Finite Group Databasebeta

← C2×C5×D4×SL(2,𝔽3) #14511333

Subgroups — C2×C5×D4×SL(2,𝔽3) #14511333

Order 1920.

1006 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 232 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
1C23×C5×SL(2,𝔽3)C212yesmaximal, complemented
2C5×D4×SL(2,𝔽3)C214yesmaximal, direct product factor
3C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×C5×D4×Q8C311yesmaximal, characteristic, complemented, Hall
5C22×C3×C5×D4—41nomaximal
6C22×C5×SL(2,𝔽3)—24no—
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C22×C5×SL(2,𝔽3)C2214yescomplemented
9C4×C5×SL(2,𝔽3)C2212yescomplemented
10C2×D4×SL(2,𝔽3)C511yesmaximal, characteristic, direct product factor, Hall
11C23×C5×Q8C2×C312yescomplemented
12C5×D4×Q8—34no—
13C5×D4×Q8C2×C314yescomplemented
14C2×C5×C4⋊Q8—31no—
15C2×C5×C22⋊Q8—32no—
16C2×C4×C5×Q8C2×C311yescharacteristic, complemented
17C2×C4×C5×D4—31no—
18C24×C3×C5—42no—
19C2×C3×C5×D4—48no—
20C2×C3×C5×D4—44no—
21C22×C3×C4×C5—41no—
22C2×C5×SL(2,𝔽3)—24no—
23C2×C5×SL(2,𝔽3)D412yesdirect product factor
24C2×C5×SL(2,𝔽3)C2311yescharacteristic
25C23×SL(2,𝔽3)C2×C512yescomplemented
26D4×SL(2,𝔽3)C2×C514yesdirect product factor
27C2×C4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
28C22×C5×Q8—24no—
29C22×C5×Q8—32no—
30C22×C5×Q8—34no—
31C22×C5×Q8—64no—
32C22×C5×Q8C22×C311yescharacteristic
33C22×C5×Q8C22×C314yescomplemented
34C22×C5×D4A411yescharacteristic
35C23×C4×C5—32no—
36C5×C4⋊Q8—34no—
37C5×C4⋊Q8—34no—
38C5×C22⋊Q8—38no—
39C5×C22⋊Q8—38no—
40C4×C5×Q8—32no—
41C4×C5×Q8C22×C312yescomplemented
42C4×C5×D4—34no—
43C4×C5×D4—34no—
44C2×C5×C4⋊C4—31no—
45C2×C5×C4⋊C4—32no—
46C2×C5×C4⋊C4—31no—
47C2×C5×C22⋊C4—32no—
48C2×C42×C5—31no—
49C2×D4×Q8C3×C511yescharacteristic, complemented, Sylow
50C23×C3×C5—41no—
51C23×C3×C5—48no—
52C23×C3×C5—44no—
53C23×C3×C5—84no—
54C23×C3×C5—84no—
55C3×C5×D4—416no—
56C2×C3×C4×C5—44no—
57C2×C3×C4×C5—42no—
58C5×SL(2,𝔽3)C2×D411yescharacteristic, direct product factor
59C22×C3×D4—41no—
60C22×SL(2,𝔽3)—24no—
61C22×SL(2,𝔽3)C22×C511yescharacteristic
62C22×SL(2,𝔽3)C22×C514yescomplemented
63C4×SL(2,𝔽3)C22×C512yescomplemented
64C24×C5C2×A412yes—
65C2×C5×Q8—24no—
66C2×C5×Q8—34no—
67C2×C5×Q8—31no—
68C2×C5×Q8—32no—
69C2×C5×Q8—34no—
70C2×C5×Q8—64no—
71C2×C5×Q8—64no—
72C2×C5×Q8—68no—
73C2×C5×Q8—64no—
74C2×C5×Q8—68no—
75C2×C5×Q8C3×D412yescomplemented
76C2×C5×Q8C23×C311yescharacteristic
77C2×C5×D4SL(2,𝔽3)18yesdirect product factor
78C2×C5×D4C2×A414yes—
79C22×C4×C5—31no—
80C22×C4×C5—34no—
81C22×C4×C5—34no—
82C22×C4×C5—31no—
83C22×C4×C5—32no—
84C22×C4×C5—64no—
85C22×C4×C5—64no—
86C22×C4×C5C2×A411yescharacteristic
87C5×C4⋊C4—32no—
88C5×C4⋊C4—32no—
89C5×C4⋊C4—38no—
90C5×C4⋊C4—34no—
91C5×C22⋊C4—38no—
92C42×C5—32no—
93C42×C5—32no—
94C23×Q8C2×C3×C512yescomplemented
95D4×Q8—34no—
96D4×Q8C2×C3×C514yescomplemented
97C2×C4⋊Q8—31no—
98C2×C22⋊Q8—32no—
99C2×C4×Q8C2×C3×C511yescharacteristic, complemented
100C2×C4×D4—31no—
101C22×C3×C5—42no—
102C22×C3×C5—42no—
103C22×C3×C5—48no—
104C22×C3×C5—42no—
105C22×C3×C5—41no—
106C22×C3×C5—84no—
107C22×C3×C5—84no—
108C22×C3×C5—816no—
109C3×C4×C5—44no—
110C24×C3—42no—
111C2×C3×D4—48no—
112C2×C3×D4—44no—
113C22×C3×C4—41no—
114C2×SL(2,𝔽3)—24no—
115C2×SL(2,𝔽3)C5×D412yesdirect product factor
116C2×SL(2,𝔽3)C23×C511yescharacteristic
117C23×C5—24no—
118C23×C5—24no—
119C23×C5C2×SL(2,𝔽3)18yescomplemented
120C23×C5C22×A414yes—
121C23×C5C22×A411yescharacteristic
122C5×Q8—32no—
123C5×Q8—31no—
124C5×Q8—32no—
125C5×Q8—64no—
126C5×Q8—64no—
127C5×Q8—68no—
128C5×Q8C2×C3×D411yescharacteristic, complemented
129C5×D4C2×SL(2,𝔽3)116yesdirect product factor
130C2×C4×C5—32no—
131C2×C4×C5—32no—
132C2×C4×C5—31no—
133C2×C4×C5—31no—
134C2×C4×C5—32no—
135C2×C4×C5—34no—
136C2×C4×C5—64no—
137C2×C4×C5—68no—
138C2×C4×C5—64no—
139C2×C4×C5—64no—
140C2×C4×C5—62no—
141C2×C4×C5C2×SL(2,𝔽3)14yescomplemented
142C2×C4×C5C22×A412yes—
143C22×Q8—24no—
144C22×Q8—32no—
145C22×Q8—34no—
146C22×Q8—64no—
147C22×Q8C22×C3×C511yescharacteristic
148C22×Q8C22×C3×C514yescomplemented
149C22×D4C5×A411yescharacteristic
150C23×C4—32no—
151C4⋊Q8—34no—
152C4⋊Q8—34no—
153C22⋊Q8—38no—
154C22⋊Q8—38no—
155C4×Q8—32no—
156C4×Q8C22×C3×C512yescomplemented
157C4×D4—34no—
158C4×D4—34no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—32no—
161C2×C4⋊C4—31no—
162C2×C22⋊C4—32no—
163C2×C42—31no—
164C2×C3×C5—41no—
165C2×C3×C5—41no—
166C2×C3×C5—41no—
167C2×C3×C5—44no—
168C2×C3×C5—88no—
169C23×C3—48no—
170C23×C3—41no—
171C23×C3—44no—
172C23×C3—84no—
173C23×C3—84no—
174C3×D4—416no—
175C2×C3×C4—44no—
176C2×C3×C4—42no—
177SL(2,𝔽3)C2×C5×D411yescharacteristic, direct product factor
178C22×C5—24no—
179C22×C5—24no—
180C22×C5—216no—
181C22×C5D4×A412yes—
182C22×C5C22×SL(2,𝔽3)12yes—
183C22×C5C22×SL(2,𝔽3)18yescomplemented
184C22×C52−1+4⋊C312yes—
185C22×C5C23×A411yescharacteristic
186C4×C5—31no—
187C4×C5—32no—
188C4×C5—31no—
189C4×C5—64no—
190C4×C5—62no—
191C4×C5C22×SL(2,𝔽3)14yescomplemented
192C24C2×C5×A412yes—
193C2×Q8—24no—
194C2×Q8—34no—
195C2×Q8—31no—
196C2×Q8—32no—
197C2×Q8—34no—
198C2×Q8—64no—
199C2×Q8—64no—
200C2×Q8—68no—
201C2×Q8—64no—
202C2×Q8—68no—
203C2×Q8C3×C5×D412yescomplemented
204C2×Q8C23×C3×C511yescharacteristic
205C2×D4C5×SL(2,𝔽3)18yesdirect product factor
206C2×D4C2×C5×A414yes—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—32no—
210C22×C4—34no—
211C22×C4—34no—
212C22×C4—64no—
213C22×C4—64no—
214C22×C4C2×C5×A411yescharacteristic
215C4⋊C4—32no—
216C4⋊C4—32no—
217C4⋊C4—38no—
218C4⋊C4—34no—
219C22⋊C4—38no—
220C42—32no—
221C42—32no—
222C3×C5—41noHall
223C22×C3—42no—
224C22×C3—42no—
225C22×C3—48no—
226C22×C3—42no—
227C22×C3—41no—
228C22×C3—84no—
229C22×C3—816no—
230C22×C3—84no—
231C3×C4—44no—
232C2×C5—28no—
233C2×C5D4×SL(2,𝔽3)14yesdirect product factor
234C2×C5C2×D4×A411yescharacteristic
235C2×C5C23×SL(2,𝔽3)11yescharacteristic
236C2×C5C2×2−1+4⋊C311yescharacteristic
237C23—24no—
238C23—24no—
239C23C2×C5×SL(2,𝔽3)18yescomplemented
240C23C22×C5×A414yes—
241C23C22×C5×A411yescharacteristic
242Q8—32no—
243Q8—31no—
244Q8—32no—
245Q8—64no—
246Q8—64no—
247Q8—68no—
248Q8C2×C3×C5×D411yescharacteristic, complemented
249D4C2×C5×SL(2,𝔽3)116yesdirect product factor
250C2×C4—32no—
251C2×C4—32no—
252C2×C4—31no—
253C2×C4—31no—
254C2×C4—32no—
255C2×C4—34no—
256C2×C4—64no—
257C2×C4—64no—
258C2×C4—68no—
259C2×C4—64no—
260C2×C4—62no—
261C2×C4C2×C5×SL(2,𝔽3)14yescomplemented
262C2×C4C22×C5×A412yes—
263C2×C3—41no—
264C2×C3—41no—
265C2×C3—41no—
266C2×C3—44no—
267C2×C3—88no—
268C5C2×D4×SL(2,𝔽3)11yescharacteristic, direct product factor, Sylow
269C22—24no—
270C22—24no—
271C22—216no—
272C22C5×D4×A412yes—
273C22C22×C5×SL(2,𝔽3)12yes—
274C22C22×C5×SL(2,𝔽3)18yescomplemented
275C22C5×2−1+4⋊C312yes—
276C22C23×C5×A411yescharacteristic
277C4—31no—
278C4—32no—
279C4—31no—
280C4—64no—
281C4—62no—
282C4C22×C5×SL(2,𝔽3)14yescomplemented
283C3—41noSylow
284C2—28no—
285C2C5×D4×SL(2,𝔽3)14yesdirect product factor
286C2C2×C5×D4×A411yescharacteristic
287C2C23×C5×SL(2,𝔽3)11yescharacteristic
288C2C2×C5×2−1+4⋊C311yescharacteristic
subgroupG/H
center, socleC23×C5C22×A4
commutator, cosocleC2×Q8C23×C3×C5
FrattiniC22C23×C5×A4
FittingC2×C5×D4×Q8C3
hypercenterC22×C5×D4A4
hypocenterQ8C2×C3×C5×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C23×C5 ◁ C22×C5×D4

Complements

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

NH ≅ G/Ntype
1. C23×C5×SL(2,𝔽3)C2semidirect
2. C5×D4×SL(2,𝔽3)C2direct (N × H)
3. C2×C4×C5×SL(2,𝔽3)C2semidirect
4. C2×C5×D4×Q8C3semidirect
8. C22×C5×SL(2,𝔽3)C22semidirect
9. C4×C5×SL(2,𝔽3)C22semidirect
10. C2×D4×SL(2,𝔽3)C5direct (N × H)
11. C23×C5×Q8C2×C3semidirect
13. C5×D4×Q8C2×C3semidirect
16. C2×C4×C5×Q8C2×C3semidirect
23. C2×C5×SL(2,𝔽3)D4direct (N × H)
25. C23×SL(2,𝔽3)C2×C5semidirect
26. D4×SL(2,𝔽3)C2×C5direct (N × H)
27. C2×C4×SL(2,𝔽3)C2×C5semidirect
33. C22×C5×Q8C22×C3semidirect
41. C4×C5×Q8C22×C3semidirect
49. C2×D4×Q8C3×C5semidirect
58. C5×SL(2,𝔽3)C2×D4direct (N × H)
62. C22×SL(2,𝔽3)C22×C5semidirect
63. C4×SL(2,𝔽3)C22×C5semidirect
75. C2×C5×Q8C3×D4semidirect
77. C2×C5×D4SL(2,𝔽3)direct (N × H)
94. C23×Q8C2×C3×C5semidirect
96. D4×Q8C2×C3×C5semidirect
99. C2×C4×Q8C2×C3×C5semidirect
115. C2×SL(2,𝔽3)C5×D4direct (N × H)
119. C23×C5C2×SL(2,𝔽3)semidirect
128. C5×Q8C2×C3×D4semidirect
129. C5×D4C2×SL(2,𝔽3)direct (N × H)
141. C2×C4×C5C2×SL(2,𝔽3)semidirect
148. C22×Q8C22×C3×C5semidirect
156. C4×Q8C22×C3×C5semidirect
177. SL(2,𝔽3)C2×C5×D4direct (N × H)
183. C22×C5C22×SL(2,𝔽3)semidirect
191. C4×C5C22×SL(2,𝔽3)semidirect
203. C2×Q8C3×C5×D4semidirect
205. C2×D4C5×SL(2,𝔽3)direct (N × H)
233. C2×C5D4×SL(2,𝔽3)direct (N × H)
239. C23C2×C5×SL(2,𝔽3)semidirect
248. Q8C2×C3×C5×D4semidirect
249. D4C2×C5×SL(2,𝔽3)direct (N × H)
261. C2×C4C2×C5×SL(2,𝔽3)semidirect
268. C5C2×D4×SL(2,𝔽3)direct (N × H)
274. C22C22×C5×SL(2,𝔽3)semidirect
282. C4C22×C5×SL(2,𝔽3)semidirect
285. C2C5×D4×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
C23×C5×SL(2,𝔽3)2
C5×D4×SL(2,𝔽3)2
C2×C4×C5×SL(2,𝔽3)2
C2×C5×D4×Q83
C22×C3×C5×D44
C2×D4×SL(2,𝔽3)5