Finite Group Databasebeta

← #10617729⋊C3 #14514027

Subgroups — #10617729⋊C3 #14514027

Order 1920.

530 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×C5×#41⋊C3C211yesmaximal, characteristic
2#66161⋊C3C212yesmaximal
3#65904⋊C3C212yesmaximal, complemented
4SL(2,𝔽3)×Dic10C212yesmaximal, complemented
5#10617729C311yesmaximal, characteristic, complemented, Hall
6C2×C5×#41⋊C3C2211yescharacteristic
7C3×#4015—41nomaximal
8C4×C5×SL(2,𝔽3)C2211yescharacteristic
9C4×C5×SL(2,𝔽3)C2211yescharacteristic
10Dic5×SL(2,𝔽3)C2214yes—
11#3503⋊C3—51nomaximal, Hall
12C4×C5×#41C2×C311yescharacteristic
13#66161C2×C312yes—
14#65904C2×C312yes—
15Q8×Dic10C2×C312yes—
16#65877—32no—
17D4×Dic10—32no—
18#65853—32no—
19#65804—31no—
20C2×#4015—31no—
21C2×C3×Dic10—42no—
22C5×#41⋊C3Q812yescomplemented
23C2×C5×SL(2,𝔽3)C2311yescharacteristic
24C3×C42×C5—41no—
25C3×C4⋊Dic5—42no—
26C3×C4⋊Dic5—42no—
27Q8×SL(2,𝔽3)—52no—
28#557⋊C3—52no—
29#520⋊C3—52no—
30C4×#41⋊C3D511yescharacteristic
31C2×C5×#41C22×C311yescharacteristic
32C22×Dic10—32no—
33C4×C5×Q8C22×C311yescharacteristic
34C4×C5×Q8C22×C311yescharacteristic
35C4×C5×D4—31no—
36C4×C5×D4—31no—
37C5×#117—31no—
38C2×C42×C5—31no—
39Q8×Dic5C22×C314yes—
40D4×Dic5—34no—
41#4072—32no—
42C2×C4⋊Dic5—32no—
43C2×C4⋊Dic5—32no—
44#4070—32no—
45#4070—32no—
46#4036—34no—
47#4034—34no—
48#4024—38no—
49#4016—32no—
50#4015—32no—
51#4015—32no—
52#4015—31no—
53#4015—32no—
54#4015A411yescharacteristic
55C4×Dic10—32no—
56C4×Dic10—32no—
57#3503—51noSylow
58C2×C3×C4×C5—41no—
59C2×C3×C4×C5—41no—
60C2×C3×C4×C5—41no—
61C2×C3×Dic5—44no—
62C3×Dic10—84no—
63C5×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
64C2×#41⋊C3C2×D511yescharacteristic
65C3×C4⋊Q8—201no—
66C4×SL(2,𝔽3)—54no—
67C4×SL(2,𝔽3)C2×D511yescharacteristic
68C4×SL(2,𝔽3)C2×D511yescharacteristic
69C5×#41—32no—
70C5×#41C3×Q812yescomplemented
71C2×C5×Q8C23×C311yescharacteristic
72C2×C5×D4—31no—
73C22×C4×C5—31no—
74C22×C4×C5—31no—
75C22×C4×C5—31no—
76C22×Dic5—34no—
77C2×Dic10—32no—
78C2×Dic10—34no—
79C2×Dic10—124no—
80C2×Dic10C2×A412yes—
81C5×C4⋊C4—31no—
82C5×C4⋊C4—31no—
83C5×C22⋊C4—32no—
84C42×C5—31no—
85C42×C5—31no—
86C42×C5—31no—
87C42×C5C2×A411yescharacteristic
88C22⋊Dic5—34no—
89C4⋊Dic5—34no—
90C4⋊Dic5—32no—
91C4⋊Dic5—32no—
92C4⋊Dic5—34no—
93C4⋊Dic5C2×A412yes—
94C4⋊Dic5C2×A412yes—
95#683—34no—
96#683—34no—
97C4×Dic5—34no—
98Q82—52no—
99#557—52no—
100#554—152no—
101D4×Q8—152no—
102#544—152no—
103#533—151no—
104C2×C4⋊Q8—151no—
105#520—52no—
106C4×#41C3×D511yescharacteristic
107C22×C3×C5—41no—
108C3×C4×C5—42no—
109C3×C4×C5—42no—
110C3×C4×C5—42no—
111C3×Dic5—84no—
112C2×C3×Q8—202no—
113#41⋊C3Dic1012yescomplemented
114C2×SL(2,𝔽3)C22×D511yescharacteristic
115C3×C4⋊C4—202no—
116C3×C4⋊C4—202no—
117C3×C42—41no—
118C23×C5—31no—
119C5×Q8—31no—
120C5×Q8C2×C3×Q811yescharacteristic
121C5×D4—32no—
122C5×D4—32no—
123C2×C4×C5—31no—
124C2×C4×C5—32no—
125C2×C4×C5—32no—
126C2×C4×C5—31no—
127C2×C4×C5—31no—
128C2×C4×C5—62no—
129C2×C4×C5—62no—
130C2×C4×C5C22×A411yescharacteristic
131C2×C4×C5C22×A411yescharacteristic
132C2×C4×C5C22×A411yescharacteristic
133C2×Dic5—34no—
134C2×Dic5—124no—
135C2×Dic5C22×A414yes—
136Dic10—24no—
137Dic10—64no—
138Dic10—124no—
139C2×#41C2×C3×D511yescharacteristic
140C22×Q8—152no—
141C4⋊Q8—51no—
142C4⋊Q8—152no—
143C4⋊Q8—152no—
144C4⋊Q8—152no—
145C4⋊Q8—154no—
146C4⋊Q8—151no—
147#125—152no—
148#125—154no—
149C22⋊Q8—158no—
150C22⋊Q8—152no—
151C22⋊Q8—152no—
152C4×Q8—54no—
153C4×Q8—152no—
154C4×Q8—152no—
155C4×Q8C2×C3×D511yescharacteristic
156C4×Q8C2×C3×D511yescharacteristic
157C4×D4—31no—
158C4×D4—31no—
159C4×D4—154no—
160#117—31no—
161#117—152no—
162C2×C4⋊C4—152no—
163C2×C4⋊C4—152no—
164C2×C42—31no—
165C2×C3×C5—41no—
166C2×C3×C5—41no—
167C2×C3×C5—41no—
168C3×Q8—404no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—204no—
173SL(2,𝔽3)C2×Dic1011yescharacteristic, complemented
174C22×C5—32no—
175C22×C5—61no—
176C22×C5—61no—
177C22×C5C23×A411yescharacteristic
178C4×C5—31no—
179C4×C5—31no—
180C4×C5—61no—
181C4×C5—61no—
182C4×C5Q8×A412yes—
183C4×C5#99412yes—
184C4×C52−1+4⋊C312yes—
185Dic5—24no—
186Dic5—64no—
187#41—32no—
188#41C3×Dic1012yescomplemented
189C2×Q8—52no—
190C2×Q8—154no—
191C2×Q8—152no—
192C2×Q8—604no—
193C2×Q8C22×C3×D511yescharacteristic
194C2×D4—31no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—154no—
199C4⋊C4—31no—
200C4⋊C4—31no—
201C4⋊C4—52no—
202C4⋊C4—52no—
203C4⋊C4—154no—
204C4⋊C4—154no—
205C4⋊C4—154no—
206C4⋊C4—154no—
207C4⋊C4—152no—
208C4⋊C4—152no—
209C22⋊C4—32no—
210C22⋊C4—154no—
211C42—31no—
212C42—31no—
213C42—31no—
214C42—154no—
215C42D5×A411yescharacteristic
216C3×C5—41noHall
217C22×C3—41no—
218C3×C4—42no—
219C3×C4—42no—
220C3×C4—42no—
221C3×C4—404no—
222C2×C5—62no—
223C2×C5C2×Q8×A411yescharacteristic
224C2×C5C2×#99411yescharacteristic
225C2×C5C2×2−1+4⋊C311yescharacteristic
226C23—31no—
227Q8—31no—
228Q8—104no—
229Q8—304no—
230Q8—604no—
231Q8C2×C3×Dic1011yescharacteristic
232D4—32no—
233D4—32no—
234C2×C4—31no—
235C2×C4—32no—
236C2×C4—32no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—54no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—154no—
243C2×C4—604no—
244C2×C4C2×D5×A411yescharacteristic
245C2×C4C2×D5×A411yescharacteristic
246C2×C4C2×D5×A411yescharacteristic
247C2×C3—41no—
248C2×C3—41no—
249C2×C3—41no—
250C5#3503⋊C311yescharacteristic, complemented, Sylow
251C22—32no—
252C22—61no—
253C22—61no—
254C22C22×D5×A411yescharacteristic
255C4—31no—
256C4—31no—
257C4—61no—
258C4—61no—
259C4—104no—
260C4—304no—
261C4A4×Dic1012yes—
262C4C5⋊#99412yes—
263C4#4150⋊C312yes—
264C3—41noSylow
265C2—62no—
266C2C2×A4×Dic1011yescharacteristic
267C2C2×C5⋊#99411yescharacteristic
268C2C2×#4150⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC4×C5×#41C2×C3
socleC22×C5C23×A4
prime radicalC2×C5×#41⋊C3C22
hypercenterC42D5×A4
hypocenterC5×Q8C2×C3×Q8
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
3. #65904⋊C3C2semidirect
4. SL(2,𝔽3)×Dic10C2semidirect
5. #10617729C3semidirect
22. C5×#41⋊C3Q8semidirect
63. C5×SL(2,𝔽3)C2×Q8semidirect
70. C5×#41C3×Q8semidirect
113. #41⋊C3Dic10semidirect
173. SL(2,𝔽3)C2×Dic10semidirect
188. #41C3×Dic10semidirect
250. C5#3503⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×C5×#41⋊C32
#66161⋊C32
#65904⋊C32
SL(2,𝔽3)×Dic102
#106177293
C3×#40154
#3503⋊C35