Finite Group Databasebeta

← C2×C4×C5×GL(2,𝔽3) #14511061

Subgroups — C2×C4×C5×GL(2,𝔽3) #14511061

Order 1920.

766 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 124 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×C5×GL(2,𝔽3)C211yesmaximal, characteristic
2C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×C5×#859C211yesmaximal, characteristic, complemented
4C4×C5×GL(2,𝔽3)C214yesmaximal, direct product factor
5C2×C4×C5×SD16—31nomaximal, Hall
6C22×C4×C5×D3—41nomaximal
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C2×C5×GL(2,𝔽3)C414yesdirect product factor
9C2×C5×GL(2,𝔽3)C2212yes—
10C4×C5×SL(2,𝔽3)C2212yescomplemented
11C5×#859C2212yescomplemented
12C2×C4×GL(2,𝔽3)C511yesmaximal, characteristic, direct product factor, Hall
13C22×C5×SD16—31no—
14C2×C4×C5×Q8D311yescharacteristic, complemented
15C2×C4×C5×D4—31no—
16C4×C5×SD16—34no—
17C4×C5×SD16—34no—
18C2×C5×#106—31no—
19C2×C5×#103—31no—
20C2×C5×#102—31no—
21C2×C4×C5×C8—31no—
22C23×C5×D3—41no—
23C22×C3×C4×C5—41no—
24C22×C5×Dic3—41no—
25C2×C4×C5×D3—44no—
26C2×C4×C5×D3—48no—
27C2×C5×SL(2,𝔽3)C2×C412yescomplemented
28C2×C5×SL(2,𝔽3)C2311yescharacteristic
29C5×GL(2,𝔽3)C2×C414yesdirect product factor
30C22×GL(2,𝔽3)C2×C511yescharacteristic
31C2×C4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
32C2×#859C2×C511yescharacteristic, complemented
33C4×GL(2,𝔽3)C2×C514yesdirect product factor
34C22×C5×Q8C2×D311yescharacteristic
35C22×C5×D4—31no—
36C23×C4×C5—61no—
37C2×C5×SD16—32no—
38C2×C5×SD16—34no—
39C2×C5×SD16—34no—
40C2×C5×SD16—32no—
41C22×C5×C8—31no—
42C22×C5×C8—31no—
43C4×C5×Q8—32no—
44C4×C5×Q8C2×D312yescomplemented
45C4×C5×D4—34no—
46C4×C5×D4—62no—
47C2×C5×C4⋊C4—31no—
48C2×C5×C4⋊C4—31no—
49C2×C5×C22⋊C4—61no—
50C2×C42×C5—31no—
51C5×#106—34no—
52C5×#103—32no—
53C5×#103—32no—
54C5×#102—34no—
55C4×C5×C8—34no—
56C2×C4×SD16—31noSylow
57C23×C3×C5—41no—
58C22×C5×D3—44no—
59C22×C5×D3—42no—
60C22×C5×D3—44no—
61C22×C5×D3—44no—
62C2×C3×C4×C5—44no—
63C2×C3×C4×C5—42no—
64C2×C5×Dic3—42no—
65C2×C5×Dic3—44no—
66C4×C5×D3—416no—
67C5×SL(2,𝔽3)C22×C411yescharacteristic, complemented
68C22×C4×D3—41no—
69C22×SL(2,𝔽3)C22×C511yescharacteristic
70C2×GL(2,𝔽3)C4×C514yesdirect product factor
71C2×GL(2,𝔽3)C22×C512yes—
72C4×SL(2,𝔽3)C22×C512yescomplemented
73#859C22×C512yescomplemented
74C24×C5—61no—
75C2×C5×Q8—31no—
76C2×C5×Q8—32no—
77C2×C5×Q8C4×D312yescomplemented
78C2×C5×Q8C22×D311yescharacteristic
79C2×C5×D4—32no—
80C2×C5×D4—34no—
81C2×C5×D4—61no—
82C2×C5×D4—62no—
83C22×C4×C5—31no—
84C22×C4×C5—31no—
85C22×C4×C5—61no—
86C22×C4×C5—64no—
87C22×C4×C5—124no—
88C22×C4×C5S411yescharacteristic
89C5×SD16—38no—
90C5×SD16—34no—
91C5×SD16—34no—
92C2×C5×C8—34no—
93C2×C5×C8—32no—
94C2×C5×C8—32no—
95C2×C5×C8—62no—
96C5×C4⋊C4—32no—
97C5×C4⋊C4—32no—
98C5×C4⋊C4—61no—
99C5×C4⋊C4—61no—
100C5×C22⋊C4—64no—
101C42×C5—32no—
102C42×C5—32no—
103C22×SD16—31no—
104C2×C4×Q8C5×D311yescharacteristic, complemented
105C2×C4×D4—31no—
106C4×SD16—34no—
107C4×SD16—34no—
108C2×#106—31no—
109C2×#103—31no—
110C2×#102—31no—
111C2×C4×C8—31no—
112C22×C3×C5—42no—
113C22×C3×C5—42no—
114C22×C3×C5—42no—
115C22×C3×C5—41no—
116C2×C5×D3—44no—
117C2×C5×D3—44no—
118C2×C5×D3—416no—
119C2×C5×D3—44no—
120C3×C4×C5—44no—
121C5×Dic3—44no—
122C23×D3—41no—
123C22×C3×C4—41no—
124C22×Dic3—41no—
125C2×C4×D3—44no—
126C2×C4×D3—48no—
127C2×SL(2,𝔽3)C2×C4×C512yescomplemented
128C2×SL(2,𝔽3)C23×C511yescharacteristic
129GL(2,𝔽3)C2×C4×C514yesdirect product factor
130C23×C5—64no—
131C23×C5—62no—
132C23×C5—122no—
133C23×C5—122no—
134C23×C5C2×S411yescharacteristic
135C5×Q8—32no—
136C5×Q8—31no—
137C5×Q8—61no—
138C5×Q8C2×C4×D311yescharacteristic, complemented
139C5×D4—34no—
140C5×D4—62no—
141C5×D4—64no—
142C2×C4×C5—32no—
143C2×C4×C5—32no—
144C2×C4×C5—31no—
145C2×C4×C5—31no—
146C2×C4×C5—32no—
147C2×C4×C5—62no—
148C2×C4×C5—62no—
149C2×C4×C5—122no—
150C2×C4×C5—128no—
151C2×C4×C5GL(2,𝔽3)14yesdirect product factor
152C2×C4×C5C2×S412yes—
153C5×C8—34no—
154C5×C8—62no—
155C22×Q8C2×C5×D311yescharacteristic
156C22×D4—31no—
157C23×C4—61no—
158C2×SD16—34no—
159C2×SD16—32no—
160C2×SD16—34no—
161C2×SD16—32no—
162C22×C8—31no—
163C22×C8—31no—
164C4×Q8—32no—
165C4×Q8C2×C5×D312yescomplemented
166C4×D4—34no—
167C4×D4—62no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C22⋊C4—61no—
171C2×C42—31no—
172#106—34no—
173#103—32no—
174#103—32no—
175#102—34no—
176C4×C8—34no—
177C2×C3×C5—44no—
178C2×C3×C5—41no—
179C2×C3×C5—41no—
180C2×C3×C5—41no—
181C5×D3—48no—
182C23×C3—41no—
183C22×D3—44no—
184C22×D3—42no—
185C22×D3—44no—
186C22×D3—44no—
187C2×C3×C4—44no—
188C2×C3×C4—42no—
189C2×Dic3—42no—
190C2×Dic3—44no—
191C4×D3—416no—
192SL(2,𝔽3)C22×C4×C511yescharacteristic, complemented
193C22×C5—64no—
194C22×C5—128no—
195C22×C5—122no—
196C22×C5—122no—
197C22×C5C4×S412yes—
198C22×C5C2×GL(2,𝔽3)12yes—
199C22×C5#98512yes—
200C22×C5C22×S411yescharacteristic
201C4×C5—31no—
202C4×C5—31no—
203C4×C5—32no—
204C4×C5—62no—
205C4×C5—122no—
206C4×C5C2×GL(2,𝔽3)14yesdirect product factor
207C24—61no—
208C2×Q8—31no—
209C2×Q8—32no—
210C2×Q8C4×C5×D312yescomplemented
211C2×Q8C22×C5×D311yescharacteristic
212C2×D4—32no—
213C2×D4—34no—
214C2×D4—61no—
215C2×D4—62no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—61no—
219C22×C4—64no—
220C22×C4—124no—
221C22×C4C5×S411yescharacteristic
222SD16—38no—
223SD16—34no—
224SD16—34no—
225C2×C8—32no—
226C2×C8—34no—
227C2×C8—32no—
228C2×C8—62no—
229C4⋊C4—32no—
230C4⋊C4—32no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C22⋊C4—64no—
234C42—32no—
235C42—32no—
236C3×C5—41noHall
237C22×C3—42no—
238C22×C3—42no—
239C22×C3—42no—
240C22×C3—41no—
241C2×D3—44no—
242C2×D3—44no—
243C2×D3—44no—
244C2×D3—416no—
245C3×C4—44no—
246Dic3—44no—
247C2×C5—124no—
248C2×C5C4×GL(2,𝔽3)14yesdirect product factor
249C2×C5C2×C4×S411yescharacteristic
250C2×C5C22×GL(2,𝔽3)11yescharacteristic
251C2×C5C2×#98511yescharacteristic
252C23—64no—
253C23—62no—
254C23—122no—
255C23—122no—
256C23C2×C5×S411yescharacteristic
257Q8—32no—
258Q8—31no—
259Q8—61no—
260Q8C2×C4×C5×D311yescharacteristic, complemented
261D4—34no—
262D4—62no—
263D4—64no—
264C2×C4—32no—
265C2×C4—32no—
266C2×C4—31no—
267C2×C4—31no—
268C2×C4—32no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—122no—
272C2×C4—128no—
273C2×C4C5×GL(2,𝔽3)14yesdirect product factor
274C2×C4C2×C5×S412yes—
275C8—34no—
276C8—62no—
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—44no—
281D3—48no—
282C5C2×C4×GL(2,𝔽3)11yescharacteristic, direct product factor, Sylow
283C22—64no—
284C22—122no—
285C22—122no—
286C22—128no—
287C22C4×C5×S412yes—
288C22C2×C5×GL(2,𝔽3)12yes—
289C22C5×#98512yes—
290C22C22×C5×S411yescharacteristic
291C4—31no—
292C4—31no—
293C4—32no—
294C4—62no—
295C4—122no—
296C4C2×C5×GL(2,𝔽3)14yesdirect product factor
297C3—41noSylow
298C2—124no—
299C2C4×C5×GL(2,𝔽3)14yesdirect product factor
300C2C2×C4×C5×S411yescharacteristic
301C2C22×C5×GL(2,𝔽3)11yescharacteristic
302C2C2×C5×#98511yescharacteristic
subgroupG/H
center, hypercenterC22×C4×C5S4
commutator, hypocenterSL(2,𝔽3)C22×C4×C5
FrattiniC22C22×C5×S4
FittingC2×C4×C5×Q8D3
socleC23×C5C2×S4
prime radicalC22×C5×GL(2,𝔽3)C2
cosocleC2×SL(2,𝔽3)C23×C5
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
2. C2×C4×C5×SL(2,𝔽3)C2semidirect
3. C2×C5×#859C2semidirect
4. C4×C5×GL(2,𝔽3)C2direct (N × H)
8. C2×C5×GL(2,𝔽3)C4direct (N × H)
10. C4×C5×SL(2,𝔽3)C22semidirect
11. C5×#859C22semidirect
12. C2×C4×GL(2,𝔽3)C5direct (N × H)
14. C2×C4×C5×Q8D3semidirect
27. C2×C5×SL(2,𝔽3)C2×C4semidirect
29. C5×GL(2,𝔽3)C2×C4direct (N × H)
31. C2×C4×SL(2,𝔽3)C2×C5semidirect
32. C2×#859C2×C5semidirect
33. C4×GL(2,𝔽3)C2×C5direct (N × H)
44. C4×C5×Q8C2×D3semidirect
67. C5×SL(2,𝔽3)C22×C4semidirect
70. C2×GL(2,𝔽3)C4×C5direct (N × H)
72. C4×SL(2,𝔽3)C22×C5semidirect
73. #859C22×C5semidirect
77. C2×C5×Q8C4×D3semidirect
104. C2×C4×Q8C5×D3semidirect
127. C2×SL(2,𝔽3)C2×C4×C5semidirect
129. GL(2,𝔽3)C2×C4×C5direct (N × H)
138. C5×Q8C2×C4×D3semidirect
151. C2×C4×C5GL(2,𝔽3)direct (N × H)
165. C4×Q8C2×C5×D3semidirect
192. SL(2,𝔽3)C22×C4×C5semidirect
206. C4×C5C2×GL(2,𝔽3)direct (N × H)
210. C2×Q8C4×C5×D3semidirect
248. C2×C5C4×GL(2,𝔽3)direct (N × H)
260. Q8C2×C4×C5×D3semidirect
273. C2×C4C5×GL(2,𝔽3)direct (N × H)
282. C5C2×C4×GL(2,𝔽3)direct (N × H)
296. C4C2×C5×GL(2,𝔽3)direct (N × H)
299. C2C4×C5×GL(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
C22×C5×GL(2,𝔽3)2
C2×C4×C5×SL(2,𝔽3)2
C2×C5×#8592
C4×C5×GL(2,𝔽3)2
C2×C4×C5×SD163
C22×C4×C5×D34
C2×C4×GL(2,𝔽3)5