Finite Group Databasebeta

← C2×C4×C5⋊GL(2,𝔽3) #14513725

Subgroups — C2×C4×C5⋊GL(2,𝔽3) #14513725

Order 1920.

766 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 105 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×#91184C211yesmaximal, characteristic, complemented
4C4×C5⋊GL(2,𝔽3)C214yesmaximal, direct product factor
5C2×C4×#688—31nomaximal, Hall
6C22×C4×D15—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
11#91184C2212yescomplemented
12C2×C4×GL(2,𝔽3)—51nomaximal, Hall
13C2×C4×C5×Q8D311yescharacteristic, complemented
14C22×#688—31no—
15C2×C4×D20—31no—
16C2×#3967—31no—
17C4×#688—34no—
18C4×#688—34no—
19C2×#3941—31no—
20C2×#3940—31no—
21C2×C4×#168—31no—
22C23×D15—41no—
23C22×C3×C4×C5—41no—
24C22×Dic15—41no—
25C2×C4×D15—44no—
26C2×C4×D15—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)—51no—
31C2×C4×SL(2,𝔽3)D511yescharacteristic, complemented
32C2×#859—51no—
33C4×GL(2,𝔽3)—54no—
34C22×C5×Q8C2×D311yescharacteristic
35C22×D20—31no—
36C22×C4×D5—61no—
37C4×C5×Q8—32no—
38C4×C5×Q8C2×D312yescomplemented
39C2×C5×C4⋊C4—31no—
40C2×C42×C5—31no—
41C2×#688—32no—
42C2×#688—34no—
43C2×#688—34no—
44C2×#688—32no—
45C2×#685—61no—
46C2×C4⋊Dic5—31no—
47C22×#168—31no—
48C22×#168—31no—
49C4×D20—34no—
50C4×D20—62no—
51#3967—32no—
52#3967—32no—
53#3941—34no—
54#3940—34no—
55C4×#168—34no—
56C2×C4×SD16—151noSylow
57C23×C3×C5—41no—
58C22×D15—44no—
59C22×D15—42no—
60C22×D15—44no—
61C22×D15—44no—
62C2×C3×C4×C5—44no—
63C2×C3×C4×C5—42no—
64C2×Dic15—42no—
65C2×Dic15—44no—
66C4×D15—416no—
67C5×SL(2,𝔽3)C22×C411yescharacteristic, complemented
68C22×C4×D3—201no—
69C22×SL(2,𝔽3)C2×D511yescharacteristic
70C2×GL(2,𝔽3)—54no—
71C2×GL(2,𝔽3)—52no—
72C4×SL(2,𝔽3)C2×D512yescomplemented
73#859—52no—
74C23×D5—61no—
75C2×C5×Q8—31no—
76C2×C5×Q8—32no—
77C2×C5×Q8C4×D312yescomplemented
78C2×C5×Q8C22×D311yescharacteristic
79C22×C4×C5—31no—
80C22×C4×C5—31no—
81C22×C4×C5S411yescharacteristic
82C22×Dic5—61no—
83C2×D20—32no—
84C2×D20—34no—
85C2×D20—62no—
86C2×D20—61no—
87C2×C4×D5—64no—
88C2×C4×D5—124no—
89C5×C4⋊C4—32no—
90C5×C4⋊C4—61no—
91C42×C5—32no—
92C42×C5—32no—
93#688—38no—
94#688—34no—
95#688—34no—
96#685—64no—
97C4⋊Dic5—32no—
98C4⋊Dic5—61no—
99C2×#168—32no—
100C2×#168—34no—
101C2×#168—32no—
102C2×#168—62no—
103C22×SD16—151no—
104C2×C4×Q8D1511yescharacteristic, complemented
105C2×C4×D4—151no—
106C4×SD16—154no—
107C4×SD16—154no—
108C2×#106—151no—
109C2×#103—151no—
110C2×#102—151no—
111C2×C4×C8—151no—
112C22×C3×C5—42no—
113C22×C3×C5—42no—
114C22×C3×C5—42no—
115C22×C3×C5—41no—
116C2×D15—44no—
117C2×D15—44no—
118C2×D15—416no—
119C2×D15—44no—
120C3×C4×C5—44no—
121Dic15—44no—
122C23×D3—201no—
123C22×C3×C4—41no—
124C22×Dic3—201no—
125C2×C4×D3—204no—
126C2×C4×D3—208no—
127C2×SL(2,𝔽3)C4×D512yescomplemented
128C2×SL(2,𝔽3)C22×D511yescharacteristic
129GL(2,𝔽3)—54no—
130C23×C5C2×S411yescharacteristic
131C22×D5—64no—
132C22×D5—62no—
133C22×D5—122no—
134C22×D5—122no—
135C5×Q8—32no—
136C5×Q8—31no—
137C5×Q8—61no—
138C5×Q8C2×C4×D311yescharacteristic, complemented
139C2×C4×C5—32no—
140C2×C4×C5—32no—
141C2×C4×C5—31no—
142C2×C4×C5—31no—
143C2×C4×C5—32no—
144C2×C4×C5—62no—
145C2×C4×C5GL(2,𝔽3)14yescomplemented
146C2×C4×C5C2×S412yes—
147C2×Dic5—62no—
148C2×Dic5—122no—
149D20—34no—
150D20—62no—
151D20—64no—
152C4×D5—128no—
153#168—34no—
154#168—62no—
155C22×Q8C2×D1511yescharacteristic
156C22×D4—151no—
157C23×C4—301no—
158C2×SD16—152no—
159C2×SD16—154no—
160C2×SD16—154no—
161C2×SD16—152no—
162C22×C8—151no—
163C22×C8—151no—
164C4×Q8—32no—
165C4×Q8C2×D1512yescomplemented
166C4×D4—154no—
167C4×D4—302no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—151no—
170C2×C22⋊C4—301no—
171C2×C42—31no—
172#106—154no—
173#103—152no—
174#103—152no—
175#102—154no—
176C4×C8—154no—
177C2×C3×C5—44no—
178C2×C3×C5—41no—
179C2×C3×C5—41no—
180C2×C3×C5—41no—
181D15—48no—
182C23×C3—41no—
183C22×D3—202no—
184C22×D3—204no—
185C22×D3—204no—
186C22×D3—204no—
187C2×C3×C4—44no—
188C2×C3×C4—42no—
189C2×Dic3—202no—
190C2×Dic3—204no—
191C4×D3—2016no—
192SL(2,𝔽3)C2×C4×D511yescharacteristic, complemented
193C22×C5C4×S412yes—
194C22×C5C2×GL(2,𝔽3)12yes—
195C22×C5#98512yes—
196C22×C5C22×S411yescharacteristic
197C2×D5—64no—
198C2×D5—128no—
199C2×D5—122no—
200C2×D5—122no—
201C4×C5—31no—
202C4×C5—31no—
203C4×C5—32no—
204C4×C5—62no—
205C4×C5C2×GL(2,𝔽3)14yescomplemented
206Dic5—122no—
207C24—301no—
208C2×Q8—31no—
209C2×Q8—32no—
210C2×Q8C4×D1512yescomplemented
211C2×Q8C22×D1511yescharacteristic
212C2×D4—152no—
213C2×D4—154no—
214C2×D4—301no—
215C2×D4—302no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—301no—
219C22×C4—304no—
220C22×C4—604no—
221C22×C4C5⋊S411yescharacteristic
222SD16—158no—
223SD16—154no—
224SD16—154no—
225C2×C8—152no—
226C2×C8—154no—
227C2×C8—152no—
228C2×C8—302no—
229C4⋊C4—32no—
230C4⋊C4—61no—
231C4⋊C4—152no—
232C4⋊C4—301no—
233C22⋊C4—304no—
234C42—32no—
235C42—32no—
236C3×C5—41noHall
237C22×C3—42no—
238C22×C3—42no—
239C22×C3—42no—
240C22×C3—41no—
241C2×D3—204no—
242C2×D3—204no—
243C2×D3—204no—
244C2×D3—2016no—
245C3×C4—44no—
246Dic3—204no—
247C2×C5C4×GL(2,𝔽3)14yescomplemented
248C2×C5C2×C4×S411yescharacteristic
249C2×C5C22×GL(2,𝔽3)11yescharacteristic
250C2×C5C2×#98511yescharacteristic
251D5—124no—
252C23—304no—
253C23—302no—
254C23—602no—
255C23—602no—
256C23C2×C5⋊S411yescharacteristic
257Q8—32no—
258Q8—31no—
259Q8—61no—
260Q8C2×C4×D1511yescharacteristic, complemented
261D4—154no—
262D4—302no—
263D4—304no—
264C2×C4—32no—
265C2×C4—32no—
266C2×C4—31no—
267C2×C4—31no—
268C2×C4—32no—
269C2×C4—62no—
270C2×C4—302no—
271C2×C4—602no—
272C2×C4—608no—
273C2×C4C5⋊GL(2,𝔽3)14yesdirect product factor
274C2×C4C2×C5⋊S412yes—
275C8—154no—
276C8—302no—
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—44no—
281D3—208no—
282C5C2×C4×GL(2,𝔽3)11yescharacteristic, complemented, Sylow
283C22—304no—
284C22—602no—
285C22—602no—
286C22—608no—
287C22C4×C5⋊S412yes—
288C22C2×C5⋊GL(2,𝔽3)12yes—
289C22#9195512yes—
290C22C22×C5⋊S411yescharacteristic
291C4—31no—
292C4—31no—
293C4—32no—
294C4—62no—
295C4—602no—
296C4C2×C5⋊GL(2,𝔽3)14yesdirect product factor
297C3—41noSylow
298C2—604no—
299C2C4×C5⋊GL(2,𝔽3)14yesdirect product factor
300C2C2×C4×C5⋊S411yescharacteristic
301C2C22×C5⋊GL(2,𝔽3)11yescharacteristic
302C2C2×#9195511yescharacteristic
subgroupG/H
center, hypercenterC22×C4C5⋊S4
commutator, hypocenterC5×SL(2,𝔽3)C22×C4
FrattiniC22C22×C5⋊S4
FittingC2×C4×C5×Q8D3
socleC23×C5C2×S4
prime radicalC22×C5⋊GL(2,𝔽3)C2
cosocleC2×C5×SL(2,𝔽3)C23
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C5×SL(2,𝔽3)C2semidirect
3. C2×#91184C2semidirect
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. #91184C22semidirect
13. 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)D5semidirect
38. C4×C5×Q8C2×D3semidirect
67. C5×SL(2,𝔽3)C22×C4semidirect
72. C4×SL(2,𝔽3)C2×D5semidirect
77. C2×C5×Q8C4×D3semidirect
104. C2×C4×Q8D15semidirect
127. C2×SL(2,𝔽3)C4×D5semidirect
138. C5×Q8C2×C4×D3semidirect
145. C2×C4×C5GL(2,𝔽3)semidirect
165. C4×Q8C2×D15semidirect
192. SL(2,𝔽3)C2×C4×D5semidirect
205. C4×C5C2×GL(2,𝔽3)semidirect
210. C2×Q8C4×D15semidirect
247. C2×C5C4×GL(2,𝔽3)semidirect
260. Q8C2×C4×D15semidirect
273. C2×C4C5⋊GL(2,𝔽3)direct (N × H)
282. C5C2×C4×GL(2,𝔽3)semidirect
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×#911842
C4×C5⋊GL(2,𝔽3)2
C2×C4×#6883
C22×C4×D154
C2×C4×GL(2,𝔽3)5