Finite Group Databasebeta

← C3×#64866 #11746643

Subgroups — C3×#64866 #11746643

Order 960.

284 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 72 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×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
2C2×C3×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3C3×D5×C22⋊C4C211yesmaximal, characteristic
4#64866C311yesmaximal, characteristic, direct product factor, Hall
5C23×C3×D5C2211yescharacteristic
6C22×C3×AGL(1,𝔽5)—21no—
7C22×C3×AGL(1,𝔽5)C412yescomplemented
8C2×C3×C4×D5—21no—
9C3×C22⋊AGL(1,𝔽5)—21no—
10C3×C22⋊AGL(1,𝔽5)—21no—
11C3×C22⋊AGL(1,𝔽5)—21no—
12C3×C22⋊AGL(1,𝔽5)C412yescomplemented
13C3×C5×C22⋊C4C411yescharacteristic, complemented
14C3×C22⋊Dic5C411yescharacteristic, complemented
15C3×#685—21no—
16C3×#342—51nomaximal, Hall
17C2×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic
18C2×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic
19D5×C22⋊C4C2×C311yescharacteristic
20C23×C3×C5C2×C411yescharacteristic
21C22×C3×D5—42no—
22C22×C3×D5C2×C411yescharacteristic
23C22×C3×D5C2×C411yescharacteristic
24C22×C3×D5D411yescharacteristic
25C22×C3×D5D411yescharacteristic
26C22×C3×D5D411yescharacteristic
27C22×C3×D5Q811yescharacteristic
28C2×C3×AGL(1,𝔽5)—22no—
29C2×C3×AGL(1,𝔽5)—22no—
30C2×C3×AGL(1,𝔽5)—42no—
31C2×C3×AGL(1,𝔽5)—42no—
32C2×C3×C4×C5—21no—
33C2×C3×Dic5—21no—
34C3×C4×D5—42no—
35C2×C3×C22⋊C4—51no—
36C2×C3×C22⋊C4—51no—
37C2×C3×C22⋊C4—51no—
38C23×D5C22×C311yescharacteristic
39C22×AGL(1,𝔽5)—21no—
40C22×AGL(1,𝔽5)C3×C412yescomplemented
41C2×C4×D5—21no—
42C22⋊AGL(1,𝔽5)—21no—
43C22⋊AGL(1,𝔽5)—21no—
44C22⋊AGL(1,𝔽5)—21no—
45C22⋊AGL(1,𝔽5)C3×C412yescomplemented
46C5×C22⋊C4C3×C411yescharacteristic, complemented
47C22⋊Dic5C3×C411yescharacteristic, complemented
48#685—21no—
49#342—51noSylow
50C22×C3×C5—41no—
51C22×C3×C5C4211yescharacteristic
52C22×C3×C5C22⋊C411yescharacteristic
53C22×C3×C5C4⋊C411yescharacteristic
54C2×C3×D5—22no—
55C2×C3×D5—22no—
56C2×C3×D5—22no—
57C2×C3×D5—41no—
58C2×C3×D5—41no—
59C2×C3×D5—41no—
60C2×C3×D5C22⋊C411yescharacteristic
61C2×C3×D5C22⋊C411yescharacteristic
62C2×C3×D5C4⋊C411yescharacteristic
63C2×C3×D5C4⋊C411yescharacteristic
64C3×AGL(1,𝔽5)—42no—
65C3×AGL(1,𝔽5)—42no—
66C3×C4×C5—41no—
67C3×Dic5—41no—
68C24×C3—51no—
69C22×C3×C4—52no—
70C22×C3×C4—101no—
71C22×C3×C4—101no—
72C3×C22⋊C4—52no—
73C3×C22⋊C4—51no—
74C3×C22⋊C4—101no—
75C3×C22⋊C4—101no—
76C3×C22⋊C4—101no—
77C3×C22⋊C4—101no—
78C3×C22⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
79C23×C5C2×C3×C411yescharacteristic
80C22×D5—42no—
81C22×D5C2×C3×C411yescharacteristic
82C22×D5C2×C3×C411yescharacteristic
83C22×D5C3×D411yescharacteristic
84C22×D5C3×D411yescharacteristic
85C22×D5C3×D411yescharacteristic
86C22×D5C3×Q811yescharacteristic
87C2×AGL(1,𝔽5)—22no—
88C2×AGL(1,𝔽5)—22no—
89C2×AGL(1,𝔽5)—42no—
90C2×AGL(1,𝔽5)—42no—
91C2×C4×C5—21no—
92C2×Dic5—21no—
93C4×D5—42no—
94C2×C22⋊C4—51no—
95C2×C22⋊C4—51no—
96C2×C22⋊C4—51no—
97C2×C3×C5—21no—
98C2×C3×C5—21no—
99C2×C3×C5—21no—
100C2×C3×C5#9511yescharacteristic
101C3×D5—21no—
102C3×D5—21no—
103C3×D5—21no—
104C3×D5#9912yes—
105C23×C3—51no—
106C23×C3—51no—
107C23×C3—51no—
108C23×C3—51no—
109C23×C3—51no—
110C23×C3—51no—
111C23×C3—202no—
112C23×C3C2×AGL(1,𝔽5)11yescharacteristic
113C2×C3×C4—21no—
114C2×C3×C4—102no—
115C2×C3×C4—102no—
116C2×C3×C4—101no—
117C2×C3×C4—202no—
118C2×C3×C4—202no—
119C2×C3×C4—202no—
120C22×C5—41no—
121C22×C5C3×C4211yescharacteristic
122C22×C5C3×C22⋊C411yescharacteristic
123C22×C5C3×C4⋊C411yescharacteristic
124C2×D5—22no—
125C2×D5—22no—
126C2×D5—22no—
127C2×D5—41no—
128C2×D5—41no—
129C2×D5—41no—
130C2×D5C3×C22⋊C411yescharacteristic
131C2×D5C3×C22⋊C411yescharacteristic
132C2×D5C3×C4⋊C411yescharacteristic
133C2×D5C3×C4⋊C411yescharacteristic
134AGL(1,𝔽5)—42no—
135AGL(1,𝔽5)—42no—
136C4×C5—41no—
137Dic5—41no—
138C24—51no—
139C22×C4—52no—
140C22×C4—101no—
141C22×C4—101no—
142C22⋊C4—52no—
143C22⋊C4—51no—
144C22⋊C4—101no—
145C22⋊C4—101no—
146C22⋊C4—101no—
147C22⋊C4—101no—
148C22⋊C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
149C3×C5#34211yescharacteristic, complemented, Hall
150C22×C3—41no—
151C22×C3—51no—
152C22×C3—51no—
153C22×C3—51no—
154C22×C3—51no—
155C22×C3—102no—
156C22×C3—102no—
157C22×C3—102no—
158C22×C3—201no—
159C22×C3—201no—
160C22×C3—201no—
161C22×C3C4×AGL(1,𝔽5)11yescharacteristic
162C22×C3C4⋊AGL(1,𝔽5)11yescharacteristic
163C22×C3C22⋊AGL(1,𝔽5)11yescharacteristic
164C3×C4—41no—
165C3×C4—201no—
166C3×C4—202no—
167C3×C4—202no—
168C2×C5—21no—
169C2×C5—21no—
170C2×C5—21no—
171C2×C5C3×#9511yescharacteristic
172D5—21no—
173D5—21no—
174D5—21no—
175D5C3×#9912yes—
176C23—51no—
177C23—51no—
178C23—51no—
179C23—51no—
180C23—51no—
181C23—51no—
182C23—202no—
183C23C2×C3×AGL(1,𝔽5)11yescharacteristic
184C2×C4—21no—
185C2×C4—102no—
186C2×C4—102no—
187C2×C4—101no—
188C2×C4—202no—
189C2×C4—202no—
190C2×C4—202no—
191C2×C3—21no—
192C2×C3—21no—
193C2×C3—21no—
194C2×C3—52no—
195C2×C3—101no—
196C2×C3—101no—
197C2×C3—101no—
198C2×C3(C2×AGL(1,𝔽5))⋊C411yescharacteristic
199C5C3×#34211yescharacteristic, complemented, Sylow
200C22—41no—
201C22—51no—
202C22—51no—
203C22—51no—
204C22—51no—
205C22—102no—
206C22—102no—
207C22—102no—
208C22—201no—
209C22—201no—
210C22—201no—
211C22C3×C4×AGL(1,𝔽5)11yescharacteristic
212C22C3×C4⋊AGL(1,𝔽5)11yescharacteristic
213C22C3×C22⋊AGL(1,𝔽5)11yescharacteristic
214C4—41no—
215C4—201no—
216C4—202no—
217C4—202no—
218C3#6486611yescharacteristic, direct product factor, Sylow
219C2—21no—
220C2—21no—
221C2—21no—
222C2—52no—
223C2—101no—
224C2—101no—
225C2—101no—
226C2C3×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
subgroupG/H
centerC2×C3(C2×AGL(1,𝔽5))⋊C4
commutatorC22×C5C3×C42
FrattiniC23C2×C3×AGL(1,𝔽5)
FittingC3×C5×C22⋊C4C4
socleC2×C3×C5#95
prime radicalC23×C3×D5C22
cosocleC23×D5C22×C3
hypercenterC3×C22⋊C4AGL(1,𝔽5)
hypocenterC5C3×#342
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #64866C3direct (N × H)
7. C22×C3×AGL(1,𝔽5)C4semidirect
12. C3×C22⋊AGL(1,𝔽5)C4semidirect
13. C3×C5×C22⋊C4C4semidirect
14. C3×C22⋊Dic5C4semidirect
40. C22×AGL(1,𝔽5)C3×C4semidirect
45. C22⋊AGL(1,𝔽5)C3×C4semidirect
46. C5×C22⋊C4C3×C4semidirect
47. C22⋊Dic5C3×C4semidirect
78. C3×C22⋊C4AGL(1,𝔽5)semidirect
148. C22⋊C4C3×AGL(1,𝔽5)semidirect
149. C3×C5#342semidirect
199. C5C3×#342semidirect
218. C3#64866direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊AGL(1,𝔽5)2
C3×D5×C22⋊C42
#648663
C3×#3425