Finite Group Databasebeta

← #14512754

Subgroups — #14512754

Order 1920.

414 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 55 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
1#65904⋊C3C211yesmaximal, characteristic, complemented
2#11752110C211yesmaximal, characteristic, complemented
3C4×C5×GL(2,𝔽3)C211yesmaximal, characteristic
4Dic5×GL(2,𝔽3)C212yesmaximal
5#11751595C212yesmaximal, complemented
6#10611363—31nomaximal, Hall
7C2×C5×GL(2,𝔽3)C2211yescharacteristic
8C4×C5×SL(2,𝔽3)C2211yescharacteristic
9D3×C4⋊Dic5—41nomaximal
10Dic5×SL(2,𝔽3)C2212yes—
11#91184C2212yes—
12C5×#859C2211yescharacteristic
13#85418—51nomaximal, Hall
14#65904D311yescharacteristic, complemented
15#65877—31no—
16C4×C5×SD16—31no—
17SD16×Dic5—32no—
18#65310—31no—
19#65296—31no—
20#65147—32no—
21#65088—32no—
22#65052—32no—
23#64969—31no—
24C8×Dic10—31no—
25C2×C4×C5×D3—41no—
26C2×C5×SL(2,𝔽3)C2311yescharacteristic
27C2×D3×Dic5—42no—
28C5×GL(2,𝔽3)Q812yescomplemented
29C4⋊Dic15—41no—
30C3×C4⋊Dic5—41no—
31Dic3⋊Dic5—42no—
32#557⋊C3—51no—
33C4×GL(2,𝔽3)—52no—
34C4×GL(2,𝔽3)D511yescharacteristic
35#5391—51no—
36#5389—52no—
37C2×C5×SD16—31no—
38C4×C5×Q8C2×D311yescharacteristic
39C4×C5×D4—31no—
40Q8×Dic5C2×D312yes—
41D4×Dic5—32no—
42C2×C4⋊Dic5—61no—
43#4070—61no—
44#4036—32no—
45#4036—32no—
46#4024—62no—
47#4015—31no—
48#4015—31no—
49C4×Dic10—31no—
50C5×#106—31no—
51C5×#103—31no—
52C5×#102—31no—
53C4×C5×C8—31no—
54#3967—32no—
55#3964—32no—
56#3949—32no—
57#3949—32no—
58#3946—32no—
59C8×Dic5—32no—
60#3939—32no—
61#3939—32no—
62C4⋊#168—31no—
63#3381—151noSylow
64C22×C5×D3—41no—
65C2×C3×C4×C5—41no—
66C2×Dic15—42no—
67C2×C5×Dic3—41no—
68C4×C5×D3—44no—
69C2×C3×Dic5—42no—
70C5×SL(2,𝔽3)C2×Q811yescharacteristic
71D3×Dic5—84no—
72C2×GL(2,𝔽3)C2×D511yescharacteristic
73D3×C4⋊C4—201no—
74C4×SL(2,𝔽3)—52no—
75C4×SL(2,𝔽3)C2×D511yescharacteristic
76#859—52no—
77#859C2×D511yescharacteristic
78C2×C5×Q8C22×D311yescharacteristic
79C2×C5×D4—31no—
80C22×C4×C5—61no—
81C22×Dic5—62no—
82C2×Dic10—31no—
83C2×Dic10—61no—
84C5×SD16—32no—
85C5×SD16—32no—
86C2×C5×C8—31no—
87C2×C5×C8—31no—
88C5×C4⋊C4—31no—
89C5×C4⋊C4—31no—
90C5×C22⋊C4—61no—
91C42×C5—31no—
92C22⋊Dic5—62no—
93C4⋊Dic5—32no—
94C4⋊Dic5—32no—
95C4⋊Dic5—61no—
96C4⋊Dic5—61no—
97C4⋊Dic5—62no—
98C4⋊Dic5S411yescharacteristic
99#683—32no—
100#683—62no—
101C4×Dic5—32no—
102C2×#168—32no—
103#557—51no—
104#554—151no—
105#499—152no—
106#498—151no—
107#479—152no—
108#478—152no—
109#477—151no—
110#474—151no—
111C8×Q8—151no—
112C4×SD16—31no—
113C4×SD16—152no—
114C22×C3×C5—41no—
115C2×C5×D3—42no—
116C2×C5×D3—42no—
117C2×C5×D3—42no—
118C3×C4×C5—42no—
119Dic15—82no—
120C3×Dic5—82no—
121C5×Dic3—42no—
122C2×C4×D3—41no—
123C2×C4×D3—202no—
124C2×SL(2,𝔽3)C22×D511yescharacteristic
125GL(2,𝔽3)Dic1012yescomplemented
126C3×C4⋊C4—201no—
127C4⋊Dic3—201no—
128Dic3⋊C4—202no—
129C23×C5—61no—
130C5×Q8—31no—
131C5×Q8D3×Q811yescharacteristic
132C5×D4—32no—
133C5×D4—61no—
134C2×C4×C5—31no—
135C2×C4×C5—31no—
136C2×C4×C5—61no—
137C2×C4×C5—122no—
138C2×C4×C5C2×S411yescharacteristic
139C2×Dic5—32no—
140C2×Dic5—62no—
141C2×Dic5—242no—
142C2×Dic5C2×S412yes—
143Dic10—62no—
144Dic10—241no—
145C5×C8—32no—
146C5×C8—61no—
147#168—62no—
148C2×SD16—31no—
149C4⋊Q8—151no—
150C4⋊Q8—151no—
151#125—152no—
152#125—152no—
153C22⋊Q8—302no—
154C22⋊Q8—301no—
155C4×Q8—52no—
156C4×Q8—151no—
157C4×Q8D3×D511yescharacteristic
158C4×D4—31no—
159C4×D4—152no—
160C2×C4⋊C4—301no—
161#107—152no—
162#107—152no—
163#106—31no—
164#106—152no—
165#106—152no—
166C4⋊C8—151no—
167C4⋊C8—152no—
168#103—31no—
169#103—152no—
170#102—31no—
171#102—152no—
172C4×C8—31no—
173C4×C8—152no—
174C2×C3×C5—41no—
175C2×C3×C5—41no—
176C2×C3×C5—41no—
177C5×D3—44no—
178C22×D3—41no—
179C2×C3×C4—41no—
180C2×C3×C4—202no—
181C2×Dic3—41no—
182C2×Dic3—202no—
183C4×D3—44no—
184C4×D3—404no—
185SL(2,𝔽3)C2×Dic1011yescharacteristic
186C22×C5—62no—
187C22×C5—121no—
188C22×C5—121no—
189C22×C5C22×S411yescharacteristic
190C4×C5—31no—
191C4×C5—31no—
192C4×C5—61no—
193C4×C5—121no—
194C4×C5#98512yes—
195Dic5—22no—
196Dic5—62no—
197Dic5—122no—
198C2×Q8—151no—
199C2×Q8—301no—
200C2×Q8C2×D3×D511yescharacteristic
201C2×D4—31no—
202C22×C4—61no—
203C22×C4—302no—
204SD16—32no—
205SD16—32no—
206C2×C8—31no—
207C2×C8—31no—
208C2×C8—152no—
209C4⋊C4—31no—
210C4⋊C4—31no—
211C4⋊C4—51no—
212C4⋊C4—152no—
213C4⋊C4—152no—
214C4⋊C4—152no—
215C4⋊C4—301no—
216C4⋊C4—302no—
217C4⋊C4—302no—
218C4⋊C4—301no—
219C22⋊C4—61no—
220C22⋊C4—302no—
221C42—31no—
222C42—152no—
223C3×C5—41noHall
224C22×C3—41no—
225C2×D3—42no—
226C2×D3—42no—
227C2×D3—42no—
228C3×C4—42no—
229C3×C4—402no—
230Dic3—42no—
231Dic3—402no—
232C2×C5—122no—
233C2×C5Q8×S411yescharacteristic
234C2×C5C2×#98511yescharacteristic
235C2×C5#592611yescharacteristic
236C23—61no—
237Q8—31no—
238Q8—302no—
239Q8—1201no—
240Q8D3×Dic1011yescharacteristic
241D4—32no—
242D4—61no—
243C2×C4—31no—
244C2×C4—31no—
245C2×C4—52no—
246C2×C4—61no—
247C2×C4—122no—
248C2×C4—152no—
249C2×C4—302no—
250C2×C4—1202no—
251C2×C4D5×S411yescharacteristic
252C8—32no—
253C8—61no—
254C8—302no—
255C2×C3—41no—
256C2×C3—41no—
257C2×C3—41no—
258D3—44no—
259C5#8541811yescharacteristic, complemented, Sylow
260C22—62no—
261C22—121no—
262C22—121no—
263C22C2×D5×S411yescharacteristic
264C4—31no—
265C4—31no—
266C4—61no—
267C4—102no—
268C4—121no—
269C4—302no—
270C4—602no—
271C4#9189212yes—
272C3—41noSylow
273C2—122no—
274C2S4×Dic1011yescharacteristic
275C2C2×#9189211yescharacteristic
276C2#1175678511yescharacteristic
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. #65904⋊C3C2semidirect
2. #11752110C2semidirect
5. #11751595C2semidirect
14. #65904D3semidirect
28. C5×GL(2,𝔽3)Q8semidirect
125. GL(2,𝔽3)Dic10semidirect
259. C5#85418semidirect

Maximal subgroups

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

subgroupindex
#65904⋊C32
#117521102
C4×C5×GL(2,𝔽3)2
Dic5×GL(2,𝔽3)2
#117515952
#106113633
D3×C4⋊Dic54
#854185