Finite Group Databasebeta

← C2×C5×#520⋊C3 #14511316

Subgroups — C2×C5×#520⋊C3 #14511316

Order 1920.

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

normal subgroups only

This group has 790 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
1C22×C5×#41⋊C3C211yesmaximal, characteristic
2C5×#520⋊C3C214yesmaximal, direct product factor
3C2×C4×C5×SL(2,𝔽3)C212yesmaximal, complemented
4C2×C5×#520C311yesmaximal, characteristic, complemented, Hall
5C2×C5×#41⋊C3C414yescomplemented
6C2×C5×#41⋊C3C2212yes—
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×C5×C4⋊C4—41nomaximal
9C4×C5×SL(2,𝔽3)C2214yescomplemented
10C2×#520⋊C3C511yesmaximal, characteristic, direct product factor, Hall
11C22×C5×#41C2×C311yescharacteristic
12C5×#520—34no—
13C5×#520C2×C314yescomplemented
14C2×C4×C5×Q8C2×C312yes—
15C2×C4×C5×D4—32no—
16C2×C5×#117—31no—
17C22×C5×C4⋊C4—31no—
18C22×C3×C4×C5—41no—
19C22×C3×C4×C5—42no—
20C5×#41⋊C3C2×C414yescomplemented
21C2×C5×SL(2,𝔽3)C2×C412yescomplemented
22C2×C5×SL(2,𝔽3)C2311yescharacteristic
23C3×C5×C4⋊C4—44no—
24C22×#41⋊C3C2×C511yescharacteristic
25#520⋊C3C2×C514yesdirect product factor
26C2×C4×SL(2,𝔽3)C2×C512yescomplemented
27C2×C5×#41—32no—
28C2×C5×#41—34no—
29C2×C5×#41C3×C414yescomplemented
30C2×C5×#41C22×C312yes—
31C22×C5×Q8C22×C311yescharacteristic
32C22×C5×D4—31no—
33C23×C4×C5—31no—
34C23×C4×C5—32no—
35C4×C5×Q8—34no—
36C4×C5×Q8C22×C314yes—
37C4×C5×D4—38no—
38C4×C5×D4—38no—
39C5×#117—34no—
40C5×#117—34no—
41C2×C5×C4⋊C4—32no—
42C2×C5×C4⋊C4—34no—
43C2×C5×C4⋊C4—31no—
44C2×C5×C4⋊C4—34no—
45C2×C5×C4⋊C4A411yescharacteristic
46C2×C5×C22⋊C4—32no—
47C2×C42×C5—32no—
48C2×#520C3×C511yescharacteristic, complemented, Sylow
49C23×C3×C5—41no—
50C2×C3×C4×C5—44no—
51C2×C3×C4×C5—42no—
52C2×C3×C4×C5—44no—
53C2×C3×C4×C5—84no—
54C5×SL(2,𝔽3)C22×C411yescharacteristic, complemented
55C2×#41⋊C3C4×C514yescomplemented
56C2×#41⋊C3C22×C512yes—
57C22×SL(2,𝔽3)C22×C511yescharacteristic
58C2×C3×C4⋊C4—41no—
59C4×SL(2,𝔽3)C22×C514yescomplemented
60C24×C5—31no—
61C5×#41—38no—
62C5×#41—38no—
63C5×#41—34no—
64C5×#41C2×C3×C414yescomplemented
65C2×C5×Q8—31no—
66C2×C5×Q8—32no—
67C2×C5×Q8C2×C3×C412yes—
68C2×C5×Q8C23×C311yescharacteristic
69C2×C5×D4—32no—
70C2×C5×D4—34no—
71C2×C5×D4—32no—
72C2×C5×D4—34no—
73C22×C4×C5—31no—
74C22×C4×C5—34no—
75C22×C4×C5—34no—
76C22×C4×C5—32no—
77C22×C4×C5—32no—
78C22×C4×C5—32no—
79C22×C4×C5—38no—
80C22×C4×C5—124no—
81C22×C4×C5C2×A412yes—
82C22×C4×C5C2×A411yescharacteristic
83C5×C4⋊C4—34no—
84C5×C4⋊C4—34no—
85C5×C4⋊C4—34no—
86C5×C4⋊C4C2×A414yes—
87C5×C22⋊C4—38no—
88C42×C5—34no—
89C42×C5—34no—
90C22×#41C2×C3×C511yescharacteristic
91#520—34no—
92#520C2×C3×C514yescomplemented
93C2×C4×Q8C2×C3×C512yes—
94C2×C4×D4—32no—
95C2×#117—31no—
96C22×C4⋊C4—31no—
97C22×C3×C5—42no—
98C22×C3×C5—42no—
99C22×C3×C5—42no—
100C22×C3×C5—41no—
101C3×C4×C5—44no—
102C3×C4×C5—84no—
103C22×C3×C4—41no—
104C22×C3×C4—42no—
105#41⋊C3C2×C4×C514yescomplemented
106C2×SL(2,𝔽3)C2×C4×C512yescomplemented
107C2×SL(2,𝔽3)C23×C511yescharacteristic
108C3×C4⋊C4—44no—
109C23×C5—34no—
110C23×C5—32no—
111C23×C5—62no—
112C23×C5—62no—
113C23×C5C22×A411yescharacteristic
114C5×Q8—32no—
115C5×Q8—31no—
116C5×Q8—32no—
117C5×Q8C22×C3×C411yescharacteristic
118C5×D4—34no—
119C5×D4—34no—
120C5×D4—38no—
121C2×C4×C5—24no—
122C2×C4×C5—32no—
123C2×C4×C5—32no—
124C2×C4×C5—31no—
125C2×C4×C5—31no—
126C2×C4×C5—34no—
127C2×C4×C5—38no—
128C2×C4×C5—34no—
129C2×C4×C5—34no—
130C2×C4×C5—64no—
131C2×C4×C5—128no—
132C2×C4×C5C4×A414yes—
133C2×C4×C5C22×A414yes—
134C2×C4×C5C22×A412yes—
135C2×#41—32no—
136C2×#41—34no—
137C2×#41C3×C4×C514yescomplemented
138C2×#41C22×C3×C512yes—
139C22×Q8C22×C3×C511yescharacteristic
140C22×D4—31no—
141C23×C4—31no—
142C23×C4—32no—
143C4×Q8—34no—
144C4×Q8C22×C3×C514yes—
145C4×D4—38no—
146C4×D4—38no—
147#117—34no—
148#117—34no—
149C2×C4⋊C4—32no—
150C2×C4⋊C4—34no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—34no—
153C2×C4⋊C4C5×A411yescharacteristic
154C2×C22⋊C4—32no—
155C2×C42—32no—
156C2×C3×C5—41no—
157C2×C3×C5—41no—
158C2×C3×C5—41no—
159C2×C3×C5—44no—
160C23×C3—41no—
161C2×C3×C4—44no—
162C2×C3×C4—42no—
163C2×C3×C4—44no—
164C2×C3×C4—84no—
165SL(2,𝔽3)C22×C4×C511yescharacteristic, complemented
166C22×C5—34no—
167C22×C5—62no—
168C22×C5—62no—
169C22×C5—68no—
170C22×C5C2×C4×A412yes—
171C22×C5#99412yes—
172C22×C52−1+4⋊C312yes—
173C22×C5C23×A411yescharacteristic
174C4×C5—24no—
175C4×C5—31no—
176C4×C5—32no—
177C4×C5—31no—
178C4×C5—64no—
179C4×C5C2×C4×A414yes—
180C24—31no—
181#41—38no—
182#41—38no—
183#41—34no—
184#41C2×C3×C4×C514yescomplemented
185C2×Q8—31no—
186C2×Q8—32no—
187C2×Q8C2×C3×C4×C512yes—
188C2×Q8C23×C3×C511yescharacteristic
189C2×D4—32no—
190C2×D4—34no—
191C2×D4—32no—
192C2×D4—34no—
193C22×C4—31no—
194C22×C4—34no—
195C22×C4—34no—
196C22×C4—32no—
197C22×C4—32no—
198C22×C4—32no—
199C22×C4—38no—
200C22×C4—124no—
201C22×C4C2×C5×A412yes—
202C22×C4C2×C5×A411yescharacteristic
203C4⋊C4—34no—
204C4⋊C4—34no—
205C4⋊C4—34no—
206C4⋊C4C2×C5×A414yes—
207C22⋊C4—38no—
208C42—34no—
209C42—34no—
210C3×C5—41noHall
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—41no—
215C3×C4—44no—
216C3×C4—84no—
217C2×C5—64no—
218C2×C5#520⋊C314yesdirect product factor
219C2×C5C22×C4×A411yescharacteristic
220C2×C5C2×#99411yescharacteristic
221C2×C5C2×2−1+4⋊C311yescharacteristic
222C23—34no—
223C23—32no—
224C23—62no—
225C23—62no—
226C23C22×C5×A411yescharacteristic
227Q8—32no—
228Q8—31no—
229Q8—32no—
230Q8C22×C3×C4×C511yescharacteristic
231D4—34no—
232D4—34no—
233D4—38no—
234C2×C4—24no—
235C2×C4—32no—
236C2×C4—32no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—34no—
240C2×C4—38no—
241C2×C4—34no—
242C2×C4—34no—
243C2×C4—64no—
244C2×C4—128no—
245C2×C4C4×C5×A414yes—
246C2×C4C22×C5×A414yes—
247C2×C4C22×C5×A412yes—
248C2×C3—41no—
249C2×C3—41no—
250C2×C3—41no—
251C2×C3—44no—
252C5C2×#520⋊C311yescharacteristic, direct product factor, Sylow
253C22—34no—
254C22—62no—
255C22—62no—
256C22—68no—
257C22C2×C4×C5×A412yes—
258C22C5×#99412yes—
259C22C5×2−1+4⋊C312yes—
260C22C23×C5×A411yescharacteristic
261C4—24no—
262C4—31no—
263C4—32no—
264C4—31no—
265C4—64no—
266C4C2×C4×C5×A414yes—
267C3—41noSylow
268C2—64no—
269C2C5×#520⋊C314yesdirect product factor
270C2C22×C4×C5×A411yescharacteristic
271C2C2×C5×#99411yescharacteristic
272C2C2×C5×2−1+4⋊C311yescharacteristic
subgroupG/H
center, socleC23×C5C22×A4
commutator, hypocenterQ8C22×C3×C4×C5
FrattiniC22C23×C5×A4
FittingC2×C5×#520C3
prime radicalC22×C5×#41⋊C3C2
cosocleC2×Q8C23×C3×C5
hypercenterC2×C5×C4⋊C4A4
LCSG ▷ Q8
UCSC1 ◁ C23×C5 ◁ C2×C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#520⋊C3C2direct (N × H)
3. C2×C4×C5×SL(2,𝔽3)C2semidirect
4. C2×C5×#520C3semidirect
5. C2×C5×#41⋊C3C4semidirect
9. C4×C5×SL(2,𝔽3)C22semidirect
10. C2×#520⋊C3C5direct (N × H)
13. C5×#520C2×C3semidirect
20. C5×#41⋊C3C2×C4semidirect
21. C2×C5×SL(2,𝔽3)C2×C4semidirect
25. #520⋊C3C2×C5direct (N × H)
26. C2×C4×SL(2,𝔽3)C2×C5semidirect
29. C2×C5×#41C3×C4semidirect
48. C2×#520C3×C5semidirect
54. C5×SL(2,𝔽3)C22×C4semidirect
55. C2×#41⋊C3C4×C5semidirect
59. C4×SL(2,𝔽3)C22×C5semidirect
64. C5×#41C2×C3×C4semidirect
92. #520C2×C3×C5semidirect
105. #41⋊C3C2×C4×C5semidirect
106. C2×SL(2,𝔽3)C2×C4×C5semidirect
137. C2×#41C3×C4×C5semidirect
165. SL(2,𝔽3)C22×C4×C5semidirect
184. #41C2×C3×C4×C5semidirect
218. C2×C5#520⋊C3direct (N × H)
252. C5C2×#520⋊C3direct (N × H)
269. C2C5×#520⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×#41⋊C32
C5×#520⋊C32
C2×C4×C5×SL(2,𝔽3)2
C2×C5×#5203
C2×C3×C5×C4⋊C44
C2×#520⋊C35