Finite Group Databasebeta

← #10617212⋊C3 #14512531

Subgroups — #10617212⋊C3 #14512531

Order 1920.

354 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 60 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#10617212C311yesmaximal, characteristic, complemented, Hall
4C2×D5×SL(2,𝔽3)C2211yescharacteristic
5AGL(1,𝔽5)×SL(2,𝔽3)—22no—
6C4×C5×SL(2,𝔽3)C411yescharacteristic, complemented
7C3×(C2×AGL(1,𝔽5))⋊C4—41nomaximal
8Dic5×SL(2,𝔽3)C411yescharacteristic, complemented
9#2351⋊C3—51nomaximal, Hall
10C2×Q8×AGL(1,𝔽5)C2×C312yes—
11C4×Q8×D5C2×C311yescharacteristic
12#65712—32no—
13#65689—31no—
14C4×C4⋊AGL(1,𝔽5)—31no—
15C22×C3×AGL(1,𝔽5)—42no—
16C2×C3×C4×D5—41no—
17C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
18D5×SL(2,𝔽3)D411yescharacteristic
19D5×SL(2,𝔽3)Q811yescharacteristic
20C2×C4×SL(2,𝔽3)—51no—
21C2×C4×SL(2,𝔽3)—52no—
22C2×Q8×D5C22×C311yescharacteristic
23Q8×AGL(1,𝔽5)—22no—
24Q8×AGL(1,𝔽5)—62no—
25C2×C4⋊AGL(1,𝔽5)—32no—
26C2×C4⋊AGL(1,𝔽5)—32no—
27C2×C4×AGL(1,𝔽5)—32no—
28C4×C5×Q8C3×C411yescharacteristic, complemented
29Q8×Dic5C3×C411yescharacteristic, complemented
30D5×C4⋊C4—31no—
31#4033—31no—
32C42×D5—31no—
33C4×Dic10—31no—
34(C2×AGL(1,𝔽5))⋊C4—32no—
35(C2×AGL(1,𝔽5))⋊C4—31no—
36(C2×AGL(1,𝔽5))⋊C4A411yescharacteristic
37#2351—51noSylow
38C22×C3×D5—41no—
39C2×C3×AGL(1,𝔽5)—44no—
40C2×C3×AGL(1,𝔽5)—82no—
41C2×C3×AGL(1,𝔽5)—82no—
42C2×C3×C4×C5—41no—
43C2×C3×Dic5—41no—
44C3×C4×D5—82no—
45C5×SL(2,𝔽3)C4⋊C411yescharacteristic, complemented
46C22×SL(2,𝔽3)—51no—
47C4×SL(2,𝔽3)—51no—
48C4×SL(2,𝔽3)—102no—
49C4×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic, complemented
50C3×#95—201no—
51C22×AGL(1,𝔽5)—32no—
52C22×AGL(1,𝔽5)C2×A412yes—
53C2×C5×Q8C2×C3×C411yescharacteristic
54Q8×D5—31no—
55Q8×D5—31no—
56Q8×D5C3×D411yescharacteristic
57Q8×D5C3×Q811yescharacteristic
58C2×C4×D5—31no—
59C2×C4×D5—31no—
60C2×C4×D5C2×A411yescharacteristic
61C2×Dic10—31no—
62C4⋊AGL(1,𝔽5)—62no—
63C4⋊AGL(1,𝔽5)—62no—
64C4⋊AGL(1,𝔽5)—64no—
65C4×AGL(1,𝔽5)—62no—
66C4×AGL(1,𝔽5)—62no—
67C5×C4⋊C4—31no—
68C42×C5—31no—
69C4⋊Dic5—31no—
70#683—32no—
71C4×Dic5—31no—
72C4×Dic5—31no—
73C4×Dic5—31no—
74C2×C4×Q8—51no—
75C2×C4×Q8—52no—
76#389—152no—
77#382—151no—
78C4×C4⋊C4—151no—
79C22×C3×C5—41no—
80C2×C3×D5—41no—
81C2×C3×D5—41no—
82C2×C3×D5—42no—
83C2×C3×D5—42no—
84C3×AGL(1,𝔽5)—84no—
85C3×C4×C5—81no—
86C3×Dic5—81no—
87C22×C3×C4—201no—
88C22×C3×C4—202no—
89C2×SL(2,𝔽3)—51no—
90C2×SL(2,𝔽3)—51no—
91C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
92C22×D5C22×A411yescharacteristic
93C2×AGL(1,𝔽5)—22no—
94C2×AGL(1,𝔽5)—22no—
95C2×AGL(1,𝔽5)—62no—
96C2×AGL(1,𝔽5)—62no—
97C2×AGL(1,𝔽5)—62no—
98C2×AGL(1,𝔽5)#41⋊C314yes—
99C5×Q8—31no—
100C5×Q8C3×C4⋊C411yescharacteristic
101C2×C4×C5—31no—
102C2×C4×C5—31no—
103C2×C4×C5C4×A411yescharacteristic
104C2×Dic5—31no—
105C2×Dic5—31no—
106C2×Dic5C4×A411yescharacteristic
107C4×D5—22no—
108C4×D5—31no—
109C4×D5—31no—
110C4×D5—31no—
111C4×D5—31no—
112C4×D5—62no—
113Dic10—32no—
114Dic10—31no—
115Dic10—31no—
116C22×Q8—51no—
117C4×Q8—51no—
118C4×Q8—102no—
119C4×Q8—151no—
120C4×Q8—151no—
121C4×Q8—302no—
122C4×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
123C2×C4⋊C4—151no—
124C2×C4⋊C4—152no—
125C2×C4⋊C4—152no—
126C2×C42—151no—
127C2×C42—152no—
128#95—51no—
129#95—151no—
130#95—152no—
131C2×C3×C5—41no—
132C2×C3×C5—41no—
133C2×C3×C5—41no—
134C3×D5—42no—
135C3×D5—42no—
136C23×C3—201no—
137C2×C3×C4—41no—
138C2×C3×C4—201no—
139C2×C3×C4—204no—
140C2×C3×C4—402no—
141C2×C3×C4—402no—
142C2×C3×C4—402no—
143SL(2,𝔽3)C4⋊AGL(1,𝔽5)11yescharacteristic, complemented
144C22×C5C2×C4×A411yescharacteristic
145C2×D5D4×A411yescharacteristic
146C2×D5Q8×A411yescharacteristic
147C2×D5C2×#41⋊C312yes—
148C2×D52−1+4⋊C312yes—
149AGL(1,𝔽5)—24no—
150AGL(1,𝔽5)—122no—
151C4×C5—21no—
152C4×C5—31no—
153C4×C5—31no—
154C4×C5—61no—
155Dic5—21no—
156Dic5—31no—
157Dic5—31no—
158Dic5—61no—
159C2×Q8—51no—
160C2×Q8—51no—
161C2×Q8—151no—
162C2×Q8—151no—
163C2×Q8—151no—
164C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
165C22×C4—52no—
166C22×C4—51no—
167C22×C4—151no—
168C22×C4—151no—
169C22×C4—152no—
170C4⋊C4—31no—
171C4⋊C4—151no—
172C4⋊C4—152no—
173C4⋊C4—304no—
174C4⋊C4—302no—
175C4⋊C4—302no—
176C42—31no—
177C42—151no—
178C42—151no—
179C42—151no—
180C42—302no—
181C42—302no—
182C3×C5—41noHall
183C22×C3—41no—
184C22×C3—202no—
185C22×C3—202no—
186C22×C3—201no—
187C22×C3—201no—
188C3×C4—81no—
189C3×C4—401no—
190C3×C4—404no—
191C2×C5A4×C4⋊C411yescharacteristic
192C2×C5C4×#41⋊C311yescharacteristic
193C2×C5#519⋊C311yescharacteristic
194D5#548⋊C312yes—
195D5#557⋊C312yes—
196C23—51no—
197Q8—31no—
198Q8—151no—
199Q8—151no—
200Q8—152no—
201Q8C3×C4⋊AGL(1,𝔽5)11yescharacteristic
202C2×C4—31no—
203C2×C4—31no—
204C2×C4—54no—
205C2×C4—51no—
206C2×C4—102no—
207C2×C4—102no—
208C2×C4—102no—
209C2×C4—151no—
210C2×C4—151no—
211C2×C4—151no—
212C2×C4—151no—
213C2×C4—151no—
214C2×C4—151no—
215C2×C4—302no—
216C2×C4—302no—
217C2×C4—302no—
218C2×C4—302no—
219C2×C4A4×AGL(1,𝔽5)11yescharacteristic
220C2×C3—41no—
221C2×C3—41no—
222C2×C3—41no—
223C2×C3—202no—
224C2×C3—202no—
225C5#2351⋊C311yescharacteristic, complemented, Sylow
226C22—51no—
227C22—51no—
228C22—52no—
229C22—52no—
230C22C2×A4×AGL(1,𝔽5)11yescharacteristic
231C4—21no—
232C4—31no—
233C4—31no—
234C4—61no—
235C4—104no—
236C4—101no—
237C4—151no—
238C4—151no—
239C4—301no—
240C4—602no—
241C3—41noSylow
242C2—52no—
243C2—52no—
244C2A4×C4⋊AGL(1,𝔽5)11yescharacteristic
245C2#66262⋊C311yescharacteristic
246C2AGL(1,𝔽5)×#41⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×AGL(1,𝔽5)
commutatorC2×C5×Q8C2×C3×C4
FittingC4×C5×Q8C3×C4
socleC22×C5C2×C4×A4
prime radicalC2×D5×SL(2,𝔽3)C22
cosocleC2×Q8×D5C22×C3
hypercenterC2×C4A4×AGL(1,𝔽5)
hypocenterC5×Q8C3×C4⋊C4
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #10617212C3semidirect
6. C4×C5×SL(2,𝔽3)C4semidirect
8. Dic5×SL(2,𝔽3)C4semidirect
28. C4×C5×Q8C3×C4semidirect
29. Q8×Dic5C3×C4semidirect
45. C5×SL(2,𝔽3)C4⋊C4semidirect
49. C4×SL(2,𝔽3)AGL(1,𝔽5)semidirect
122. C4×Q8C3×AGL(1,𝔽5)semidirect
143. SL(2,𝔽3)C4⋊AGL(1,𝔽5)semidirect
225. C5#2351⋊C3semidirect

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
#106172123
C3×(C2×AGL(1,𝔽5))⋊C44
#2351⋊C35