Finite Group Databasebeta

← C2×C5×#519⋊C3 #14511315

Subgroups — C2×C5×#519⋊C3 #14511315

Order 1920.

806 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 184 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
1C23×C5×SL(2,𝔽3)C211yesmaximal, characteristic
2C5×#519⋊C3C214yesmaximal, direct product factor
3C2×C4×C5×SL(2,𝔽3)C212yesmaximal, complemented
4C2×C5×#519C311yesmaximal, characteristic, complemented, Hall
5C22×C5×SL(2,𝔽3)C414yescomplemented
6C22×C5×SL(2,𝔽3)C2211yescharacteristic
7C22×C5×SL(2,𝔽3)C2212yes—
8C2×C3×C5×C22⋊C4—41nomaximal
9C4×C5×SL(2,𝔽3)C2214yescomplemented
10C2×#519⋊C3C511yesmaximal, characteristic, direct product factor, Hall
11C23×C5×Q8C2×C311yescharacteristic
12C5×#519—34no—
13C5×#519C2×C314yescomplemented
14C2×C4×C5×Q8—32no—
15C2×C4×C5×Q8C2×C312yescomplemented
16C2×C5×#117—31no—
17C2×C5×#117—31no—
18C24×C3×C5—41no—
19C22×C3×C4×C5—42no—
20C2×C5×SL(2,𝔽3)C2×C412yes—
21C2×C5×SL(2,𝔽3)C2×C414yescomplemented
22C2×C5×SL(2,𝔽3)C2311yescharacteristic
23C3×C5×C22⋊C4—44no—
24C23×SL(2,𝔽3)C2×C511yescharacteristic
25#519⋊C3C2×C514yesdirect product factor
26C2×C4×SL(2,𝔽3)C2×C512yescomplemented
27C22×C5×Q8—31no—
28C22×C5×Q8—32no—
29C22×C5×Q8—34no—
30C22×C5×Q8C3×C414yescomplemented
31C22×C5×Q8C22×C311yescharacteristic
32C22×C5×Q8C22×C312yes—
33C23×C4×C5—31no—
34C4×C5×Q8—38no—
35C4×C5×Q8—38no—
36C4×C5×Q8—34no—
37C4×C5×Q8C22×C314yescomplemented
38C5×#117—34no—
39C5×#117—34no—
40C5×#117—34no—
41C5×#117—34no—
42C2×C5×C4⋊C4—32no—
43C2×C5×C4⋊C4—32no—
44C2×C5×C22⋊C4—31no—
45C2×C5×C22⋊C4A411yescharacteristic
46C2×C42×C5—32no—
47C2×C42×C5—32no—
48C2×#519C3×C511yescharacteristic, complemented, Sylow
49C23×C3×C5—41no—
50C23×C3×C5—44no—
51C23×C3×C5—42no—
52C23×C3×C5—84no—
53C2×C3×C4×C5—44no—
54C2×C3×C4×C5—84no—
55C5×SL(2,𝔽3)C22×C411yescharacteristic
56C22×SL(2,𝔽3)C4×C514yescomplemented
57C22×SL(2,𝔽3)C22×C511yescharacteristic
58C22×SL(2,𝔽3)C22×C512yes—
59C2×C3×C22⋊C4—41no—
60C4×SL(2,𝔽3)C22×C514yescomplemented
61C24×C5C2×A411yescharacteristic
62C2×C5×Q8—32no—
63C2×C5×Q8—34no—
64C2×C5×Q8—38no—
65C2×C5×Q8—31no—
66C2×C5×Q8—32no—
67C2×C5×Q8—34no—
68C2×C5×Q8—38no—
69C2×C5×Q8—32no—
70C2×C5×Q8—34no—
71C2×C5×Q8C2×C3×C412yes—
72C2×C5×Q8C2×C3×C414yescomplemented
73C2×C5×Q8C23×C311yescharacteristic
74C22×C4×C5—34no—
75C22×C4×C5—34no—
76C22×C4×C5—32no—
77C22×C4×C5—32no—
78C22×C4×C5—31no—
79C22×C4×C5—31no—
80C22×C4×C5—32no—
81C22×C4×C5C2×A412yes—
82C5×C4⋊C4—34no—
83C5×C4⋊C4—34no—
84C5×C4⋊C4—38no—
85C5×C22⋊C4—34no—
86C5×C22⋊C4C2×A414yes—
87C42×C5—38no—
88C42×C5—34no—
89C42×C5—34no—
90C23×Q8C2×C3×C511yescharacteristic
91#519—34no—
92#519C2×C3×C514yescomplemented
93C2×C4×Q8—32no—
94C2×C4×Q8C2×C3×C512yescomplemented
95C2×#117—31no—
96C2×#117—31no—
97C22×C3×C5—42no—
98C22×C3×C5—44no—
99C22×C3×C5—44no—
100C22×C3×C5—41no—
101C22×C3×C5—84no—
102C22×C3×C5—88no—
103C3×C4×C5—84no—
104C24×C3—41no—
105C22×C3×C4—42no—
106C2×SL(2,𝔽3)C2×C4×C512yes—
107C2×SL(2,𝔽3)C2×C4×C514yescomplemented
108C2×SL(2,𝔽3)C23×C511yescharacteristic
109C3×C22⋊C4—44no—
110C23×C5—24no—
111C23×C5C4×A414yes—
112C23×C5C22×A412yes—
113C23×C5C22×A411yescharacteristic
114C5×Q8—32no—
115C5×Q8—34no—
116C5×Q8—31no—
117C5×Q8—38no—
118C5×Q8—32no—
119C5×Q8—34no—
120C5×Q8C22×C3×C411yescharacteristic
121C2×C4×C5—24no—
122C2×C4×C5—34no—
123C2×C4×C5—38no—
124C2×C4×C5—32no—
125C2×C4×C5—34no—
126C2×C4×C5—32no—
127C2×C4×C5—34no—
128C2×C4×C5—32no—
129C2×C4×C5—31no—
130C2×C4×C5—31no—
131C2×C4×C5—34no—
132C2×C4×C5—64no—
133C2×C4×C5C22×A414yes—
134C22×Q8—31no—
135C22×Q8—32no—
136C22×Q8—34no—
137C22×Q8C3×C4×C514yescomplemented
138C22×Q8C22×C3×C511yescharacteristic
139C22×Q8C22×C3×C512yes—
140C23×C4—31no—
141C4×Q8—38no—
142C4×Q8—38no—
143C4×Q8—34no—
144C4×Q8C22×C3×C514yescomplemented
145#117—34no—
146#117—34no—
147#117—34no—
148#117—34no—
149C2×C4⋊C4—32no—
150C2×C4⋊C4—32no—
151C2×C22⋊C4—31no—
152C2×C22⋊C4C5×A411yescharacteristic
153C2×C42—32no—
154C2×C42—32no—
155C2×C3×C5—41no—
156C2×C3×C5—42no—
157C2×C3×C5—44no—
158C2×C3×C5—84no—
159C23×C3—44no—
160C23×C3—41no—
161C23×C3—42no—
162C23×C3—84no—
163C2×C3×C4—44no—
164C2×C3×C4—84no—
165SL(2,𝔽3)C22×C4×C511yescharacteristic
166C22×C5—24no—
167C22×C5—28no—
168C22×C5C2×C4×A414yes—
169C22×C5C2×C4×A412yes—
170C22×C52−1+4⋊C314yes—
171C22×C5C23×A411yescharacteristic
172C4×C5—24no—
173C4×C5—31no—
174C4×C5—32no—
175C4×C5—34no—
176C4×C5—31no—
177C4×C5—64no—
178C24C2×C5×A411yescharacteristic
179C2×Q8—32no—
180C2×Q8—34no—
181C2×Q8—38no—
182C2×Q8—31no—
183C2×Q8—32no—
184C2×Q8—34no—
185C2×Q8—38no—
186C2×Q8—32no—
187C2×Q8—34no—
188C2×Q8C2×C3×C4×C512yes—
189C2×Q8C2×C3×C4×C514yescomplemented
190C2×Q8C23×C3×C511yescharacteristic
191C22×C4—34no—
192C22×C4—34no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—32no—
196C22×C4—32no—
197C22×C4—32no—
198C22×C4C2×C5×A412yes—
199C4⋊C4—34no—
200C4⋊C4—34no—
201C4⋊C4—38no—
202C22⋊C4—34no—
203C22⋊C4C2×C5×A414yes—
204C42—38no—
205C42—34no—
206C42—34no—
207C3×C5—41noHall
208C22×C3—42no—
209C22×C3—44no—
210C22×C3—44no—
211C22×C3—41no—
212C22×C3—84no—
213C22×C3—88no—
214C3×C4—84no—
215C2×C5—24no—
216C2×C5#519⋊C314yesdirect product factor
217C2×C5C22×C4×A411yescharacteristic
218C2×C5C2×2−1+4⋊C312yes—
219C23—24no—
220C23C4×C5×A414yes—
221C23C22×C5×A412yes—
222C23C22×C5×A411yescharacteristic
223Q8—32no—
224Q8—34no—
225Q8—31no—
226Q8—38no—
227Q8—32no—
228Q8—34no—
229Q8C22×C3×C4×C511yescharacteristic
230C2×C4—24no—
231C2×C4—32no—
232C2×C4—34no—
233C2×C4—38no—
234C2×C4—34no—
235C2×C4—34no—
236C2×C4—32no—
237C2×C4—31no—
238C2×C4—32no—
239C2×C4—31no—
240C2×C4—34no—
241C2×C4—64no—
242C2×C4C22×C5×A414yes—
243C2×C3—41no—
244C2×C3—42no—
245C2×C3—44no—
246C2×C3—84no—
247C5C2×#519⋊C311yescharacteristic, direct product factor, Sylow
248C22—24no—
249C22—28no—
250C22C2×C4×C5×A414yes—
251C22C2×C4×C5×A412yes—
252C22C5×2−1+4⋊C314yes—
253C22C23×C5×A411yescharacteristic
254C4—24no—
255C4—31no—
256C4—32no—
257C4—34no—
258C4—31no—
259C4—64no—
260C3—41noSylow
261C2—24no—
262C2C5×#519⋊C314yesdirect product factor
263C2C22×C4×C5×A411yescharacteristic
264C2C2×C5×2−1+4⋊C312yes—
subgroupG/H
center, socleC23×C5C22×A4
commutator, hypocenterQ8C22×C3×C4×C5
FrattiniC22C23×C5×A4
FittingC2×C5×#519C3
prime radicalC23×C5×SL(2,𝔽3)C2
cosocleC2×Q8C23×C3×C5
hypercenterC2×C5×C22⋊C4A4
LCSG ▷ Q8
UCSC1 ◁ C23×C5 ◁ C2×C5×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#519⋊C3C2direct (N × H)
3. C2×C4×C5×SL(2,𝔽3)C2semidirect
4. C2×C5×#519C3semidirect
5. C22×C5×SL(2,𝔽3)C4semidirect
9. C4×C5×SL(2,𝔽3)C22semidirect
10. C2×#519⋊C3C5direct (N × H)
13. C5×#519C2×C3semidirect
15. C2×C4×C5×Q8C2×C3semidirect
21. C2×C5×SL(2,𝔽3)C2×C4semidirect
25. #519⋊C3C2×C5direct (N × H)
26. C2×C4×SL(2,𝔽3)C2×C5semidirect
30. C22×C5×Q8C3×C4semidirect
37. C4×C5×Q8C22×C3semidirect
48. C2×#519C3×C5semidirect
56. C22×SL(2,𝔽3)C4×C5semidirect
60. C4×SL(2,𝔽3)C22×C5semidirect
72. C2×C5×Q8C2×C3×C4semidirect
92. #519C2×C3×C5semidirect
94. C2×C4×Q8C2×C3×C5semidirect
107. C2×SL(2,𝔽3)C2×C4×C5semidirect
137. C22×Q8C3×C4×C5semidirect
144. C4×Q8C22×C3×C5semidirect
189. C2×Q8C2×C3×C4×C5semidirect
216. C2×C5#519⋊C3direct (N × H)
247. C5C2×#519⋊C3direct (N × H)
262. C2C5×#519⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C5×SL(2,𝔽3)2
C5×#519⋊C32
C2×C4×C5×SL(2,𝔽3)2
C2×C5×#5193
C2×C3×C5×C22⋊C44
C2×#519⋊C35