Finite Group Databasebeta

← C2×SL(2,𝔽3)×D20 #14514014

Subgroups — C2×SL(2,𝔽3)×D20 #14514014

Order 1920.

1006 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×D5×SL(2,𝔽3)C212yesmaximal, complemented
2C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×D20C214yesmaximal, direct product factor
4C2×Q8×D20C311yesmaximal, characteristic, complemented, Hall
5C22×C3×D20—41nomaximal
6C22×C5×SL(2,𝔽3)C2211yescharacteristic
7C2×D5×SL(2,𝔽3)—24no—
8C2×D5×SL(2,𝔽3)C2214yescomplemented
9C4×C5×SL(2,𝔽3)C2212yescomplemented
10C2×D4×SL(2,𝔽3)—51nomaximal, Hall
11C22×Q8×D5C2×C312yescomplemented
12C2×C4×C5×Q8C2×C311yescharacteristic, complemented
13Q8×D20—34no—
14Q8×D20C2×C314yescomplemented
15C2×#4043—32no—
16C2×C4×D20—31no—
17C2×#4015—31no—
18C23×C3×D5—42no—
19C22×C3×C4×C5—41no—
20C2×C3×D20—48no—
21C2×C3×D20—44no—
22C2×C5×SL(2,𝔽3)D412yescomplemented
23C2×C5×SL(2,𝔽3)C2311yescharacteristic
24D5×SL(2,𝔽3)—24no—
25C23×SL(2,𝔽3)—52no—
26D4×SL(2,𝔽3)—54no—
27C2×C4×SL(2,𝔽3)D511yescharacteristic, complemented
28C22×C5×Q8C22×C311yescharacteristic
29C2×Q8×D5—24no—
30C2×Q8×D5—34no—
31C2×Q8×D5—64no—
32C2×Q8×D5C22×C314yescomplemented
33C22×D20A411yescharacteristic
34C22×C4×D5—32no—
35C22×Dic10—32no—
36C4×C5×Q8—32no—
37C4×C5×Q8C22×C312yescomplemented
38C2×C5×C4⋊C4—31no—
39C2×C42×C5—31no—
40C2×#685—32no—
41C2×C4⋊Dic5—32no—
42C2×C4⋊Dic5—31no—
43#4043—38no—
44#4043—38no—
45C4×D20—34no—
46C4×D20—34no—
47#4015—34no—
48#4015—34no—
49C2×D4×Q8—51noSylow
50C23×C3×C5—41no—
51C22×C3×D5—48no—
52C22×C3×D5—44no—
53C22×C3×D5—84no—
54C22×C3×D5—84no—
55C2×C3×C4×C5—44no—
56C2×C3×C4×C5—42no—
57C3×D20—416no—
58C5×SL(2,𝔽3)C2×D411yescharacteristic, complemented
59C22×C3×D4—201no—
60C22×SL(2,𝔽3)—54no—
61C22×SL(2,𝔽3)—104no—
62C22×SL(2,𝔽3)C2×D511yescharacteristic
63C4×SL(2,𝔽3)C2×D512yescomplemented
64C23×D5C2×A412yes—
65C2×C5×Q8—31no—
66C2×C5×Q8—32no—
67C2×C5×Q8C3×D412yescomplemented
68C2×C5×Q8C23×C311yescharacteristic
69C22×C4×C5—31no—
70C22×C4×C5—31no—
71C22×C4×C5C2×A411yescharacteristic
72C22×Dic5—32no—
73Q8×D5—24no—
74Q8×D5—68no—
75Q8×D5—68no—
76Q8×D5—64no—
77C2×D20SL(2,𝔽3)18yesdirect product factor
78C2×D20C2×A414yes—
79C2×C4×D5—34no—
80C2×C4×D5—34no—
81C2×C4×D5—64no—
82C2×C4×D5—64no—
83C2×Dic10—34no—
84C2×Dic10—34no—
85C2×Dic10—64no—
86C2×Dic10—64no—
87C5×C4⋊C4—32no—
88C5×C4⋊C4—32no—
89C42×C5—32no—
90C42×C5—32no—
91#685—38no—
92C4⋊Dic5—38no—
93C4⋊Dic5—34no—
94C23×Q8—52no—
95D4×Q8—54no—
96D4×Q8—154no—
97C2×C4⋊Q8—151no—
98C2×C22⋊Q8—152no—
99C2×C4×Q8C3×D511yescharacteristic, complemented
100C2×C4×D4—151no—
101C22×C3×C5—42no—
102C22×C3×C5—42no—
103C22×C3×C5—42no—
104C22×C3×C5—41no—
105C2×C3×D5—48no—
106C2×C3×D5—84no—
107C2×C3×D5—84no—
108C2×C3×D5—816no—
109C3×C4×C5—44no—
110C24×C3—202no—
111C2×C3×D4—204no—
112C2×C3×D4—208no—
113C22×C3×C4—41no—
114C2×SL(2,𝔽3)—104no—
115C2×SL(2,𝔽3)D2012yesdirect product factor
116C2×SL(2,𝔽3)C22×D511yescharacteristic
117C23×C5C22×A411yescharacteristic
118C22×D5—24no—
119C22×D5—24no—
120C22×D5C2×SL(2,𝔽3)18yescomplemented
121C22×D5C22×A414yes—
122C5×Q8—32no—
123C5×Q8—31no—
124C5×Q8—32no—
125C5×Q8C2×C3×D411yescharacteristic, complemented
126C2×C4×C5—32no—
127C2×C4×C5—32no—
128C2×C4×C5—31no—
129C2×C4×C5—31no—
130C2×C4×C5—32no—
131C2×C4×C5—62no—
132C2×C4×C5C2×SL(2,𝔽3)14yescomplemented
133C2×C4×C5C22×A412yes—
134C2×Dic5—34no—
135C2×Dic5—64no—
136D20C2×SL(2,𝔽3)116yesdirect product factor
137C4×D5—64no—
138C4×D5—64no—
139C4×D5—68no—
140Dic10—68no—
141Dic10—64no—
142Dic10—64no—
143C22×Q8—54no—
144C22×Q8—104no—
145C22×Q8—152no—
146C22×Q8—154no—
147C22×Q8—304no—
148C22×Q8C2×C3×D511yescharacteristic
149C22×D4—51no—
150C23×C4—152no—
151C4⋊Q8—154no—
152C4⋊Q8—154no—
153C22⋊Q8—158no—
154C22⋊Q8—158no—
155C4×Q8—32no—
156C4×Q8C2×C3×D512yescomplemented
157C4×D4—154no—
158C4×D4—154no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—152no—
161C2×C4⋊C4—151no—
162C2×C22⋊C4—152no—
163C2×C42—31no—
164C2×C3×C5—41no—
165C2×C3×C5—41no—
166C2×C3×C5—41no—
167C2×C3×C5—44no—
168C3×D5—88no—
169C23×C3—41no—
170C23×C3—204no—
171C23×C3—208no—
172C23×C3—404no—
173C23×C3—404no—
174C3×D4—2016no—
175C2×C3×C4—44no—
176C2×C3×C4—42no—
177SL(2,𝔽3)C2×D2011yescharacteristic, direct product factor
178C22×C5D4×A412yes—
179C22×C5C22×SL(2,𝔽3)12yes—
180C22×C52−1+4⋊C312yes—
181C22×C5C23×A411yescharacteristic
182C2×D5—24no—
183C2×D5—24no—
184C2×D5—216no—
185C2×D5C22×SL(2,𝔽3)18yescomplemented
186C4×C5—32no—
187C4×C5—31no—
188C4×C5—31no—
189C4×C5—62no—
190C4×C5C22×SL(2,𝔽3)14yescomplemented
191Dic5—64no—
192C24—52no—
193C2×Q8—31no—
194C2×Q8—32no—
195C2×Q8—104no—
196C2×Q8—154no—
197C2×Q8—154no—
198C2×Q8—304no—
199C2×Q8—304no—
200C2×Q8—308no—
201C2×Q8—304no—
202C2×Q8—308no—
203C2×Q8C3×D2012yescomplemented
204C2×Q8C22×C3×D511yescharacteristic
205C2×D4—58no—
206C2×D4—54no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—154no—
210C22×C4—154no—
211C22×C4—152no—
212C22×C4—304no—
213C22×C4—304no—
214C22×C4D5×A411yescharacteristic
215C4⋊C4—32no—
216C4⋊C4—32no—
217C4⋊C4—158no—
218C4⋊C4—154no—
219C22⋊C4—158no—
220C42—32no—
221C42—32no—
222C3×C5—41noHall
223C22×C3—42no—
224C22×C3—42no—
225C22×C3—42no—
226C22×C3—41no—
227C22×C3—208no—
228C22×C3—404no—
229C22×C3—4016no—
230C22×C3—404no—
231C3×C4—44no—
232C2×C5D4×SL(2,𝔽3)14yescomplemented
233C2×C5C2×D4×A411yescharacteristic
234C2×C5C23×SL(2,𝔽3)11yescharacteristic
235C2×C5C2×2−1+4⋊C311yescharacteristic
236D5—28no—
237C23—58no—
238C23—54no—
239C23—104no—
240C23—104no—
241C23C2×D5×A411yescharacteristic
242Q8—32no—
243Q8—31no—
244Q8—32no—
245Q8—304no—
246Q8—304no—
247Q8—308no—
248Q8C2×C3×D2011yescharacteristic, complemented
249D4—516no—
250C2×C4—32no—
251C2×C4—32no—
252C2×C4—31no—
253C2×C4—31no—
254C2×C4—32no—
255C2×C4—62no—
256C2×C4—154no—
257C2×C4—304no—
258C2×C4—304no—
259C2×C4—308no—
260C2×C4—304no—
261C2×C4D5×SL(2,𝔽3)14yescomplemented
262C2×C4C2×D5×A412yes—
263C2×C3—41no—
264C2×C3—41no—
265C2×C3—41no—
266C2×C3—44no—
267C2×C3—408no—
268C5C2×D4×SL(2,𝔽3)11yescharacteristic, complemented, Sylow
269C22—58no—
270C22—104no—
271C22—104no—
272C22—1016no—
273C22A4×D2012yes—
274C22C2×D5×SL(2,𝔽3)12yes—
275C22#4150⋊C312yes—
276C22C22×D5×A411yescharacteristic
277C4—31no—
278C4—32no—
279C4—31no—
280C4—62no—
281C4—304no—
282C4C2×D5×SL(2,𝔽3)14yescomplemented
283C3—41noSylow
284C2—108no—
285C2SL(2,𝔽3)×D2014yesdirect product factor
286C2C2×A4×D2011yescharacteristic
287C2C22×D5×SL(2,𝔽3)11yescharacteristic
288C2C2×#4150⋊C311yescharacteristic
subgroupG/H
centerC23C2×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FrattiniC22C22×D5×A4
FittingC2×C4×C5×Q8C2×C3
socleC23×C5C22×A4
hypercenterC22×C4D5×A4
hypocenterC5×Q8C2×C3×D4
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×D5×SL(2,𝔽3)C2semidirect
2. C2×C4×C5×SL(2,𝔽3)C2semidirect
3. SL(2,𝔽3)×D20C2direct (N × H)
4. C2×Q8×D20C3semidirect
8. C2×D5×SL(2,𝔽3)C22semidirect
9. C4×C5×SL(2,𝔽3)C22semidirect
11. C22×Q8×D5C2×C3semidirect
12. C2×C4×C5×Q8C2×C3semidirect
14. Q8×D20C2×C3semidirect
22. C2×C5×SL(2,𝔽3)D4semidirect
27. C2×C4×SL(2,𝔽3)D5semidirect
32. C2×Q8×D5C22×C3semidirect
37. C4×C5×Q8C22×C3semidirect
58. C5×SL(2,𝔽3)C2×D4semidirect
63. C4×SL(2,𝔽3)C2×D5semidirect
67. C2×C5×Q8C3×D4semidirect
77. C2×D20SL(2,𝔽3)direct (N × H)
99. C2×C4×Q8C3×D5semidirect
115. C2×SL(2,𝔽3)D20direct (N × H)
120. C22×D5C2×SL(2,𝔽3)semidirect
125. C5×Q8C2×C3×D4semidirect
132. C2×C4×C5C2×SL(2,𝔽3)semidirect
136. D20C2×SL(2,𝔽3)direct (N × H)
156. C4×Q8C2×C3×D5semidirect
177. SL(2,𝔽3)C2×D20direct (N × H)
185. C2×D5C22×SL(2,𝔽3)semidirect
190. C4×C5C22×SL(2,𝔽3)semidirect
203. C2×Q8C3×D20semidirect
232. C2×C5D4×SL(2,𝔽3)semidirect
248. Q8C2×C3×D20semidirect
261. C2×C4D5×SL(2,𝔽3)semidirect
268. C5C2×D4×SL(2,𝔽3)semidirect
282. C4C2×D5×SL(2,𝔽3)semidirect
285. C2SL(2,𝔽3)×D20direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D5×SL(2,𝔽3)2
C2×C4×C5×SL(2,𝔽3)2
SL(2,𝔽3)×D202
C2×Q8×D203
C22×C3×D204
C2×D4×SL(2,𝔽3)5