL(2,𝔽3)14yes—
99C2×AGL(1,𝔽5)#41⋊C314yes—
100C5×Q8—31no—
101C5×Q8C3×C4211yescharacteristic, complemented
102C2×C4×C5—31no—
103C2×C4×C5—31no—
104C2×C4×C5C4×A411yescharacteristic
105C2×Dic5—31no—
106C2×Dic5—31no—
107C2×Dic5C4×A411yescharacteristic
108C4×D5—32no—
109C4×D5—32no—
110C4×D5—62no—
111C4×D5C2×SL(2,𝔽3)12yes—
112C4×D5#41⋊C312yes—
113Dic10—32no—
114Dic10—32no—
115C22×Q8—51no—
116C4×Q8—54no—
117C4×Q8—51no—
118C4×Q8—151no—
119C4×Q8—151no—
120C4×Q8—154no—
121C4×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
122C2×C4⋊C4—151no—
123C2×C4⋊C4—152no—
124C2×C42—51no—
125C2×C42—151no—
126C2×C42—152no—
127C2×C42—152no—
128C2×C42—151no—
129#95—152no—
130C2×C3×C5—41no—
131C2×C3×C5—41no—
132C2×C3×C5—41no—
133C3×D5—42no—
134C3×D5—42no—
135C23×C3—201no—
136C2×C3×C4—41no—
137C2×C3×C4—202no—
138C2×C3×C4—202no—
139C2×C3×C4—201no—
140C2×C3×C4—204no—
141C2×C3×C4—204no—
142C2×C3×C4—204no—
143SL(2,𝔽3)C4×AGL(1,𝔽5)11yescharacteristic, direct product factor
144C22×C5C2×C4×A411yescharacteristic
145C2×D5C2×C4×A412yes—
146C2×D5C22×SL(2,𝔽3)11yescharacteristic
147C2×D5C2×#41⋊C312yes—
148C2×D5C2×#41⋊C311yescharacteristic
149AGL(1,𝔽5)—64no—
150AGL(1,𝔽5)C4×SL(2,𝔽3)18yesdirect product factor
151C4×C5—31no—
152C4×C5—31no—
153C4×C5—61no—
154C4×C5C4×SL(2,𝔽3)12yescomplemented
155Dic5—32no—
156Dic5—61no—
157Dic5C4×SL(2,𝔽3)12yescomplemented
158C2×Q8—52no—
159C2×Q8—151no—
160C2×Q8—152no—
161C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
162C22×C4—52no—
163C22×C4—51no—
164C22×C4—151no—
165C22×C4—151no—
166C22×C4—152no—
167C4⋊C4—31no—
168C4⋊C4—151no—
169C4⋊C4—152no—
170C4⋊C4—154no—
171C4⋊C4—154no—
172C42—31no—
173C42—54no—
174C42—151no—
175C42—151no—
176C42—154no—
177C42—154no—
178C42—151no—
179C42—304no—
180C42—302no—
181C3×C5—41noHall
182C22×C3—41no—
183C22×C3—201no—
184C22×C3—201no—
185C22×C3—202no—
186C22×C3—202no—
187C3×C4—42no—
188C3×C4—202no—
189C3×C4—208no—
190C2×C5C42×A411yescharacteristic
191C2×C5C2×C4×SL(2,𝔽3)11yescharacteristic
192C2×C5C4×#41⋊C311yescharacteristic
193D5C2×C4×SL(2,𝔽3)12yes—
194D5C4×#41⋊C312yes—
195C23—51no—
196Q8—31no—
197Q8—152no—
198Q8—152no—
199Q8C3×C4×AGL(1,𝔽5)11yescharacteristic, complemented
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—54no—
203C2×C4—54no—
204C2×C4—54no—
205C2×C4—51no—
206C2×C4—52no—
207C2×C4—52no—
208C2×C4—152no—
209C2×C4—152no—
210C2×C4—151no—
211C2×C4—154no—
212C2×C4—151no—
213C2×C4—302no—
214C2×C4—302no—
215C2×C4—302no—
216C2×C4A4×AGL(1,𝔽5)11yescharacteristic
217C2×C3—41no—
218C2×C3—41no—
219C2×C3—41no—
220C2×C3—202no—
221C2×C3—202no—
222C5C42×SL(2,𝔽3)11yescharacteristic, complemented, Sylow
223C22—52no—
224C22—51no—
225C22—51no—
226C22—52no—
227C22C2×A4×AGL(1,𝔽5)11yescharacteristic
228C4—31no—
229C4—31no—
230C4—58no—
231C4—52no—
232C4—61no—
233C4—152no—
234C4—304no—
235C4—301no—
236C4AGL(1,𝔽5)×SL(2,𝔽3)12yesdirect product factor
237C3—41noSylow
238C2—52no—
239C2—52no—
240C2C4×A4×AGL(1,𝔽5)11yescharacteristic
241C2C2×AGL(1,𝔽5)×SL(2,𝔽3)11yescharacteristic
242C2AGL(1,𝔽5)×#41⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2×C4A4×AGL(1,𝔽5)
commutator, hypocenterC5×Q8C3×C42
FrattiniC22C2×A4×AGL(1,𝔽5)
FittingC4×C5×Q8C3×C4
socleC22×C5C2×C4×A4
prime radicalC2×D5×SL(2,𝔽3)C22
cosocleC2×Q8×D5C22×C3
LCSG ▷ C5×Q8
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C4×Q8×AGL(1,𝔽5)C3semidirect
6. AGL(1,𝔽5)×SL(2,𝔽3)C4direct (N × H)
7. C4×C5×SL(2,𝔽3)C4semidirect
8. Dic5×SL(2,𝔽3)C4semidirect
24. Q8×AGL(1,𝔽5)C3×C4semidirect
30. C4×C5×Q8C3×C4semidirect
31. Q8×Dic5C3×C4semidirect
46. C5×SL(2,𝔽3)C42semidirect
51. C4×SL(2,𝔽3)AGL(1,𝔽5)direct (N × H)
67. C4×AGL(1,𝔽5)SL(2,𝔽3)direct (N × H)
101. C5×Q8C3×C42semidirect
121. C4×Q8C3×AGL(1,𝔽5)semidirect
143. SL(2,𝔽3)C4×AGL(1,𝔽5)direct (N × H)
150. AGL(1,𝔽5)C4×SL(2,𝔽3)direct (N × H)
154. C4×C5C4×SL(2,𝔽3)semidirect
157. Dic5C4×SL(2,𝔽3)semidirect
199. Q8C3×C4×AGL(1,𝔽5)semidirect
222. C5C42×SL(2,𝔽3)semidirect
236. C4AGL(1,𝔽5)×SL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×AGL(1,𝔽5)×SL(2,𝔽3)2
C4×D5×SL(2,𝔽3)2
C4×Q8×AGL(1,𝔽5)3
C2×C3×C4×AGL(1,𝔽5)4
C42×SL(2,𝔽3)5