Finite Group Databasebeta

← #10617208⋊C3 #14512530

Subgroups — #10617208⋊C3 #14512530

Order 1920.

386 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×AGL(1,𝔽5)×SL(2,𝔽3)C212yesmaximal
2C4×D5×SL(2,𝔽3)C211yesmaximal, characteristic
3#10617208C311yesmaximal, characteristic, complemented, Hall
4C2×D5×SL(2,𝔽3)C2211yescharacteristic
5AGL(1,𝔽5)×SL(2,𝔽3)C414yescomplemented
6C4×C5×SL(2,𝔽3)C411yescharacteristic, complemented
7C3×(C2×AGL(1,𝔽5))⋊C4—41nomaximal
8Dic5×SL(2,𝔽3)C411yescharacteristic, complemented
9#73167—51nomaximal, Hall
10C2×Q8×AGL(1,𝔽5)C2×C312yes—
11C4×Q8×D5C2×C311yescharacteristic
12C4⋊C4×AGL(1,𝔽5)—32no—
13C4×C4⋊AGL(1,𝔽5)—31no—
14#65686—31no—
15C22×C3×AGL(1,𝔽5)—42no—
16C2×C3×C4×D5—41no—
17C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
18D5×SL(2,𝔽3)C2×C412yes—
19C2×C4×SL(2,𝔽3)—52no—
20C2×C4×SL(2,𝔽3)—51no—
21C2×Q8×D5C22×C311yescharacteristic
22Q8×AGL(1,𝔽5)—34no—
23Q8×AGL(1,𝔽5)C3×C414yescomplemented
24C2×C4⋊AGL(1,𝔽5)—32no—
25C2×C4×AGL(1,𝔽5)—32no—
26C2×C4×AGL(1,𝔽5)—32no—
27C4×C5×Q8C3×C411yescharacteristic, complemented
28Q8×Dic5C3×C411yescharacteristic, complemented
29D5×C4⋊C4—31no—
30#4033—31no—
31C42×D5—31no—
32C4×Dic10—31no—
33(C2×AGL(1,𝔽5))⋊C4—32no—
34(C2×AGL(1,𝔽5))⋊C4—31no—
35(C2×AGL(1,𝔽5))⋊C4A411yescharacteristic
36#2276—51noSylow
37C22×C3×D5—41no—
38C2×C3×AGL(1,𝔽5)—44no—
39C2×C3×AGL(1,𝔽5)—82no—
40C2×C3×AGL(1,𝔽5)—82no—
41C2×C3×C4×C5—41no—
42C2×C3×Dic5—41no—
43C3×C4×D5—82no—
44C5×SL(2,𝔽3)C4211yescharacteristic, complemented
45C22×SL(2,𝔽3)—51no—
46C4×SL(2,𝔽3)—54no—
47C4×SL(2,𝔽3)—51no—
48C4×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic, complemented
49C3×#95—201no—
50C22×AGL(1,𝔽5)—32no—
51C22×AGL(1,𝔽5)C2×A412yes—
52C2×C5×Q8C2×C3×C411yescharacteristic
53Q8×D5—32no—
54Q8×D5C2×C3×C412yes—
55C2×C4×D5—31no—
56C2×C4×D5—31no—
57C2×C4×D5C2×A411yescharacteristic
58C2×Dic10—31no—
59C4⋊AGL(1,𝔽5)—34no—
60C4⋊AGL(1,𝔽5)—34no—
61C4×AGL(1,𝔽5)—34no—
62C4×AGL(1,𝔽5)—34no—
63C4×AGL(1,𝔽5)—122no—
64C5×C4⋊C4—31no—
65C42×C5—31no—
66C4⋊Dic5—31no—
67#683—32no—
68C4×Dic5—31no—
69C4×Dic5—31no—
70C4×Dic5—31no—
71C2×C4×Q8—52no—
72C2×C4×Q8—51no—
73C4×C4⋊C4—151no—
74C4×C4⋊C4—152no—
75#376—151no—
76C22×C3×C5—41no—
77C2×C3×D5—42no—
78C2×C3×D5—41no—
79C2×C3×D5—42no—
80C2×C3×D5—41no—
81C3×AGL(1,𝔽5)—84no—
82C3×C4×C5—81no—
83C3×Dic5—81no—
84C22×C3×C4—201no—
85C22×C3×C4—202no—
86C2×SL(2,𝔽3)—52no—
87C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
88C22×D5C22×A411yescharacteristic
89C2×AGL(1,𝔽5)—22no—
90C2×AGL(1,𝔽5)—22no—
91C2×AGL(1,𝔽5)—34no—
92C2×AGL(1,𝔽5)—62no—
93C2×AGL(1,𝔽5)—62no—
94C2×AGL(1,𝔽5)C4×A414yes—
95C5×Q8—31no—
96C5×Q8C3×C4211yescharacteristic
97C2×C4×C5—31no—
98C2×C4×C5—31no—
99C2×C4×C5C4×A411yescharacteristic
100C2×Dic5—31no—
101C2×Dic5—31no—
102C2×Dic5C4×A411yescharacteristic
103C4×D5—22no—
104C4×D5—32no—
105C4×D5—32no—
106C4×D5—61no—
107C4×D5—61no—
108Dic10—32no—
109Dic10—32no—
110C22×Q8—51no—
111C4×Q8—54no—
112C4×Q8—51no—
113C4×Q8—151no—
114C4×Q8—151no—
115C4×Q8—154no—
116C4×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
117C2×C4⋊C4—151no—
118C2×C4⋊C4—152no—
119C2×C42—151no—
120C2×C42—152no—
121C2×C42—152no—
122#95—51no—
123#95—152no—
124#95—151no—
125C2×C3×C5—41no—
126C2×C3×C5—41no—
127C2×C3×C5—41no—
128C3×D5—42no—
129C3×D5—42no—
130C23×C3—201no—
131C2×C3×C4—41no—
132C2×C3×C4—201no—
133C2×C3×C4—204no—
134C2×C3×C4—402no—
135C2×C3×C4—402no—
136C2×C3×C4—402no—
137SL(2,𝔽3)C4×AGL(1,𝔽5)11yescharacteristic, complemented
138C22×C5C2×C4×A411yescharacteristic
139C2×D5C2×C4×A412yes—
140C2×D5#99411yescharacteristic
141C2×D52−1+4⋊C311yescharacteristic
142C2×D52−1+4⋊C312yes—
143AGL(1,𝔽5)—24no—
144AGL(1,𝔽5)—64no—
145C4×C5—21no—
146C4×C5—31no—
147C4×C5—31no—
148C4×C5—61no—
149Dic5—21no—
150Dic5—32no—
151Dic5—61no—
152C2×Q8—52no—
153C2×Q8—151no—
154C2×Q8—152no—
155C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
156C22×C4—52no—
157C22×C4—51no—
158C22×C4—151no—
159C22×C4—151no—
160C22×C4—152no—
161C4⋊C4—31no—
162C4⋊C4—151no—
163C4⋊C4—152no—
164C4⋊C4—154no—
165C4⋊C4—154no—
166C42—31no—
167C42—151no—
168C42—151no—
169C42—154no—
170C42—154no—
171C42—151no—
172C42—602no—
173C3×C5—41noHall
174C22×C3—41no—
175C22×C3—202no—
176C22×C3—202no—
177C22×C3—201no—
178C22×C3—201no—
179C3×C4—81no—
180C3×C4—401no—
181C3×C4—404no—
182C2×C5C42×A411yescharacteristic
183C2×C5#519⋊C311yescharacteristic
184C2×C5#520⋊C311yescharacteristic
185D5#519⋊C312yes—
186D5#520⋊C312yes—
187C23—51no—
188Q8—31no—
189Q8—152no—
190Q8—152no—
191Q8C3×C4×AGL(1,𝔽5)11yescharacteristic
192C2×C4—31no—
193C2×C4—31no—
194C2×C4—54no—
195C2×C4—51no—
196C2×C4—102no—
197C2×C4—102no—
198C2×C4—102no—
199C2×C4—152no—
200C2×C4—152no—
201C2×C4—151no—
202C2×C4—154no—
203C2×C4—151no—
204C2×C4—301no—
205C2×C4—301no—
206C2×C4—302no—
207C2×C4—302no—
208C2×C4A4×AGL(1,𝔽5)11yescharacteristic
209C2×C3—41no—
210C2×C3—41no—
211C2×C3—41no—
212C2×C3—202no—
213C2×C3—202no—
214C5#7316711yescharacteristic, complemented, Sylow
215C22—52no—
216C22—52no—
217C22—51no—
218C22—51no—
219C22C2×A4×AGL(1,𝔽5)11yescharacteristic
220C4—21no—
221C4—31no—
222C4—31no—
223C4—61no—
224C4—104no—
225C4—101no—
226C4—152no—
227C4—304no—
228C4—301no—
229C3—41noSylow
230C2—52no—
231C2—52no—
232C2C4×A4×AGL(1,𝔽5)11yescharacteristic
233C2#66262⋊C311yescharacteristic
234C2#66266⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×AGL(1,𝔽5)
commutator, hypocenterC5×Q8C3×C42
FittingC4×C5×Q8C3×C4
socleC22×C5C2×C4×A4
prime radicalC2×D5×SL(2,𝔽3)C22
cosocleC2×Q8×D5C22×C3
hypercenterC2×C4A4×AGL(1,𝔽5)
LCSG ▷ C5×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #10617208C3semidirect
5. AGL(1,𝔽5)×SL(2,𝔽3)C4semidirect
6. C4×C5×SL(2,𝔽3)C4semidirect
8. Dic5×SL(2,𝔽3)C4semidirect
23. Q8×AGL(1,𝔽5)C3×C4semidirect
27. C4×C5×Q8C3×C4semidirect
28. Q8×Dic5C3×C4semidirect
44. C5×SL(2,𝔽3)C42semidirect
48. C4×SL(2,𝔽3)AGL(1,𝔽5)semidirect
116. C4×Q8C3×AGL(1,𝔽5)semidirect
137. SL(2,𝔽3)C4×AGL(1,𝔽5)semidirect
214. C5#73167semidirect

Maximal subgroups

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

subgroupindex
C2×AGL(1,𝔽5)×SL(2,𝔽3)2
C4×D5×SL(2,𝔽3)2
#106172083
C3×(C2×AGL(1,𝔽5))⋊C44
#731675