Finite Group Databasebeta

← C3×#65780 #11752313

Subgroups — C3×#65780 #11752313

Order 960.

516 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 128 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×C3×D4×D5C211yesmaximal, characteristic
2C2×C3×C22⋊AGL(1,𝔽5)C214yesmaximal, complemented
3C2×C3×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C2×C3×C4×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
5#65780C311yesmaximal, characteristic, direct product factor, Hall
6C23×C3×D5C2212yes—
7C22×C3×AGL(1,𝔽5)C2212yes—
8C22×C3×AGL(1,𝔽5)C2212yes—
9C2×C3×C5×D4C411yescharacteristic, complemented
10C2×C3×C22⋊D5C412yescomplemented
11C3×D4×D5—22no—
12C3×D4×D5—22no—
13C2×C3×D20C411yescharacteristic, complemented
14C2×C3×C4×D5C2211yescharacteristic
15C3×C22⋊AGL(1,𝔽5)—28no—
16C3×C4⋊AGL(1,𝔽5)—22no—
17C3×C4×AGL(1,𝔽5)—22no—
18C3×#390—51nomaximal, Hall
19C2×D4×D5C2×C311yescharacteristic
20C2×C22⋊AGL(1,𝔽5)C2×C314yescomplemented
21C2×C4⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
22C2×C4×AGL(1,𝔽5)C2×C311yescharacteristic, complemented
23C23×C3×C5C2×C412yes—
24C22×C3×D5—24no—
25C22×C3×D5—22no—
26C22×C3×D5—22no—
27C22×C3×D5—24no—
28C22×C3×D5C2×C412yes—
29C22×C3×D5C2311yescharacteristic
30C2×C3×AGL(1,𝔽5)—22no—
31C2×C3×AGL(1,𝔽5)—22no—
32C2×C3×AGL(1,𝔽5)—22no—
33C2×C3×AGL(1,𝔽5)—24no—
34C2×C3×AGL(1,𝔽5)D414yescomplemented
35C3×C5×D4—22no—
36C2×C3×C4×C5C2×C411yescharacteristic
37C3×C22⋊D5—24no—
38C2×C3×Dic5C2×C411yescharacteristic
39C3×D20—22no—
40C3×C4×D5D412yes—
41C3×C4×D5D412yes—
42C22×C3×D4—51no—
43C2×C3×C4⋊C4—51no—
44C2×C3×C22⋊C4—54no—
45C2×C3×C42—51no—
46C23×D5C22×C312yes—
47C22×AGL(1,𝔽5)C22×C312yes—
48C22×AGL(1,𝔽5)C22×C312yes—
49C2×C5×D4C3×C411yescharacteristic, complemented
50C2×C22⋊D5C3×C412yescomplemented
51D4×D5—22no—
52D4×D5—22no—
53C2×D20C3×C411yescharacteristic, complemented
54C2×C4×D5C22×C311yescharacteristic
55C22⋊AGL(1,𝔽5)—28no—
56C4⋊AGL(1,𝔽5)—22no—
57C4×AGL(1,𝔽5)—22no—
58#390—51noSylow
59C22×C3×C5—22no—
60C22×C3×C5—24no—
61C22×C3×C5C22×C411yescharacteristic
62C2×C3×D5—22no—
63C2×C3×D5—24no—
64C2×C3×D5—44no—
65C2×C3×D5—42no—
66C2×C3×D5—42no—
67C2×C3×D5C2×D411yescharacteristic
68C2×C3×D5C2×D412yes—
69C2×C3×D5C2×D411yescharacteristic
70C2×C3×D5#4112yes—
71C3×AGL(1,𝔽5)—24no—
72C3×AGL(1,𝔽5)—42no—
73C3×C4×C5C22⋊C412yescomplemented
74C3×Dic5C22⋊C412yescomplemented
75C24×C3—52no—
76C2×C3×D4—52no—
77C2×C3×D4—51no—
78C2×C3×D4—102no—
79C2×C3×D4—102no—
80C2×C3×D4AGL(1,𝔽5)11yescharacteristic, complemented
81C22×C3×C4—51no—
82C22×C3×C4—52no—
83C22×C3×C4—52no—
84C3×C4⋊C4—102no—
85C3×C22⋊C4—108no—
86C3×C42—102no—
87C23×C5C2×C3×C412yes—
88C22×D5—24no—
89C22×D5—22no—
90C22×D5—22no—
91C22×D5—24no—
92C22×D5C2×C3×C412yes—
93C22×D5C23×C311yescharacteristic
94C2×AGL(1,𝔽5)—22no—
95C2×AGL(1,𝔽5)—22no—
96C2×AGL(1,𝔽5)—22no—
97C2×AGL(1,𝔽5)—24no—
98C2×AGL(1,𝔽5)C3×D414yescomplemented
99C5×D4—22no—
100C2×C4×C5C2×C3×C411yescharacteristic
101C22⋊D5—24no—
102C2×Dic5C2×C3×C411yescharacteristic
103D20—22no—
104C4×D5C3×D412yes—
105C4×D5C3×D412yes—
106C22×D4—51no—
107C2×C4⋊C4—51no—
108C2×C22⋊C4—54no—
109C2×C42—51no—
110C2×C3×C5—42no—
111C2×C3×C5C2×C22⋊C411yescharacteristic
112C2×C3×C5C4×D412yes—
113C3×D5—42no—
114C3×D5#12112yes—
115C3×D5#12411yescharacteristic
116C3×D5#12711yescharacteristic
117C23×C3—51no—
118C23×C3—52no—
119C23×C3—104no—
120C23×C3—102no—
121C23×C3—102no—
122C23×C3—104no—
123C23×C3C2×AGL(1,𝔽5)12yes—
124C3×D4—22no—
125C3×D4—104no—
126C3×D4—102no—
127C2×C3×C4—51no—
128C2×C3×C4—52no—
129C2×C3×C4—52no—
130C2×C3×C4—54no—
131C2×C3×C4—102no—
132C2×C3×C4—102no—
133C2×C3×C4—104no—
134C2×C3×C4—102no—
135C2×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
136C22×C5—22no—
137C22×C5—24no—
138C22×C5C22×C3×C411yescharacteristic
139C2×D5—22no—
140C2×D5—24no—
141C2×D5—44no—
142C2×D5—42no—
143C2×D5—42no—
144C2×D5C2×C3×D411yescharacteristic
145C2×D5C2×C3×D412yes—
146C2×D5C2×C3×D411yescharacteristic
147C2×D5C3×#4112yes—
148AGL(1,𝔽5)—24no—
149AGL(1,𝔽5)—42no—
150C4×C5C3×C22⋊C412yescomplemented
151Dic5C3×C22⋊C412yescomplemented
152C24—52no—
153C2×D4—52no—
154C2×D4—51no—
155C2×D4—102no—
156C2×D4—102no—
157C2×D4C3×AGL(1,𝔽5)11yescharacteristic, complemented
158C22×C4—51no—
159C22×C4—52no—
160C22×C4—52no—
161C4⋊C4—102no—
162C22⋊C4—108no—
163C42—102no—
164C3×C5#39011yescharacteristic, complemented, Hall
165C22×C3—24no—
166C22×C3—22no—
167C22×C3—52no—
168C22×C3—52no—
169C22×C3—51no—
170C22×C3—51no—
171C22×C3—104no—
172C22×C3—102no—
173C22×C3—204no—
174C22×C3—202no—
175C22×C3—202no—
176C22×C3C22×AGL(1,𝔽5)11yescharacteristic
177C3×C4—52no—
178C3×C4—104no—
179C3×C4—202no—
180C3×C4C22⋊AGL(1,𝔽5)12yescomplemented
181C2×C5—42no—
182C2×C5C2×C3×C22⋊C411yescharacteristic
183C2×C5C3×C4×D412yes—
184D5—42no—
185D5C3×#12112yes—
186D5C3×#12411yescharacteristic
187D5C3×#12711yescharacteristic
188C23—51no—
189C23—52no—
190C23—104no—
191C23—102no—
192C23—102no—
193C23—104no—
194C23C2×C3×AGL(1,𝔽5)12yes—
195D4—22no—
196D4—104no—
197D4—102no—
198C2×C4—51no—
199C2×C4—52no—
200C2×C4—52no—
201C2×C4—54no—
202C2×C4—102no—
203C2×C4—102no—
204C2×C4—104no—
205C2×C4—102no—
206C2×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
207C2×C3—42no—
208C2×C3—52no—
209C2×C3—51no—
210C2×C3—51no—
211C2×C3—202no—
212C2×C3D4×AGL(1,𝔽5)12yes—
213C2×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
214C5C3×#39011yescharacteristic, complemented, Sylow
215C22—24no—
216C22—22no—
217C22—52no—
218C22—52no—
219C22—51no—
220C22—51no—
221C22—104no—
222C22—102no—
223C22—204no—
224C22—202no—
225C22—202no—
226C22C22×C3×AGL(1,𝔽5)11yescharacteristic
227C4—52no—
228C4—104no—
229C4—202no—
230C4C3×C22⋊AGL(1,𝔽5)12yescomplemented
231C3#6578011yescharacteristic, direct product factor, Sylow
232C2—42no—
233C2—52no—
234C2—51no—
235C2—51no—
236C2—202no—
237C2C3×D4×AGL(1,𝔽5)12yes—
238C2C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22×C3C22×AGL(1,𝔽5)
commutatorC22×C5C22×C3×C4
FrattiniC22C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×D4C4
socleC22×C3×C5C22×C4
prime radicalC2×C3×D4×D5C2
cosocleC22×D5C23×C3
hypercenterC2×C3×D4AGL(1,𝔽5)
hypocenterC5C3×#390
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C22⋊AGL(1,𝔽5)C2semidirect
3. C2×C3×C4⋊AGL(1,𝔽5)C2semidirect
4. C2×C3×C4×AGL(1,𝔽5)C2semidirect
5. #65780C3direct (N × H)
9. C2×C3×C5×D4C4semidirect
10. C2×C3×C22⋊D5C4semidirect
13. C2×C3×D20C4semidirect
20. C2×C22⋊AGL(1,𝔽5)C2×C3semidirect
21. C2×C4⋊AGL(1,𝔽5)C2×C3semidirect
22. C2×C4×AGL(1,𝔽5)C2×C3semidirect
34. C2×C3×AGL(1,𝔽5)D4semidirect
49. C2×C5×D4C3×C4semidirect
50. C2×C22⋊D5C3×C4semidirect
53. C2×D20C3×C4semidirect
73. C3×C4×C5C22⋊C4semidirect
74. C3×Dic5C22⋊C4semidirect
80. C2×C3×D4AGL(1,𝔽5)semidirect
98. C2×AGL(1,𝔽5)C3×D4semidirect
150. C4×C5C3×C22⋊C4semidirect
151. Dic5C3×C22⋊C4semidirect
157. C2×D4C3×AGL(1,𝔽5)semidirect
164. C3×C5#390semidirect
180. C3×C4C22⋊AGL(1,𝔽5)semidirect
214. C5C3×#390semidirect
230. C4C3×C22⋊AGL(1,𝔽5)semidirect
231. C3#65780direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×D52
C2×C3×C22⋊AGL(1,𝔽5)2
C2×C3×C4⋊AGL(1,𝔽5)2
C2×C3×C4×AGL(1,𝔽5)2
#657803
C3×#3905