Finite Group Databasebeta

← Dic10×#226 #14512596

Subgroups — Dic10×#226 #14512596

Order 1920.

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

normal subgroups only

This group has 390 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
2#11752110C211yesmaximal, characteristic
3C4×C5×#226C211yesmaximal, characteristic
4#11751598C212yesmaximal
5Dic5×#226C212yesmaximal
6Q16×Dic10—31nomaximal, Hall
7C2×C5×#226C2211yescharacteristic
8C4×C5×SL(2,𝔽3)C2211yescharacteristic
9Dic3×Dic10—41nomaximal
10Dic5×SL(2,𝔽3)C2212yes—
11#91184C2212yes—
12C5×#859C2211yescharacteristic
13Q8×#226—51nomaximal, Hall
14Q8×Dic10—31no—
15Q8×Dic10D311yescharacteristic
16C4×C5×Q16—31no—
17Q16×Dic5—32no—
18#65310—31no—
19#65310—31no—
20#65163—32no—
21#65083—32no—
22#65083—32no—
23#64971—31no—
24C8×Dic10—31no—
25C2×C3×Dic10—41no—
26C2×C5×SL(2,𝔽3)C2311yescharacteristic
27C5×#226Q812yescomplemented
28C4⋊Dic15—41no—
29C4×C5×Dic3—41no—
30#6668—42no—
31Dic3×Dic5—42no—
32Q8×SL(2,𝔽3)—51no—
33#5391—51no—
34#5391—52no—
35C4×#226—52no—
36C4×#226D511yescharacteristic
37C2×C5×Q16—31no—
38C4×C5×Q8—31no—
39C4×C5×Q8C2×D311yescharacteristic
40Q8×Dic5—32no—
41Q8×Dic5C2×D312yes—
42#4034—32no—
43#4034—32no—
44#4034—62no—
45#4015—31no—
46#4015—31no—
47#4015—61no—
48C4×Dic10—31no—
49C4×Dic10—61no—
50C5×#107—31no—
51C5×#103—31no—
52C5×#103—31no—
53C4×C5×C8—31no—
54#3967—32no—
55#3967—32no—
56#3950—32no—
57#3950—32no—
58#3946—32no—
59C8×Dic5—32no—
60#3939—32no—
61#3939—32no—
62C4⋊#168—31no—
63Q8×Q16—151noSylow
64C2×C3×C4×C5—41no—
65C2×Dic15—42no—
66C2×C5×Dic3—41no—
67C2×C5×Dic3—41no—
68C2×C3×Dic5—42no—
69C3×Dic10—44no—
70C5×SL(2,𝔽3)C2×Q811yescharacteristic
71C2×#226C2×D511yescharacteristic
72Q8×Dic3—201no—
73C4×SL(2,𝔽3)—52no—
74C4×SL(2,𝔽3)C2×D511yescharacteristic
75#859—52no—
76#859C2×D511yescharacteristic
77C2×C5×Q8—31no—
78C2×C5×Q8C22×D311yescharacteristic
79C2×Dic10—61no—
80C2×Dic10—62no—
81C2×Dic10—61no—
82C2×Dic10S411yescharacteristic
83C5×Q16—32no—
84C5×Q16—32no—
85C2×C5×C8—31no—
86C2×C5×C8—31no—
87C5×C4⋊C4—31no—
88C5×C4⋊C4—31no—
89C5×C4⋊C4—61no—
90C42×C5—31no—
91C42×C5—61no—
92C4⋊Dic5—31no—
93C4⋊Dic5—32no—
94C4⋊Dic5—32no—
95C4⋊Dic5—62no—
96C4⋊Dic5—61no—
97#683—32no—
98#683—62no—
99C4×Dic5—32no—
100C4×Dic5—62no—
101C2×#168—32no—
102Q82—51no—
103Q82—151no—
104#500—151no—
105#500—152no—
106#477—152no—
107#477—152no—
108#477—151no—
109#477—151no—
110C8×Q8—151no—
111C4×Q16—31no—
112C4×Q16—152no—
113C22×C3×C5—41no—
114C3×C4×C5—42no—
115Dic15—82no—
116C3×Dic5—44no—
117C5×Dic3—42no—
118C5×Dic3—81no—
119C2×C3×Q8—201no—
120C2×SL(2,𝔽3)C22×D511yescharacteristic
121#226Dic1012yesdirect product factor
122C4⋊Dic3—202no—
123C4⋊Dic3—201no—
124C4×Dic3—41no—
125C4×Dic3—202no—
126C5×Q8—31no—
127C5×Q8—32no—
128C5×Q8—61no—
129C5×Q8D3×Q811yescharacteristic
130C2×C4×C5—31no—
131C2×C4×C5—31no—
132C2×C4×C5—61no—
133C2×C4×C5—61no—
134C2×C4×C5C2×S411yescharacteristic
135C2×Dic5—32no—
136C2×Dic5—62no—
137C2×Dic5C2×S412yes—
138Dic10—241no—
139Dic10—242no—
140Dic10—241no—
141Dic10#22614yesdirect product factor
142C5×C8—32no—
143C5×C8—61no—
144#168—62no—
145C2×Q16—31no—
146C4⋊Q8—151no—
147C4⋊Q8—151no—
148C4⋊Q8—152no—
149C4⋊Q8—152no—
150C4⋊Q8—301no—
151C4⋊Q8—302no—
152C4×Q8—31no—
153C4×Q8—52no—
154C4×Q8—151no—
155C4×Q8—152no—
156C4×Q8—301no—
157C4×Q8D3×D511yescharacteristic
158#107—31no—
159#107—152no—
160#107—152no—
161#107—152no—
162#107—152no—
163C4⋊C8—151no—
164C4⋊C8—152no—
165#103—31no—
166#103—31no—
167#103—152no—
168#103—152no—
169C4×C8—31no—
170C4×C8—152no—
171C2×C3×C5—41no—
172C2×C3×C5—41no—
173C2×C3×C5—41no—
174C3×Q8—204no—
175C2×C3×C4—41no—
176C2×C3×C4—202no—
177C2×Dic3—41no—
178C2×Dic3—41no—
179C2×Dic3—202no—
180SL(2,𝔽3)C2×Dic1011yescharacteristic
181C22×C5C22×S411yescharacteristic
182C4×C5—31no—
183C4×C5—31no—
184C4×C5—62no—
185C4×C5—61no—
186C4×C5—121no—
187C4×C5C2×#22612yes—
188Dic5—62no—
189Dic5—122no—
190Dic5C2×#22614yes—
191C2×Q8—31no—
192C2×Q8—51no—
193C2×Q8—301no—
194C2×Q8—301no—
195C2×Q8—302no—
196C2×Q8C2×D3×D511yescharacteristic
197Q16—32no—
198Q16—32no—
199C2×C8—31no—
200C2×C8—31no—
201C2×C8—152no—
202C4⋊C4—31no—
203C4⋊C4—31no—
204C4⋊C4—61no—
205C4⋊C4—152no—
206C4⋊C4—152no—
207C4⋊C4—152no—
208C4⋊C4—151no—
209C4⋊C4—302no—
210C4⋊C4—302no—
211C4⋊C4—301no—
212C42—31no—
213C42—61no—
214C42—152no—
215C42—302no—
216C3×C5—41noHall
217C22×C3—41no—
218C3×C4—42no—
219C3×C4—204no—
220Dic3—42no—
221Dic3—81no—
222Dic3—402no—
223C2×C5C22×#22611yescharacteristic
224C2×C5Q8×S411yescharacteristic
225C2×C5#592511yescharacteristic
226Q8—31no—
227Q8—32no—
228Q8—54no—
229Q8—61no—
230Q8—1201no—
231Q8—1201no—
232Q8—1202no—
233Q8D3×Dic1011yescharacteristic
234C2×C4—31no—
235C2×C4—31no—
236C2×C4—52no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—152no—
240C2×C4—302no—
241C2×C4D5×S411yescharacteristic
242C8—32no—
243C8—61no—
244C8—302no—
245C2×C3—41no—
246C2×C3—41no—
247C2×C3—41no—
248C5Q8×#22611yescharacteristic, complemented, Sylow
249C22C2×D5×S411yescharacteristic
250C4—31no—
251C4—31no—
252C4—54no—
253C4—62no—
254C4—61no—
255C4—121no—
256C4—302no—
257C4—602no—
258C4D5×#22612yes—
259C3—41noSylow
260C2S4×Dic1011yescharacteristic
261C2C2×D5×#22611yescharacteristic
262C2#1175677511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×S4
commutator, prime radical, cosocleC2×C5×SL(2,𝔽3)C23
FittingC4×C5×Q8C2×D3
socleC22×C5C22×S4
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
27. C5×#226Q8semidirect
121. #226Dic10direct (N × H)
141. Dic10#226direct (N × H)
248. C5Q8×#226semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×Dic102
#117521102
C4×C5×#2262
#117515982
Dic5×#2262
Q16×Dic103
Dic3×Dic104
Q8×#2265