Finite Group Databasebeta

← Dic10×GL(2,𝔽3) #14512712

Subgroups — Dic10×GL(2,𝔽3) #14512712

Order 1920.

510 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 63 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
1SL(2,𝔽3)×Dic10C211yesmaximal, characteristic, complemented
2#11752105C211yesmaximal, characteristic, complemented
3C4×C5×GL(2,𝔽3)C211yesmaximal, characteristic
4Dic5×GL(2,𝔽3)C212yesmaximal
5#11751608C212yesmaximal, complemented
6SD16×Dic10—31nomaximal, Hall
7C2×D3×Dic10—41nomaximal
8C2×C5×GL(2,𝔽3)C2211yescharacteristic
9C4×C5×SL(2,𝔽3)C2211yescharacteristic
10Dic5×SL(2,𝔽3)C2212yes—
11#91184C2212yes—
12C5×#859C2211yescharacteristic
13Q8×GL(2,𝔽3)—51nomaximal, Hall
14Q8×Dic10D311yescharacteristic, complemented
15D4×Dic10—31no—
16C4×C5×SD16—31no—
17SD16×Dic5—32no—
18#65309—31no—
19#65297—31no—
20#65144—32no—
21#65080—32no—
22#65050—32no—
23#64969—31no—
24C8×Dic10—31no—
25C2×Dic30—41no—
26C2×C4×C5×D3—41no—
27C2×C3×Dic10—41no—
28C2×C5×SL(2,𝔽3)C2311yescharacteristic
29C2×#1204—42no—
30C2×D3×Dic5—42no—
31D3×Dic10—48no—
32C5×GL(2,𝔽3)Q812yescomplemented
33Q8×SL(2,𝔽3)—51no—
34C4×GL(2,𝔽3)—52no—
35C4×GL(2,𝔽3)D511yescharacteristic
36#5386—51no—
37#5386—52no—
38C22×Dic10—61no—
39C2×C5×SD16—31no—
40C4×C5×Q8C2×D311yescharacteristic
41C4×C5×D4—31no—
42Q8×Dic5C2×D312yes—
43D4×Dic5—32no—
44#4070—61no—
45#4034—32no—
46#4034—32no—
47#4024—62no—
48#4015—31no—
49#4015—31no—
50C4×Dic10—31no—
51C5×#106—31no—
52C5×#103—31no—
53C5×#102—31no—
54C4×C5×C8—31no—
55#3967—32no—
56#3964—32no—
57#3949—32no—
58#3949—32no—
59#3946—32no—
60C8×Dic5—32no—
61#3940—32no—
62#3940—32no—
63C4⋊#168—31no—
64Q8×SD16—151noSylow
65C22×C5×D3—41no—
66C2×C3×C4×C5—41no—
67C2×Dic15—42no—
68Dic30—44no—
69C2×C5×Dic3—41no—
70C4×C5×D3—44no—
71C2×C3×Dic5—42no—
72C3×Dic10—44no—
73C5×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
74#1204—48no—
75D3×Dic5—48no—
76C2×D3×Q8—201no—
77C2×GL(2,𝔽3)C2×D511yescharacteristic
78C4×SL(2,𝔽3)—52no—
79C4×SL(2,𝔽3)C2×D511yescharacteristic
80#859—52no—
81#859C2×D511yescharacteristic
82C2×C5×Q8C22×D311yescharacteristic
83C2×C5×D4—31no—
84C22×C4×C5—61no—
85C22×Dic5—62no—
86C2×Dic10—61no—
87C2×Dic10—62no—
88C2×Dic10—61no—
89C2×Dic10—124no—
90C2×Dic10S411yescharacteristic
91C5×SD16—32no—
92C5×SD16—32no—
93C2×C5×C8—31no—
94C2×C5×C8—31no—
95C5×C4⋊C4—31no—
96C5×C4⋊C4—31no—
97C5×C22⋊C4—61no—
98C42×C5—31no—
99C22⋊Dic5—62no—
100C4⋊Dic5—31no—
101C4⋊Dic5—32no—
102C4⋊Dic5—32no—
103C4⋊Dic5—61no—
104#683—32no—
105#683—62no—
106C4×Dic5—32no—
107C2×#168—32no—
108Q82—51no—
109D4×Q8—151no—
110#498—151no—
111#498—152no—
112#476—152no—
113#476—151no—
114#475—152no—
115#475—151no—
116C8×Q8—151no—
117C4×SD16—31no—
118C4×SD16—152no—
119C22×C3×C5—41no—
120C2×C5×D3—42no—
121C2×C5×D3—42no—
122C2×C5×D3—42no—
123C3×C4×C5—42no—
124Dic15—44no—
125C3×Dic5—44no—
126C5×Dic3—42no—
127C2×C3×Q8—201no—
128D3×Q8—208no—
129C2×C4×D3—41no—
130C2×C4×D3—202no—
131C2×Dic6—202no—
132C2×Dic6—201no—
133C2×SL(2,𝔽3)C22×D511yescharacteristic
134GL(2,𝔽3)Dic1012yesdirect product factor
135C23×C5—61no—
136C5×Q8—31no—
137C5×Q8D3×Q811yescharacteristic, complemented
138C5×D4—32no—
139C5×D4—61no—
140C2×C4×C5—31no—
141C2×C4×C5—31no—
142C2×C4×C5—61no—
143C2×C4×C5—122no—
144C2×C4×C5C2×S411yescharacteristic
145C2×Dic5—32no—
146C2×Dic5—62no—
147C2×Dic5—124no—
148C2×Dic5C2×S412yes—
149Dic10—122no—
150Dic10—124no—
151Dic10—241no—
152Dic10GL(2,𝔽3)14yesdirect product factor
153C5×C8—32no—
154C5×C8—61no—
155#168—62no—
156C22×Q8—301no—
157C2×SD16—31no—
158C4⋊Q8—151no—
159C4⋊Q8—151no—
160C4⋊Q8—152no—
161C4⋊Q8—152no—
162C22⋊Q8—302no—
163C22⋊Q8—301no—
164C4×Q8—52no—
165C4×Q8—151no—
166C4×Q8D3×D511yescharacteristic
167C4×D4—31no—
168C4×D4—152no—
169#106—31no—
170#106—152no—
171#106—152no—
172#106—152no—
173#106—152no—
174C4⋊C8—151no—
175C4⋊C8—152no—
176#103—31no—
177#103—152no—
178#102—31no—
179#102—152no—
180C4×C8—31no—
181C4×C8—152no—
182C2×C3×C5—41no—
183C2×C3×C5—41no—
184C2×C3×C5—41no—
185C5×D3—44no—
186C22×D3—41no—
187C3×Q8—204no—
188C2×C3×C4—41no—
189C2×C3×C4—202no—
190C2×Dic3—41no—
191C2×Dic3—202no—
192C4×D3—44no—
193C4×D3—208no—
194Dic6—208no—
195Dic6—204no—
196SL(2,𝔽3)C2×Dic1011yescharacteristic, complemented
197C22×C5—62no—
198C22×C5—121no—
199C22×C5—121no—
200C22×C5C22×S411yescharacteristic
201C4×C5—31no—
202C4×C5—31no—
203C4×C5—61no—
204C4×C5—121no—
205C4×C5C2×GL(2,𝔽3)12yes—
206Dic5—62no—
207Dic5—122no—
208Dic5C2×GL(2,𝔽3)14yes—
209C2×Q8—51no—
210C2×Q8—301no—
211C2×Q8—302no—
212C2×Q8—301no—
213C2×Q8—604no—
214C2×Q8C2×D3×D511yescharacteristic
215C2×D4—31no—
216C22×C4—61no—
217C22×C4—302no—
218SD16—32no—
219SD16—32no—
220C2×C8—31no—
221C2×C8—31no—
222C2×C8—152no—
223C4⋊C4—31no—
224C4⋊C4—31no—
225C4⋊C4—152no—
226C4⋊C4—152no—
227C4⋊C4—152no—
228C4⋊C4—151no—
229C4⋊C4—301no—
230C4⋊C4—302no—
231C22⋊C4—61no—
232C22⋊C4—302no—
233C42—31no—
234C42—152no—
235C3×C5—41noHall
236C22×C3—41no—
237C2×D3—42no—
238C2×D3—42no—
239C2×D3—42no—
240C3×C4—42no—
241C3×C4—204no—
242Dic3—42no—
243Dic3—204no—
244C2×C5—122no—
245C2×C5C22×GL(2,𝔽3)11yescharacteristic
246C2×C5Q8×S411yescharacteristic
247C2×C5#592411yescharacteristic
248C23—61no—
249Q8—31no—
250Q8—54no—
251Q8—604no—
252Q8—602no—
253Q8—1201no—
254Q8D3×Dic1011yescharacteristic, complemented
255D4—32no—
256D4—61no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—52no—
260C2×C4—61no—
261C2×C4—122no—
262C2×C4—152no—
263C2×C4—302no—
264C2×C4—604no—
265C2×C4D5×S411yescharacteristic
266C8—32no—
267C8—61no—
268C8—302no—
269C2×C3—41no—
270C2×C3—41no—
271C2×C3—41no—
272D3—44no—
273C5Q8×GL(2,𝔽3)11yescharacteristic, complemented, Sylow
274C22—62no—
275C22—121no—
276C22—121no—
277C22C2×D5×S411yescharacteristic
278C4—31no—
279C4—31no—
280C4—54no—
281C4—61no—
282C4—121no—
283C4—302no—
284C4—602no—
285C4D5×GL(2,𝔽3)12yes—
286C3—41noSylow
287C2—122no—
288C2S4×Dic1011yescharacteristic
289C2C2×D5×GL(2,𝔽3)11yescharacteristic
290C2#1175677311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC4×C5×Q8C2×D3
socleC22×C5C22×S4
prime radicalC2×C5×GL(2,𝔽3)C22
hypercenterC2×C4D5×S4
hypocenterC5×SL(2,𝔽3)C2×Q8
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. SL(2,𝔽3)×Dic10C2semidirect
2. #11752105C2semidirect
5. #11751608C2semidirect
14. Q8×Dic10D3semidirect
32. C5×GL(2,𝔽3)Q8semidirect
73. C5×SL(2,𝔽3)C2×Q8semidirect
134. GL(2,𝔽3)Dic10direct (N × H)
137. C5×Q8D3×Q8semidirect
152. Dic10GL(2,𝔽3)direct (N × H)
196. SL(2,𝔽3)C2×Dic10semidirect
254. Q8D3×Dic10semidirect
273. C5Q8×GL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×Dic102
#117521052
C4×C5×GL(2,𝔽3)2
Dic5×GL(2,𝔽3)2
#117516082
SD16×Dic103
C2×D3×Dic104
Q8×GL(2,𝔽3)5