Finite Group Databasebeta

← SL(2,𝔽3)×C4⋊AGL(1,𝔽5) #14512529

Subgroups — SL(2,𝔽3)×C4⋊AGL(1,𝔽5) #14512529

Order 1920.

386 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 74 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
3Q8×C4⋊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)—22no—
7C4×C5×SL(2,𝔽3)C411yescharacteristic, complemented
8Dic5×SL(2,𝔽3)C411yescharacteristic, complemented
9C4⋊C4×SL(2,𝔽3)—51nomaximal, Hall
10C2×Q8×AGL(1,𝔽5)C2×C312yes—
11C4×Q8×D5C2×C311yescharacteristic
12#65712—32no—
13#65688—31no—
14C4×C4⋊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)D411yescharacteristic
20D5×SL(2,𝔽3)Q811yescharacteristic
21C2×C4×SL(2,𝔽3)—51no—
22C2×C4×SL(2,𝔽3)—52no—
23C2×Q8×D5C22×C311yescharacteristic
24Q8×AGL(1,𝔽5)—22no—
25Q8×AGL(1,𝔽5)—62no—
26C2×C4⋊AGL(1,𝔽5)—32no—
27C2×C4⋊AGL(1,𝔽5)—31no—
28C2×C4⋊AGL(1,𝔽5)—32no—
29C2×C4⋊AGL(1,𝔽5)A411yescharacteristic
30C2×C4×AGL(1,𝔽5)—32no—
31C4×C5×Q8C3×C411yescharacteristic, complemented
32Q8×Dic5C3×C411yescharacteristic, complemented
33D5×C4⋊C4—31no—
34#4033—31no—
35C42×D5—31no—
36C4×Dic10—31no—
37(C2×AGL(1,𝔽5))⋊C4—32no—
38Q8×C4⋊C4—51noSylow
39C22×C3×D5—41no—
40C2×C3×AGL(1,𝔽5)—44no—
41C2×C3×AGL(1,𝔽5)—82no—
42C2×C3×AGL(1,𝔽5)—82no—
43C2×C3×C4×C5—41no—
44C2×C3×Dic5—41no—
45C3×C4×D5—42no—
46C3×C4×D5—42no—
47C5×SL(2,𝔽3)C4⋊C411yescharacteristic, complemented
48C22×SL(2,𝔽3)—51no—
49C2×C3×C4⋊C4—201no—
50C4×SL(2,𝔽3)—51no—
51C4×SL(2,𝔽3)—102no—
52C4×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic, complemented
53C22×AGL(1,𝔽5)—32no—
54C22×AGL(1,𝔽5)C2×A412yes—
55C2×C5×Q8C2×C3×C411yescharacteristic
56Q8×D5—31no—
57Q8×D5—31no—
58Q8×D5C3×D411yescharacteristic
59Q8×D5C3×Q811yescharacteristic
60C2×C4×D5—31no—
61C2×C4×D5—31no—
62C2×C4×D5C2×A411yescharacteristic
63C2×Dic10—31no—
64C4⋊AGL(1,𝔽5)—62no—
65C4⋊AGL(1,𝔽5)—62no—
66C4⋊AGL(1,𝔽5)—62no—
67C4⋊AGL(1,𝔽5)—122no—
68C4⋊AGL(1,𝔽5)SL(2,𝔽3)14yesdirect product factor
69C4×AGL(1,𝔽5)—62no—
70C4×AGL(1,𝔽5)—62no—
71C5×C4⋊C4—31no—
72C42×C5—31no—
73C4⋊Dic5—31no—
74#683—32no—
75C4×Dic5—31no—
76C4×Dic5—31no—
77C4×Dic5—31no—
78C2×C4×Q8—51no—
79C2×C4×Q8—52no—
80#389—152no—
81#384—151no—
82C4×C4⋊C4—151no—
83C22×C3×C5—41no—
84C2×C3×D5—41no—
85C2×C3×D5—41no—
86C2×C3×D5—41no—
87C2×C3×D5—41no—
88C2×C3×D5—41no—
89C2×C3×D5—41no—
90C3×AGL(1,𝔽5)—84no—
91C3×C4×C5—42no—
92C3×Dic5—42no—
93C22×C3×C4—201no—
94C22×C3×C4—202no—
95C2×SL(2,𝔽3)—51no—
96C2×SL(2,𝔽3)—51no—
97C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
98C3×C4⋊C4—204no—
99C22×D5C22×A411yescharacteristic
100C2×AGL(1,𝔽5)—22no—
101C2×AGL(1,𝔽5)—22no—
102C2×AGL(1,𝔽5)—62no—
103C2×AGL(1,𝔽5)—62no—
104C2×AGL(1,𝔽5)—62no—
105C2×AGL(1,𝔽5)C2×SL(2,𝔽3)14yes—
106C5×Q8—31no—
107C5×Q8C3×C4⋊C411yescharacteristic, complemented
108C2×C4×C5—31no—
109C2×C4×C5—31no—
110C2×C4×C5C4×A411yescharacteristic
111C2×Dic5—31no—
112C2×Dic5—31no—
113C2×Dic5C4×A411yescharacteristic
114C4×D5—31no—
115C4×D5—31no—
116C4×D5—31no—
117C4×D5—31no—
118C4×D5—61no—
119C4×D5—61no—
120C4×D5C2×SL(2,𝔽3)12yes—
121C4×D5#41⋊C312yes—
122Dic10—32no—
123Dic10—31no—
124Dic10—31no—
125C22×Q8—51no—
126C4×Q8—51no—
127C4×Q8—102no—
128C4×Q8—151no—
129C4×Q8—151no—
130C4×Q8—302no—
131C4×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
132C2×C4⋊C4—51no—
133C2×C4⋊C4—151no—
134C2×C4⋊C4—151no—
135C2×C4⋊C4—152no—
136C2×C4⋊C4—152no—
137C2×C42—151no—
138C2×C42—152no—
139#95—152no—
140C2×C3×C5—41no—
141C2×C3×C5—41no—
142C2×C3×C5—41no—
143C3×D5—41no—
144C3×D5—41no—
145C3×D5—41no—
146C3×D5—41no—
147C23×C3—201no—
148C2×C3×C4—41no—
149C2×C3×C4—202no—
150C2×C3×C4—202no—
151C2×C3×C4—201no—
152C2×C3×C4—204no—
153C2×C3×C4—402no—
154C2×C3×C4—402no—
155SL(2,𝔽3)C4⋊AGL(1,𝔽5)11yescharacteristic, direct product factor
156C22×C5C2×C4×A411yescharacteristic
157C2×D5D4×A411yescharacteristic
158C2×D5C22×SL(2,𝔽3)11yescharacteristic
159C2×D5Q8×A411yescharacteristic
160C2×D5C2×#41⋊C311yescharacteristic
161C2×D5#99411yescharacteristic
162C2×D52−1+4⋊C311yescharacteristic
163AGL(1,𝔽5)—24no—
164AGL(1,𝔽5)—122no—
165C4×C5—31no—
166C4×C5—31no—
167C4×C5—61no—
168C4×C5C4×SL(2,𝔽3)12yescomplemented
169Dic5—31no—
170Dic5—31no—
171Dic5—61no—
172Dic5C4×SL(2,𝔽3)12yescomplemented
173C2×Q8—51no—
174C2×Q8—51no—
175C2×Q8—151no—
176C2×Q8—151no—
177C2×Q8—151no—
178C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
179C22×C4—51no—
180C22×C4—52no—
181C22×C4—151no—
182C22×C4—151no—
183C22×C4—152no—
184C4⋊C4—31no—
185C4⋊C4—54no—
186C4⋊C4—151no—
187C4⋊C4—152no—
188C4⋊C4—302no—
189C4⋊C4—302no—
190C4⋊C4—302no—
191C4⋊C4—602no—
192C42—31no—
193C42—151no—
194C42—151no—
195C42—151no—
196C42—302no—
197C42—302no—
198C3×C5—41noHall
199C22×C3—41no—
200C22×C3—201no—
201C22×C3—201no—
202C22×C3—201no—
203C22×C3—201no—
204C22×C3—201no—
205C22×C3—201no—
206C3×C4—42no—
207C3×C4—202no—
208C3×C4—404no—
209C2×C5A4×C4⋊C411yescharacteristic
210C2×C5C2×C4×SL(2,𝔽3)11yescharacteristic
211C2×C5#520⋊C311yescharacteristic
212D5D4×SL(2,𝔽3)11yescharacteristic
213D5C4⋊(#41⋊C3)11yescharacteristic
214D5#557⋊C311yescharacteristic
215D5Q8×SL(2,𝔽3)11yescharacteristic
216C23—51no—
217Q8—31no—
218Q8—151no—
219Q8—151no—
220Q8—152no—
221Q8C3×C4⋊AGL(1,𝔽5)11yescharacteristic, complemented
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—51no—
225C2×C4—52no—
226C2×C4—52no—
227C2×C4—54no—
228C2×C4—102no—
229C2×C4—102no—
230C2×C4—151no—
231C2×C4—151no—
232C2×C4—151no—
233C2×C4—151no—
234C2×C4—151no—
235C2×C4—151no—
236C2×C4—301no—
237C2×C4—301no—
238C2×C4—302no—
239C2×C4—302no—
240C2×C4—302no—
241C2×C4A4×AGL(1,𝔽5)11yescharacteristic
242C2×C3—41no—
243C2×C3—41no—
244C2×C3—41no—
245C2×C3—201no—
246C2×C3—201no—
247C2×C3—201no—
248C2×C3—201no—
249C5C4⋊C4×SL(2,𝔽3)11yescharacteristic, complemented, Sylow
250C22—51no—
251C22—51no—
252C22—51no—
253C22—51no—
254C22—51no—
255C22—51no—
256C22C2×A4×AGL(1,𝔽5)11yescharacteristic
257C4—31no—
258C4—31no—
259C4—52no—
260C4—61no—
261C4—104no—
262C4—151no—
263C4—151no—
264C4—301no—
265C4—602no—
266C4AGL(1,𝔽5)×SL(2,𝔽3)12yescomplemented
267C3—41noSylow
268C2—51no—
269C2—51no—
270C2—51no—
271C2—51no—
272C2A4×C4⋊AGL(1,𝔽5)11yescharacteristic
273C2C2×AGL(1,𝔽5)×SL(2,𝔽3)11yescharacteristic
274C2#66266⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×AGL(1,𝔽5)
commutatorC2×C5×Q8C2×C3×C4
FittingC4×C5×Q8C3×C4
socleC22×C5C2×C4×A4
prime radicalC2×D5×SL(2,𝔽3)C22
cosocleC2×Q8×D5C22×C3
hypercenterC2×C4A4×AGL(1,𝔽5)
hypocenterC5×Q8C3×C4⋊C4
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. Q8×C4⋊AGL(1,𝔽5)C3semidirect
7. C4×C5×SL(2,𝔽3)C4semidirect
8. Dic5×SL(2,𝔽3)C4semidirect
31. C4×C5×Q8C3×C4semidirect
32. Q8×Dic5C3×C4semidirect
47. C5×SL(2,𝔽3)C4⋊C4semidirect
52. C4×SL(2,𝔽3)AGL(1,𝔽5)semidirect
68. C4⋊AGL(1,𝔽5)SL(2,𝔽3)direct (N × H)
107. C5×Q8C3×C4⋊C4semidirect
131. C4×Q8C3×AGL(1,𝔽5)semidirect
155. SL(2,𝔽3)C4⋊AGL(1,𝔽5)direct (N × H)
168. C4×C5C4×SL(2,𝔽3)semidirect
172. Dic5C4×SL(2,𝔽3)semidirect
221. Q8C3×C4⋊AGL(1,𝔽5)semidirect
249. C5C4⋊C4×SL(2,𝔽3)semidirect
266. C4AGL(1,𝔽5)×SL(2,𝔽3)semidirect

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